Haynes–Shockley experiment

The Haynes–Shockley experiment was a classic experiment that demonstrated that diffusion of minority carriers in a semiconductor could result in a current. The experiment was reported in a paper by Shockley, Pearson, and Haynes in 1949.^[1]^[2] The experiment can be used to measure carrier mobility, carrier lifetime, and diffusion coefficient.^[3]

In the experiment, a piece of semiconductor gets a pulse of holes, for example, as induced by voltage or a short laser pulse.

To see the effect, we consider a n-type semiconductor with the length d. We are interested in determining the 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:

$j_e=-\mu_n n E-D_n \frac{\partial n}{\partial x}$

$j_p=%2B\mu_p p E-D_p \frac{\partial p}{\partial x}$

where the first part is the drift current and the second is the diffusion current.

We consider the continuity equation:

$\frac{\partial n}{\partial t}=\frac{-(n-n_0)}{\tau_n}-\frac{\partial j_e}{\partial x}$

$\frac{\partial p}{\partial t}=\frac{-(p-p_0)}{\tau_p}-\frac{\partial j_p}{\partial x}$

The electrons and the holes recombine with the time $\tau$ .

We define $p_1=p-p_0$ and $n_1=n-n_0$ so the upper equations can be rewritten as:

$\frac{\partial p_1}{\partial t}=D_p \frac{\partial^2 p_1}{\partial x^2}-\mu_p p \frac{\partial E}{\partial x}- \mu_p E \frac{\partial p_1}{\partial x}-\frac{p_1}{\tau_p}$

$\frac{\partial n_1}{\partial t}=D_n \frac{\partial^2 n_1}{\partial x^2}%2B\mu_n n \frac{\partial E}{\partial x}%2B \mu_n E \frac{\partial n_1}{\partial x}-\frac{n_1}{\tau_n}$

In a simple approximation, we can consider the electric field to be constant between the left and right electrodes and neglect $\frac{\partial E}{\partial x}$ . However, as electrons and holes diffuse at a different speed, the material has a local electric charge, inducing an inhomogeneous electric field which can be calculated with Gauss's law:

$\frac{\partial E}{\partial x}= \frac{\rho}{\epsilon \epsilon_0}=\frac{e_0 ((p-p_0)-(n-n_0))}{\epsilon \epsilon_0} = \frac{e_0 (p_1-n_1)}{\epsilon \epsilon_0}$

We make the following change of variables: $p_1 = n_\text{mean}%2B\delta$ , $n_1 = n_\text{mean}-\delta$ , and suppose $\delta$ to be much smaller than $n_\text{mean}$ . The two initial equations write:

$\frac{\partial n_\text{mean}}{\partial t}=D_p \frac{\partial^2 n_\text{mean}}{\partial x^2}-\mu_p p \frac{\partial E}{\partial x}- \mu_p E \frac{\partial n_\text{mean}}{\partial x}-\frac{n_\text{mean}}{\tau_p}$

$\frac{\partial n_\text{mean}}{\partial t}=D_n \frac{\partial^2 n_\text{mean}}{\partial x^2}%2B\mu_n n \frac{\partial E}{\partial x}%2B \mu_n E \frac{\partial n_\text{mean}}{\partial x}-\frac{n_\text{mean}}{\tau_n}$

Thanks to Einstein relation $\mu=e\beta D$ , these two equations can be combined:

$\frac{\partial n_\text{mean}}{\partial t}=D^* \frac{\partial^2 n_\text{mean}}{\partial x^2}- \mu^* E \frac{\partial n_\text{mean}}{\partial x}-\frac{n_\text{mean}}{\tau^*},$

where for $D^*$ , $\mu^*$ and $\tau^*$ holds:

$D^*=\frac{D_n D_p(n%2Bp)}{p D_p%2BnD_n}$ , $\mu^*=\frac{\mu_n\mu_p(n-p)}{p\mu_p%2Bn\mu_n}$ and $\frac{1}{\tau^*}=\frac{p\mu_p\tau_p%2Bn\mu_n\tau_n}{\tau_p\tau_n(p\mu_p%2Bn\mu_n)}.$

Considering n>>p or $p\rightarrow 0$ (that is a fair approximation for a semiconductor with only few holes injected), we see that $D^*\rightarrow D_p$ , $\mu^*\rightarrow \mu_p$ and $\frac{1}{\tau^*}\rightarrow \frac{1}{\tau_p}$ . The semiconductor behaves as if there were only holes traveling in it.

The final equation for the carriers is:

$n_\text{mean}(x,t)=A \frac{1}{\sqrt{4\pi D^* t}} e^{-t/\tau^*} e^{-\frac{(x%2B\mu^*Et-x_0)^2}{4D^*t}}$

This can be interpreted as a 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 $\mu, D$ and $\tau$ can be obtained from the shape of the signal.

$\mu^*=\frac{d}{E t_0}$

$D^*=(\mu^* E)^2 \frac{(\delta t)^2}{16 t_0}$

References

^ Shockley, W. and Pearson, G. L., and Haynes, J. R. (1949). "Hole injection in germanium – Quantitative studies and filamentary transistors". Bell System Technical Journal 28: 344–366.
^ Jerrold H. Krenz (2000). Electronic concepts: an introduction. Cambridge University Press. p. 137. ISBN 9780521662826. http://books.google.com/books?id=Le9zdVoMEOEC&pg=PA137.
^ Ajay Kumar Singh (2008). Electronic Devices And Integrated Circuits. PHI Learning Pvt. Ltd.. p. 119. ISBN 9788120331921. http://books.google.com/books?id=2aqtlybkFE0C&pg=PA119.

External links

Applet simulating the Haynes–Shockley experiment