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In [[semiconductor physics]], the '''Haynes–Shockley experiment''' was an experiment that demonstrated that diffusion of [[minority carrier]]s in a [[semiconductor]] could result in a [[current (electrical)|current]]. The experiment was reported in a short paper by Haynes and [[William Shockley|Shockley]] in 1948,<ref>{{cite doi|10.1103/PhysRev.75.691}}</ref> with a more detailed version published by Shockley, Pearson, and Haynes in 1949.<ref>
{{cite journal
| journal = Bell System Technical Journal
| title = Hole injection in germanium – Quantitative studies and filamentary transistors
| author = Shockley, W. and Pearson, G. L., and Haynes, J. R.
| volume = 28
| pages = 344–366
| year = 1949
}}</ref><ref>
{{cite book
| title = Electronic concepts: an introduction
| edition =
| author = Jerrold H. Krenz
| publisher = Cambridge University Press
| year = 2000
| isbn = 978-0-521-66282-6
| page = 137
| url = http://books.google.com/books?id=Le9zdVoMEOEC&pg=PA137
}}</ref>
The experiment can be used to measure carrier [[Electron mobility|mobility]], [[carrier lifetime]], and [[diffusion coefficient]].

In the experiment, a piece of semiconductor gets a pulse of [[Electron hole|hole]]s, for example, as induced by voltage or a short [[laser]] pulse.

==Equations==

To see the effect, we consider a [[n-type semiconductor]] with the length ''d''. We are interested in determining the [[Electron mobility|mobility]] of the carriers, [[diffusion constant]] and [[relaxation time]]. In the following, we reduce the problem to one dimension.

The equations for electron and hole currents are:

:<math>j_e=+\mu_n n E+D_n \frac{\partial n}{\partial x}</math>

:<math>j_p=+\mu_p p E-D_p \frac{\partial p}{\partial x}</math>

where the ''j''s are the [[current density|current densities]] of electrons (''e'') and holes (''p''), the ''μ''s the charge carrier mobilities, ''E'' is the [[electric field]], ''n'' and ''p'' the number densities of charge carriers, the ''D''s are [[diffusion coefficient]]s, and ''x'' is position. The first term of the equations is the [[drift current]], and the second term is the [[diffusion current]].

==Derivation==

We consider the [[continuity equation]]:

:<math>\frac{\partial n}{\partial t}=\frac{-(n-n_0)}{\tau_n}-\frac{\partial j_e}{\partial x}</math>

:<math>\frac{\partial p}{\partial t}=\frac{-(p-p_0)}{\tau_p}-\frac{\partial j_p}{\partial x}</math>
Subscript 0s indicate equilibrium concentrations. The electrons and the holes recombine with the carrier lifetime τ.

We define
:<math>p_1=p-p_0\,,\quad n_1=n-n_0</math>

so the upper equations can be rewritten as:

:<math>\frac{\partial p_1}{\partial t}=D_p \frac{\partial^2 p_1}{\partial x^2}-\mu_p p \frac{\partial E}{\partial x}-
\mu_p E \frac{\partial p_1}{\partial x}-\frac{p_1}{\tau_p}</math>
:<math>\frac{\partial n_1}{\partial t}=D_n \frac{\partial^2 n_1}{\partial x^2}+\mu_n n \frac{\partial E}{\partial x}+
\mu_n E \frac{\partial n_1}{\partial x}-\frac{n_1}{\tau_n}</math>

In a simple approximation, we can consider the electric field to be constant between the left and right electrodes and neglect ∂''E''/∂''x''. However, as electrons and holes diffuse at different speeds, the material has a local electric charge, inducing an inhomogeneous electric field which can be calculated with [[Gauss's law]]:

:<math>\frac{\partial E}{\partial x}= \frac{\rho}{\epsilon \epsilon_0}=\frac{e_0 ((p-p_0)-(n-n_0))}{\epsilon \epsilon_0} = \frac{e_0 (p_1-n_1)}{\epsilon \epsilon_0}</math>

where ε is permittivity, ε0 the permittivity of free space, ρ is charge density, and ''e''0 elementary charge.

Next, change variables by the substitutions:
:<math>p_1 = n_\text{mean}+\delta\,,\quad n_1 = n_\text{mean}-\delta\,,</math>
and suppose δ to be much smaller than <math>n_\text{mean}</math>. The two initial equations write:

:<math>\frac{\partial n_\text{mean}}{\partial t}=D_p \frac{\partial^2 n_\text{mean}}{\partial x^2}-\mu_p p \frac{\partial E}{\partial x}-
\mu_p E \frac{\partial n_\text{mean}}{\partial x}-\frac{n_\text{mean}}{\tau_p}</math>

:<math>\frac{\partial n_\text{mean}}{\partial t}=D_n \frac{\partial^2 n_\text{mean}}{\partial x^2}+\mu_n n \frac{\partial E}{\partial x}+
\mu_n E \frac{\partial n_\text{mean}}{\partial x}-\frac{n_\text{mean}}{\tau_n}</math>

Using the [[Einstein relation (kinetic theory)|Einstein relation]] <math>\mu=e\beta D</math>, where β is the inverse of the product of [[temperature]] and the [[Boltzmann constant]], these two equations can be combined:

:<math>\frac{\partial n_\text{mean}}{\partial t}=D^* \frac{\partial^2 n_\text{mean}}{\partial x^2}-
\mu^* E \frac{\partial n_\text{mean}}{\partial x}-\frac{n_\text{mean}}{\tau^*},</math>

where for ''D''*, μ* and τ* holds:
:<math>D^*=\frac{D_n D_p(n+p)}{p D_p+nD_n}</math>, <math>\mu^*=\frac{\mu_n\mu_p(n-p)}{p\mu_p+n\mu_n}</math> and <math>\frac{1}{\tau^*}=\frac{p\mu_p\tau_p+n\mu_n\tau_n}{\tau_p\tau_n(p\mu_p+n\mu_n)}.</math>

Considering ''n'' >> ''p'' or ''p'' → 0 (that is a fair approximation for a semiconductor with only few holes injected), we see that ''D''* → ''Dp'', μ* → μp and 1/τ* → 1/τp. The semiconductor behaves as if there were only holes traveling in it.

The final equation for the carriers is:

:<math>n_\text{mean}(x,t)=A \frac{1}{\sqrt{4\pi D^* t}} e^{-t/\tau^*} e^{-\frac{(x+\mu^*Et-x_0)^2}{4D^*t}}</math>

This can be interpreted as a [[Dirac delta function]] that is created immediately after the pulse. Holes then start to travel towards the electrode where we detect them. The signal then is [[Gaussian curve]] shaped.

Parameters μ, ''D'' and τ can be obtained from the shape of the signal.

:<math>\mu^*=\frac{d}{E t_0}</math>

:<math>D^*=(\mu^* E)^2 \frac{(\delta t)^2}{16 t_0}</math>

where ''d'' is the distance drifted in time ''t''0, and ''δt'' the [[pulse width]].

==See also==

*[[Alternating current]]
*[[Conduction band]]
*[[Convection–diffusion equation]]
*[[Direct current]]
*[[Drift current]]
*[[Electron gas]]
*[[Random walk]]

==References==

{{reflist}}

==External links==
* [http://www.benfold.com/sse/hs.html Applet simulating the Haynes–Shockley experiment]
* [http://www.youtube.com/watch?v=zYGHt-TLTl4 Video explaining the original experiment]
* [http://www.labtrek.it/proHSuk.html Educational approach to the HS experiment]

{{DEFAULTSORT:Haynes-Shockley experiment}}
[[Category:Semiconductors]]

Generalized Wiener process - Revision history

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