Klein tunneling in quantum geometric semimetals

TL;DR

This work analyzes Klein tunneling in quadratic band-touching semimetals, revealing a cooperative interplay between intrinsic mass asymmetry and quantum geometry. By employing a generic isotropic QBT model and a delta-function barrier limit, it demonstrates that mass asymmetry sets the angular transport landscape while quantum geometry, captured by the maximum Hilbert–Schmidt distance $d_{max}$, universally tunes transmission through pseudospin mismatch and shifts in Fabry–Pérot resonances. A delta-barrier solution provides an exact, geometry-only channel for transmission, establishing a clear route for band-geometry engineering to control conductance. The results highlight the importance of including geometric degrees of freedom alongside band structure to describe quantum transport, with implications for angle-selective transport and ballistic electron optics in QBT materials.

Abstract

Klein tunneling stands as a fundamental probe of relativistic quantum transport in two-dimensional materials. We investigate this phenomenon in quadratic band-touching systems, where the Hilbert-Schmidt quantum distance plays a central role in the underlying mechanism. By employing a generic parabolic model, we systematically disentangle the cooperative effects of intrinsic mass asymmetry and tunable quantum geometry. We demonstrate that mass asymmetry sets the overall transmission profile, including the angular distribution and the resonance channels. In contrast, we show that quantum geometry provides a universal parameter that modulates tunneling efficiency by tuning the quantum distance, while leaving the energy dispersion unchanged. Specifically, quantum geometry plays a dual role: it governs the overall transmission amplitude through pseudospin mismatch, while its interplay with Fabry-Perot interference induces observable shifts in resonance angles. Our findings reveal that incorporating quantum geometry alongside band structure is essential for a complete description of quantum transport.

Figures (5)

• Figure 1: Schematic illustration of quantum tunneling and pseudospin textures in a QBT system. Pseudospin orientation of propagating quasiparticle states is plotted for three distinct regions: incident ($x<0$), barrier ($0<x<a$), and transmitted ($a<x$). The red arrows indicate the direction of the pseudospin. The barrier has height $u_0$ and width $a$; $E$ is the incoming particle energy. The pseudospin of the upper band rotates in the opposite direction compared to that of the lower band. (a) Mass asymmetric system with $d_{\mathrm{max}}=1$. (b) Mass symmetric system with $d_{\mathrm{max}}=0.5$. (c) Mass asymmetric system with $d_{\mathrm{max}}=0$. The insets in (a-c) plot the pseudospin orientation as a function of the incidence angle for each corresponding $d_{\mathrm{max}}$ value. (d) Enlarged plot of the pseudospin orientation as a function of incidence angle for $d_{\mathrm{max}}=1, 0.5, \text{and } 0$. Red and blue arrows indicate the pseudospin for the upper and lower bands, respectively. For $d_{\mathrm{max}}=1$, the pseudospins of both bands are aligned (overlap), so only the red arrow is shown.
• Figure 2: (a) QBT band structures for different values of $M\alpha$, where $M\alpha$ is the mass asymmetry, which is product of the mass asymmetry parameter $\alpha$ and the relative effective mass $M$: (a1) $M \alpha = 0$, (a2) $0 < M \alpha < 1/4$, (a3) $1/4 \leq M \alpha$, (a4) $-1/4 < M \alpha < 0$, (a5) $M \alpha \leq -1/4$. The dashed line represents the zero-energy line ($\varepsilon=0$). (b) Contour plot of the transmission probability $T_{\mathrm{max}}$ (representing the maximum value of $T$ across all incidence angles) as a function of mass asymmetry parameter $M \alpha$ and the barrier height $M u_0$. Blue and red dots indicate examples of unit transmission ($T_{\mathrm{max}}=1$) points shown in Fig. 2(c) and (d). Here, [unit] = $[m_\mathrm{BLG}/2 \cdot \mathrm{meV}]$ with the effective mass of bilayer graphene $m_{\mathrm{BLG}} =2.0122 \times 10^{-10}(\text{meVs}^2/\text{m}^2)$. (c, d) Transmission probability $T$ as a function of incidence angle $\phi$ for (c) $(M \alpha,Mu_0)= (0.102,30~[\text{unit}])$ and (d) $(M \alpha,Mu_0)= (0.035,50~[\text{unit}])$, corresponding to the blue and red dots in Fig. 2(b), respectively. These results are calculated for $ME = 17~[\text{unit}]$ and $a = 100~ \text{nm}$.
• Figure 3: (a) Transmission probability $T$ as a function of incidence angle $\phi$ for $d_\mathrm{max} = 1$ (blue), $0.9$ (red), and $0.8$ (green), respectively. The angles $\phi_1$ and $\phi_2$ indicate the incidence directions at which perfect transmission occurs for $d_\mathrm{max} = 1$. (b) Transmission probability $T$ at two representative incidence angles, $\phi_1$ (red) and $\phi_2$ (black). As $d_\mathrm{max}$ decreases, the transmission is suppressed in both cases, but with distinct decay profiles, indicating angle-dependent sensitivity to the underlying quantum geometry. (c) Transmission probability $T$ as a function of $d_{\mathrm{max}}$ at two incidence angles near the representative perfect transmission angles: $\phi_1 + \epsilon_1$ (red) and $\phi_2 - \epsilon_2$ (black). Here, $\epsilon_1 = 0.03 \phi_1$ and $\epsilon_2 = 0.2 \phi_2$. These results are calculated for $M\alpha=0$, $Mu_0=50~[\text{unit}]$, $ME = 17~[\text{unit}]$ and $a = 100~ \text{nm}$.
• Figure 4: Transmission probability $T$ as a function of incidence angle $\phi$ for $d_{\mathrm{max}} = 1$ (blue), $0.9$ (red), and $0.8$ (green), respectively. (a) $(M \alpha,Mu_0)= (0.102,30~[\text{unit}])$ and (b) $(M \alpha,Mu_0)= (0.035,50~[\text{unit}])$, corresponding to Figs. 2(c) and (d), respectively. These results are calculated for $ME = 17~[\text{unit}]$ and $a = 100~ \text{nm}$. Here, $\phi_1$ and $\phi_2$ correspond to the transmission peak angles for $d_{\max}=1$ and $d_{\max}=0.8$, respectively.
• Figure C1: (a)[b] Contour plot of the transmission probability $T_{\mathrm{max}}$ as a function of the barrier height $M u_0$(barrier width $a$) and the barrier width $a$(mass asymmetry parameter $M \alpha$).