interbandtransitionsin bulkse/c momentum matrix element ...bulutay/573/notes/ders_10.pdf · c....

C. Bulutay Lecture 10Topics on Semiconductor Physics

� Interband Transitions in Bulk Se/c

� Momentum Matrix Element

In This Lecture:

� Momentum Matrix Element

� Polarization dependence

� Interband Transitions in Quantum Wells

� Intraband Transitions in Bulk & QWs


Interband Transitions in Bulk Se/c

As the photon momentum is

negligible compared to

electronic crystal momenta,

the transitions are (almost)

For parabolic bands:

the transitions are (almost)

vertical (kop=0)

[See, next page]

Ref: Singh


Momentum

Matrix Elements

vanishes when kop=0 due to

Bloch fn orthogonality

,

* 3

, ,

, ,

Predominantly we are interested in transitions between CB and VB

so that ,

ˆ ˆ

where

1 ( ) ( ); similarly for

op

c

c

c kc c

ik r

c k k

ik r

c k k

i f c

e p e e p d r

r e u rV

υυ

υ

υ

ψ ψ

ψ ψ

⋅

⋅

→ →

⋅ = ⋅

=

∫

�

� �

� �

� �

� �

� �

� �

3*

, ,

( )

( ) ( ) opcc

ik rik r ik r

c c k k

r

d rp u r e e k e u r

V

υ

υ

υυ υ υ

⋅− ⋅ ⋅= ∫��

� �

�

� � �ℏ

Treat slowly-varying

(envelopes) and

the cell-periodic

parts separatelyunit cell volume xtal volume

3*

, ,

3( )*

, ,

( ) ( )

( ) ( )

opc

c

c op

c

ik rik r ik r

c k k

i k k k

c c k k

V

d ru r e e u r e

i V

d rp u r u r e

i

υ

υ

υ

υ

υ

υ υ

⋅− ⋅ ⋅

− + + ⋅

Ω

+ ∇

= ∇ Ω

∫

∫

∫

��

� �

� � �

� �

�ℏ� �

�ℏ� � �

,

3

3*

, , ,( ) ( )

k k kc op

c op c

r

V

c k k k c k k

d r

V

d rp u r u r

i

υ

υ υ

δ

υ υδ

+

+Ω

= ∇ Ω

∫

∫

� � �

�

� � � � �

��

�ℏ� � �


Electric dipole forbidden transitions

When certain pcv transition matrix element vanishes (due to some symmetry

reason etc.) this is termed as a electric dipole-forbidden-transition.

In this case higher-order contributions such as electric quadrupole and magnetic

dipole transitions become important. Compared to the electric dipole transitions

they are reduced in strength by a factor of (lattice constant/wavelength of light)2,

that requires very high frequencies (UV to X-rays)...that requires very high frequencies (UV to X-rays)...


Conduction Band:

,

Valence Bands (only HH, LH):

3 3 1 , ( ) ,

2 2 2

3 3 1

iS iS

X iY

↑ ↓

−= + ↑

( )

( )

0

Momentum-matrix parameter:

CB to HH Transitions:

3 3ˆ ˆ , ,

2 2 2

3 3 , 0,

2 2

3 3ˆ ˆ , ,

x x y z

x

x

mP iS p X iS p Y iS p Z P

PiS p x iy

iS p

PiS p x iy

= = = =

↑ = − +

↓ =

↓ − = −

ℏ

�

�

�

WATCH OUT:

No coupling of

the z-polarized

light between

Polarization Dependence

Recall se/c band edge states:

3 3 1 , ( ) ,

2 2 2

3 1 1 2 , ( ) ,

2 2 36

3 1 1 2 , ( )

2 2 36

X iY

X iY Z

X iY Z

− = − ↓

−= + ↓ + ↑

− = − ↑ + ↓

( )3 3 ˆ ˆ , ,2 2 2

3 3 , 0,

2 2

CB to LH Transition

xPiS p x iy

iS p

↓ − = −

↑ − =

�

�

( )

( )

s:

3 1 2ˆ , ,

2 2 3

3 1ˆ ˆ , ,

2 2 6

3 1 2ˆ , ,

2 2 3

3 1ˆ ˆ , ,

2 2 6

x

x

x

x

iS p P z

PiS p x iy

iS p P z

PiS p x iy

↑ =

↓ = − +

↓ − =

↑ − = −

�

�

�

�

light between

CB & HH


�For a cubic xtal what differentiates z from x or y?

�Recall that in defining the expansion basis vectors we assumed

electron wavevector to be along z direction

�For that reason we are also using the z-projection of the spin

Reflections on the polarization dependence

�Does that give enough support for singling out z from x or y direction?

�After all that’s just for the sake of formulation, say a convention

�Away from k=0 HH & LH become mixed

�So only at k=0 we could talk about such a selectivity

�But at k=0 we lose any sense of direction of the k-vector!


To Remind you the LK Hamiltonian

Ref: Chuang


Averaging over the polarization for bulk

�These considerations suggest us to consider unpolarized light

�Equivalently we shall consider electron wavevector to pointalong a general direction and average the matrix elementover the solid angle

Let the electron wavevector to be along a direction (θ,Φ):

For illustration consider CB-HH transition:

averaging over the solid angle


CB:

HH:

Ref: Chuang

Note that for ease of calculation we keep the spin parts in the new (rotated) coordinate system…


Consider, for instance optical transition from the CB of one spin, say

to either of the HH bands ; one of them is already zero

Ref: Chuang


Bulk Momentum Matrix Element for Unpolarized Light

Kane’s parameter,(not a surprise)

where

(not a surprise)

Alternatively, an energy parameter Ep can be defined as:

20 0

2

2, so that

6p b p

m mE P M E= =

ℏ

where

Ref: Chuang


The other polarizations, spin, and LH band

2

'

Same result is obtained for

ˆ ˆ ˆ ˆ For or (cubic symmetry)

For the other spin component of the CB,

bM

e y e z

iS

= =

↓

*

*

2'

' '

For the transition between the LH band (per spin),

3 1 3 ,

2 2iS ex iS ex↓ + ↓

*

2'

1,

2 2−


Joint Density of States (also called reduced DOS)

This is an important piece that appears inside total transition rate expressions

Single Parabolic Band DOS:

Ec

( )st1 BZ

2 2

*

* 3/ 2

( ) ( )

For a parabolic band: 2

m m

k

c

dos

N E E E k

kE E

m

σ

δ∈

= −

− =

−

∑ ∑�

�

ℏ

( )

*

st

* 3/ 2

2 3

2 2

* *

1

1 BZ

( )( ) 2 ,

1 1Between two parabolic CB and VB:

2

( ) ( ) ( )

r

dos c

m

g

e h

m

c c

k

m E EN E

kE

m m

N E k E k

N

υ υσ

π

ω

ω δ ω∈

−=

− = +

= − +∑ ∑�

ℏ

ℏℏ

��

� �ℏ ℏ

* 3/ 2

2 3

( )( ) 2

r g

c

m Eυ

ωω

π

−=

ℏℏ

ℏ

Joint DOS of CB-VB:

Eg ωℏ


Absorption Rate (Final Expression)

With all these ingredients the bulk absorption rate for unpolarized lightbecomes:

JDOS

( )2

2

2

0

2 ( )ph

abs b c

e nW M N

mυ

πω

ωε=

ℏℏ

ℏ

* 3/ 2

2 3

( )( ) 2

r g

c

m EN υ

ωω

π

−=

ℏℏ

ℏ

JDOS


Radiative e-h Recombination Time: Emission

ωℏ

In the case of interband recombination rate of an e with a holeat the same k state, we integrate over all possible photon states

3D total photon DOS

0

For =0,

1Associated e-h radiative recombination time is

ph em spon

spon

n W W

Wτ

→

=

ωℏ


Interband Transitions in Quantum Wells

transitions between subbands derived from different bulk bands

CB1

CB2

HH1HH2

HH3LH1

Subband Wavefunctions

HH3LH1

Normalization

area Well

width

Due to mixing

in the VB

3D to 2D: Optical transitions are affected in two ways

� Form of JDOS

� Momentum matrix element; anisotropy is now genuine

Ref: Singh


Momentum Matrix Element in QWs

In going from 3D to 2D:

env. fn. overlap in-plane overlap

Cartesian component

Ref: Singh

env. fn. overlapalong growth dir.

[ Other term, acting on ( ) leaves 0 at the same state]n vma c cp g z u u kυ =�

Unlike 3D, polarization dependence exists in 2D

NotationTE (to growth axis): Electric field in QW plane

TM (to growth axis): Electric field along growth axis

in-plane overlap


Let the QW growth axis be z axis

TE (Optical electric field in xy plane)

Optical dipole matrix element is averaged over the azimuthal angle

From both HH bands to

Same results for

the other CB spins

From both LH bands to

Same results for

the other CB spins

not considered

not considered

Ref: Chuang


TM (Optical electric field along z axis)

Sum

Rules

Ref: Chuang


Back to Absorption Rate in QWs

JDOS in 2D:

Ref: Singh

Observe that even-odd parity transitions are not allowed due tovanishing of this overlap


Indirect Interband Transitions in Bulk

Common Indirect Se/c: Si, Ge, C, AlAs, GaP, AlP, SiC, AlN (zb)

2nd Order Matrix Element:

Intermediate

virtual state

summed overNo energy

cons.

Weight of these two

pathways 21

i nE E∝

−With photon energies smaller than the direct band gap intermediate transitions can occur since energy need not be conserved

Ref: Singh


Pathways which require phonon emission/absorption

Form of the matrix elements:

direct optical transitions

e-phonon scattering matrix elements due to optical phonon intervalley scatteringwith the associated matrix element:

abs.

em.

: Deformation potential

: Mass density

: Intervalley phonon frequency

( ) : phonon occupancy (BE distr.)

ij

ij

ij

D

n

ρ

ω

ω

direct optical transitions

Ref: Singh


For parabolic bands, the absorption rate results in:# equivalent

valleys

Photon-related Photon-related

matrix element

Note the contrast in Wabs

Direct Bandgap:

Indirect Bandgap:

In amorphous se/c, k-conservation requirement is relaxed (no periodicity, xtal momentum not a good quantum label)

This results in higher absorption coefficientRef: Singh


Intraband Transitions in Bulk Se/c

�As each band at a k-state is single-valued 1st order vertical intraband transitions are not possible

�Intraband transitions must involve some secondmechanism (phonon, ionized imp, defects...) tomechanism (phonon, ionized imp, defects...) toensure momentum conservation

�Intraband transitions are also known as free carrierabsorption and are effective in the cladding layers of lasers


Drude Model (to explain free carrier absorption)

w/o scattering no net energy xfer;e’s oscillate back and forth within the band

By introducing a scattering mechanism, energy gained by the ein one cycle will be partially lost in the form of, say phonon

* * * 2

0 0cos( )m x m x m eE tγ ω ω+ + =ɺɺ ɺ

in one cycle will be partially lost in the form of, say phononemission by the electron.

If the mobility is large (weak scattering) absorption coefficientbecomes small

2

1( )

1

α ωω

µ

∝

∝

ℏ

mobility


Intraband Transitions in Quantum Wells

CB2

CB1

�Since a number of subbands may originate from the same bulk band, certain inter-subband transitions (CB1-CB2) may betermed as intraband transitions in QWs

�Such inter-subband transitions have greatimportance for far infrared detectors andimportance for far infrared detectors andforms the basis of Quantum Cascade Lasers

orthogonal Approximately same for the CB

Ref: Singh


Momentum Matrix Element:

If the polarization lies on the QW plane, then due tothe orthogonality of the remaining envelope parts(g1,g2), pif=0

�Thus for EM wave polarized in the plane of the QW, inter-subband transition rate is zero (This can be relaxed under strong mixing of thecell-periodic parts as in the VB.)

� For EM wave polarized along the QW growth axis (say z), we get

Brings g1 to the same parity with g2 Ref: Singh

interbandtransitionsin bulkse/c momentum matrix element ...bulutay/573/notes/ders_10.pdf · c....

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