Electron and Optical Phonon Temperatures in Electrically Biased Graphene

We examine the intrinsic energy dissipation steps in electrically biased graphene channels. By combining in-situ measurements of the spontaneous optical emission with a Raman spectroscopy study of the graphene sample under conditions of current flow, we obtain independent information on the energy distribution of the electrons and phonons. The electrons and holes contributing to light emission are found to obey a thermal distribution, with temperatures in excess of 1500 K in the regime of current saturation. The zone-center optical phonons are also highly excited and are found to be in equilibrium with the electrons. For a given optical phonon temperature, the anharmonic downshift of the Raman $G$ mode is smaller than expected under equilibrium conditions, suggesting that the electrons and high-energy optical phonons are not fully equilibrated with all of the phonon modes.

Figure 1

(a) Source-drain current-voltage characteristics at zero gate bias of the $3.6\times 1.6\mu {\mathrm{m}}^2$ graphene channel shown in the top inset (the scale bar is $2\mu \mathrm{m}$). The lower inset shows the back-gate dependence of the channel conductivity ($\sigma$). (b) Spectral radiance of the same graphene sample as a function of the electrical power per unit area $P$ dissipated in the channel (symbols). The spectra are fit to Planck’s law (solid lines). In the fits, the thickness of the ${\mathrm{SiO}}_2$ layer was fixed at $280\pm 2\mathrm{nm}$. The dips around 2.33 eV are artifacts arising from the notch filter used for the Raman measurements [Fig. 2]. (c) Radiance for photon energies from 1.33–2.59 eV as a function of the electronic temperature ($T_{\mathrm{el}}$) deduced from the fits in (b). The solid line is a fit based on Planck’s law.

Figure 2

(a) Broadband Raman spectra of the same graphene channel as in Fig. 1 under zero- and high- source-drain biases. The inset is a closeup showing the weak $D$ mode. (b) Anti-Stokes and (c) Stokes Raman $G$ modes (symbols). The solid lines are Voigt fits. The spectra are vertically offset for clarity. All spectra were acquired with the same integration time. The dissipated electrical power per unit area ($P$) and the optical phonon temperature ($T_{\mathrm{op}}$) are indicated.

Figure 3

Electronic ($T_{\mathrm{el}}$, squares) and optical phonon ($T_{\mathrm{op}}$, circles) temperatures as a function of the dissipated electrical power. The dashed line is a guide to the eye, based on a scaling of the temperature as $\sqrt{P}$.

Figure 4

(a) Frequency (${\omega }_G$) and (b) Lorentzian full width at half maximum (${\Gamma }_G$) of the $G$ mode feature as a function of $T_{\mathrm{op}}$. Data are extracted from Voigt fits, taking into account a constant Gaussian contribution ($13{\mathrm{cm}}^{-1}$) due to our setup spectral resolution. The thick solid lines in (a),(b) are the theoretically predicted temperature dependences of ${\omega }_G$ and ${\Gamma }_G$ under equilibrium conditions [21]. The dashed line in (a) is a linear extrapolation of the results in Ref. 21, neglecting higher-order anharmonic terms. To compute the solid line in (b), we considered a residual charge density of ${10}^{12}{\mathrm{cm}}^{-2}$ and an anharmonic decay rate of $2.5{\mathrm{cm}}^{-1}$ at $T=300\mathrm{K}$ [25].