Transmission and conductance across junctions of isotropic and anisotropic three-dimensional semimetals

Abstract

We compute the transmission coefficients and zero-temperature conductance for chiral quasiparticles propagating through various geometries, which consist of junctions of three-dimensional nodal point semimetals. In the first scenario, we consider a potential step with two Rarita–Schwinger–Weyl or two birefringent semimetals, which are tilted with respect to the other on the two sides of the junction. The second set-up consists of a junction between a doped Dirac semimetal and a ferromagnetic Weyl semimetal, where an intrinsic magnetization present in the latter splits the doubly-degenerate Dirac node into a pair of Weyl nodes. A scalar potential is also applied in the region where the Weyl semimetal phase exists. Finally, we study sandwiches of Weyl/multi-Weyl semimetals, with the middle region being subjected to both scalar and vector potentials. Our results show that a nonzero transmission spectrum exists where the areas, enclosed by the Fermi surface projections (in the plane perpendicular to the propagation axis) of the incidence and transmission regions, overlap. Such features can help engineer unidirectional carrier propagation, topologically protected against impurity backscattering, because of the chiral nature of the charge carriers.

Boundary conditions and continuity of probability flux

Let us consider the low-energy expansion of the Hamiltonian around a nodal point, which can be described as

$$\begin{aligned} \mathcal {H} ({\textbf{k}} ) = \textbf{d} ({\textbf{k}}) \cdot \varvec{\mathcal {S}}\,, \end{aligned}$$

(1)

where $\textbf{d} ({\textbf{k}}) =\lbrace d_x({\textbf{k}}), \, d_y({\textbf{k}}), \, d_z({\textbf{k}}_z) \rbrace$ represents three functions of the momenta, and $\varvec{\mathcal {S}}$ is a three-vector, made up of the matrices $S_j$ obeying the algebra $\left[ S_j, \,S_k \right] =i\,\epsilon _{jkl} \,S_l$ [with $j, k, l \in \lbrace x,\, y,\, z \rbrace$], that represents the pseudospin degrees of freedom. Here, we have dealt with the following cases:

• (1) For the RSW semimetal, $\varvec{\mathcal {S}}$ is given by the $4\times 4$ pseudospin-3/2 matrices ${\textbf{J}} =\lbrace J_x, \, J_y,\, J_y\rbrace$ shown in Eq. (2.2).

• (2) For the pseudospin-1/2 Weyl/multi-Weyl quasiparticles, $\varvec{\mathcal {S}} = \varvec{\sigma }$, where $\varvec{\sigma } = \lbrace \sigma _x, \, \sigma _y,\, \sigma _y \rbrace$ represents the three $2\times 2$ Pauli matrices. The above form for $\varvec{\mathcal {S}}$ is clear when we note that for the multi-Weyl case, the Hamiltonian around a single node can be written as

  $$\begin{aligned} \mathcal {H} ( {\textbf{k}}) = \frac{v_\perp \, k_\perp ^J }{k_0^{J-1 }} \left[ \cos (J \,\phi _k ) \, \sigma _x + \sin (J\, \phi _k) \, \sigma _y \right] + \chi \,v_z \, k_z \, \sigma _z , \end{aligned}$$

  where $\phi _k = \arctan (k_y,k_x)$ and $k_\perp = \sqrt{k_x^2 + k_y^2 }\,$.

• (3) For the birefringent and Dirac semimetals, we have $\varvec{\mathcal {S}}\propto \varvec{\sigma } \otimes \varvec{\sigma }$, which appears as a tensor product of the Pauli matrices in two different pseudospin-1/2 spaces.

Now for the cases considered in this manuscript, we have $d_z ({\textbf{k}}) = v_z \, k_z$ (where $v_z$ is the Fermi velocity along the z-direction), and the quasiparticles are moving along the z-axis when they encounter junctions (which are aligned perpendicular to the z-axis). In general, the potential V(z) varies across the junction, which we take to be a finite value on either side. To capture this situation, we need to consider the Hamiltonian with the z-part written in position space:

$$\begin{aligned} \tilde{\mathcal {H}}&= \mathcal {H} ( k_x, k_y, k_z \rightarrow -i \,\partial _z ) + V(z) \,,\quad V(z) = {\left\{ \begin{array}{ll} V_1 &{} \text { for } z \le 0 \\ V_2 &{} \text { for } z > 0 \end{array}\right. }\,. \end{aligned}$$

(2)

Then the eigenvalue equation is $\left( \tilde{\mathcal {H}} -E \right) \tilde{\Psi }_E (k_x, k_y, z) = 0$, where $\tilde{\Psi }_E$ is an eigenfunction of $\tilde{\mathcal {H}}$ with eigenvalue E. It is a first order differential equation in z and we intend to find solutions which are finite, continuous, and differentiable over the entire interval $z \in (-\infty , \infty )$ [64]. In other words, all solutions of a differential equation must be differentiable implying that continuity is a requirement for differentiability. Compare this when the Hamiltonian has a second derivative (e.g., describing a Schrodinger particle) and the differential equation reduces to the problem of the Sturm–Liouville type [64]. For a junction located at $z=0$, we integrate the above equation over an infinitesimal interval $2\,\epsilon$ about $z =0$, and subsequently take the limit $\epsilon \rightarrow 0$ as follows:

$$\begin{aligned}&\lim _{\epsilon \rightarrow 0} \int _{-\epsilon }^\epsilon dz \left[ d_x \,{\mathcal {S}}_x + d_y \,{\mathcal {S}}_y + v_{z} \,{\mathcal {S}}_z\, \partial _{z} + V(z) - E \right] \tilde{\Psi }_E (k_x,k_y, z) = 0 \,, \end{aligned}$$

(3)

This leads to the boundary conditions

$$\begin{aligned} v_{z_1} \, \tilde{\Psi }_E^{(1)} (k_x,k_y, z) \big \vert _{z = 0^{-}} = v_{z_2} \, \tilde{\Psi }_E^{(2)} (k_x,k_y, z) \big \vert _{z = 0^{+}} \,, \end{aligned}$$

(4)

where the subscripts and the superscripts $a \in \lbrace 1, \, 2 \rbrace$ refer to the materials on the left and right of the junction, respectively. If we have a 2N-component wavefunction, we get 2N equations. Note that unlike the Schrödinger equation case, we do not have any more boundary conditions. In particular, we do not have a condition for the continuity of the derivative of the wavefunction at the boundary (and this scenario is similar to the case of the relativistic Dirac equation case). This is a fundamental equation and, naturally, holds for all cases discussed in the paper.

Let us assume that $\left[ \tilde{{\mathcal {H}}}^{(a)} - V(z) \right]$ represents a 2N-band system with the energy eigenvalues given by

$$\begin{aligned} \varepsilon _{a,n}^s (k_x, k_y, k_{z_a}) = s\,c_{a,n} \, \sqrt{ \left( d_x^{(a)} \right) ^2 + \left( d_y^{(a)} \right) ^2 + v_{z_a}^2 \, k_{z_a}^2 } \text { for } n \in [1, N] \text { and } s=\pm \,, \end{aligned}$$

(5)

where the set $\lbrace c_{a,n} \rbrace$ consists of N real numbers for each value of a. Let $\lbrace \psi _{a,n}^s (k_x, k_y, k_{z_a} )\rbrace$ represent the set of orthonormal wavefunctions associated with $\lbrace \varepsilon _{a,n}^s \rbrace$. We also have 2N solutions for the momentum along the z-axis, for a given set of values for the perpendicular momenta $\lbrace k_x, \,k_y \rbrace$, expressed by $\pm k_{z_a, n}$, where

$$\begin{aligned} k_{z_a, n} = \frac{1}{v_{z_a}} \, \sqrt{ \left( \frac{ E - V_a }{c_{a,n} } \right) ^2 - \left( d_x^{(a)} \right) ^2-\left( d_z^{(a)} \right) ^2 } \,. \end{aligned}$$

(6)

We assume that $E > V_1$ is the energy of an incident quasiparticle propagating toward the positive z-axis.

Using the above notations, we take $\Psi _E^{(1)}$ and $\Psi _E^{(2)}$ to be linear combinations of plane wave solutions propagating along the z-axis. Suppose the quasiparticles are incident from the left of the junction with the plane wave momentum $k_{z_1, n_0}$. Then we have

$$\begin{aligned} \tilde{\Psi }_E^{(1)} (k_x, k_y, z)&= \psi _{a,n_0}^+ (k_x, k_y, k_{z_1, n_0}) \, e^{i\,k_{z_1, n_0} \, z} + \sum _{n=1}^{N} r_n \, {\psi _{a,n}^+ (k_x, k_y, -k_{z_1, n}) \, e^{ - i\,k_{z_1, n} \, z}} \,, \nonumber \\ \tilde{\Psi }_E^{(2)} (k_x, k_y, z)&= \sum _{n=1}^{N} t_n \, {\psi _{a,n}^+ (k_x, k_y, k_{z_2, n}) \, e^{ i\,k_{z_2 , n} \, z}} \left[ 1 - \Theta \big ( |\textrm{Im} \,k_{z_2 , n} | \big )\right] . \end{aligned}$$

(7)

The 2N unknown coefficients $\lbrace r_n, t_n \rbrace$ can be determined by the 2N equations obtained from Eq. (4). Note that a $t_n$ is defined only for a propagating wave, i.e., when $k_{z_2, n}$ is real.

The carrier current for each of systems is then along the z-axis, and is captured by the probability flux operator $\hat{j}_z= \partial _{k_z} \mathcal {H} ({\textbf{k}}) = v_z \, {\mathcal {S}}_z.$ The expectation value of the probability flux along the z-axis is then given by [32]

$$\begin{aligned} \langle j_z \rangle = v_z \,\tilde{\Psi }_E^\dagger \, {\mathcal {S}}_z \,\tilde{\Psi }_E \, . \end{aligned}$$

(8)

Due to the time-independence of the problem, the continuity of the probability flux reduces to $\partial _z \langle j_z \rangle = 0$. This condition translates into the statement that

$$\begin{aligned}\langle j_z^{(1)} \rangle \Big \vert _{z<0 } &= \langle j_z^{(2)} \rangle \Big \vert _{z> 0} \Rightarrow v_{z_1} \langle \tilde{\Psi }_E^{(1)} (k_x,k_y, z)| \, {\mathcal {S}}_z \, | \tilde{\Psi }_E^{(1)} (k_x,k_y, z) \rangle ^2 \Big \vert _{z < 0}\nonumber \\& = v_{z_2} \langle \tilde{\Psi }_E^{(1)} (k_x,k_y, z)| \, {\mathcal {S}}_z \, | \tilde{\Psi }_E^{(1)} (k_x,k_y, z) \rangle ^2 \Big \vert _{z > 0}. \end{aligned}$$

(9)

Noting that

$$\begin{aligned} v_{z_a} \left[ \psi _{a,n}^s (k_x, k_y, k_{z_a}) \, e^{i \, k_{z_a} \, z}\right] ^\dagger \, {\mathcal {S}}_z \left[ \psi _{a,n}^s (k_x, k_y, k_{z_a}) \, e^{i \, k_{z_a} \, z}\right] = \frac{ v_{z_a} ^2 k_{z_a} }{ \varepsilon _{n, a}^s } \equiv \partial _{k_{z_a}} \varepsilon _{a,n}^s \,, \end{aligned}$$

(10)

Eq. (9) reduces to

$$\begin{aligned}&\partial _{ k_{z_1}} \varepsilon _{1,n_0}^+ \Big \vert _{k_{z_1} = k_{z_1, n_0}} - \sum _n |r_n|^2 \, \partial _{ k_{z_1}} \varepsilon _{1,n}^+ \Big \vert _{k_{z_1} = k_{z_1, n}} = \sum _n |t_n|^2 \, \partial _{ k_{z_2}} \varepsilon _{2,n}^+ \Big \vert _{k_{z_2} = k_{z_2, n}} \nonumber \\ \Rightarrow&\quad 1- {{\mathcal {R}}} = {{\mathcal {T}}} \,, \end{aligned}$$

(11)

where

$$\begin{aligned}&{{\mathcal {R}}} = \sum _n \left( \frac{\partial _{ k_{z_1}} \varepsilon _{1,n}^+ \Big \vert _{k_{z_1} = k_{z_1, n}}}{\partial _{ k_{z_1}} \varepsilon _{1,n_0}^+ \Big \vert _{k_{z_1} = k_{z_1, n_0}}} \right) |r_n|^2 \text { and } {{\mathcal {T}}} = \sum _n \left( \frac{ \partial _{ k_{z_2}} \varepsilon _{2,n}^+ \Big \vert _{k_{z_2} = k_{z_2, n}}}{\partial _{ k_{z_1}} \varepsilon _{1,n_0}^+ \Big \vert _{k_{z_1} = k_{z_1, n_0}}} \right) | t_n|^2 \end{aligned}$$

(12)

are the reflection and transmission probabilities, respectively. This is how the continuity of the probability flux comes into play.