Haynes–Shockley experiment

Derivation

We consider the continuity equation:

${\displaystyle \frac{\mathrm{\partial }n}{\mathrm{\partial }t}=\frac{-(n-n_0)}{{\tau }_n}+\frac{\mathrm{\partial }{j}_e}{\mathrm{\partial }x}}$

${\displaystyle \frac{\mathrm{\partial }p}{\mathrm{\partial }t}=\frac{-(p-p_0)}{{\tau }_p}-\frac{\mathrm{\partial }{j}_p}{\mathrm{\partial }x}}$

Subscript 0s indicate equilibrium concentrations. The electrons and the holes recombine with the carrier lifetime τ.

We define

${\displaystyle p_1=p-p_0,n_1=n-n_0}$

so the upper equations can be rewritten as:

${\displaystyle \frac{\mathrm{\partial }{p}_1}{\mathrm{\partial }t}=D_p\frac{{\mathrm{\partial }}^2{p}_1}{\mathrm{\partial }{x}^2}-{\mu }_{p}p\frac{\mathrm{\partial }E}{\mathrm{\partial }x}-{\mu }_{p}E\frac{\mathrm{\partial }{p}_1}{\mathrm{\partial }x}-\frac{p_1}{{\tau }_p}}$

${\displaystyle \frac{\mathrm{\partial }{n}_1}{\mathrm{\partial }t}=D_n\frac{{\mathrm{\partial }}^2{n}_1}{\mathrm{\partial }{x}^2}+{\mu }_{n}n\frac{\mathrm{\partial }E}{\mathrm{\partial }x}+{\mu }_{n}E\frac{\mathrm{\partial }{n}_1}{\mathrm{\partial }x}-\frac{n_1}{{\tau }_n}}$

In a simple approximation, we can consider the electric field to be constant between the left and right electrodes and neglect ∂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:

${\displaystyle \frac{\mathrm{\partial }E}{\mathrm{\partial }x}=\frac{\rho }{ϵ{ϵ}_0}=\frac{e_0((p-p_0)-(n-n_0))}{ϵ{ϵ}_0}=\frac{e_0(p_1-n_1)}{ϵ{ϵ}_0}}$

where ε is permittivity, ε₀ the permittivity of free space, ρ is charge density, and e₀ elementary charge.

Next, change variables by the substitutions:

${\displaystyle p_1=n_{\text{mean}}+\delta ,n_1=n_{\text{mean}}-\delta ,}$

and suppose δ to be much smaller than ${\displaystyle n_{\text{mean}}}$. The two initial equations write:

${\displaystyle \frac{\mathrm{\partial }{n}_{\text{mean}}}{\mathrm{\partial }t}=D_p\frac{{\mathrm{\partial }}^2{n}_{\text{mean}}}{\mathrm{\partial }{x}^2}-{\mu }_{p}p\frac{\mathrm{\partial }E}{\mathrm{\partial }x}-{\mu }_{p}E\frac{\mathrm{\partial }{n}_{\text{mean}}}{\mathrm{\partial }x}-\frac{n_{\text{mean}}}{{\tau }_p}}$

${\displaystyle \frac{\mathrm{\partial }{n}_{\text{mean}}}{\mathrm{\partial }t}=D_n\frac{{\mathrm{\partial }}^2{n}_{\text{mean}}}{\mathrm{\partial }{x}^2}+{\mu }_{n}n\frac{\mathrm{\partial }E}{\mathrm{\partial }x}+{\mu }_{n}E\frac{\mathrm{\partial }{n}_{\text{mean}}}{\mathrm{\partial }x}-\frac{n_{\text{mean}}}{{\tau }_n}}$

Using the Einstein relation ${\displaystyle \mu =e\beta D}$, where β is the inverse of the product of temperature and the Boltzmann constant, these two equations can be combined:

${\displaystyle \frac{\mathrm{\partial }{n}_{\text{mean}}}{\mathrm{\partial }t}=D^{\ast }\frac{{\mathrm{\partial }}^2{n}_{\text{mean}}}{\mathrm{\partial }{x}^2}-{\mu }^{\ast }E\frac{\mathrm{\partial }{n}_{\text{mean}}}{\mathrm{\partial }x}-\frac{n_{\text{mean}}}{{\tau }^{\ast }},}$

where for D*, μ* and τ* holds:

${\displaystyle D^{\ast }=\frac{D_n{D}_p(n+p)}{p{D}_p+n{D}_n}}$, ${\displaystyle {\mu }^{\ast }=\frac{{\mu }_n{\mu }_p(n-p)}{p{\mu }_p+n{\mu }_n}}$ and ${\displaystyle \frac{1}{{\tau }^{\ast }}=\frac{p{\mu }_p{\tau }_p+n{\mu }_n{\tau }_n}{{\tau }_p{\tau }_n(p{\mu }_p+n{\mu }_n)}.}$

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

The final equation for the carriers is:

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

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.

${\displaystyle {\mu }^{\ast }=\frac{d}{E{t}_0}}$

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

where d is the distance drifted in time t₀, and δt the pulse width.