Crystals Caught Doping: Metallic Wigner Crystals in Rhombohedral Graphene

Abstract

Nearly a century after Wigner's initial proposal, electron crystals are now a topic of intense experimental and theoretical interest. However, most proposed crystalline phases are commensurate and therefore become insulating in the presence of even weak pinning. In this work we discuss when a commensurate Wigner crystal will spontaneously self dope and develop itinerant carriers, giving rise to an incommensurate and thus metallic Wigner crystal (MWC). We develop a general criterion for the instability of the commensurate crystal which involves the competition between the charge gap at commensurability and a ``packing bias'' whose sign selects whether electron or hole doping is preferred. We then apply these insights to rhombohedral multilayer graphene, where calculations for commensurate crystals reveal instabilities towards self-doping. Carrying out self-consistent Hartree-Fock over the landscape of incommensurate crystals reveals the phase diagram, where a broad MWC phase appears directly adjacent to an insulating Wigner crystal phase. Recent observations of an island of reversed Hall conductance near a putative Wigner crystal phase in rhombohedral graphene are naturally explained by our theory.

Figures (10)

• Figure 1: (a) Schematic band structures of an insulating Wigner crystal (WC) and a metallic Wigner crystal (MWC), shifted such that at the same chemical potential $\mu$ shown in the plot they correspond to the same electron density. Vertical lines indicate respective crystalline Brillouin zone edges. (b) Cartoon dependence of energy on the density deviation defined in Eq. \ref{['eq:density_deviation']}. Insulating WCs generally have a cusp in $E(\delta n)$ at $\delta n=0$ corresponding to its charge gap when pinned. Incommensurate WCs have $\delta n^*\neq 0$. (c) Phase diagram of rhombohedral four-layer graphene at low density within SCHF, where crystalline phases including insulating WCs and MWCs compete with the non-crystalline metal (NCM) shown in gray, whose $|\rho_{\boldsymbol{G}}|A_{uc}<0.05$ for $\boldsymbol{G}$ in the first shell. The colors show $\delta n^*$, the optimal deviation from commensurate density, defined in Eq. \ref{['eq:density_deviation']}.
• Figure 2: Properties of the insulating Wigner crystal at $(n, u_D)=(0.4e12\per cm\squared,45meV)$ (left) and the metallic Wigner crystal at the same density and $u_D=60meV$ (right). (a,b) Non-interacting dispersion over a high symmetry path of the commensurate crystalline Brillouin zone. The lowest band is highlighted in red. A window of energy corresponding to the interaction strength $U=e^2\sqrt{n}/\epsilon_r\epsilon_0$ is shaded purple. (c,d) $E(\delta n)/N$ computed as a function of the density mismatch $\delta n=n-n_{uc}$. The optimal value $\delta n^*$ for which $E$ is minimized is shown with a red star. On the left, $\delta n^*$ is $0$, indicating a commensurate insulating WC, and on the right, $\delta n^*=-0.025e12\per cm\squared$, indicating an incommensurate MWC. (e,f) SCHF band structures. The lowest band is highlighted in red, with solid circles indicating occupied states and empty circles indicating unoccupied states. On the right, the MWC exhibits a hole pocket at the $\gamma$ point of its crystalline BZ.
• Figure 3: (Top) Evolving fermiology of the metallic Wigner crystal at $u_D=60meV$ and density $n=(0.375, 0.425, 0.475)\times$e12cm (left to right). Black hexagons show the Brillouin zone corresponding to optimal $n_{uc}=n - \delta n^*$. (Bottom) Optimal deviation $\delta n^*$ from commensurate density, Eq. \ref{['eq:density_deviation']}, as a function of density at $u_D=60meV$.
• Figure 4: (a) The charge gap $\Delta$ of the SCHF ground state at commensurate filling $\delta n = 0$ as a function of $(n,u_D)$. A smaller charge gap at commensuration leads to either MWCs or NCMs. The phase boundary corresponding to the commensurate insulating WC is shown in red. (b) The instability $I$, Eq. \ref{['eq:instability']}, for the commensurate SCHF state as a function of $(n,u_D)$. A positive (negative) sign of $I$ signals instability to electron (hole) doping. (c) Comparison of packing bias $k_0$ and $\pm \Delta /2$ as a function of density at $u_D = 55meV$. $k_0$ changes rapidly while $\pm \Delta/2$ remains roughly constant in $n$, illuminating the delicate competition between $\Delta$ and $k_0$. (d) Map of the single particle heuristic $W=(E_{\gamma}-E_{\kappa^+})/U$, where $E_k$ corresponds to the lowest band after folding the non-interacting band structure on to a commensurate crystalline Brillouin zone and $U$ is the interaction scale $e^2\sqrt{n}/\epsilon_r\epsilon_0$. Large $W$ suggests instability towards an MWC.
• Figure 5: Time-dependent Hartree-Fock (TDHF) spectrum of the metallic Wigner crystal. The orange line denotes the acoustic phonon branch. Purple dashed lines mark $2k_F$ for a circular Fermi surface of the corresponding density. Parameters match Fig. \ref{['fig:MWC_properties']}(f).