Mobility in disordered semiconductors
In earlier sections we introduced the density of states (DoS) and the multiple trapping model, where carriers can occupy either extended states (free carriers) or localized states (traps). This picture naturally leads to the realization that mobility in a disordered semiconductor cannot be treated as a constant material parameter. Instead, it depends on how many carriers are free and how many are trapped at any given position, energy, and time. Understanding this dependence is essential, because transport in disordered materials is governed not only by the intrinsic mobility of free carriers but also by the dynamics of trapping and release. This section shows how the model accounts for these effects and why the resulting “effective mobility” is both carrier-density dependent and time-dependent.
Mobility and carrier density
In crystalline semiconductors, carrier mobility is often approximated as a constant. In disordered materials, however, carriers can either be free, with finite mobilities \(\mu_e^0\) and \(\mu_h^0\), or trapped, with zero mobility. The average mobility therefore depends on the ratio of free to trapped carriers:
\[ \mu_e(n) = \frac{\mu_e^0 \, n_{\mathrm{free}}}{n_{\mathrm{free}} + n_{\mathrm{trap}}}. \]
If all electrons are free, the mobility equals \(\mu_e^0\); if all are trapped, the effective mobility is zero. In practice, the fraction of free carriers changes with carrier density, so the mobility varies across the device and under different bias or illumination conditions. This dependence is crucial: without it, the model would miss the dominant transport physics of disordered semiconductors.
Why does density matter?
Mobility determines how efficiently carriers move through the device and are collected at the contacts. In trap-dominated materials, the effective mobility drops whenever a significant fraction of carriers are trapped. Correctly capturing this density dependence is essential for predicting J–V curves, recombination rates, and transient responses.
Mobility as a dynamic quantity
Because the free–trapped carrier balance depends on operating conditions, mobility is also a dynamic quantity. Transient techniques such as CELIV or ToF highlight this clearly. In a CELIV simulation, for example, the effective mobility \(\mu_e(n)\) decreases during the negative voltage ramp: as carriers are extracted, fewer remain free, and the apparent mobility drops. If one then applies the standard CELIV analysis equation to extract a single value, the result will not match either the input mobility \(\mu_e^0\) or the instantaneous values of \(\mu_e(n)\) during the ramp.
This illustrates a general principle: in disordered semiconductors, mobility is not a fixed constant but a property
that evolves with time, voltage, illumination, and measurement method. The model therefore outputs effective mobilities
\(\mu_e(n)\) and \(\mu_h(p)\) as functions of position and time, stored in
mu_n_ft.dat, mu_p_ft.dat, dynamic_mue.dat, and dynamic_muh.dat.
These values reflect the actual transport conditions inside the device.
The practical implication is that when quoting a “mobility” for a disordered semiconductor, it is meaningful only under the operating conditions of interest. For example, in an organic solar cell the most relevant mobilities are those under 1 Sun illumination, near the maximum power point of the J–V curve. Using a single constant number taken out of context can be misleading.