Wigner localization in a graphene quantum dot with a mass gap

In spite of unscreened Coulomb interactions close to charge neutrality, relativistic massless electrons in graphene allegedly behave as noninteracting particles. A clue to this paradox is that both interaction and kinetic energies scale with particle density in the same way. In contrast, in a dilute gas of nonrelativistic electrons, the different scaling drives the transition to Wigner crystal. Here we show that Dirac electrons in a graphene quantum dot with a mass gap localize in the manner of Wigner for realistic values of device parameters. Our theoretical evidence relies on many-body observables obtained through the exact diagonalization of the interacting Hamiltonian, which allows us to take all electron correlations into account. We predict that the experimental signatures of Wigner localization are the suppression of the fourfold periodicity of the filling sequence and the quenching of excitation energies, both of which may be accessed through Coulomb blockade spectroscopy. Our findings are relevant to other carbon-based nanostructures exhibiting a mass gap.

Figure 1

Graphene QD defined by electrostatic gates. (a) Proposed setup. The top gate ($V_T$) defines the dot, while source ($V_S$), drain ($V_D$), and back ($V_G$) gates allow for Coulomb blockade spectroscopy. (b) Radial QD confinement potential. The interaction between graphene and substrate opens a mass gap $2\Delta$ in the QD energy spectrum. (c) Lowest noninteracting QD energy levels in the conduction band. Black [red (gray)] lines label states in the $K$ ($K^{\prime }$) valley. (d) Real part of sublattice-resolved envelopes whose energies are labeled by the square and triangle symbols in panel (c).

Figure 2

Coulomb blockade linear spectroscopy. Chemical potential $\mu \left(N\right)$ vs electron number $N$ for different background dielectric constants $\epsilon$, with radius $R=250$ Å. Inset: Charging energy $\Delta \mu \left(N\right)$ vs $N$. Lines are guides to the eye. $\Delta \mu$ may be measured as electrons are added to the quantum dot one by one tuning the backgate shown in Fig. 1.

Figure 3

Emergence of radial correlations in the wave function. One-body density $〈n\left(\mathbf{r}\right)〉$ vs radial coordinate $\rho$ for different values of dielectric constant $\epsilon$ and electron number $N$, with radius $R=1250$ Å. Realistically screened mutual interactions push electrons against the QD potential wall.

Figure 4

Suppression of exchange interactions. Spin-resolved density $〈n_{\sigma }\left(\mathbf{r}\right)〉$ vs radial coordinate $\rho$ for different values of dielectric constant $\epsilon$ (left panel, $R=500$ Å) and radius $R$ (right panel, $\epsilon =5$), with $N=5$ and spin projection $S_z=1/2$. Solid and dashed lines point to $\langle n_{\uparrow }\rangle$ and $\langle n_{\downarrow }\rangle$, respectively. Wigner localization depletes the probability weight in the regions halfway between an electron and its neighbors and hence suppresses exchange interactions, inducing large spin degeneracies.

Figure 5

Polygonal Wigner molecules. (a) Three-dimensional contour plots of pair correlation functions $P(\mathbf{r},{\mathbf{r}}_0)$ for $\epsilon =2$ and $R=2250$ Å. Black dots point to the locations ${\mathbf{r}}_0$ of fixed electrons. (b) Pair correlation function $P(\mathbf{r},{\mathbf{r}}_0)$ vs angle $\phi$ with $\left|\mathbf{r}\right|=\left|{\mathbf{r}}_0\right|$ for four electrons and different values of dielectric constant $\epsilon$, with $R=500$ Å. Inset: corresponding contour plots of $P(\mathbf{r},{\mathbf{r}}_0)$ in the $xy$ plane. Increasing the interaction strength leads to the formation of the correlation hole as well as the development of angular correlations, which enforce a square Wigner molecule.

Figure 6

Excitation spectrum of a Wigner molecule. Low-lying excitation energies vs dielectric constant $\epsilon$ for $N=4$ and $R=500$ Å. Numbers label degeneracies of selected multiplets. Insets: density $n\left(\mathbf{r}\right)$ vs radial coordinate $\rho$ averaged over the ground state (black curve) and the first excited multiplet [red (gray) curve]. The Wigner-molecule ground state is highly degenerate, as localized electrons may independently flip their (iso)spins.