Thermal transport in compensated semimetals: Effect of electron-electron scattering on Lorenz ratio

It is well known that the electronic thermal conductivity of clean compensated semimetals can be greatly enhanced over the electric conductivity by the availability of an ambipolar mechanism of conduction, whereby electrons and holes flow in the same direction experiencing negligible Coulomb scattering as well as negligible impurity scattering. This enhancement—resulting in a breakdown of the Wiedemann-Franz law with an anomalously large Lorenz ratio—has been recently observed in two-dimensional monolayer and bilayer graphene near the charge neutrality point. In contrast to this, three-dimensional compensated semimetals such as ${\mathrm{WP}}_2$ and Sb are typically found to show a reduced Lorenz ratio. We investigate the reasons for this difference, focusing on the low-temperature regime where the electron-electron scattering is expected to dominate over other scattering mechanisms. We show that the different regimes of Fermi statistics (nondegenerate electron-hole liquid in graphene versus degenerate electron-hole liquid in compensated semimetals) are not sufficient to explain the reduction of the Lorenz ratio in the latter. We propose that the solution of the puzzle lies in the large separation of electron and hole pockets in momentum space, which allows compensated semimetals to sustain sizable regions of electron-hole accumulation near the contacts. These accumulations suppress the ambipolar conduction mechanism and effectively split the system into two independent electron and hole conductors. We present a quantitative theory of the crossover from ambipolar to unipolar conduction as a function of the size of the electron-hole accumulation regions, and show that it naturally leads to a sample-size-dependent thermal conductivity.

Figure 1

(a) Schematic illustration of the difference between heat and charge current in a charge-neutral system. A thermal current can be set up in a semimetal simply by letting electrons and holes drift with equal speeds in the same direction (upper row). Electric field causes electrons and holes to drift in opposite directions (lower row). (b) Experimental observation of the enhanced Lorenz ratio in monolayer graphene (the solid lines are guides to the eye); data were reproduced from Ref. [3]. (c) Experimental observation of the reduced Lorenz ratio in ${\mathrm{WP}}_2$; data were reproduced from Refs. [20, 21]. The insets in (b) and (c) depict the low-energy bands in graphene systems (dashed curves for monolayer and solid curves for bilayer) and in a compensated semimetal, respectively. In both cases, the chemical potential is taken as the zero of the energy.

Figure 2

(a) Schematics of the thermal conductivity measurement. (b) Spatial distribution of the electron and hole components of the electric current. ${\ell }_D$ is the diffusion length. The ambipolar transport region is shaded. The red dashed lines represent the enhanced density of electrons and holes near the contacts.

Figure 3

Intrinsic unipolar and ambipolar thermal resistivities (as defined in the text) as well as the intrinsic electric resistivity as functions of temperature (scaled with $T_F$). In the presence of disorder the total electric and thermal resistivities, respectively, become ${\rho }_{\mathrm{el}}+{\rho }_{\mathrm{el},\mathrm{dis}}$ and ${\rho }_{\mathrm{th},\mathrm{uni}/\mathrm{ambi}}+{\rho }_{\mathrm{th},\mathrm{dis}}$ (not shown in the figure), where ${\rho }_{\mathrm{el},\mathrm{dis}}$ is constant at low $T$ (impurity-dominated regime) and linearly scales with $T$ when phonons are relevant. Making use of the Wiedemann-Franz law for a noninteracting disordered system, we have assumed ${\rho }_{\mathrm{el},\mathrm{dis}}/{\rho }_{\mathrm{th},\mathrm{dis}}={\pi }^2/3$ (in reduced units).

Figure 4

Lorenz ratio as a function of (a) temperature for different values of $\alpha \left(\ell \right)$ from 0 (perfect ambipolar regime) to 1 (perfect unipolar regime) with a step of 0.01 and (b) as a function of $\alpha \left(\ell \right)$ (in logarithmic scale) for different temperatures as labeled. The strength of the charge impurity- and phonon-limited resistivities were defined through the hydrodynamicity parameters ${\mathrm{\Gamma }}_{\mathrm{imp}}^2={\rho }_{\mathrm{el},\mathrm{imp}}/{\rho }_{\mathrm{el}}(T\gg T_F)$ and ${\mathrm{\Gamma }}_{\mathrm{ph}}^2={\rho }_{\mathrm{el},\mathrm{ph}}/{\rho }_{\mathrm{el}}(T\gg T_F)$, respectively. The inset in (b) shows $\alpha \left(\ell \right)$, Eq. (5), as a function of $\ell /{\ell }_D$. The onset temperature for phonons is taken as $T_{\mathrm{ph}}=25$ K. In the hydrodynamic regime, i.e., $0\lesssim T\lesssim 25$ K in this figure, the Lorenz ratio deviates from its standard (impurity/phonon)-limited value $L_0$. Due to the ambipolar constraint $j_e+j_h=0$, the violation of the Lorenz ratio is a large enhancement while with the unipolar situation ($j_e=j_h=0$) the Lorenz number is drastically reduced below $L_0$.

Figure 5

2D plot of the Lorenz ratio as a function of $\ell /{\ell }_D$ (sample length scaled with the diffusion length) and $T$. Different transport regimes are indicated on the figure. The onset temperature of phonons $T_{\mathrm{ph}}=25$ K and the hydrodynamicity parameters ${\mathrm{\Gamma }}_{\mathrm{imp}}=0.01$ and ${\mathrm{\Gamma }}_{\mathrm{ph}}=3$ were chosen to be the same as in Fig. 4. The enhancement of the Lorenz ratio (in the ambipolar hydrodynamic regime) is relevant for graphene systems and is a consequence of electron-hole scattering, which selectively enhances the electric resistivity. The reduced Lorenz number (in the unipolar hydrodynamic regime) is relevant for compensated semimetals, where the electron and hole bands are well separated in momentum space.