1.4.2 Spin Current and Spin-related Force

To start with a general formalism, you can consider the Schrödinger equ-ation for a particle moving in an external SU(2)gauge field

where Ψ is a two-component wavefunction and η refers to the coupling strength. The gauge potential corresponding to the Rashba and Dressel-haus spin-orbit couplings, the Zeeman term, and the sheer strain field for a system in x-y plane are given by

where B≡(Bx, By, Bz)stands for the magnetic field, α and β refer to the Rashba and Dresselhaus coupling strengths, and γ is related to the sheer strain field, respectively. Clearly, the Hamiltonian(1.60)is the special case of(1.62)with vanishing β, γ and B.

Let be the spin operators, and then the local spin density is given by

Using the method similar to the derivation of the continuity equation for the charge current, you obtain a continuity-like equation from(1.62)

where the spin-current density is defined as

with the velocity operator given by

and ησa=which is just the spin operator if η=ħ/2. In order to avoid ambiguities, the indices i, j, k, µ, ν refer to the components of a vector in spatial space while a, b, c refer to those in intrinsic space(Lie algebra space, or spin space). Moreover, a vector in spatial space is denoted by a bold face while that in intrinsic space is specified by an overhead arrow, i.e., =(S1, S2, S3), ==(J1, J2, J3), J3=(), etc.

Using the above formalism, you can easily derive the formula of the force exserted on spin density and spin current. Similar to the field-strength tensor of the electromagnetic field which is defined by Fµν=∂µAν-∂νAµ, the field strength tensor of the SU(2)field is defined by Fµν=σa, whose components are given by =∂µ-∂ν-ηϵabc. Analogous to the Lorentz force evaluated by jµFµiin electromagnetism, the general form of this force fican be written out(I omit it here, and the details can be found in the reference)from .

For the familiar case α=β=γ=0 and B=0, you have

which corresponds to the force due to inhomogeneity of the magnetic field that has been employed in Stern-Gerlach apparatus. If the Rashba and Dresselhaus coupling strengths are uniform (α and β are constants) and the magnetic field is absent, the formula is simplified to

Clearly, the forces arising from the Rashba and Dresselhaus couplings are along the opposite directions. The magnitudes of the forces are related to the perpendicular component (a=3) of the spin current only. The direction of the force is perpendicular to the direction of spin current. Spin procession and deflection of the motion caused by the aforementioned force play an essential role in spin related quantum device.

On spintronics

Spintronics is now a promising research field, in which the quantum me-chanical principle of the electron spin is used for switching purpose instead of its charge. The main advantages of this concept are lower power con-sumption and fast switching. A typical example of spintronics device is the spintransistor proposed by Datta and Das in 1990. In such a device, the ferromagnetic(FM)source and drain contacts are used as spin injector and receiver. In between the source and drain, the spin orientation is con-trolled by the Rashba effect(see Fig.1.9)with the help of a gate electrode.So the spin of the electron that arrives at the receiver can be managed to be in parallel or antiparallel with the magnetization of the drain. This provides a useful route to control the current through the device.

Problems

1. Work out all the derivation procedures to get the plane-wave solu-tion of Dirac equation for free particles. Show the wavefunctions corresponding to the positive eigenenergy Epand negative one-Ep are given by

where Ep= , u†u=1 and N ±= (2πħ)-3/2[(Ep± mc2)/Ep]1/2.

2. Consider the motion of an electron in a Coulomb potential V(r)=-. Solve the Dirac equation exactly and show the energy eigen-values are given by

which gives rise to the superfine structure of an atomic spectrum. Interested readers can try to solve this problem.

3. Consider the motion of an electron in a uniform magnetic field B along the z-axis. Solve the most general four-component positive energy eigenfunctions and show the energy eigenvalues are given by

where n=0,1,2, . . .

4. Consider a Coulomb potential V(r)=-. Use the perturbation theory to calculate the contributions of the Darwin term , rela-tivistic correction and spin-orbit coupling and to show they are respectively

Sum up those terms to obtain the result of relativistic corrections to energy levels for hydrogen-like atoms

5. For 2DEG with Rashba spin-orbit coupling, the Hamiltonian is given by

Solve the corresponding Schrödinger equation and plot out the energy-momentum relations E=E(kx, ky).

6. The honeycomb lattice L can be regarded as a superposition of two triangular sublattices LAand LB, i.e., L=LA∪LB, of which the basis vectors are a1= (Fig.1.10. ) and a2= (0, a). It is plotted in

Clearly, these two sublattices are connected by b1= (,0), b2=( ), and b3=(), i.e., r∈LAand LB∋r′=r+bi (i=1,2,3). Thus the tight-binding model on the honeycomb lattice can be written as

where (CA)denotes creation(annihilation)operator of a fermion on sublattice LA, while (CB) denotes creation (annihilation) operator of a fermion on sublattice LB. Solve this problem by means of Fourier transform to obtain the band structure, and plot out the result for m=0 to recover Fig.1.4.