Hui Zhao, PhD , University of Kansas
One important aspect of charge carrier dynamics in graphene is the role played by hot carriers. It has been shown that the mean-free path of carriers in graphene is several 100 nm even at room temperature. Therefore, even in devices with a channel length as long as of 1 µm, injected carriers only undergo few or even no phonon scattering events during the transport. In transport measurements performed with an applied voltage on the order of a fraction of one volt, the carriers can have a temperature of several thousand kelvins. To develop nanoscale devices of graphene, it is necessary to understand and control hot carrier dynamics.
In the first year of this project, we have used ultrafast laser techniques to study two of the key issues on hot carrier dynamics in graphene: the diffusion and the energy relaxation of hot carriers.
1. Hot carrier diffusion in graphene
Diffusion of hot carriers is studied on three types of graphene samples: epitaxial graphene, CVD graphene, and reduced graphene oxide. All types of graphene are of great technological relevance. The experimental approach is rather straightforward. Carriers are first excited with a 100-fs pump laser pulse that is incident normal to the graphene layer, with a spot size of 1.6 µm. The spatial density profile of excited carriers is initially thin, but after a short time, the carriers diffuse out of the excitation spot, which results in a broadening of the profile. In this process, electrons and holes move as pairs due to the Coulomb attraction between them. For a classical diffusion process with a Gaussian initial profile, the squared width increases linearly with time, with a slope determined by the diffusion coefficient.
We monitor the carrier diffusion process by using a time-delayed probe pulse with a spot size of 1.2 µm. The differential transmission of the probe pulse, that is the change of the transmission caused by the carriers, is proportional to the carrier density. Hence, by measuring the differential transmission as a function of the probe delay (the delay time between the probe and the pump pulses) and the probe location with respect to the pump spot, we can monitor the spatiotemporal evolution of the carrier density, and measure the diffusion coefficient. We found that the carrier diffusion coefficients of about 40000, 11000, and 6000 squared centimeter per second for the CVD, epitaxial, and thermally reduced samples. We found that the diffusion coefficient only changes slightly when the sample temperature changes from 10 to 300 Kelvin. At room temperature, the diffusion coefficient in these graphene samples is two to three orders of magnitude higher than the conventional semiconductors.
For a thermalized system, the diffusion coefficient is related to the mobility by Einstein’s relation. Due to the ultrafast thermalization process in graphene, the carriers can be treated as a thermalized system. However, the diffusion occurs during the energy relaxation process. During the time range of the measurement, the temperature of the hot carriers changes from 4300 to 2900 Kelvin. To estimate the mobility corresponding to the measured diffusion coefficients, we use an average temperature in this range of 3600 Kelvin. The deduced mobility for the epitaxial graphene samples is 35000 squared centimeters per volt second. The result is reasonably consistent with previously reported values.
2. Energy relaxation and optical phonon emission of hot carriers in graphene
Recently, energy relaxation of the hot carriers in graphene has been studied by several groups by fitting the decay of a differential transmission signal. This tool can be very valuable as it can give much insight into the time scales and mechanisms behind the relaxation of carriers, but it also has its limitations. First, the decay time is often comparable to the temporal width of the probe pulse. Second, a slow decay component related to carrier recombination is often seen, making the decay multiple exponential. Both of these factors limit the accuracy of such measurements in determining the energy relaxation time of excited carriers.
We used a two-color probe scheme to accurately measure the energy relaxation rate of hot carriers in graphene samples. First, carriers are injected by a pump pulse. By precisely overlapping the two probes in time, and measuring the time between which the differential transmission signals peak with each probe, we are able to monitor the density of carriers at two different energies for various times after excitation. Under the assumption that the carriers rapidly thermalize after excitation, this can be interpreted as an average carrier energy decrease from 0.72 to 0.36 eV in approximately 47 fs with a carrier density of 2.3E13 per squared centimeters. This corresponds to an energy relaxation rate of about 8 meV/fs. Since the energy relaxation of carriers is mainly caused by the emission of 195 meV G-mode optical phonons, we deduce an optical phonon emission time of about 25 fs. Furthermore, we found that the optical phonon emission time decreases from about 50 to about 20 fs, and the energy relaxation rate increases from 4 to 10 meV/fs, as the carrier density is increased from 1.5E12 to 3E13 per squared centimeters. The observed density dependence is inconsistent with the phonon bottleneck effect that was previously observed.