Anisotropic Andreev reflection in semi-Dirac materials

We now turn to the scattering problem in a NS junction extending along the $x$-axis. In practice, at the boundary between the N and S regions, the defects and lattice mismatch are inevitable during the fabrication of device, which may profoundly influence the transport properties. To take into account the proposed effects, we introduce a interfacial barrier $U(x)\equiv U_{0}\Theta(x)\Theta(d-x)$, and take the limit of $U_{0}\rightarrow\infty$ and $d\rightarrow 0$ with $U_{0}d/(\hbar v)\equiv Z$ being finite. We note that in the current direction, the low-energy excitations in SDMs are similar as that in Dirac materials, thus the scattering amplitudes can be calculated by the standard procedure employed in related Dirac-material-based NS junctions Beenakker (2006, 2008); Bhattacharjee and Sengupta (2006); Ludwig (2007); Linder and Yokoyama (2014); Li (2016); Efetov and Efetov (2016). By solving

at the boundary of $x=0$, we can obtain $r_{he(ee)}$ and $t_{eq(hq)}$, which are the reflection amplitude for the AR (normal reflection) and the transmission amplitude for the electron(hole)-like quasiparticle, respectively. The details of basis scattering states $\psi^{+}_{eq(hq)}$ and $\psi^{\pm}_{e,h}$ are, respectively, given by Eqs. (26b) and  (28b). The transfer matrix ${\cal M}=e^{i\sigma_{x}\tau_{0}Z}$, with $\tau_{0}$ a $2\times 2$ identity matrix operating on the Nambu space.

In the limit regime of $\mu_{S}\gg\max(\mu_{N},E,\Delta_{0})$, the reflection amplitudes are, respectively, given by

where $k^{\pm}_{e(h)}=k_{e(h)}\pm i\eta k^{2}_{y}/(\hbar v)$, $k_{e(h)}={\mathrm{sgn}[\varepsilon_{+(-)}]}\sqrt{\varepsilon_{+(-)}^{2}-\eta^{2}k^{4}_{y}}/(\hbar v)$, and $\beta=\cos^{-1}(E/\Delta_{0})\Theta(\Delta_{0}-E)-i\cosh^{-1}(E/\Delta_{0})\Theta(E-\Delta_{0})$. As can be seen, both $r_{ee}$ and $r_{he}$ are independent of $Z$, implying that in the limit of $\mu_{S}\gg\max(\mu_{N},E,\Delta_{0})$, the transport properties along the $x$-axis direction are insensitive to the interfacial barrier.

Resorting to the reflection amplitudes, the probabilities for the normal reflection (NR) and AR processes can be, respectively, defined as

where the particle current density operator ${\cal J}_{x}\equiv\frac{-i}{\hbar}[x,H_{\mathrm{BdG}}]=v\sigma_{x}\tau_{z}$, with $\tau_{z}$ the Pauli matrix operating on the Nambu space.

Taking advantage of the Blonder-Tinkham-Klapwijk (BTK) formula Blonder et al. (1982), the zero-temperature differential conductance along the $x$-direction is given by

where $V$ indicates the bias voltage and $W$ is the width along the $y$-direction of the junction. To normalize the conductance, we define $G^{x}_{0}(eV)=\frac{2e^{2}W}{\pi h}\sqrt{|eV+\mu_{N}|/\eta}$, which is the conductance along the $x$-direction of a SDM-based NN junction in the ballistic limit.

In a NS junction along the $y$-axis, the low-energy excitations of the N region disperse quadratically in the current direction. For electron(hole)-like excitations with $|k_{x}|\leq|E+(-)\mu_{N}|/(\hbar v)$, Eq. (3) determines four possible scattering modes, including two propagating modes with $k_{y}=\pm q_{e(h)}$ and two evanescent ones with $k_{y}=\pm i\kappa_{e(h)}$. The related parameters are given by

with $s_{e(h)}={\mathrm{sgn}}[\varepsilon_{+(-)}]$. Although the evanescent modes do not contribute to the current, they need to be included in the wave functions to get correct boundary conditions. With this in mind, for a propagating electron-like incident mode, the wave function in the N region should be parameterized as

with $\tilde{r}_{ee(he),1}$ and $\tilde{r}_{ee(he),2}$ the reflection amplitudes of propagating and evanescent modes in the NR(AR) processes, respectively. The detailed structures of related basis scattering states are given by Eq. (30d). In the S region, the wave function takes the form of

where the associated basis scattering states are shown in Eq. (31b). The coefficients $\tilde{t}_{eq(hq),1}$ and $\tilde{t}_{eq(hq),2}$, respectively, represent the transmission amplitudes for the propagating and evanescent electron(hole)-like quasiparticles.

To model the effects resulting from the interfacial defects and lattice mismatch, we introduce a interfacial barrier modeled by $\tilde{U}(y)\equiv\tilde{U}_{0}\delta(y)$, with $\delta(y)$ being the Delta function. In doing so, the boundary conditions can be formulated as

where ${\tilde{\cal M}}=\partial_{y}-\frac{\tilde{U}_{0}}{\eta}\sigma_{y}\tau_{0}$. For brevity of notation, we introduce a parameter $\chi\equiv\frac{\tilde{U}_{0}}{\eta q^{F}_{S}}$ to characterize the interfacial barrier strength, with $q^{F}_{S}=\sqrt{\mu_{S}/\eta}$. The probabilities for the NR and AR processes are, respectively, defined as

with the particle current density operator ${\cal J}_{y}\equiv\frac{-i}{\hbar}[y,H_{\mathrm{BdG}}]=-\frac{2i\eta}{\hbar}\sigma_{y}\tau_{z}\partial y$. We note that the probabilities of evanescent modes $\tilde{R}_{ee,2}$ and $\tilde{R}_{he,2}$ are vanishing and do not contribute to the current. Therefore, in accordance with the BTK formula Blonder et al. (1982), the zero-temperature differential conductance along the $y$-direction can be written as

where $L$ is the width along the $x$-direction of the junction. It is convenient to normalize the conductance by $G^{y}_{0}(eV)=\frac{2e^{2}L}{\pi h}\frac{|eV+\mu_{N}|}{\hbar v}$, which represents the conductance in the $y$-direction of a SDM-based NN junction in the ballistic limit.

To understand the underlying physics behind the transport properties, it is instructive to reveal the pseudo-spin textures in the N regions of SDMs-based NS junctions. To do so, we define the pseudo-spin vector as $\textbf{\emph{P}}\equiv\langle\bm{\sigma}\otimes\tau_{0}\rangle$, which is a unit vector consisting of the expectation values of operator $\bm{\sigma}\otimes\tau_{0}$ in normalized basis scattering states, where $\bm{\sigma}\equiv(\sigma_{x},\sigma_{y})$ San-Jose et al. (2009); Schomerus (2010); Majidi and Zareyan (2011, 2011); Habe (2019). Specifically, the pseudo-spin carried by a propagating electron-like mode can be formulated as

Accordingly, the pseudo-spin components $P_{e,x}$ and $P_{e,y}$ depend, respectively, linearly and quadratically on the momentum, inheriting the intrinsic anisotropy of SDMs. Fig. 2 gives the schematic plots for the pseudo-spin textures of electron-like conduction band. As can be seen, $P_{e,y}$ is always along the positive $y$-direction, regardless of the sign of $k_{y}$, this scenario is profoundly different from that in Dirac materials San-Jose et al. (2009); Schomerus (2010); Majidi and Zareyan (2011, 2011); Habe (2019); Breunig et al. (2018); Uchida et al. (2014); Bai and Yang (2020). As will be proposed in Sec. III.2, the unique pseudo-spin textures provide illuminating elucidations for the subgap transport properties in SDMs-based NS junctions.

In this subsection, we proceed to analyze the numerical results and discuss the differential conductance in the subgap regime. What we mainly concentrate on are the manifestations of the intrinsic anisotropy in the subgap transport properties, implemented by comparing the AR configurations and subgap differential conductance along the $x$- and $y$-axes. To ensure the validity of the mean-field approximation, we assume the S regions are heavily doped to the regime of $\mu_{S}\gg\Delta_{0}$, and we set $\mu_{S}=500\Delta_{0}$ in the numerical calculation for definiteness.

As a starting point, we focus on the bias-voltage-dependent subgap differential conductance, and analyze the results in terms of the unique pseudo-spin textures. In a NS junction extending along the $x$-axis, the zero-bias conductance is insensitive to the momentum-mismatch (or alternatively, the ratio of $\mu_{N}/\mu_{S}$). As shown in Fig. 3, the differential conductance always exhibits a zero-bias peak, regardless of the value of $\mu_{N}$. Remarkably, for any nonvanishing $\mu_{N}$, $G^{x}(0)/G^{x}_{0}(0)$ takes the same value of $\frac{8}{5}$. On the contrary, the $G^{y}(0)/G^{y}_{0}(0)$ strongly depends on the momentum-mismatch in the NS junction extending along the $y$-axis. As can be seen in Fig. 3, the subgap differential conductance almost vanishes in the presence of large momentum-mismatch (e.g., $\mu_{N}/\Delta_{0}=0,0.1,0.5,1$). Therefore, the behavior of zero-bias conductance is highly orientation-dependent. This scenario can be understood by virtue of the anisotropic pseudo-spin textures illustrated in Fig. 2. In the NS junction along the $x$-direction, for a small $k_{y}$ (i.e., with a small incident angle), the pseudo-spin carried by an electron-like incident mode is nearly opposite to that of the normally reflected one. Consequently, when $k_{y}$ is small enough, the conservation of pseudo-spin results in the prohibition of NR, and thus leads to the enhancement of AR in the subgap regime, even in the presence of strong momentum-mismatch. Moreover, since the conductance is mainly contributed by the modes with small $k_{y}$, the zero-bias conductance is insensitive to the momentum-mismatch. However, in the NS junction extending along the $y$-direction, a couple of incident and normally reflected modes possess the same pseudo-spin, so that the back scattering is enabled. Therefore, the subgap differential conductance in the $y$-direction is strongly suppressed by increasing the momentum-mismatch.

Parenthetically, although the configuration of $G^{x}(eV)/G^{x}_{0}(eV)$ shown in Fig. 3 is similar as that in associated NS junctions based on graphene Beenakker (2006) and silicene Linder and Yokoyama (2014); Li (2016), there is a significant difference on the value of $G^{x}(0)/G^{x}_{0}(0)$. For $eV=0$, according to Eqs. (6b) and (7b), the reflection probabilities reduces into $R_{ee}|_{eV=0}=\eta^{2}k^{4}_{y}/\mu^{2}_{N}$ and $R_{he}|_{eV=0}=(\mu^{2}_{N}-\eta^{2}k^{4}_{y})/\mu^{2}_{N}$, respectively. Substituting the results into Eq. (8), we arrive at $G^{x}(0)/G^{x}_{0}(0)=\sqrt{\frac{\eta}{\mu_{N}}}\int^{\sqrt{\mu_{N}/\eta}}_{0}(2-2\eta^{2}k^{4}_{y}/\mu_{N}^{2})dk_{y}=\frac{8}{5}$, differing from the value of $\frac{4}{3}$ in graphene- and silicene-based NS junctions Beenakker (2006); Linder and Yokoyama (2014); Li (2016). This consequence can be ascribed to the unique band structure of SDMs intermediate the linear and quadratic spectra.

The anisotropic aspect of the subgap differential conductance also manifests itself in the distinct crossover behaviors from the retro-AR to specular-AR in the two representative NS junctions. In the NS junction with a weakly doped N region satisfying $\mu_{N}<\Delta_{0}$, the subgap differential conductance vanishes at $eV=\mu_{N}$, due to the absence of AR for any incident angles. Furthermore, in the regimes of $eV<\mu_{N}$ and $eV>\mu_{N}$, the subgap differential conductance is dominated by the retro-AR and specular-AR, respectively. Therefore, the point of $eV=\mu_{N}$ serves as the crossover boundary between the retro-AR and specular-AR in the subgap regime. In the NS junction along the $x$-axis, as the bias-voltage increases, the subgap differential conductance exhibits a clear crossover from retro-AR to specular-AR, as depicted by Fig. 4. However, as illustrated in Fig. 4, the boundary between retro-AR and specular-AR is ambiguous in the NS junction extending along the $y$-axis. This scenario can be understood from the unique pseudo-spin textures. In the NS junction extending in the $y$-axis, the incident and normally reflected modes carry the same pseudo-spin, thus the NR is favorable, especially in the presence of large momentum-mismatch. Consequently, as illustrated in Fig. 3 and the inset therein, for $\mu_{N}\leq\Delta_{0}\ll\mu_{S}$, the AR process is strongly suppressed so that $G^{y}(eV)/G^{y}_{0}(eV)\sim G^{y}(\mu_{N})/G^{y}_{0}(\mu_{N})=0$ in the subgap regime, thus blurring the boundary between the retro-AR and specular-AR.

We now turn to the effects of the interfacial barriers on the subgap differential conductance. For the transport along the $x$-axis, the pseudo-spin carried by the incident and normally reflected modes are nearly opposite to each other when the incident angle is small enough, resulting in the prohibition of NR. As a consequence, the zero-bias differential conductance just oscillates with $Z$ without a decaying profile, as charted out by Fig. 5. In the case of the NS junction extending along the $y$-axis, the back scattering is available since the incident and normally reflected modes host the same pseudo-spin. Therefore, the AR is strongly suppressed by increasing the interfacial barrier strength, leading to the exponential decaying profile in the $\chi$-dependent $G^{y}(0)/G^{y}_{0}(0)$ shown in Fig. 5. In this regard, the influences of the interfacial barrier on the subgap transport are orientation-dependent, reflecting the anisotropic aspect of SDMs.