Quantum Hall Response to Time-Dependent Strain Gradients in Graphene

Mechanical deformations of graphene induce a term in the Dirac Hamiltonian that is reminiscent of an electromagnetic vector potential. Strain gradients along particular lattice directions induce local pseudomagnetic fields and substantial energy gaps as indeed observed experimentally. Expanding this analogy, we propose to complement the pseudomagnetic field by a pseudoelectric field, generated by a time-dependent oscillating stress applied to a graphene ribbon. The joint Hall-like response to these crossed fields results in a strain-induced charge current along the ribbon. We analyze in detail a particular experimental implementation in the (pseudo)quantum Hall regime with weak intervalley scattering. This allows us to predict an (approximately) quantized Hall current that is unaffected by screening due to diffusion currents.

Figure 1

(a) Graphene ribbon oriented along the armchair direction, with a uniaxial stretch generating a larger deformation at its narrow (top) end. The resulting strain gradient is designed via the shape function $W(y)$ to create a uniform pseudomagnetic field $\overrightarrow{B}$ (Ref. [22]). (b) A time varying (oscillating) stress component generates an additional pseudoelectric field $\overrightarrow{E}$. The orientations of both $\overrightarrow{B}$ and $\overrightarrow{E}$ are opposite for electrons in the $K$ and $K^{\prime }$ valleys. (c) Illustration of the valley symmetric drift dynamics considering half an oscillating period so that $\overrightarrow{E}$ has a fixed sign.

Figure 2

(a) Strain tensor component ${\epsilon }_{yy}(x,y)$ calculated by comsol. We set the vertical stretch $\mathrm{\Delta }y$ to induce a maximum strain ${\epsilon }_{yy}$ of 20% (red). (b) Color plot of the calculated pseudomagnetic field $(\nabla \times \overrightarrow{A}{)}_z$ and arrow plot of the pseudovector potential calculated from Eq. (1), determining the strength and direction of the pseudoelectric field for ac strain. (c) Profiles of $B$, showing a relatively uniform $\approx 3\mathrm{T}$ over a microscale region (blue), and of $A_x$, allowing one to extract the pseudovoltage of 0.1 mV (see text).

Figure 3

(a) Plot of typical PLLs (or LLs) and edge states. (b) Tilting PLLs due to bulk pseudoelectric field caused by ac strain. A finite pseudovoltage difference $V_{\omega }$ between the edges is indicated. ($c$) Strong intervalley scattering causes equilibration within each edge.

Figure 4

Schematics of QH vs pseudo-QH response as a function of density near the Dirac point. (Black dashed line) Two-terminal quantized conductance for external magnetic field $B_{\mathrm{ext}}$ and a voltage $V_{\mathrm{ext}}$ applied through a chemical potential difference between the terminals. (Blue) Two-terminal conductance for a pseudomagnetic field $B$ and voltage $V_{\mathrm{ext}}$; backscattering due to counterpropagating valleys leads to deviations from quantization. (Red) Alternating current response to simultaneous pseudo-$B$ and pseudo-$E$ fields generating an internal pseudovoltage difference $V_{\omega }$. The current is approximately quantized at the plateaus, and strongly suppressed at the plateau transitions due to screening of the pseudo-$E$ field [30]. (Inset) DOS consisting of extended states at the PLL energies surrounded by localized states.