Charge Carrier Mobility in Disordered Semiconductors

In crystalline semiconductors such as silicon or GaAs, charge carrier mobility is typically treated as a material parameter determined by scattering mechanisms and only weakly dependent on carrier density. In disordered semiconductors, however, the presence of localized states leads to trap-limited transport, so the effective mobility is governed by the balance between free and trapped carriers.

In the multiple trapping picture introduced in earlier sections, carriers occupy either extended states (mobile) or localized states (traps). The effective mobility therefore depends on the fraction of carriers that are free at a given position, energy, and time. Transport is controlled not only by the intrinsic mobility of free carriers but also by the dynamics of trapping and release, leading to a mobility that depends on carrier density and evolves during device operation.

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.

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.