Solid-state analog of gravitational redshift: Transport signatures of massless Dirac fermions in tilted Dirac cone heterostructures

Abstract

A Dirac electron passing through a heterojunction with spatially variable tilt experiences an effective curved spacetime. In this work, we show how this experience is manifested in its conductance. We investigate the propagation of electron waves in a two-dimensional tilted Dirac cone heterostructure where tilt depends on the coordinate z along the junction. The resulting Dirac equation in an emergent curved spacetime for the spinor ψ(z) can be efficiently solved using a fourth-order Runge-Kutta numerical method by a transformation to a suitable spinor φ where the resulting Dirac cone looks locally upright. The spatial texture of the tilt induces oscillatory behaviors in key physical quantities such as the norm |φ(z)|2 of the wave function, polar and azimuthal angles Θ(z) and Φ(z) of the pseudospin, and the integrated transmission τ where oscillation wavelengths get shorter (longer) in stronger (weaker) tilt regions. Such an oscillatory behavior that is reminiscent of gigantic gravitational redshift is an indicator of an underlying spacetime metric that can be probed in tunneling experiments. We derive analytical approximations for the position-depdendent wave numbers Δkz(z) that explain the redshift patterns and corroborate it with numerical simulations. For a tilt bump spread over length scale ℓ, upon increasing ℓ, the amplitude of redshifted oscillations reduces whereas the number of peaks increases. The scale invariance of the Dirac equation allows us to probe these aspects of ℓ dependence by a voltage sweep in transmission experiments. Smooth variations of the tilt reduces impedance mismatch of the electron waves, thereby giving rise to very high transmission rates. This concept can be used in combination with a sigmoid-shaped tilt texture for a gigantic redshift or blueshift engineering of the transmitted waves, depending on whether the sigmoid is downswing or upswing. © 2025 authors. Published by the American Physical Society.