Generalized balance equations for charged particle transport via localized and delocalized states: Mobility, generalized Einstein relations, and fractional transport

A generalized phase-space kinetic Boltzmann equation for highly nonequilibrium charged particle transport via localized and delocalized states is used to develop continuity, momentum, and energy balance equations, accounting explicitly for scattering, trapping and detrapping, and recombination loss processes. Analytic expressions detail the effect of these microscopic processes on mobility and diffusivity. Generalized Einstein relations (GER) are developed that enable the anisotropic nature of diffusion to be determined in terms of the measured field dependence of the mobility. Interesting phenomena such as negative differential conductivity and recombination heating and cooling are shown to arise from recombination loss processes and the localized and delocalized nature of transport. Fractional transport emerges naturally within this framework through the appropriate choice of divergent mean waiting time distributions for localized states, and fractional generalizations of the GER and mobility are presented. Signature impacts on time-of-flight current transients of recombination loss processes via both localized and delocalized states are presented.

Figure 1

Phase-space diagram illustrating the collision, trapping, detrapping, and recombination processes [16].

Figure 2

The impact of free and trapped particle recombination on current transients for an ideal time-of-flight experiment as modeled by Eq. (16). Nondimensionalization has been performed using the material thickness $L$, trap-free transit time $t_{\mathrm{tr}}\equiv L/W$, and initial current $j\left(0\right)=eN\left(0\right)/t_{\mathrm{tr}}$. For these plots we define the diffusion coefficient $D{t}_{\mathrm{tr}}/L^2=0.02$, the initial impulse location $x_0/L=1/3$, and the trapping rate ${\nu }_{\mathrm{trap}}{t}_{\mathrm{tr}}={10}^2$. We choose an exponential distribution of trapping times, $\varphi \left(t\right)={\nu }_{\mathrm{detrap}}{\mathrm{e}}^{-{\nu }_{\mathrm{detrap}}\mathrm{t}}$, with the mean trapping time chosen as ${\left({\nu }_{\mathrm{detrap}}{t}_{\mathrm{tr}}\right)}^{-1}=0.03$.

Figure 3

Plots of drift velocity, Eq. (42), and mean energy, Eq. (44), against electric field for a situation in which negative differential conductivity arises. All quantities have been nondimensionalized with respect to the mean energy without a field applied, ${\varepsilon }^{☆}\equiv \frac{3}{2}{k}_{\mathrm{B}}{T}_{\mathrm{eff}}\left({\varepsilon }^{☆}\right)$. Specifically, we have chosen to nondimensionalize using $W^{☆}\equiv \sqrt{\frac{{\varepsilon }^{☆}}{m}}$ and $E^{☆}\equiv \frac{m{\nu }_{\mathrm{eff}}\left({\varepsilon }^{☆}\right)}{e}{W}^{☆}$. For this figure, we consider a constant collision frequency, ${\nu }_{\mathrm{coll}}\left(\varepsilon \right)=1$, and a trapping frequency that approximates a step function, $R{\nu }_{\mathrm{trap}}\left(\varepsilon \right)=\frac{1}{2}\left\{1+tanh\left[5\left(\varepsilon -{\varepsilon }_{\mathrm{thresh}}\right)\right]\right\}\approx H\left(\varepsilon -{\varepsilon }_{\mathrm{thresh}}\right)$, turning on at the threshold energy ${\varepsilon }_{\mathrm{thresh}}=6$. In addition, Maxwellian temperatures have been chosen such that $k_{\mathrm{B}}{T}_{\mathrm{coll}}=1$ and $k_{\mathrm{B}}{T}_{\mathrm{detrap}}=5$.

Figure 4

The impact of free and trapped particle recombination on current transients for an ideal time-of-flight experiment as modeled by Eq. (16) for the case of dispersive transport. Nondimensionalization has been performed using the material thickness $L$, trap-free transit time $t_{\mathrm{tr}}\equiv L/W$, and initial current $j\left(0\right)=eN\left(0\right)/t_{\mathrm{tr}}$. For these plots we define the diffusion coefficient $D{t}_{\mathrm{tr}}/L^2=0.02$, the initial impulse location $x_0/L=1/3$, and the trapping rate ${\nu }_{\mathrm{trap}}{t}_{\mathrm{tr}}={10}^4$. For dispersive transport to occur we have chosen to describe trapping times by the heavy-tailed distribution (112) with a trap severity of $\alpha =1/2$. This corresponds specifically to the distribution $\varphi \left(t\right)=\frac{1}{2t}\left(\frac{\sqrt{\pi }}{2}\frac{\mathrm{erf}\sqrt{{\nu }_{0}t}}{\sqrt{{\nu }_{0}t}}-{\mathrm{e}}^{-{\nu }_{0}t}\right)$, where we have chosen ${\nu }_0{t}_{\mathrm{tr}}=5\times {10}^5$. The exponents of the power-law regimes are indicated with arrows. Such regimes, especially at late times, can be indicative of dispersive transport.