Experimental Determination of the Energy per Particle in Partially Filled Landau Levels

We describe an experimental technique to measure the chemical potential $\mu$ in atomically thin layered materials with high sensitivity and in the static limit. We apply the technique to a high quality graphene monolayer to map out the evolution of $\mu$ with carrier density throughout the $N=0$ and $N=1$ Landau levels at high magnetic field. By integrating $\mu$ over filling factor $\nu$, we obtain the ground state energy per particle, which can be directly compared to numerical calculations. In the $N=0$ Landau level, our data show exceptional agreement with numerical calculations over the whole Landau level without adjustable parameters as long as the screening of the Coulomb interaction by the filled Landau levels is accounted for. In the $N=1$ Landau level, a comparison between experimental and numerical data suggests the importance of valley anisotropic interactions and reveals a possible presence of valley-textured electron solids near odd filling.

Figure 1

Measurement setup. (a) Optical image of the device; scale bar is $10\mu \mathrm{m}$. (b) Measurement schematic. Static gate voltages are applied to the top gate ($v_t$), bottom gate ($v_b$), and detector monolayer ($v_d$). ${\sigma }_d$ is measured by applying an ac voltage ${\tilde{v}}_d$ at frequency $f_1=13.77\mathrm{Hz}$ to one of the internal contacts and measuring the current at another with a DL1211 amplifier. An additional ac voltage $({\tilde{v}}_t)$ is applied to the top gate at $f_2=110\mathrm{Hz}$. Demodulating the current at $f_2-f_1$ with an SR860 lock-in amplifier produces a signal proportional to $d{\sigma }_d/d{v}_t$, which vanishes at a conductivity minimum independent of contact resistance. We use a feedback loop to continuously adjust a static voltage $\delta v_t$ in order to maintain $d{\sigma }_d/d{v}_t=0$. Under these conditions, $\delta \mu =-c_t\delta v_t/c_0$. (c) ${\sigma }_d$ (blue) and $\delta {\sigma }_d/\delta v_t$ (red) as a function of $\delta v_t$. (d) $\mu (n)$ at $B=0\mathrm{T}$ (red) and 0.2 T (blue), measured at $T=15\mathrm{mK}$. (e) Density of states $dn/d\mu$ calculated by numerical differentiation of data in panel (d). The ZLL is split by a sublattice gap [27, 28] of ${\mathrm{\Delta }}_{AB}=6.9\mathrm{meV}$. (f) $n$-dependent $v_F$ measured by fitting $B=0\mathrm{T}$ data to ${\mu }^2=({\mathrm{\Delta }}_{AB}/2{)}^2+(\hslash v_F\sqrt{\pi |n|}{)}^2$ with ${\mathrm{\Delta }}_{AB}$ fixed and $v_F$ a free function of $n$. The red curve is a fit to theoretical models [29, 30, 31] of Fermi velocity renormalization by Coulomb interactions.

Figure 2

Chemical potential and energy per flux quantum in the $N=0\mathrm{LL}$. (a) $\mu (\nu )$ within the ZLL measured at $B=14\mathrm{T}$ and nominal $T=15\mathrm{mK}$. Blue regions indicate domains of $\nu$ where the charging time of the sample exceeds the measurement time of $\sim 1\sec$ (see Fig. S5). (b) $\mu$ at $B=18\mathrm{T}$ and nominal $T=40\mathrm{mK}$ for low ${\nu }^{*}$, measured relative to $\nu =-1$ (orange), $\nu =0$ (blue), and $\nu =1$ (red). The cyan and purple curves are calculated $\mu$ for a Wigner crystal with screened and unscreened Coulomb interactions, respectively, taking ${\epsilon }_{\mathrm{hBN}}=4.0$ and ${\alpha }_G=1.85$; shaded ranges reflect uncertainty in those parameters as described in main text. The data are offset so that $\mu ({\nu }^{*})=0$. (c) Numerically calculated [36] total ground state energy of the $N=0$ LL after accounting for the screened Coulomb interactions. (d) Comparison of experimentally determined (solid lines) and numerically calculated (dark blue crosses) $\tilde{E}$. Both experimental and numerical data have a linear-in-${\nu }^{*}$ background subtracted so that $\tilde{E}$ vanishes at integer ${\nu }^{*}$. Data were taken at $B=18\mathrm{T}$ and $T=40\mathrm{mK}$.

Figure 3

Multicomponent effect and lattice scale interaction in the $N=1\mathrm{LL}$. (a) $\mu$ in the $N=1$ LL at $T=15\mathrm{mK}$ and $B=13\mathrm{T}$. (b) $\mu$ measured near ${\nu }_0=-2$, $-4$, and $-6$. Solid lines are $\mu$ calculated from the Wigner crystal model with parameters identical to those used in Fig. 2. (c) $\mu$ near ${\nu }_0=-3$ and $-5$. The solid lines showing the Wigner crystal model do not match the data, suggesting the importance of valley merons [44] near these fillings. (d) Comparison of experimentally determined $\tilde{E}$ with numerical simulations for $-3<\nu <-2$. One-component numerical calculations underestimate the experimental result by a significant margin. Including both valley components as well as the contribution of lattice scale anisotropies as in Eq. (1) with $g_z=g_{xy}=0.1(a/{\ell }_B)E_C$ can restore agreement to within $100\mu eV\approx 2.5\times {10}^{-3}{E}_C$.