k.

p Theory & Applications

 How to measure the band structure?
Angle-resolved photoemission spectroscopy (ARPES)

Ø energy conservation:

Ø momentum conservation:

V 0 can be determined by
examining only the electrons
Ø Three parmeters emitted perpendicular to the
surface

The fingerprint of the band structure: E(ki)

 The origin of bands in solids

electrons must occupy The splitting of the ü The lower band (valance band)
different energies due single energy levels is formed by the overlap
to Pauli exclusion form bands of between the occupied orbitals
principle. allowed and ü The upper conduction band:
is formed by the contribution
forbidden energies of the unoccupied orbitals
 The band structure calculations

Methods to calculate the band structure:

Ø Ab Initio Pseudopotentials
Ø Density Functional Theory (DFT)
Ø k.p theory
 Outline

Ø k.p method Kane's model

Ø Band structure of GaAs
Ø 14-band model
Ø Thirty-band model

Ø Band structure the wide band gap semiconductor

Ø Application: shift current calculation and related
problem
Ø The Coulomb matrix elements in the frame of k.p
and related challenge
 Symmetry dependence
Ø Band structure calculations dependence on the crystal symmetry

Wurtzite Crystal:
Zinc Blende Crystal (wide-bandgap semiconductors)
GaAs, InAs, GaP, AlAs GaN, ZnO, AlN, CdSi, CdS...etc

14 & 30 band-models 8 band-model

 k.p method: Schrödinger equation
Ø The state of an electron in crystal is described by the SE:

periodic potential

but

to be expanded in the term of the periodic functions

k.p method: Bloch's function expansion

periodic wavefunctions: the solutions of

the Schrödinger equation at k=0

Ø substituting the last equation into the first SE, we have:

by integrating over the unit cell

the eigenstate problem: (next slide)

 The k.p Hamlitonian matrix
The eigenstate problem:

eigenvalues eigenvectors

with and

Ø This Hamlitonian matrix can be diagonalized to obtain the set of

eigenvalues and the eigenvectors.
 Extended Kane’s model

Ø Here, the spin-orbit interaction is taken into account

the Schrödinger equation reads

due to the spin orbit-interaction

to be expanded in the terms of
atomic orbitals

atomic orbital-type
The Hamlitonian matrix of Kane’s model

The eigenstate problem:

can be obtained via the definitions of

the contributed orbitals
(basis functions)
 14-band model: Bulk-GaAs
Ø Expand the Bloch function via 14 atomic orbital-type functions:
14
uk (r ) = ∑ a nk u n (r )
n =1

atomic orbital-type
Ø Recall the angular momentum eigenfunctions (spherical harmonics)

1 1
l = 0 orbital s Y00 (θ , ϕ ) = ≡ S
2 π
1 3 1 3 z
l = 1 orbital p Y00 (θ , ϕ ) = cosθ = ≡ Z
2 2π 2 2π r
1 3 1 3 x − iy
Y0−1 (θ , ϕ ) = sin θ e −iϕ = ≡ X −i Y
2 2π 2 2π r
1 3 1 3 x + iy
Y0+1 (θ , ϕ ) = − sin θ eiϕ = − ≡ −( X + i Y )
2 2π 2 2π r
 The eigenfunctions of j and its component jz
j, j z

p*-type p-type

1 ⎛ X +i Y ⎞ 6 p*-type
1 ⎛ X' +i Y' ⎞ 9
u = 3
,+ 3 ⎜⎜ ⎟⎟
u1 = 3
,+ 32 =− ⎜⎜ ⎟⎟ 2 2 v =− states
2 c'
2⎝ 0 2⎝ 0 ⎠
⎠ E
1 ⎛ − 2 Z' ⎞ 1 ⎛ −2 Z ⎞
2
u = 3
,+ 1
=− ⎜ ⎟ u10 = 3
,+ 12 =− ⎜ ⎟
6 ⎜⎝ X + i Y ⎟
2
6 ⎜⎝ X ' + i Y ' ⎟
2 2 c' v
⎠ ⎠
ε 01−6 = Egʹ
1 ⎛ X' +i Y' ⎞ 1 ⎛ X +i Y ⎞
u3 = 3
,− 12 = ⎜ ⎟ u11 = 3
,− 12 = ⎜ ⎟ CB
6 ⎜⎝ 2 Z ' ⎟
2
6 ⎜⎝ 2 Z ⎟
c' 2 v
⎠ ⎠
1 ⎛ 0 ⎞ 1 ⎛ 0 ⎞
u4 = 3
,− 32 = ⎜ ⎟ u12 = 3
,− 32
2 c'
2 ⎜⎝ X ' − i Y ' ⎟
⎠
2 v
= ⎜
2 ⎜⎝ X − i Y
⎟
⎟ ε 06,7 = Eg
⎠
1 ⎛ Z' ⎞ 2 s*-type
u5 = 1
,+ 12 =− ⎜ ⎟ 1 ⎛ Z ⎞
13
u = 1
,+ 1
=− ⎜ ⎟ states
2 c'
3 ⎜⎝ X ' + i Y ' ⎟
⎠ 2 2 v
3 ⎜⎝ X + i Y ⎟ ε 08−14 = 0
⎠
1 ⎛ X ' −i Y' ⎞
X −i Y ⎞ VB
u6 = 1
,− 12 =− ⎜ ⎟ 14 1 ⎛
2 c'
3 ⎜⎝ − Z ' ⎟⎠ u = 1
,− 12 =− ⎜ ⎟ 6 p-type
3 ⎜⎝ − Z ⎟⎠
2 v
states k=
7 ⎛S ⎞ 8 ⎛ 0 ⎞ 0
s*-type u = 1
2 ,+ 1
2 c = ⎜⎜ ⎟⎟ u = 1
2 ,− 1
2 c = ⎜⎜ ⎟⎟
⎝ 0 ⎠ ⎝S ⎠
Ø R. Winkler, Spin-orbit coupling effects in two-dimensional electron and hole systems,
Springer Tracts in Modern physics Vol. 191, 2003.
 The Kan's parameters
⎛ n ! 2k 2 ⎞ ! !
H nn' = ⎜⎜ ε 0 + ⎟⎟ δ nn' + k ⋅ un pˆ un ' + 2 2
un σ ⋅ (∇U × pˆ ) un ' GaAs
⎝ 2m0 ⎠ m0 4m0 c
Eg (eV) 1.52
momentum matrix elements spin-orbit
! ! ! matrix E’g 4.49
S pˆ x X = S pˆ y Y = S pˆ z Z ≡ P
m0 m0 m0 elements Δ 0.34
! ! !
S pˆ x X ʹ = S pˆ y Y ʹ = S pˆ z Z ʹ ≡ iPʹ Δ’ 0.17
m0 m0 m0
! ! ! Δ- -0.05
X pˆ y Z ʹ = Y pˆ z X ʹ = Z pˆ x Y ʹ ≡ Q
m0 m0 m0 P (eV.nm) 1.05
− 3i! − 3i! − 3i!
2 2
X (∇U × p )y Z = 2 2
Y (∇U × p )z X = Z (∇U × p )x Y ≡ Δ P’ 0.5
4m0 c 4m0 c 4m02 c 2
− 3i! − 3i! − 3i! Q 0.82
2 2
X ʹ (∇U × p )y Z ʹ = 2 2
Y ʹ (∇U × p )z X ʹ = Z ʹ (∇U × p )x Y ʹ ≡ Δʹ
4m0 c 4m0 c 4m02 c 2
m*/m0 0.067
− 3i! − 3i! − 3i!
X (∇ U × p ) y Z ʹ = Y (∇ U × p )z X ʹ = Z (∇U × p )x Y ʹ ≡ iΔ γ1 6.85
4m02 c 2 4m02 c 2 4m02 c 2
P, P’, Q, Δ, Δ’, Δ-, Eg, and E’g are elementary γ2 2.1
parameters of the “bare” extended Kane model γ3 2.9
Band structure of GaAs (14-band model )
ü Anisotropic dispersion
ü Spin splitting
6
6 6

5
5

4
4 4
Energy(eV)

Energy(eV)
3

Energy(eV)
2 2
2

1
1

0 0
0
-1
-1
-2 -2
-3 -2 -1 0 1 2 3 -2
-3 -2 -1 0 1 2 3

k[110](nm-1)
-2 0 2
k[001](nm-1)
k[111](nm-1)

The accuracy of the 14-band model is limited to a range of roughly 0.5 eV

above and below the semiconducting band gap, thus limiting their application!
Ø Phys. Rev. B 76, 205113 (2007)
 Thirty-band model
Ø New basis function were included
( )
Sv : sp3 s*

Dz = 3 z 2 − r 2
d-like orbitals were added first time by Cardona
(
Dx = 3 x 2 − y 2 )
Without spin: 15 basis functions
Cardona M and Pollak F 1966 Phys. Rev. 142 530

With spin: B30= 14+8X2=30 basis functions

Richard S, Aniel F and Fishman G 2004 Phys. Rev. B 70 235204

B30 = B14 ! ⎧⎨ Sv , Su , Dz , Dx , X d , yd , Z z , Sq ⎫⎬ ⊗
⎩ ⎭
{↑, ↓}
 The parameters of the 30-band model
⎛ n ! 2k 2 ⎞ ! !
H nn' = ⎜⎜ ε 0 + ⎟⎟ δ nn' + k ⋅ un pˆ un ' + 2 2
un σ ⋅ (∇U × pˆ ) un '
⎝ 2 m0 ⎠ m0 4 m0 c

Richard S, Aniel F and Fishman G 2004 Phys. Rev. B 70 235204

 The parameters of the 30-band model

Richard S, Aniel F and Fishman G 2004 Phys. Rev. B 70 235204

 Additional parameters
Ø Additional matrix elements and
spin-orbit coupling

Ø due to simulation: all these elements are

taken null except P and Δ’
Richard S, Aniel F and Fishman G 2004 Phys. Rev. B 70 235204
Band structure of GaAs (30-band model )
ü Anisotropic dispersion
ü Spin splitting
6

2
Energy(eV)

-2

-4

-6
L Γ X U, K Γ
Ø Perfect agreement with the real structure
 Wurtzite semiconductor
For a wide bandgap such GaN and ZnO, one can ignore the interaction between the VB
and CB and rewrite the 8 X8 matrix as:
Ø 2X2 matrix for conduction band (parabolic dispersion)
Ø 6X6 matrix for the VB (Chang and Chuang Hamlitonian)

0,0
0,04 Γ−Μ Dirction Γ−Α Dirction
Γ−Α Dirction Γ−Μ Dirction
0,02

0,00

Energy (eV)
Energy (eV)

-0,02

-0,04 GaN ZnO

-0,1
-0,06

-0,08

-0,10
-0,10 -0,05 0,00 0,05 0,10 -0,15 -0,10 -0,05 0,00 0,05 0,10 0,15
-1 -1
k(A ) k(A )

üstrong anisotropic dispersion

ü Spin splitting
üCrystal field splitting
S. L. Chuang and C. S. Chang, Phys. Rev B 54, 2491, (1996).
 Application: shift-current calculations
Multi level SBE:

30X30 shift current calculation for bulk-GaAs Density matrix:

2e-07

⎛ x11 . . . xknm ⎞⎟
⎜ k
1.5e-07 Valance bands contribution ⎜ . xk22 . . . ⎟
⎜ ⎟
⎜ . . . . . ⎟
Shift _z[arb. units]

1e-07 ⎜ . . . . . ⎟
interband contribution ⎜⎜ mn nn ⎟⎟
Conduction bands contribution ⎝ xk . . . xk ⎠
5e-08
Ø off diagonal matrix: shift current
Ø diagonal matrix: injection current
0
-200 -100 0 100 200 300 400
Time[fs]

Ø The total shift current obtained by 30-band model is less than

that obtained for the 14-band model
 Optical matrix elements

Interband:
14: From L 1=3 to L 2 =10
Linear apsorption

30: From L 1=1 to L 2 =30

Detwining (meV)

Ø Optical matrix elements are required to be

investigated in the 30-band-model
 k.p and the Coulomb matrix elements
Ø Starting from the second quantization :

with and

atomic orbital-type
Ø Transformation to be over the relative and over the center of mass coordinates
Ø Based on the atomic localization
Ø Momentum conservation and Fourier transform

k.p wavevectors
 Computational Challenge
Calculation of exciton absorption for an exciton absorption with
with spherical symmetry k.p frame

Challenge: in the k.p frame,

Linear apsorption

the Coulomb matrix

element for the bulk
semiconductor to be
calculated with an
anisotropic bands
(e.g. 3D cartesian grids)

Detwining (meV)

For 100 k point in each direction, we need to solve

14X14X(100)3 differential equations
196 millions differential equations
 summary
Ø k.p theory is used to calculate the band structure and the optical
properties of solids

Ø k.p model with spin-orbit interaction is described via the Kane’s model

Ø The accuracy of the 14-band model of zinc Crystal is limited to a range of

roughly 0.5 eV above and below the semiconducting band gap

Ø The band structure calculation of the 30-band model exhibit a perfect

agreement with real structure

Ø The interaction between the VB and CB in the wide-band semiconductors can

be ignored. Addition to the SO coupling, CF occur in this materials

Ø Together with SBE, the k.p can be used to calculate the shift current in the
bulk GaAs

Ø The behavior of the optical matrix element in the 30-band model is still not
understood

ØT he numerical evaluation of the Coulomb matrix elements is still challenging

 Matlab codes for the band structure, and the optical matrix elements
of 3D-GaAs semiconductor Based 30 k.p model
(Full code are available: salmanm2@cardiff.ac.uk )

Format
clc
ee=1.60217657e-19;
z=sqrt(-1);
hb=.658229;
xm0=(9.11*10)/(1*1.783*9.);
beta=.5*(hb.^2)/xm0;
alpha=sqrt(beta);

x=0;
tem=0.0;
e0=1.519;
e1=4.488;
d0=0.341;
d1=0.171;
dm=-0.05;
p0=1.0493;
p1=0.478;
q=0.8165;
gLc=1/(1.0);
gL1=6.85;
gL2=2.1;
gL3=2.9;
ck=-.00034;
gc=gLc-(2*xm0)/ (3*h b*h b)* (p 0*p0* (2/ e0+1/ (e0+ d0))-p1* p1*(1/(e 1-e 0)+2 /(e1 +d1 -
e0))+(4./3)*p0*p1*d m*(1/(e0*(e1+d1 -e0 ))-1/ ((e 0+d0 )* (e1-e0))));
g1=gL1-(2*xm0)/(3* hb* hb )*(p0*p 0/e0+ q*q /(e1 +d1 )+q* q/e1 +(2. /3)* p0* p1*d m/ (e0* (e1+ d1)));
g2=gL2-(2*xm0)/(3* hb* hb )*(.5 *(p0*p0 /e0 -q* q/e1 )+(1./3 )*p0* p1* dm/(e0 *(e1+ d1 )));
g3=gL3-(2*xm0)/(3* hb* hb )*(.5 *(p0*p0 /e0+ q*q/ e1)+ (1./ 3)* p0*p 1*d m/ (e0* (e1+d 1)));