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 --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

The improvement of the bond orbital 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 BOM

Bond Orbital Model

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

Zinc-blende Lattice Structure:

The BOM matrix elements: where ： The interaction parameters E s and E p : on-site parameters E ss, E sx, E xx, E xy, and E zz : the nearest-neighbor interaction parameters

The BOM matrix: where with H(k)=

Taking Taylor-expansion on the BOM matrix: (up to the second order) where and H(k)= - - -

Relations between BOM parameters and Luttinger parameters VBMCVBM /3

Bulk Bandstructure: (001)-orientation

Superlattice Bandstructure: (001)-orientation

The orthogonal transformation matrix: where the angles and are the polar and azimuthal angles of the new growth axis relative to the primary crystallographic axes.

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

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

The second-neighbor bond orbital (SBO) model: Where and H(k)=

Bulk Bandstructure: With the Second Nearest Neighbor Interaction:

Bulk Bandstructure in the Antibonding Orbital Model:

Bond Orbital Model with Microscopic Effects

 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 interfaces --(InGa)As/InP with (InGa)-P and In-As heterobonds at the interfaces

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

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) ~ = ( ), = ( ), = ( ), = ( ), (R) + ),

The potential term of the MBOM:  a potential matrix form, but not a scalar potential V  V 4X4 (R z )=V

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

Orientation Dependence of Interface Inversion Asymmetry Effect on InGaAs/InP Quantum Wells

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

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    M for the (InGa)P-like interface; and N+1    A B A B A B C D C D C D    N for 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.

Where R z is the z component of lattice site r, i.e., R=R // +R z Ž, 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

(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 strictly written as for the (3,5) and (4,6) matrix elements and 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.

The 71-Å-wide (21 monolayers) (111) InGaAs/InP QW:  The same order of atomic planes as the (001) QW  A and

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

(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-Å-wide (35 monolayers) (110) InGaAs/InP QW:  = = =- + and = -  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    M for the left interface and N    A B A B C D C D       B A B A D C D C    N+1 for the right 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

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

Spin Splitting of the Lowest Conduction Subband: ((110) InGaAs/InP Quantum Well)

Symmetry point group of QWs. MicroscopicBOM Bulk TdTd OhOh CAQW(001) D 2d D 4h NCAQW(001) C 2v D 4h NCAQW(111) C 3v D 3d NCAQW(110) C 1h or C 1 D 2h

Dresselhaus-like Spin Splitting

 Dresselhaus effect: The degeneracy bands of the zinc-blends bulk are lifted 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 Effect on (Anti)crossing Behavior and Semiconductor-semimetal Transition in InAs/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 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.

(Anti)crossing Behavior of InAs/GaSb Superlattice

(001) 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 Method

 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]

 the kinetic term of the usual k ˙ p Hamiltonian [in the basis ) can be written as C T 0 T 0 0 S R R P+Q B P-Q - C -B P+Q where the superscript * means Hermitian conjugate, P=E v –[(2E xx + E zz )/3]a 2 k 2, Q= – (E xx – E zz ),a 2 (k 2 - )/12, R=E c –E ss a 2 k 2 S= – E sx a (k x + ), T= E sx /, B= E xy a 2 ( – ), C=[( E xx – E zz )( – ) / 4 – E xy ], and E c =E s + 12E ss, E v =E p + 8E xx + 4 E zz.

 the time-independent equation can be expressed as a function of k z, that is ]F=EF With the replacement of k z 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 F=EF The k.p finite difference method

Optimum Step Length in the KPFD Method

 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 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 state function. The reason of the optimum step length = =

 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 nearest- 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.

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

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

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

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

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

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

Anisotropic Optical Matrix Elements in Quantum Wells with Various Substrate Orientations

The (11N) 4  4 Luttinger Hamiltonian at the Brillouin-zone center (k 1 =k 2 =0) H k. p (k 1 =k 2 =0)= (E p +8E xx +4E zz )- + where a is the lattice constant,  is the angle between the z and X 3 axes, which is equal to

 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.

Anisotropic Optical Matrix Elements (in the (11N)-orientation