Journal of the Korean Physical Society, Vol. 64, No. 11, June 2014, pp. 1713∼1720

A Novel Approach to the Evaluation of the Interface Roughness ScatteringForm Factor in Intersubband Transitions

Nguyen Thanh Tien∗ and Pham Thi Bich Thao

College of Natural Sciences, Can Tho University, 3-2 Road, Can Tho, Vietnam

Le Tuan

School of Engineering Physics, Hanoi University of Science and Technology, 1 Dai Co Viet Road, Hanoi, Vietnam

(Received 6 January 2014, in final form 25 March 2014)

We propose a modification of the interface roughness (IFR) scattering form factor in intersubbandtransitions. We properly derived a formula for the form factor for IFR scattering in terms of theintegrals of the envelope wave functions. This new form factor has a more global natural than theold one (proposed by Ando) and may be suitable for a wide range of applications. In this paper,we calculate and compare the absorption linewidths by applying of the old form factor and thenew one. Different from previous calculations, for the same surface profile (Δ, Λ), the interfaceroughness scattering absorption linewidth calculated with the new form factor is twice as great asthat calculated with the old one. Our numerical calculations may better explain the experimentalresults for the well-width dependence of the intersubband absorption linewidth.

PACS numbers: 78.67.De, 68.35.Ct, 78.40.Fy, 78.30.FsKeywords: Absorption linewidth, Intersubband transitions, Interface roughness scattering, Quantum well,Transition form factorDOI: 10.3938/jkps.64.1713

I. INTRODUCTION

Intersubband transitions in semiconductor quantumwells (QWs) have been widely applied to optoelectronicdevices, such as quantum cascade lasers [1–4] and QWinfrared photodetectors [5, 6]. The uniqueness of theseapplications lies in the many degrees of freedom for de-signing QW active layers and surrounding structures, en-abling device operation at various photon energies be-low the original band gaps of the host materials. Here,the main parameters for changing a transition energyin QWs are the well width L and the barrier potentialVb. Recently, the authors [7] attempted to modify theenergy and the linewidth of intersubband transitions byusing the hetero-insertion of a submonolayer into semi-conductor quantum wells. The energy and the linewidthof an intersubband transition are closely related to eachother in optoelectronic device design. In narrow QWs,especially at low temperatures, the two parameters (Land Vb) also affect the absorption/emission linewidth,which is usually governed by IRF scattering [8–13]. Inother words, IFR plays the dominant role in controllingthe linewidth. We proposed an efficient method for in-dividual single-valued estimation of the two parameters

∗E-mail: nttien@ctu.edu.vn; Fax: +84-710-3832062

of an interface profile from optical data [14]. We alsobelieve that the IFR scattering is dominant in severalcoexisting scattering mechanisms, so, the IFR scatteringform factor in intersubband transitions needs to be con-sidered carefully. In previous studies [7,9,12], this formfactor was usually calculated through the local value ofthe wave function at the barrier. This led to significanterrors if the calculated value of the wave function at thebarrier was inaccurate. In the present paper, we proposea method for a generalized, more accurate calculationof the IFR scattering form factor. We will derive a for-mula in which the IFR scattering form factor is given interms of quantities that are insensitive to the values ofthe wave functions at the interface. This form factor willbe proper for cases that use approximate wave functionsand the finite barrier model with the band bending effect[15,16].

The paper is organized as follows: In Section II, webriefly present the basic equations for calculating thelinewidth of the intersubband optical absorption withina microscopic theory and formulate the IFR scatteringform factor in detail. In Section III, we compare the ab-sorption linewidths obtained by applying of the old andthe new form factors by numerically. Lastly, a summaryis given in Section IV.

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II. THEORETICAL CALCULATION

1. Basic Consideration for Calculating theLinewidth of Intersubband Optical Absorption inQWs

We consider the case when only the ground subbandis occupied by electrons and the light’s energy is close tothe energy separation between the two lowest subbands�ω ∼ E10 = E1−E0 in the QWs. The absorption of lightpolarized in the growth direction (z) is proportional tothe real part of the dynamical 2D conductivity [17]. Thiswas derived by using the microscopic theory of Ando [18,19]. For single- particle excitation, it reads as follows:

Re σzz(ω) =e2f10

2mz

∫dE

m∗

π�

× f(E)Γ(E)

(�ω − E10)2 + Γ2(E). (1)

Here, e is the elementary charge, � is the reduced Planckconstant, mz and m∗ are the out-of-plane and the in-plane effective masses of the electron, f10 is the oscillatorstrength for the transition E0 → E1, E10 = E1 − E0

is the intersubband energy separation (the ground stateand the 1st excited state), f(E) is the Fermi distributionfunction, 2Γ(E) means the full width at half maximum(FWHM) of the Lorentzian lineshape with energy E, andΓ(E) is the relaxation rate due to scattering processesgiven by [9,12,20]

Γ(E) = 12 [Γintra(E) + Γinter(E)]. (2)

Here, we only consider the IFR scattering in square QWs,so the first term Γintra(E) = Γ−

intra(E) + Γ+intra(E) arises

from intrasubband transition processes with

Γ∓intra(E) =

m∗(ΔΛ)2

�2(F∓

00−F∓11)

2

∫ π

0

dθ e−q2Λ2/4, (3)

and the second term Γinter(E) = Γ−inter(E) + Γ+

inter(E)arises from intersubband transition process with

Γ∓inter(E) =

m∗(ΔΛ)2

�2(F∓

01)2

∫ π

0

dθ e−q2Λ2/4. (4)

Δ and Λ are two roughness parameters for the well-barrier interfaces, θ is the in-plane scattering angle, q

and q are the absolute values of the in-plane scatter-ing vectors in the intrasubband and the intersubbandscattering processes, respectively, F∓

mn are the scatteringform factors, and ∓ stands for the interfaces z = −L/2and z = +L/2.

In many previous studies [7,9,12], those form factorswere usually calculated through the local value of thewave function at the barrier. For a symmetric squareQW (centered at z = 0) of well width L and potentialbarrier height V0, the following holds:

F∓mn = V0ξm(∓L

2)ξn(∓L

2), (m, n = 0, 1). (5)

The values of F∓mn depend on the values of the wave

functions at −L/2 and L/2. Those values have a localnature. Thus, the value of the form factor is incorrectif the determination of the value of the wave function atthe barriers inaccurate. To overcome this problem, in thenext section, we propose a novel approach for calculatingof the form factors more accurately.

2. The Interface Roughness Scattering FormFactor

We propose a new form factor formula that is moreglobal and more reasonable to apply to many differenttypes of potential wells, particularly the types of wellsfor which the band bending effects have to be considered[21]. We begin with the Schrodinger equation for the mstate,

�2

2mz

d2ξm(z)dz2

= −Emξm(z) + V (z)ξm(z), (6)

where V (z) is the effective confining potential along thez direction. The most basic form of V (z) given by

V (z) = Vb(z) + VH(z). (7)

The first term in this equation is the potential barrier.The second term is the Hartree potential, which is ob-tained from the Poisson equation [22]. Multiplying two

sides of the Eq. (6) bydξn(z)

dz, we get

�2

2mz

d2ξm(z)dz2

dξn(z)dz

= −Emξm(z)dξn(z)

dz+ V (z)ξm(z)

dξn(z)dz

. (8)

In a similar way, the Schrodinger equation for the n state is as follows:

�2

2mz

d2ξn(z)dz2

dξm(z)dz

= −Enξn(z)dξm(z)

dz+ V (z)ξn(z)

dξm(z)dz

. (9)

A Novel Approach to the Evaluation of the Interface Roughness· · · – Nguyen Thanh Tien et al. -1715-

Adding Eqs. (8) and (9) side by side, we have

�2

2mz

(d2ξm(z)dz2

dξn(z)dz

+d2ξn(z)

dz2

dξm(z)dz

)= −

(Emξm(z)

dξn(z)dz

+ Enξn(z)dξm(z)

dz

)

+ V (z)(ξm(z)

dξn(z)dz

+ ξn(z)dξm(z)

dz

). (10)

On the other hand, we know that

d2ξm(z)dz2

dξn(z)dz

+d2ξn(z)

dz2

dξm(z)dz

=d

dz

(dξm(z)dz

dξn(z)dz

), (11)

En = Em + Enm, (12)

and

ξm(z)dξn(z)

dz+ ξn(z)

dξm(z)dz

=d

dz

(ξm(z)ξn(z)

). (13)

Thus, Eq. (10) can be rewritten as

�2

2mz

d

dz

(dξm(z)dz

dξn(z)dz

)= − Em

d

dz

(ξm(z)ξn(z)

)− Enmξn(z)

dξm(z)dz

+d

dz

(V (z)ξm(z)ξn(z)

)− ξm(z)ξn(z)

dV

dz. (14)

Integrating the two sides of Eq. (14) from ∓∞ to zi with −L

2< zi <

L

2and i = 0, 1, we get

�2

2mz

dξm(z)dz

dξn(z)dz

∣∣∣zi

∓∞= − Emξm(z)ξn(z)

∣∣∣zi

∓∞− Enm

∫ zi

∓∞dzξn(z)

dξm(z)dz

+ V (z)ξm(z)ξn(z)∣∣∣zi

∓∞−

∫ zi

∓∞dzξm(z)ξn(z)

dV

dz. (15)

On the other hand,

dV

dz=

dVH

dz+

dVb

dz=

dVH

dz+

d

dz

[V0δ

(z − L

2

)+ V0δ

(− z − L

2

)], (16)

and we know thatdξm(z)

dz

∣∣∣∓∞

= 0, ξm(z)∣∣∣∓∞

= 0,dξn(z)

dz

∣∣∣∓∞

= 0, and ξn(z)∣∣∣∓∞

= 0.

We now consider a square potential well, finite height, for which potential of the barrier can be expressed in theform

Vb(z) =

⎧⎪⎨⎪⎩

0 if |z| ≤ L

2V0 if |z| ≥ L

2.

(17)

Substituting Eqs. (16) and (17) in Eq. (15), we obtain

�2

2mz

dξm

dz(zi)

dξn

dz(zi) = Emξm(zi)ξn(zi) − Enm

∫ zi

∓∞dzξn(z)

dξm(z)dz

+ [VH(z)

+ V0(z)]ξm(zi)ξn(zi) +∫ zi

∓∞dzξm(z)ξn(z)

dVH

dz. (18)

We can write the IFR scattering transition form factor from the state m to the state n as

F∓mn = V0ξm(∓L

2)ξn(∓L

2) =

[Em − VH(zi)

]ξm(zi)ξn(zi) +

�2

2mz

dξm

dz(zi)

dξn

dz(zi)

+ Enm

∫ zi

∓∞dzξn(z)

dξm(z)dz

+∫ zi

∓∞dzξm(z)ξn(z)

dVH

dz, (19)

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with

Enm = En − Em. (20)

From Eqs. (15) and (16), we find that the second term on the right-hand side of Eq. (19) is equal to 0 when at leastone of the two indices m or n is 0. Therefore, Eq. (19) becomes

F∓mn = V0ξm(∓L

2)ξn(∓L

2) =

[Em − VH(zi)

]ξm(zi)ξn(zi) + Enm

∫ zi

∓∞dzξn(z)

dξm

dz+

∫ zi

∓∞dzξm(z)ξn(z)

dVH

dz. (21)

We also note that the choice of zi can be optional, but it

must satisfy the conditions −L

2≤ zi ≤ L

2(Vb(zi) = 0).

An optimal choice is zi = z0, z1. Here, z0 is the peakposition of the basic wave function, and z1 is the valueof the first excited wave function in the well (Vb(z) = 0).

The advantages of using this new transition form fac-tor are several. First, it is exact and is applicable tobound states in a QW of any potential barrier height V0.This is also convenient for asymmetric QWs. Second, itis convenient for use in the case of infinite-barrier QWsbecause its right-hand side remains definite whereas theone in the local representation, Eq. (5), becomes in-definite at the ideal limit: V0 → ∞ and ζm(∓L/2) → 0.Third, it enables a reduction in the errors associated withthe use of approximate wave functions in a realistic QWmodel, because it involves integral quantities. Fourth, itis more convenient for calculating of sophisticated phe-nomena, such as dynamic screening by mobile particles,many-body effects and band bending effects.

3. Calculations for a Symmetric Quantum Well

To illustrate and compare the two methods, we per-form calculations for a symmetric quantum well. Weconsider the case when only the ground subband in QWsis occupied by electrons and the light’s energy is close tothe energy separation between the two lowest subbands�ω ∼ E10 = E1 −E0 (� is the reduced Planck constant).For a symmetric square QW (centered at z = 0) of wellwidth L and potential barrier height V0, the wave func-tions are given as follows [6]:

For the ground state,

ζ0(z)= C0

⎧⎨⎩

eκ0(z+L/2) cos(k0L/2), if z < −L/2cos(k0z), if |z| ≤ L/2e−κ0(z−L/2) cos(k0L/2), if z > L/2,

(22)

C0 =1√

L/2 + (Vb/κ0E0) cos2(k0L/2), (23)

with

cos(k0L/2) − mbzk0

mczκ0

sin(k0L/2) = 0. (24)

Fig. 1. (Color online) Normalized ground state wave func-tions ζ0(z) for various well widths L = 70, 90, and 110 A(the labels are showed in the figure). The arrows indicate thevalues of the wave functions at z = −L/2, ζ0(−L/2). Theother parameters are presented in Section III.

For the first excited state,

ζ1(z)= C1

⎧⎨⎩

−eκ1(z+L/2) sin(k1L/2), if z < −L/2sin(k1z), if |z| ≤ L/2e−κ1(z−L/2) sin(k1L/2) if z > L/2,

(25)

C1 =1√

L/2 + (Vb/κ1E1) sin2(k1L/2)(26)

with

cos(k1L/2) +mc

zκ1

mbzk1

sin(k1L/2) = 0. (27)

Here mc/bz is the out of-plane effective masses of the

electron in the channel and the barrier, respectively.The wave number in the channel is k0,1 =

√2mc

zE0,1/�,and in the barrier, it is κ0,1 =

√2mb

z(Vb − E0,1)/�.

Figure 1 shows the ground state wave functions fol-lowing Eqs. (22), (23) and (24) for various well widthsL = 70, 90, and 110 A. Figure 1 reveals that the electron

A Novel Approach to the Evaluation of the Interface Roughness· · · – Nguyen Thanh Tien et al. -1717-

Fig. 2. (Color online) Normalized first excited state wavefunctions ζ1(z) for various well widths L = 70, 90, and 110 A(the labels are showed in the figure). The arrows indicate thevalues of the wave functions at z = −L/2, ζ1(−L/2). Theother parameters are presented in Section III.

gas spreads out in the well in all three cases. The largerthe width of the quantum well is, the more the electrongas spreads out. The values ζ0(−L/2) monotonically de-crease when L is increased because we do not take intoaccount the role of the doping in band bending.

Figure 2 shows the first excited state wave functionsfollowing Eqs. (25)−(27) for various well widths L =70, 90, and 110 A. Figure 2 reveals that the value of thewave function ζ1(z) at −L/2, ζ1(−L/2) increases whenL increases, but its amplitude monotonically decreaseswhen L increases because we only consider symmetricquantum wells.

In this study, we do not take into account the role ofthe doping in band bending [12], so the Hartree potentialis done only to the electrons in the potential well. Wesolve the Poisson equation with the boundary conditionsfor the Hartree potential such that [15] ∂VH(z=−∞)

∂z =∂VH(z=∞)

∂z = 0. We obtained the following results for theHartree potential VH(z):

VH(z)= −4πe2

εnsC0

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

e2κ0(z+L/2)

4κ20

cos2(k0L/2), if z < −L/2

z2

4 − cos(2k0z)8k2

0+ 1

8κ20− L2

16 , if |z| ≤ L/2

e−2κ0(z−L/2)

4κ20

cos2(k0L/2), if z > L/2.

(28)

Here ns is a sheet density of electrons, ε is the dielectric constant of the material in well. Thus,

dVH(z)dz

= −4πe2

εnsC0

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

e2κ0(z+L/2)

2κ0cos2(k0L/2), if z < −L/2

z2 + sin(2k0z)

4k0, if |z| ≤ L/2

−e−2κ0(z−L/2)

2κ0cos2(k0L/2), if z > L/2.

(29)

III. NUMERICAL RESULTS ANDDISCUSSION

To illustrate the method, we present the numerical re-sults of the theoretical calculations of the energy broad-ening and the absorption linewidth. It is worth men-tioning that the lineshape described by Eq. (1) may beinterpreted [19,23] as a superposition of Lorentzian line-shapes with different energies distributed following theFermi function. Therefore, the absorption linewidth γmay be defined in a good approximation by the averageof the FWHMs with a weight f(E), so γ ≈ 2Γ, where[19]

Γ =∫

dEf(E)Γ(E)(∫

dEf(E))−1

. (30)

The sample used in this calculation was an undopedAl0.3Ga0.7As/GaAs square single quantum well. Fornumerical calculation, we need to specify the materialparameters as inputs. The material parameters listed inRefs. 8 and 12 as follows:i) the barrier height: V0 = 210 meV;ii) the effective mass: m∗

c/m0 = 0.0665, m∗b/m0 =

0.09155;iii) the static dielectric: ε = 12.91;iv) the sheet electron concentration: ns ∼ 6 × 1011

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Fig. 3. (Color online) Energy broadening due to interfaceroughness scattering plotted as a function of the in-plane ki-netic energy E (Ando’s approach and our approach). Γintra isthe contribution from the difference in the intrasubband scat-terings between the ground and the first excited states. Γinter

is the contribution from the difference between intersubbandscatterings. The inset figure only shows the effective energybroadening, 2Γop, versus E for the two approaches.

Fig. 4. (Color online) Well-width dependence of the calcu-lated absorption linewidth at 4 K (only considering interfaceroughness scattering). The dashed curves show the absorp-tion linewidths according to the approach of Ando. The solidcurves show the one according to our approach. The intersub-band absorption linewidth (Inter), the intrasubband absorp-tion linewidth (Intra) and the effective absorption linewidthdue to interface roughness (IFR) are noted in the figure. Theexperimental absorption linewidths at low temperature areshown as solid circles [8]. The inset figure only shows the cal-culated effective absorption linewidth for the two approaches.

Fig. 5. (Color online) Calculated transition form factorsas a function of well width at 4 K. Fmn are transition formfactors with the corresponding scattering processes (0 −→ 0,0 −→ 1 and 1 −→ 1).

cm−2 for various well widths in the range L = 75−110 A.

At first, we study the energy broadening due toIFR scattering. Figure 3 shows the calculated energybroadening due to IFR scattering as a function of thein-plane kinetic energy E according to Eqs. (2)−(5) and(21). The roughness parameters were chosen as Δ = 3A and Λ = 85 A [9,24], and the quantum well widthas L = 80 A. From Fig. 3, we may draw the followingconclusions:i) The difference between the values of Γinter(E) for thetwo different approaches is small.ii) The values of Γintra(E) are significantly differentfor the two different approaches. Γintra(E), accordingto the new way, is found to be much larger than itscounterpart. However, both Γintra(E) (according to thetwo different approaches) monotonically decrease whenthe in-plane kinetic energy (E) increase.

Next, we investigate the well-width dependence of in-tersubband absorption linewidth by varying the inter-face roughness. Figure 4 shows the calculated linewidthversus well width at low temperature (T =4 K) in therange L = 70 − 120 A. We set the roughness parame-ters as Δ = 3 A and Λ = 85 A which is the Gaussianroughness profile [15]. Figure 4 shows clearly that theintersubband absorption linewidth is small. The intra-subband absorption linewidth causes a significant contri-bution to the effective absorption linewidth due to theinterface roughness. This indicates that an accurate cal-culation of the contributions of the wave functions is im-portant. The calculated results in our approach fit theexperimental results of Campman et al. well [8]. Un-like the opinion of Takeya Unuma et al. [12], we believethat the role of phonon scattering is not important in

A Novel Approach to the Evaluation of the Interface Roughness· · · – Nguyen Thanh Tien et al. -1719-

Fig. 6. (Color online) Absorption linewidths due to the in-terface roughness scattering plotted as functions of the rough-ness amplitude Δ. The linewidths due to intersubband andintrasubband transitions are also shown in the figure.

very low-temperature systems. Because, Fig. 4 in Ref12 shows that the value of 2Γop(E) (energy broadeningdue to LO phonon scattering) is very small although thetemperature of this sample is 300 K.

To clarify the role of the two transitions (intrasubbandand intersubband) in the two different approach, we plotthe form factors versus the well width following Eqs. (5)and (21) in Fig. 5. Figure 5 shows the value of the tran-sition form factors (F00, F11 and F01). The values ofthose form factors are significantly different for the twodifferent calculations. Those values also follow differenttrends, even trends in the opposite directions.

Finally, in Fig. 6, we display the calculated absorptionlinewidths as functions of the roughness amplitude Δ.From Fig. 6, we may conclude the following:i) The absorption linewidth increases with increasingroughness amplitude Δ. Thus when roughness amplitudeis increasing, the intrasubband transitions scattering in-tensity due to IFR also increases. In low temperaturesamples, the absorption linewidth is strongly dominatedby intrasubband transitions.ii) The absorption linewidths calculated by using our for-mula are larger than the linewidths calculated by usingthe old method.

IV. CONCLUSIONS

In summary, we have proposed a novel formula forcalculating of the interface roughness scattering formfactor in intersubband transitions. We have numericallycalculated the absorption linewidth for the undopedAl0.3Ga0.7As/GaAs square quantum well.

Our numerical results show that the interface rough-ness scattering absorption linewidth obtained by usingthe new form factor is twice that obtained by usingAndo’s approach. The new form factor may also bebetter for explaining the experimental well-width depen-dence of the intersubband optical absorption linewidth.We also recognize that intrasubband scattering makesan important contribution to the linewidth due to theinterface roughness. Our form factor formula may besuccessfully used to calculated the asymmetric potentialform and the band bending effect.

ACKNOWLEDGMENTS

The authors would like to thank Prof. Doan NhatQuang (Institute of Physics, Vietnam Academy of Sci-ence and Technology) for his inspiring discussions at ear-lier stages of the present study. This research is fundedby Vietnam’s Ministry of Education and Training undergrant number B2012-16-12.

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