1、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IntroductionWhy is the choosing the BOM?a hybrid or link between the k.p and the tight-binding methodscombining the virtues of the two above approaches -the computational effort is comparable to the k.p m
2、ethod -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 geometriesThe improvement of the bond orbital
3、model(BOM):the(hkl)-oriented BOM Hamiltonian the BOM Hamiltonian with the second-neighbor interaction the BOM in the antibonding orbital framework the BOM with microscopic interface perturbation(MBOM)the k.p formalism from the BOMBond Orbital ModelWhat is the bond orbital model?a tight-binding-like
4、framework with the s-and p-like basis orbital the interaction parameters directly related to the Luttinger parametersZinc-blende Lattice Structure:2/a)1,1,0(),1,0,1(),0,1,1(The BOM matrix elements:,BOM)(Hk)(e,jijjRRkwhere：The interaction parameters,Es and Ep:on-site parametersEss,Esx,Exx,Exy,and Ezz
5、:the nearest-neighbor interaction parametersThe BOM matrix:where)k(IEEHsssss),2/acos2/acos2/acos2/acos2/acos2/a(cos4)(zyzxyxkkkkkkIk),2/acos2/acos2/acos2/a(cos2/asin4kkkkkiEHzyxsxs),2/acos2/acos2/acos2/a(cos2/acos)(4)k(kkkkkEEIEEHzyxzzxxzzp)(2/asin2/asin4kkEHxywithssHsxHsyHszH*sxHxxHxyHxzH*syHxyHyyH
6、yzH*szHxzHyzHzzHH(k)=Taking Taylor-expansion on the BOM matrix:(up to the second order)where,12EEEsssc,4E8EEEzzxxpV,2/a)EE(2xxzz1,2/a)EE(2xxzz2.aE2xy3andH(k)=kaEE22ssczsxakE4ixsxakE4i2221kkExvyxkk3zxkk3ysxakE4iyxkk32221kkEyvzykk3zxkk3zykk32221kkEzvxsxakE4i-ysxakE4i-zsxakE4i-Relations between BOM par
7、ameters and Luttinger parametersVBMCVBM32/)8/X12(EE022sxhlgR 32/)8/X12(EE022sxhlgR g2xy03xyE/)E166ER24/X)/(3EE16Eg2sx01xxhlR8/XEExxzzhl2/XE12EExxphlvggcEERmm1264EE2sx00ss/303xy6ER021xx)4(ER021zz)8(ER01p12EERvBulk Bandstructure:(001)-orientationSuperlattice Bandstructure:(001)-orientationThe orthogon
8、al transformation matrix:cossinsincoscosTsinsincossincoscos0sinwhere the angles and are the polar and azimuthal angles of the new growth axis relative to the primary crystallographic axes.)/(tan221lkh)/(tan1hkBulk InAs Bandstructure:(111),(110),(112),(113),and(115)-orientationInAs/GaSb Superlattice
9、Bandstructure:(111),(110),(112),(113),and(115)-orientationThe second-neighbor bond orbital(SBO)model:WhereandT)EE(TEExx)1(zzszz),(zyxC)VV(CVExx)2(zzszz),2/akcos()2/ak)cos(2/aksin(4Tzyxxi),2/akcos()2/ak)cos(2/aksin(4Tzxyyi),2/akcos()2/ak)cos(2/aksin(4Tzzyxi),2/akcos()2/ak)cos(2/akcos(4Tzyxxx),2/akcos
10、()2/ak)cos(2/akcos(4Tzxyyy),2/akcos()2/ak)cos(2/akcos(4Tzzyxz),2/aksin()2/aksin(4Tyxxy),2/aksin()2/aksin(4Tzxxz),2/aksin()2/aksin(4Tzyyz,2/)TTT(Tzzyyxxs),aksin(2Sxxi),aksin(2Syyi),aksin(2Szzi),akcos(2Cxx),akcos(2Cyy),akcos(2Czz.CCCCzyxssssssssCVTEExsxxsxSVTEysxysxSVTEzsxzsxSVTE*xsx*xsxSVTE(2)x(1)xpE
11、EExyxyTExzxyTE*ysx*ysxSVTExyxyTE(2)y(1)ypEEEyzxyTE*zsx*zsxSVTExzxyTEyzxyTE(2)z(1)zpEEEH(k)=Bulk Bandstructure:With the Second Nearest Neighbor Interaction:Bulk Bandstructure in the Antibonding Orbital Model:Bond Orbital Model with MicroscopicEffects For the common atom(CA)heterostructure eg:(AlGa)As
12、/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 interfaces -(InGa)As/InP with(InGa)-P and In-As heterobonds at the interfacesThe(001)InAs/GaSb superlattice：the planes of atoms are stacked in the growth direction
13、 as followsGa Sb Ga Sb In As In Asfor the one interface;and In As In As Ga Sb Ga Sbfor the next interface.The extracting of microscopic information:the s-and p-like bond orbitals expanded in terms of the tetrahedral(anti)bonding orbitalsand instead of scalar potential by potential operatorthis is th
14、e so-called modified bond orbital model(MBOM),aSR=(+),1,aR2,aR3,aR4,aR21=(+-),bXR211,bR2,bR3,bR4,bR=(-+-),bYR211,bR2,bR3,bR4,bR=(-+),bZR211,bR2,bR3,bR4,bR(R)+),41)(iiUVia,Ria,R)()(RiVib,Rib,RThe potential term of the MBOM:a potential matrix form,but not a scalar potential VVVVVVVVVVzxzsyxysxxxssxssz
15、R)(44VVVVVVVVzzzyyzyyxzxyszsy V4X4(Rz)=V+)(44ZRVU21000)(44ZRVV21V2100V21000V21V2100InAs/GaSb Superlattice Bandstructure:(calculated with the BOM and MBOM)Orientation Dependence of InterfaceInversion Asymmetry Effect on InGaAs/InP Quantum WellsInversion asymmetry effect:the microscopic crystal struct
16、ure:Dresselhaus effect the macroscopic confining potential:Rashba effect the inversion asymmetry between two interfaces:NCA heterostructures -the zero-field spin splitting -in-plane anisotropyThe 73-wide(25 monolayers)(001)InGaAs/InP QW:Aand the planes of atoms are stacked in the growth direction as
17、 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.,4321bbbbRRRRR21212121X,4321bbbbRRRRR21212121Y,4321aaaaRRR
18、RR21212121S,4321bbbbRRRRR21212121ZWhere Rz is the z component of lattice site r,i.e.,R=R/+Rz,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(Z66VU2100000U2100000v
19、32iv210000v21v32i0000v21v32i0000v21v32i0000(001)InGaAs/InP Quantum Well Bandstructure:(calculated with the BOM and MBOM)Spin Splitting of the Lowest Conduction Subband:(001)InGaAs/InP Quantum Well)When the in-plane wave vector moves around the circle(=0 2),the mixing elements in Eq.(4.2)should be st
20、rictly written as)22(exp)(321)(2cos2(sin321iVVifor the(3,5)and(4,6)matrix elements and)22(exp)(321)(2cos2(sin321iVVifor 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(expi)22(expi4/34/74/4/5The 7
21、1-wide(21 monolayers)(111)InGaAs/InP QW:The same order of atomic planes as the(001)QW A and,X,432bbbRRRR626161,Y,32bbRRR2121.,Z,4321bbbbRRRRR32132132123,4321aaaaRRRRR21212121S the heterobonds in the 111 growth direction:the heterobonds are the remaining three bonds other than the bond along the 111
22、direction:)R(Z66Vv2100000v2100000000000U410000000000U41000000)R(Z66V000000000000000000000000000000U43U43vvv21v21(111)InGaAs/InP Quantum Well Bandstructure:(calculated with the BOM and MBOM)Spin Splitting of the Lowest Conduction Subband:(111)InGaAs/InP Quantum Well)The 73-wide(35 monolayers)(110)InG
23、aAs/InP QW:=+=-+-=-+and =-S,R211,aR212,aR213,aR214,aRX,R211,bR212,bR213,bR214,bRY,R212,bR213,bRZ,R211,bR214,bRacross 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
24、interface,where N=M+35 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(Z66V000000000U410000000000U41000v81v621v83v381v381v83v381v81v621v381v621v621(110)InGaAs/InP Quantum Well Bandstructure:(calculated with the BOM and MBOM
25、)Spin Splitting of the Lowest Conduction Subband:(110)InGaAs/InP Quantum Well)Symmetry point group of QWs.MicroscopicBOMBulkTdOhCAQW(001)D2dD4hNCAQW(001)C2vD4hNCAQW(111)C3vD3dNCAQW(110)C1h or C1D2hDresselhaus-like Spin Splitting Dresselhaus effect:The degeneracy bands of the zinc-blends bulk are lif
26、ted except for the wave vector along the and directions,and this is the so-called Dresselhaus effect.Subband Structure of(110)InAs/GaSb Superlattice:(calculated with the BOM and MBOM)MBOM Bandstructure of InAs/GaSb Superlattice(grown on the(001),(111),(113),and(115)-orientation)Microscopic Interface
27、 Effect on(Anti)crossing Behavior andSemiconductor-semimetal Transition inInAs/GaSb Superlattices 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 insig
28、ht 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 wi
29、ll 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.(Anti)crossing Behavior of InAs/GaSb Superlattice(001)
30、Semimetal Phenomenon:(calculated with the BOM and MBOM)(111)Semimetal Phenomenon:(calculated with the BOM and MBOM)(110)Semimetal Phenomenon:(calculated with the BOM and MBOM)k.p Finite Difference Methodthe BOM eigenfunctions must be Bloch functions,which can be expressed as where the notation is us
31、ed 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
32、R to the origin and (see chapter 2)is the interaction parametertaking the Taylor-expansion on the BOM Hamiltonian and omitting terms higher than the second order in k,the general kp formalism is easily obtained,whose matrix elements can be written as 11,1,eNRkRkRiR,kkk,H,)(H,BOM)(e,jijjRRk)(RRRjR,j2
33、11)(H,kpk2)(jRk).(,jjiRRkthe kinetic term of the usual kp Hamiltonian in the basis )can be written as,23232121u),232323212321uuuCT0T00S0000000)(kHpk*S*T3/S3/*S*TS*B*C3/S*C3/*S*SRRP+QBP-QP-Q*B-C-BP+Qwhere the superscript*means Hermitian conjugate,P=Ev(2Exx+Ezz)/3a2k2,Q=(Exx Ezz),a2(k2-)/12,R=Ec Essa2
34、k2S=Esxa(kx+),T=Esx /,B=Exya2(),C=(Exx Ezz)()/4 Exy ,andEc=Es+12Ess,Ev=Ep+8Exx+4 Ezz.2k3z22iyik8 izka6xkyik3/zk2xk2xkiyxkk3/a2the time-independent equation can be expressed as a function of kz,that is F=EF With the replacement of kz by ,this equation can be expressed as =and =the Schrdinger equation
35、 can be written in the layer-orbital basis as where is the interaction between and layers 2201000zzkHkHHZi/22201 000ZHZiHHF=EF)(lZZZFhFFll211)(22lZZZF,2211hFFFlll0FHFHFH1111,lllllllll,Hlll lThe k.p finite difference methodOptimum Step Length in the KPFD Methodthe -dependent terms of the kp Hamiltoni
36、an 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 calculation,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 stat
37、e function.The reason of the optimum step lengthzkzR)cos1(22222zzzzRkmRmk)2(222zzzzRikRikzeemR=zzzzzzRkRPPksin),(2zzzzRikRikzzeeiRP=zkZi/),(211llzlzFFihPZFiP)2(221122222llllFFFmhZFmlthe Schrdinger equation solved by the KPFD method can be written as where is the interaction between and layers;the in
38、terger 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 nearest-neighbor bond orbitals belong respectively to(2n+1)layers,which are easily classified according to the longitudinal component of the bond-orbital posit
39、ion 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,n1,jlljlljjljllllllll,HllMulti-step Length in the(110)KPFD Method:Multi-step Length in the(112)KPFD Method:M
40、ulti-step Length in the(113)KPFD Method:Multi-step Length in the(115)KPFD Method:InAs Bulk Bandstructure(calculated with the kp and SBO method)InAs Bulk Bandstructure(calculated with the SBO and KPSFD method)Anisotropic Optical Matrix Elements inQuantum Wells with Various SubstrateOrientationsThe(11
41、N)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 1000010000100001)EE2(34k4azzxx232)32-sin23-sin2)(EE(42zzxx)32-sin3-sin4(E42xy1000010000100001)sin23-)(1sin31)(E2E(E2x
42、yzzxx0cos2sin0cos200sinsin00cos20sincos20).2N/N(cos21the optical transition matrix element between the conduction and the valence bands can be written as where is the momentum operator and 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/,VieCM,22|22|MMMM2|M2M101Anisotropic Optical Matrix Elements(in the(11N)-orientation
