= Photoconductor transient with quadratic recombination
A <photoconductor transient with quadratic recombination> is a carrier-density model in which the loss rate grows as the square of the mobile excess density. It explains how nonlinear loss can produce different turn-on and turn-off waveforms without assigning unrelated exponential lifetimes to the two directions. A <photoconductor> converts illumination-dependent <electrical conductivity> into a signal under external bias; the manufacturer's primary PbS description establishes that conversion mechanism.<How does a PbS photoconductive detector work? by Shimadzu Scientific Instruments>[band excitation and external resistance readout]{ref} The quadratic loss law below is an explicitly stipulated model, not a fitted specification of that detector.
Assume a spatially uniform excess density $n\ge0$, generation rate $g\ge0$, constant coefficient $\beta>0$ and an output proportional to $n$ at fixed mobility and bias:
$$
\dot n=g-\beta n^2.
$$
Density has units $\mathrm{m^{-3}}$, $g$ has units $\mathrm{m^{-3}s^{-1}}$ and $\beta$ has units $\mathrm{m^3/s}$. A two-carrier recombination rate proportional to the product of equal electron and hole densities motivates a quadratic term in an appropriate regime. Background carriers, other loss processes, contacts and trapping can invalidate that specialization.
For constant $g>0$, set $n_\infty=\sqrt{g/\beta}$ and $k=\beta n_\infty$. Separating variables, or differentiating the following expression, gives the solution for any finite nonnegative initial density $n_0$:
$$
\boxed{n(t)=n_\infty\frac{n_0+n_\infty\tanh(kt)}{n_\infty+n_0\tanh(kt)}.}
$$
For $n_0=0$ it reduces to $n_\infty\tanh(kt)$. For $n_0=n_\infty$ it stays at equilibrium, and for $n_0>n_\infty$ it approaches the same equilibrium from above. In the dark, $g=0$ instead yields
$$
\boxed{n(t)=\frac{n_0}{1+\beta n_0t},\qquad
\frac1{n(t)}=\frac1{n_0}+\beta t.}
$$
The reciprocal-density plot is linear in this model, while a semilogarithmic decay plot is curved. The dark half-decay time is $1/(\beta n_0)$, so it depends on the starting illumination.
The <small-signal lifetime under quadratic recombination> is different again. Linearize about a nonzero background $\overline n=\sqrt{\overline g/\beta}$:
$$
\delta\dot n=\delta g-2\beta\overline n\,\delta n,
\qquad \tau_{\rm small}=\frac1{2\beta\overline n},\qquad
\frac{\delta n(\omega)}{\delta g(\omega)}=\frac1{2\beta\overline n+i\omega}.
$$
A locally exponential small-signal response can therefore coexist with a strongly nonexponential large dark transient. One measured time constant cannot be substituted for all three quantities.
For an original example, stipulate $\beta=10^{-15}\,\mathrm{m^3/s}$ and $\overline n=10^{18}\,\mathrm{m^{-3}}$. Then $\overline g=10^{21}\,\mathrm{m^{-3}s^{-1}}$, dark half-decay time $1\,\mathrm{ms}$, small-signal lifetime $0.5\,\mathrm{ms}$ and small-signal half-power frequency about $318.3\,\mathrm{Hz}$. Four times the background generation doubles equilibrium density and doubles that frequency in this model.
A proposed test measures small modulation about several backgrounds separately from complete turn-off traces, while controlling actual detector bias, temperature and readout response. Examine reciprocal-density linearity only after justifying how the measured current maps to density. A successful fit supports the tested rate-law approximation; it does not uniquely identify the microscopic <electron-hole recombination> mechanism. Temperature and spectral sensitivity also vary in real PbS detectors and must be held or measured independently.<How does a PbS photoconductive detector work? by Shimadzu Scientific Instruments>[temperature-dependence paragraph]{ref}
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