Super-Planckian Electron Cooling in a van der Waals Stack

Radiative heat transfer (RHT) between macroscopic bodies at separations that are much smaller than the thermal wavelength is ruled by evanescent electromagnetic modes and can be orders of magnitude more efficient than its far-field counterpart, which is described by the Stefan-Boltzmann law. In this Letter, we present a microscopic theory of RHT in van der Waals stacks comprising graphene and a natural hyperbolic material, i.e., hexagonal boron nitride (hBN). We demonstrate that RHT between hot carriers in graphene and hyperbolic phonon polaritons in hBN is extremely efficient at room temperature, leading to picosecond time scales for the carrier cooling dynamics.

Figure 1

(a) A schematic view of the physical system: hot carriers in graphene (red spherical balls) efficiently radiate energy into phonon-polariton modes in nearby hyperbolic crystal slabs (semitransparent green parallelepipeds). (b),(c) Color plots of the dimensionless function $\mathcal{Z}(q,\omega )$ in Eq. (5) for an electron temperature $T_{\mathrm{e}}=300\mathrm{K}$ and top (bottom) hBN thickness $d^{\prime }=9\mathrm{nm}$ ($d=27\mathrm{nm}$). All other parameters are reported in Ref. [43]. (b) $n=5.0\times 10^{12}{\mathrm{cm}}^{-2}$. (c) $n=1.0\times 10^{12}{\mathrm{cm}}^{-2}$. Dashed lines indicate the edges of the graphene particle-hole continua [42]. In panel (b), the bottom edge of the interband continuum is not present since it occurs well above the hBN reststrahlen bands for $n=5\times 10^{12}{\mathrm{cm}}^{-2}$. The quantity $\mathcal{Z}(q,\omega )$ is maximum at the location of the poles of the dressed Coulomb interaction $V_{\mathit{q},\omega }$ (standing hBN phonon polaritons [19, 20]) and at the zeros of the dynamical dielectric function $ϵ(q,\omega )$ (plasmon-phonon polaritons [15, 21]).

Figure 2

(a) The cooling time ${\tau }^{*}$ (2) as a function of the electron density $n$, for different values of the electron temperature $T_e$. (b) The cooling time ${\tau }^{*}$ as a function of $T_e$ for different values of $n$.

Figure 3

(a) Intraband (solid line), interband (short-dashed line), and plasmon-phonon polariton (long-dashed line) contributions to the cooling rate $1/{\tau }^{*}$. The separate contributions to $1/{\tau }^{*}$ are plotted as functions of electron density. (b) A comparison between cooling times for RHT into hBN (blue line) and a nonhyperbolic crystal with identical phonon properties (grey-shaded area). RHT into hyperbolic phonon polaritons is clearly much more efficient. All data for ${\tau }^{*}$ have been calculated by setting $T_e=T_L=300\mathrm{K}$.

Figure 4

Cooling dynamics into hyperbolic phonon polaritons. We present typical results of the numerical solution of the complete heat equation, Eq. (1), where we retained the full dependence of $\tau (T_e,T_L)$ on $T_e(t)$. These data have been obtained by setting $T_e(0)=1000\mathrm{K}$, $T_L=1\mathrm{K}$ (solid line) and $T_L=300\mathrm{K}$ (dashed line), and $n=10^{12}{\mathrm{cm}}^{-2}$. We clearly see that, for $T_L=300\mathrm{K}$, equilibration with the lattice occurs exponentially fast.