Anomalous Thermodiffusion of Electrons in Graphene

We reveal a dramatic departure of electron thermodiffusion in solids relative to the commonly accepted picture of the ideal free-electron gas model. In particular, we show that the interaction with the lattice and impurities, combined with a strong material dependence of the electron dispersion relation, leads to counterintuitive diffusion behavior, which we identify by comparing a two-dimensional electron gas (2DEG) and single-layer graphene. When subject to a temperature gradient $\nabla T$, thermodiffusion of massless Dirac fermions in graphene exhibits an anomalous behavior with electrons moving along $\nabla T$ and accumulating in hot regions, in contrast to normal electron diffusion in a 2DEG with parabolic dispersion, where net motion against $\nabla T$ is observed, accompanied by electron depletion in hot regions. These findings bear fundamental importance for the understanding of the spatial electron dynamics in emerging materials, establishing close relations with other branches of physics dealing with electron systems under nonuniform temperature conditions.

Figure 1

(a) Illustration of a two dimensional system illuminated by a focused continuous-wave laser beam to locally heat electrons (red region). (b),(c) Energy dispersion and group velocity of (b) free electrons in a 2DEG and (c) MDFs in graphene. The electron diffusion direction is determined by the temperature dependence of the diffusivity coefficient $\mathcal{D}=⟨v^2⟩⟨\tau ⟩$, which displays the qualitative behavior summarized in (a) (right text), as derived from the electron group velocity in (b) and (c). Normal and anomalous electron diffusion are predicted in the 2DEG and graphene, respectively.

Figure 2

Electron diffusion coefficient $\mathcal{D}$ in (a) a 2DEG and (b) graphene as a function of temperature. Each curve only considers one scattering mechanism caused by impurities (black), acoustic phonons (blue), or optical phonons (red). The temperature dependence of $\mathcal{D}$ supports the conclusion anticipated in Fig. 1. The Fermi level and impurity density are assumed to be $E_F=0.2\mathrm{eV}$ and $n_i=2\times {10}^{11}{\mathrm{cm}}^{-2}$ in both materials (see main text for other parameters).

Figure 3

Thermoelectric field acting on uniform (a)–(d) 2DEG and (e)–(h) graphene films subject to a temperature Gaussian distribution $T(x)=1700[\mathrm{K}]e^{-x^2/w^2}+300[\mathrm{K}]$ with $w=1\mu \mathrm{m}$ [see insets in (c),(g)]. We consider different models for scattering: (a),(e) a constant scattering rate, (b),(f) impurity scattering, (c),(g) acoustic-phonon scattering, and (d),(h) optical-phonon scattering. Electron diffusion is determined by the total thermoelectric field ${\overrightarrow{\mathcal{E}}}_{\mathrm{tot}}$ (black curves), which is the sum of the electromotive field associated with the Seebeck effect ${\overrightarrow{\mathcal{E}}}_{\mathrm{emf}}$ (red curves) and an effective field ${\overrightarrow{\mathcal{E}}}_{\mu }$ (blue curves), arising from the temperature dependence of the chemical potential. The force $-e{\overrightarrow{\mathcal{E}}}_{\mathrm{tot}}$ drives electron diffusion opposite and along the temperature gradient in the 2DEG and graphene, respectively, for all scattering mechanisms considered (see signs of ${\overrightarrow{\mathcal{E}}}_{\mathrm{tot}}$), except when assuming an unrealistic constant relaxation time in graphene [see (e)], which leads to ${\overrightarrow{\mathcal{E}}}_{\mathrm{tot}}=0$. All parameters are the same as in Fig. 2.

Figure 4

(a) Illustration of a Gaussian temperature distribution in a 2D system: $T(x)=1700[\mathrm{K}]e^{-x^2/w^2}+300[\mathrm{K}]$ with $w=1\mu \mathrm{m}$, produced by focused laser heating (inset). (b),(c) Steady-state electron density in (b) a 2DEG and (c) graphene under the temperature distribution in (a). Here, $n_0$ is the unperturbed electron density, determined by the Fermi level $E_F=0.2\mathrm{eV}$. (d) Amplitude (color scale) and orientation (arrows) distributions of the static electric fields ${\overrightarrow{\mathcal{E}}}_{\text{Coul}}$ generated by the thermally excited nonuniform electron charges [color in (b),(c)]. The 2DEG and graphene are located at $z=0$. The three scattering mechanisms by impurities and phonons shown in Fig. 3 are all included (i.e., ${\tau }^{-1}={\tau }_{\text{im}}^{-1}+{\tau }_{\text{ac}}^{-1}+{\tau }_{\text{op}}^{-1}$) using the same calculation parameters.