Recently, graphene have attracted a lot of studies because graphene has Dirac points structure at each corners of Brilloin zone. Therefore, to find out new structure with Dirac cone by artificial fabrication in semiconductors can be addressed an important issue.
In realistic case we must consider the spin-orbit interaction for our case (see chapter1).
Another interesting topic is quantum spin Hall state which has a special property for electrons transmission with certain spin helix.
1.1.1 spin-orbit coupling in solid-state system
Electron spin, the only internal degree of freedom of electrons, follows naturally from the Dirac equation when Dirac tried to put wave function in a covariant form, when space and time appear on equal footing. A nonrelativistic limit of the Dirac equation gives rise to the spin-orbit interaction term, that has been found great success in atomic energy spectra. The spin-orbit interaction, in vacuum can be expressed by [2]
HSO = − e~
4m20c2 σ · (E × p) = ~
4m20c2 σ · (∇V × p) , (1.1)
CHAPTER 1. INTRODUCTION
where m0 is the free electron mass, ~ the Planck constant and c the light speed of light.
The physical mechanism of HSO can be interpreted: an electron moving in an electric potential region sees, in its frame of reference, an effective magnetic field which couples with the electron spin through the magnetic moment of the electron spin. Through this effective magnetic field, which certainly depends on the orbital motion of the electron, the SOI is established. This physical picture holds in semiconductor too, when V (r) could be the periodic potential of the host lattice and also the impurities.
The k·p model is often applied to the electron energy band calculation in semiconduc-tor, when the states in the vicinity of the band edges. Furthermore, within the envelope function approximation (EFA), the energy band can be characterized by effective masses.
The SOI in semiconductors requires, first of all, an effective electric field in the material.
Such effective electric field can be contributed from the build-in crystal field where the crystal has bulk inversion asymmetric (BIA) the so-called Dresselhaus SOI, or structural inversion asymmetry (SIA), the so-called Rashba SOI. The BIA is found in zinc-blende structure and the SIA in asymmetric quantum wells (QWs) or heterostructures.
Based on the effective mass approximation, the effect of all the fast-varying atomic potential has been incorporated into the effective mass. Slower varying V (r), with varia-tion length scale much greater than the lattice spacing, is found to contribute to SOI with a much greater SO coupling constant λ. For a central potential V (r) = V (r) in vacuum, the SO coupling is
~
where L is the orbital angular momentum, σ is the vector Pauli matrices and λvac =
−~2/(4m20c2) ≈ −3.72 × 10−6˚A2.
CHAPTER 1. INTRODUCTION
But in a semiconductor, also for a central potential V (r) = V (r), the SO coupling is
HSO = −λ
For a 2DEG, the SOI becomes
HSO = −λ
Here P is the momentum matrix element between s- and p-orbitals, Eg is the energy band gap, and ∆0 represents the SOI energy split to the spin split-off hole band [3, 4]. Of particular interest is that λ = 120 ˚A2 in InAs, which is seven order of magnitude greater than λvac [3, 5].
Qualitatively, this large enhancement of SO coupling constant can be explained in the following. With λvac ∝ m12
Comparing with 120 ˚A2
3.73×10−6˚A2 = 32 × 106, such hand waving argument has captured the essential physical origin of the great enhancement.
CHAPTER 1. INTRODUCTION
1.1.2 Quantum Hall effect
The integer quantum Hall state (von Klitzing, Dorda, and Pepper, 1980; Prange and Girvin, 1987), occurs when free electrons are confined to two dimensions by applying a strong magnetic field. The quantization of the electrons circular orbits with cyclotron frequency ωc leads to quantized Landau levels with energy εm = ~ωc(m + 1/2). If N Landau levels are filled and the rest are empty, then an energy gap separates the occupied and empty states just as in an insulator. Unlike an insulator, though, an x-direction electric field causes the cyclotron orbits to drift, leading to a Hall current characterized by the quantized Hall conductivity,
σxy = Ne2/h. (1.2)
This precision is a manifestation of the topological nature of σxy Landau levels can be viewed as a band structure. Since the generators of translations do not commute with one another in a magnetic field, electronic states cannot be labeled by momentum. However, if a unit cell with area 2π~c/eB enclosing a flux quantum is defined, then lattice translations restore the commutation relation, so Blochs theorem allows electron states to be labeled by 2D crystal momentum k. In the absence of a periodic potential, the energy levels are simply the k independent of Landau levels Em(k) = εm. In the presence of a lattice periodic potential, the energy levels will disperse with k. This leads to a band structure that looks identical to that of an ordinary insulator.
1.1.3 Quantum spin Hall effect
The quantum spin Hall state is a state of matter proposed to exist in special, two-dimensional, semiconductors with spin-orbit coupling. The quantum spin Hall state of matter is the cousin of the integer quantum Hall state, but, unlike the latter, it does not require the application of a large magnetic field. The quantum spin Hall state does not break any discrete symmetries (such as time-reversal or parity symmetry). The first
CHAPTER 1. INTRODUCTION
proposal for the existence of a quantum spin Hall state was developed by Kane and Mele [6] who adapted an earlier model for graphene by Haldane [7] which exhibits an integer quantum Hall effect. The Kane and Mele model is two copies of the Haldane model such that the spin up electron exhibits a chiral integer quantum Hall Effect while the spin down electron exhibits an anti-chiral integer quantum Hall effect. It has been recently proposed [8] and subsequently experimentally realized [9] in mercury (II) telluride (HgTe) semiconductors.
Overall the Kane-Mele model has a charge-Hall conductance of exactly zero but a spin-Hall conductance of exactly σxyspin = (in units of4πe ) Independently, a quantum spin Hall model was proposed by Bernevig and Zhang [10] in an intricate strain architec-ture which engineers, due to spin-orbit coupling, a magnetic field pointing upwards for spin-up electrons and a magnetic field pointing downwards for spin-down electrons. The main ingredient is the existence of spin-orbit coupling, which can be understood as a momentum-dependent magnetic field coupling to the spin of the electron.
Strictly speaking, the models with spin-orbit coupling do not have a quantized spin Hall conductance σxyspin 6= 2. Those models are more properly referred as topological insulator which is an example of topologically ordered states.
CHAPTER 1. INTRODUCTION