Graphene is neither Relativistic nor Non-Relativistic case: Thermodynamics Aspects

TL;DR

The paper argues that electrons in graphene do not obey standard non-relativistic or ultra-relativistic hydrodynamics. It develops a graphene-specific hydrodynamic framework (GHD) with dispersion $E=pv_g$, deriving thermodynamic quantities from grand-canonical statistics and Fermi-Dirac integrals, and comparing 3D and 2D cases against NR and UR limits. Key results show distinct scaling for graphene thermodynamics (e.g., $\epsilon$, $P$, $n$) and reveal a clear DF-to-FL transition as $\mu/T$ varies, with a notable enhancement of specific heat in the Dirac regime that may relate to violations of the Wiedemann-Franz law. The work thus clarifies that graphene operates in an intermediate regime, with implications for interpreting experiments and drawing parallels to high-energy systems like quark matter in different temperature-chemical-potential domains.

Abstract

Discovery of electron hydrodynamics in graphene system has opened a new scope of analytic calculations in condensed matter physics, which was traditionally well cultivated in science and engineering as a non-relativistic hydrodynamics and in high energy nuclear and astro physics as relativistic hydrodynamics. Electrons in graphene follow neither non-relativistic nor relativistic hydrodynamics and thermodynamics. Present article has gone through systematic microscopic calculations of thermodynamical quantities like pressure, energy density, etc. of electron-fluid in graphene and compared with corresponding estimations for non-relativistic and ultra-relativistic cases. Identifying the Dirac fluid and Fermi liquid domains, we have sketched the transition of temperature and Fermi energy dependency of electron thermodynamics for graphene and other cases. An equivalent transition for quark matter is also discussed. The most exciting part is the general expression of specific heat, whose Fermi to Dirac fluid domain transition can be realized as a transition from a solid-based to a fluid-based picture. This understanding may be connected to the experimentally observed Wiedemann-Franz Law violation in the Dirac fluid domain of graphene system.

Figures (7)

• Figure 1: The Condensed Matter Physics (CMP) and High Energy Physics (HEP) locations and their Dirac Fluid (DF) and Fermi Liquid (FL) domains in $T$-$\mu$ plane.
• Figure 2: The energy density in different domains to $\epsilon_{SB}$ with $\mu/T$ for (a) 3D and (b) 2D cases.
• Figure 3: The pressure in different domains to $P_{SB}$ with $\mu/T$ for (a) 3D and (b) 2D cases.
• Figure 4: The number density in different domains to $n_{SB}$ with $\mu/T$ for (a) 3D and (b) 2D cases.
• Figure 5: The ratio of the energy density and number density for (a) 3D and (b) 2D cases.