Photoconductor response time

Photoconductor response time describes how quickly the electrical signal follows a change in illumination under stated bias, temperature, background and readout conditions. A rise time, fall time and modulation bandwidth summarize different features and require definitions. The manufacturer’s PbS description notes that sensitivity, response speed and dark resistance vary with temperature, so a timing number is not independent of operating conditions.^[ref]

An original one-lifetime model takes excess carrier density $n_e$ to obeyWith constant mobility and fixed detector bias, suppose the excess current is proportional to $n_e$. After a step from zero to generation $g_0$, the normalized signal is $1-e^{-t/\tau }$. The times to reach ten and ninety percent of its final value are $-\tau \ln 0.9$ and $-\tau \ln 0.1$, givingThe time from zero to ninety percent is instead $\tau \ln 10$. A quoted rise time without its thresholds is therefore incomplete.

For small sinusoidal generation changes, the normalized frequency response isThe half-power frequency is $f_c=1/(2\pi \tau )$, so this single-pole model gives $t_{10\text{–}90}{f}_c=\ln 9/(2\pi )\approx 0.3497$. This relationship is a consequence of the stated model, not a universal identity for arbitrary detectors.

For $\tau =100\mu \mathrm{s}$, the rise time is about $219.7\mu \mathrm{s}$ and the cutoff about $1.592\mathrm{kHz}$. At $1\mathrm{kHz}$, the signal amplitude is about $0.8467$ of its low-frequency value and its phase lags by $32.14^{\circ }$. The signal can therefore have a substantial phase delay before reaching the half-power point.

A readout adds its own dynamics. If the detector response is cascaded with a separate first-order electronic response of the same time constant, the overall magnitude is $1/[1+(2\pi f\tau {)}^2]$. Its half-power frequency is $\sqrt{\sqrt{2}-1}/(2\pi \tau )$, about $0.6436$ times the single-pole cutoff. Assigning that combined bandwidth entirely to the material would give an incorrect carrier-lifetime estimate.

Photoconductor rise and fall asymmetry tests another limitation of the simple model: linear one-lifetime dynamics predict equal normalized rise and fall behavior, but trap occupancy, changing background or a nonlinear circuit can produce different transients. A long relaxation tail should be measured rather than compressed into an unsupported single time constant.

A proposed experiment switches or modulates a monitored optical source, records detector current and actual bias, and measures ten-to-ninety and ninety-to-ten crossings after settling the initial illumination state. Repeat the modulation sweep with a characterized faster readout or independently measured amplifier response. Vary temperature and optical background while holding the wavelength and spot fixed. Agreement between step and frequency-response fits tests the one-pole approximation; disagreement identifies missing dynamics without by itself proving which physical mechanism caused them.

Photoconductor rise and fall asymmetry means that a photoconductor's normalized turn-on and turn-off transients have different shapes or threshold-crossing durations under stated illumination and electrical conditions. Define the levels first: a ten-to-ninety-percent rise and a ninety-to-ten-percent fall exclude different portions of a long tail than a zero-to-ninety-percent measurement. Detector temperature is an essential condition because the primary PbS description reports changes in response speed, dark resistance and spectral sensitivity with temperature.^[ref]

The simplest linear carrier model predicts symmetry. For excess carrier density $n$ satisfying $\dot{n}=g-n/\tau$, a step to $g_0$ gives $n/n_{\infty }=1-e^{-t/\tau }$; switching off after equilibration gives $n/n_{\infty }=e^{-t/\tau }$. Both threshold durations equal $\tau \ln 9$. Thus different measured durations are evidence against that complete one-pole measurement model, rather than a result it already predicts.

A photoconductor transient with quadratic recombination supplies an illuminating alternative model. Stipulate constant mobility, fixed detector bias, zero dark excess density andUnder constant $g_0$, let $n_{\infty }=\sqrt{g_0/\beta }$ and ${\tau }_{\ast }=1/(\beta n_{\infty })$. Starting at zero gives $n/n_{\infty }=\tanh (t/{\tau }_{\ast })$. Starting at $n_{\infty }$ and setting $g=0$ gives $n/n_{\infty }=(1+t/{\tau }_{\ast }{)}^{-1}$. Direct substitution verifies both solutions.

For an output proportional to $n$, the resulting durations areWith stipulated ${\tau }_{\ast }=1\mathrm{ms}$, the rise takes $1.372\mathrm{ms}$ and the fall $8.889\mathrm{ms}$, despite having no separate fitted rise and fall constants. This is a derived nonlinear electron-hole recombination model, not measured behavior or a specification of the cited PbS or CdS detector. Its long algebraic tail also shows why a single exponential fit may depend strongly on the fitted interval.

The readout can itself introduce asymmetry. A resistance divider has output $V_s{R}_{L}G/(1+R_{L}G)$ as a function of detector electrical conductance $G$. Equal exponential rise and fall in $G$ need not give equal normalized threshold durations after this nonlinear conversion. Carrier dynamics, the electrical transfer curve and an amplifier's response must therefore be separated before assigning a material lifetime.

A proposed experiment switches a monitored optical source between two reproducible levels, measures actual detector bias and current, waits for the same initial equilibrium, and repeats in both directions. Compare current-derived conductance with divider output, then vary background, temperature and step size. A slow reference source or readout can imitate a detector tail. Asymmetry alone does not uniquely identify trapping or any other microscopic mechanism.

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.^[ref] The quadratic loss law below is an explicitly stipulated model, not a fitted specification of that detector.

Assume a spatially uniform excess density $n\ge 0$, generation rate $g\ge 0$, constant coefficient $\beta >0$ and an output proportional to $n$ at fixed mobility and bias:Density has units ${\mathrm{m}}^{-3}$, $g$ has units ${\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$ 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 $n_0$: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 yieldsThe 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 $\overline{n}=\sqrt{\overline{g}/\beta }$: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 $\overline{g}=10^{21}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$, 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.^[ref]