[Paper Review] Quasi-Relativistic Doppler Effect and Non-Reciprocal Plasmons in Graphene

Dan S. Borgnia, Trung V. Phan+1 more|arXiv (Cornell University)|Dec 30, 2015

Plasmonic and Surface Plasmon Research18 references22 citations

TL;DR

This paper proposes a nonmagnetic, linear mechanism for strong nonreciprocity in graphene plasmons using the quasi-relativistic Doppler effect induced by a DC current. By exploiting high carrier drift velocities (up to 0.5v), the method induces asymmetric plasmon dispersion, enabling one-way propagation and enabling tunable plasmonic isolators and resonant mode splitting without optical pumping or magnetic fields.

ABSTRACT

Strong optical nonreciprocity at the nanoscale, relying on extreme one-way modes and backscattering suppression, can enable fundamentally new approaches in optoelectronics and plasmonics. Of special interest is achieving nonreciprocity in systems devoid of magnetic couplings. We describe a new approach based on the plasmonic Doppler effect which takes place for plasmons propagating in the presence of an electrical DC current. Large carrier drift velocities reachable in high-mobility electron systems, such as graphene, can enable strongly nonreciprocal or even fully one-way modes. Striking effects such as mode isolation and one-way transmission in DC-current-controlled Mach-Zehnder interferometers provide clear manifestations of plasmonic nonreciprocity. Phenomena such as plasmon resonance splitting into a doublet, induced by a DC current, afford new ways to generate and exploit unidirectionally propagating plasmon modes.

Motivation & Objective

• To achieve strong nonreciprocity in nanoscale plasmonic systems without relying on magnetic fields or optical nonlinearity.
• To exploit the quasi-relativistic Doppler effect in high-mobility 2D electron systems like graphene to break time-reversal symmetry in a linear, tunable way.
• To demonstrate experimentally accessible platforms—such as Mach-Zehnder interferometers and resonant couplers—for detecting and utilizing nonreciprocal plasmon modes.
• To show that plasmon dispersion becomes highly asymmetric at drift velocities approaching the Fermi velocity, leading to mode isolation and one-way transmission.
• To establish that nonreciprocity arises from the Doppler shift of plasmon modes due to carrier flow, with no need for nonlinearities or external pumping.

Proposed method

• The authors derive a hydrodynamic model for plasmon modes in a current-carrying graphene sheet, incorporating the relativistic-like dispersion of Dirac fermions and the Coulomb interaction.
• They solve the linearized Vlasov-Poisson system for small perturbations around a drifting Fermi distribution, leading to a dispersion relation that explicitly includes the Doppler shift via the drift velocity u.
• The key equation is the dispersion relation (11): $( ilde{ u} - s_{+}k)( ilde{ u} - s_{-}k) = rac{1 - ar{u}^2}{1 - rac{1}{2}ar{u}^2} v^2 k_* k$, where $s_{/pm}$ are the group velocities in the presence of flow.
• The model shows that the Doppler shift becomes order-one at $u ightarrow v$, leading to strong asymmetry between co- and counter-propagating modes.
• They propose two experimental platforms: (1) a current-tunable Mach-Zehnder interferometer where the phase shift is controlled by a DC current in one arm, and (2) a resonant coupling setup that splits the plasmon resonance under current bias.
• The analysis is validated in both hydrodynamic and collisionless regimes, with the former showing weak damping ($ au_{ ext{e-e}} ightarrow 20 ext{ fs}$) at room temperature.

Experimental results

Research questions

• RQ1 Can strong nonreciprocity in graphene plasmons be achieved without magnetic fields or optical nonlinearity?
• RQ2 How does a DC current induce a Doppler shift in plasmon modes, and what is the resulting asymmetry in the dispersion relation?
• RQ3 What is the maximum achievable nonreciprocity in terms of frequency shift and mode isolation at realistic drift velocities?
• RQ4 Can the plasmonic Doppler effect be harnessed to build tunable, backscattering-free plasmonic devices such as isolators?
• RQ5 How does the nonreciprocity depend on carrier density, temperature, and drift velocity in the hydrodynamic regime?

Key findings

• The plasmonic Doppler effect induces a frequency shift of order $v k$ at $u ightarrow v$, leading to strong nonreciprocity with $ ilde{ u}_{-k} eq ilde{ u}_k$.
• At $u ightarrow 0.5v$, the Doppler shift reaches order-one in units of $v k_*$, resulting in a significant asymmetry in the plasmon dispersion relation.
• One of the counter-propagating modes softens and eventually disappears at high drift velocities, leading to one-way propagation and mode isolation.
• The Mach-Zehnder interferometer setup enables current-tunable phase control and one-way transmission, with the interference condition dependent on the DC current in one arm.
• Resonant coupling under current bias leads to a splitting of the plasmon resonance into a doublet, providing a direct detection method for unidirectional modes.
• The hydrodynamic model predicts weakly damped plasmon modes with scattering times of ~20 fs at room temperature, supporting propagation over relevant lengthscales.