Slow imbalance relaxation and thermoelectric transport in graphene

We compute the electronic component $\left(\kappa \right)$ of the thermal conductivity and the thermoelectric power $\left(\alpha \right)$ of monolayer graphene within the hydrodynamic regime, taking into account the slow rate of carrier population imbalance relaxation. Interband electron-hole generation and recombination processes are inefficient due to the nondecaying nature of the relativistic energy spectrum. As a result, a population imbalance of the conduction and valence bands [i.e., a nonequilibrium state with ${\mu }_e+{\mu }_h\ne 0$, where ${\mu }_e$ $\left({\mu }_h\right)$ denotes the electron (hole) chemical potential] is generically induced upon the application of a thermal gradient. We show that the thermoelectric response of a graphene monolayer depends upon the ratio of the sample length to an intrinsic length scale $l_Q$ set by the imbalance relaxation rate. At the same time, we incorporate the crucial influence of the metallic contacts required for the thermopower measurement (under open circuit boundary conditions) since carrier exchange with the contacts also relaxes the imbalance. These effects are especially pronounced for clean graphene, where the thermoelectric transport is limited exclusively by intercarrier collisions. For specimens shorter than $l_Q$, the population imbalance extends throughout the sample; $\kappa$ and $\alpha$ asymptote toward their zero imbalance relaxation limits. In the opposite limit of a graphene slab longer than $l_Q$, at nonzero doping $\kappa$ and $\alpha$ approach intrinsic values characteristic of the infinite imbalance relaxation limit. Samples of intermediate (long) length in the doped (undoped) case are predicted to exhibit an inhomogeneous temperature profile, while $\kappa$ and $\alpha$ grow linearly with the system size. In all cases except for the shortest devices, we develop a picture of bulk electron and hole number currents that flow between thermally conductive leads, where steady-state recombination and generation processes relax the accumulating imbalance. Our analysis incorporates, in addition, the effects of (weak) quenched disorder.

Figure 1

Kinematical constraints for imbalance relaxation. The left figure shows the Feynman diagram for the typical two-particle decay process given by Eq. (1.1); ${\epsilon }_i$ and ${\mathbf{p}}_i$ respectively denote the energy and momentum of the $i\text{th}$ electron or hole. The right figure depicts momentum conservation for this process. The length of the dashed path $L_f\equiv \left|{\mathbf{p}}_2\right|+\left|{\mathbf{p}}_3\right|+\left|{\mathbf{p}}_4\right|$ traced out by the sum of “decay product” momenta is always greater than or equal to the length $L_i\equiv \left|{\mathbf{p}}_1\right|$ of the “parent” particle momentum: $L_f\ge L_i$. If the particle energy spectrum takes the form $\epsilon \left(\mathbf{p}\right)={\left|\mathbf{p}\right|}^{\beta }$, then the depicted decay process is kinematically forbidden for $\beta <1$. For $\beta =1$, only forward scattering is allowed.

Figure 2

(Color online) Schematic setup for the two terminal thermopower and thermal conductivity measurements. The graphene slab of length $L$ is denoted by the dashed gray line. The contacts at $x=0,L$ are held at temperatures $T=T_1,T_2$, respectively, via external thermostats. The two contacts are electrically bridged by a highly resistive wire represented in the figure by the thick black interconnect. The development of a thermoelectric response $V\left(L\right)-V\left(0\right)\ne 0$ in the graphene induces a voltage drop $\Delta V_w\equiv V_w^L-V_w^0$ in the wire, which drives the current $J$. Measurement of $J$ (via galvanometer) and knowledge of the wire resistance allows computation of the graphene thermopower, which is simply the ratio of $\Delta V_w$ to the temperature drop $\Delta T=T_1-T_2$, in the limit of infinite wire resistance.

Figure 3

This figure depicts contact (“built-in”) potentials $V_{\text{bi}}$ that arise through equilibration of graphene and the wire interconnect (voltmeter) depicted in Fig. 2. (a) and (b) depict the electrochemical equilibration of electron-doped graphene and the metallic interconnect, while (c) and (d) show the same for the hole-doped case. All chemical potentials are measured from the position of the Dirac point; in this figure, we have assumed that ${\mu }_w>0$, i.e., the work function of the metal is less than the electron affinity of the graphene. The notion of a negative hole chemical potential $-{\mu }_w$ for the carriers in the wire is easily understood via (c): before electrochemical equilibration, the chemical potential difference ${\mu }_h-\left(-{\mu }_w\right)$ drives holes in the graphene to “float up” toward the hole Fermi sea in the metal. Given the assumption of local electroneutrality throughout both the graphene and metal, the tunneling holes must recombine with electrons on the surface of the metal, leading to the accumulation of a dipole charge layer at the boundary. As a result, a static built-in voltage $V_{\text{bi}}=-\frac{1}{e}\left({\mu }_w+{\mu }_h\right)$ develops between the wire and graphene which precisely offsets the intrinsic chemical potential difference. Of course, this built-in voltage is not directly measurable with a voltmeter.

Figure 4

The top graph (a log-log plot) depicts the qualitative form of the thermal “conductivity” $\kappa \equiv L{G}_{\text{th}}/W$ [Eq. (3.4)] versus sample length $L$ for an undoped graphene strip possessing both slow imbalance relaxation and weak quenched disorder. Three manners of functional behavior for $\kappa$ are demarcated with dotted, solid, and dashed line segments; for each of these, the bottom plots show representative spatial profiles of the temperature $T\left(x\right)-\overline{T}=\Psi \left(x\right)$ [Eq. (2.40)] and number current density $J_n\left(x\right)$ [Eq. (2.43)]. For system sizes $l_Q\lesssim L\lesssim l_{\infty }$ (solid curves), $\kappa$ grows linearly with $L$, while the temperature profile is inhomogeneous; this regime is also characterized by the maximal particle flux $J_n$, as measured at the device center $x=L/2$. By contrast, in the short $\left(L⪡l_Q\right)$ and long $\left(L⪢l_{\infty }\right)$ sample size limits, respectively designated by dotted and dashed curves in the figure, $\kappa$ saturates to ${\kappa }_{\circ }$ and ${\kappa }_{\infty }$, respectively; here, the temperature profile asymptotes to a linear gradient. The assumption of infinite contact resistivity [Eq. (3.3)] ensures that $J_n\left(x\right)$ vanishes at $x=0,L$.