Hot-electron noise properties of graphene-like systems

We study the hot-electron noise properties of two-dimensional materials with a graphene-like energy dispersion under a strong applied electric field which drives the system far from equilibrium. Calculations are based on a Boltzmann–Green-function method within a two-relaxation-time approximation that allows for both inelastic scattering coming from electron-phonon scattering and elastic scattering coming from electron-impurity scattering. The steady-state distribution function is used to calculate the average current and the low-frequency spectral density for current fluctuations (noise) in the nonequilibrium steady-state. We find that as the electric field strength increases, the noise decreases from its equilibrium thermal noise value. This is in contrast with semiconductors with a quadratic energy–wave-vector dispersion where the noise increases in a constant-relaxation-time model with the square of the electric field due to the Joule heating of the electron gas by the electric field. We have also studied these properties for an electronic dispersion with a gap introduced into the Dirac spectrum. The inclusion of the gap in the electronic dispersion causes an initial increase in the noise as a function of external electric field due to the heating of the electron gas for large gap values. At high electric fields, the noise decreases with increasing electric field as in the case of gapless dispersion at higher fields.

Figure 1

The diagram shows the effect of the two distinct scattering mechanisms. Inelastic scattering allows for change in energy by scattering between states of different energy while elastic scattering corresponds to scattering between different momentum states with the same energy.

Figure 2

Comparison between contour plots for the steady-state distribution function $f\left(\vec{k}\right)$ on varying the scattering-rate ratio $R$ and field $E$ for a fixed electron density $n={10}^{11}{\text{cm}}^{-2}$ at $T=300\text{K}$. (a) $E=0$ kV/cm, $R=0$, (b) $E=1$ kV/cm, $R=0$, (c) $E=10$ kV/cm, $R=0,$ (d) $E=0$ kV/cm, $R=5$, (e) $E=1$ kV/cm, $R=5,$ and (f) $E=10$ kV/cm, $R=5$, for Dirac-like dispersion. The plots indicate that the presence of a strong elastic scattering mechanism takes momentum from the field direction and transfers it over the constant-energy surface. Note: The color-bar scales are different for different fields.

Figure 3

Steady-state distribution vs $k_x$ at fixed $k_y=0$ for different electric field strengths at $T=300$ K. For number density $n={10}^{11}{\text{cm}}^{-2}$. (a) Scattering ratio $R=0$, meaning the presence of only inelastic scattering; (b) scattering ratio $R=5$, meaning the presence of strong elastic scattering. For comparison purposes the case of a higher carrier density $n={10}^{12}{\text{cm}}^{-2}$ is considered. (c) and (d) show the equilibrium Fermi-Dirac distribution and the effects of scattering-rate ratio $R$ on the steady-state distribution.

Figure 4

Comparison between contour plots for the steady-state distribution function $f\left(\vec{k}\right)$ on varying the scattering-rate ratio $R$ and field $E$ for a fixed electron density $n={10}^{11}{\text{cm}}^{-2}$ at $T=300\text{K}$. (a) $E=0$ kV/cm, $R=0$, (b) $E=1$ kV/cm, $R=0$, (c) $E=10$ kV/cm, $R=0$, (d) $E=0$ kV/cm, $R=5$, (e) $E=1$ kV/cm, $R=5$, and (f) $E=10$ kV/cm, $R=5$, for gapped Dirac-like dispersion $\left(\Delta =60\text{meV}\right)$. The plots indicate that the presence of a strong elastic scattering mechanism takes momentum from the field direction and transfers it over the constant-energy sufrace. Note: The color-bar scales are different for different fields.

Figure 5

(a) Average $x$ component of velocity, and (b) zero-frequency noise versus electric field for a Dirac-like dispersion $\epsilon \left(\vec{k}\right)=\hslash v_{f}k$ on varying the value of $R$ (the ratio of elastic scattering rate to inelastic scattering rate) where $n={10}^{11}{\text{cm}}^{-2}$ and $T=300$ K. The plots for the case of a higher carrier density $n={10}^{12}{\text{cm}}^{-2}$ look extremely similar. Note: $S_o=4N{e}^2{v}_f^2{\tau }_{\mathrm{in}}/L^2$.

Figure 6

Toy diagram to show the occupancy of states in the Dirac cone under the effect of electric field and scattering-rate ratio. (a) $E=0$ kV/cm where the distribution is the equilibrium distribution function. (b) $E=10$ kV/cm and $R=0$, meaning a moderate field and just inelastic scattering, and the distribution drifts in the direction of the field, occupying large-$k_x$ states. (c) $E=10$ kV/cm and $R=5$, meaning a moderate field and the presence of large elastic scattering along with inelastic scattering, such that the distribution drifts in the direction of the field; however, the elastic scattering reduces the spread in the field direction.

Figure 7

(a) Average $x$ component of velocity, and (b) zero-frequency noise versus electric field for a gapped Dirac-like dispersion $\epsilon \left(\vec{k}\right)=\sqrt{{\Delta }^2/4+{\left(\hslash v_{f}k\right)}^2}$ on varying the value of $R$ (the ratio of the elastic scattering rate to the inelastic scattering rate) where $n={10}^{11}{\text{cm}}^{-2}$ and $T=300$ K. The value of the gap is $\Delta =60\text{meV}$. Note: $S_o=4N{e}^2{v}_f^2{\tau }_{\mathrm{in}}/L^2$.

Figure 8

(a) Average $x$ component of velocity, and (b) zero-frequency noise versus electric field for a gapped Dirac-like dispersion $\epsilon \left(\vec{k}\right)=\sqrt{{\Delta }^2/4+{\left(\hslash v_{f}k\right)}^2}$ on varying the value of $R$ (the ratio of the elastic scattering rate to the inelastic scattering rate) where $n={10}^{11}{\text{cm}}^{-2}$ and $T=300$ K. The value of gap is $\Delta =1000\text{meV}$. Note: $S_o=4N{e}^2{v}_f^2{\tau }_{\mathrm{in}}/L^2$.