Klein tunneling in quantum geometric semimetals

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.

💡 Research Summary

This paper presents a comprehensive theoretical study of Klein tunneling in two‑dimensional semimetals that possess a quadratic band‑touching (QBT) point. While Klein tunneling has been extensively explored in monolayer graphene (massless Dirac fermions) and bilayer graphene (massive chiral fermions), QBT materials form a broader family that can exhibit intrinsic effective‑mass asymmetry and a tunable quantum‑geometric structure of the Bloch wavefunctions. The authors develop a generic isotropic QBT Hamiltonian that contains three independent parameters: the conduction‑band mass m_u, the valence‑band mass m_l, and an inter‑band coupling d. The latter is directly related to the Hilbert–Schmidt quantum distance d_HS between eigenstates at different momenta; its maximum value d_max (0 ≤ d_max ≤ 1) quantifies the “quantum geometry” of the two‑band system. Importantly, d_max does not appear in the energy dispersion ε(k)=ℏ²k²/2m_{u,l}, so it represents a purely geometric degree of freedom that can be varied independently of the band curvature.

The transport problem is set up with a rectangular potential barrier of height u₀ and width a aligned along the x‑direction. Translational invariance in y guarantees conservation of k_y, and the spinor wavefunction in each region is expressed as a superposition of propagating and evanescent modes. By imposing continuity of the full spinor at the two interfaces, the transmission probability T=|a₃/a₁|² is obtained. The authors also derive an exact analytical expression for a δ‑function barrier, which isolates the influence of d_max on the transmission amplitude.

A central focus of the work is the role of effective‑mass asymmetry, quantified by the dimensionless parameter α ≡ (1/m_u + 1/m_l)/4. Three regimes emerge:

1. Symmetric case (α = 0) – This reproduces the bilayer‑graphene model with d_max = 1. The pseudospin winding leads to anti‑Klein tunneling at normal incidence (T = 0) and perfect‑transmission resonances at specific oblique angles.
2. Asymmetric Klein‑tunneling regime (0 < |α| < ¼) – The conduction and valence bands have opposite curvature but unequal magnitudes. Snell’s law, k₁ sin φ = k₂ sin θ, governs the refraction of the quasiparticles. As |α| approaches ¼, the critical angle φ_c shrinks, narrowing the angular window for transmission. Moreover, Fabry‑Perot (FP) resonances, which require constructive interference inside the barrier (k_{2x} a = π n), become inaccessible because the required incidence angle exceeds the physical range.
3. Blocked regime (|α| ≥ ¼) – Both effective masses acquire the same sign, destroying the electron–hole conversion essential for Klein tunneling; transmission is essentially suppressed.

The authors map the maximum transmission T_max (the peak of T(φ) over all incidence angles) as a function of (α, u₀) for fixed energy and barrier width. The color map reveals sharp boundaries where T_max drops to zero, confirming the optical‑analogy explanation. Increasing the barrier width a adds more FP resonance conditions, expanding the region where T_max = 1, as shown in supplemental calculations.

Quantum geometry, encoded in d_max, plays a distinct but equally crucial role. A larger d_max aligns the pseudospins of the incident electron and the hole‑like state inside the barrier, reducing the pseudospin mismatch and enhancing the overall transmission amplitude. Conversely, reducing d_max weakens this alignment, leading to a universal suppression of T even when the FP phase condition is satisfied. The analytical δ‑function result explicitly displays the factor (1 − d_max²) multiplying the transmission amplitude, confirming that d_max acts as a “geometric transmission coefficient” independent of the dispersion.

The interplay between α and d_max is most striking in the shift of FP resonance angles. While the FP condition k_{2x} a = π n determines a set of candidate angles, the actual peaks of T(φ) are displaced by an amount proportional to the quantum‑geometric phase accumulated at the interfaces. This displacement is observable in the angular‑resolved transmission plots and provides a direct experimental signature of quantum‑geometric effects.

In summary, the paper establishes two complementary mechanisms governing Klein tunneling in QBT semimetals:

• Effective‑mass asymmetry (α) controls the shape of the transmission profile—how many resonances appear, where they are located, and whether a transmission window exists at all.
• Quantum‑geometric distance (d_max) controls the overall amplitude of transmission by modulating pseudospin overlap, and it fine‑tunes the resonance angles through interface phase shifts.

Both parameters must be considered for a complete description of relativistic quantum transport in quadratic‑band systems. The findings suggest that engineering d_max—through strain, layer stacking, or symmetry‑breaking perturbations—offers a practical route to tailor carrier transmission, opening avenues for quantum‑geometric design of electronic devices such as angle‑selective filters, resonant tunneling transistors, and low‑dissipation interconnects in emerging two‑dimensional materials.