Hot-carrier cooling length and cooling time in graphene: direct determination beyond the 2-micron lower bound and their dependence on carrier density, temperature, and excitation power

When graphene absorbs light, the photoexcited carriers rapidly thermalize among themselves to an effective electron temperature far above the lattice temperature, and energy leaves this electronic system only slowly through electron-phonon coupling. The spatial decay of the hot-electron temperature is set by a cooling length $\xi=\sqrt{D\tau}$, where $D$ is the electronic thermal diffusivity and $\tau$ the electron-lattice cooling time; $\xi$ is the distance over which hot carriers transport energy before equilibrating with the lattice. In the originating experiment (Gabor et al., Science 334, 648, 2011), dual-gated monolayer and bilayer graphene p-n junction devices at T = 40 K showed nonlocal, hot-carrier-dominated intrinsic photoresponse, but device dimensions allowed only a lower bound $\xi>2\,\mu\mathrm{m}$ on the cooling length. Later work has measured electron-lattice cooling times across femtosecond-to-nanosecond scales and identified the dominant cooling channels (disorder-assisted acoustic-phonon supercollision cooling, optical-phonon emission, and out-of-plane coupling to hyperbolic phonon polaritons in hexagonal boron nitride), and individual cooling lengths have been extracted in several platforms (approximately 0.7-1.1 microns at 10 K in graphene on SiO2, about 140 nm at room temperature in CVD graphene, and about 2.68 microns in WSe2-encapsulated graphene). The open problem is a direct, systematic spatial determination of the cooling length in clean graphene devices as a function of carrier density, lattice temperature, and excitation power, and of the maximum distance over which hot carriers can transport energy before thermalizing with the lattice (the largest achievable value of this cooling length).

Background

Graphene is a single atomic layer of carbon whose charge carriers have a linear, electron-hole-symmetric (Dirac-like) energy dispersion and a vanishing density of states at the charge-neutrality point. When graphene absorbs a photon, the excited electron-hole pair shares its energy with the other carriers through fast electron-electron scattering, so the electronic population can be described by an effective electron temperature $T_e$ well above the lattice temperature $T_L$; such an out-of-equilibrium population is called a hot-carrier population. Energy leaves this electronic system only through comparatively weak electron-phonon coupling, a hot-carrier bottleneck that is unusually pronounced in graphene because of its low electronic density of states. If a temperature difference exists between two regions with different Seebeck coefficients $S$ (the thermovoltage developed per unit temperature difference), a thermoelectric voltage and current appear; when the temperature gradient is produced by absorbed light this is called the photothermoelectric (PTE) effect. In dual-gated graphene p-n junctions, a global bottom gate and a local top gate, insulated by a dielectric (in the source experiment, a hexagonal boron nitride, hBN, flake), tune the sign and density of carriers on each side of the junction independently. Gabor et al. (2011) scanned a focused 850-nm laser spot over such monolayer and bilayer graphene devices and found photocurrent extending over several microns from the illuminated p-n interface toward the metal contacts, together with striking six-fold photovoltage patterns versus the two gate voltages. These observations established that the intrinsic photoresponse of graphene is dominated by nonlocal transport of a long-lived, spatially distributed hot-carrier population rather than by charge separation in a built-in electric field (the photovoltaic effect). The spatial profile of the electronic temperature follows a steady-state heat-diffusion equation with a decay term, $d^2T/dx^2=(T-T_L)/\xi^2$, where the cooling length $\xi=\sqrt{\kappa/(\gamma C_{\rm el})}=\sqrt{D\tau}$ is set by the electronic thermal conductivity $\kappa$ (related to the electrical conductivity through the Wiedemann-Franz law), the electronic heat capacity $C_{\rm el}$, the electron-lattice cooling rate $\gamma$, the thermal diffusivity $D=\kappa/C_{\rm el}$, and the electron-lattice cooling time $\tau=1/\gamma$. Physically, $\xi$ is the length over which hot carriers carry thermal energy away from an absorbed photon before that energy is transferred to lattice vibrations. Theory available at the time predicted very slow cooling: in the disorder-free (clean) limit, acoustic-phonon cooling gives a cooling power growing as $T_e^4$ (Bistritzer and MacDonald, 2009), whereas disorder relaxes momentum conservation and enables faster so-called supercollision cooling with cooling power growing as $T_e^3$ (Song, Reizer and Levitov, 2012). Fits of the heat-equation solutions to the measured photocurrent profiles gave $\xi>2\,\mu\mathrm{m}$ at $T=40$ K, but because the devices themselves were only a few microns across, only this lower bound, not the cooling length itself, could be established. Subsequent work showed that $\xi$ is not a fixed material constant: it depends on disorder, on the substrate and encapsulation (energy can leak out of plane, for example into hyperbolic phonon polaritons, which are hybrid light-matter lattice modes of hBN, or into nearby metal gates), on carrier density, temperature, and absorbed power, and it directly sets the responsivity and speed of graphene PTE photodetectors, thermoelectric energy harvesters, and hot-electron calorimeters and bolometers.

Problem Statement

Determine, directly and quantitatively, the hot-carrier cooling length $\xi$ (the characteristic spatial decay length of the hot-electron temperature, $\xi=\sqrt{D\tau}$ with $D$ the electronic thermal diffusivity and $\tau$ the electron-lattice cooling time) and the corresponding cooling time $\tau$ in graphene, going beyond the lower bound $\xi>2\,\mu\mathrm{m}$ established at $T=40$ K in dual-gated monolayer and bilayer graphene p-n junction devices, whose limited dimensions permitted only a bound (Gabor et al., 2011). Concretely: (i) measure the actual cooling length in devices large enough that the hot-carrier temperature profile decays before reaching the contacts, and thereby establish the maximum distance over which hot carriers can transport energy before thermalizing with the lattice (the largest achievable value of $\xi$), including, but not limited to, the clean, weakly disordered and dielectric-engineered limits; and (ii) characterize how $\xi$ and $\tau$ depend on carrier density (gate voltage), lattice temperature, and excitation power. The determination also identifies which of the cooling channels that carry different dependences dominate the measured $\tau$ in the device studied: in-plane electronic heat diffusion to the contacts, disorder-assisted acoustic-phonon (supercollision) cooling, optical-phonon emission, and out-of-plane coupling to substrate or encapsulation phonon modes (for example hyperbolic phonon polaritons in hBN).

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