Thermopower in hBN/graphene/hBN superlattices

Thermoelectric effects are highly sensitive to the asymmetry in the density of states around the Fermi energy and can be exploited as probes of the electronic structure. We experimentally study thermopower in high-quality monolayer graphene, within heterostructures consisting of complete hBN encapsulation and 1D edge contacts, where the graphene and hBN lattices are aligned. When graphene is aligned to one of the hBN layers, we demonstrate the presence of additional sign reversals in the thermopower as a function of carrier density, directly evidencing the presence of the single-aligned moiré superlattice. We show that the temperature dependence of the thermopower enables the assessment of the role of built-in strain variation and van Hove singularities and hints at the presence of Umklapp electron-electron scattering processes. As the thermopower peaks around the neutrality point, this allows to probe the energy spectrum degeneracy. Further, when graphene is double aligned with the top and bottom hBN crystals, the thermopower exhibits features evidencing multiple cloned Dirac points caused by the differential super-moiré superlattice. For both cases we evaluate how well the thermopower agrees with Mott's equation. Finally, we show the same moiré superlattice device can exhibit a temperature-driven thermopower reversal from positive to negative and vice versa, by controlling the carrier density. The study of thermopower provides an alternative approach to study the electronic structure of 2D superlattices, whilst offering opportunities to engineer the thermoelectric response on these heterostructures.

Figure 1

Charge transport and thermopower of the encapsulated single-aligned graphene/hBN device. (a) Resistivity as a function of normalized carrier density $n/n_0$ at different temperatures $T$. The secondary neutrality points appear at carrier density $-n_0$ and $n_0$ ($n_0\approx 2.2\times {10}^{-12}{\mathrm{cm}}^{\text{–}2}$). Inset shows an optical microscopy image of the device. Scale bar is 5 µm. (b) Temperature dependence of the excess resistivity $\mathrm{\Delta }\rho$ relative to that at 16.5 K at different fixed carrier densities. The dashed lines depicts the quadratic temperature dependence expected for the electron-electron Umklapp scattering from the edges of the moiré mini Brillouin zone. (c) Arrhenius plot of the conductivity at the main neutrality point ${\sigma }_{\text{min}}$ used to estimate the width of the apparent band gap ($E_{\text{g}}=12\pm 2\mathrm{eV}$) extracted from the slope of the activation dependence (dashed line). (d) Thermopower $S$ as a function of the normalized carrier density $n/n_0$ at the same temperatures as in (a). The dashed red curve is the prediction from Eq. (3) for unaligned graphene with an energy-independent scattering time ($m=0$) at $T=210\mathrm{K}$. The three vertical shaded bands mark the sign reversal regions associated with the corresponding charge neutrality points. (e) Position of the thermopower minima and maxima associated with each of the charge reversal regions shown in (d) and marked by the same data point symbols, plotted as a function of temperature. Dashed curves represent the expected value of $\pm \mathrm{\Delta }/2$ given by Eq. (2) with the appropriate superlattice degeneracy $g_{\text{sl}}$ carrier density uncertainties $\delta n$ and $\delta n_0$. (f) Thermopower at selected carrier densities marked by arrows in (d) as a function of dimensionless temperature $T/T_{\text{F}}$. The dashed (dotted) line is the prediction from Eq. (3) for $m=1$ ($m=0$).

Figure 2

Thermoelectric measurements and approximation with Mott's equation. [(a)–(c)] The measured thermopower $S$ as a function of the normalized carrier density $n/n_0$ at three device temperatures $T$ (black curves) compared with the model based on Mott's equation in Eq. (1) with a “nonaligned” DoS (magenta curves) and “aligned” DoS (red curves). (d) Resistivity $\rho$ as a function of the normalized density $n/n_0$ at $T=30\mathrm{K}$ for the high-mobility double-aligned hBN/graphene/hBN sample. The arrows point out the charge density at which the additional super-moiré Dirac peaks appear ($n^{\prime }=\pm 0.135n_0$, where $n_0=4.58\times {10}^{12}{\mathrm{cm}}^{-2}$). (e) The measured thermopower $S$ as a function of the normalized carrier density $n/n_0$ at $T=30\mathrm{K}$, (black curve) for the double-aligned hBN/graphene/hBN sample compared with the predicted value given by Mott's equation in Eq. (1) with a “nonaligned” DoS (red curve) with the corresponding carrier density uncertainties $\delta n$, $\delta n^{\prime }$, and $\delta n_0$.

Figure 3

Temperature dependence of thermopower reversal and of thermopower modulation at each neutrality point. (a) Thermopower $S$ as a function of temperature at two fixed carrier densities commensurate with the holes regime zero crossing point ($-0.4n_0$) and electron regime zero crossing point ($0.7n_0$). We normalize for $n_0$ at the SNP (2.2 ×10¹² cm^–2). (b) The difference between the respective local maximum and minimum thermopower near each neutrality point $\mathrm{\Delta }S$ as a function of temperature. The open symbols represent estimates where the measurement could not extend the carrier density to the maximum (hSNP) or minimum (eSNP) due to risk of damaging the device by gating at higher global temperature, so the values shown are those obtained from the highest carrier density measurement achieved.