Does carrier pulse width vary with diffusion coefficient?

[HS-experiment]

Hello Physics Forums

I’m doing some numerical studies on diffusion. I began with monte carlo simulations on gas diffusion, and more recently I’ve started to dabble in carrier diffusion inside semiconductors.

It looks like diffusion in semiconductors is a lot more amenable to experimental study than gas diffusion, taking for instance the Haynes Shockley experiment. I’ve focused on reproducing the Haynes Shockley experiment in an MC simulation in 1 dimension. I have one issue however.

It seems to me that the pulse width at distance X from injection should be constant regardless of the diffusion coefficient (neglecting carrier recombination). The diffusion coefficient D is related to carrier mobility by the Einstein relationD = u * kT

u = mobility

k = Boltzmann constant

T = absolute temperatureMobility is the ratio of the drift velocity Vd to the strength of the electric field. This indicates that if diffusivity is doubled, the drift velocity would also be doubled. So the pulse width is proportional to ( X / Vd ) * D. It seems to me it is constant because any change in Vd results in a proportional change to D.

My simulation however indicates that pulse width at distance X will be different depending on D. If the pulse travels distance X in a medium of diffusion coefficient 1, the pulse width will be smaller by a factor of sqrt(2) compared to a pulse that travels distance X with a diffusion coefficient 2.
Which appears to makes sense, if you have lower diffusion, the pulse width won't broaden as much. However, it will take twice as long, leaving more time to diffuse.

So, should pulse width really remain constant at distance X from injection, no matter the diffusion coefficient? Or am I misinterpreting the Einstein relation in this context?

Your input is really appreciated! :)

 

[Henryk]

Actually it is not the pulse width, but the square of the pulse width that is equal to $\sigma^2 = 2\cdot D \cdot\Delta t = 2\cdot D \frac{X}{V_d}$
This is the pulse width in the space domain and yes, it will be independent on the diffusion coefficient because D is proportional to V_d.
However, the width in time domain, the width will be proportional to the drift time.

 

[HS-experiment]

Actually it is not the pulse width, but the square of the pulse width that is equal to $\sigma^2 = 2\cdot D \cdot\Delta t = 2\cdot D \frac{X}{V_d}$

Of course, that equation makes it very clear. t is of course equal to X/Vd. In this sense the width of the pulse after t seconds will be identical whether or not there is any drift. Due to the relation between mobility and diffusion coefficient, it is also equivalent in the space domain.

I think that my simulation, because it has been converted from a dilute gas monte-carlo, doesn't quite get the proportionality constant (D~Vd) right. I will have to change it until this is the case.

Thank you for your input!