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Drift mobility

An important characterizaton tool is the measurement of the drift mobility. This type of measurement is typically carried out by injecting pulses of high charge carrier density at one end of the sample which can be achieved either by a pulsed electron beam, or by a flash lamp which creates electron-hole pairs. The carriers are swept through the sample by drift in the field of an applied bias voltage. In the case of flash lamp excitation, the polarity can select between electron or hole transport. Note that the mobility in a conductivity measurement is different from the one in a drift experiment because in the latter the charge carriers are injected rather than thermally excited. Another thing to note is the fact that the carriers are swept through the film before they have much chance to undergo recombination. Thus, the whole injected charge contributes to the signal.

A very first and simple model for a temperature dependent drift mobility takes into account a single trap state. Trap states are distinguished from dopants by the fact that they are normally neutral and become charged when they trap a charge carrier. As the name suggests, trapped charge carriers are no longer mobile, hence they do not take part in the transport. The drift mobility is then determined by the equilibrium share of charge carriers that are thermally released from the trap, similar to occupational statistics of dopants discussed in the chapter on dopant ionization.

$\mu_D = \mu_0 \frac{n_{free}}{n_{total}} = \mu_0 \frac{n_{free}}{n_t + n_{free}}$

The difference is that useful dopants are shallow and therefore mostly ionized at room temperature whereas traps do not release the carriers so easily because they are generally deeper in the gap. The following deals with electron transport and an electron trap close to the conduction band, but the treatment is equally applicable to holes. We assume that the available density of mobile states is given by he density of states N_C in the conduction band, multiplied by a width kT, whereas the density of trap states located at E_t is given by n_t. This yields approximately:

$\mu_d \approx \mu_0 \frac{N_C kT}{n_t} \cdot e^{-\frac{E_t}{kT}}$

Drift mobility of PE-CVD grown amorphous silicon. Black open sqares: [LeComber-1970prl], full circles: [Moore-1977apl], open circles: [Tiedje-1981prl]. Black symbols signify nominally undoped films, blue symbols denote n-type samples, red symbols represent p-type material.

The figure above summarizes some published data of drift mobilities. LeCombers data (black) exhibit a change of slope at a temperature where he observed also a kink in the conductivity. LeComber interpreted the data with a well defined trap state 0.2 eV below the conduction band, and transition into an activated hopping regime at lower temperature. Such a transition between two different activation energies in the drift mobility was also found in doped samples [Moore-1977apl]. However, a later investigation by Tiedje failed to reproduce this behaviour [Tiedje-1981prl]. By that time also the idea of a well defined trapping centre had come under dispute.

Band tails

Charge transport in the presence of band tail states was first treated by [Tiedje-1981prl]. We will follow this treatment because it is essentially still accepted today. The model assumes a multitude of trap states that tail off from a high density close to the band edges towards low density further in the band gap. Charge transport is then imagined as a series of trapping and release events into a variety of traps with different depths.

A guess as good as any would be that the trap density N_t tails off exponentially with respect to the distance E from the conduction band edge. The distribution is described by a characteristic energy kT₀ (note the upwards directed energy scale).

$N_t(E)=N_C e^{\frac{E}{kT_0}}$

The model keeps the idea that trapped charges do not contribute to the charge transport, regardless how shallow the actual trap state is, or how dense the traps become. This energy dependence of the mobility is therefore often called mobility edge. Tiedje's model assumes that the probabiltiy of trapping a charge is the same for all traps. The occupation of the trap states is thus simply a competition between capture and thermal release. Deep lying traps will gradually be filled, the closer they are to the band gap, the higher is the probability that the trapped charge can escape. Thus, there is an intermediate energy E_t where the density of occupied traps assumes its maximum value N_trap.

$N_{trap}=\frac{N_t(E_t)\cdot kT}{1-T/T_0} + N_t(E_t)\cdot kT_0}$

We can imagine that eventually the low lying traps fill more and more. Thus, the dominant energy sinks deeper E_t but at the same time the value of the density N_trap is reduced because of the exponential tail. In order to take into account this time dependence, Tiedje introduces an attempt frequency which counts how often a charge attempts to leave from its trap. This time is thermally activated being longer for deep lying traps.

$t=\nu^{-1} e^{-\frac{E_t}{kT}}$

Here comes a bold step: After determining the dominant trap energy and further assuming an activated release time, everything is put into the equation for the single trap level. The result is a time dependent drift mobility which is actually observed when the drift experiments are carried out at different bias voltages.

$\mu_d \approx \mu_0 \alpha (1-\alpha) \cdot (\nu t)^{-(1-\alpha)}$

The quantity α denotes the ratio T/T₀. Tiedje determined characteristic temperatures T₀ of 500 K and 312 K for the valence and the conduction band tail, respectively. The corresponding energies kT₀ are 43 and 27 meV, respectively.

The band tail slopes are also accessible by measurements of the absorption; in addition to the signature of the band gap absorption which we expect for energies above 1.75 eV, there is an exponential tail towards lower energies, the so called Urbach tail. Redfield [Redfield-1982ssc] showed that the slope of this tail (in a logarithmic plot) is related to the broader of the two tails, i.e. the the valence band tail slope. The absorption strengh is related to both band tails. The band tail slopes have also been measured by photoelecton spectroscopy, giving similar values [Jackson-1985prb]

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Last updated March 31st, 2010