optical properties of cylindrical nanowiresductor cylindrical nanowires. the optical properties are...

Optical properties of cylindrical nanowires

L.A. Haverkate; L.F. Feiner

15th December 2006

Abstract

A theoretical analysis is presented of the optical absorption of III-V semicon-ductor cylindrical nanowires. The optical properties are described by meansof the dielectric function, calculated for band-to-band transitions close tothe band gap.

We have treated the electronic structure using effective mass theory,taking the degeneracy of the valence band into account.

A strong polarization anisotropy is found which is due to quantum con-finement, in agreement with atomistic methods. We show that the effectivemass approach provides a fast and flexible tool to analyze the diameter de-pendent properties of nanowires for a wide range of semiconductor materials.

In addition we discuss the effect of classical Mie scattering and show thatit is negligible in the quantum regime.

Contents

Introduction 6

I Classical theory of light scattering by a cylinder 8

1 General solution 91.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 Maxwell equations . . . . . . . . . . . . . . . . . . . . 91.1.2 Boundary conditions . . . . . . . . . . . . . . . . . . . 10

1.2 Mie’s formal solution for circular cylinders . . . . . . . . . . . 111.3 Scattering problem . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.1 Scattering coefficients, general solution . . . . . . . . . 151.4 Far field theory . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4.1 Far field approximation . . . . . . . . . . . . . . . . . 171.4.2 Poynting vector and electromagnetic energy rates . . . 181.4.3 Cross sections and efficiencies . . . . . . . . . . . . . . 20

2 Small dielectric cylinders 232.1 Coefficients in Rayleigh approximation . . . . . . . . . . . . . 232.2 Fields inside the wire . . . . . . . . . . . . . . . . . . . . . . . 252.3 Efficiency, polarization anisotropy and - contrast in Rayleigh

approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.1 Polarization anisotropy, polarization contrast . . . . . 28

2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4.1 Efficiencies and polarization anisotropy at oblique in-

cidence . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.2 Efficiencies and polarization anisotropy as a function

of wavelength . . . . . . . . . . . . . . . . . . . . . . . 36

II Absorption 40

3 Electronic properties 413.1 The k · p method . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.1 Top valence bands in III-V semiconductors . . . . . . 44

3

Contents

3.2 Effective mass approximation . . . . . . . . . . . . . . . . . . 463.2.1 Crystal Hamiltonian in envelope representation . . . . 473.2.2 Top valence bands in III-V semiconductors . . . . . . 48

3.3 Envelope description for infinite cylinders . . . . . . . . . . . 493.3.1 Hole in III-V semiconductor nanowires . . . . . . . . . 513.3.2 Electron in III-V semiconductor nanowires . . . . . . 53

3.4 Hole dispersion around kz = 0 . . . . . . . . . . . . . . . . . . 543.4.1 Solutions at the wire zone center . . . . . . . . . . . . 543.4.2 Hole dispersion around kz = 0 for |fz| = 1

2 , (−) . . . . 563.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5.1 Hole energy bands of III-V material nanowires . . . . 593.5.2 Hole wave functions of III-V material nanowires . . . 613.5.3 Band gap in III-V material nanowires . . . . . . . . . 64

4 EM transition matrix 674.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1.1 Radiation matter interaction . . . . . . . . . . . . . . 674.1.2 EM transition matrix . . . . . . . . . . . . . . . . . . 68

4.2 Bloch representation . . . . . . . . . . . . . . . . . . . . . . . 704.2.1 Total wavefunction in Bloch functions . . . . . . . . . 704.2.2 EM transition matrix in Bloch functions . . . . . . . . 71

4.3 Reformulation of transition matrix element . . . . . . . . . . 744.3.1 EM field in dipole approximation . . . . . . . . . . . . 744.3.2 EM field including Mie scattering . . . . . . . . . . . . 754.3.3 Polarization anisotropy of the transition matrix . . . . 75

4.4 Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.4.1 Polarization selection rules . . . . . . . . . . . . . . . 774.4.2 Selection rules on the envelope wavefunctions . . . . . 80

4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.5.1 Dipole approximation . . . . . . . . . . . . . . . . . . 834.5.2 EM field including Mie scattering . . . . . . . . . . . . 87

5 Dielectric function nanowire 915.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.1.1 Atomic polarizability approach . . . . . . . . . . . . . 915.1.2 Transition rate method . . . . . . . . . . . . . . . . . 945.1.3 Dielectric function expressed in reduced effective mass 96

5.2 Dielectric function for finite group transitions . . . . . . . . . 975.3 Polarization anisotropy nanowire . . . . . . . . . . . . . . . . 985.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4.1 Estimation kz dependence of |Tcv|2 . . . . . . . . . . 1005.4.2 Polarization anisotropy and R dependence . . . . . . . 1025.4.3 Material dependence . . . . . . . . . . . . . . . . . . . 1045.4.4 Effect of the dielectric background . . . . . . . . . . . 105

4

Contents

Conclusions 107

A Hole wavefunctions for different kz 112

B Polarization selection rules 116

C Interband matrix elements 119

D Reference articles 122

Bibliography 125

5

Introduction

In the last several years semiconductor nanowires have attracted much in-terest, due to the tunability of their fundamental optical and electronicproperties. Techniques for the growth of nanostructures have been develo-ped and high quality III-V semiconductor nanowires with a length of severalmicrons and a lateral size of only a few nanometers have been obtained. Re-cent experiments have shown a large polarization anisotropy in such wires[1][2]. For example, Figure 1 shows the photoluminescence and excitationspectra of an InP nanowire on a flat gold surface [2]. The radius of this wirewas ∼ 15 nm and the measured polarization anisotropy is fully explained bythe dielectric mismatch between the wire and the surrounding.

Figure 1: Experimental results for optical absorption [2]. a) Photolu-minescence image (CCD camera, incident laser light polarized parallelto the wire axis) and b) Excitation spectra for parallel (‖) and perpen-dicular (⊥) polarized incident light of an InP nanowire on a flat goldsurface. The length of the wire is ∼ 2 µm, its radius ∼ 15 nm and thewavelength of the exciting laser beam is 457.9 nm. The emitted lightwas unpolarized in order to take only the polarization anisotropy in theabsorption process into account.

However, next to this classical effect of dielectric contrast it is expectedthat quantum confinement starts to contribute significantly for decreasing

6

wire radius. This quantum effect has already been observed in the shift inthe fundamental band gap [3], but based on atomistic theories[4][5] it is alsopredicted that quantum confinement causes drastic changes in the polariza-tion anisotropy of nanostructures.

In this paper we will analyze the optical absorption properties of III-Vsemiconductor cylindrical nanowires using effective mass theory. Within thisapproach it is possible to describe the optical and electronic properties forvarying wire thickness and for a wide range of semiconductor materials. Con-trary, ab initio methods using a fully atomistic description (tight-binding,pseudo-potential,...) are limited with respect to the dimensions of the nano-system since with increasing number of atoms the calculations become moreand more complex, or even impossible. Although the effective mass appro-ach generally is less accurate, it thus provides a relative fast and flexible toolto simulate real nanowires, with dimensions which are technically feasibleat the present day.

Despite the large amount of papers on the subject of nanowires, or evennanostructures in general, little attention has been paid to the effects ofclassical scattering. Usually it is assumed that the wavelength of the incidentlight is sufficiently larger than the wire radius in order to neglect the spatialvariance of the electromagnetic (EM) field within the wire, which justifiesconsidering the response of the nanowire to the incident light in the dipolelimit. For increasing wire radius, however, the wave behavior of the EMfield cannot simply be neglected any more. Therefore, it is one of the mainquestions in this thesis if this so called Mie-scattering already starts to playa significant role in the quantum confinement regime.

For this purpose we have to know the local response field inside the wire.But historically most of the work in classical scattering theory was dedicatedto measurable quantities far from the scattering objects, driven by the largeinterest from application fields as astronomy and meteorology. In Part I,Chapters 1 and 2, we therefore start with a classical theory describing thescattering of light by an infinite cylindrical structure. In particular we deriveexplicit expressions for the EM field inside the wire by using a procedureoriginally developed by Mie [6].

In Part II we subsequently focus on the effects of quantum confinementby means of a corrected description of the dielectric function of cylindricalnanowires. In Chapter 3 the electronic properties of nanowires made fromIII-V compounds are discussed, Chapter 4 treats the EM matrix element forband-to-band transitions between the top Γ8 valence bands and the lowestlying Γ6 conduction band in III-V semiconductor nanowires and finally inChapter 5 the dielectric function and polarization anisotropy of a nanowireare obtained including the quantum confinement corrections by the band-to-band transitions.

7

Part I

Classical theory of lightscattering by a cylinder

8

Chapter 1

General solution

In this chapter classical theory is treated which describes the scattering oflight by an infinite cylinder at arbitrary angle of incidence and wire radius.In the first sections general theory is discussed and specified to the case ofan infinite cylinder by using a procedure originally developed by Mie [6]. Insection the theory will be put in an applicable form by deriving measurablequantities (cross sections, efficiency factors) in the far field region.

1.1 General theory

1.1.1 Maxwell equations

The scattering of light at oblique incidence by an infinite cylinder needs afull, formal treatment, in particular when the solution has to be expandedin the wire radius.The starting-point of the full problem is Maxwell’s theory. Assuming thelight waves to be periodic with time dependence e−iωt, the charge densityρ equal to zero and the magnetic permeability µ equal to 1, the Maxwellequations are:

∇×H = −ik0m2E, (1.1)

∇×E = ik0H, (1.2)∇ ·H = 0, (1.3)

∇ · (m2E) = 0, (1.4)

where

k0 =ω

c=

2π

λ0, (1.5)

is the wave number in vacuum and

m2 = ε +4πiσ

ω. (1.6)

9

Chapter 1. General solution

The parameter m is the complex refractive index of the medium at thefrequency ω of the light waves and consists of an optic part and an electricpart. The former is associated with ε, the dielectric constant, the latterwith the conductivity σ, which is taken to be zero since the electrical partis beyond the scope of this paper. Both parts are complex and depend onthe circular frequency ω of the light waves.

It should be noted that in general m is a tensor and moreover depends onthe position in the medium. For the applications considered in this paper themedium is assumed to be homogeneous and in that case m is a constant. Wewill also assume here that m is a scalar. As a consequence, from (1.1)-(1.4),the field vectors E and H satisfy the vector wave equation:

∆A + k20m

2A = 0. (1.7)

As a consequence the rectangular components of E and H satisfy the scalarwave equation

∆ψ + k20m

2ψ = 0, (1.8)

which has plane wave solutions with the propagation constant equal to k0m.This shows that the wave is damped if m has a negative imaginary part andin that case absorption takes place.

1.1.2 Boundary conditions

In case of a sharp boundary between two homogeneous media (1 and 2) theintegral representation of the Maxwell equations (1.1) and (1.2) gives theboundary conditions on the tangential components of the fields, after a wellknown limiting process (Jackson [12], page 16):

n× (H2 −H1) = 0, (1.9)n× (E2 −E1) = 0, (1.10)

where n is the normal to the boundary.In the same way the Maxwell equations (1.3) and (1.4) lead to the boundaryconditions on the normal components:

n · (m22E2 −m1

2E1) = 0, (1.11)n · (H2 −H1) = 0. (1.12)

The tangential and normal boundary conditions are not independent. Forinstance, boundary condition (1.10) can be derived from (1.12), Maxwellequation (1.3) and applying the limiting procedure. In the same way it canbe shown that (1.9) and (1.11) are dependent on each other. Therefore it issufficient to look only at the tangential components.

10

1.2. Mie’s formal solution for circular cylinders

1.2 Mie’s formal solution for circular cylinders

In order to solve the boundary value problem exactly the coordinate systemshould be the one in which the scalar wave equation is separable in thecoordinates. In case of circular cylinders these coordinates are (ρ, φ, z),where the cylinder axis coincides with the z -axis (see Figure 1.1). As acondition for this separability the cylinder length L has to be assumed muchlarger then its diameter:

L À 2R, (1.13)

where R denotes the cylinder radius. In this case the cylinder can be seenas infinitely long and then it is possible to use the following formal solutiondeveloped by Mie [6]. If ψ satisfies the scalar wave equation (1.8), defineMψ and Nψ as

Mψ = ∇× (ez · ψ), (1.14)mk0Nψ = ∇×Mψ. (1.15)

Then both Mψ and Nψ satisfy the vector wave equation (1.7), and theelementary solutions of Maxwell’s equations can be expressed as

E = M v + iNu, (1.16)H = mMu − imN v, (1.17)

where u and v are the two independent solutions of the scalar wave equation.The scalar wave equation (1.8) in cylindrical coordinates for a homoge-

neous medium with complex refractive index m is(

∂2

∂ρ2+

1ρ

∂2

∂ρ+

1ρ2

∂2

∂φ2+

∂2

∂z2+ m2k2

0

)ψ = 0, (1.18)

and its solutions can be found by separating the variables. The resultingdifferential equation for the ρ coordinate is the Bessel equation, which hastwo independent solutions: Jn, the integral order Bessel function and Nn,the integral order Neumann function. This means that the solutions of (1.8)can be found by an appropriate superposition of:

ϕn = Zn(ρ√

m2k20 − g2)ei(gz−ωt)einφ, (1.19)

with n an integer, Zn any Bessel function of order n and g arbitrary. Incylindrical coordinates Mϕn and Nϕn are then derived as:

Mϕn =

inρ

− ∂∂ρ

0

ϕn, mk0Nϕn =

ig ∂∂ρ−ngρ

m2k20 − g2

ϕn, (1.20)

11


on the basis of cylindrical unit vectors eρ, eφ, ez. Consequently, with un =Anϕn and vn = Bnϕn for certain An, Bn and taking the sum over all n, thecomponents of E and H are

Eρ =∞∑

n=−∞

in

ρvn − g

mk0

∂un

∂ρ, (1.21)

Hρ =∞∑

n=−∞

inm

ρun +

g

k0

∂vn

∂ρ, (1.22)

normal to the cylinder surface and

Eφ =∞∑

n=−∞−∂vn

∂ρ− ing

mk0ρun, (1.23)

Ez =∞∑

n=−∞

i(m2k20 − g2)

mk0un, (1.24)

Hφ =∞∑

n=−∞−m

∂un

∂ρ+

ing

k0ρvn, (1.25)

Hz =∞∑

n=−∞− i(m2k2

0 − g2)k0

vn (1.26)

tangential to the cylinder surface.

1.3 Scattering problem

With the above formal solution it is now possible to solve the general scatte-ring problem of an arbitrary polarized plane electromagnetic wave incidentobliquely on a circular cylinder of infinite length.For oblique incidence the direction of propagation of the incident wave ma-kes an angle θ with the normal to the z -axis, see Figure 1.1. Furthermorethe cylinder is assumed to be surrounded by vacuum and the refractive in-dex of the cylinder is equal to m. In case of a surrounding homogeneousmedium with refractive index m1 the solutions are of the same form if mis considered as the refractive index of the cylinder relative to the medi-um: m = m2

m1. With the above definitions the incident wave, depicted in

Figure 1.1 is represented by the scalar wave function

ψ0 = E0 e−i(k0x cos θ+k0z sin θ+ωt)

= E0 e−i(hz+ωt)∞∑

n=−∞(−i)nJn(lρ)eınφ, (1.27)

where

h ≡ k0 sin θ, (1.28)

l ≡ k0 cos θ =√

k20 − h2. (1.29)

12

1.3. Scattering problem

Z

X

Y

EO I

Esca

Eint

k

EO II

HO II

k

HO I

Figure 1.1: Definition of the coordinates for scattering by a circularcylinder. The incident waves are showed, including the correspondingincident fields: E0 I , H0 I in Case I and E0 II , H0 II in Case II. Theangle of incidence is defined by θ .

Equation (1.27) represents a wave travelling in the −ex direction if θ equalszero. Note that the last expression in (1.27) is an expansion in Bessel func-tions and has the required form of (1.19). In this way also the scatteredwave and internal wave (inside the cylinder) can be formed from a superpo-sition of functions of the form (1.19). Finiteness at the origin requires thatJn(ρ

√m2k2

0 − h2) is the radial function describing the internal wave, whereh is given by (1.28) because of continuity at the boundary ( (1.9) and (1.10)).The last argument also holds for the scattered wave, which is described bythe first Hankel function H

(1)n (ρ

√k2

0 − h2), describing an outgoing wave atlarge distances from the cylinder.

Following the procedure of Van de Hulst [7] and Kerker [8], the polarizedincident wave has to be resolved into two components:

• Case I: a Transverse Magnetic (TM) mode. The magnetic field ofthe incident wave is perpendicular to the cylinder axis (Figure 1.1).This mode is described by choosing un = 1

il E0 (−i)n ϕn (withg = −h, Zn = Jn ) and vn = 0 in (1.21)- (1.26). This choice alsofixes the orientation of the incident electric field: E0 I = E0 (cos θez −sin θex) e−i(hz+lx+ωt). The factor 1

il is just a normalization constant,for further details see Bohren and Huffman [11].

• Case II: a Transverse Electric (TE) mode.The electric field is perpen-dicular to the cylinder axis. Now un = 0 and vn = 1

il E0(−i)nϕn andthe incident field is given by E0 II = E0 ey e−i(hz+lx+ωt).

For an arbitrary elliptically polarized incident wave the solutions can befound by an appropriate superposition of Case I and Case II. The decompo-

13


sition of the incident wave does not necessarily mean that the scattered andinternal waves resolve in the same way. This can be explained by lookingclosely at the general expressions of the scalar fields inside and outside thecylinder. These are:

Case I

ρ > R unI = E0 Fn Jn(lρ)− bnIH(1)n (lρ) , (1.30)

vnI = E0 Fn anIH(1)n (lρ) , (1.31)

ρ < R unI = E0 Fn dnIJn(jρ) , (1.32)vnI = E0 Fn cnIJn(jρ) , (1.33)

Case II

ρ > R unII = E0 Fn bnIIH(1)n (lρ) , (1.34)

vnII = E0 Fn Jn(lρ)− anIIH(1)n (lρ) , (1.35)

ρ < R unII = E0 Fn dnIIJn(jρ) (1.36)vnII = E0 Fn cnIIJn(jρ) , (1.37)

where

Fn ≡ 1il

e−i(hz+ωt)(−i)neinφ (1.38)

and

j ≡√

m2k20 − h2. (1.39)

Unlike the incident waves, which are chosen to be TM or TE, the solutionsfor the scattered and internal scalar waves are in general decomposed intotwo components:

• A solution with the same orientation as the incident wave (TM or TE),contained in unI (Case I) and vnII (Case II) respectively.

• A ”cross mode” with an opposite orientation, TE (v1) in Case I andTM (uII) in Case II.

Only in case of normal incidence, θ = 0, the cross terms turn out to be zeroand the scattered and internal waves resolve in the same way as the incidentwave (see paragraph below).

As stated in section 1.2, the scalar wave expressions (1.30)-(1.37) alsodetermine the fields inside and outside the cylinder in the various cases. Theincident, scattered and internal fields are denoted with E0, Esca and Eint,respectively.As an example, combining the expression for the scattered scalar wave inCase I (the second term in (1.30)) with the equations for the field compo-nents (1.21)-(1.26) one finds for the scattered electric field Esca in cylindercoordinates:

14

1.3. Scattering problem

Esca I = E0

∞∑n=−∞

Fn

anI

inρ

− ∂∂ρ

0

H(1)

n (lρ) + i(−bnI)

−ihk0

∂∂ρ

nhk0ρl2

k

H(1)

n (lρ)

.

(1.40)

The scattered electric field Esca II in Case II can be obtained from (1.40)by replacing anI by −anII and −bnI by bnII .

1.3.1 Scattering coefficients, general solution

The coefficients anI , bnI , cnI and dnI (anII , bnII , cnII and dnII) are in generalfunctions of the angle of incidence θ and the wire radius R. They can bedetermined by the fact that the boundary conditions (1.11)-(1.12) requirecontinuity of the tangential components of E and H. As a consequence theequations (1.23)-(1.26) have to be continuous at R = ρ. These conditionslead in both cases to four linear algebraic equations which can be solved forthe four coefficients:

Case I

anI(R, θ) =ı sin θ n(m2 − 1)N−1

n −O−1n

lR( jk0

)2Ln − (m2 + 1) jk0

Dn + L−1n (Cn −m2D2

n) ,

bnI(R, θ) =H

(1) ′n (lR)( j

k0)2Kn −m2 j

k0Dn+ H

(1)n (lR)− j

k0DnKn + m2D2

n − CnH

(1)n (lR)Mn( j

k0)2Ln − (m2 + 1) j

k0Dn + L−1

n (Cn −m2D2n)

,

cnI(R, θ) =l2

j2

anI(R, θ)H(1)

n (lR)Jn(jR)

,

dnI(R, θ) =ml2

j2

Jn(lR)Jn(jR)

− bnI(R, θ)H(1)n (lR)

Jn(jR)

, (1.41)

Case II:

anII(R, θ) =H

(1) ′n (lR)( j

k0)2Kn − j

k0Dn+ H

(1)n (lR)m2 j

k0DnKn + m2D2

n − CnH

(1)n (lR)Mn( j

k0)2Ln − (m2 + 1) j

k0Dn + L−1

n (m2D2n − Cn)

,

bnII(R, θ) = −anI(R, θ),

cnII(R, θ) =l2

j2

Jn(lR)Jn(jR)

− bnII(R, θ)H(1)n (lR)

Jn(jR)

,

dnII(R, θ) =ml2

j2

anII(R, θ)H(1)

n (lR)Jn(jR)

, (1.42)

15


where

Cn ≡ (m2 − 1)2 n2 tan2 θ

j2R2, (1.43)

Dn ≡ cos θJ′n(jR)

Jn(jR), (1.44)

Kn ≡ J′n(lR)

Jn(lR), Ln ≡ H

(1) ′n (lR)

H(1)n (lR)

, (1.45)

Mn ≡ H(1) ′n (lR)Jn(lR)

, Nn ≡ H(1)n (lR)

Jn(lR), (1.46)

On ≡ H(1) ′n (lR)J ′

n(lR)(1.47)

and the functions l (1.29) and j (1.39) depend on θ. Despite of the differentform, equations (1.30)-(1.37) are the same as derived by Bohren [11]. Theygive the complete, formal solution for the scattering problem of a planeelectromagnetic wave incident obliquely on a circular cylinder of infinitelength. In principle the electromagnetic fields and the intensities can beobtained by calculating the full expansion of (1.30)-(1.33) for Case I ((1.34)-(1.37) for Case II) and subsequently use these expressions to calculate thefields (1.21-1.26). However, in practice it is impossible to get an exactanalytic solution and a numerical procedure is the only way to solve thefull problem.

In the special case of a normal incident wave (θ = 0), the scatteringcoefficients (1.30)-(1.37) reduce to:

Case I

anI(R, 0) = 0,

bnI(R, 0) =mJn(k0R)J

′n(mk0R)− J

′n(k0R)Jn(mk0R)

mH(1)n (k0R)J ′

n(mk0R)−H(1) ′n (k0R)Jn(mk0R)

,

cnI(R, 0) = 0,

dnI(R, 0) =H

(1) ′n (k0R)Jn(k0R)−H

(1)n (k0R)J

′n(k0R)

mH(1) ′n (k0R)Jn(mk0R)−m2H

(1)n (k0R)J ′

n(mk0R),(1.48)

Case II

anII(R, 0) =Jn(k0R)J

′n(mk0R)−mJ

′n(k0R)Jn(mk0R)

H(1)n (k0R)J ′

n(mk0R)−mH(1) ′n (k0R)Jn(mk0R)

,

bnII(R, 0) = 0,

cnII(R, 0) =H

(1) ′n (k0R)Jn(k0R)−H

(1)n (k0R)J

′n(k0R)

m2H(1) ′n (k0R)Jn(mk0R)−mH

(1)n (k0R)J ′

n(mk0R),

dnII(R, 0) = 0. (1.49)

16

1.4. Far field theory

As stated before, the cross terms disappear, which means that all waves inCase I are TM and all waves in Case II TE.

1.4 Far field theory

In principle the scattering theory derived in the previous sections is comple-te and everything one wants to know can be derived from it. However, inorder to make predictions about measurable quantities, in this section thetheory will be put in an applicable form.It is important to realize that usually the experimental measurements are do-ne at a large distance from the scattering object(s), so in the first paragraphgeneral expressions for the fields in this region are derived. Subsequent-ly measurable quantities (cross sections, efficiency factors) are defined andapplied to the situation of scattering by an infinite cylinder. The theorydepicted here is derived in a detailed form by Bohren and Huffman [11].More intuitive approaches are found in [7],[8].

1.4.1 Far field approximation

As stated in section 1.3 the scattered wave is associated with the first Hankelfunction, based on the fact that the wave has to be an outgoing wave. At lar-ge distances from the cylinder the first Hankel function can be approximatedby its asymptotic expression:

H(1)n (z) ∼

√2πz

eiz(−i)ne−iπ/4, | z |À n2. (1.50)

This is the only ingredient needed to approximate the scattered part of thefields at large distances from the wire.Consider for this purpose equation (1.40) for the scattered electrical field.In the far field approximation (lρ À 1) the Hankel functions in this expres-sion are approximated by (1.50). After elaboration of the derivatives andneglecting all terms ∼ 1

lρ√

lρ, which fall of much faster then the terms ∼ 1√

lρ,

this results in:

Esca I ∼ −E0e−iπ/4

√2

πlρei(lρ−hz−ωt)

∞∑n=−∞

(−1)neinφ [anI eφ + bnI(sin θeρ + cos θ)ez] .

(1.51)

This is the result for an incident wave with the magnetic field perpen-dicular to the wire axis (Case I). For Case II, when the incident field is TE(the electric field perpendicular to the wire axis) the asymptotic expressionof the scattered field has the same form, apart from changing anI into −anII

and −bnI into bnII .

17


Equation (1.51) shows that the surfaces of constant phase, or wavefronts,of the scattered wave obey

ρ cos θ − z sin θ = C, C ∈ R, (1.52)

which represents cones of half-angle θ and apexes at z = −C/ sin θ. Inclu-ding the e−iωt factor, the scattered wave can be visualized as a cone slidingdown the cylinder [11].

1.4.2 Poynting vector and electromagnetic energy rates

One of the most important properties of electromagnetic (EM) waves is theflux of EM energy through a certain area. In the case of light scattering ata particle not only the magnitude of this flux has to be specified, but alsoits direction. This is given by the Poynting vector S = c

8πReE × H∗,which defines the time-averaged flux of energy crossing a unit area. As aconsequence the rate of EM energy crossing a plane surface A, with normalunit vector n, is equal to

∫S · n dA.

For a surface A which encloses a volume V the net rate W at which EMenergy crosses the boundary A is defined as

W = −∮

AS · n dA, n ≡ unit normal outward toA. (1.53)

This is a definition in the sense that the minus sign ensures that W is positiveif there is a net rate of EM energy flowing into the volume V (S · n < 0),so in the case of absorption of EM energy in the volume.

Denoting the incident and scattered EM fields in the same way as before,the Poynting vector at any point outside the particle can be written in thesefields as:

S =c

8πRe(E0 + Esca)× (H∗

0 + H∗sca) = S0 + Ssca + Sext ,

where

S0 =c

8πReE0 ×H∗

0 , Ssca =c

8πReEsca ×H∗

sca , (1.54)

Sext =c

8πReE0 ×H∗

sca + Esca ×H∗0) .

The decomposition in (1.54) nicely shows that, next to the expected Poyn-ting vectors of the incident (S0) and scattered (Ssca) fields, a term Sext

arises which describes the interaction between the incident and scatteredwaves.

To be more precise, it turns out that Sext represents the removal of ener-gy from the incident light waves, the extinction. Consider for this purposean imaginary sphere of radius a and surface A around a particle of finitesize . The rate of energy Wabs absorbed within the sphere equals the energy

18


rate absorbed by the particle because the surrounding medium is supposedto be non-absorbing. Wabs is given by equation (1.53), now with ea theoutward unit normal to the sphere and may be decomposed in:

Wabs = W0 −Wsca + Wext ,

where

W0 = −∮

AS0 · ea dA , Wsca =

∮

ASsca · ea dA , (1.55)

Wext = −∮

ASext · ea dA .

The choice of the minus signs here again ensures that all the energy ratesare positive, note in particular Ssca · ea > 0. Furthermore the energy rateW0 associated with the incident wave vanishes for a non-absorbing medium,so

Wext = Wabs + Wsca , (1.56)

which shows that Wext indeed represents the extinction, namely the sum ofthe energy scattering rate and energy absorbing rate.

In case of an infinite cylinder the imaginary sphere in the precedingargumentation has to be replaced by an imaginary surrounding cylinder ofinfinite length. Now it is convenient to look at the rate of EM energy flow perunit length , since this quantity is finite. Furthermore, infinite cylinders don’texist except as an idealization, so the statements here have to be carefullyapplied to the situation of a cylinder long compared with is diameter, asused in the previous sections. This is possible if edge effects are negligible,such that there is no net contribution to Wabs from the ends of the imaginarycylinder.In that case, denoting r as the radius of the constructed cylinder and a lengthL equal to the length of the cylinder, the expressions for the scattering andextinction rates of EM energy per unit length become

Wsca/L =∫ 2π

0(Ssca)ρ ρ dφ |ρ = r ,

Wext/L =∫ 2π

0(Sext)ρ ρ dφ |ρ = r , (1.57)

with (Ssca)ρ and (Sext)ρ the (positive) radial components of the expressionsderived in (1.54). On physical grounds the absorption energy rate has tobe independent of r if the medium outside the cylinder is non absorbing.Indeed, with the far field solution of (1.51), the r dependence in (1.57) dropsout.

19


1.4.3 Cross sections and efficiencies

In stead of using the energy rates it is more convenient to take the normalizedforms of them: cross sections, or, better, efficiency factors. The former aresurfaces, defined as

Cabs =Wabs

I0, Csca =

Wsca

I0, Cext =

Wext

I0, (1.58)

where I0 = c8π |E0|2 is the incident intensity. Dividing these optical cross

sections by the geometrical cross section G, dimensionless efficiency factorsare found:

Qabs =Cabs

G, Qsca =

Csca

G, Qext =

Cext

G. (1.59)

Note that equation (1.56) has a synonym in terms of efficiency factors:

Qext = Qsca + Qabs . (1.60)

For a circular cylinder with radius R and length L the geometrical cross sec-tion equals 2RL. Note that the efficiency factors indeed are dimensionless.

With the far field scattered electric field (1.51) and a similar expressionfor the scattered magnetic field now it is a matter of patience to derive:

Qsca I =1

πx

∫ 2π

0(|T11(π − φ)|2 + |T12(π − φ)|2) dφ

=2x

|b0I |2 + 2

∞∑

n=1

(|bnI |2 + |anI |2)

, (1.61)

Qext I =2x

ReT11(π = φ)

=2x

Re

b0I + 2

∞∑

n=1

(bnI)

, (1.62)

Qsca II =1

πx

∫ 2π

0(|T22(π − φ)|2 + |T21(π − φ)|2) dφ

=2x

|a0II |2 + 2

∞∑

n=1

(|bnII |2 + |anII |2)

, (1.63)

Qext II =2x

ReT22(π = φ)

=2x

Re

a0II + 2

∞∑

n=1

(anII)

, (1.64)

20


where

T11(π − φ) ≡∞∑

n=−∞bnIe

−in(π−φ),

T22(π − φ) ≡∞∑

n=−∞anIIe

−in(π−φ),

T12(π − φ) ≡∞∑

n=−∞anIe

−in(π−φ),

T21(π − φ) ≡∞∑

n=−∞bnIIe

−in(π−φ) (1.65)

are the four components of the amplitude scattering matrix T, as defined byKerker [8] and Bohren & Huffman [11]. 1 2

Two general features can be mentioned from (1.61)-(1.64), the efficiencyfactors for light falling obliquely on a cylinder long compared to its radius:

• The efficiencies are expansions in the size parameter kR of the particlein question.

• The extinction quantities only depend on the scattering amplitudes inthe forward direction (φ = π), while it contains the effect of scatteringin all directions by the particle. This is a particular form of the opticaltheorem and a intuitive explanation is given in [7], [11].

The efficiencies Qabs, Qsca and Qext are the main quantities which can bemeasured in optical experiments. If the resolution in a particular experimentis high enough, also the differential efficiencies dQsca/dφ can be estimated,which are given by

dQsca I/dφ =1

πx(|T11(π − φ)|2 + |T12(π − φ)|2),

dQsca II/dφ =1

πx(|T22(π − φ)|2 + |T21(π − φ)|2). (1.66)

They specify the angular distribution of the scattered light.

It is important to note that the efficiencies defined here in principle cantake values larger then unity, contrary to what one should expect from themeaning of the word ”efficiencies”. In particular it can be shown that in thegeometrical limit, i.e. if all the dimensions of the scattering object are much

1The expressions for Qext are derived after quite a lot of algebraic work [11], it can bedone faster by using the optical theorem in advance [7].

2The transformation of φ to π − φ comes from the definition of the incident wave: asin [11] the incident wave is in the −ex direction, while Van de Hulst [7] and Kerker [8]use the opposite and no transformation is needed.

21


larger then the wavelength, the extinction efficiency approach the limitingvalue two. This is rather peculiar, because it suggests that the object remo-ves twice the energy that is incident on it. This so called extinction paradoxis resolved by taking also diffraction into account: the edge deflects rays inits neighborhood which from a geometrical view would have passed undis-turbed. In this way the incident wave is influenced beyond the geometricalsize of the scattering particle.

22

Chapter 2

Small dielectric cylinders

As stated in chapter 1, it is not possible to express the general solution of thescattering problem explicitly as a function of the material properties (die-lectric constant, wire radius), geometric configuration (angle of incidence,radial distance from cylinder) and the wave number of the incident light. Inthis chapter the special case of cylindrical wires with radius small comparedto the wavelength of the incident light will be treated. It will be shown thatin this approximation it is possible to get an analytic solution. In section 2.4numerical results are given for InP.

2.1 Coefficients in Rayleigh approximation

When the radius of the cylinder is sufficiently small compared to the wave-length of the incident light, the Bessel functions appearing in the scatteringcoefficients can be expanded in terms of kR. To be precise, sufficiently smallmeans the following condition:

|m|x ¿ 1, (2.1)

where

x ≡ k0R (2.2)

is defined as the size parameter of the circular cylinder. This condition isphysically based on the two Rayleigh assumptions:

• The wave behavior of the incident field can be neglected with respectto the size of the particle: x ¿ 1. This implies the external field canbe considered as an homogeneous field.

• The applied field should penetrate so fast into the particle that thestatic polarization is established in a time t short compared to theperiod T , so t/T ¿ 1. Since the velocity inside the cylinder is c/mand the wave period T = 1/ck this assumption is satisfied by (2.1).

23

Chapter 2. Small dielectric cylinders

With condition (2.1) the following expressions for the Bessel and firstHankel functions can be used:

J0(z) ' 1− z2

4 , J ′0(z) ' −z

2,

J1(z) ' −J ′0(z) , J ′1(z) ' 12− 3

16z2 ,

H0(z) ' 1 + 2iπ γ + 2i

π log z2 , H ′

0(z) ' 2i

π

1z

,

H1(z) ' −H ′0(z) , H ′

1(z) ' 2i

π

1z2

, (2.3)

where | z | ¿ 1, H denotes the first Hankel function and γ is Euler’s con-stant.The Hankel functions in 2.3 are expanded to zeroth order in z, because thisis sufficient for a second order approximation of the scattering coefficients.With these expansions it can be easily shown that the scattering coefficients(1.41) and (1.42) up to second order in the size parameter x are approxima-ted by:

Case I

a0I(x, θ) = 0,

a1I(x, θ) =πx2

4(m2 − 1)(m2 + 1)

sin θ +O(x4),

b0I(x, θ) = − iπx2

4(m2 − 1) cos2 θ +O(x4),

b1I(x, θ) = − iπx2

4(m2 − 1)(m2 + 1)

sin2 θ +O(x4), (2.4)

Case II

a0II(R, θ) = O(x4),

a1II(R, θ) = − iπx2

4(m2 − 1)(m2 + 1)

+O(x4),

b0II(R, θ) = 0,

b1II(R, θ) = −πx2

4(m2 − 1)(m2 + 1)

sin θ +O(x4), (2.5)

This result is in agreement with the earlier work of Wait [9], [10] andalso gives the expressions derived by Van de Hulst [7] and Kerker [8] fornormal incidence (θ = 0) . The internal coefficients for the fields inside thewire, cnI , dnI , cnII and dnII are not explicitly shown here because they havea complex form. They follow directly from equations (1.41) and (1.42).In principle now it is possible to proceed further and use equations (2.4) and(2.5) for the approximation of the fields (equations (1.21)-(1.26)). However,it is really important to be careful, because an expansion of the fields tosecond order in the size parameter needs more and further expanded coeffi-cients then showed above.

24

2.2. Fields inside the wire

2.2 Fields inside the wire

To our best knowledge the expressions for the fields inside a dielectric cy-linder have only been derived in the dipole limit x → 0 [7][13]. This is arelatively small result compared to the the huge amount of research donein the far field region outside the scattering object, where a lot of interestin particular for applications in meteorology and astronomy worked as adriving force.

In the dipole approximation, also used by Wang and Lieber [1], theincident field is really taken to be homogeneous. It is a special, strongerform of the Rayleigh approximation discussed above, since the second ordercorrections now are completely neglected. In this way the expressions forthe fields are independent of the size parameter and are derived in terms ofthe incident fields as [7][13]:

Eint ‖ = E0 ‖ , (2.6)

Eint ⊥ =2

1 + m2E0⊥ . (2.7)

Here we will extend this solution to finite values of the size parameterx. Before doing this care has to be taken by expanding the coefficients, asnoted before. This is because the internal fields also depend implicitly on thesize parameter via ρ, apart from the explicit dependence via the coefficients.This implicit dependence can be split up in two parts:

• The internal fields are expressed in terms of Jn(jρ), see (1.30)-(1.37).In second order this results in a ρ dependence by (2.3).

• Some of the components of the fields (1.21)-(1.26) have an extra 1/ρdependence.

Taking this into account for a second order approximation of the internalfields one needs the internal coefficients cnI , dnI , cnII and dnII up to thefollowing orders in x:

n 0 1 2 3 ...order 3 2 1 0 ...

Now it is a matter of mathematics to get the solution of the fields insi-de the wire up to second order. Since the expressions for oblique incidenceare too complex to show in an illuminating way, only the results for normalincidence are showed below.

Starting with Case I, where for normal incidence Eint I ρ(x, 0) and Eint I φ(x, 0)are directly zero (see end of section 1.3), the z component of the electric field

25


becomes

Eint ‖ z (x, 0) = E0 e−iωt

1− imk0ρ cosφ− m2k20ρ

2

2cos2 φ +

14(m2 − 1)(1− ρ2

R2) x2 + (2.8)

14(m2 − 1)

(−2γ + iπ − 2 log

x

2

)x2

+O(x3).

As required, this solution reduce to (2.6) in the dipole limit x → 0(note: mk0ρ = x ρ

R → 0). Also the limit m → 1 provides the desired resultEint I = E0 I : for m = 1 there is no optical difference between inside andoutside any more.Furthermore, the part between brackets in equation (2.8) can be divided inthree parts, each written on a different line here and each with a differentphysical background:

• The first part (including the E0 e−iωt term in front of the brackets)expresses the original wave behavior of the incident field: it is theexpansion of e−imkρ cos φ up to second order.

• As the cylinder radius increases, the effect of optical focusing gets amore important role. This is described by the second part: it has itsmaximum in the middle of the cylinder and falls off quadratically tozero at ρ = R. Note that this term is quadratic in the size parameter.

• Also the third part is quadratic in x, but in contrast to the second termconstant over the wire. It has its origin in the expansion of H0(kρ), see(2.3). It is a rather striking expression: the iπ part in it can be seen asa constant phase shift of the field. Roughly speaking it is responsiblefor a correction on the absorption: taking the absolute value squaredthis iπ part mixes with the complex part of m and decreases the fluxof energy crossing the cylinder.Note that the log x

2 part together with the x2 after the brackets isfinite: limx→0 x2 log x = 0. It gives positive contribution to Iint =|Eint I z (x, 0)|2 that can be large enough to get a value for Iint/I0

larger then unity. This is an example of the extinction paradox, aswill be explained further in the next sections where a link will bemade between internal quantities and the external efficiency factors.

The components of the electric field in Case II show mainly the samefeatures as mentioned above. Remember from section 1.3 that the internalelectric field is perpendicular to the wire axis (TE) at normal incidence, soEint II z(x, 0) = 0. The other two components become

26

2.3. Efficiency, polarization anisotropy and - contrast in Rayleigh approximation

Eint II ρ (x, 0) = sinφ2

(m2 + 1)E0 e−iωt

1− imk0ρ cosφ− m2k2

0ρ2

2cos2 φ +

18(m2 − 1)(1− ρ2

R2) x2 + (2.9)

14

(m2 − 1)(m2 + 1)

(12− 2γ + iπ − 2 log

x

2

)x2

+O(x3),

Eint II φ (x, 0) = cosφ2

(m2 + 1)E0 e−iωt

1− imk0ρ cosφ− m2k2

0ρ2

2cos2 φ +

38(m2 − 1)(1− ρ2

R2) x2 + (2.10)

14

(m2 − 1)(m2 + 1)

(−(m2 +

12)− 2γ + iπ − 2 log

x

2

)x2

+O(x3).

Again, the terms on the first line describe the original wave behavior ofthe incident field. The incident TE field was taken to be in the positive ey

direction and decomposing this in cylindrical coordinates one gets indeedthe first terms in the expressions (2.9) and (2.10).

2.3 Efficiency, polarization anisotropy and - con-trast in Rayleigh approximation

Contrary to the quantities inside the wire, in the far field region it is possi-ble to get simple analytic expressions in Rayleigh approximation at obliqueincidence. For expansion of the scattering and extinction efficiencies (1.61)-(1.64) to third order in the size parameter x, the coefficients have to beestimated to second and fourth order for Qsca and Qext, respectively.

The third order term for Qext I and Qext II are too extended to showhere, but are included in all calculations of the next section.With this in mind the efficiencies (1.61)- (1.64) are approximated by:

Qsca I (x, θ) = |m2 − 1|2

cos4 θ +2 sin2 θ(1 + sin2 θ)

|m2 + 1|2

π2x3

8+O(x5) ,

(2.11)

Qsca II (x, θ) =|m2 − 1|2|m2 + 1|2

2− cos2 θ

π2x3

4+O(x5) , (2.12)

Qext I (x, θ) = Imm2 − 1

cos2 θ +4 sin2 θ

|m2 + 1|2

πx

2+O(x3), (2.13)

Qext II (x, θ) =Imm2 − 1|m2 + 1|2 2πx +O(x3) . (2.14)

27


As required, in the limit m → 1 all efficiencies become zero: the refractiveindex is the same everywhere and there is no scattering any more. Also inthe dipole limit x → 0 the efficiencies become zero: in the far field regionthe scattered field can be neglected with respect to the incident field.

The solutions (2.11)-(2.14) depend in a specific way on the angle ofincidence θ, which will be illustrated in the next paragraph. It has its originin the boundary conditions (1.9)-(1.12) on the fields.Apart from this the limit θ → π

2 gives an extra requirement: in this limitthe difference between Case I and Case II has to vanish as can be arguedwith symmetry arguments. This can be seen by taking θ = π

2 in Figure 1.1.In both cases the incident fields E0 and H0 become perpendicular to thewire axis and by rotational symmetry around this axis Case I and Case IIdescribe the same situation. Indeed, taking the limit θ → π

2 the efficienciesin Case I and Case II are equal:

Qsca I (x,π

2) = Qsca II (x,

π

2) =

|m2 − 1|2|m2 + 1|2

π2x3

2, (2.15)

Qext I (x,π

2) = Qext II (x,

π

2) =

Imm2 − 1|m2 + 1|2 2πx +O(x3) . (2.16)

It is a limit in the sense that an incident wave in the same direction as thewire axis (θ = π

2 ) needs a special treatment. How to consider light incidenton the endpoints of an (relatively) infinite cylinder? In fact this is the situ-ation of wave guiding, which will not be treated in this paper.

Furthermore, the solutions (2.11)-(2.14) can be compared to literaturefor θ = 0. At normal incidence they are in agreement with the efficiencyfactors derived by Van de Hulst [7] and Kerker [8]:

Qsca I (x, 0) = |m2 − 1|2 π2x3

8+O(x5) , (2.17)

Qsca II (x, 0) =|m2 − 1|2|m2 + 1|2

π2x3

4+O(x5) , (2.18)

Qext I (x, 0) = Imm2 − 1πx

2+O(x3), (2.19)

Qext II (x, 0) =Imm2 − 1|m2 + 1|2 2πx +O(x3) . (2.20)

2.3.1 Polarization anisotropy, polarization contrast

In order to express the difference between incident TM (Case I) and TE(Case II) waves properly, it is common to define a polarization anisotropyρ [1] [26]. Up to now, for dielectric cylinders this quantity has mainly been

28

2.3. Efficiency, polarization anisotropy and - contrast in Rayleigh approximation

estimated by looking at the internal fields at normal incidence and in thedipole limit. Denoting the polarization anisotropy in this special case byρint, it is defined by:

ρint ≡ |Eint I |2 − |Eint II |2|Eint I |2 + |Eint II |2 , (2.21)

where |Eint|2 = Iint indicates the internal intensity in dipole approximation.By using the fields in dipole approximation (2.6) and (2.7) this yields

ρint =|m2 + 1|2 − 4|m2 + 1|2 + 4

. (2.22)

In principle one could proceed further by using the expressions (2.8), (2.9)and (2.10) to get the expanded intensities and so an expansion of the polari-zation anisotropy ρint (x, 0) inside the wire , but in fact this last step requiresfar too much calculations: also the direction of the field has to be taken into account properly. Even harder, the intensity (or Poynting vector) startsto depend on the position in the wire.

Instead, it is much easier to calculate the polarization anisotropy bymaking use of the efficiency factors in the far field region. In terms of theextinction efficiencies, the extinction polarization anisotropy is defined by

ρext ≡ Qext I −Qext II

Qext I + Qext II, (2.23)

A formal prove that this ratio in general equals the polarization anisotropyinside the wire, ρext = ρint, is complicated and will not be given here. In-stead, the equality can be explained by the following argument: the internalfields are modified with respect to the incident field both by scattering andabsorption, so ρint contains the relative difference of the total removal ofenergy. This is nothing else than the relative difference in extinction bet-ween the two cases, which is described by ρext.Contrary to the general case, it is easy to prove the equality at normalincidence in the dipole limit. Using equations (2.19) and (2.20), ρext isapproximated by

ρext (x, 0) =|m2 + 1|2 − 4|m2 + 1|2 + 4

+O(x2) = ρint +O(x2) , (2.24)

so taking the dipole limit x → 0 on both sides yields ρext = ρint.Next to the extinction polarization anisotropy defined above, it is also

insightful to define scattering and absorption polarization anisotropies. Sin-ce scattering, absorption and extinction are related to each other by (1.60)only the scattering polarization anisotropy will be treated here. It is definedby

ρsca ≡ Qsca I −Qsca II

Qsca I + Qsca II. (2.25)

29


Using (2.17) and (2.18) the scattering polarization anisotropy at normalincidence is expanded in the size parameter by

ρsca (x, 0) =|m2 + 1|2 − 2|m2 + 1|2 + 2

+O(x2) , (2.26)

Although often used, the polarization anisotropy is not a quite usefulquantity to work with in practice. This will be illustrated in the next sec-tion, but at this stage it is anticipated by introducing a new quantity thatdescribes the difference between the case of incident TM waves and incidentTE waves. It is called the polarization contrast and defined by

Cext ≡ Qext I

Qext II, (2.27)

Csca ≡ Qsca I

Qsca II, (2.28)

for the total removal of EM energy and for scattering, respectively.Again, using the expanded efficiencies (2.17)-(2.20) the approximations

at normal incidence are

Cext (x, 0) =|m2 + 1|2

4+O(x2) , (2.29)

Csca (x, 0) =|m2 + 1|2

2+O(x2) . (2.30)

Note that the limit m → 1 does not work for the scattering polarizationanisotropies and contrasts any more.

30

2.4. Results

2.4 Results

As an illustration of the results in the previous sections it is insightful tochoose a particular material, InP for example. In particular it is interestingfor which radius (or size parameter) the Rayleigh approximation is valid.Recall that the complex refractive index is the only material property ap-pearing in the classical theory derived here, apart from R. It depends onthe circular frequency ω of the incident light, see section 1.1 : the materialresponds to the incident periodic EM field and this response depends on thefrequency and so on the wavelength of the incident light.For InP this is illustrated in Figure 2.1. Actually the figure shows the realand imaginary parts ε′ and ε′′ of the complex dielectric function ε [11], whichis related to the complex refractive index by

m2 ≡ ε ≡ ε′ + i ε′′. (2.31)

Absorption and scattering are more simply described by these optical ”con-stants”, so from now on all quantities are discussed in terms of ε.

350 400 450 500 550 600Λ

0

2.5

5

7.5

10

12.5

15

17.5

Ε', Ε''for InP

Ε'

Ε''

Figure 2.1: Bulk values of the real and imaginary part ε′ and ε′′ of thecomplex dielectric function ε for InP, as a function of λ0. It is calculatedby interpolating between 32 (optic) experimental values in this interval.

From Figure 2.1 it becomes directly clear that showing the efficienciesas a function of the dimensionless size parameter x is misleading: it reallymatters if x is changed by varying the wave number or the radius. At dif-ferent wave numbers also ε has changed, only over a narrow range at smallvalues of k the optical constants can be considered as constant.

However, in literature it is common to show the efficiency as a functionof x at a fixed ε, mainly because it is the most convenient way. This is also

31


the starting point in this paper, see Figure 2.2. For six fixed values of λ0 , sofor six different values of ε, the extinction efficiencies in Case I and Case IIup to third order in x at normal incidence are plotted as a function of x.

0 0.01 0.02 0.03 0.04 0.05x

0

0.2

0.4

0.6

0.8

1

1.2

1.4

QIext

HaL: QIext, Θ = 0

354 nm

376 nm

395 nm

422 nm

508 nm

608 nm

0 0.01 0.02 0.03 0.04 0.05x

0

0.005

0.01

0.015

QIIext

HbL: QIIext, Θ = 0

354 nm

376 nm

395 nm

422 nm

508 nm

608 nm

Figure 2.2: Extinction efficiencies Qext I and Qext II at normal incidenceas a function of the size parameter x = kR. In every plot k, and so ε,is fixed. The corresponding values for the wavelength are showed in theleft corner. The efficiencies are expanded up to third order in x.

In the illustrated domain the linear terms (2.19) and (2.20) in the ex-tinction efficiencies dominate and the total removal of EM energy from theincident beam increases with the size parameter.The slope of this relation depends on the wavelength and in a quite differentway for the two cases. It is explained by looking closely to the dependenceon the complex dielectric function:

• In Case I the slope increases to a maximum around λ0 = 400 nmafter which it decreases for increasing wavelength. This is caused bythe factor Imm2 − 1 = ε′′ in (2.19), see Figure 2.1.

• In Case II the slope decreases in the whole domain for increasing λ0.The increase of the denominator of Imm2−1

|m2+1|2 ∝ ε′′(ε′)2+(ε′′)2 in (2.20)

dominates the increase of the numerator to ∼ 400 nm. Afterwardsε′′ falls off so fast that the slope remains decreasing for increasingwavelength.

It even seems from the figure that the third order terms can be neglectedin the given domain. Nevertheless, a closer look gives the opposite: forincreasing x the third order terms start to give significant corrections. Thisis illustrated in Figure 2.3: here the correction by the third order terms withrespect to (2.19) and (2.20) are shown in percentages.

The figure reveals that also the magnitude of the deviation depends onthe wavelength: in both cases the influence of the third order correctionbecomes larger for increasing wavelength. This is explained with the same

32

2.4. Results

0 0.01 0.02 0.03 0.04 0.05x

0

5

10

15

20

%HaL: dev. QIext, Θ = 0

354 nm

376 nm

395 nm

422 nm

508 nm

608 nm

0 0.01 0.02 0.03 0.04 0.05x

0

2

4

6

8

%

HbL: dev. QIIext, Θ = 0

354 nm

376 nm

395 nm

422 nm

508 nm

608 nm

Figure 2.3: Deviation of linear behavior in percentages of Qext I andQext II at normal incidence, as a function of x.

kind of arguments as for the extinction factors itself, but it is omitted heresince the third order corrections are not shown explicitly.

In the same line as for the extinction efficiencies also the scattering andabsorption efficiencies can be illustrated. Since these quantities are depen-dent of each other only Qsca I and Qsca II are showed here. Figure 2.4 showsthat for small x the extinction is completely dominated by the absorption:the contribution of the scattering to the total extinction (Figure 2.2) is onlyabout 1%. This is due to the absence of a first order term in the expansion

0 0.01 0.02 0.03 0.04x

0

0.002

0.004

0.006

0.008

0.01

0.012

QIsca

HaL: QIsca, Θ = 0

354 nm

376 nm

395 nm

422 nm

508 nm

608 nm

0 0.01 0.02 0.03 0.04x

0

0.00002

0.00004

0.00006

0.00008

QIIsca

HbL: QIIsca, Θ = 0

354 nm

376 nm

395 nm

422 nm

508 nm

608 nm

Figure 2.4: Scattering efficiencies Qsca I and Qsca II at normal incidenceas a function of x. In every plot k, and so ε, is fixed. The correspondingvalues for the wavelength are showed in the left corner. The efficienciesare expanded to third order in x.

of Qsca compared to Qext, see equations (2.17)-(2.20).The correction to the third order terms in Figure 2.4 are depicted in

Figure 2.5. This is the deviation caused by the x5 terms in percentages.The calculation of these x5 terms is only done for scattering, since in thiscase the coefficients are needed to x4 while for extinction one needs also thesixth order terms.

33


0 0.01 0.02 0.03 0.04x

0

1

2

3

4

%HaL: err. QIsca, Θ = 0

354 nm

376 nm

395 nm

422 nm

508 nm

608 nm

0 0.05 0.1 0.15 0.2x

0

1

2

3

4

5

%

HbL: err. QIIsca, Θ = 0

354 nm

376 nm

395 nm

422 nm

508 nm

608 nm

Figure 2.5: Correction to the third order expansion of Qsca I and Qsca II

by the x5 terms in percentages; again at normal incidence, as a functionof x.

The above illustrations are useful for determining the range of the sizeparameter in which a certain approximation is valid. For instance, Figure 2.5shows that for a maximal deviation of 5% the third order approximationholds to x ∼ 0.03 and x ∼ 0.2 for Qsca I and Qsca II , respectively.

However, this way of displaying is quite awkward for investigating thewavelength dependence: the different curves belong to different values of k.Actually, keeping the wire radius fixed requires looking at a smaller x valueby going to a curve at higher wavelength.

Also the determination of the limiting R values will not work properly.A size parameter x ∼ 0.03 at λ0 = 400 nm gives R ∼ 2.2 nm, but it would bemuch more convenient to get these values as a function of the wavelength.Before doing this, the next paragraph will illustrate the dependence on theangle of incidence.

2.4.1 Efficiencies and polarization anisotropy at oblique in-cidence

The most dominant feature that will appear by displaying the efficienciesas a function of the angle of incidence θ is the symmetry requirement forθ = π

2 , as explained in section 2.3.Starting with extinction, it is important to note that also the x3 terms

are encountered in Figure 2.6. This means for instance that the first orderterm (2.14), which is independent of θ, is corrected a little bit by the thirdorder term. The scattering efficiencies are shown in Figure 2.7. In bothcases the cylinder radius is fixed, R = 2 nm. Figure 2.6 illustrates the θdependence of Qext I and Qext II as given in equations (2.13) and (2.14).The difference between Qext I and Qext II at θ = 0 is explained by the largedenominator in (2.14): |m2 +1|2 ' 200. For increasing θ, Qext I decreases toa limiting value which is equal to Qext II at θ = π

2 : the first term in (2.13)

34

2.4. Results

falls off to zero, while the second term increases leading to Qext I ' Qext II

at θ = π2 . Also the scattering efficiencies, shown in Figure 2.7, are the same

at θ = π2 . As discussed before this is due to the fact that the TM and TE

case describe the same physical situation at θ = π2 .

0 Π8

Π

43 Π

8

Π

2

Θ

0

0.2

0.4

0.6

0.8

QIext

HaL: QIext, R = 2 nm

354 nm

376 nm

395 nm

422 nm

508 nm

608 nm

0 Π8

Π

43 Π

8

Π

2

Θ

00.0020.0040.0060.0080.010.0120.014

QIIext

HbL: QIIext, R = 2 nm

354 nm

376 nm

395 nm

422 nm

508 nm

608 nm

Figure 2.6: Extinction efficiencies Qext I and Qext II as a function of theangle of incidence θ for a fixed cylinder radius R = 2 nm. In every plotk, and so ε, is fixed. The efficiencies are expanded to third order in x.

0 Π8

Π

43 Π

8

Π

2

Θ

00.00250.0050.0075

0.010.01250.0150.0175

QIsca

HaL: QIsca, R = 2 nm

354 nm

376 nm

395 nm

422 nm

508 nm

608 nm

0 Π8

Π

43 Π

8

Π

2

Θ

0

0.00005

0.0001

0.00015

QIIsca

HbL: QIIsca, R = 2 nm

354 nm

376 nm

395 nm

422 nm

508 nm

608 nm

Figure 2.7: Scattering efficiencies Qsca I and Qsca II as a function of θfor R = 2 nm. In every plot k, and so ε, is fixed. The efficiencies areexpanded to third order in x.

In order to look more closely to the difference between TM and TEwaves, the extinction polarization anisotropy (2.23) as well as the scatteringpolarization anisotropy (2.25) corresponding to the expanded efficiencies atR = 2nm are depicted in Figure 2.8. Indeed, ρsca and ρext become zero in thelimit θ → π

2 . At normal incidence the polarization anisotropies reach theirmaximum value, around 0.985 for extinction as well as for scattering. Butthe distinction between the curves for different wavelength is hard to extractfrom the figures. Also the difference between extinction and scattering isnot illustrated clearly.

As stated in section 2.3 it is more convenient to look at the polarization

35


0 Π8

Π

43 Π

8

Π

2

Θ

0

0.2

0.4

0.6

0.8

1Ρsca

HaL: Ρ sca for R = 2 nm

354 nm

376 nm

395 nm

422 nm

508 nm

608 nm

0 Π8

Π

43 Π

8

Π

2

Θ

0

0.2

0.4

0.6

0.8

1

Ρext

HaL: Ρ extfor R = 2 nm

354 nm

376 nm

395 nm

422 nm

508 nm

608 nm

Figure 2.8: Polarization anisotropy ρext and scattering polarization ani-sotropy ρsca as a function of θ for R = 2 nm. In every plot k, and so ε,is fixed. The factors are expanded up to second order in x.

contrast, equations (2.27) and (2.27). For R = 2 nm this is displayed inFigure 2.9. Now the difference between scattering and extinction becomesclear: at normal incidence the depolarization for scattering is significantlarger then for extinction. By rotating the angle of incidence to θ = π

2 thescattering polarization anisotropy also decreases faster to the limiting valuezero. This effect is also visible in Figure 2.8, but less clearly.

0 Π8

Π

43 Π

8

Π

2

Θ

0

50

100

150

200

250

Csca

C sca for R = 2 nm

354 nm

376 nm

395 nm

422 nm

508 nm

608 nm

0 Π8

Π

43 Π

8

Π

2

Θ

020406080100120140

Cext

C extfor R = 2 nm

354 nm

376 nm

395 nm

422 nm

508 nm

608 nm

Figure 2.9: Polarization contrast Cext and scattering polarization con-trast Csca as a function of θ for R = 2 nm. In every plot k, and so ε, isfixed. The factors are expanded up to second order in x.

2.4.2 Efficiencies and polarization anisotropy as a functionof wavelength

As stated in section 2.4, the most physically accurate picture of the scatte-ring process is obtained by showing the efficiencies and polarization aniso-tropies as a function of the wavelength (or wave number).Apart from the practical points made in the previous sections, this kind

36

2.4. Results

of figures also contain far more information: for every wavelength a set ofoptical constants has to be used. This is done by interpolating betweenmeasured data points for the complex dielectric function, see Figure 2.1. Inthis way the response of the dielectric cylinder as a function of the frequencyof the incident EM field is obtained.

350 400 450 500 550 600Λ

0

0.5

1

1.5

2

2.5

QIext

HaL: QIext, Θ = 0

1 nm

2 nm

3 nm

4 nm

5 nm

350 400 450 500 550 600Λ

0

0.01

0.02

0.03

QIIext

HbL: QIIext, Θ = 0

1 nm

2 nm

3 nm

4 nm

5 nm

Figure 2.10: Extinction efficiencies Qext I and Qext II at normal inci-dence as a function of the wavelength at constant cylinder radii. Thecorresponding five R values are showed in the right corner. The effi-ciencies are expanded to third order in x. The plots are calculated byinterpolating between thirty-two (optic) experimental values of ε in thisinterval.

350 375 400 425 450 475 500Λ

0

0.05

0.1

0.15

0.2

0.25

0.3

QIsca

HaL: QIsca, Θ = 0

1 nm

2 nm

3 nm

4 nm

5 nm

350 400 450 500 550 600Λ

0

0.00025

0.0005

0.00075

0.001

0.00125

0.0015

QIIsca

HbL: QIIsca, Θ = 0

1 nm

2 nm

3 nm

4 nm

5 nm

Figure 2.11: Scattering efficiencies Qsca I and Qsca II at normal incidenceas a function of the wavelength at constant cylinder radii. The efficienciesare expanded to third order in x.

For the extinction and scattering efficiencies at normal incidence this isdepicted in Figure 2.10 and Figure 2.11, respectively. The domain of thewavelength is limited: the high and low frequency regions are omitted. Thefigures show the dependence of the wavelength at five fixed values of R.The dependence of the cylinder radius is visible, especially for extinction:the extinction efficiencies increase linearly with increasing radius by goingfrom one curve to the next in Figure 2.10.

37


The shape of the curves has been explained above: they where alreadyvisible in Figure 2.2 and Figure 2.4, but not so clearly. Now the particulardependence of (2.19) and (2.20) on the complex dielectric function is reallyvisible. For instance, Figure 2.10 nicely shows that the behavior of theimaginary part ε′ is completely reflected in the extinction efficiency.

350 400 450 500 550 600Λ

0

5

10

15

20

%

HaL: err. QIsca,Θ=0

1 nm

2 nm

3 nm

4 nm

5 nm

350 400 450 500 550 600Λ

0

0.25

0.5

0.75

1

1.25

1.5

%

HbL: err. QIIsca,Θ=0

1 nm

2 nm

3 nm

4 nm

5 nm

Figure 2.12: Correction to the third order expansion of Qsca I and Qsca II

by the x5 terms in percentages; again at normal incidence, as a functionof λ0 at constant R.

In the same way as in Figure 2.5, the correction to the third order termsin Figure 2.11 is illustrated in Figure 2.12. This is the deviation caused bythe x5 terms given as a percentage. For a cylinder radius below 2 nm theused expansion is accurate to 5%. For larger radii it really depends on thewavelength if the approximation is acceptable.

350 400 450 500 550 600Λ

0.98

0.982

0.984

0.986

0.988

0.99

0.992

Ρsca

HaL: Ρ sca , Θ = 0

1 - 5 nm

350 400 450 500 550 600Λ

0.96

0.965

0.97

0.975

0.98

0.985

Ρext

HbL: Ρ ext, Θ = 0

1 nm

2 nm

3 nm

4 nm

5 nm

Figure 2.13: Polarization anisotropy ρext and scattering polarization ani-sotropy ρsca at normal incidence as a function of the wavelength at con-stant cylinder radii. The corresponding five R values are showed in theright corners. Figure (a) shows that ρsca is independent of the wire ra-dius. The polarization anisotropy ρext is expanded including the secondorder term, ρsca up to second order in x.

The polarization ratio and scattering polarization ratio corresponding toFigure 2.10 and Figure 2.11 are shown in Figure 2.13. Figure 2.13 (a) shows

38

2.4. Results

that ρsca expanded up to second order in x is independent of the wire radius,see (2.26). For extinction also the second order is included. Figure 2.13 (b)shows that in that case ρext depends on the wire radius: in particular forwavelengths larger then 400 nm the polarization ratio becomes larger. Inother words, increasing the wire radius implies a bigger difference betweenthe case of an incident wave with the magnetic field perpendicular to thewire axis (TM) and the case where the electric field is perpendicular (TE).Remember that this result only applies for small R, results for larger radiusor size parameter are showed in [7] [8] [11].

It is really interesting to compare the obtained polarization anisotropyρext with the depolarization in the dipole limit ρint (2.22), also used in [1][26].This is shown in Figure 2.14. The blue curve shows the case when for ρint

in the dipole limit only the real part of the bulk ε is taken into account.The difference with the the red curve, indicating ρint for the complete ε,becomes dramatically large for λ below 400 nm, and remains significant forthe other wavelengths.

350 400 450 500 550 600Λ

0.86

0.88

0.9

0.92

0.94

0.96

0.98

Ρ

HaL: Ρ ext, Ρ int; , Θ = 0

Ρ ext,R = 2 nm

Ρ ext,R = 5 nm

Ρ int, dip.app.

Ρ int, dip.app., Ε'

350 400 450 500 550 600Λ

0

20

40

60

80

100

120

140

CextCint

HbL: C ext, C int; Θ = 0

C ext,R = 2 nm

C ext,R = 5 nm

C int, dip.app.

C int, dip.app., Ε'

Figure 2.14: Comparison of the internal polarization anisotropy/contrastin the dipole limit with ρext / Cext. In blue curve, indicating ρint, onlythe real part of ε is taken into account. The red curve shows ρint for thecomplete ε .

The yellow and green curve show that increasing the wire radius to R =5nm already gives a significant difference between the solution of the dipolelimit and the expanded polarization anisotropy ρext. Figure 2.14 (b) showsthe same results in terms of the contrasts.

39

Part II

Absorption

40

Chapter 3

Electronic properties

In this chapter the electronic properties of nanowires made from III-V com-pounds are discussed. The results are based on more detailed studies whichcan be found in basic semiconductor books [14] [15] and articles by Luttinger[16], Sercel [17] and Marechal [18].The electronic band structure and wave functions in a nanowire are calcu-lated using the effective mass approximation. This method is in particularconvenient to study the optical properties of a semiconductor structure, be-cause analytic expressions for the band dispersion, effective mass and elec-tron/hole wavefunctions around high symmetry points can be obtained.Before turning to the nanowire, in section 3.1 first the band dispersion inbulk material will be derived. Next to general theory, the specific situationof the degenerate top valence band in III-V semiconductor materials will betreated. It has its specific importance in the next chapters and will thereforealso be the guideline in the other sections of this chapter: in section 3.2 theeffective mass theory for bulk systems is treated, sections 3.3 and 3.4 sum-marize the envelope description in case of an infinite nanowire and explicitresults for InP and InAs are found in section 3.5.

3.1 The k · p method

There are various ways to determine the electronic bands of a semiconductor.Global dispersion relations of bulk materials are available (pseudo-potentialtechniques, tight binding) but in a lot of cases they are unnecessary.In particular, for describing the optical properties of a semiconductor struc-ture it is often sufficient to know the band dispersion in a small range aroundthe band extremes. This is achieved by the k · p method, which differs fromthe procedures mentioned above in the fact that, next to the band gaps, alsothe oscillator strengths of the transitions are used as input. In the k ·p me-thod the band dispersion around any point ka is obtained by extrapolationfrom the k = ka energy gaps and optical matrix elements, using either de-

41

Chapter 3. Electronic properties

generate or non-degenerate perturbation theory . The input data at k = ka

can be obtained from experimental results, typically at the high symmetrypoints of the crystal.

Starting point is the one-electron Schrodinger equation describing themotion of an electron in an averaged potential V (r), which is obtained fromthe Hamiltonian of a perfect crystal containing N unit cells after usualassumptions such as the Born-Oppenheimer and mean field approximation.The potential V (r) is assumed to reflect the periodicity of the perfect crystal:

V (r + R) = V (r), (3.1)

where R are the lattice vectors. Including the spin-orbit interaction theHamiltonian describing the unperturbed semiconductor becomes

H0 =p2

2m0+ V (r) +

~4c2m2

0

(σ ×∇V ) · p, (3.2)

where m0 denotes the free electron mass and σ are the Pauli spin matrices.The relativistic character of the spin-orbit term is reflected by the 1

c2depen-

dence.Note that the total Hamiltonian, including the spin-orbit interaction, is in-variant under a translation by R. H0 thus commutes with the translationoperator of the crystal and has Bloch functions as solutions. After nor-malizing over the whole crystal, containing N unit cells, these are definedas:

ψnk(r) = N− 12 eik·runk(r), (3.3)

where the unk’s have the periodicity of the lattice, are normalized over oneunit cell and k lies in the first Brillouin zone.

The Bloch functions (3.3) form a complete and orthonormal set. Nextto this, the Bloch solutions ψn0 = N− 1

2 un0 at k = 0 are also periodic. Oncethese, or to be more precise, the corresponding interband matrix elementsand energies εn ≡ εn(0) are known, the energy dispersion around the zonecenter (k = 0) can be derived using perturbation theory.In principle, this argumentation can be extended to any point k = ka,provided the transition matrix elements and energies at k = ka are known.This result has been widely discussed in literature [14] [15] [16] [17] [18],here only the results around the zone center are summarized.

Assuming the band structure has an extremum (almost) at the zonecenter and taking k sufficiently small 1, the dispersion relation for a non-degenerate band (apart from spin) is given by

εn(k) = εn +~2k2

2m∗n

, (3.4)

1Sufficiently small means that the corresponding energy difference εn(k)− εn remainsmuch smaller then the band edge differences εn− εn′ and that the terms linear k are smallenough to be neglected.

42

3.1. The k · p method

where m∗n is the effective mass of the band,

1m∗

n

=1

m0+

2m2

0 k2

∑

n′ 6=n

| πnn′ · k |2εn − εn′

(3.5)

and πnn′ are the the interband matrix elements at the zone center:

πnn′ ≡ 〈un0 | p +~

4m0c2σ ×∇V | un′0〉 (3.6)

=∫

d3r u∗n0(r)(

p +~

4m0c2σ ×∇V

)un′0(r). (3.7)

With the same assumptions, for a degenerate band relation (3.4) is replacedby

hj,j′(k) = εj δj,j′ +~2k2

21

m∗jj′

(3.8)

with

1m∗

jj′=

1m0

δj,j′ +2

m20 k2

∑

εn′ 6=εj

(k · πjm)(k · πmj′)εj − εn′

. (3.9)

Here j denotes the degeneracy and the summation over n′ describes the cou-pling between the group of degenerate states and the other bands. Contraryto the case of a non-degenerate band, one is left with a matrix hj,j′ (k) whichhas to be diagonalized in order to get the dispersion relation(s).

It should be noted that within the notation used here the tensor behaviorof the effective mass is neglected. In general, the coupling between k andthe interband matrix elements causes the effective mass to be non isotropicand inclusion of this effect is achieved by the substitution

1m∗

n

k2 −→∑

αβ

1

m∗ αβn

kαkβ , (3.10)

1

m∗ αβn

=1

m0δαβ +

2m2

0

∑

n′ 6=n

παn′nπβ

nn′

εn − εn′(3.11)

in (3.4) and a similar one in case of a degenerate band. The surfaces ofconstant energy belonging to this effective mass tensor are not spheres anymore, but warped in certain directions, depending on the symmetry proper-ties of the band under consideration.

Furthermore, there are three remarks important to be made at this sta-ge. First, in most of the cases the summation over the bands n′ in (3.5)and (3.9) can be executed over a limited number of values. For large ener-gy differences εn − εn′ the contribution of n′ to the effective mass becomes

43


relatively unimportant. Also the interband matrix elements in the numera-tor reduces the number of bands that contribute to the effective mass. Aswill be explicitly shown in subsection 4.4.1, the matrix elements are subjectto selection rules which are determined by the symmetry properties of thebands in question. Most of the matrix elements become zero by this kind ofsymmetry arguments.

Secondly, including the spin-orbit interaction in case of a non-degenerateband makes little practical difference, since its effect is absorbed in the inter-band matrix elements which are determined by experiment. For a degenerateband this is different, because the spin-orbit interaction in general lifts thedegeneracy and will cause small splitting between the bands.

A last important remark has to be made concerning the limitations of thek·p method as depicted here, up to second order in k. The above results relyon the assumption that εn(k)−εn remains much smaller then the band edgedifferences εn − εn′ (and a similar assumption in case of degenerate bands),which is not necessarily satisfied, e.g. in semiconductor compounds with anarrow band gap. Instead of expanding beyond second order in the sameframework, a commonly used approach [14] [15] [17] initiated by Kane [19]solves this problem by diagonalizing the group of neighboring bands exactlyand afterwards treating the coupling with the well separated other bands ina second order perturbation. However, in the remaining part of this paperit is assumed that the bands under investigation are well separated from theothers, i.e. splitting terms as the band gap Eg and spin-orbit splitting ∆0

are assumed to be sufficiently large.

3.1.1 Top valence bands in III-V semiconductors

In principle the above theoretical statements now can be applied to anyband, or group of bands, once the energy and the interband matrix elementsare known. Here the band structure of the six fold degenerate (includingspin) top valence band at the Γ point (k = 0) in III-V semiconductor com-pounds will be summarized. However, the explicit diagonalization of thematrix (3.8) will be performed further on in the envelope function frame-work since this is the most convenient way when the nanowire structure isanticipated.

Starting with symmetry considerations, it is well known [14][15] that thetop valence bands in III-V materials have Γ4 like symmetry, apart from spin.The corresponding spatial parts of the valence band wavefunctions at k = 0are p-like, which means that they are triply degenerate and transform underrotations like the three components of a vector. Including spin this leads tosix band edge Bloch functions, which are denoted by |X〉|σ〉, |Y 〉|σ〉, |Z〉|σ〉,with σ =↑,↓.

The one-electron Hamiltonian H0 is diagonalized by linear combinationsof these band edge Bloch functions. Rewriting the spin orbit term in (3.2)

44

3.1. The k · p method

as

Hs.o. = λL′ · σ, (3.12)

with L′ the angular momentum of the atomic states and treating Hs.o. as asmall perturbation2, this term is diagonalized by the eigenfunctions of thetotal angular momentum J = L′ + σ of the atomic states. Subsequentlythe total Hamiltonian H0 can be expressed in the transformed zeroth ordereigenfunctions |j, jz〉, with j the eigenvalues of J and jz the eigenvalues ofits projection Jz along the z axis. These are defined as

∣∣32 , 3

2

⟩= − 1√

2|X + iY 〉| ↑ 〉, (3.13)

∣∣32 , 1

2

⟩= − 1√

6|X + iY 〉| ↓ 〉+

√23 |Z〉| ↑ 〉, (3.14)

∣∣32 ,−1

2

⟩= 1√

6|X − iY 〉| ↑ 〉+

√23 |Z〉| ↓ 〉, (3.15)

∣∣32 ,−3

2

⟩= 1√

2|X − iY 〉| ↓ 〉 (3.16)

for the j = 32 quadruplet and

∣∣12 , 1

2

⟩= − 1√

3|X + iY 〉| ↓ 〉+ |Z〉| ↑ 〉, (3.17)

∣∣12 ,−1

2

⟩= − 1√

3|X − iY 〉| ↑ 〉 − |Z〉| ↓ 〉, (3.18)

for the two j = 12 states. The last ones are split from the j = 3

2 states by thespin-orbit interaction, with a magnitude ∆0 = 3

2λ. For ∆0 sufficiently large,such that the matrix elements which couple the j = 3

2 and j = 12 bands are

negligible compared to ∆0, the 6 × 6 Hamiltonian can be decoupled into a4× 4 and a 2× 2 matrix.In most III-V semiconductors, the 4 × 4 matrix of the j = 3

2 states cor-responds to the top most valence band. Assuming the spin-orbit couplinglarge enough, in this paper the valence band dispersion will be derived bydiagonalizing the Γ8 Hamiltonian of these j = 3

2 states. This is achieved inthe same framework as used by Sercel [17] and as in [18]; the explicit resultsare given in section 3.2.As stated above, corrections to this approach can be found by including thesplit-off (Γ7) band of the j = 1

2 states and possibly also the lowest con-duction band, which usually has Γ6 symmetry. The last one in general isless important since in most III-V semiconductors the spin-orbit splitting ismuch smaller than the band gap Eg. Including more bands will improve theresults, but makes the calculations harder. Focussing on the dispersion forsmall k around the zone center, it is assumed that these corrections can beneglected in first instance.

2λ is small because of the relativistic character of the spin-orbit interaction

45


3.2 Effective mass approximation

Suppose an infinite system which is built from the perfect crystal, and adisturbance δV which has to be restricted by specific properties, as will beexplained below. The Schrodinger equation (S.E.) of the system is given by

(H0 + δV ) |Ψ〉 = E |Ψ〉. (3.19)

In principle the solutions of the S.E. can be found by expanding Ψ in termsof the complete orthonormal set of Bloch functions, but without making anyfurther approximation this requires an extensive job since the disturbanceδV breaks the translational symmetry of the crystal.

The problem is solved much easier by assuming δV to be slowly varyingover one unit cell and making use of the band parameters of the unperturbedsystem, equations (3.4) and (3.8). This approach is known as the effectivemass approximation. It can be derived either by utilizing Bloch functions, orin the context of the more localized Wannier functions. Here the Wannierfunctions are used. They are related to the Bloch functions by Fouriertransformation and defined by

anR(r) = N− 12

∑

k

e−ik·Rψnk(r). (3.20)

Note that the Wannier functions are indexed by the lattice vector R, re-flecting the localized character. They form a complete, orthonormal setjust as the Bloch functions and depend on the difference between r and R:anR(r) = an(r −R).Using the complete and orthonormal set of Wannier functions, the solutionΨ(r) of (3.19) is expanded as:

Ψ(r) =∑

jR

Fj(R)ajR(r), (3.21)

where j sums over the j degenerate bands and thus includes only one bandn in the non degenerate case. The functions Fj(R) are known as the en-velope wave functions: as will be shown below, they describe wave packets,extended over (a part of) the crystal and are the envelopes of the atomisticvariations caused by the Wannier functions.

In order to convert the total Hamiltonian H0+δV in (3.19) into operatorsacting on the Wannier functions, it is stated here that k and R are conjugateoperators in the sense that

R ←→ i∇k and k ←→ −i∇R, (3.22)

in the limit of large N . Note that R now is treated as a continuous variable,which is justified by the large N limit, i.e. the size of the semiconductorcompound is much larger than the distance between the atoms.

46

3.2. Effective mass approximation

Using this result and assuming that δV is a slowly varying function withrespect to a lattice vector, it can be shown [14][18] that the S.E. (3.19)reduces to a Schrodinger equation for the envelope functions:

εn(−i∇R) + δV (R)Fn(R) = EFn(R) (3.23)

in case of a non degenerate band n and∑

j′hj,j′(−i∇R) + δV (R)Fj′(R) = EFj(R) (3.24)

for a degenerate band. For a given band, equation (3.23) ((3.24)) describesthe motion of a particle with effective mass m∗

n (m∗jj′) in a potential δV .

Note that the total wave function of this particle, moving in the perturbedcrystal, is obtained from the solutions of (3.23)/(3.24) by multiplying withthe Wannier functions as in (3.21).

3.2.1 Crystal Hamiltonian in envelope representation

The above envelope framework initially was derived in the context of im-purity states, but as stated by Sercel [17], the procedure can also be usedto develop a representation of the unperturbed Hamiltonian H0 which an-ticipates a centrosymmetric or cylindrical heterostructure. Instead of theWannier representation (3.21), the solution is expanded in the zone centerBloch functions |uj〉 by the ansatz

|Ψ〉 =∑

j

|Fj〉 |uj〉, (3.25)

which is justified if the energy difference εj − εn′ between the degeneratebands and al others is sufficiently large such that unk ' un0.The notation used in (3.25) stresses the fact that the envelope functions Fj

act in a different space as the zone center Bloch functions, this is shownin more detail in chapter 4 considering the transition matrix element. TheBloch functions are defined within a unit cell, while the envelopes are exten-ded over a sufficiently large group of lattice points. It should be mentionedagain that the assumption unk ' un0 is essential in this context.

In addition to the assumptions in section 3.13, an extra approximationhas to be made here concerning the anisotropy, in order to profit fully fromthe envelope representation. Conform the situation in most of the III-Vsemiconductor materials, it is assumed that the anisotropic terms in theHamiltonian can be neglected, at least as a first order approximation. Inthis spherical approximation the lower cubic terms causing the warping of

3I.e. k sufficiently small and energy gaps such as ∆0 and Eg large enough to neglectthe coupling of the band(s) under investigation with the others.

47


the bands are set to zero by a restriction on the involved Kohn-Luttingerparameters: γ2 = γ3[17][18].

As an upshot, adopting the spherical approximation amounts to replacethe space group Td of the crystal with the full rotational group. Now thecrystal Hamiltonian is invariant under rotations and additional operatorscan be found which share the same basis of eigenstates. In a cylindricalrepresentation these operators are Pz and Fz, where Pz is the envelopemomentum along the z-axis and Fz denotes the total angular momentumalong the z axis:

Fz = Jz + Lz, (3.26)

with Lz the z component of the envelope angular momentum L. The zcomponent of the total angular momentum is a conserved operator, or inother words, Fz commutes with the crystal Hamiltonian. Consequently,the eigenvalue fz of Fz is a good quantum number and the Hamiltonian isdiagonal with respect to Fz.

3.2.2 Top valence bands in III-V semiconductors

This is illustrated in more detail by narrowing the focus again to the situa-tion of the top Γ8 valence bands in III-V semiconductors.Following the same approach as in section 3.1, j in (3.25) sums over the jz

values of the j = 32 quadruplet and |uj〉 = |32 , jz〉. The envelope functions

|Fj〉 now are represented as |kz; k, m〉, where kz is the eigenvalue of Pz, k de-notes the radial wavenumber and m ε Z are the eigenvalues of the envelopeangular momentum Lz. Making use of Lz = Fz−Jz, the envelope functionsin the cylindrical representation are of the form Jfz−jz(kρ)ei(fz−jz)φeikzz,where Jn(z) is a Bessel function. With 〈ρ φ z|kz; k, fz − jz〉 the envelopefunctions in cylindrical coordinates, this results in

〈ρφ z|kz; k, fz − jz〉 |32 , jz〉 ∝ Jfz−jz(kρ)ei(fz−jz)φeikzz |32 , jz〉. (3.27)

as a basis for the solution (3.25), which is is orthogonal in fz, jz, k and kz.The Hamiltonian HΓ8

Fzof the the top Γ8 valence band in III-V semiconductors

now is expressed in this basis by [17][18]

HΓ8Fz

=

p + q s r 0s p− q 0 rr 0 p− q −s0 r −s p + q

, (3.28)

48

3.3. Envelope description for infinite cylinders

where the basis is ordered with respect to jz as 32 , 1

2 ,−12 ,−3

2 and p, q, rand s are given by

p + q = − ~2

2m0((γ1 + γ2)k2 + (γ1 − 2γ2)k2

z) , (3.29)

p− q = − ~2

2m0((γ1 − γ2)k2 + (γ1 + 2γ2)k2

z) , (3.30)

r =~2

2m0

√3γ2k

2 , (3.31)

s =~2

2m02√

3γ2kkz . (3.32)

This Hamiltonian has two different eigenvalues, corresponding to a heavyhole (HH) and a light hole (LH) band which are degenerate at the zonecenter:

εHH = − ~2

2m0(γ1 − 2γ2)(k2

HH + k2z), (3.33)

εLH = − ~2

2m0(γ1 + 2γ2)(k2

LH + k2z). (3.34)

Both bands are doubly degenerate and the unnormalized eigenvectors aregiven by

|HH1〉 =

k2HH+4k2

z√3k2

HH2kzkHH

10

, |HH2〉 =

01

− 2kzkHH

k2HH+4k2

z√3k2

HH

, (3.35)

|LH1〉 =

−√32kzkLH

10

, |LH2〉 =

01

− 2kzkLH

−√3

, (3.36)

with respect to the basis given in (3.27), ordered as 32 , 1

2 ,−12 ,−3

2 withrespect to jz.

3.3 Envelope description for infinite cylinders

In principle it is possible to apply the effective mass approximation in thecontext of the geometrical configuration of a nanowire. However, as pointedout in [14][18], care has to be taken concerning the foundation of the theo-retical framework developed in the previous sections.

49


In the first place, the effective mass approximation relies on the assump-tion that the potential is a slowly varying function over a unit cell. Imposingthe wire configuration by taking

δV (r) = −V0Θ(R− ρ), (3.37)

with Θ the Heaviside function, this requires the wire radius R to be suf-ficiently large. Intuitively this makes sense directly, for if there are just afew atoms within the wire, the potential change at the boundary of the wirecannot be neglected any more with respect to the interatomic distances a.To be more precise, by rewriting the potential (3.37) in Fourier space, it canbe seen [14][18] that the entire concept of an effective mass is only useful ifaR ¿ 1, the limit in which only the Fourier components δV (k) around thezone center contribute significantly.

Secondly, in the theory of section 3.2 the atomic wavefunctions are assu-med to be the same everywhere. If the effective mass approximation is nottreated in a suitable form, it thus fails to describe in a proper way the he-terostructure situation with two or more completely different environments(e.g. a semiconductor compound in vacuum) and consequently drastic chan-ges in the atomic wavefunctions.This problem is solved in a general way by assuming that every different en-vironment can be described in a large part independently of the other(s)[18].The bands in the different systems are subsequently related to each otherby matching bands with the same symmetry. Instead of equation (3.21), thecorresponding wavefunction is assumed to be of the form

Ψ(r) =∑

jR

Fj(R)a(s)jR(r), (3.38)

where s indicates that for the atomic functions, in this case in Wannier re-presentation, the solutions are taken far in the corresponding system s.

Even if the environments are large enough to approximate them mainlyas bulk systems in this way, it still remains a problem to match the wa-vefunctions of the different systems near the boundaries. For example, itcannot be expected that the atoms around the interface of different systemssimply are positioned at the lattice points of a perfect crystal. The atomicfunctions on both sides of the boundary are not orthogonal to each otherand in case of a semiconductor structure in vacuum, there are even no atomsoutside the structure any more. Another practical point is the oxidation ofthe structure, resulting in a system probably better described as a core shellstructure.

However, the neglect of these effects concerning changes in the atomicwavefunctions and matching of the boundary can be justified by the spatial

50


extent of the envelope functions: by taking an infinite potential well for thenanowire geometry, the envelope functions fall off to zero at the boundary.In this model the atomic functions outside the wire are of no importanceand possible fluctuations around the boundary are neglected because theoverlapping envelope is almost zero. More problems are expected when V0

is finite. In this case the envelope function leaks with a certain extent intothe region outside the wire and the change in atomic wavefunctions plays amore important role.

In the present paper the potential V0 in (3.37) is assumed to be largeenough to consider it as representing an infinite potential well. In case ofan infinite cylinder structure, this leads to a boundary condition on theenvelopes of bulk wavefunctions:

Fj (ρ = R, φ, z) = 0, ∀ j, φ, z. (3.39)

The solutions for this boundary condition can be labeled with a set of quan-tum numbers, say λ, where λ will be specified for the valence band in pa-ragraph 3.4.1 and for the conduction band in paragraph 3.3.2. In addition,for a given band λ the wavefunctions depend on the wavenumber kz.

Denoting the complete labeling with λ kz one obtains a normalizati-on condition for the envelope functions Fλ kz ,j if the total wavefunctionΨλ kz(r) = Cλ kz

∑jR Fλ kz ,j(R)ajR(r) is normalized to unity:

∫dr |Ψλ kz(r)|2 = C2

λ kz

∑

j, j′

∑

R, R′F ∗

λ kz ,j′(R′)Fλ kz ,j(R)

∫dr a∗

j′R′(r)ajR(r)

= C2λ kz

∑

j, j′

∑

R, R′F ∗

λ kz , j′(R′)Fλ kz , j(R)δR′,Rδj′,j

= C2λ kz

∑

j

∑

R

|Fλ kz , j(R)|2 = 1, (3.40)

from which the normalization constant Cλ kz is obtained. In the third stepin equation (3.40) the orthonormality of the Wannier functions is used. Thesummation over R can be replaced by an integral in the same context asequation (3.22). The same normalization condition is obtained using theBloch representation (3.25).

3.3.1 Hole in III-V semiconductor nanowires

With the above remarks in mind consider again the top valence bands of III-V semiconductors. The wire geometry is imposed by the infinite potentialwell:

δV (ρ) =∞, ρ > R,

0, ρ ≤ R.(3.41)

51


This leads to the boundary condition on the envelopes (3.39), with j =jz = 3

2 , 12 ,−1

2 ,−32. Since the bulk heavy- and light hole solutions (3.35)

and (3.36) have four components which cannot be zero simultaneously, thisrequirement (3.39) can be satisfied only if the total wavefunction is a su-perposition of the four bulk heavy- and light hole eigenstates for a givenfz. Consequently, apart from normalization constant (3.40) the envelopewavefunctions are determined by

Fλ kz ,jz(ρ, φ, z) = (vHH1|HH1〉jz + vHH2|HH2〉jz)Jfz−jz(kHHρ) + (3.42)

(vLH1|LH1〉jz + vLH2|LH2〉jz)Jfz−jz(kLHρ) ei(fz−jz)φeikzz ,

where |HH1〉-|LH2〉 are the bulk eigenstates given in (3.35) and (3.36)and vHH1, vHH2, vLH1, vLH2 are the coefficients which satisfy Fλ kz ,jz(ρ =R, φ, z) = 0. The boundary condition for jz = 3

2 , 12 ,−1

2 ,−32 results in the

determinant equation

0 =

Jf− 32(kLH)Jf− 1

2(kLH)Jf+ 1

2(kHH)Jf+ 3

2(kHH)

+ Jf− 32(kHH)Jf− 1

2(kHH)Jf+ 1

2(kLH)Jf+ 3

2(kLH)

+ 3Jf− 32(kLH)Jf− 1

2(kHH)Jf+ 1

2(kHH)Jf+ 3

2(kLH) (3.43)

+ 4k2z

kLHkHH

Jf− 3

2(kLH)Jf− 1

2(kHH)Jf+ 1

2(kLH)Jf+ 3

2(kHH)

+Jf− 32(kHH)Jf− 1

2(kLH)Jf+ 1

2(kHH)Jf+ 3

2(kLH)

+ (k2LH+4k2

z)(k2HH+4k2

z)

3k2Lk2

HHJf− 3

2(kHH)Jf− 1

2(kLH)Jf+ 1

2(kLH)Jf+ 3

2(kHH),

which is a relation for the allowed energies. Here the wire radius R is ab-sorbed in the wave numbers by kHH → kHHR, kLH → kLHR and kz → kzR.From now on kHH , kLH and kz denote these dimensionless ”wavenumbers”,unless stated otherwise.

Together with the constraint obtained from the equation for the energy,

εHH = εLH = E, (3.44)

where the bulk energies εHH and εLH are given by (3.33), the determinantequation (3.43) fixes the radial wavenumbers kHH(kz) and kLH(kz) for a gi-ven kz. Using these solutions of kHH(kz) and kLH(kz), also the coefficientsvHH1-vLH2 are obtained from the boundary equation (3.39).

Before turning to more explicit results in the next sections, the followinggeneral remarks are important to keep in mind. First, the determinantequation (3.43) is invariant under the inversion fz → −fz, which reflects thetime-reversal symmetry of the Hamiltonian HΓ8

Fzgiven in (3.28): the total

angular momentum reverses direction if t → −t. Consequently, the energysolutions E are doubly degenerate in fz and the corresponding wavefunctionsturn into each other under fz → −fz.

52


Secondly it should be noted that the wavenumber kz, giving the disper-sion in the z direction where the electron (hole) is still free to move, cannotbe separated from the lateral terms in the envelope wavefunction (3.42).The radial wavenumbers kHH and kLH are functions of kz, so the dispersi-on in the z direction in general depends on the lateral distribution of thewavefunction.

Furthermore, for a given kz the set of equations (3.43), (3.44) has to besolved numerically. Only in special cases the energy E and hole wavefuncti-on reduce to relative simple analytical expressions. In the next section, firstsome analytical results at kz = 0 are summarized. Subsequently an expres-sion for the effective mass of a hole in a III-V nanowires will be derived byexpansion around kz = 0, the wire zone center.

3.3.2 Electron in III-V semiconductor nanowires

Up till now only the situation of the degenerate top valence bands in III-Vsemiconductors was discussed. Since also the conduction band propertiesare needed in the remaining of this paper and because it is also illustrativeto consider a nondegenerate example which is much easier to handle, he-re the electron dispersion and wavefunctions of the lowest lying conductionband in III-V semiconductors are treated shortly.

Again an infinite confinement is assumed, as given in equation (3.41).Since the conduction band is non-degenerate, the S.E. for the electron isgiven by (3.23) and the envelope function for ρ < R is given by

Fλ kz(ρ, φ, z) = CλJlz(klzρ)eilzφeikzz, (3.45)

with Cλ the normalization constant.

Assuming also the warping sufficiently small, i.e. adopting the sphericalapproximation by taking an uniform effective mass m∗

c for the conductionband, the energy dispersion becomes

~2

2m∗c

(k2lz + k2

z) = E. (3.46)

The boundary condition (3.39) now simply gives klz ,n = jlz,n

R , the allo-wed values of klz which are independent of kz. Here jlz ,n is the nth zero ofthe Bessel function Jlz(x)

It is convenient to introduce a notation which summarizes the labeling ofthe conduction subbands, as derived in the current framework. In the pre-sent case, the total Hamiltonian already is diagonal in the envelope angularmomentum, so the conduction subbands are labeled with |lz|.

53


Following the notation of [4], the irreducible representation of the con-duction subbands in cylinder configuration is characterized by

C(±)|lz |, n , (3.47)

where (±) denotes the parity, n the nth solution at this parity and theabsolute value of the envelope angular momentum is taken because of thedegeneracy in lz.

3.4 Hole dispersion around kz = 0

As discussed in [17] [18], the top valence band Hamiltonian HΓ8Fz

, given by(3.28), decouples into two 2×2 blocks at the wire zone center, kz = 0. Bothblocks have solutions which are characterized by parity : the correspondingBessel functions are only even or only odd under ρ → −ρ.

Apart from a general discussion, in this section the focus will be narrowedto an exceptional case: the odd solutions for |fz| = 1

2 . In this case it ispossible to derive a transparent equation for the effective mass in the zdirection by Taylor expansion around kz = 0.

3.4.1 Solutions at the wire zone center

At the wire zone center, kz = 0, one obtains from the energy equation (3.44):

kHH = βkLH , β ≡√

γ1 + 2γ2

γ1 − 2γ2=

√m∗

HH

m∗LH

, (3.48)

where m∗HH and m∗

LH are the effective masses of the heavy and light holebulk bands, respectively. Also the boundary condition simplifies at kz = 0.By block diagonalizing (3.28) it is found that the four heavy- and light holewavefunctions decouples into two groups: either

vHH2 = vLH2 = 0 , vLH1 = α1vHH1, (3.49)

or

vHH1 = vLH1 = 0 , vLH2 = α2vHH2. (3.50)

Here α1 and α2 are determined by the determinant equation (3.43) at thewire zone center, which decouple into two mutually excluding determinants

13

Jfz− 32(kHH)

Jfz− 32(kLH)

= −Jfz+ 1

2(kHH)

Jfz+ 12(kLH)

≡ α1, (3.51)

or

13

Jfz+ 32(kHH)

Jfz+ 32(kLH)

= −Jfz− 1

2(kHH)

Jfz− 12(kLH)

≡ α2, (3.52)

54

3.4. Hole dispersion around kz = 0

so the only possible solutions indeed are given by (3.49) and (3.50). Notethat the inversion symmetry of fz is revealed by the two determinants: using

J−n(z) = (−1)nJn (z), (3.53)

it easy to show that (3.49) turns into (3.50) under fz → −fz.

Energy equality (3.48), together with either (3.49) or (3.50) determinesthe energy at the zone center. The different energy bands and correspondingwavefunctions are characterized by parity at the wire zone center: for agiven fz ε Z +1

2 the solution corresponding to (3.49)/(3.50) contains onlyeven/odd (odd/even) Bessel functions. Note that Bessel functions transformunder inversion in ρ in the same way as their label (i.e. under z → −z,Jn(z) → Jn (z) if n is even, Jn(z) → −Jn (z) if n is odd).

As long as kz = 0, parity thus is a good quantum number and this is stillapproximately the case for kz close to 0. Consequently, the energy bandsand corresponding valence subbands are labelled with +/− for respectivelyeven/odd solutions at kz = 0. Note that for a given parity there are differentsolutions labelled by n.

At this point it is convenient to specify the labeling of the valence sub-bands further. As stated in paragraph 3.2.1, in a cylindrical representationthe total Hamiltonian HΓ8 is diagonal in Fz, which implies that the sub-bands are also labeled with |fz| (the absolute value is taken because of thedegeneracy in fz → −fz). Consequently, the complete set of solutions forthe Γ8 valence band in the infinite cylinder configuration is characterized bythe quantum numbers fz, (±), nth solution at this parity. This irreduciblerepresentation of the valence subbands is indicated with

E(±)|fz |, n, (3.54)

where, contrary to the notation in [4], n denotes the nth solution at a par-ticular parity.

In general, even at kz = 0 the original bulk heavy- and light hole solutionsare coupled to each other in a nanowire. However, it turns out that the oddsolutions at |fz| = 1

2 form an exceptional group. The determinant equation(either (3.49) or (3.50)) in this case reduces to

J1 (kHH)J1 (kLH) = 0, (3.55)

with Bessel zeros j1,n as solutions:

kHH =j1,n

R, kLH =

j1,n

βR(3.56)

55


or

kHH = βj1,n

R, kLH =

j1,n

R, (3.57)

where kHH , kLH are the original wavenumbers, so R is written out explicitely.As can be seen from (3.51) or (3.52), the relevant coefficient α1/α2 is zerofor these solutions. Since the other heavy -, light hole pair already is exclu-ded (equation (3.49) or (3.50)) the odd wavefunctions at kz = 0, |fz| = 1

2consequently are pure heavy - or light hole like. Note that for the light holesolutions both α1 and α2 should be inverted.

3.4.2 Hole dispersion around kz = 0 for |fz| = 12, (−)

The confinement by the infinite wire geometry, resulting in the determinantequation (3.43), reduces the dimensions in which the electron (hole) is freeto move to one. The dispersion relation in this direction (z) becomes morecomplex than the quadratic dispersion of the two original bulk bands, due tothe fact that kz cannot be separated from the lateral terms in the envelopewavefunction in case of the degenerate III-V top valence band. However,for the odd solutions at |fz| = 1

2 it is possible to approximate the dispersionaround the wire zone center with an effective mass.

For this purpose, the first step is to expand kHH(kz) and kLH(kz) up tosecond order in kz. Recall that the Γ8 band minimum is assumed to be atat the zone center, so

∂E

∂kz

∣∣∣∣kz=0

= 0. (3.58)

Utilizing this assumption, one finds for the lateral wavenumbers, by Taylorexpansion around kz = 0,

k2HH (kz) ' a2

HH + b2HHk2

z , k2LH (kz) ' a2

LH + b2LHk2

z . (3.59)

Consequently, expanding up to second order in kz, kHH and the correspon-ding Bessel function are approximated by

kHH (kz) ' aHH +b2

HH

2aHH

k2z , (3.60)

Jn (kHH (kz)) ' Jn (aHH) +b2

HH

2aHH

J ′n (aHH)k2z (3.61)

and the expressions for kLH are similar.

56

3.4. Hole dispersion around kz = 0

Before expanding (3.43) in this way, first it can be simplified for |fz| = 12

by using the Bessel function property (3.53):

0 =43k2

LHk2HHJ1(kLH)J1(kHH) J0(kLH)J2(kHH) + 3J0(kHH)J2(kLH)+

4kLHkHH

J2

1 (kLH)J0(kHH)J2(kHH) + J21 (kHH)J0(kLH)J2(kLH)

k2

z +43(k2

LH + k2HH)J1(kLH)J1(kHH)J0(kLH)J2(kHH)k2

z , (3.62)

where it should be mentioned that kHH = 0 (or kLH = 0) is not a solution[18]. Expanding the determinant equation up to second order with (3.59)-(3.61), the zeroth order part of the first line in (3.62) gives the solutions atkz = 0:

0 = J1(aLH)J1(aHH) J0(aLH)J2(aHH) + 3J0(aHH)J2(aLH) , (3.63)

where the term between the brackets in (3.63) corresponds to the even so-lutions.

In order to simplify the quadratic term in the expansion of (3.62), aHH

and aLH should be fixed by either the even or the odd solutions in (3.63).For the odd solutions this results in a transparent equation for the dispersionaround the wire zone center. Imposing J1 (aHH)J1 (aLH) = 0 and utilizing aproperty of the Bessel functions,

J ′n(z) = ∓Jn±1 (z)± n

zJn(z), (3.64)

the quadratic term becomes

0 = aLHaHH

13

b2LH2aLH

J0(aLH)[∓J1±1 (aLH)± 1

aLHJ1(aLH)

]J1(aHH)J2(aHH)+

13

b2HH2aHH

J0(aLH)J1(aLH)[∓J1±1 (aHH)± 1

aHHJ1(aHH)

]J2(aHH) +

b2HH2aHH

J0(aHH)[∓J1±1 (aHH)± 1

aHHJ1(aHH)

]J1(aLH)J2(aLH) +

b2LH2aLH

J0(aHH)J1(aHH)[∓J1±1 (aLH)± 1

aLHJ1(aLH)

]J2(aLH) + (3.65)

1aLHaHH

(J2

1 (aLH)J0(aHH) J2(aHH) + J21 (aHH)J0(aLH) J2(aLH)

)k2

z .

The final step is to specify the odd solution further by choosing eitherJ1(aLH) = 0 or J1(aHH) = 0. Here the discussion is restricted to aHH =j1,n, which corresponds to the lowest, heavy hole like energy bands.4 Afterchoosing the convenient signs in (3.65), the determinant equation in thiscase reduces to an expression for the expansion factor bHH of the heavy holelateral wavenumber:

b2HH = −

2aLH

J1(aLH)13J0(aLH)− J2(aLH)

= −2βj1,n

J1(j1,n

β )13J0(

j1,n

β )− J2(j1,n

β ), (3.66)

4Note again that for the odd solutions there is no heavy -, light hole mixing any more.

57


where the second expression follows from aLH = 1β aHH , see (3.48).

Expanding also the energy equation EUh

= −(γ1 − 2γ2)(k2HH(kz) + k2

z)(equation (3.44)) using (3.59)-(3.61), the odd |fz| = 1

2 heavy hole bands areapproximately given by

E

Uh= −(γ1 − 2γ2)j1,n − (γ1 − 2γ2)(1 + b2

HH)(kzR)2, (3.67)

where bHH is given in (3.66) and the dependence on the wire radius is expli-citly shown by defining an energy unit Uh:

Uh ≡ ~2

2m0R2 . (3.68)

This results in an expression for the effective mass of a heavy hole in theodd |fz| = 1

2 energy bands of an infinite wire:

m∗HH, z = m0 (γ1 − 2γ2)−1(1 + b2

HH)−1, (3.69)

where z is the only direction in which the hole is still free to move.

58

3.5. Results

3.5 Results

As an illustration of the above theory, in this section numerical results aregiven on the basis of specific examples. In particular it is insightful to com-pare III-V materials with different properties, in this case different Kohn-Luttinger parameters. Hence, the III-V compounds InP and InAs are in-vestigated, their Kohn-Luttinger parameters given in Table 3.1 are takenfrom [20]. Note that the values of γ3 are not needed here: the theoretical

γ1 γ2

InP 5.08 1.60InAs 20.0 8.5

Table 3.1: Kohn-Lutinger parameters for InP and InAs

framework rests on the assumption γ3 = γ2.

3.5.1 Hole energy bands of III-V material nanowires

The first seven hole energy bands of InP and InAs nanowires are shown infigure 3.1. They are calculated from equations (3.43) and (3.44). The blackline in the graphs corresponds to the reference band −(γ1 + 2γ2)k2

z withkLH = 0, where all bands end because there are no solutions if kLH ≤ 0. TheR dependence is absorbed in the units along the axes, kzR and E R2, withR in nm, kzR dimensionless and E R2 in eV nm2.

0 1 2 3 4 5 6kz R

-6

-5

-4

-3

-2

-1

0

ER

2He

Vnm

2L

InP

fz = 12, H+L 1fz = 12, H-L 1

fz = 32, H+L 1fz = 32, H-L 1

fz = 32, H+L 2fz = 12, H+L 2

fz = 12, H-L 2

0 0.5 1 1.5 2 2.5 3 3.5kz R

-6

-5

-4

-3

-2

-1

0

ER

2He

Vnm

2L

InAs

fz = 12, H+L 1fz = 12, H-L 1

fz = 32, H+L 1fz = 32, H-L 1

fz = 32, H+L 2fz = 12, H+L 2

fz = 12, H-L 2

Figure 3.1: Hole energy bands for InP and InAs. The bands are labeledby absolute total angular momentum in the z-direction, |fz|, and parity,denoted with (+)nth (nth even solution) and (−)nth (odd). The blackline in the graphs corresponds to the reference band (γ1 + 2γ2)k2

z withkLH = 0. The R dependence is absorbed in the units along the axes,with wire radius R in nm.

Using the notation given in (3.54) and (3.47), the representation of thefirst seven hole subbands of InP and InAs nanowires are shown in Table 3.2.

59


The subbands vi are ordered with respect to their zone center offset, seeFigure 3.1. For convenience, also the representation of first two electronsubbands c1 and c2 are given, within the framework of Subsection 3.3.2.

Subband InP InAs

v1 E(+)12,1

E(−)12,1

v2 E(−)12,1

E(+)12,1

v3 E(+)32,1

E(+)32,1

v4 E(−)32,1

E(+)12,2

v5 E(+)32,2

E(−)32,1

v6 E(+)12,2

E(−)12,2

v7 E(−)12,2

E(+)32,2

c1 C(+)0, 1 C

(+)0, 1

c2 C(−)1, 1 C

(−)1, 1

Table 3.2: Irreducible representation of the first seven hole subbands vi

and first two electron subbands cj for InP and InAs nanowires. Thecharacterization is also valid for conduction subbands calculated in afinite potential well.

Around kz = 0 the hole band dispersion can be approximated by thequadratic expressions given in Table 3.3. In general, for |fz| = 1

2 and oddparity these numerical results are in good agreement with the analyticalexpansion given in (3.67). For instance, for InP the effective mass in the zdirection m∗

HH, z corresponding to the values in Table 3.3 are 3.45 m0 and11.49m0 for the first even and first odd subband, while the analytical ex-pression (3.69) gives 3.39m0 and 10.20 m0, respectively.

Comparing the two materials, the following remarks are supported byFigure 3.1, Table 3.2 and Table 3.3:

60

3.5. Results

hole state InP InAs|fz| = 1

2 , (+) 1 −0.75− 0.14(kzR)2 −2.14− 1.87(kzR)2

|fz| = 12 , (−) 1 −1.05− 0.29(kzR)2 −1.68 + 0.77(kzR)2

|fz| = 12 , (+) 2 −2.16− 0.07(kzR)2 −3.91− 0.72(kzR)2

|fz| = 12 , (−) 2 −3.53− 0.087(kzR)2 −5.62− 0.35(kzR)2

|fz| = 32 , (+) 1 −1.38 + 0.31(kzR)2 −2.95 + 0.44(kzR)2

|fz| = 32 , (−) 1 −1.79− 0.59(kzR)2 −4.07− 0.45(kzR)2

|fz| = 32 , (+) 2 −2.08− 0.16(kzR)2 −6.52 + 1.29(kzR)2

Table 3.3: Numerical results for the hole energy E R2 (eV nm2), fittedto k2

zR2 around the wire zone center

• The shape of a particular band, including its zone center energy, ismaterial dependent. It depends on the magnitude of the gamma’s byγ1 − 2γ2, but also their ratio γ1

γ2is a deciding quantity. Consequently,

the corrections to the band gap Eg caused by the confinement arematerial dependent. For the present two examples the zone centerband gaps of InAs are more shifted by the infinite wire configuration.

• Next to the shape of the individual bands, also their mutual orderingis material dependent. This means that the parity of the lowest ly-ing band (and the others) can differ depending on the material. Forexample, the lowest lying band is even for InP and odd for InAs.

Furthermore, it should be noted that the results are valid for all R, withthe only requirement that R should be sufficient large in order to consider theconfinement potential (3.41) as a slowly varying function with respect to theunit cell dimensions. This means that in the limit R →∞ the confinementcorrection on the band gap has to disappear, which is indeed the case as canbe concluded from the 1/R2 dependence.

3.5.2 Hole wave functions of III-V material nanowires

As pointed out in section 3.3, the wavenumber along the cylinder axis kz isnot independent of the lateral part of the hole wavefunction. As a conse-quence, the total wavefunction for a particular band depends in a non trivialway on kz and should be calculated from (3.42) for every value of kz separa-tely. Here the results of this procedure are summarized by focussing, next tothe kz dependence, on three other subjects: the parity of the wavefunctions,the invariance under total z-angular momentum reversion and the influenceof material properties (Kohn-Luttinger parameters).

Starting with parity and the invariance under fz → −fz, Figure 3.2shows the φ = 0 radial part of the envelope wavefunction, decomposedinto the different jz components at the same value of kz and for the same

61


material. The graphs in the first row correspond to the two E(+)12,2

solutions,

those in the second row to the solutions with representation E(−)32,1

. The total

envelope function is normalized using equation (3.40) and the wire radius Ris absorbed in the dimensionless unit ρ

R .

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 32 , kzR = 2.66667 , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = -32 , kzR = 2.66667 , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 2.66667 , H+L 2

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = -12 , kzR = 2.66667 , H+L 2

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

Figure 3.2: Radial (φ = 0, eikzz omitted) part of the normalized holeenvelope functions, decomposed in the different jz components: the jz =32 components are given in yellow, jz = 1

2 in green, jz = − 12 in blue and

jz = − 32 in red. The first row gives the solutions for |fz| = 1

2 , +(2),the second row for |fz| = 3

2 , −(1). The Kohn-Luttinger parameters aretaken from InP as given in Table 3.1 and kz R = 2.67 is fixed.

Figure 3.2 illustrates that for a given subband E(±)|fz |,n the solution at fz

for a particular jz is the same (apart from minus sign) as at −fz for −jz.Moreover, as expected the two total wavefunctions turn into each other

under time reversal, because under fz = lz + jz → −fz = −lz − jz anycomponent Jl(kρ) |j, jz〉 → J−l(kρ) |j, −jz〉, so besides jz → −jz the oddsolutions reverse sign, as shown in Figure 3.2.

The wavefunctions in Figure 3.2 are calculated away from the wire zonecenter, at kzR = 1.5 10−9 with R in nm. As a consequence, next to the jz

components with the parity of the zone center, also other jz componentsappear which have the opposite parity. For example, in the first graph thedominant jz = 1

2 component (green curve) and the jz = −32 component (red

curve) are the evolved even components which are present at the zone cen-ter, while the jz = 3

2 (orange) and jz = −12 (blue) curves are odd in ρ → −ρ.

62

3.5. Results

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 1.25 , H+L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 1.5 , H+L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0.75 , H+L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 1. , H+L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0.25 , H+L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0.5 , H+L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0. , H+L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0.125 , H+L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

Figure 3.3: Radial part of the |fz| = 12 , + (1) hole envelope wavefuncti-

ons for InAs. The value of kzR changes from 0 in the first picture to themaximum value 1.5 (at the end of the band) in the last graph.

This is a general property: at the wire zone center the total wavefunctionconsist only of either even or odd components, while away from kz = 0 alsosignificant contributions with the other parity arise.

63


The variation of the wavefunction as a function of kz is illustrated in moredetail in Figure 3.3. It shows the radial part of the E

(+)12,1

hole wavefunctions

for InAs for different values of kz, given at the top of each graph.In Appendix A also the first odd solution, E

(−)12,1

, for InAs is shown in

this way; the illustrations can be compared to the results in case of InP,Figure 10 and Figure 11.

The following remarks are revealed by Figure 3.3 and the figures inAppendix A:

• As noted before, at kz = 0 the total hole wavefunction is either evenor odd under inversion ρ → −ρ. Remarkable is that the shape of thefunctions depends only a little on the choice of material.

• Increasing kz slowly, the shapes of the different jz components changein a continuous way: an extra graph between the first and the secondwould give a result in between.

• Comparing the results for InP and InAs, it can be seen that for agiven band the altering of the hole wavefunctions by increasing kz ismaterial dependent. Actually, the amount of change depends on theshape of the corresponding band: a smaller effective mass correspondswith a faster change in the hole wavefunctions around kz = 0, whichcan be checked for the present examples with the help of Table 3.3.

• At the end of a particular band, one of the jz components disappears,so in general there are three or fewer jz components which contributeto the total wavefunction at the end of a band.

The above results have some important consequences, in particular con-cerning the calculation of the absorption matrix elements over the entireband, see next chapter. Since the different jz components of the hole wa-vefunction change just slightly over a particular band, it suffices to choosea suitable small number of kz points by which the hole wavefunctions atneighboring points are approximated.

3.5.3 Band gap in III-V material nanowires

Finally, it is illustrative to estimate the effect of the infinite confinement inthe present model by comparing the bulk band gap Eg with the confinementenergy E conf

v1→c1 of the fundamental transition v1 → c1 between the highestlying hole state v1 and lowest electron state c1 at the wire zone center.

The values of effective mass of the Γ6 conduction band and the bulkband gap Eg for InP and InAs are given in Table 3.4. Using the values

64

3.5. Results

InP InAsm∗

c/m0 0.0795 0.026Eg (eV ) 1.4236 0.417V0 (eV ) 4.28 4.93

Table 3.4: The band gap Eg, potential well V0 and effective masses ofthe Γ6 bulk conduction band for InP and InAs

of the effective mass, the first row in Table 3.5 gives the energy of thelowest conduction subband c1 at the wire zone center. The second row

InP InAs

Ec1 (eV ) 2.77 R−2 8.48R−2

E confv1→e1 (eV ) 3.52 R−2 10.16R−2

Table 3.5: The energy in eV at kz = 0 of the lowest conduction subband

c1 and the confinement energy E confv1→c1

of the fundamental transitionv1 → c1 for InP and InAs in an infinite wire confinement, as derivedwith the model described in this paper. R denotes the wire radius.

in Table 3.5 shows the confinement energy E confv1→c1(R) of the fundamental

transition v1 → c1. Note that E confv1→c1(R) does not include the bulk Eg, it is

defined as

E transvi→cj

(R) = Eg + Eci(R)− Evi(R) ≡ Eg + E confvi→cj

(R). (3.70)

However, the assumption of an infinite potential well is too strong. Thedifference between the vacuum level and the conduction band edge (electronaffinity) is in the order of electron volts for III-V materials and taking thisfiniteness into account leads to significant corrections, in particular for theconduction subbands[18][21].

Here the discussion will be restricted to correcting the conduction sub-band c1 for InP and InAs, given in Table 3.5, with a reduction factor dueto the finite potential well. For the valence subbands it is expected that thecorrection is less crucial, in the first place because the correction is smallerfor bulk bands with a higher effective mass. Another reason is that, apartfrom the extra energy difference by the band gap, the difference with thevacuum level becomes larger for deeper lying subbands, in contrast to theconduction band states.

In the finite potential well model, the dependence on the wire radiusbecomes more complicated then the simple R−2 dependence in the infini-te case. Actually, the electronic properties depend on the dimensionless

65


quantity RlV

, where lV is defined as

lV =

√~2

2m∗V. (3.71)

For the conduction band, the potential V equals the vacuum level offset V0

which is given for InP and InAs in Table 3.4.

InP InAs

Ec1 (eV ) 0.07 0.13E conf

v1→c1 (eV ) 0.11 0.20

Table 3.6: The energy in (eV ) at kz = 0 of the lowest conduction sub-

band c1 and the confinement energy E confv1→c1

of the fundamental transi-tion v1 → c1 for InP and InAs at R = 4.83 nm and R = 4.85 nm, res-pectively. These are the corrected results of Table 3.5: the conductionsubbands are calculated in the finite potential wells given in Table 3.4.

The corresponding energies of the lowest conduction subband c1 in thefinite wire configuration are shown in Table 3.6. These values of Ec1 can becompared to those for the infinite potential well case at R = 4.8 nm: 0.2 eVand 0.36 eV for InP and InAs, respectively. Similarly, for the confinementenergy E conf

v1→c1 , the values in the infinite potential well case are 0.15 eV and0.43 eV for InP and InAs, respectively. The difference between the twomodels is larger for InAs due to the smaller effective mass of the conductionband.

66

Chapter 4

EM transition matrix

In this chapter the EM matrix element for band-to-band transitions betweenthe top Γ8 valence bands and the lowest lying Γ6 conduction band in III-Vsemiconductor nanowires will be derived. Section 4.1 contains some generaltheory, in section 4.2 the matrix element is developed further in the Blochrepresentation, section 4.3 gives explicit expressions for the band-band ma-trix elements and section 4.4 treats the selection rules on the intersubbandtransitions. Explicit results for InP and InAs are shown in section 4.5.

4.1 General theory

4.1.1 Radiation matter interaction

In this paper the interaction between the external EM field and the electronswithin the semiconductor system is described using a macroscopic, semi-classical approach. In this method the EM field is treated classically, whilethe semiconductor material is described quantum mechanically in the spiritof the previous chapter. Next to the assumption that this semi-classical pic-ture approximates the more realistic QED model, it is also assumed that thesemiconductor heterostructure can be described using macroscopic Maxwellequations, i.e. where the different parts in the system are characterized bymacroscopically averaged quantities, such as the dielectric function.

As will be discussed in more detail in Chapter 5, it is unclear if thismacroscopic framework still holds if the size of the system is reduced to na-noscale, when the dimensions of the system are not large any more comparedto the microscopic (atomic) distances. Then a microscopic semiclassical the-ory would be a more realistic approach [22].

However, proceeding with the macroscopic semiclassical approach, thegauge freedom in choosing the scalar potential φ and vector field A is used

67

Chapter 4. EM transition matrix

by taking the Coulomb gauge,

∇ ·A = 0. (4.1)

Then the transverse part of the electric field equals −1c

∂A∂t . Note that the

longitudinal part −∇φ is absorbed in the matter part of the semiconductorHamiltonian: the averaged potential V (r) in equation (3.1) contains the fullCoulomb interaction between the particles. Further details are found in [22].

In this scheme the Hamiltonian representing the radiation-matter inter-action is given by

Hr−m = − e

m0c

∑

i

A(ri) · pi, (4.2)

where i labels the N electrons in the material and m0 the free electronmass. Taking m0 instead of an effective mass m∗

i is justified for interbandtransitions [15]. Furthermore, since the EM field is typically is small, theterm of order |A|2 is neglected in (4.2).

4.1.2 EM transition matrix

Treating the time dependent EM interaction (4.2) as a small perturbation,it induces a transition between initial state |Ψi〉 and final state |Ψf 〉, whichare eigenstates of the semiconductor system in the absence of the EM field.The probability that the unperturbed system |Ψi〉 transforms under theabsorption/emission of light to |Ψf 〉 is proportional to the transition matrixelement

Mfi = 〈Ψf |Hr−m|Ψi〉. (4.3)

In case of a degenerate initial and/or final state, with |Ψim〉 and |Ψfn〉 them- and n-fold degenerate initial and final states, this degeneracy is takeninto account by

|Mfi|2 =∑m,n

|〈Ψfn |Hr−m|Ψim〉|2, (4.4)

i.e. the different possible transitions at the same energy are summed asprobabilities. If the degeneracy is lifted by a perturbation which breaks aparticular symmetry, say an external magnetic field, then |Ψim〉 and |Ψim′ 〉are separated in energy and their matrix elements can be distinguished asbelonging to different transitions |Mfi| and |Mfi′ |.

In this paper the absorption process is considered, in the specific caseof direct band-to-band transitions in nanowires. As initial state the many

68

4.1. General theory

body groundstate Ψ0 of the undoped semiconductor nanowire is taken, withall valence bands completely filled and the conduction bands empty:

Ψ0(r1, r2, .., rN ) = A(∏

i

Ψλikzi (ri),

), (4.5)

where the product of one particle states is anti-symmetrized by the operatorA and λi denote all valence bands.

As final state an excited state is constructed by taking an electron outof one of the top most valence bands,

Ψexcited(r1, r2, .., rN ) = AΨλj = c kzj (rj)

∏

i 6=j

Ψλikzi (ri)

, (4.6)

where j = c indicates that the electron is placed into one of the empty con-duction bands. In the remainder this notation will be used to distinguishwavefunctions belonging to conduction bands from valence band wavefunc-tions Ψλikzi(ri).

One can proceed further by forming the final exciton state from a linearcombination of the excited states (4.6). However, in this paper only theband-to-band transitions from the groundstate () to a particular excitedstate (4.6) are considered. By writing down the antisymmetrization operatorexplicitly in terms of the permutation operator P , A = 1√

N !

∑P (−1)P P , the

transition matrix (4.3) for these specific initial and final states becomes

Mc v = − e

m0c〈Ψexcited |

∑

i

A(ri) · pi |Ψ0〉 = − e

m0c

1N !

∑

PP ′(−1)P+P ′

∫dr1 . . . drN ×

P(Ψ∗

λl = c kzl(rl)

∏k 6=l Ψ∗

λkkzk(rk)

)∑i A(ri) · pi P

′(∏

j Ψλjkzj (rj))

. (4.7)

In order to simplify this expression, we recall the orthonormality relations∫

dr Ψ∗λi = c kzi

(ri)Ψλjkzj (rj) = 0, (4.8)∫

dr Ψ∗λikzi

(ri)Ψλj kzj (rj) = δλi,λj δkzi,kzj , (4.9)

where (4.8) is due to the orthogonality of the zone center atomic functions,s-like and p-like for conduction and valence band wavefunctions, respecti-vely. Equation (4.9) is explained by the orthogonality of (3.27) and thenormalization (3.40) of the total wavefunction.

It can be shown [15] that, utilizing (4.8) and (4.9), the transition matrixelement (4.7) simplifies to

Mc v = − e

m0 c

∫dr Ψ∗

λckzc(r) A(r) · p Ψλvkzv(r), (4.10)

69


in other words, Mc v is the transition matrix for an electron in one particu-lar state Ψλvkzv in a valence band λv which is excited to a conduction bandstate Ψλckzc by the EM radiation.

It is easy to check the simplification (4.10) of equation (4.7) if N = 2:from the 8 terms only two identical terms are nonzero and they cancel the2! in the denominator.

4.2 Bloch representation

The transition matrix (4.10) is developed further by making use of either theWannier or the Bloch representation for the wavefunctions. Here the Blochscheme is chosen, for the practical reason that zone center Bloch functionsare much better known. An other disadvantage of expanding in Wannierfunctions lies in the fact that Wannier functions are extended over a regionlarger then a unit cell, which makes it less defensible to approximate theEM field as constant over the relevant intervals of integration [18].

4.2.1 Total wavefunction in Bloch functions

Instead of expanding the wavefunction in zone center Bloch functions by ap-plying ansatz (3.25), the Bloch representation is generalized by expandingthe wavefunction in k space using the Fourier transform of the Wannierfunctions, equation (3.20).

Defining the Fourier transform of the envelope function Fλkz ,j (k′) asFλkz ,j (k′) ≡ N− 1

2∑

R Fλkz ,j (R) e−ik′·R, the total wavefunction Ψλ kz(r)in this way ecomes

Ψλ kz(r) =∑

jR

Fλkz ,j (R) ajR(r) = N− 12

∑

jR

∑

k′Fλkz ,j (R) e−ik′·Rψnk′(r)

= N− 12

∑

jk′

N− 1

2

∑

R

Fλkz ,j (R) e−ik′·R

eik′·rujk′(r)

= N− 12

∑

jk′Fλkz ,j (k′) eik′·rujk′(r), (4.11)

where in the second line the normalized Bloch functions (3.3) are used.

This expression is specified further in the infinite wire configuration bydecomposing the envelope Fλ kz , j in its radial part and the plane wave alongthe cylinder axis,

Fλ kz , j(R) = χλ kz , j (R⊥)eikzZ

√M

, (4.12)

70

4.2. Bloch representation

where the envelope is normalized in the Z direction by denoting the numberof atoms in this direction with M . Note that the not normalized lateral partχλ kz , j (R⊥) in general depends on kz, as concluded in the previous section.

Utilizing equation (4.12), the Fourier transform of the envelope functionFλkz ,j (k′) is decomposed as

Fλkz ,j (k′) = N− 12

∑

R⊥

χλ kz , j (R⊥) e−ik′⊥·R⊥

∑

Z

ei(kz−k′z)Z

√M

= ξλkz ,j (k′⊥) δkz ,k′z , (4.13)

where

ξλkz ,j (k′⊥) ≡

(N

M

)− 12 ∑

R⊥

χλ kz , j (R⊥) e−ik′⊥·R⊥ (4.14)

is the Fourier transform of the radial part χλ kz , j (R⊥).

Inserting (4.13) in equation (4.11), the total wavefunction expanded inBloch functions uj (k′

⊥,kz) reads

Ψλ kz(r) = N− 12

∑

jk′⊥

ξλkz ,j (k′⊥) ei(k′

⊥·r⊥+kzz)uj (k′⊥,kz)(r). (4.15)

Within the framework developed in the previous chapter, this expressionis valid around the zone center of any bulk band. This general form thusalso describes the total wavefunction in a confined conduction band, wherethe sum over j usually disappears because this band is non degenerate inmost of the III-V materials. Furthermore, equation (4.15) is also valid foran arbitrary strength of the confinement V0 (as in equation (3.37)).

4.2.2 EM transition matrix in Bloch functions

Utilizing the general expression (4.15) both for the valence and conducti-on band wavefunction, the transition matrix (4.10) is expanded in Blochfunctions by

Mc v = − e

m0 c

∫dr Ψ ∗

λckzc(r) A(r) · p Ψλvkzv(r)

= − e

m0 c

∑

jcjv

∑

k′⊥c k′⊥v

ξ∗λckzc,jc(k′⊥c) ξλvkzv,jv (k′⊥v) × (4.16)

1N

∫dr e−i(k′⊥c·r⊥+kzcz) ujc(k

′⊥c,kzc)(r) A(r) · p ei(k′⊥v ·r⊥+kzvz) ujv(k′⊥v,kzv)(r),

where the labeling ⊥ indicates that the corresponding vector lies in the planeperpendicular to the wire axis, as in section 4.2.1.

71


For the moment, we narrow the focus to the last line in equation (4.16).The integral over the entire space r can be split up into N integrals over aunit cell,

∫drf(r) =

∑R

∫Ω0

dr′f(R+r′). Using the commutation relation[p, eik·r ] = −~k eik·r, the third line in (4.16) becomes

1N

∑

R

e−i(k′⊥c−k′⊥v)·R⊥ e−i(kzc−kzv)Z × (4.17)

∫

Ω0

dr e−i(k′⊥c−k′⊥v)·r⊥ e−i(kzc−kzv)zujc(k′⊥c,kzc)(r) A(R + r) · (p + ~k) ujv(k′⊥v ,kzv)(r).

Further progress is made by assuming the variation of the EM waves to besmall over a unit cell, i.e.

A(R + r) ' A(R), (4.18)

This approximation is justified by the fact that the wavelength λ0 of theEM radiation typically is much larger then the interatomic distances a0. Asin the case of the effective mass approximation, where the restriction on δV(slowly varying over one unit cell) leads to a separation between atomic wa-vefunctions and ”macroscopic” envelope functions, the envelope and Blochparts of the transition matrix elements will factor into separate integrals dueto assumption (4.18).

In order to show this explicitly, reconsider expression (4.17) and includethe following:

• Assuming the EM field constant over a unit cell leads to the eliminationof the ~k term, because in this case the integral contains just twoorthogonal Bloch functions.

• In the case of an infinite cylinder the EM field is of the form

A(R) = A(R⊥)eiqzZ (4.19)

To be more precise, in Section 1.3 it was derived that qz = −k0 sin θ,with θ the angle of incidence compared to a plane perpendicular tothe wire axis.

• It is more convenient to rewrite kc and kv in terms of the total mo-mentum K of the system by defining

K ≡ kc − kv; (4.20)k ≡ kv. (4.21)

72

4.2. Bloch representation

Now (4.17) is simplified by1N

∑

R⊥

A(R⊥)e−iK′⊥·R⊥

∑

Z

e−i(Kz−qz)Z · (4.22)

∫

Ω0

dr e−iK′⊥·r⊥ e−iKzzujc(K

′⊥+k′⊥,Kz+kz)(r) p ujv(k′⊥,kz)(r),

With the notion that under assumption (4.18) qz is much smaller then areciprocal lattice vector Giz and that Kz lies in the first Brillouin zone,the summation over M lattice points results in a momentum conservationrelation in the Z direction:∑

Z

e−i(Kz−qz)Z = Mδqz ,Kz . (4.23)

This simplifies (4.22) further into

δqz ,Kz A(K ′⊥) ·

∫

Ω0

dr e−iK′⊥·r⊥ e−iKzzujc(K

′⊥+k′⊥,Kz+kz)(r) p ujv(k′⊥,kz)(r),

where A(K⊥) is defined as the Fourier transform of A(R⊥),

A(K⊥) ≡ M

N

∑

R⊥

A(R⊥)e−iK⊥·R⊥ . (4.24)

Furthermore, since the envelope functions are also assumed to be slowly va-rying over a unit cell, now it is possible to factor the envelope and Blochparts of the transition matrix elements into separate integrals. Any wave-vector κ in (4.16) is much smaller then a reciprocal lattice vector Gi becauseit is assumed that R À a0 and κ ∼ 1

R and Gi ∼ 1a0

. Under this conditionujκ ∼ uj0 and e−iK′

⊥·r⊥ e−iKzz ∼ 1. Inserting (4.24) into (4.16) and using∑

k′⊥K′⊥

ξ∗λckzc,jc(k′⊥ + K ′

⊥)A(K ′⊥)ξλvkzv ,jv(k

′⊥) =

∑R⊥ χ∗λckzc,jc

(R⊥)A(R⊥)χ∗λvkzv ,jv(R⊥)

the final form of the transition matrix element in Bloch functions is obtained:

Mc v = − e

m0 cδkzc,kzv

∑

jcjv

∑

R⊥

χ∗λckzc,jc(R⊥)A(R⊥)χ∗λvkzv ,jv

(R⊥) ·∫

Ω0

dr ujc0(r) p ujv0(r). (4.25)

This result is also obtained by using ansatz (3.25), a simplified expansionof the wavefunction in Bloch states which is justified if the wire radius Ris sufficiently large, i.e. the Fourier related k values are so small that thecorresponding energy difference εj(k) − εj remains much smaller then theband edge differences εj − εn′ .

It should be noted that the degeneracy of the hole and electron energybands is not taken into account in (4.25). It can be included in the sameway as by replacing (4.3) with (4.4), as will be done further on when thematrix elements are calculated explicitly.

73


4.3 Reformulation of transition matrix element

In this section the transition matrix element (4.25) is reconsidered by in-cluding the degeneracy of the conduction and valence subbands in case oftransitions between the top Γ8 valence bands and the lowest lying Γ6 con-duction band in III-V semiconductors. Apart from the trivial degeneracyin ± kz, the transitions are degenerate in the quantum numbers ± lz c and±σ of the conduction subbands and in ± fz v of the valence subbands, seeChapter 3.

The EM field is specified further by considering two approximations withrespect to the wire radius as derived in Part I. First, the spatial variationof the EM field across the wire diameter is entirely neglected, i.e. the tran-sition matrix element is formulated in the dipole limit, where λ0 À R soA(R⊥) ' A. Secondly, a spatial variation of the EM field is taken intoaccount by applying the scattering fields (2.8), (2.9) and (2.10), which areexpansions up to second order in 2π

λ0R for normal incident light.

Recall that the Coulomb gauge is chosen by deriving the transition ma-trix element, so the vector potential is related to the transverse electric fieldby E = −1

c∂A∂t . In case of absorption this yields

E = − iω

cA, (4.26)

with ω the frequency of the EM field.

4.3.1 EM field in dipole approximation

In Part I it was shown that the strength of the EM field inside the wire de-pends on the polarization of the incident light. In the dipole approximationthis resulted in (2.6) and (2.7) for polarization parallel and perpendicular tothe wire axis, respectively. In order to separate this polarization anisotropyfrom the dielectric mismatch, a matrix element Tc v is defined by

|Mc v(kz)|2 ≡ (e

m0ω)2|E

2|2|Tc v(kz)|2, (4.27)

where kz = kzc = kzv and E is the strength of the electric field inside thewire in the dipole approximation, E ≡ Eε.

Including the degeneracy, from equation (4.4) and (4.25) one obtains forthe transition matrix Tc v between valence and conduction subband v and c:

|Tc v(kz)|2 =∑

σ ldzcfdzv

∣∣∣∣∣∣∑

jz

〈χlzc |χfzv;jz(kz) 〉〈Sσ| ε · p |32jz 〉∣∣∣∣∣∣

2

, (4.28)

74

4.3. Reformulation of transition matrix element

where ldzc denotes lzc = |lzc| , −|lzc|, the degeneracy at a given |lzc|. A si-milar definition applies to fd

zv. The notation of the atomic part is explainedin (4.34). Furthermore, 〈χlzc |χfzv ;jz(kz) 〉 =

∫dR⊥χ∗lzc

(R⊥)χfzvkz ;jz(R⊥),where the replacement of the summation

∑R⊥ by an integral is justified

since in the effective mass approximation the dimensions of the wire are as-sumed to be much larger then interatomic distances. The additional step sizeby going from summation to integration is eliminated by the normalizationof the wavefunctions.

Note that indeed the strength of the internal field is separated fromTcv. This classical penetration effect is absorbed in E, as shown in equation(4.28).

4.3.2 EM field including Mie scattering

In case of a spatially varying EM field the separation in (4.27) is not possibleany more. Instead, the EM field has to be integrated between the conductionand valence subband envelope functions over the wire cross section areaπR2, as shown in (4.25). However, from equations (2.8), (2.9) and (2.10)one finds that the reduction factor caused by the penetration into the wirestill can be divided out, just as in the dipole limit. Thus, by separatingout the factor E = E0 in case of polarization parallel to the wire axis andE = 2

1+εE0 at perpendicular polarization, the matrix element Tc v containsonly the scattering (optical focusing) and expansion terms due to the wavebehavior of the EM field inside the wire. Denoting E′(R⊥) as the electricfield without the penetration strength, E E′(R⊥) ≡ E(R⊥), the right handside of (4.28) now is replaced with

|Tc v(kz; ε,R)|2 =∑

σ ldzc,fdzv

∣∣∣∣∣∣∑

jz

〈χlzc,n|E′|χfzv,n;jz(kz) 〉 · 〈Sσ|p |32jz 〉∣∣∣∣∣∣

2

.(4.29)

Note that, contrary to the dipole limit, the direction of the internal EMfield is in general different from that of the incident field . Furthermore,due to the scattering field the matrix elements now depend on the dielectricfunction and the wire radius.

4.3.3 Polarization anisotropy of the transition matrix

At this point it is instructive to define a polarization anisotropy purely ori-ginating from the matrix elements. As demonstrated by equation (4.28), inthe dipole limit it is possible to separate the anisotropy caused by the die-lectric mismatch from that which is due to the polarization in the transitionmatrix. In other words, in principle one is able to determine the polari-zation anisotropy caused by the transition matrix elements alone once itis (experimentally) possible to eliminate the polarization anisotropy of the

75


dielectric mismatch, for instance by a sufficient increase of the intensity ofthe incident field in the perpendicular case or by changing the surrounding(for instance if the nanowire is covered by an oxide). In analogy with (2.21),the polarization anisosotropy ρTcv of the matrix element alone is defined as

ρTcv ≡ |Tcv, ‖ |2 − |Tcv,⊥|2|Tcv, ‖|2 + |Tcv,⊥|2 , (4.30)

where Tcv, ‖ = Tcv, z is the matrix element corresponding to a polarizationparallel to the wire and Tcv,⊥ denotes the perpendicular case.

4.4 Selection rules

The interband transition matrix (4.25) is investigated in more detail byconsidering the different kind of selection rules it imposes. A selection ruleoriginates from an underlying symmetry of the system under considerationand generally disappears if the symmetry on which it relies is broken. Bya selection rule some transitions are ”selected” to be allowed, while othersare said to be forbidden.

As stated in section 4.3, a spatially varying EM field has to be treateddifferently then the more common dipole approximation. The dipole ap-proximation is crucial for the selection rules originating from the envelopepart of the transition matrix element (4.25). Away from this limit the va-riation of the EM field starts to break the symmetry of the matrix elementbetween the envelope parts of the electron and hole wavefunctions.

On the other hand, the selection rules originating from the atomic likematrix element of the momentum operator p are independent of E, providedthat the field can be considered as constant over a unit cell, an approxima-tion which was made earlier.

Leaving the discussion of a spatially varying EM field to paragraph 4.5.2,the selection rules in the dipole approximation fall into three different clas-ses:

• Polarization selection rules

• Selection rules on the lz-angular momentum of the envelope wavefunc-tion

• Parity selection rules.

The polarization rules are caused by the atomic like matrix element of themomentum operator p, while the selection rules on the lz-angular momen-tum of the envelope wavefunction and the related parity selection rules are

76

4.4. Selection rules

due to the envelope part of the transition matrix (4.25).

Explicit investigation of the transition matrix requires a restriction to amore specific case. The selection rules depend on which particular system isconsidered and subsequently which symmetry properties are valid. Thereforethe focus will be on the band-to-band transitions between the top mostvalence bands and the lowest lying conduction band in III-V materials. Inthis case the matrix element is given by (4.28), or by (4.29) in case of aspatially varying EM field.

4.4.1 Polarization selection rules

In this paragraph the polarization selection rules in case of transitions bet-ween the top Γ8 valence bands and the lowest lying Γ6 conduction band inIII-V semiconductors are derived. However, since the theory strongly relieson the results of atomic physics, it is instructive to summarize these shortlyin advance.

Suppose an atomic system which is built from the orthonormal base|η j m〉, where the quantum numbers j and m come from an angular mo-mentum operator J and η refers to other possible quantum numbers whichcomplete the basis of the system in consideration. Then for any vectoroperator V applies

〈η j m′|V+|η j m〉 = 0 if m′ −m 6= 1,

〈η j m′|V−|η j m〉 = 0 if m′ −m 6= −1,

〈η j m′|Vz|η j m〉 = 0 if m′ −m 6= 0, (4.31)

where V+ ≡ Vx + iVy and V− ≡ Vx − iVy, as usual.In case of the absorption of light, where p is the vector operator of inte-

rest, this result gets its physical interpretation if one realizes that a photoncarries spin 1, so m = 1, 0,−1: in this case (4.31) is just a consequenceof the conservation of angular momentum.

Turning to the optical transitions in III-V semiconductors, rememberthat an analogy was made between the band edge Bloch states and atomicfunctions, see paragraph 3.1.1. Now it becomes more clear what is actuallymeant with ”atomic-like”: in the k · p method the optical matrix elementsare used as input. Without specifying the precise form of a particular bandedge Bloch function, it can be argued that its symmetry properties are thesame as a particular atomic function. All what remains is to determine(experimentally) the optical matrix elements.

Concerning transitions between the Γ8 valence bands and the Γ6 con-

77


duction band, the only nonzero matrix elements are given by

− i

m0〈S|px|X〉 = − i

m0〈S|py|Y 〉 = − i

m0〈S|pz|Z〉 = P, (4.32)

where |S〉 denotes the band edge Bloch function of the conduction bandwhich is s-like. The magnitude P of the matrix elements in (4.32) is relatedto the Kane matrix element Ep by

EP = 2m0P2, (4.33)

which can be determined experimentally for each particular III-V bulk se-miconductor material.

As an upshot, the polarization selection rules in semiconductor materialsresult from restrictions imposed on the matrix element of the momentumoperator by the symmetry properties of the atomic-like Bloch states. Ge-nerally, from group theory it is known that a matrix element 〈ψ1|O|ψ2〉 isonly nonzero if the symmetry S1 of ψ1 is the same as one of the irreduciblerepresentations of the direct product O ⊗ S2, where O denotes the symme-try of the operator O. Indeed, analyzing the symmetry properties of themomentum operator, one finds that p-like and s-like states lead to the onlynonzero matrix elements given in (4.32)[14].

Narrowing the focus to band-to-band transitions between the top Γ8 va-lence bands and the lowest lying Γ6 conduction band in III-V semiconduc-tors, the the atomic part of the transition matrix (4.25) is specified furtherby

∫

Ω0

dr ujc0(r) p ujv0(r) = 〈S σ |p |32 jz 〉, (4.34)

again with σ = ↑, ↓ and jz ε 32 , 1

2 ,−12 ,−3

2. This matrix can be calculated interms of the only nonzero matrix elements (4.32) by using the decompositionof |32 jz 〉 in the states |X〉,|Y 〉 and |Z〉, as given in (3.13).

Table 4.1 shows the result for the unpolarized interband matrix element〈S σ | px + py + pz |32 jz 〉.

Here the matrix elements Pu are introduced for convenience,

〈S|pu|U〉 ≡ Pu, u = x, y, z, U = X, Y, Z. (4.35)

In terms of the Kane matrix elements, equations (4.32) and (4.33), Ta-ble 4.2 gives the quantitative results of the polarization dependence of thematrix element 1

(m0Ep)12〈S σ | pu |32 jz 〉. This result plays a dominant role

by determining the polarization anisotropy in the EM transition matrix ele-ments, see paragraph 4.3.3.

78


px + py + pz |32 32 〉 |32 1

2 〉 |32 − 12 〉 |32 − 3

2 〉

〈S ↑ | − 1√2Px − i√

2Py

√2√3Pz

1√6Px − i√

6Py 0

〈S ↓ | 0 − 1√6Px − i√

6Py

√2√3Pz

1√2Px − i√

2Py

Table 4.1: Result for the unpolarized interband matrix elements

〈S σ | px + py + pz | 32 jz 〉 in terms of Pu ≡ 〈S|pu|U〉. For a particu-lar transition, the spin σ of the conduction band electron is shown inthe left column and the Bloch angular momentum jz belonging to thevalence band in the first row.

For instance, Table 4.2 clearly demonstrates that the ratio of∑

σ |〈S σ | pz |32 jz =±1

2 〉|2 and∑

σ |〈S σ | px |32 jz = ±12 〉|2 equals 4. The polarization selection

rule has an even stronger effect on the valence subband states which aredominated by terms with jz = ±3

2 : in this case the matrix element of pz

is strictly zero. A strictly zero matrix element of a particular transitionand direction of the momentum operator is said to be polarization forbid-den, or polF in short notation. This qualitative result is summarized inTable 4.3, which shows the allowed polarizations. Here x, y denotes the al-lowed polarizations perpendicular to the wire axis and z the allowed parallel

1

(m0Ep)12

pu |32 32 〉 |32 1

2 〉 |32 − 12 〉 |32 − 3

2 〉

px − i2 0 i

2√

30

〈S ↑ | py12 0 1

2√

30

pz 0 i√3

0 0

px 0 − i2√

30 i

2

〈S ↓ | py 0 12√

30 1

2

pz 0 0 i√3

0

Table 4.2: Selection rules on the atomic-like interband matrix elements1

(m0Ep)12〈S σ | pu | 32 jz 〉.

79


polarization. Another feature which will be extracted from Table 4.2 further

ε · p |32 32 〉 |32 1

2 〉 |32 − 12 〉 |32 − 3

2 〉

〈S ↑ | x, y z x, y polF

〈S ↓ | polF x, y z x, y

Table 4.3: Selection rules on the atomic-like interband matrix elements

〈S σ | ε · p | 32 jz 〉. The allowed polarizations are denoted with x, y andz, where x, y are perpendicular to the wire axis and z gives the parallelpolarization. Polarization forbidden transitions are denoted with polF .

on is that the matrix elements corresponding to polarizations perpendicularto the wire axis are the same, i.e. |Tcv, x|2 = |Tcv, y|2, which is a consequenceof the rotational symmetry around the wire axis in the dipole approximation.

Finally, it is instructive to come back to the selection rules in the atomiccase. In fact, the results in Table 4.2 lead to the same selection rules as givenin (4.31). For this purpose, first note that Table 4.2 can be formulated inan algabraic way by

〈Sσ|px|32 jz 〉 = i(m0Ep)12

√16|jz| (δσ−jz ,1 − δσ−jz ,−1),

〈Sσ|py|32 jz 〉 = (m0Ep)12

√16|jz| (δσ−jz ,1 + δσ−jz ,−1),

〈Sσ|pz|32 jz 〉 = i(m0Ep)12

√23|jz| δσ,jz . (4.36)

In terms of p+ ≡ px + ipy and p− ≡ px − ipy the matrix elements of px andpy yield

〈Sσ|p+|32 jz 〉 = i(m0Ep)12

√23|jz| δσ−jz ,1,

〈Sσ|p−|32 jz 〉 = (m0Ep)12

√23|jz| δσ−jz ,−1. (4.37)

Together with the matrix element of pz in (4.36), these are the same selectionrules as in the atomic case, now originating from the conservation of angularmomentum on the Bloch part of the transition matrix.

4.4.2 Selection rules on the envelope wavefunctions

The selection rules on the envelope part of the transition matrix are not asgeneral as the polarization selection rules derived in paragraph 4.4.1. While

80


the polarization selection rules originate from the bulk, atomic like matrixelement of the momentum operator, the selection rules on the envelope partare dependent on the configuration of the system (wire radius, length) andalso depend on whether the EM field can be considered in the dipole limit.

Transition Polarization Class |jz|

C0, 1 → E 12,n x, y, z MP 1

2

C0, 1 → E 32,n x, y SP 3

2

C0, 1 → E 52,n − lF −

C1, 1 → E 12,n x, y, z MP 1

2

C1, 1 → E 32,n x, y, z MP 1

2

C1, 1 → E 52,n x, y SP 3

2

Table 4.4: Summary of the polarization and class for the lowest band-to-band transitions in C∞ nanowires with a constant EM field. Polarizationperpendicular to the wire axis is denoted with x, y, parallel polarizationwith z. The envelope angular momentum forbidden transitions are de-noted with lF while SP and MP denote the polarization class, singleand mixed polarization respectively. The last column gives the allowedvalues of |jz|.

Proceeding with the transitions between the Γ8 valence bands and theΓ6 conduction band in III-V semiconductors in the dipole limit, equation(4.28), the φ part of 〈χlzc,n|χfzv,n;jz(kz) 〉 gives

12π

∫ 2π

0dφ e−i(lzc−(fzv−jz))φ = δ lzc,fzv−jz , (4.38)

which can be considered as a selection rule on the envelope angular momen-tum, since lzv = fzv − jz. As a direct consequence, transitions for whichlzc − fzv 6= −3

2 ,−12 , 1

2 , 32 are l-angular momentum forbidden (lF ). This

result can be found in Table 4.4, which shows a combination of the pola-rization and l-angular momentum selection rules. The last row gives theallowed values of |jz|.

81


The constraint (4.38) also leads to a selection rule which is a consequenceof the parity of the subbands at the wire zone center. An even (odd) wave-function corresponds to an even (odd) Bessel function Jlz and by (4.38) lzc

and lzv = fzv−jz has to be the same, i.e. the parity of the valence subbandwavefunction has to match with the parity of the conduction subband. Inother words, transitions

E (±) → C (∓) (4.39)

are parity forbidden (pF) at kz = 0. Away from the zone center this se-lection rule generally is broken: the valence subband wavefunctions are notcharacterized by parity any more.

82

4.5. Results

4.5 Results

In this section the above theoretical framework is applied to specific examples.Again InP and InAs are chosen since those III-V materials are two kind ofextremes regarding their electronic properties. As a matter of fact, alt-hough this is also the case for the Kane matrix element Ep, the effect of thisdifference will be small since Ep is hardly material dependent, see Table 4.5.

InP InAsEp (eV ) 20.7 21.5

Table 4.5: The Kane matrix element Ep for InP and InAs

Since a spatially varying EM field requires a different approach comparedto the theory derived in the dipole limit, this case is treated separately inparagraph 4.3.2.

4.5.1 Dipole approximation

The topmost illustration in Figure 4.1 (a) and Table 4.6 show the nume-

ETrans HeVL, R = 9.96 nm

ÈΡT

vcÈ

ÈTv

cÈ2Ha

rb.u

nitsL


0.49 0.5 0.5 0.51 0.52 0.530

0.20

0.40

0.60

0.80

1.00

Ρ > 0Ρ < 0

0.48 0.49 0.5 0.5 0.51 0.52 0.530

0.20

0.40

0.60

0.80

1.00

0.02

0.04

0.06

0.08

0.10

0.12

0.140.65 0.68 0.72 0.75 0.78 0.82

aL : kzR = 0.

z - pol.y - pol.v2

->

c1

v3->

c1

v7->

c1

v4->

c1

0.62 0.65 0.68 0.72 0.75 0.78 0.82

0.02

0.04

0.06

0.08

0.10

0.12

0.14bL : kzR = 0.45

v2->

c1

v1->

c1

v3->

c1

v5->

c1

v7->

c1

v4->

c1

v6->

c1

Figure 4.1: Matrix elements |Tcv,‖|2 (first row, white bars) and |Tcv,⊥|2(black) in arbitrary units and corresponding polarization anisotropy ρTcv

(second row) of the first 7 transitions vi → c1 for InAs at a) : kzR = 0and b) : kzR = 0.45. The colors are the same as in Figure 3.1; for therepresentations see Table 4.6. The energy scale for all graphs is given attwo values of R, indicated at the top and bottom of the figure.

83


rical results at kz = 0 for InAs of the matrix elements perpendicular andparallel to the wire axis, |Tcv, y|2 and |Tcv, z|2 respectively. The upper graphFigure 4.1 (b) gives the same results away from the wire zone center, atkzR = 0.45. The matrix elements are calculated for wavefunctions validin an infinite potential well both for the valence and the conduction sub-bands and so the values are independent of R. Contrary to the energies, thecorrection on the subband wavefunctions by taking a finite potential intoaccount is expected to be negligible, provided the dimensionless quantity lVin (3.71) is not too small.

The last two columns in Table 4.6 give the transition energy and corres-ponding wavelength. In this case the conduction subbands are calculated inthe finite potential wells given in Table 3.4, at R = 4.85 nm.

Transition Representation Class |Tcv, y|2 |Tcv, z|2 Etrans (eV ) λtrans (nm)

v1 → c1 E(−)12,1→ C

(+)0, 1 pF − − 0.616 2011

v2 → c1 E(+)12,1→ C

(+)0, 1 MP 2.94 10−2 11.77 10−2 0.636 1949

v3 → c1 E(+)32,1→ C

(+)0, 1 SP 1.23 10−2 − 0.670 1850

v4 → c1 E(+)12,2→ C

(+)0, 1 MP 6.15 10−3 24.6 10−3 0.712 1754

v5 → c1 E(−)32,1→ C

(+)0, 1 pF − − 0.718 1726

v6 → c1 E(−)12,2→ C

(+)0, 1 pF − − 0.784 1581

v7 → c1 E(+)32,2→ C

(+)0, 1 SP 8.39 10−2 − 0.818 1515

Table 4.6: Interband matrix elements |Tcv, y|2 and |Tcv, z|2 of the firstseven transitions vi → c1 for InAs at kz = 0. The parity forbiddentransitions are denoted with pF , SP and MP refer to single and mixedpolarization, respectively. The last two columns show the transitionenergy and corresponding wavelength calculated at R = 4.85 nm withthe conduction subband in the finite potential well model.

Furthermore, Table 4.6 summarizes the representation and class of thefirst seven transitions vi → c1 as derived in the previous sections. In Figu-re 4.1 the transitions are indicated in the same colors as used in Figure 3.1.The second row in Figure 4.1 gives the polarization anisotropy ρTcv corres-ponding to the matrix elements alone, as defined by equation (4.30).

84

4.5. Results

A closer investigation and comparison with the theory leads to the fol-lowing conclusions:

• At the zone center only transitions between subbands with the sameparity are allowed. Away from the zone center, the parity selection isbroken since the hole states are not characterized by parity any more.

• The polarization anisotropy is completely determined by the polariza-tion rules. All E

(+)12

→ C(+)0 transitions show a polarization of 0.6,

which is in agreement with the analytical result of paragraph 4.4.1,where a polarization contrast |Tcv, ‖|2

|Tcv,⊥|2 of 4 was determined for |jz| = 12 .

Indeed only these states contribute to the matrix elements, since in thedipole limit it is required that |jz| = |fzv| by the l-angular momentumselection rule on E 1

2→ C0. Also the results of E

(+)32

→ C(+)0 are as

expected: for the |jz| = 32 states a parallel polarization is forbidden

(polF).

• The matrix elements are independent of the wire radius. It should benoted that the finite confinement for the conduction subbands is nottaken into account, but this correction is expected to be small. On theother hand, the transition energies strongly depend on the wire radius.Therefore, the energy scale is given at two values of R in Figure 4.1,R = 4.85 nm and R = 9.96 nm indicated at the top and bottomof the figure, respectively. Since the correction is substantial for thetransition energies, these are calculated with the conduction subbandin the finite potential well model.

• The present effective mass approach can be compared successfully withthe results derived in an atomistic approach. In Appendix D, Figure 14the band-to-band matrix elements for an R = 4.8 nm InAs wire areshown based on an atomistic, empirical pseudo-potential plane-wavemethod [4]. It is noted that the C∞ v representations given in Table IIof the article differ from those derived in the present paper and are inconflict with the basic symmetry considerations of Chapter 4. Withthis in mind, comparing Figure 4.1 with FIG. 3 in [4] it is concludedthat also in the more accurate atomistic approach the E

(+)32

→ C(+)0, 1

transitions are completely y-polarized. Considering the polarizationanisotropy of the E

(+)12

→ C(+)0, 1 transitions, the deviations from 0.6 in

FIG. 3 in [4] are explained by the possible corrections of including thesplit-off band, or even diagonalizing the full 8× 8 Hamiltonian of thethree Γ8 valence bands and Γ6 conduction band in the present effectivemass approximation. This also explains that some pF transitions in

85


[4] still have a small contribution at kz = 0. For the shifts in thetransition energies the additional argument holds that taking also finiteconfinement for the valence subbands into account may change thevalence energies slightly.

ETrans HeVL, R = 10. nm

ÈΡT

vcÈ

ÈTv

cÈ2Ha

rb.u

nitsL


1.45 1.46 1.460

0.20

0.40

0.60

0.80

1.00

Ρ > 0Ρ < 0

1.46 1.46 1.47 1.47 1.480

0.20

0.40

0.60

0.80

1.00

0.02

0.04

0.06

0.08

0.10

0.12

0.141.53 1.55 1.58

aL : kzR = 0.

z - pol.y - pol.v1

->

c1

v3->

c1

v5->

c1v6->

c1

1.55 1.58 1.6 1.62 1.65

0.02

0.04

0.06

0.08

0.10

0.12

0.14bL : kzR = 0.93

v1->

c1

v2->

c1v3->

c1

v4->

c1v5->

c1v6->

c1

v7->

c1Figure 4.2: Matrix elements |Tcv,‖|2 and |Tcv,⊥|2 and corresponding po-larization anisotropy ρTcv of the first 7 transitions vi → c1 for InP ata) : kzR = 0 and b) : kzR = 0.93.The colors are the same as in FIgure 3.1, for the representations seeTable 4.7. The energy scale for all graphs is given at two values of R,indicated at the top and bottom of the figure.

The above results of InAs can be compared with InP, see Figure 4.2 andTable 4.7. This leads to the following conclusions:

• The large difference in the transition energies, for instance the tran-sition E

(+)12

→ C(+)0, 1 corresponds to Etrans = 0.636 eV for InAs and

Etrans = 1.529 eV for InP, is caused by the difference in the bulk bandgap Eg, see Table 3.4. It is slightly reduced by the difference in confi-nement energies for the electron subband, see Table 3.6: since InP hasa larger conduction band effective mass m∗

c than InAs, its confinementenergy of the conduction subbands is lower.

• As already stated in Chapter 3, the ordering of the valence subbands ismaterial dependent, due to the differences in the heavy- and light holeeffective masses. For instance the first two transitions E

(−)12

→ C(+)0, 1

86

4.5. Results

Transition Representation Class |Tcv, y|2 |Tcv, z|2 Etrans (eV ) λtrans (nm)

v1 → c1 E(+)12,1→ C

(+)0, 1 MP 3.4 10−2 13.7 10−2 1.529 811

v2 → c1 E(−)12,1→ C

(+)0, 1 pF − − 1.542 804

v3 → c1 E(+)32,1→ C

(+)0, 1 SP 9.1 10−2 − 1.556 797

v4 → c1 E(−)32,1→ C

(+)0, 1 pF − − 1.574 788

v5 → c1 E(+)32,2→ C

(+)0, 1 SP 1.2 10−2 − 1.586 782

v6 → c1 E(+)12,2→ C

(+)0, 1 MP 3. 10−4 12. 10−4 1.590 780

v7 → c1 E(−)12,2→ C

(+)0, 1 pF − − 1.648 752

Table 4.7: Interband matrix elements |Tcv, y|2 and |Tvc, z|2 of the firstseven transitions vi → c1 for InP at kz = 0. The parity forbiddentransitions are denoted with pF , SP and MP refer to single and mixedpolarization, respectively. The last two columns show the transitionenergy and corresponding wavelength calculated at R = 4.83 nm withthe conduction subband in the finite potential well model.

and E(+)12

→ C(+)0, 1 are ordered the other way around in case of InP.

As an important consequence, the parity selection rule works on thelowest possible transition in case of InAs, while for InP this selectionrule at kz = 0 works on the second possible transition.

• Next to the energies, also the transition strengths are material depen-dent. This is caused by the different heavy- and light hole effectivemasses which result in different valence subband wavefunctions.

4.5.2 EM field including Mie scattering

In Figure 4.3 the numerical results are shown of the matrix elements |Tcv,‖|2and |Tcv,⊥|2 and corresponding polarization anisotropy ρTcv for InAs, inclu-ding the effects of spatial variation of the EM field up to second order. Thepenetration strength, which is also present in the dipole limit, is not takeninto account and the results are obtained by using equation (4.29). Sincethe matrix elements now depend on the wire radius by the scattering field,

87


R = 4.85 nm is fixed. Contrary to the expressions for the expanded electricfield at normal incidence in Part I, which were given in cylindrical coor-dinates, here there components in cartesian coordinates are used. As canbe concluded from (2.9) and (2.10), for polarization perpendicular to thewire axis the internal field does not have the same direction as the incidentfield. However, contributions from the other direction (say x) to the matrixelements are negligible.


ÈΡT

vcÈ

ÈTv

cÈ2Ha

rb.u

nitsL


0.62 0.65 0.68 0.72 0.75 0.78 0.820

0.20

0.40

0.60

0.80

1.00

Ρ > 0Ρ < 0

0.62 0.65 0.68 0.72 0.75 0.78 0.820

0.20

0.40

0.60

0.80

1.00

0.02

0.04

0.06

0.08

0.10

0.12

0.140.62 0.65 0.68 0.72 0.75 0.78 0.82

aL : kzR = 0.

z - pol.y - pol.

v2->

c1

v1->

c1

v3->

c1

v5->

c1

v7->

c1

v4->

c1

v6->

c1

0.62 0.65 0.68 0.72 0.75 0.78 0.82

0.02

0.04

0.06

0.08

0.10

0.12

0.14bL : kzR = 0.45

v2->

c1

v1->

c1

v3->

c1

v5->

c1

v7->

c1

v4->

c1

v6->

c1

Figure 4.3: Matrix elements |Tcv,‖|2 and |Tcv,⊥|2 and corresponding po-larization anisotropy ρTcv of the first 7 transitions vi → c1 for InAs, cal-culated including the scattering terms in the EM field at R = 4.85 nm.The corresponding energy scale is given at the top and bottom of thefigure.

Before proceeding further by analyzing the results and comparing themwith the dipole limit, at this point it is of particular importance to note thatthe bulk value of the dielectric function is used by calculating the matrixelements. As will be discussed in more detail in Chapter 5, this can only givea first estimation since a proper calculation of the matrix elements requiresa self-consistent determination of the dielectric function.

Nevertheless, starting with a qualitative approach, at first sight Figu-re 4.3 seems to be not very different from the results in the dipole limit,Figure 4.1. The selection rules in the dipole limit still determine almostcompletely the strength of the matrix elements and away from the zone cen-ter the corresponding polarization anisotropy ρTcv . At kz = 0, however,

88

4.5. Results

the parity selection rule is broken and the resulting transition strengths,regardless of their magnitude, are subject to a selection rule which requiresa strictly zero matrix element |Tcv,‖|2 in case of the |fzv| = 1

2 subbands.The explanation is subtle. First recall from paragraph 3.4.1 that the

E(−)12

valence subbands are heavy hole like. At the wire zone center, for

fzv = 12 the corresponding lateral part of the envelope wavefunction (the jz

component) is given by

χ 12,jz

(ρ, φ) = |HH1〉jzJ 12−jz

(j1,nρ

R) ei( 1

2−jz)φ, (4.40)

apart from a normalization constant and with j1,n the nth zero of J1(x) = 0.Here |HH1〉 is given in (3.35), with kz = 0.

Turning to the band-to-band transitions E(−)12,1

→ C(+)0, 1 , the transition

matrix for the ∼ ρ cosφ term in the internal field, see (2.9) and (2.10), is ofthe form

∑σ

∣∣∣∣∣∣∑

jz

〈χlzc=0| ρ cosφ |χ 12,jz〉〈Sσ|py|32jz〉

∣∣∣∣∣∣

2

, (4.41)

again only considering fz = +12 and focussing on the perpendicular (y)

polarization. The integral over φ in the envelope part of (4.41) is onlynonzero if jz = 3

2 or jz = −12 and the integral over ρ gives the same value

in these two cases, apart from a minus sign in case of jz = 32 (coming from

J−1(x) = −J1(x)). Absorbing the value of the integral over ρ in a constantc, which also includes the overall normalization of the wavefunctions, andusing (4.40), equation (4.41) equals

∑σ

∣∣∣−c 2π|HH1〉 32〈Sσ|py|32 3

2〉+ c 2π|HH1〉− 12〈Sσ|py|32 − 1

2〉∣∣∣2

= |c|2(2π)2∑

σ

∣∣∣∣−1√3〈Sσ|py|32 3

2〉+ 〈Sσ|py|32 − 12〉

∣∣∣∣2

= |c|2(2π)2∣∣∣∣−

1√3

12

+1

2√

3

∣∣∣∣2

= 0, (4.42)

where in the first equality the |HH1〉jz are inserted utilizing (3.35) at kz =0, while for the second equality the polarization selection rules, given inTable 4.2, are used.

Since the only nonzero contributions for transitions from E(−)12

to the

first conduction subband are coming from the ρ cosφ term (all others areparity forbidden), equation (4.42) clearly demonstrates that for perpendicu-lar (y) polarization the fzv = +1

2 part of this transition is strictly forbidden.

89


In a similar way it can be shown that this is also the case for fzv = −12 , so

we conclude that the transitions E(−)12,1→ C

(+)0, 1 at perpendicular incidence

are polarization forbidden, if the envelope angular momentum selection rule4.38 is changed by the ρ cosφ term in the scattering field.

Turning to the quantitative aspects of taking the spatial variation intoaccount, it can be concluded that the corresponding corrections are toosmall to overcome the parity selection rule at kz = 0 significantly. Thisis a general result: in Appendix C the effect of the Mie scattering is alsoinvestigated for InP at different R. The maximal contribution at kz = 0 of|Tcv,‖|2 for the E

(−)12,1→ C

(+)0, 1 transitions is of the order 10−6 and a similar

order was found for the matrix elements of the E(−)32,1→ C

(+)0, 1 transitions.

For the transitions already present in the dipole approximation it canbe concluded from Figure 4.2, Figure 12 and Figure 13, that the scatteringreduces the strength slightly, with a comparable amount for both |Tcv,‖|2and |Tcv,⊥|2. While the polarization anisotropy of the matrix elements moreor less remains the same, the strength of the transition thus reduces at largerR. As an example, for ε = 12 and a transition wavelength of 900 nm, thereduction is estimated with the expansion parameter |ε| 12 k0R to be about1% at R = 5nm and 6 % at R = 10 nm, which is indeed confirmed by theresults in Appendix C.

90

Chapter 5

Dielectric function nanowire

Finally, in this chapter the dielectric function and polarization anisotropyof III-V semiconductor nanowires are obtained including the quantum con-finement corrections by the band-to-band transitions between the top Γ8

valence bands and the lowest lying Γ6 conduction band. Section 5.1.3 con-siders general theory about the dielectric response of a particular group ofinterband transitions. In section 5.2 a total dielectric function is formulatedincluding the background response of all transitions which are not connec-ted with the quantum confinement. Subsequently, in section 5.3 the totalpolarization anisotropy of a nanowire is derived and in section 5.4 explicitresults are given for InP and InAs.

5.1 General theory

The transition matrix elements, derived in the previous chapter for band-to-band transitions, are of crucial importance for determining the opticalresponse of a system to an incident EM field. There are mainly two approa-ches for determining the macroscopic dielectric function from the quantummechanical oscillator strengths, one based on the atomic polarizability, theother using the absorption transition rate. Both methods rely on the assump-tion that the EM field can be considered in the dipole limit, i.e. E(R⊥) ' E.Next to this it is required that the semiconductor heterostructure can be de-scribed using macroscopic Maxwell equations, i.e. where the different partsin the system are characterized by macroscopically averaged quantities.

In the following both approaches are applied to the nanowire case, inparticular concerning the band-to-band transitions.

5.1.1 Atomic polarizability approach

In order to derive the dielectric function of the nanowire by using the ato-mic polarizability, the transition matrix has to be expressed in the position

91

Chapter 5. Dielectric function nanowire

operator x instead of the momentum operator p.

Already at this stage the dipole approximation is of crucial importance.If E(R⊥) = E the electric field can be taken out of the integral as in(4.27). Omitting the subband characterization for convenience by denotingthe conduction and valence subband wavefunctions with |Ψc〉 and |Ψv〉 nowit suffices to find the relation between 〈Ψc|p |Ψv〉 and 〈Ψc|x |Ψv〉. Utilizingthe commutation relation

i~m0

p = [x,H], (5.1)

where H is the one-electron crystal Hamiltonian in the wire configurationderived in Chapter 3, one obtains

〈Ψc|p |Ψv〉 =m0

i~〈Ψc|[x,H]|Ψv〉

=im0E

transcv (kz)~

〈Ψc|x |Ψv〉, (5.2)

where

Etranscv (kz) ≡ ~ωcv(kz) ≡ Eg + Ec(kz)−Ev(kz), (5.3)

is the transition energy. The R dependence is omitted for the moment.

Following Ziman[23], the dipole moment is proportional to the local field,

〈−ex(t) 〉 = α′(ω)E(t) (5.4)

where α′ denotes the real part of the atomic polarizability. Also this onlymakes sense in the dipole approximation, the proportionality is integratedout in case of a spatially varying field. From equations (5.2), (4.27) and(4.28) the atomic polarizability for the band-to-band transitions betweenthe Γ8 and Γ6 subbands considered in this paper is given by

α′(ω) =1N

e2

m0

∑cv

∑

kz

2m0 ~ωcv(kz)

|Tc v(kz)|2 1ω2

cv(kz)− ω2, (5.5)

where contrary to the one atom model in [23] here a factor 1N appears since

already N atoms are taken into account in (5.5) by the sum over the Ninitial valence subband states.

The formal prove of equations (5.4) and (5.5) requires time-dependentperturbation theory and in the present case this means that in the excitedstate (4.6) the time dependence has to be included by solving the correspon-ding time-dependent Schrodinger equation. Here we make an analogy withan atomic system, since the basic arguments are the same in the two pictu-res. Denote a ground-state orbital with |Φ0〉 corresponding to a ground-state

92

5.1. General theory

energy ε0, from which an electron can be excited to higher orbitals |Φj〉 withenergy εj . Assuming the electric field in the x direction for simplicity1, theelectron wavefunction |Ψ(t)〉 at a particular moment can be written as

|Ψ(t)〉 = |Φ0〉e−iε0t/~ +∑

j

cj(t)|Φj〉e−iεjt/~, (5.6)

where the coefficients cj(t) ∝ eEx 〈Φj |x |Φ0〉 are the solutions of the time-dependent Schrodinger equation. The expectation value of the dipole mo-ment −〈Ψ(t)| ex |Ψ(t)〉 consequently becomes proportional to the electricfield as in (5.4), where the atomic polarizability in the present case is givenby

α′(ω) =e2

m

∑

j

fj

(εj − ε0)2 − ω2, (5.7)

with fj = 2m~2 ~(εj − ε0) |〈Φj |x|Φ0〉|2 the oscillator strength of these atomic

transitions.Turning back to the nanowire, using (5.2) it is easy to show that the

oscillator strength of a transition v → c at kz corresponding to (5.5) equals

fcv(kz) =2

m0 ~ωcv(kz)|Tc v(kz)|2, (5.8)

Classically, the oscillator strength is the number of oscillators with frequencyωcv(kz). Indeed, the quantity (5.8) is dimensionless. Quantum mechanicallyit has to satisfy the Thomas-Reiche-Kuhn sum rule, in the present case∑

cv

∑kz

fcv(kz) = N .

Once the real part of a linear response is known, the imaginary part isuniquely determined by the Kramers-Kronig relations which are a conse-quence of the causality condition. This leads to

α(ω) =1N

e2

m0

∑cv

∑

kz

fcv(kz)(

1ω2

cv(kz)−ω2 + iπ2ω δ(ω − ωcv)

). (5.9)

Apart from the oscillator strength, this is basically the same result as ob-tained in the one atom model [4].

Since the atomic dipole moment (5.4) described in this way is the sa-me for all the N electrons involved in transitions between the Γ8 and Γ6

subbands, the total dipole moment per unit volume P is given by

P =N

VαE. (5.10)

1An electric field represented by Exe−iωt corresponds to a real electric field with am-plitude 2Ex, so take E(t) = Ex(eiωt + e−iωt)

93


All what remains is to determine the relation between this macroscopicallyaveraged quantity and the dielectric function. Again assuming a linear res-ponse, the dielectric displacement D equals

D = εE = E + 4πP , (5.11)

which leads to

ε− 1 =4πN

Vα. (5.12)

Combining this with (5.9) the imaginary part of the dielectric function isobtained:

ε′′(ω) =2π2e2

m0ωV

∑cv

∑

kz

fcv(kz)δ(ω − ωcv(kz)), (5.13)

where V is the volume πR2L of the nanowire.In the above derivation it is assumed that the local field which excites an

electron equals the macroscopic field obtained from the Maxwell equations.It is well known [8][12][23] that a more realistic result requires inclusion ofthe Lorentz correction, which has to be reconsidered when dealing with ananostructure. In the present paper this correction is not taken into account.

5.1.2 Transition rate method

In the second approach the quantum mechanically determined transitionprobability is related to the macroscopic power loss in the wire due to theabsorption process. The method is explained here shortly since it is exten-ded more easily to the case of a varying EM field.

Starting with the transition matrix Mc v, the probability Pcv for a transi-tion between subbands c and v is obtained from from Fermi’s Golden Rule,

Pcv(kz) =2π

~|Mc v(kz)|2 δ(~ω −Etrans

cv (kz)). (5.14)

The total transition rate P subsequently equals∑

kz

∑cv Pcv(kz) and mul-

tiplying with the energy ~ω in each photon this equals the power loss in thewire volume due to absorption. With equations (4.27) and (4.28), in thedipole limit this results in

Power loss =2π

ω

e2

m20

|E|2∑

cv,kz

|Tc v(kz)|2 δ(~ω − Etranscv (kz)). (5.15)

In Part 1 this quantity already was derived macroscopically for an infinitecylinder by

Wext = CextI0 = Cextc

8π|2E0|2 = Qext

c

8π|2E0|22RL, (5.16)

94

5.1. General theory

see equations (1.58) and (1.59). In the second step it is taken into accountthat an electric field represented by E0e

−iωt corresponds to a real electricfield with amplitude 2E0.

Utilizing the efficiency factors (2.17)-(2.20) in the dipole limit, with thenotion that the scattering contribution can be neglected in that case, thetotal power loss Wabs due to absorption equals

Wabs = ε′′Vω

2π|E|2, (5.17)

where E is defined in the same way as in (4.27).Finally this gives

ε′′(ω) =1V

(2πe

m0ω

)2 ∑cv

∑

kz

|Tc v(kz)|2 δ(~ω −Etranscv (kz)), (5.18)

which is the same result as in (5.13), here shown with fcv explicitly writtenout.

As in the case of the polarizability approach, the current derivation relieson the dipole approximation, but here it is more easy to see the consequencesof allowing a spatial variation of the EM field compared to the scale ofR. Starting with the classical absorption rate, equation (5.17) in fact is anexpansion in mk0R up to first order, with an additional factor R coming fromthe geometrical cross section. Increasing R and subsequently mk0R requiresa further expansion than (5.17), here denoted with Wabs(ε,R). In case ofthe quantum mechanically determined transition probability, in (5.14) nowequation (4.29) has to be used instead of (4.28).

Equalizing the two transition rates yields

Wabs(ε,R) = ~ω∑

kz

∑cv

Pcv(kz; ε, R) (5.19)

=2π

ω

e2

m20

|E|2∑

cv,kz

|Tc v(kz; ε,R)|2 δ(~ω −Etranscv (kz, R)),

with the R, ε dependence is indicated, also for Etranscv . Together with the

Kramers-Kronig relations, (5.19) has to be solved self-consistently in orderto find ε′(ω) and ε′′(ω).

As a final remark, in the above expressions (5.13) and (5.18) for ε′′(ω)in the dipole approximation and even more (5.19) as a relation for ε(ω) incase of a spatial varying EM field, it is implicitly assumed that the dielectricfunction is a constant, not depending on the wire radius. A correct descrip-tion, however, requires allowing ε(R), also in the dipole limit: the inclusionof the Lorentz correction has to be reconsidered within a microscopic, se-miclassical approach, when dealing with nanoscale systems which are not

95


”microscopically large” any more: the resonant states are extended over thewhole volume by the envelope part of the wavefunctions and consequentlythe induced polarization is position dependent.

As a first step, however, in this paper the dielectric function is approxi-mated with a constant, which depends on the orientation of the incident fieldbut is calculated in a local approach, not including the spatial variation.

5.1.3 Dielectric function expressed in reduced effective mass

The dielectric function derived in the previous section is expressed in a moreeasily applicable form by taking advantage of the effective mass approxima-tion.

For this purpose, first note that the sum over kz in (5.18) is replacedby an integral over the first Brillouin zone if L → ∞. Using L = Ma andtaking the reciprocal distance 1

M2πa into account, (5.18) is replaced with

ε′′(ω) =2

R2

e2

m20ω

2

∑cv

∫ πa

−πa

dkz|Tc v(kz)|2 δ(~ω −Etranscv (kz)), (5.20)

where the integral is taken over the first Brillouin zone.

Apart from the correction by the finite potential well on the conductionsubbands, the simple k2

z dependence of Etranscv (kz) derived in the effective

mass approximation results in

dkz

dEtranscv (kz)

=(

2µ∗cv z

~2

) 12 1

2√

Etranscv (kz)− Etrans

cv

, (5.21)

where

1µ∗cv, z

≡ 1m∗

c

+1

m∗v, z

(5.22)

is the reduced effective mass in the z-direction of the nanowire obtainedfrom the results in Chapter 3 and Etrans

cv ≡ Etranscv (0).

With the density of states expression (5.21) the integral part in (5.20)equals

∫dE

1√E − Etrans

cv

|Tc v(E)|2 δ(~ω − E) (5.23)

and after evaluating the δ distribution finally one obtains

ε′′(ω) =2

R2

(2µ∗cv z~2

) 12(

em0ω

)2 ∑cv |Tc v(~ω)|2 1√

~ω−Etranscv

, (5.24)

where |Tc v(~ω)|2 is given by (4.28).

96

5.2. Dielectric function for finite group transitions

In the last step of the above derivation it is more realistic to add abroadening term, for instance by replacing the δ function with a Gaussiandistribution, since in practice never an infinite sharp line is seen. The quan-tum efficiency is always reduced by the radiative relaxation of the levels, orby impurities for instance at the surface of the nanowire.

5.2 Dielectric function for finite group transitions

Although the results in the previous sections apply to band-to-band transi-tions between the Γ8 valence and Γ6 conduction subbands in a cylindricalnanowire, up till now we did not specify precisely how to consider the sumover v and c. In principle, the real part of the dielectric function at a fre-quency ω away from any absorption peak is built up from all responses withfrequencies ω′ > ω, i.e. from all atomic oscillators which are able to followthe oscillation of the EM field. In practice, however, it is hardly feasible tocalculate all contributions, for example those of bulk bands lying deep in aparticular system, or even all relevant symmetry points of the highest lyingbands.

Nevertheless, focussing again on the nanowire, there is a way to takeonly a particular group of transitions into account explicitly without neglec-ting the others entirely, provided the bulk dielectric function εbulk is knownreasonably well. Denoting this group with c and v, referring to the notationabove, a background dielectric function εbg can be defined which is basicallythe bulk dielectric function, from which the c, v group is projected out. Inother words, denoting εw

cv as the dielectric response of one particular transi-tion c, v in the nanowire and εbulk

cv as its contribution at the same frequencyin the bulk material, the total dielectric function in the wire configurationis given by

εw(ω) = εbg(ω) +∑cv

εwcv(ω), (5.25)

with

εbg = εbulk −∑cv

εbulkcv . (5.26)

As a first approximation, away from a bulk absorption peak εbg is approxima-ted with εbulk, the bulk dielectric function. Contrary to common literature[4][5][14][15], where εbg usually is not taken into account at all, this approachwill be retained carefully in the remaining of this paper.

97


5.3 Polarization anisotropy nanowire

Closely related to the discussion above, the common approach [4][5][14][15]to derive the polarization anisotropy of an infinite cylinder in the dipoleapproximation is insufficient when the quantum confinement becomes es-sential. In the usual procedure the polarization anisotropy is estimated byconsidering the absorption coefficient of the nanowire. In contrast, by usingthe efficiency factors, here an approach is followed in which the correct die-lectric function εw(ω) appears in a natural way. Also the dipole limit canbe taken more precisely.

By defining the relative difference δ⊥ between the macroscopically de-termined internal field at parallel and perpendicular polarization by

δ2⊥(ω, x) ≡

∣∣∣∣2

1 + ε⊥w(ω, x)

∣∣∣∣2

, (5.27)

the efficiency factors derived in Part 1 are summarized for a nanowire atnormal incidence by

Qext‖ (ω, x) = ε′′‖w (ω, x)πx

2+ O(x3), (5.28)

Qext⊥(ω, x) = ε′′⊥w(ω, x)πx

2δ2⊥(ω, x) + O(x3), (5.29)

where the expansion parameter x is defined as

x ≡ k0R, (5.30)

while ε′′‖w(ω, x) and ε′′⊥w(ω, x) are the dielectric functions of the nanowi-re at parallel and perpendicular incidence, respectively. Consequently, thepolarization anisotropy due to the extinction is given by

ρext(ω, x) =Qext‖ (ω, x) − Qext⊥(ω, x)Qext‖ (ω, x) + Qext⊥(ω, x)

=ε′′‖w (ω, x) − ε′′⊥w(ω, x) δ2

⊥(ω, x) + O(x2)

ε′′‖w (ω, x) + ε′′⊥w(ω, x) δ2⊥(ω, x) + O(x2)

. (5.31)

Taking the wire dipole limit by neglecting terms of higher order in x, thisleads to

ρdip(ω,R) =ε′′‖w(ω,R)− ε′′⊥w(ω, R) δ2

⊥(ω, R)

ε′′‖w(ω,R) + ε′′⊥w(ω, R) δ2⊥(ω, R)

, (5.32)

where the R dependence is explicitly shown. Note that the scattering processin the wire dipole limit is negligible compared to absorption. It should be

98

5.3. Polarization anisotropy nanowire

stressed again that for practical purpose it is more convenient to use apolarization contrast, in the present case denoted with

Cdip(ω, R) =ε′′‖w(ω, R)

ε′′⊥w(ω, R) δ2⊥(ω,R)

. (5.33)

Utilizing the framework of section 5.2 it is of particular interest to con-sider these results in two special cases.

Starting with the one which leads in a natural way to the result in theusual procedure, if the transitions are investigated at frequencies ω whereε′′bulk ∼ 0 and ε′bulk is large compared to both the real and imaginary partof

∑cv εw

cv(ω) , then the background contribution to ε′′w(ω) is negligible andδ⊥ is approximated with the bulk value. In this case

ρdip ∼∑cv

|Tcv,‖|2 − |Tcv,⊥|2 δ2⊥

|Tcv,‖|2 − |Tcv,⊥|2 δ2⊥

, (5.34)

which indeed equals the usual expression [14][4][5].Secondly, take a macroscopically small, but microscopically large R which

allows to neglect the quantum corrections, but still satisfies the dipole li-mit λ0 À R. Considering (5.25) and (5.26), we conclude that in this casethe quantum correction

∑cv εw

cv(ω)−∑cv εbulk

cv (ω) becomes negligible, whichleads back to the classical result (2.22), as required.

99


5.4 Results

Using the results of the previous chapters, in this section the above theore-tical framework is applied to the specific examples InP and InAs. Initiallythe focus will be on the dielectric response purely due to the band-to-bandtransitions between the Γ8 valence and Γ6 conduction subbands in nanowiresby neglecting the imaginary part ε′′bg of the background dielectric functionin (5.25). Hereby the following aspects will be discussed in more detail:

• The effect of the kz dependence of the transition matrix on the ima-ginary part ε′′w of the nanowire dielectric function .

• The polarization anisotropy (4.30) of the matrix elements alone and in-cluding the polarization due to the dielectric mismatch only by takingε′bg into account, as in (5.34).

• The R dependence of ε′′w and corresponding polarization anisotropy(5.34).

• Material dependence and comparison with literature [4][5].

Finally, in paragraph 5.4.2, ε′′bg is taken into account by calculating thecorrect expression for the polarization contrast, corresponding to equation(5.33) and equivalent to the polarization anisotropy given in (5.34). Allresults are obtained in the dipole approximation.

5.4.1 Estimation kz dependence of |Tcv|2

In Chapter 4 it was shown that the transition matrix elements depend onkz in a nontrivial way.

0.65 0.7 0.75 0.8 0.85Energy HeVL

1

2

3

4

Ε'' w

ire

y-pol.z-pol aL: kz = 0.

v2v3

v7v4

0.65 0.7 0.75 0.8 0.85Energy HeVL

1

2

3

4

Ε'' w

ire

y-pol.z-pol bL: kz = 0.45

v2

v1v3

v7v4

Figure 5.1: Complex part of the dielectric function εw at parallel (z) andperpendicular (y) polarization for InAs and R = 4.85 nm, fixing |Tcv|2at a) : kzR = 0 and b) : kzR = 0.45. Only the first 7 transitions vi → c1

are taken into account, the corresponding peaks are labeled in the samecolors as in Figure 4.1. The background response ε′′bg is neglected.

100

5.4. Results

As a first simple trial, Figure 5.1 shows the imaginary part of the die-lectric function εw for InAs and R = 4.85 nm , obtained by fixing |Tcv|2 atits value at a) : kzR = 0 and b) : kzR = 0.45 in equation (5.24). The singu-larities at ~ω − Etrans

cv = 0 are broadened in a qualitative way by adding adisplacement of 0.004 eV to ~ω in the denominator of (5.24). Although thisis not the common way to include the broadening, it qualitatively gives thethe same results as in the usual procedure, where the Dirac distribution isreplaced for instance with a Gaussian. Since the non-radiative decay whichcauses the broadening is not estimated yet it makes little practical difference.

Focussing on the first peaks in Figure 5.1, it is concluded that simplytaking |Tcv(0)|2 in (5.24) not only quantitatively, but also qualitatively failssince it does not include the contribution of the first peak, corresponding tothe transition E

(−)12,1→ C

(+)0, 1 which is parity forbidden at kzR = 0, but not

at finite kzR.

In Figure 5.2 the kz dependence of |Tcv|2 is taken into account properly.At each fixed value of ~ω equation (5.24) is calculated using the correctvalue of |Tcv|2, which means that for every point in the figure the correctvalence subband wavefunction is used in (4.28). Only the first two peaksare shown.

0.62 0.64 0.66 0.68 0.7 0.72Energy HeVL

0

1

2

3

4

Ε'' w

ire

y-pol.

z-pol

v2

v1

Figure 5.2: Parallel (dots) and perpendicular (line) contributions to εw

of the first two transitions vi → c1 for InAs and R = 4.85 nm. At eachfixed value of ~ω equation (5.24) is calculated using the correct value of|Tcv|2, which means that for every point in the figure the correct valencesubband wavefunction is included in (4.28).

Since it is desirable to avoid such an extensive calculation, from nowon the simpler approach of taking |Tcv|2 constant is used, but at a finitekz value such as in Figure 5.1 b). In this way the results of Figure 5.2are reproduced qualitatively: the peaks which are parity forbidden (pF)

101


at kz = 0 are included. Quantitatively, the slope corresponding to oneparticular transition is underestimated in most of the cases, since except forthe pF transitions, the matrix elements become smaller for larger kz. Theheight of the peaks will be corrected with respect to the strength of thetransitions which are already present at kz = 0. As a consequence, the pFtransition peaks are slightly overestimated in the following paragraphs.

Furthermore, it is important to note that the tails of the different transi-tion contributions to ε′′w only represent a first rough qualitative estimation.In the effective mass approach the bulk bands are assumed to be quadraticand as stated in Chapter 3 this rests on the assumption that k is sufficientlysmall. In fact, a particular transition stops contributing when it reaches theboundary of the Brillouin zone, but in the present procedure this point isnot reached at all since the hole subbands derived in Chapter 3 have a finiteextent.

5.4.2 Polarization anisotropy and R dependence

Utilizing the above mentioned procedure, Figure 5.3 a) again illustrates theimaginary part of the dielectric function εw for InAs and R = 4.85 nm ,now obtained by fixing |Tcv|2 at kzR = 0.45 and correcting the height ofthe peaks with respect to the strength of the transitions which are alreadypresent at kz = 0. As stated before, the background response ε′′bg will beneglected up till paragraph 5.4.4. The remarks about broadening remainthe same.

0.65 0.7 0.75 0.8 0.85Energy HeVL

1

2

3

4

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ire

y-pol.z-polv2

v1v3

v7v4

0.65 0.7 0.75 0.8 0.85Energy HeVL

-0.5

-0.25

0

0.25

0.5

0.75

1

Ρ

Ρ T v c

Ρ bulk

Ρ wire

Figure 5.3: a): Complex part of the dielectric function εw at parallel(z) and perpendicular (y) polarization for InAs and R = 4.85 nm, fixing|Tcv|2 at kzR = 0.45 and correcting the height of the peaks with respectto the strength of the transitions which are already present at kz =0. Only the first 7 transitions vi → c1 are taken into account, thecorresponding peaks are labeled in the same colors as in Figure 4.1. b):Polarization anisotropy, calculated from a) alone (dotted), only frombulk mismatch (line) and both together (red line).

In Chapter 4 it already was shown that v1 → c1 is the only pF transitionwhich contributes significantly. Indeed this is confirmed by Figure 5.3.

102

5.4. Results

Secondly, the influence of the valence subband effective mass on thereduced effective mass µ∗cv z (5.22) is negligible: in all relevant situationsconsidered here, the effective mass of the conduction subband c1, also in thefinite potential well model, is much smaller then m∗

v, z.As in Chapter 4, the transition energies include the correction by the fi-

nite potential well model on the conduction subbands. The correction on thematrix elements is not taken into account, since it can be argued to be small.

Figure 5.3 b) gives the polarization anisotropy ρTcv due to the matrixelements alone (dotted line), ρbulk only from bulk mismatch (black line) andboth together (red line). Starting from the left, up till the appearance ofthe peak v3, ρTcv has a constant value of 0.6, as expected (compare Figu-re 4.1 b)): for all kz both transitions are subject in the same way to theenvelope angular momentum and polarization selection rules described inthe previous chapter. The peak v3 changes ρTcv drastically: in the caseof the E 3

2,n → C0, 1 transitions the perpendicular component is polarizati-

on forbidden. This is also the case for the v7 peak, here the polarizationanisotropy of the matrix elements alone becomes even negative.

Compared to the bulk value ρbulk due to the dielectric mismatch, whichis almost a constant in the region of interest, it is concluded that ρTcv cau-ses giant changes in the total polarization anisotropy of the wire. This willbecome even more visible in paragraph 5.4.4 considering the polarizationcontrast.

0.52 0.54 0.56 0.58 0.6 0.62Energy HeVL

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2

3

4

Ε'' w

ire

y-pol.z-pol

v2

v1v3

v7v4

0.52 0.54 0.56 0.58 0.6 0.62Energy HeVL

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Ρ

Ρ T v c

Ρ bulk

Ρ wire

Figure 5.4: a): Complex part of the dielectric function εw at parallel(z) and perpendicular (y) polarization for InAs and R = 7.5 nm. b):Polarization anisotropy, calculated from a) alone (dotted), only frombulk mismatch (line) and from both together (red line).

The statements about the R dependence of ε′′w and corresponding polari-zation anisotropy (5.34) are summarized if one compares Figure 5.3 with thesame results at R = 7.5 nm, Figure 5.4. Without including the backgrounddielectric response εbg, the dielectric function of the wire behaves formallyas ε′′w ∝ 1

R2 while the polarization anisotropy ρTcv remains the same. Howe-

103


ver, the different transitions come closer to each other for increasing R, withtheir mutual distances also behaving like 1

R2 in the infinite-well approxima-tion. Moreover, more and more higher transitions which are not taken intoaccount here enter the energy region of interest. Apart from the remarksabout the tails corresponding to the different transitions one should thuskeep in mind that a correct picture of the higher energy part includes morepeaks, for instance coming from vi → c2 transitions.

5.4.3 Material dependence

For InP the present effective mass approach can be compared with resultsobtained from an tight-binding approach [5].

1.54 1.56 1.58 1.6 1.62 1.64Energy HeVL

0.2

0.4

0.6

0.8

Ε'' w

ire

y-pol.z-pol

v1

v2

v3

v4v5

v6

1.54 1.56 1.58 1.6 1.62 1.64Energy HeVL

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Ρ

Ρ T v c

Ρ bulk

Ρ wire

Figure 5.5: a): Complex part of the dielectric function εw at parallel(z) and perpendicular (y) polarization for InP and R = 4.83 nm, fixing|Tcv|2 at kzR = 0.6. Only the first 7 transitions vi → c1 are taken intoaccount. b): Polarization anisotropy, calculated from a) alone (dotted),only from bulk mismatch (line) and from both together (red line).

Comparing Figure 5.5 with Figure 15 in Appendix D and noting thatthe present results are obtained with the parameters at T = 0 K only forthe first seven transitions vi → c1, we conclude that the effective massapproximation, without including the split off Γ8 band and assuming aninfinite potential well in case of the valence subbands, successfully describesthe overall features of ε′′w . Remarkable is the difference in the lowest twotransition(s): in FIG 3 of [5] the band edge optical transition is fully z-polarized. This polarization selection cannot be explained from an effectivemass approach at all considering the E

(+)12

→ C(+)0, 1 transitions, but we note

that also the more accurate atomistic approach [4] contradicts this strictselection found in [5], see paragraph 4.5.1.

With the effective mass approach it is relatively easy to change the ma-terial parameters and wire radius. In Figure 5.6 the imaginary part ofthe dielectric function εw is shown at R = 10 nm both for InAs and InP.Comparing the two materials it is concluded that a particular band-to-bandtransition has a significantly larger contribution in the dielectric function

104

5.4. Results

0.48 0.49 0.5 0.51 0.52 0.53 0.54Energy HeVL

0.25

0.5

0.75

1

1.25

1.5

1.75

Ε'' w

ire

y-pol.z-polaL: InAs

v2

v3

v7

v4

1.46 1.47 1.48 1.49Energy HeVL

0.25

0.5

0.75

1

1.25

1.5

1.75

Ε'' w

ire

y-pol.z-polbL: InP

v1 v3v4

Figure 5.6: Complex part of the dielectric function εw at parallel (z) andperpendicular (y) polarization for a): InAs and b): InP at R = 10 nm.The pF transitions are neglected by fixing |Tcv|2 at kzR = 0.

of InAs. This is directly caused by the larger band gap of InP since thetransition probability ∝ 1

ω2 .

5.4.4 Effect of the dielectric background

Finally, in this paragraph the effect of the background response εbg is esti-mated by looking at the polarization contrast. For this purpose, comparethe results shown in Figure 5.7 and Figure 5.8.

0.52 0.54 0.56 0.58 0.6 0.62Energy HeVL

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50

75

100

125

150

175

200

C

aL: InAs

C bulkC wire

1.48 1.5 1.52 1.54Energy HeVL

40

60

80

100

120

140

160

C

bL: InP

C bulk

C wire

Figure 5.7: Polarization contrast Cbulk (black line), only due to thedielectric mismatch, and Cwire (red line), including the polarization ani-sotropy caused by quantum confinement, for a:) InAs and b:) InP, atR = 7.5 nm. The dielectric background is not taken into account pro-perly: ε′′bg is set to zero.

First of all, we recover the result that the effect of the quantum confi-nement is huge if εbg is not taken into account: both for InP and InAs Fi-gure 5.7 shows a maximum enhancement by a factor 4 due to the quantumconfinement, as already predicted in paragraph 4.4.1. However, in generalthis overestimates the polarization anisotropy substantially.

After including εbg in a proper way, see Figure 5.8, the effect of thequantum confinement is still large for InAs (maximum enhancement ' 2),

105


0.52 0.54 0.56 0.58 0.6 0.62Energy HeVL

30

40

50

60

70

80

90

100

C

aL: InAsC bulk

C wire

1.48 1.5 1.52 1.54Energy HeVL

40

42

44

46

48

C

bL: InPC bulk

C wire

Figure 5.8: Polarization contrast Cwire compared to Cbulk for a): InAsand b): InP, at R = 7.5 nm using the correct dielectric function of thewire: εw(ω) = εbg(ω) +

∑cv εw

cv(ω). It is assumed that εbg ' εbulk.

but considerably reduced for InP (maximum enhancement ' 1.2). Theseresults are far more conform reality and reflect the difference between InPand InAs: as shown in the previous section the confinement correction toεw is much smaller in the case of InP.

Furthermore, for both materials the effect of the quantum confinementdisappears if R is increased sufficiently. This is only achieved if εwire isconsidered in the right way: εw(ω) = εbg(ω) +

∑cv εw

cv(ω). To be moreprecise, taking all transitions c, v corresponding to one particular bulk tran-sition into account,

∑cv εw

cv(ω) ' ∑cv εbulk

cv (ω) for sufficiently large R andconsequently εw(ω) ' εbulk(ω). So for R → ∞ the quantum confinementcorrection

∑cv εw

cv(ω) equals the contribution of the same group of transiti-ons projected in the bulk system.

106

Summary and Conclusions

In this paper we analyzed the optical absorption of III-V semiconductor cy-lindrical nanowires with the aim to get a theory which describes the opticalproperties for arbitrary wire thickness and for a wide range of semiconductormaterials.

In Part I we started with a classical theory describing the scattering oflight by an infinite cylindrical structure. At arbitrary angle of incidence, inChapter 1 general expressions were found for the cross sections and corres-ponding efficiency factors, which are measurable quantities in the region farfrom the cylindrical wire.

Subsequently, in Chapter 2 we focussed on the case of cylindrical wiressmall compared to the wavelength of the incident light and derived analy-tic solutions explicitly as a function of the material properties (dielectricconstant, wire radius R), geometric configuration (angle of incidence) andwavenumber (k0) of the incident light. Next to reproducing the well knownresults in the dipole limit, we extended the theory by Mie expansion of theEM field inside cylinder up to second order in the dimensionless parameterk0R. We concluded that for increasing cylinder radius, besides the wavebehavior of the EM field inside the wire, the effect of optical focussing getsa more important role.

Furthermore, numerical results of the efficiencies and corresponding po-larization anisotropy are given for InP in the region between 350 and 600 nmand wire radii up till 5 nm. Hereby the nanowire is treated classically bytaking the bulk value of the dielectric function.

In Part II we have included the effects of quantum confinement by meansof a corrected description of the dielectric function of cylindrical nanowires.

For this purpose, in Chapter 3 we first derived the electronic structureusing effective mass theory. This method utilizes the already well knownbulk energy gaps and optical matrix elements at the band extreme. The re-sulting bulk dispersion obtained from the crystal Hamiltonian H0 is treatedas a kinetic energy term, which after including the cylindrical confinementpotential and assuming the wire radius R large compared to interatomicdistances, results in a one-particle Schrodinger equation acting on the enve-

107

lope of the nanowire wavefunction. Neglecting the small anisotropic termsin H0 by applying the spherical approximation, the total Hamiltonian of thenanowire becomes diagonal with respect to the total angular momentumalong the z axis Fz = Jz + Lz, with Lz and Jz the z-projection of the en-velope angular momentum and the total angular momentum of the atomicstates, respectively. Consequently the eigenvalue fz of Fz is a good quantumnumber.

In case of the Γ point valence band in III-V semiconductors we neglectedthe split-off band and diagonalized the 4× 4 Hamiltonian HΓ8

Fzof the heavy

and light hole band in the basis jz v = 32 , 1

2 ,−12 ,−3

2. At kz = 0, the wirezone center, HΓ8

Fzdecouples into two 2 × 2 blocks with solutions which are

characterized by parity : the envelopes are even or odd under ρ → −ρ.Assuming an infinite confinement potential, the complete set of solutions

for the Γ8 valence band in a cylindrical nanowire is thus characterized with|fz|, the parity (±) and n, denoting the nth solution at this parity. Awayfrom the zone center, the valence subband wavefunctions depend in a nontrivial way on kz by the the lateral part of the envelope function

For the Γ6 conduction band the Hamiltonian is already diagonal in Lz

and consequently the irreducible representation of the conduction subbandsis given by the eigenvalue lz of Lz and again (±), n. We have taken thefiniteness of the potential well into account by including a reduction factorto the conduction subband energies calculated in an infinite confinementmodel, which results in a large correction since the bulk conduction bandhas a relative small effective mass. Contrary to the energies, no correctionis made to the subband wavefunctions since in this case the difference withthe finite potential well model is expected to be negligible.

In Chapter 4 we analyzed the radiation-matter interaction − em0cA · p

between the external electromagnetic (EM) field and the electrons withinthe semiconductor system using a macroscopic, semiclassical approach: wetreated the EM field classically, while the semiconductor nanostructure isdescribed in the spirit of Chapter 3. Similar to the separation of the nanowirewavefunction into an atomic part and ”macroscopic” envelope functions, theEM transition matrix factors into separate integrals: a bulk, atomic likematrix element of the momentum operator and the integral of the EM fieldbetween the envelopes. This separation relies on the natural assumptionthat the EM field varies slowly compared to atomic distances.

Both the envelope and the atomic part of the transition matrix are sub-ject to selection rules. The last ones are the semiconductor variant of thewell known selection rules on p in atomic systems. In case of the p-likeΓ8 valence states and the s-like Γ6 conduction states we concluded for the|jz| = 3

2 valence states that the atomic-like matrix elements correspondingto components of p parallel to the wire axis are polarization forbidden. Con-

108

sidering the |jz| = 12 states we found a ratio of 4 between the parallel and

perpendicular p components, respectively. Physically, these polarization se-lection rules are based on the conservation of angular momentum on theatomic part of the transition matrix.

While the polarization selection rules originate from the bulk, atomiclike matrix element of the momentum operator, the selection rules on theenvelope part depend on the configuration of the system (wire dimensions,wavelength of incident EM field). Starting with the envelope integral overthe coordinate z parallel to the wire axis, in Part I we found for infinitecylinders that the EM field always is of the form E(r⊥)eiqzz and togetherwith the ∝ eikzz dependence of the envelope functions this results in theconservation of momentum equation along the z-direction.

Considering the lateral envelope part of the transition matrix, a spatiallyvarying EM field requires a different approach than the dipole limit.

In the common dipole approximation, the valence and conduction sub-band wavefunctions (∝ eilzφ) must have the same envelope angular momen-tum. In addition, this envelope angular momentum conservation naturallyleads to parity selection at the wire zone center: transitions between stateswith different envelope parity are not allowed. Away from the zone centerthis is the only selection rule which is broken since the valence subbandwavefunctions are not characterized by parity any more. We calculated theband-to-band transitions between the first 7 valence subbands and the lo-west Γ6 conduction subband for InP and InAs at different wire radii R. Firstof all, while the transition energies strongly depend on the wire radius, thematrix elements are independent of R, approximatively even if the finiteconfinement for the conduction subband wavefunctions would be taken intoaccount. Secondly, comparing the results of InAs with InP it can be con-cluded that next to the energy of a particular transition, also its strengthis material dependent. Finally we conclude that the polarization anisotropyof the transition matrix is completely determined by the polarization selec-tion rules. At all kz, the transitions from |fz| = 3

2 valence subbands to thelowest lz = 0 conduction subband are strictly forbidden for parallel polari-zation and for |fz| = 1

2 the transitions to the lz = 0 subband have a fixedpolarization anisotropy of 0.6. These observations compare favorably withthe results derived in an atomistic, empirical pseudo-potential plane-wavemethod[4]. The deviations can be explained by the possible corrections ofincluding the split-off band, or even diagonalizing the full 8×8 Hamiltonianof the three bulk Γ8 valence bands and Γ6 conduction band in the presenteffective mass approximation.

One of the purposes of this paper was to estimate the effect of classicalscattering for wire dimensions in the quantum confinement regime. Gener-ally, a spatially varying EM field breaks the symmetry in the envelope partof the transition matrix. We analyzed this by including the Mie scatteringterms found in Part I into the envelope integral. Although the qualitative

109

features indeed are different, since new peaks arize which are parity forbid-den in the wire dipole limit, we conclude that the order of magnitude ofthese kind of corrections is too small to overcome the parity selection ruleat kz = 0 significantly. The transitions that become weakly allowed areweaker by a factor ∼ 10−5 for InP and InAs and wire radii up till 15 nm.A second effect is that the strength of transitions which are already allowedin the wire dipole limit changes, but also this effect is not large. For a InPnanowire with a radius of 10 nm the Mie correction on these already allowedstates is about 5 %. Consequently, the effect of classical Mie scattering canbe neglected in the quantum regime.

Finally, in Chapter 5 we analyzed the full optical absorption process incylindrical nanowires by deriving an expression for the nanowire dielectricfunction including the quantum confinement effects. Hereby we made a firstorder approximation by assuming that the local field acting on a particularelectron at a lattice site is the same as the averaged field obtained from themacroscopic Maxwell equations. Thus, we neglected the additional internalfield due to the induced polarization of the neighboring atoms. It is notedthat the inclusion of such a Lorentz correction has to be reconsidered withina microscopic, semiclassical approach, when dealing with nanoscale systemswhich are not ”microscopically large” any more: the resonant states areextended over the whole volume by the envelope part of the wavefunctionsand consequently the induced polarization is position dependent.

As a first step, however, the dielectric function is approximated by aconstant and, based on the results in Chapter 4, the EM field can be consi-dered in the dipole limit. In this framework we used the advantages of theeffective mass approach by deriving a simplified expression for the imagina-ry part of the nanowire dielectric function, in which the integral over kz isreplaced by an integral over the energy utilizing an explicit expression forthe 1D density of states.

In this paper we focussed on the band-to-band transitions close to theband gap of III-V materials. In order to include also the dielectric respon-se of all other transitions, a background dielectric function εbg is definedwhich is basically the bulk dielectric function εbulk, in which the group ofband-to-band transitions is projected out. As a first approximation we tookεbg ' εbulk and neglected the quantum confinement contribution of the groupband-to-band transitions to the real part of the total dielectric function.

By making some rough simplifications, for instance neglecting the kz

dependence of the transition matrix and inserting line broadening of theabsorption peaks by hand, we obtained successfully the qualitative behaviorof the nanowire dielectric function. In agreement with recent literature[4][5], we find that the dielectric response of the band-to-band transitionsis strongly polarization dependent, which completely relies on the resultsof Chapter 4. Furthermore, comparing InP with InAs it is concluded that

110

the confinement has a significantly smaller effect on the dielectric functionof InP, which is explained by the larger bulk band gap of InP resulting insmaller band-to-band transition probabilities.

Compared to the bulk polarization anisotropy due to the classical die-lectric mismatch, which is almost a constant in the region of interest, itis concluded that the polarization anisotropy due to quantum confinementcauses giant changes in the total polarization anisotropy of the wire. Inaddition, by including the dielectric background the effect of quantum con-finement disappears in a natural way if R is increased sufficiently.

After all, we have thus established that the effective mass approach pro-vides a fast and flexible tool to analyze the diameter dependent propertiesof nanowires for a wide range of semiconductor materials. Possible impro-vements of the current framework are achieved if the Γ8 spin-off band isincluded, or eventually diagonalizing the full 8× 8 Hamiltonian of the threebulk valence bands and Γ6 conduction band in the present effective massapproximation.

111

A Hole wavefunctions fordifferent kz

112

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 1. , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 1.125 , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0.5 , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0.75 , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0.25 , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0.375 , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0. , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0.125 , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

Figure 9: Radial part of the |fz| = 12 , − (1) hole envelope wavefunctions

for InAs. The value of kzR changes from 0 in the first picture to themaximum value 1.125 (at the end of the band) in the last graph.

113

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 1.66667 , H+L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 2.03333 , H+L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 1. , H+L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 1.33333 , H+L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0.333333 , H+L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0.666667 , H+L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0. , H+L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0.166667 , H+L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

Figure 10: Radial part of the |fz| = 12 , +(1) hole envelope wavefunctions

for InP. The value of kzR changes from 0 in the first picture to themaximum value 2.033 (at the end of the band) in the last graph.

114

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 1.66667 , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 2.76667 , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 1. , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 1.33333 , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0.333333 , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0.666667 , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0. , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

0 0.2 0.4 0.6 0.8 1Ρ R

-1

-0.5

0

0.5

1

1.5

Χj z

fz = 12 , kzR = 0.166667 , H-L 1

jz = 3 2jz = 1 2

jz = -1 2jz = -3 2

Figure 11: Radial part of the |fz| = 12 , − (1) hole envelope wavefunctions

for InP. The value of kzR changes from 0 in the first picture to themaximum value 2.767 (at the end of the band) in the last graph.

115

B Polarization selection rules

116

1m0〈S ↑ | ε · p |32 3

2 〉 εx εy εz

propagation ‖ to ex impossible Π√2

polF

propagation ‖ to ey −i Π√2

impossible polF

propagation ‖ to ez −i Π√2

Π√2

impossible

Table 1: Selection rules on the atomic-like interband matrix elements1

m0〈S ↑ | ε · p | 32 3

2 〉. The propagation direction of the EM-wave isdenoted in the left column. The unit directions of E are denoted with εx,εy and εz; Π is related to the Kane matrix element Ep by Ep = 2m0Π2.The polarization forbidden transitions are denoted with polF .

1m0〈S ↑ | ε · p |32 1

2 〉 εx εy εz

propagation ‖ to ex impossible polF i 2Π√6

propagation ‖ to ey polF impossible i 2Π√6

propagation ‖ to ez polF polF impossible


m0〈S ↑ | ε · p | 32 1

2 〉.

1m0〈S ↑ | ε · p |32 − 1

2 〉 εx εy εz

propagation ‖ to ex impossible Π√6

polF

propagation ‖ to ey i Π√6

impossible polF

propagation ‖ to ez i Π√6

Π√6

impossible


m0〈S ↑ | ε · p | 32 − 1

2 〉.

117

1m0〈S ↑ | ε · p |32 − 3

2 〉 εx εy εz

propagation ‖ to ex impossible polF polF

propagation ‖ to ey polF impossible polF

propagation ‖ to ez polF polF impossible


m0〈S ↑ | ε · p | 32 − 3

2 〉.The propagation direction of the EM-wave isdenoted in the left column. The unit directions of E are denoted with εx,εy and εz; Π is related to the Kane matrix element Ep by Ep = 2m0Π2.The polarization forbidden transitions are denoted with polF .

118

C Interband matrix elements

119


ÈΡT

vcÈ

ÈTv

cÈ2Ha

rb.u

nitsL


1.53 1.55 1.58 1.6 1.62 1.650

0.20

0.40

0.60

0.80

1.00

Ρ > 0Ρ < 0

1.55 1.58 1.6 1.62 1.650

0.20

0.40

0.60

0.80

1.00

0.02

0.04

0.06

0.08

0.10

0.12

0.141.53 1.55 1.58 1.6 1.62 1.65

aL : kzR = 0.

z - pol.y - pol.v1

->

c1v2->

c1v3->

c1

v4->

c1

v5->

c1v6->

c1

v7->

c11.55 1.58 1.6 1.62 1.65

0.02

0.04

0.06

0.08

0.10

0.12

0.14bL : kzR = 0.93

v1->

c1

v2->

c1v3->

c1

v4->

c1v5->

c1v6->

c1

v7->

c1

Figure 12: Matrix elements |Tcv,‖|2 and |Tcv,⊥|2 and corresponding po-larization anisotropy ρTcv of the first 7 transitions vi → c1 for InP, cal-culated including the scattering terms in the EM field at R = 4.85 nm.The corresponding energy scale is given at the top and bottom of thefigure.

120


ÈΡT

vcÈ

ÈTv

cÈ2Ha

rb.u

nitsL


1.45 1.46 1.46 1.47 1.47 1.480

0.20

0.40

0.60

0.80

1.00

Ρ > 0Ρ < 0

1.46 1.46 1.47 1.47 1.480

0.20

0.40

0.60

0.80

1.00

0.02

0.04

0.06

0.08

0.10

0.12

0.141.45 1.46 1.46 1.47 1.47 1.48

aL : kzR = 0.

z - pol.y - pol.v1

->

c1v2->

c1v3->

c1

v4->

c1

v5->

c1v6->

c1

v7->

c11.46 1.46 1.47 1.47 1.48

0.02

0.04

0.06

0.08

0.10

0.12

0.14bL : kzR = 0.93

v1->

c1

v2->

c1v3->

c1

v4->

c1v5->

c1v6->

c1

v7->

c1

Figure 13: Matrix elements |Tcv,‖|2 and |Tcv,⊥|2 and corresponding po-larization anisotropy ρTcv of the first 7 transitions vi → c1 for InP,calculated including the scattering terms in the EM field at R = 10nm.

121

D Reference articles

122

Figure 14: Califano and Zunger [4], pg. 6. In FIG. 3 the interbandmatrix elements for a R = 4.8 nm InAs wire are shown based on anatomistic, empirical pseudopotential plane-wave method. The C∞ v re-presentations given in Table II differ from those derived in the presentpaper.

123

Figure 15: Persson and Xu [5], pg. 3. In FIG. 3 ε′′w is shown for aR = 2.9 nm InAs wire based on an atomistic tight-binding approach.

124

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optical properties of cylindrical nanowiresductor cylindrical nanowires. the optical properties are...

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