Characterizing mobility in disordered semiconductors

I always look for analogies between physical systems. This often leads to dead ends but sometimes you uncover some interesting parallels that actually add to the knowledge-base of information and ideas for both systems.

As I worked out the problem of CO2 dispersion in the atmosphere, I went back and revisited the work I did on dispersive transport in amorphous semiconductors. Essentially the same math gets used on both analyses, with the same fundamental goal in mind -- that of trying to characterize the annoyingly sluggish response from an input stimulus.

For the climate case, the poor response comes from CO2 molecules wandering around aimlessly trying to find a good resting place. For the disordered semiconductor, the carrier of electricity (the electron or hole) encounters so many trapping states and scattering centers, that it effectively takes much longer for the charge to cross a region. It does have the advantage of the assist of an electric field, but the low effective transport rate makes an amorphous semiconductor such as hydrogenated amorphous silicon (a-Si:H) marginally useful for any time-sensitive applications -- yet eminently usable as a photo-voltaic.

Still, knowing the physical characteristics helps to understand the nature of the material, and could unlock some secrets beneficial to future applications of material such as polycrystalline or amorphous silicon, or any disordered semiconductor. In the future, we will make mass quantities of this material for the PV industry and we won't have the luxury of single crystal material.

The fact that dispersive transport does have the help of an electric field, makes it amenable to experimentation. By applying various electric fields, one can distinguish between a drift component and a diffusive component (of the photoelectric current, for example). With no electric field, any photo-generated carriers will wander around until they recombine. This can take relatively long times, especially in comparison to a piece of single crystal silicon. As the electric field increases, the carriers get swept out faster and the diffusion plays less of a role.

The fact that the atmosphere has no drift role apart from turbulent diffusion, means that CO2 plays the analogous part of a electronic device with generated carriers, but nowhere to remove them (alas, we have no electrodes attached to the atmosphere). So, I wanted to get a bit of insight by looking at the carrier transport problem, and as a goal, perhaps find a way to increase the removal of CO2 by something equivalent to an electric field, and particularly to ask if this could reduce the CO2 mean residence time.

I noticed one detail that I left hanging on the dispersive carrier transport problem. This had to do with the initial diffusion transient often observed. See the figure to the right (from here). You can see the transient near the start time as a quickly declining response from the initial impulse. The particular trace in the tiny inset came from a non-disordered device (perhaps from a commercial-grade photodetector), as the individual regions show sharp delineations. For a disordered material, the regions show more blurring, as shown in the following figure.

Figure 1: Fitting to the dispersive tail from previous posting. Note the missing initial transient in the curve fit in the curves in color.

I did not include the initial spike term in my initial analysis from last year, as I forgot to apply the chain rule to one set of rate equations. I had justified a transport function that had a non-linear component that propagates as the square-root of time, characteristic of diffusion. Yet to generate a current from this, one needs to differentiate this as a simple chain rule. Not too surprisingly, but perhaps non-intuitively, the derivative of a square root generates the reciprocal of the square root, which of course will spike to infinity at times close to zero. However, the accumulated amount of current generated by this spike nowhere approaches infinity, as the transient has very little width to it. Looking at it on a log-log plot, the width appears long but that occurs simply as an optical illusion.

For a pulsed light source, the entire impulse response equation boils down to a simple charge conservation problem. We know that charge builds up as the photons excite the carriers, but we only know the mean rate and we let the Maximum Entropy Principle figure out the rest. The concentrations build up as the following form, with g(t) acting as the transport growth term across a region w:

C(t) = C0 * g(t)* (1 - exp(-w/g(t))

with

g(t) = sqrt(2*D*t) + u*E*t

where D is the diffusivity, w is the active width, u is the charge mobility, and E is the electric field strength (E = Voltage/w). The total number of excited carriers is C0, and this number provides the maximum amount of current that gets collected. Common to all stochastic probability problems, the conservation of probability becomes a strong constraint.

The current derives as:

I(t) = dC(t)/dt = C0 * dg(t)/dt * (1 - exp(-w/g(t) * (1 + w/g(t))

Note the dg(t)/dt term, which I had neglected to derive completely before, keeping only the drift term (note the 1/sqrt(t) term below).

dg(t)/dt = 0.5*sqrt(D/t) + u*E

Re-plotting the original fitted curve trace with the extra chain-rule term, we can actually see the initial transient. Take a close look at the figure below, and observe how well the curve matches all the inflection points, and works over several orders of magnitude. Mystery solved.

Figure 2 : Dispersive transport which includes a term to describe the initial transient. Note the agreement of the dispersive transport model at short durations. Upper curve fits a fixed average mobility sample. For the lower curve, the average mobility depends on applied electric field strength.

You can spend all sorts of time trying to fit the curves; the more time you spend, the better an estimate you can make of the average mobility, u, and diffusivity, D. Suffice to say, no fudge factors play into the equations. If this isn't a textbook ready formula, I don't know what is.

As I said before, no one in the semiconductor industry seems to use this simple dispersive formulation, preferring to hand-wave and heuristically account for the fat-tails of the transient. Importantly, this particular impulse response function both explains the behavior seen, and derives from the most simple particle counting statistics (i.e maximum entropy randomness), so it likely serves as the most canonical model for dispersive transport in disordered materials.

Linking back to CO2

Now a curious fact presents itself. Not many people in science and engineering seem to understand disorder. If they did, somebody would have discovered this dispersion formulation. Yet they haven't (AFAIK). Billions of dollars goes into semiconductor research and I can only find several purely academic papers on anomolous diffusion and Levy flights and fractional random walks. It really is not that complicated to derive the physics behavior, if you simply assume entropic disorder.

So as it turns out, the dispersion math essentially matches that of what happens to CO2 as it enters the atmosphere. The peculiar piece in the transport that provides that initial photo-current spike acts identically to the fast rate of CO2. In other words, a fraction of charged carriers that can diffuse quickly to a recombination site (i.e. an electrode) act precisely the same as CO2 that reacts quickly and removes itself from the atmosphere. Yet the long tails in the dispersion remain, both in the disordered semiconductor, and in the disordered atmosphere. The fat-tails will kill us in atmospheric CO2 build-up, just like the fat-tails in amorphous semiconductors make it useless to use in a fast microprocessor or in a cell-phone receiver.

Now put 2 and 2 together. No wonder no one knows how to simply describe the CO2 buildup problem! Like the scientists and engineers who experiment with dispersive transport can't see the forest for the trees and thus can't come up with a simple derivation that a near layman can understand, the climate scientists also completely miss out on the obvious and have never come up with the equivalent "probability as logic" formulation.

ImpulseResponseCO2(t) = 1/(1+sqrt(t/T))

That is all there is to it.