Non-Hermitian thermoelectric transport in graphene: Tunable anomalous transmission through complex barriers

Technical Summary: Non-Hermitian Thermoelectric Transport in Graphene

Problem Statement
This work investigates thermoelectric transport in monolayer graphene across a finite complex barrier, modeled by a spatially localized potential $V(x) = U + iW$. While graphene is a paradigmatic platform for studying quantum transport via the massless Dirac-Weyl equation, standard treatments assume Hermitian Hamiltonians where probability is conserved. However, in open quantum systems, effective Hamiltonians often become non-Hermitian due to coupling with reservoirs, gain/loss mechanisms, or unresolved environmental degrees of freedom. The paper addresses the lack of a systematic analysis regarding how a rectangular complex barrier modifies the scattering matrix, probability balance, and Landauer conductance in graphene. Specifically, it seeks to determine how the imaginary part of the potential ($W$) alters the familiar Klein tunneling and Fabry-Pérot resonance phenomena, and how non-unitarity affects fundamental transport properties such as gauge invariance and thermoelectric coefficients.

Methodology
The authors employ a Landauer scattering framework combined with the exact solution of the Dirac-Weyl equation for a piecewise constant complex potential.

1. Scattering Formalism: The system is divided into three regions: two semi-infinite leads (regions I and III) and a finite barrier region (region II) of length $L$ with potential $U + iW$. The authors construct exact spinor scattering states for arbitrary incidence angles by matching boundary conditions at the interfaces.
2. Non-Unitary Scattering Matrix: They derive the full $2 \times 2$ scattering matrix ($S$) connecting incoming and outgoing amplitudes. Unlike the Hermitian case, the presence of $W \neq 0$ renders the $S$-matrix non-unitary.
3. Flux Balance: The continuity equation is modified to include a source/sink term proportional to $W$. This leads to a generalized flux-balance relation where the sum of transmission ($T$) and reflection ($R$) probabilities equals $1 + \Gamma$, where $\Gamma$ is a gain/loss coefficient determined by the net flux generated or absorbed within the barrier.
4. Thermoelectric Transport: The authors formulate a linear-response theory for charge and heat currents. Crucially, they account for the fact that in non-Hermitian systems, currents in the left and right leads are not necessarily equal and depend on how the voltage bias is partitioned between the leads. They derive explicit expressions for electrical conductance ($G$), thermal conductance ($\kappa$), Seebeck coefficient ($S$), and the figure of merit ($ZT$), treating them as linear-response slopes around a reference state that may include zero-bias offsets.

Key Contributions and Results

• Generalized Flux Balance: The study demonstrates that the imaginary part of the barrier replaces the standard unitary condition ($T+R=1$) with $T+R = 1 + \Gamma$. For $W > 0$ (gain), the total outgoing flux can exceed unity; for $W < 0$ (loss), it is suppressed. The resonant channels (Fabry-Pérot type) are selectively amplified or attenuated depending on the sign of $W$, significantly modifying the angular response.
• Breakdown of Gauge Invariance: A central finding is that the lead-resolved conductances ($G_L$ and $G_R$) become dependent on the bias partition parameter ($\eta$). In the Hermitian limit, conductance is independent of how the voltage drop is distributed. In the non-Hermitian case, $G_L \neq G_R$ unless the partition is symmetric ($\eta = 1/2$). This provides a direct signature of the breakdown of gauge invariance in the effective two-terminal response, attributed to the barrier acting as an effective source or sink of flux.
• Zero-Temperature Conductance: At $T=0$, the conductance exhibits sharp resonant peaks for $W > 0$ due to the amplification of specific angular channels, whereas $W < 0$ smooths the response. The conductance curves show a strong dependence on the bias asymmetry parameter, a feature absent in standard Hermitian graphene junctions.
• Thermoelectric Trade-off: At finite temperature, the imaginary part of the barrier acts as a tunable control parameter.
  • Gain ($W > 0$): Enhances both electrical and thermal conductances and increases the magnitude of the Seebeck coefficient. However, the increase in thermal conductance is disproportionately large, leading to a lower thermoelectric figure of merit ($ZT$).
  • Loss ($W < 0$): Suppresses thermal conductance more efficiently than electrical conductance. This results in the highest $ZT$ within the studied parameter range, despite lower overall throughput.
  • The authors note that the usual Onsager reciprocal relations ($L_{12} = -L_{21}$) do not hold generally in this non-Hermitian setting due to the presence of zero-bias currents.

Significance and Claims
The paper claims that complex barriers extend the range of transport behaviors accessible in graphene beyond the constraints of the usual Hermitian $n$-$p$-$n$ junction. By tuning the imaginary part of the potential, one can continuously move between a high-conductance regime (gain) and a high-efficiency thermoelectric regime (loss) without altering the real barrier geometry.

The authors position the complex potential not merely as a mathematical curiosity but as a practical effective description for unresolved source-sink channels or additional probes coupled to the device, particularly when a full microscopic model of the environment is unavailable. The breakdown of gauge invariance and the dependence of conductance on bias partition are identified as direct experimental signatures of such non-Hermitian effects.

The study is modest in its scope, focusing on a minimal continuum model that allows for full analytical and numerical tractability. The authors explicitly state that their thermoelectric analysis includes only the electronic contribution to transport; the lattice (phonon) thermal conductance, which is significant in graphene, is not included. Therefore, the calculated $ZT$ values represent the electronic part of the response rather than the complete device performance. The work serves as a proof of principle that non-Hermitian physics can be integrated into standard Landauer transport theory to model and control transport features in graphene.