Relativistic Wigner functions in transition metal dichalcogenides

Abstract

The Wigner function has been proven useful in many studies of transport as well as quantum coherence in experimental situations. When we deal with Dirac bands, this function becomes a matrix function. Here, we study the Wigner matrix for a situation in a transition metal dichacogenides with a dominant spin–orbit interaction. Here, we discuss $\hbox {WS}_{2}$, where the bands can be described with the Dirac equation, and a unique spin-valley coupling arises. This leads to a 2X2 Wigner matrix for either the conduction band or the valence band. The off-diagonal elements display interference phenomena from the two diagonal components.

Appendices

Appendix A

Before delving into the distribution function itself, there are several important points about the quantum system that underlie the methods used here. It is fairly well known that the Dirac equation can be recast in the form of the Klein–Gordon equation. Whether we solve the Klein–Gordon equation or the Schrödinger equation itself, we are dealing with a linear differential equation which can be characterized with an arbitrary basis set and the exact set used is irrelevant. This basis set can be transformed into any other set by a similarity transformation [56]. Hence, we can use an arbitrary basis in which each state is localized at a point in space. This basis does not need to be orthonormal, as non-orthogonal basis sets have been known for quite some time [57]. A basis set of Gaussians is such a non-orthogonal basis, but represents a viable and common set widely used.

It is known that the Boltzmann or Wigner equation can be cast into a path integral form [58,59,60,61,62]. In the presence of a constant relaxation time, this formulation demonstrates that the distribution function retains its exact form from the initial time, which is a drifted distribution function [63]. Let us write the Wigner equation of motion, in a constant electric field, as

$$\begin{aligned} \left( {\frac{\partial }{\partial t}+\mathbf{v}\cdot \nabla +e\mathbf{E}\cdot \frac{\partial }{\partial \mathbf{p}}} \right) f(\mathbf{r},\mathbf{p},t)=-\frac{f-f_0 }{\tau }. \end{aligned}$$

(A1)

Here, the motion is described in a six-dimensional phase space which is sufficient for the single particle Wigner function discussed here. The coordinate along a trajectory is taken to be s, from which it may be parameterized as [58, 60]

$$\begin{aligned} \mathbf{r}\rightarrow \mathbf{x}^{*}(s),\quad \mathbf{p}\rightarrow \mathbf{p}^{*}(s),\quad t\rightarrow s \end{aligned}$$

(A2)

for which the partial derivatives are

$$\begin{aligned} \frac{\hbox {d}{} \mathbf{x}^{*}}{\hbox {d}s}=\mathbf{v},\quad \mathbf{x}^{*}(t)=\mathbf{r} \end{aligned}$$

(A3)

and

$$\begin{aligned} \frac{\hbox {d}{} \mathbf{p}^{*}}{\hbox {d}s}=\hbox {e}{} \mathbf{E},\quad \mathbf{p}^{*}(t)=\mathbf{p}\quad . \end{aligned}$$

(A4)

Now, (A1) can be written as

$$\begin{aligned} \frac{\hbox {d}f}{\hbox {d}s}=-\frac{f-f_0 }{\tau }, \end{aligned}$$

(A5)

which has the trivial solution

$$\begin{aligned} f(s,~{\tau })~=~f_{0} . \end{aligned}$$

(A6)

Hence, along the trajectories, the form of the distribution function is precisely that of the equilibrium distribution by modified by the parameters of the steady motion along the trajectories. This has a rich tradition in transport studies, as the “drifted,” or shifted, distribution [64]. It was well known already in the nineteenth century and has a rich history [63],[65]. The spatial variation of the distribution function can be described by the basis set chosen, so that a single Gaussian represents a typical example of the particle described by the distribution function [6]. That is, we describe a typical particle in terms of one of the basis set (the Gaussians) which moves in response to a balance between the applied fields and the dissipative forces represented by the relaxation time $\tau$.

We choose to use a Gaussian packet because simulations of the equation of motion, whether from a discretized scheme [66] or from a particle Monte Carlo method [67], show that the Gaussian is observed to conserve its shape, just as the distribution is expected to do. We admit that this is only an observation. However, a rigorous proof has been given by Brodier and Ozorio de Almeida [5]. This latter proof holds for any quadratic Hamiltonian, such as our use of a constant, homogeneous electric field.

Appendix B

To begin, we write the integrand for the off-diagonal Wigner function as

$$\begin{aligned} I_{12} =\frac{1}{\pi \sigma ^{2}}\hbox {e}^{-\Lambda _x }e^{-\Lambda _y }q\quad . \end{aligned}$$

(B1)

We note the x-velocities are the same for both terms, but this is not the case for the y-components, as indicated in (6). For the x-direction, the argument of the exponential becomes

$$\begin{aligned} \Lambda _x= & {} \frac{\left( {x-v_{1x} t+\frac{{x}'}{2}} \right) ^{2}+\left( {x-v_{2x} t-\frac{{x}'}{2}} \right) ^{2}}{4\sigma ^{2}} \nonumber \\&\quad -ik_x {x}'+ik_{1x} \left( {x+\frac{{x}'}{2}} \right) -ik_{2x} \left( {x-\frac{{x}'}{2}} \right) . \nonumber \\= & {} \frac{(x-v_{1x} t)^{2}}{2\sigma ^{2}}+\frac{{x}'^{2}}{8\sigma ^{2}}+i\left( {k_x -k_{1x} } \right) {x}' \end{aligned}$$

(B2)

This result is the same as for the diagonal terms. The y-direction will be different, because of the oppositely directed velocities and wave numbers. The Gaussian parts are

$$\begin{aligned} \Lambda _y= & {} \frac{\left( {y-v_{1y} t+\frac{{y}'}{2}} \right) ^{2}+\left( {y-v_{2y} t-\frac{{y}'}{2}} \right) ^{2}}{4\sigma ^{2}} \nonumber \\&\quad -ik_y {y}'+ik_{1y} \left( {y+\frac{{y}'}{2}} \right) -ik_{2y} \left( {y-\frac{{y}'}{2}} \right) \nonumber \\= & {} \frac{\left( {y-v_{1y} t+\frac{{y}'}{2}} \right) ^{2}+\left( {y+v_{1y} t-\frac{{y}'}{2}} \right) ^{2}}{4\sigma ^{2}}\nonumber \\&\quad -ik_y {y}'+ik_{1y} \left( {y+\frac{{y}'}{2}} \right) +ik_{1y} \left( {y-\frac{{y}'}{2}} \right) \end{aligned}$$

(B3)

We can expand the various terms and arrive at the final form

$$\begin{aligned} \Lambda _y= & {} \frac{y^{2}}{2\sigma ^{2}}+2\sigma ^{2}k_y^2 +2ik_{1y} \left( {y-v_{1y} t} \right) \nonumber \\&\quad +\frac{\left( {{y}'-2v_{1y} t-4i\sigma ^{2}k_y } \right) ^{2}}{8\sigma ^{2}} \end{aligned}$$

(B4)

We have completed the square for the Gaussian integration over the primed coordinates, which is now slightly more complicated because of the extra shifts. The Wigner function for the off-diagonal terms now becomes (11).