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The Application and Calculation of Bond Orbital Model on Quantum Semiconductor 鍵鍵鍵鍵鍵鍵鍵鍵鍵鍵鍵 鍵鍵鍵鍵鍵鍵

The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

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The Application and Calculation of Bond Orbital Model on Quantum Semiconductor. 鍵結軌道理論在量子半導體之應用與計算. Introduction. Why is the choosing the BOM?. a hybrid or link between the k.p and the tight-binding methods combining the virtues of the two above approaches - PowerPoint PPT Presentation

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Page 1: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

鍵結軌道理論在量子半導體之應用與計算

Page 2: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Introduction

Page 3: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Why is the choosing the BOM? a hybrid or link between the k.p and the tight-binding methods combining the virtues of the two above approaches

--the computational effort is comparable to the k.p method

--avoiding the tedious fitting procedure like the tight-binding method

--it is adequate for ultra-thin superlattice

--the boundary condition between materials is treated in the straight-forward manner

--its flexibility to accommodate otherwise awkward geometries

Page 4: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

The improvement of the bond orbital model (BOM):

the (hkl)-oriented BOM Hamiltonian the BOM Hamiltonian with the second-neighbor interactio

n the BOM in the antibonding orbital framework the BOM with microscopic interface perturbation (MBO

M) the k.p formalism from the BOM

Page 5: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Bond Orbital Model

Page 6: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

What is the bond orbital model?

a tight-binding-like framework with the s- and p-like basis orbital

the interaction parameters directly related to the Luttinger parameters

Page 7: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Zinc-blende Lattice Structure:

2/a)1,1,0(),1,0,1(),0,1,1(

Page 8: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

The BOM matrix elements:

,BOM ')(H k )(e ,' ji

j

j RRk

where : The interaction parameters ,'

Es and Ep: on-site parametersEss, Esx, Exx, Exy, and Ezz: the nearest-neighbor interaction parameters

Page 9: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

The BOM matrix:

where)k(IEEH sssss

),2/acos2/acos2/acos2/acos2/acos2/a(cos4)( zyzxyx kkkkkkI k),2/acos2/acos2/acos2/a(cos2/asin4 kkkkkiEH zyxsxs

),2/acos2/acos2/acos2/a(cos2/acos)(4)k( kkkkkEEIEEH zyxzzxxzzp

)(2/asin2/asin4 kkEH xy

with

ssH sxH syH szH*sxH xxH

xyH

xzH*syH

xyH

yyH yzH*szH xzH yzH zzH

H(k)=

Page 10: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Taking Taylor-expansion on the BOM matrix:

(up to the second order)

where,12EEE sssc

,4E8EEE zzxxpV ,2/a)EE(λ 2

xxzz1 ,2/a)EE(λ 2

xxzz2

.aEλ 2xy3

and

H(k)=

kaEE 22ssc zsx akE4ixsx akE4i

22

21 kλkλE xv yxkkλ 3

zxkkλ 3

ysx akE4i

yxkkλ 3 22

21 kλkλE yv zykkλ 3

zxkkλ 3

zykkλ 3 22

21 kλkλE zv

xsx akE4i-ysx akE4i-zsx akE4i-

Page 11: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Relations between BOM parameters and Luttinger parameters

VBM CVBM

32/)8/X12(EE 022sx hlg R

32/)8/X12(EE 022sx hlg R

g2xy03xy E/)E166E R

24/X)/(3EE16E g2sx01xx hlR

8/XEE xxzz hl

2/XE12EE xxp hlv

ggc EER

mm 1264EE 2

sx00

ss /3

03xy 6E R021xx )4(E R 021zz )8(E R

01p 12EE Rv

Page 12: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Bulk Bandstructure:(001)-orientation

Page 13: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Superlattice Bandstructure:(001)-orientation

Page 14: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

The orthogonal transformation matrix:

cossin

sin

coscos

T

sinsin

cos

sincos

cos

0

sin

where the angles and are the polar and azimuthal angles of the new growth axis relative to the primary crystallographic axes.

)/(tan 221 lkh )/(tan 1 hk

Page 15: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Bulk InAs Bandstructure:(111), (110), (112), (113), and (115)-orientation

Page 16: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

InAs/GaSb Superlattice Bandstructure:(111), (110), (112), (113), and (115)-orientation

Page 17: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

The second-neighbor bond orbital (SBO) model:

Where

and T)EE(TEE xx

)1(zzszz

),,( zyx C)VV(CVE xx

)2(zzszz

)],2/akcos()2/ak)[cos(2/aksin(4T zyxx i

)],2/akcos()2/ak)[cos(2/aksin(4T zxyy i

)],2/akcos()2/ak)[cos(2/aksin(4T zz yxi )],2/akcos()2/ak)[cos(2/akcos(4T zyxxx

)],2/akcos()2/ak)[cos(2/akcos(4T zxyyy

)],2/akcos()2/ak)[cos(2/akcos(4Tzz yxz

),2/aksin()2/aksin(4T yxxy

),2/aksin()2/aksin(4T zxxz ),2/aksin()2/aksin(4T zyyz

,2/)TTT(T zzyyxxs ),aksin(2S xx i),aksin(2S yy i),aksin(2S zz i

),akcos(2C xx

),akcos(2C yy ),akcos(2C zz

.CCCC zyxs

sssssss CVTEE xsxxsx SVTE ysxysx SVTE zsxzsx SVTE *xsx

*xsx SVTE (2)

x(1)xp EEE xyxyTE

xzxyTE

*ysx

*ysx SVTE xyxyTE (2)

y(1)yp EEE

yzxyTE*zsx

*zsx SVTE

xzxyTE

yzxyTE

(2)z

(1)zp EEE

H(k)=

Page 18: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Bulk Bandstructure:With the Second Nearest Neighbor Interaction:

Page 19: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Bulk Bandstructure in the Antibonding Orbital Model:

Page 20: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Bond Orbital Model with MicroscopicEffects

Page 21: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

For the common atom (CA) heterostructure eg: (AlGa)As/GaAs, InAs/GaAs For the no common atom (NCA) heterostructure eg:InAs/GaSb, (InGa)/As/InP --InAs/GaSb with In-Sb and Ga-As heterobonds at the i

nterfaces --(InGa)As/InP with (InGa)-P and In-As heterobonds at

the interfaces

Page 22: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

The (001) InAs/GaSb superlattice:the planes of atoms are stacked in the growth directio

n as follows... Ga Sb Ga Sb In As In As....for the one interface;and ... In As In As Ga Sb Ga Sb....for the next interface.

Page 23: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

The extracting of microscopic information:

the s- and p-like bond orbitals expanded in terms of the tetrahedral (anti)bonding orbitals

and

instead of scalar potential by potential operator

~this is the so-called modified bond orbital model (MBOM) ~

', aSR = ( + + + ),

1,aR 2,aR 3,aR 4,aR21

= ( + - - ),', bXR21

1,bR 2,bR 3,bR 4,bR

= ( - + - ),', bYR 21

1,bR 2,bR 3,bR 4,bR

= ( - - + ),', bZR21

1,bR 2,bR 3,bR 4,bR

(R) + ),

4

1

)((ˆi

iUV ia,R ia,R )()( RiV ib,R ib,R

Page 24: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

The potential term of the MBOM: a potential matrix form, but not a scalar potential V

VV

VV

VV

VV

V

zxzs

yxys

xxxs

sxss

zR )(44

VV

VV

VV

VV

zzzy

yzyy

xzxy

szsy

V4X4(Rz)=V+ )(44 ZRV

U21

0

0

0

)(44 ZRVV

21

V21

0

0 V21

0

0

0

V21

V21

0

0

Page 25: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

InAs/GaSb Superlattice Bandstructure:(calculated with the BOM and MBOM)

Page 26: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Orientation Dependence of InterfaceInversion Asymmetry Effect on InGaAs/InP Quantum Wells

Page 27: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Inversion asymmetry effect:

the microscopic crystal structure: Dresselhaus effect

the macroscopic confining potential: Rashba effect the inversion asymmetry between two interfaces:

NCA heterostructures --the zero-field spin splitting --in-plane anisotropy

Page 28: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

The 73-Å-wide (25 monolayers) (001) InGaAs/InP QW: A

and

the planes of atoms are stacked in the growth direction as follows: M+1 C D C D C D A B A B A B Mfor the (InGa)P-like interface; and N+1 A B A B A B C D C D C D Nfor the InAs-like interface, where A=(InGa), B=As, C=In, and D=P. The Mth (or Nth) monolayer is located at the left (or right) interface, where N=M+25.

,,,,,, 4321 bbbb RRRRR 21

21

21

21X

,,,,,, 4321 bbbb RRRRR 21

21

21

21Y

,,,,,, 4321 aaaa RRRRR 21

21

21

21S

,,,,,, 4321 bbbb RRRRR 21

21

21

21Z

Page 29: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Where Rz is the z component of lattice site r, i.e., R=R//+RzŽ, and also the U (for the conduction band) and the V (for the valence band) denote the difference of potential energy between the heterobond species and the host material at the interfaces.

)R( Z66V

U21

00

0

0

0

U21

0

0

0

0

0

v32

i

v21

0

0

0

0

v21

v32

i

0

00

0 v21

v32

i

0

0

0

0 v21

v32

i

0

00

0

Page 30: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

(001) InGaAs/InP Quantum Well Bandstructure:(calculated with the BOM and MBOM)

Page 31: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Spin Splitting of the Lowest Conduction Subband:((001) InGaAs/InP Quantum Well)

Page 32: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

When the in-plane wave vector moves around the circle ( =0 2 ), the mixing elements in Eq. (4.2) should be strictly written as

)22

(exp)(321

))(2cos2(sin321

iVVi

for the (3,5) and (4,6) matrix elements and

)22

(exp)(321

))(2cos2(sin321

iVVi

for the (5,3) and (6,4) matrix elements. Therefore, the mixing strength depends on the azimuthal angle Moreover, the and terms equal to –1 for or and 1 for or .

. )22

(exp i

)22

(exp i 4/3 4/7 4/

4/5

Page 33: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

The 71-Å-wide (21 monolayers) (111) InGaAs/InP QW:

The same order of atomic planes as the (001) QW

A

and

,,,,X, 432 bbb RRRR 62

61

61

,,,Y, 32 bb RRR 21

21

.,,,,Z, 4321 bbbb RRRRR 321

321

321

23

,,,,,, 4321 aaaa RRRRR 21

21

21

21S

Page 34: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

the heterobonds in the [111] growth direction:

the heterobonds are the remaining three bonds other than the bond along the [111] direction:

)R( Z66V

v21

00

0

0

0

v21

0

00

00

0

00

000

U41

0

0

000

00000

U41

0

00

000

)R( Z66V00

0

0

00

00

00

0

00

00

0

0

000

00000

00

000U

43

U43

v

v

v21

v21

Page 35: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

(111) InGaAs/InP Quantum Well Bandstructure:(calculated with the BOM and MBOM)

Page 36: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Spin Splitting of the Lowest Conduction Subband:((111) InGaAs/InP Quantum Well)

Page 37: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

The 73-Å-wide (35 monolayers) (110) InGaAs/InP QW: = + + +

=- + + - =- +and = -

S,R21

1,aR21

2,aR21

3,aR21

4,aR

X,R21

1,bR21

2,bR21

3,bR21

4,bR

Y,R2

12,bR

21

3,bR

Z,R2

11,bR 2

14,bR

across perfect (110) interfaces, planes of atoms are arranged in the order of: M+1 D C D C B A B A C D C D A B A B Mfor the left interface and N A B A B C D C D B A B A D C D C N+1for the right interface, where N=M+35

Page 38: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

where the upper sign is used for the Mth and Nth monolayer, and the lower sign is used for the (M+1)th and (N+1)th monolayer.

)R( Z66V

0

0

0

0

0

0

0

0

0U

41

0

0

0

0

0

00

0

0

0

U41

0

0

0

v81

v621

v83

v381

v381

v83v

381

v81 v

621

v381

v621

v621

Page 39: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

(110) InGaAs/InP Quantum Well Bandstructure:(calculated with the BOM and MBOM)

Page 40: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Spin Splitting of the Lowest Conduction Subband:

((110) InGaAs/InP Quantum Well)

Page 41: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Symmetry point group of QWs.

Microscopic BOMBulk Td Oh

CAQW(001) D2d D4h

NCAQW(001) C2v D4h

NCAQW(111) C3v D3d

NCAQW(110) C1h or C1 D2h

Page 42: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Dresselhaus-like Spin Splitting

Page 43: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Dresselhaus effect:

The degeneracy bands of the zinc-blends bulk are lifted except for the wave vector along the <001> and <111> directions, and this is the so-called Dresselhaus effect.

Page 44: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Subband Structure of (110) InAs/GaSb Superlattice:(calculated with the BOM and MBOM)

Page 45: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

MBOM Bandstructure of InAs/GaSb Superlattice(grown on the (001), (111), (113), and (115)-orientation)

Page 46: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Microscopic Interface Effect on(Anti)crossing Behavior andSemiconductor-semimetal Transition inInAs/GaSb Superlattices

Page 47: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

This MBOM model is based on the framework of the bond orbital model (BOM) and combines the concept of the heuristic Hbf model to include the microscopic interface effect. The MBOM provides the direct insight into the microscopic symmetry of the crystal chemical bonds in the vicinity of the heterostructure interfaces. Moreover, the MBOM can easily calculate various growth directions of heterostructures to explore the influence of interface perturbation.

In this chapter, by applying the proposed MBOM, we will calculate and discuss the (anti)crossing behavior and the semiconductor to semimetal transition on InAs/GaSb SLs grown on the (001)-, (111)-, and (110)-oriented substrates. The effect of interface perturbation on InAs/GaSb will be studied in detail.

Page 48: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

(Anti)crossing Behavior of InAs/GaSb Superlattice

Page 49: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

(001) Semimetal Phenomenon:(calculated with the BOM and MBOM)

Page 50: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

(111) Semimetal Phenomenon:(calculated with the BOM and MBOM)

Page 51: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

(110) Semimetal Phenomenon:(calculated with the BOM and MBOM)

Page 52: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

k.p Finite Difference Method

Page 53: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

the BOM eigenfunctions must be Bloch functions, which can be expressed as

where the notation is used for an-like (=s,x,y,z) bond orbital located at a fcc lattice site R, k is the wave vector, and N is the total number of fcc lattice points.

the BOM matrix elements with the bond-orbital basis (without spin-orbit coupling) are given by (in k-space)

Where is the relative position vector of the lattice site R to the origin and (see chapter 2) is the interaction parameter

taking the Taylor-expansion on the BOM Hamiltonian and omitting terms higher than the second order in k, the general k‧p formalism is easily obtained, whose matrix elements can be written as [11]

,,1, eN

Rk Rk

R i

R,

kkk ,H,)(H ',BOM '

)(e ,' ji

j

j RRk

)( 'RRR j'R ,'

j 2

11{)(H ,' kpk

2)( jRk ).(} ,' jji RRk

Page 54: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

the kinetic term of the usual k˙p Hamiltonian [in the basis ) can be written as

,,, 23232121 u ),, 23

232321

2321 uuu

CT

0

T

0

0S0

00

0

00

0

)(kH pk

*S

*T

3/S3/*S

*TS

*B*C

3/S

*C

3/*S

*S

RR

P+Q BP-Q P-

Q *B-

C-BP+Q

where the superscript * means Hermitian conjugate,P=Ev –[(2Exx + Ezz)/3]a2k2,

Q= – (Exx – Ezz),a2(k2- )/12, R=Ec –Essa2k2

S= – Esxa (kx+ ), T= Esx / , B= Exya2( – ) , C=[( Exx – Ezz)( – ) / 4 – Exy ] , andEc=Es + 12Ess, Ev=Ep + 8Exx + 4 Ezz.

2k3 z

22i yik8i

zka 6

xk yik 3/zk2xk

2xk i yxkk 3/a 2

Page 55: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

the time-independent equation can be expressed as a function of k z, that is

]F=EF With the replacement of kz by , this equation can be expressed as

= and =

the Schrödinger equation can be written in the layer-orbital basis as

where is the interaction between and layers

2]2[0

]1[0

]0[0 zz kHkHH

Zi /

2

2]2[

0]1[

0]0[

0 ZH

ZiHH F=EF

)(lZZZF

hFF ll

211

)(2

2

lZZZF

,2

211

hFFF lll

0FHFHFH 1111 ,,, lllllllll

',H ll l 'l

The k.p finite difference method

Page 56: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Optimum Step Length in the KPFD Method

Page 57: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

the -dependent terms of the k˙p Hamiltonian can be written as

and

where is the spacing of monolayers along the growth direction the replacement of by and then treated by the finite-difference calc

ulation, we have

and

where is the pseudo-layer, the step length h is the spacing between two adjacent pseudo-layers, and F is the corresponding state function.

The reason of the optimum step lengthzk

zR

)cos1(2 2

222

zzz

z RkmRm

k

)2(2 2

2zzzz RikRik

z

eemR

=

zzz

zzz Rk

RP

Pk sin

),(2

zzzz RikRik

z

z eeiRP =

zk Zi /

),(2 11

llzlz FFihP

ZF

iP

)2(22 112

2

2

22

llll FFF

mhZF

m

l

Page 58: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

the Schrödinger equation solved by the KPFD method can be written as

where is the interaction between and layers; the interger n is 1 for the (001) and (111) samples, 2 for the (110) and (113) samples, 3 for the (112) and (115) samples, …, etc. That is to say, the on-site and 12 neares

t- neighbor bond orbitals belong respectively to (2n+1) layers, which are easily classified according to the longitudinal component of the bond- orbital position vector. The step length between the on-site layer and the nearest-, second-, or third-neighbor interaction layer is 1ML, 2ML, or 3ML spacing in the longitudinal direction, respectively.

,0FHFHFH ,,

n

1,,,

jlljll

jjljllllll

ll ,'H 'l l

Page 59: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Multi-step Length in the (110) KPFD Method:

Page 60: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Multi-step Length in the (112) KPFD Method:

Page 61: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Multi-step Length in the (113) KPFD Method:

Page 62: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Multi-step Length in the (115) KPFD Method:

Page 63: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

InAs Bulk Bandstructure(calculated with the k˙p and SBO method)

Page 64: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

InAs Bulk Bandstructure(calculated with the SBO and KPSFD method)

Page 65: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Anisotropic Optical Matrix Elements inQuantum Wells with Various SubstrateOrientations

Page 66: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

The (11N) 44 Luttinger Hamiltonian at the Brillouin-zone center (k1=k2=0)

Hk.p(k1=k2=0)=

(Ep+8Exx+4Ezz)-

+ +

+

where a is the lattice constant, is the angle between the z and X3 axes, which is equal to

1000

0100

0010

0001

)EE2(34k

4a

zzxx23

2

)

32-θsin

23-θsin2)(EE( 42

zzxx )

32-θsin3-θsin4(E 42

xy

1000

0100

0010

0001

)θsin23-θ)(1sin

31)(E2E(E 2

xyzzxx

0θcos2θsin0

θcos200θsin

θsin00θcos2

0θsinθcos20

).2N/N(cos 21

Page 67: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

the optical transition matrix element between the conduction and the valence bands can be written as

where is the momentum operator and ê is the unit polarization vector.

the in-plane optical anisotropy can be calculated as

whereand are the squared matrix elements for the polarization parallel and perpendicular to , respectively.

i/

,ˆ Vi

eCM

,22

||

22

||

MM

MM

2

||M2

M]101[

Page 68: The Application and Calculation of Bond Orbital Model on Quantum Semiconductor

Anisotropic Optical Matrix Elements

(in the (11N)-orientation