Transport coefficients of graphene: Interplay of impurity scattering, Coulomb interaction, and optical phonons

We study the electric and thermal transport of the Dirac carriers in monolayer graphene using the Boltzmann-equation approach. Motivated by recent thermopower measurements [F. Ghahari, H.-Y. Xie, T. Taniguchi, K. Watanabe, M. S. Foster, and P. Kim, Phys. Rev. Lett. 116, 136802 (2016)], we consider the effects of quenched disorder, Coulomb interactions, and electron–optical-phonon scattering. Via an unbiased numerical solution to the Boltzmann equation we calculate the electrical conductivity, thermopower, and electronic component of the thermal conductivity, and discuss the validity of Mott's formula and of the Wiedemann-Franz law. An analytical solution for the disorder-only case shows that screened Coulomb impurity scattering, although elastic, violates the Wiedemann-Franz law even at low temperature. For the combination of carrier-carrier Coulomb and short-ranged impurity scattering, we observe the crossover from the interaction-limited (hydrodynamic) regime to the disorder-limited (Fermi-liquid) regime. In the former, the thermopower and the thermal conductivity follow the results anticipated by the relativistic hydrodynamic theory. On the other hand, we find that optical phonons become non-negligible at relatively low temperatures and that the induced electron thermopower violates Mott's formula. Combining all of these scattering mechanisms, we obtain the thermopower that quantitatively coincides with the experimental data.

Figure 1

Diagrams representing the collision integrals in Eq. (2.3). The arrows indicate the flow of electric charge only and the wave vector labels correspond to incoming and outgoing fermions on the left and right of the scattering vertices, respectively. (a) Static (both short- and long-ranged) impurity scattering. (b) Coulomb collision processes that preserve electron and hole numbers separately. The label $\lambda \in \pm 1$ denotes electrons ($+1$) or holes ($-1$). (c) Carrier–optical-phonon scattering: (i) and (ii) phonon absorption; (iii) and (iv) phonon emission.

Figure 2

Impurity-only transport coefficients as functions of the charge-carrier density $n$ for various temperatures. The symbols are the numerical result obtained by the orthogonal-polynomial method and the solid curves are the analytical result obtained by Eq. (3.44). In our calculations, we use the parameters in Ref. [31]: the effective short-ranged impurity strength $\tilde{g}\sim 1.1\times {10}^{-4}$, the long-ranged impurity concentration $n_{\mathrm{imp}}=2.4\times {10}^9{\mathrm{cm}}^{-2}$, and the fine structure constant ${\alpha }_{\mathrm{int}}=0.6$. (i) Electric conductivity. The horizontal black (diagonal red) dashed line indicates the conductivity in the absence of long-ranged impurities (in the absence of short-ranged impurities) at $T=0$. (ii) Thermoelectric power. (iii) Lorenz ratio. The insert panel shows the Lorenz ratio as a function of temperature at the charge neutral point $n=0$. (iv) Lorenz ratio as a function of the fine structure constant [which determines the Thomas-Fermi wave vector (B7)] at charge neutrality $n=0$ in the absence of short-ranged impurities $\tilde{g}=0$.

Figure 3

Transport coefficients in the presence of Coulomb interactions and short-ranged disorder as functions of $\frac{\mu }{k_{\mathrm{B}}T}$. We take the short-ranged disorder strength, the fine structure constant, and the order of the polynomial basis the same as those in Fig. 2. The black squares are the numerical result. (i) Conductivity. The horizontal blue dashed line indicates the disorder-only conductivity ${\sigma }_{\mathrm{imp}}^{\left(\mathrm{s}\right)}$ in Eq. (2.4), while the red dashed line is the “Drude” component of the hydrodynamic conductivity ${\sigma }_D$ given by the first term of Eq. (3.48a). The inset panel shows the minimal conductivity at the charge neutrality as a function of the fine structure constant. A linear fit of the numerical result gives ${\alpha }_{\mathrm{int}}^2{\sigma }_{\min }\approx 0.79+9.13{\alpha }_{\mathrm{int}}$ (black dashed line). This is consistent with the unscreened result in Refs. [10, 11]. (ii) Thermoelectric power. The top red dashed curve is the ideal clean hydrodynamic result in Eq. (2.7) and the bottom blue dashed curve is the result obtained from Mott's formula. The insert panel is a semilogarithmic plot for the hydrodynamic regime. (iii) Thermal conductivity. The insert panel shows the following “synthetic” Lorenz ratio; this is a plot of the thermal conductivity for a hydrodynamic relativistic gas in the absence of impurities, normalized to the minimal conductivity at charge neutrality, Eq. (3.54). (iv) Lorenz ratio for graphene with Coulomb interactions and short-ranged disorder only. The horizontal red dashed line indicates the Wiedemann-Franz law.

Figure 4

Optical-phonon–limited transport coefficients as functions of temperature and charge density. We take the effective dimensionless electron–optical-phonon coupling strength ${\tilde{\alpha }}_{\mathrm{opt}}^2=1$ [cf. Eq. (3.14); ${\tilde{\alpha }}_{\mathrm{opt}}^2={\alpha }_{\mathrm{opt}}^2/\left(16{\pi }^2\right)]$ and the optical-phonon temperature $T_{A^{\prime }}\equiv \hslash {\omega }_{A^{\prime }}/k_{\mathrm{B}}\approx 2200\mathrm{K}$ [31]. (i) Resistivity $\rho \equiv {\sigma }^{-1}$ as a function of temperature for various charge densities. For comparison, the insert panel shows the Bose-Einstein distribution function of optical phonons [Eq. (3.13)]. (ii) Thermoelectric power ${\alpha }_{\infty }$ as a function of density for various temperatures. The solid (dashed) curves show the result in the presence (absence) of the electron-hole imbalance relaxation processes [see the diagrams (${\mathrm{c})}_{\mathsf{ii}}$ and (${\mathrm{c})}_{\mathsf{iv}}$ in Fig. 1].

Figure 5

Thermopower as a function of doping and temperature including various scattering mechanisms, and comparison to the experiment in Ref. [31]. The dotted (solid) curves are the result of theory (experiment). The bottom red and top blue dashed lines show the thermopower calculated from the experimental conductivity data using Mott's formula [31] and the ideal hydrodynamic result [Eq. (2.7)], respectively. We use the same parameters and the temperature-dependent optical-phonon-electron coupling strength as in Ref. [31]. (i) Thermopower incorporating impurities, Coulomb channels A and B, and all electron–optical-phonons scattering processes depicted in Fig. 1. The “optical” electron-hole Coulomb scattering channel C [Fig. 1${}_{\mathsf{iii}}]$, which shows a plasmon-enhancement in the RPA, is excluded by hand. (ii) Thermopower incorporating only short- and long-ranged impurities. (iii) Thermopower incorporating impurities and Coulomb channels A, B, and C, neglecting optical phonons. (iv) Thermopower incorporating all scattering mechanisms, including the Coulomb channel C. (v) Thermopower incorporating disorder, Coulomb channels A and B, and optical phonons, but neglecting the optical-phonon mediated electron-hole imbalance relaxation processes depicted in Figs. 1(${\mathrm{c})}_{\mathsf{ii}}$ and 1(${\mathrm{c})}_{\mathsf{iv}}$.

Figure 6

Elliptic and hyperbolic coordinates of the momentum transfer for Coulomb interactions. (i) Channel B. The momentum transfer $\mathbf{q}$ lays on a ellipse (red dashed curve) with two foci at the incident momenta $-\mathbf{p}$ and $\mathbf{k}$ (blue dotted lines). (ii) Channels A and C. The momentum transfer $\mathbf{q}$ lays on the two branches of a hyperbola (dashed curves) with two foci at the incident momenta $\mathbf{p}$ and $\mathbf{k}$ (blue dotted lines). The lower branch (red) corresponds to channel A and the upper branch (black) to channel C.

Figure 7

Schematic density plot for the modulus-squared of the RPA screened Coulomb interaction (B1). The screening is perfect along the forward scattering direction $\omega ={\mathsf{v}}_{\mathrm{F}}q$. This compensates the collinear singularity in the Coulomb collision integrals due to the linear dispersion [10, 11], see Eq. (B9). The red dashed line indicates the plasmon dispersion [Eq. (B11)]. Due to kinematic constraints channels A and B [Fig. 1${}_{\mathsf{i},\mathsf{ii}}]$ act in the “quasistatic” regime ${\mathsf{v}}_{\mathrm{F}}q\ge \left|\omega \right|$, while channel C [Fig. 1${}_{\mathsf{iii}}]$ acts the “optical” regime ${\mathsf{v}}_{\mathrm{F}}q\le \left|\omega \right|$.

Figure 8

Comparison between the approximate Coulomb interaction (B10) and the exact RPA result (B1). We take ${\alpha }_{\mathrm{int}}=0.6$ and $z=exp\left(\beta \mu \right)=5$ and define $s\equiv \left|\omega \right|/{\mathsf{v}}_{\mathrm{F}}q$. The solid and dashed curves are the results of Eqs. (B10) and (B1), respectively. (i) Quasistatic regime ${\mathsf{v}}_{\mathrm{F}}q\ge \left|\omega \right|$. (ii) Optical regime ${\mathsf{v}}_{\mathrm{F}}q\le \left|\omega \right|$.

Figure 9

Thomas-Fermi wave vector and plasmon dispersion as functions of density and temperature. We take ${\alpha }_{\mathrm{int}}=0.6$. (i) Thomas-Fermi wave vector [Eq. (B7)]. (ii) Plasmon dispersion [Eq. (B11)]. The dashed line depicts $\omega ={\mathsf{v}}_{\mathrm{F}}q$ for guiding the eyes.