Average free carrier mobility

In this model there are two types of electrons (holes), free electrons (holes) and trapped electrons (holes). Free electrons (holes) have a finite mobility of μ_e⁰ ( μ_h⁰) and trapped electrons (holes) can not move at all and have a mobility of zero. To calculate the average mobility we take the ratio of free to trapped carriers and multiply it by the free carrier mobility.:

$${\frac{{\mu_e^0 n_{free}}}{{n_{free}+n_{trap}}}}$$ (72)

Thus if all carriers were free, the average mobility would be μ_e⁰ and if all carriers were trapped the average mobility would be 0. It should be noted that only μ_e⁰ ( μ_h⁰) are used in the model for computation and μ_e(n) is an output parameter.

The value of μ_e⁰ ( μ_h⁰) is an input parameter to the model. This can be edited in the electrical parameter editor. The value of μ_e(n), and μ_h(p) are output parameters from the model. The value of μ_e(n), and μ_h(p) change as a function of position, within the device, as the number of both free and trapped charge carriers change as a function of position. The values of μ_e(x), and μ_h(x) can be found in mu$\_n$$\_ft$.dat and mu$\_p$$\_ft$.dat within the snapshots directory. The spatially averaged value of mobility, as a function of time or voltage can be found in the files dynamic$\_mue$.dat or dynamic$\_muh$.dat within the dynamic directory.

Were one to try to measure mobility using a technique such as CELIV or ToF, one would expect to get a value closer to μ_e(n) or μ_h(p) rather than closer to μ_e⁰ or μ_h⁰. It should be noted however, that measuring mobility in disordered materials is a difficult thing to do, and one will get a different experimental value of mobility depending upon which experimental measurement method one uses, furthermore, mobility will change depending upon the charge density profile within the device, and thus upon the applied voltage and light intensity. To better understand this, try for example doing a CELIV simulation, and plotting μ_e(n) as a function of time (Voltage). You will see that mobility reduces as the negative voltage ramp is applied, this is because carriers are being sucked out of the device. Then try extracting the mobility from the transient using the CELIV equation for extracting mobility. Firstly, the CELIV equation will give you one value of mobility, which is a simplification of reality as the value really changes during the application of the voltage ramp. Secondly, the value you get from the equation will almost certainly not match either μ_e⁰ or any value of μ_e(n). This simply highlights, the difficult of measuring a value of mobility for a disordered semiconductor and that really when we quote a value of mobility for a disordered material, it really only makes sense to quote a value measured under the conditions a material will be used. For example, for a solar cell, values of μ_e(n) and μ_h(n), would be most useful to know under 1 Sun at the P_max point on a JV curve.