Wide Critical Fluctuations of the Field-Induced Phase Transition in Graphite

In the immediate vicinity of the critical temperature ($T_c$) of a phase transition, there are fluctuations of the order parameter that reside beyond the mean-field approximation. Such critical fluctuations usually occur in a very narrow temperature window in contrast to Gaussian fluctuations. Here, we report on a study of specific heat in graphite subject to a high magnetic field when all carriers are confined in the lowest Landau levels. The observation of a BCS-like specific heat jump in both temperature and field sweeps establishes that the phase transition discovered decades ago in graphite is of the second order. The jump is preceded by a steady field-induced enhancement of the electronic specific heat. A modest (20%) reduction in the amplitude of the magnetic field (from 33 to 27 T) leads to a threefold decrease of $T_c$ and a drastic widening of the specific heat anomaly, which acquires a tail spreading to two times $T_c$. We argue that the steady departure from the mean-field BCS behavior is the consequence of an exceptionally large Ginzburg number in this dilute metal, which grows steadily as the field lowers. Our fit of the critical fluctuations indicates that they belong to the $3DXY$ universality class as in the case of the ${}^4\mathrm{He}$ superfluid transition.

Figure 1

Field dependence of $\gamma =C_{\mathrm{el}}/T$ at $T=1.6\mathrm{K}$. The vertical black line indicates the quantum limit (QL) regime. When $B>B_{\mathrm{QL}}=7.5\mathrm{T}$, holes and electrons are confined to their $n=0$ Landau levels. Open circles represent $\gamma$ in the normal state (labeled ${\gamma }_N$) deduced from temperature sweeps shown in Fig. 2. Inset: field sweeps at different temperatures.

Figure 2

Temperature dependence of $\gamma$. (a) $\gamma =C_{\mathrm{el}}/T$ as function of temperature for different magnetic fields. Curves are shifted for clarity. (b) $C_{\mathrm{el}}/{\gamma }_{N}T$ as function of $\tau =T-T_c^p/T_c^p$, where $T_c^p$ is the temperature of the peak position. At $B=27.7\mathrm{T}$, the tail of the transition extends up to twice $T_c$. (c) Specific heat anomaly for a BCS transition. The amplitude of the jump at $T_c$ is such that $\mathrm{\Delta }{C}_{\mathrm{el}}(T_c)/{\gamma }_N{T}_c=1.43$ where $\mathrm{\Delta }{C}_{\mathrm{el}}(T)=C_{\mathrm{el}}(T)-{\gamma }_{N}T$. (d) Specific heat anomaly in a Bose-Einstein condensation transition (in black) and the singularity caused by a $3DXY$ critical fluctuation (in green) like in the $\lambda$ transition in helium.

Figure 3

(a) Phase diagram of graphite showing the evolution of critical temperature ($T_c$) as a function of the inverse of the magnetic field ($B_c^{-1}$) according to different experimental probes. Anomalies in ${\mathrm{R}}_{xx}$ (black open square points) and ${\mathrm{R}}_{zz}$ (black open diamond points) (from [13]) are compared to peaks in $C_{\mathrm{el}}$ ($T_c^p$ in purple closed circles). The black line represents $T_c=T^{\star }\exp (-B^{\star }/B_c)$ and reasonably describes the evolution of $R_{xx}$ anomalies. (b) Field dependence of $\mathrm{\Delta }{C}_{\mathrm{el}}(T_c^p)/{\gamma }_N{T}_c^p$ and ${\mathrm{\Delta }}_c/k_B{T}_c$. ${\mathrm{\Delta }}_c$ is the activation gap deduced from the $c$-axis resistance measurements (see [17]). Inset: ${\mathrm{\Delta }}_c$ vs $B$.

Figure 4

Fluctuations regime of the electronic specific heat. (a) $\mathrm{\Delta }{C}_{\mathrm{el}}=C_{\mathrm{el}}-{\gamma }_{N}T$ as a function of $\tau =T-T_c/T_c$ for $\tau >0$. The dashed line correspond to a phenomenological fit $\mathrm{\Delta }{C}_{\mathrm{el}}=C_0\exp (-(\tau /{\tau }_0))$. Inset: $\ln (\mathrm{\Delta }{C}_{\mathrm{el}})$ as a function of $\tau$. (b) Field dependence of $C_0$ and ${\tau }_0^{-1}$. The red dot line is linear fit of ${\tau }_0^{-1}$. (c) Field dependence of ${\xi }_{\perp }={\ell }_B$, ${\xi }_{//}={\xi }_0$ (see text), and ${\xi }_m=({\xi }_{\perp }^2{\xi }_{//}{)}^{1/3}$. (d) Temperature dependence of ${\tau }_0$ compared to ${\tau }_G$. Inset: plot ${\tau }_0^{-1}$ vs $T_c^p$ with a linear fit (blue line). (e) Normalized plot $\mathrm{\Delta }{C}_{\mathrm{el}}$ as function of $(T-T_c^{\mathrm{ren}})/T_c^{\mathrm{ren}}{\tau }_0$, where $T_c^{\mathrm{ren}}$ is a renormalized $T_c^{\mathrm{ren}}=T_c^p(1+0.5{\tau }_0)$. The black dot line is a fit with the $3DXY$ model (see text and [17]).