A single trap level at energy in the gap of a semiconductor has electron and hole…

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Semiconductor physics and devices · Semiconductor physics and devices

Solid State Physics, Ch Semiconductors: masses, statistics, defect levels and mobility

Problem

A single trap level at energy

E_t

in the gap of a semiconductor has electron and hole capture times

\tau_n

and

\tau_p

(set by the capture cross sections, the thermal velocity and the trap density) and Shockley-Read-Hall recombination rate

\begin{aligned} U &=\frac{np-n_i^2}{\tau_p(n+n_1)+\tau_n(p+p_1)}, \\ n_1 &=n_i\,e^{(E_t-E_i)/k_BT}, \\ p_1 &=n_i\,e^{-(E_t-E_i)/k_BT}, \end{aligned}

where

E_i

is the intrinsic level, taken here as the reference energy for the trap position, and

n_1p_1=n_i^2

. ((a)) For fixed electron and hole densities

n

and

p

and equal capture times

\tau_n=\tau_p=\tau_0

, show that the denominator of

U

is minimised, and

U

therefore maximised, when the trap sits exactly at

E_t=E_i

. ((b)) Take near-intrinsic silicon at

T=300\,\mathrm{K}

(

k_BT=0.0259\,\mathrm{eV}

) carrying a small injected excess, so that

np

sits just above

n_i^2

while

n=p\approx n_i

in the denominator of

U

, and a trap with

\tau_n=\tau_p=\tau_0

sitting

0.30\,\mathrm{eV}

away from

E_i

. Compute the factor by which

U

is smaller than it would be for an otherwise identical trap sitting exactly at

E_i

, with

n

and

p

held at the same values.

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