Commensurate-Incommensurate Transition in Submonolayer $^3$He on Graphite

Abstract

We report high-precision heat-capacity measurements of submonolayer $^3$He adsorbed on highly crystalline graphite, revealing new aspects of the commensurate$-$incommensurate transition. Below 1K, two possible striped domain-wall phases emerge: $α_1$ with variable wall spacing and $α_2$ with fixed spacing. The $T$-linear heat capacity in $α_1$ arises from one-dimensional phonons along the walls. $α_2$ melts into $α_1$ at a critical density via a second-order transition, consistent with a quantum nematic (quantum liquid-crystal) state in $α_1$, and reconciling thermodynamic and prior nuclear-magnetic data.

Figures (4)

• Figure 1: (a) Phase diagram of submonolayer $^3$He on graphite inferred from previous heat-capacity measurements hering1976apparentBretz1973-vs. Open circles denote heat-capacity peak temperatures. Phases: C, $\sqrt{3} \times \sqrt{3}$ commensurate solid; $\alpha$, putative DW phase; $\beta$, DW fluid; IC, incommensurate solid. Insets: schematic structures of the C and IC solid phases; A, B, and C mark the three equivalent adsorption sites in the C phase. Unrelaxed (b) striped and (c) hexagonal arrangements of superheavy DWs. Black spheres represent atoms in the DWs, and light blue spheres denote those in the domains.
• Figure 2: Heat-capacity data for submonolayer $^3$He films on a ZYX graphite substrate at selected areal densities. Complete datasets for measured densities are provided in SM-2 SuppMat. Raw data are shown only for $\rho$ = 6.75 nm$^{-2}$; otherwise, the data are shown as smoothed curves for clarity. Squares, circles, and triangles indicate the 3 K, 1 K, and IC peaks, respectively. Dash-dotted curves reproduce data from Ref. hering1976apparent on Grafoil (see SM-3 SuppMat for density rescaling). Numbers denote areal densities in nm$^{-2}$. Inset: smallest 1 K peak detected at $\rho_{\mathrm{1}} = 6.51\ \mathrm{nm}^{-2}$.
• Figure 3: Density dependences of heat-capacity anomalies: (a) peak temperature $T_{\mathrm{peak}}$; (b) specific-heat peak height $C_{\mathrm{peak}}/N k_{\mathrm{B}}$; (c) entropy change $\Delta S$ from the 1 K peak. The proposed phase diagram of the submonolayer $^3$He on graphite is shown in color in (a); $\rho_{\mathrm{1/3}}$, $\rho_{1}$, $\rho_{2}$, $\rho_{3}$, $\rho_{\mathrm{IC}}$, and $\rho_{\text{cr}}$ denote characteristic areal densities (see text). In (b), yellow squares, green circles, and blue triangles represent the 3 K, 1 K, and IC peaks, respectively. In (a)--(c), crosses are data from Ref. hering1976apparent on Grafoil (see SM-3 SuppMat for density rescaling). Inset: temperature dependence of the entropy $S$ deduced from the present heat-capacity data at $\rho = 7.24\ \mathrm{nm}^{-2}$. (d) Effective exchange energy $J_{\chi}$ determined from the nuclear-magnetization measurement Ikegami1998-cd (see text). Solid and dashed lines are guides to the eye.
• Figure 4: Density dependence of (a) the exponent $\nu$ and (b) the coefficient $a$ obtained from power-law fits $C = a T^{\nu}$ over 0.3--0.9 K. Red circles represent results from this study, while crosses show data from Ref. hering1976apparent on Grafoil (see SM-3 SuppMat). Solid and dashed lines are guides to the eye. The top panel shows the phase diagram proposed in this study. Although Ref. hering1976apparent exhibits trends qualitatively similar to ours, its $a$ values are substantially larger, indicating strong finite-size effects in the phonon spectrum.