Photoconductor

A photoconductor is a detector whose illuminated material changes electrical conductivity, allowing an external circuit to measure the change under applied bias. It uses photoconductivity: absorbed photons can produce mobile electrons and electron holes, changing the conductive path between contacts. A simple photoresistor reads this as a light-dependent resistance. Shimadzu’s primary description explains band-gap excitation and the resistance decrease in a PbS detector.^[ref]

An original uniform-bar model makes the signal mechanism quantitative. Let the contact separation be $\ell$, the conducting area $A$, electron and hole number densities $n,p$, and their mobilities ${\mu }_n,{\mu }_p$. In the low-field ohmic model,Here $e$ is the positive elementary charge and $V_b$ the actual detector electric potential difference. If illumination changes carrier densities while mobilities stay fixed,The assumptions exclude contact barriers, strongly nonuniform illumination and a bias changed by the readout; those effects require an expanded model.

For $\ell =1\mathrm{mm}$, $A=10^{-8}{\mathrm{m}}^2$, ${\mu }_n=0.10{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})$, ${\mu }_p=0.05{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})$ and $\Delta n=\Delta p=10^{18}{\mathrm{m}}^{-3}$, the conductivity increment is about $0.0240S/m$. At fixed $V_b=5\mathrm{V}$, the predicted current increment is about $1.20\mu \mathrm{A}$. These stipulated parameters illustrate the conversion of extra carriers into a signal, rather than specifications of a named detector.

Readout loading can change that result. In a divider powered at $V_s$, with detector resistance $R_d$ in series with load $R_L$, the output across the load isFor $V_s=5\mathrm{V}$ and $R_L=10\mathrm{k\Omega }$, changing $R_d$ from $100$ to $10\mathrm{k\Omega }$ raises the output from about $0.455$ to $2.50\mathrm{V}$. The detector’s own bias falls from about $4.55$ to $2.50\mathrm{V}$. A light-dependent conductance and a measured divider voltage therefore need not have the same linearity. Record the actual bias and illuminated area when measuring spectral responsivity.

Photoconductor response time also depends on carrier storage and loss. A simple spatially uniform excess-carrier model is $d\Delta n/dt=g(t)-\Delta n/\tau$. After a step to constant generation $g_0$, initially zero excess density becomes $g_0\tau (1-e^{-t/\tau })$. Small sinusoidal modulation has response proportional to $(1+i2\pi f\tau {)}^{-1}$. With stipulated $\tau =100\mu \mathrm{s}$, the single-pole cutoff is about $1.59\mathrm{kHz}$. Real traps, contacts and electronic filters need not yield one such time constant.

A proposed characterization biases the detector through a known readout, measures dark and illuminated currents at controlled wavelength and temperature, then chops or modulates the light and measures amplitude and phase versus modulation frequency. Repeat at several powers and biases to separate detector behavior from circuit loading. Temperature control matters: the manufacturer notes that PbS spectral sensitivity, response speed and dark resistance all change with temperature.^[ref]

The name photoconductive mode for a reverse-biased photodiode describes an operating choice of a junction detector; it does not make every such device the bulk-resistance detector modeled above. Likewise, a detector’s signal increase is not by itself a claim of unlimited useful gain: background response, dark current, noise, saturation and response time remain distinct characterization questions.

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]

A photoresistor, also called a light-dependent resistor or LDR, is a two-terminal component whose resistance changes with incident light. A common photoconductive implementation uses an illuminated semiconductor between contacts; an external circuit converts its resistance change into a voltage or current signal. A manufacturer’s CdS datasheet explicitly identifies these names and describes resistance decreasing with visible illumination.^[ref]

The photoconductor mechanism is a change in conductivity through generated mobile carriers. A photoresistor’s resistance is consequently not a calibrated unit of optical power by itself. Material, wavelength distribution, temperature, illumination history and the measuring circuit help determine the observed response.

For an original illustrative model over a limited range, let $R(P)=R_0(P/P_0{)}^{-\gamma }$ with $\gamma >0$, fixed spectrum and geometry. Choose $R_0=20\mathrm{k\Omega }$ and $\gamma =0.7$. Four times the reference optical input gives $R\approx 7.579\mathrm{k\Omega }$, rather than one quarter of the original resistance. This stipulated power law illustrates a nonlinear response, not a fitted specification for the cited part.

In a divider with supply $V_s$ and fixed resistor $R_L$, measuring across $R_L$ givesAt $V_s=3.3\mathrm{V}$ and $R_L=10\mathrm{k\Omega }$, the model output rises from $1.10\mathrm{V}$ at $P_0$ to about $1.877\mathrm{V}$ at $4P_0$. The logarithmic sensitivity is greatest when $R(P)=R_L$, where it is $\gamma V_s/4$. Choosing a resistor therefore selects an operating region; it does not remove the material’s response law.

The electrical input must also be characterized. The divider seen at its output has Thevenin resistance $R(P)\parallel R_L$. A finite-input-resistance voltmeter or an analog-to-digital converter with a switched input can change the reading. Self-heating depends on the detector’s actual voltage and resistance, while photoconductor response time can make a rapidly sampled value differ from its settled value.

Photoresistor spectral calibration is essential when interpreting an illumination reading. Lux is photometrically weighted, so equal illuminance from different spectra need not produce equal absorption or equal resistance. The cited manufacturer specifies source conditions for light-resistance measurements, a waiting time for dark resistance and light-adaptation preconditioning before its tests.^[ref] Those conditions describe what was tested; they should not be silently omitted from a comparison.

A proposed characterization uses a stable monitored light source, fixed detector position and a controlled spectrum, stepping illumination in both directions and recording the settled resistance and transient. Measure actual bias and temperature, then test the intended readout separately. A component can work well as a light/dark threshold sensor while requiring substantially more characterization to estimate optical power quantitatively.

Photoresistor spectral calibration determines how a photoresistor's electrical reading depends on the illumination spectrum under specified geometry, background, temperature, bias and adaptation history. A resistance-versus-lux curve measured with one lamp is not automatically valid for another spectrum. The manufacturer's primary CdS datasheet specifies a $2854\mathrm{K}$ source for its light-resistance tests and preconditions cells with light before electrical measurements.^[ref]

The distinction is between irradiance and illuminance. With spectral irradiance $E_{\lambda }(\lambda )$, illuminance is approximatelywhere $V$ is the photopic luminous efficiency function. The detector weights wavelengths according to its own physical response, which need not be proportional to $V$. NIST's detector-based photometric realization measures that spectral response, rather than assuming an arbitrary detector reproduces the photometric weighting.^[ref]

For a locally linear small-signal model about a fixed operating state, writeHere $s_G$ is the differential spectral responsivity of electrical conductance, including the stated illumination geometry. It is not automatically a global, illumination-independent material constant. A background-dependent photoresistor spectral response requires rechecking this derivative when the operating point changes.

An original two-band example makes the mismatch quantitative. Stipulate relative photometric weights $V_a=0.8$, $V_b=0.2$ and equal detector generation weights $s_a=s_b$ for two narrow bands. Equal illuminance requires four times as much irradiance in band $b$, so its detector-weighted generation is four times greater. If, over this illustrative range, $R\propto g^{-0.7}$, the resistance ratio is $R_b/R_a=4^{-0.7}\approx 0.379$. These weights and exponent are stipulated, not measured specifications or tabulated values for named wavelengths. Equal lux can produce substantially unequal resistance.

A proposed spectral characterization uses narrow-band sources or a monochromator, a reference detector to measure incident irradiance, fixed beam geometry and controlled detector temperature. Record resistance or conductance after a defined settling time at several background levels. A separate calibrated photometer can establish illuminance; it cannot by itself establish the photoresistor's spectral weighting. Keep electrical loading and self-heating fixed while changing the source.

For a broadband transfer, characterize the reference and test spectra and test whether the measured spectral-response model predicts their relative readings. Existing spectral mismatch correction methods address the analogous transfer problem for photometers, but a nonlinear photoresistor first needs a justified operating-point model.^[ref] The result is a calibration valid for stated conditions, not a universal conversion from resistance to optical power.

A background-dependent photoresistor spectral response is the differential change of a photoresistor's output with wavelength-specific illumination, evaluated at a stated background state. It can depend on illumination level, spectral composition, bias, temperature and prior adaptation. The primary CdS datasheet specifies light-adaptation preconditioning before electrical tests, demonstrating that a measurement condition includes more than the instantaneous lux value.^[ref]

Let the steady electrical conductance be a functional $G=\mathcal{F}[E_{\lambda }]$ of incident spectral irradiance. About a background $E_{\lambda ,0}$, define the local response byThis derivative concerns small changes after specifying the operating state. A finite change or rapid modulation needs further nonlinear or dynamic characterization. Photoresistor spectral calibration establishes the required optical and electrical conditions.

An original separable model makes a useful distinction. Suppose $g=\int a(\lambda )E_{\lambda }(\lambda )d\lambda$ and $G=G_d+\kappa g^{\gamma }$, with positive $g,\kappa ,\gamma$. ThenThe amplitude changes with background, but the normalized spectral shape remains $a(\lambda )$. With $\gamma =0.7$, quadrupling $g_0$ multiplies the absolute differential response by $4^{-0.3}\approx 0.6598$. For a small one-percent generation increment, the leading fractional conductance change above the dark contribution is $0.7\%$; the exact model gives $(1.01{)}^{0.7}-1\approx 0.69895\%$.

A change in spectral shape requires more than that single separable nonlinearity. For an illustrative two-band response in dimensionless input coordinates, letThe derivatives are $\partial G/\partial E_a=1/(2\sqrt{E_a})$ and $\partial G/\partial E_b=1$. Changing the first-band background from $E_a=1$ to $4$ changes their ratio from $1/2$ to $1/4$. This toy model shows how relative spectral sensitivity can change; it is not measured behavior of a named component. Separable and nonseparable photoresistor spectral-response models should consequently be distinguished in a calibration analysis.

Resistance readout adds another operating-point factor: since $R=1/G$, $\delta R=-\delta G/G_0^2$. A resistance-divider output has yet another derivative. Comparing raw electrical slopes without converting to the same output variable can imitate a change in material response. Likewise, equal illuminance need not mean equal detector excitation, because the photopic luminous efficiency function supplies a different spectral weighting.^[ref]

A proposed apparatus combines a stable background source with a weak wavelength-selectable probe beam, a monitored reference detector and a fixed illumination geometry. At each background, measure the output derivative while decreasing probe amplitude to test the small-signal regime, then repeat after the prescribed settling or adaptation period. Normalize spectra only after retaining their absolute scale. This separates a changing overall gain from a changing spectral shape and from readout loading; it does not assume that a curve measured at one background transfers to every illumination condition.