Carrier scattering and impact ionization in bilayer graphene

Abstract

Transport properties of carriers in bilayer graphene (BLG) were studied. Several analytical models were developed for drift velocity, scattering rate and ionization coefficient of BLG for the first time. Then, the joint effect of temperature and potential difference of layers were addressed on the modeled parameters. The accuracy of the proposed models for drift velocity and scattering rate was verified by the simulation results of published works. In addition, the analytical results of ionization coefficient of BLG were compared with those of silicon.

Appendix

Plugging (7) in (6) and converting summation into integral:

$$\begin{aligned} \frac{1}{\tau_m(E)} =&\frac{2}{\hbar\pi} \biggl( \frac{D_A^2\hbar\omega _{\beta}}{\rho_m A v_s^2} \biggr) N_{\beta} \\ &{}\times \int_0^{\infty} \delta \biggl(E_c+\frac{\hbar}{2m^*}\bigl(|k|-k_g\bigr)^2-E \biggr)kdk \\ & {}\times\frac{1}{2}\int_0^{2\pi}(1-\cos \theta) (1+\cos\theta)d\theta \end{aligned}$$

(23)

According to the definition of Dirac delta function:

$$ \int\delta\bigl(x-x^{\prime}\bigr)f\bigl(x^{\prime} \bigr)dx^{\prime}=f(x) $$

(24)

the (23) can be rewritten as:

$$\begin{aligned} \frac{1}{\tau_m(E)} =&\frac{A}{2\hbar} \biggl(\frac{D_A^2\hbar\omega _{\beta}}{\rho_m A v_s^2} \biggr) N_{\beta}\sqrt{\frac {2m^*}{E-E_c}} \\ &{}\times\int_0^{\infty}\delta(s- \sqrt{E-E_c}) \biggl(\frac{\sqrt{2m^*}}{\hbar}s+k_g\biggr)ds \end{aligned}$$

(25)

where

$$ s=\frac{\hbar}{\sqrt{2m^*}}\bigl(|k|-k_g\bigr) $$

(26)