Photoconductor

1. Theory

• Definition: A photoconductor is a slab or thin film of semiconductor with two ohmic contacts whose conductivity rises when absorbed photons generate free carriers, so that the signal is the photocurrent $I_{ph}=I-I_{dark}$ of a biased resistor.
• Photoconductivity: The change $\Delta \sigma =q({\mu }_n\Delta n+{\mu }_p\Delta p)$ follows from the excess densities, which in the steady state equal the generation rate times the lifetime $\tau$, and the photon energy must exceed the band gap, $h\nu \ge E_g$, for the intrinsic effect.
• Photoconductive gain: $G=\tau /t_{tr}$ is the ratio of the carrier lifetime to the transit time between the electrodes, $t_{tr}=L^2/(\mu V)$, so that one absorbed photon can give many electrons in the circuit without avalanche multiplication.
• Materials: CdS and CdSe cover the visible, PbS the near infrared to about $3.1$ μm to $3.2$ μm when cooled, PbSe and InSb the mid infrared, and Cu-doped Ge the far infrared; research devices use black phosphorus, graphene and nanowires.
• Resistance: A CdS cell has a dark resistance of up to several MΩ and a resistance of a few hundred Ω in bright light, and a thermoelectrically cooled PbS element has a dark resistance of 0.5 MΩ to 10 MΩ.
• Figures of merit: The responsivity $\mathfrak{R}$ in $A/W$, the gain, the response time (set by the lifetime), the noise-equivalent power $\mathrm{NEP}$, the specific detectivity $D^{\ast }$ and the linear dynamic range.
• Noise: The generation-recombination noise grows with the gain, $⟨i_{GR}^2⟩=4qG{I}_{ph}\Delta f$, and adds to the Johnson noise of the element and the $1/f$ noise of the traps.
• Trade-off: A long lifetime gives a large gain and a slow response, because the same $\tau$ sets $G$ and the bandwidth $f_{3dB}=1/(2\pi \tau )$.

2. Working Principle

• Generation: A photon with $h\nu \ge E_g$ (or enough to excite an impurity level) creates an electron-hole pair, and the excess density is $\Delta n=\eta P_{opt}\tau /(h\nu WLD)$.
• Bias and readout: A bias $V$ and a series load resistor $R_L$ convert the fall of the resistance of the element into a voltage change across $R_L$.
• Carrier recycling: The field sweeps a carrier to an electrode within $t_{tr}$, the ohmic contact injects a new carrier, and the circulation continues until the pair recombines after $\tau$, which gives $G=\tau /t_{tr}$.
• Sensitization and traps: Impurity centres that replace short-lived recombination centres by long-lived ones, and traps that hold one carrier type, raise the effective lifetime and the gain by several orders of magnitude, at the cost of a slow and history-dependent response.
• Saturation: At a high photon rate the traps fill, the effective lifetime falls, and the photocurrent follows $F/(1+F/F_0)$, so the responsivity decreases with power and the power law $\Delta \sigma \propto {\Phi }^b$ has $b<1$.
• Temperature: Cooling raises the resistance and lowers the thermal noise, shifts the spectral response of PbS to longer wavelengths and slows the response.

3. Applications

• Audio opto-isolators (vactrols): Resistive cells in which the slow, history-dependent resistance of the CdS cell is the wanted behaviour for compressors and tremolo.
• Near- and mid-infrared detection: PbS and PbSe photoconductors, which work uncooled or with a thermoelectric cooler, as in the detector of a spectrophotometer and for detection near $3.8$ μm.
• Far-infrared astronomy and spectroscopy: Cu-doped Ge photoconductors, among the best far-infrared detectors.

Name	Formula	Related Formulae	Variables/Constants
Photoconductivity and carrier density	$\Delta \sigma =q\left({\mu }_n\Delta n+{\mu }_p\Delta p\right),\Delta n=\frac{\eta P_{opt}\tau }{h\nu WLD}$	$I_{ph}={\sigma }_{ph}EWD=q\mu \Delta nEWD$ $R_d=\frac{L^2}{q\left({\mu }_n+{\mu }_p\right)\eta \Phi \tau }$	$\Delta \sigma :photoconductivity$ ${\mu }_n,{\mu }_p:electronandholemobilities$ $\Delta n,\Delta p:excesscarrierdensities$ $\eta :quantumefficiency$ $P_{opt}:opticalpower$ $\tau :carrierlifetime$ $\begin{array}{r}h\nu :photonenergy(h\nu \ge E_{g}forthe\\ intrinsiceffect)\end{array}$ $\begin{array}{r}W,L,D:width,lengthandthicknessof\\ thechannel\end{array}$ $E:electricfieldinthechannel$ $\Phi :photonrate(s^{-1})$ $\begin{array}{r}{R}_d:resistanceofthephoto-generated\\ conductancewith\Delta n=\Delta p(thedark\\ conductanceneglected)\end{array}$
Photoconductive gain	$G=\frac{\tau }{t_{tr}},t_{tr}=\frac{L}{\mu E}=\frac{L^2}{\mu V}$	$G=\frac{\tau \left({\mu }_n+{\mu }_p\right)V}{L^2}=\frac{\tau }{t_{tr,n}}\left(1+\frac{{\mu }_p}{{\mu }_n}\right)$ $G=\frac{{\tau }_{n,eff}}{t_{tr,n}}\left(1+G^{\prime }\frac{{\mu }_p}{{\mu }_n}\right),G^{\prime }=\frac{\Delta p}{\Delta n}$	$\begin{array}{r}G:photoconductivegain(electronsinthe\\ circuitperabsorbedphoton)\end{array}$ $t_{tr}:transittimebetweentheelectrodes$ $V:biasvoltage$ $t_{tr,n}:transittimeoftheelectrons$ ${\tau }_{n,eff}:effective(trap-extended)lifetime$ $\begin{array}{r}{G}^{\prime }:ratiooftheexcessholeandelectron\\ densities\end{array}$ $\begin{array}{r}(for\tau =1ms,\mu =100\mathrm{c}{\mathrm{m}}^2/Vs,V=10V\\ andL=100\mu \mathrm{m}(assumed)thetransittime\\ is0.1\mu \mathrm{s}andG=10^4,computed)\end{array}$
Photocurrent and responsivity	$I_{ph}=\eta q\Phi G,\mathfrak{R}=\frac{I_{ph}}{P_{opt}}=\frac{\eta q\lambda }{hc}G$	$\mathfrak{R}=\eta G\frac{\lambda \left(\mu m\right)}{1.24}A/W$ $I_{ph}=q\eta \frac{{\tau }_0}{t_{tr}}\frac{F}{1+F/F_0}$	$\mathfrak{R}:responsivity$ $\lambda :wavelength$ $\begin{array}{r}c:speedoflight,theresponsivityrises\\ inproportionto\lambda uptothecutoff{\lambda }_c=\\ hc/E_g\end{array}$ $F:photonabsorptionrate$ $F_0:rateatwhichthetrapssaturate$ $\begin{array}{r}{\tau }_0:lifetimewithoutsaturation(the\\ Hornbeck-Haynesform),(for\eta =0.8,G=\\ 100and\lambda =0.55\mu \mathrm{m}(assumed)\mathfrak{R}=35.5\\ A/W,computed)\end{array}$
Gain-bandwidth product	$f_{3dB}=\frac{1}{2\pi \tau },G{f}_{3dB}=\frac{1}{2\pi t_{tr}}=\frac{\mu V}{2\pi L^2}$	$t_{tr}=\frac{L^2}{\mu V}$ $f_{3dB}=\frac{1}{2\pi {\tau }_0}$	$\begin{array}{r}{f}_{3dB}:bandwidthsetbythecarrierlifetime\\ (derived),theproductG{f}_{3dB}doesnot\\ dependon\tau andrisesas1/L^2\end{array}$ $\begin{array}{r}(for{\tau }_0=0.13ms,L=2\mu \mathrm{m},\mu =300\\ \mathrm{c}{\mathrm{m}}^2/VsandV=0.1Vthetransittimeis\\ 1.3ns,f_{3dB}=1.2kHzandtheproduct120\\ MHz,computed)\end{array}$
Dependence on the light level	$\Delta \sigma \propto {\Phi }^b,R=A{E}^{-\gamma },\gamma =\frac{\log \left(R_{10}/R_{100}\right)}{\log \left(100/10\right)}$	$b=1\left(\mathrm{monomolecular}\right),b=\frac{1}{2}\left(\mathrm{bimolecular}\right)$ $\gamma =\frac{d}{d+1+{\beta }_v},v\sim {\mathcal{E}}^{{\beta }_v}$	$\begin{array}{r}b:exponentofthephotoconductivity(0.5\\ <b<1foramixtureoflinearand\\ quadraticrecombination)\end{array}$ $A,\gamma :constantsofaphotoresistor$ $E:illuminance$ $R_{10},R_{100}:resistancesat10and100lux$ $d:dimension$ $\begin{array}{r}{\beta }_v:exponentofthecarriervelocity\\ againstthelocalfield\mathcal{E}({\beta }_v=1gives\\ 3/5and{\beta }_v\to 0gives3/4ford=3and2/3\\ ford=2),(R_{10}=20k\Omega and{R}_{100}=5k\Omega \\ (assumed)give\gamma =0.60,computed)\end{array}$
Trap and recombination kinetics	$\frac{dn}{dt}=F+P{n}_t-R_t\left(N_t-n_t\right)n-R_r\left(n+n_t\right)n,\frac{d{n}_t}{dt}=R_t\left(N_t-n_t\right)n-P{n}_t$	$g\Phi =R_r\left(n+n_t\right)n$ ${\sigma }_{PC}\left(t\right)={\sigma }_{PC,0}{e}^{-{\left(t/{\tau }_{str}\right)}^{{\beta }_s}}$	$n:densityofconductionelectrons$ $n_t:densityoftrappedelectrons$ $N_t:trapdensity$ $F:generationrate$ $P:releaseratefromthetraps$ $R_t:trappingcoefficient$ $R_r:recombinationcoefficient$ $g\Phi :generationinthesteadystate$ ${\tau }_{str}:timeconstant$ $\begin{array}{r}{\beta }_s:stretchingexponent(0.43forthe\\ short-timedecayofaCdSfilm),theslow\\ decaycausesthepersistent\\ photoconductivity\end{array}$
Noise of the photoconductor	$⟨i_{GR}^2⟩=4qG{I}_{ph}\Delta f,⟨i_J^2⟩=\frac{4k_{B}T\Delta f}{R_d}$	$⟨i_N^2⟩=⟨i_{GR}^2⟩+⟨i_J^2⟩+⟨i_{1/f}^2⟩$ $G{I}_{ph}=\frac{k_{B}T}{q{R}_d}$	$i_{GR}:generation-recombinationnoisecurrent$ $\Delta f:noisebandwidth$ $i_J:Johnsonnoisecurrent$ $k_B:Boltzmannconstant$ $T:temperature$ $R_d:resistanceoftheelement$ $\begin{array}{r}{i}_{1/f}:1/fnoisefromtrappingand\\ detrapping,with⟨i_{1/f}^2⟩\propto \Delta f/f^{{\alpha }_f},{\alpha }_f\approx 1\end{array}$ $\begin{array}{r}thesecondrelatedformulaisthe\\ photocurrentatwhichthetwonoise\\ currentsareequal(derived),(25.9nAfor\\ R_d=1M\Omega at300K,computed)\end{array}$
Noise-equivalent power and detectivity	$\mathrm{NEP}=\frac{i_N}{\mathfrak{R}},D^{\ast }=\frac{\sqrt{A\Delta f}}{\mathrm{NEP}}$	$D_{BLIP}^{\ast }=\frac{\lambda }{2hc}\sqrt{\frac{\eta }{{\Phi }_B}}$ $\mathrm{LDR}=10{\log }_{10}\left(\frac{P_{sat}}{\mathrm{NEP}}\right)\mathrm{dB}$	$NEP:noise-equivalentpower(W/\sqrt{\mathrm{Hz}})$ $D^{\ast }:specificdetectivity(\mathrm{cm}\mathrm{H}{\mathrm{z}}^{1/2}/W)$ $A:area$ ${\Phi }_B:backgroundphotonfluxdensity$ $BLIP:background-limitedperformance$ $P_{sat}:saturationpower$ $\begin{array}{r}LDR:lineardynamicrange,(for\eta =0.5,\lambda \\ =5\mu \mathrm{m}and{\Phi }_B=1.23\times 10^{16}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}thelimit\\ is8.0\times 10^{10}\mathrm{cm}\mathrm{H}{\mathrm{z}}^{1/2}/W,computed;a\\ photodiodehasthefactor\sqrt{2}larger)\end{array}$

Important Questions

Ques: Why does a photoconductor have gain, and what sets it?

Each absorbed photon creates an electron-hole pair, and the bias sweeps the faster carrier to its electrode in the transit time $t_{tr}=L^2/(\mu V)$. The ohmic contact supplies a new carrier at the other end to keep the channel neutral, so a carrier circulates until the pair recombines after the lifetime $\tau$, and one photon gives $G=\tau /t_{tr}$ electrons in the circuit; with two mobile carriers $G=\tau ({\mu }_n+{\mu }_p)V/L^2$. For $\tau =1$ ms, $\mu =100\mathrm{c}{\mathrm{m}}^2/Vs$, $V=10$ V and $L=100$ μm (assumed) the transit time is 0.1 μs and $G=10^4$ (computed). A black-phosphorus device with ${\tau }_0=0.13$ ms and a transit time of 1.3 ns had ${\tau }_0/t_{tr}\approx 10^5$. No multiplication is involved; the gain is the ratio of a long lifetime to a short transit time. Reference: Black Phosphorus Mid-Infrared Photodetectors with High Gain, arXiv:1603.07346; Graphene-based light sensing: fabrication, characterisation, physical properties and performance, arXiv:1809.04862; Leiden Observatory lecture notes Intrinsic Noise, DTL 02 2017 (home.strw.leidenuniv.nl/~brandl/DOL).

Ques: Does a photoconductor have gain intrinsically, or only through traps?

The classical recycling result $G=\tau /t_{tr}$ assumes a uniform excess density $\Delta n=g\tau$ and equal excess densities of electrons and holes. arXiv:1511.03118 argues that a metal contact forces $\Delta n=0$ at the interface, so that an ideal semiconductor has a gain of at most about 1, or the ratio ${\mu }_{maj}/{\mu }_{min}$, whatever $\tau$ and $t_{tr}$ are, and that the large measured gains come from defects and surface states that trap one carrier type and leave $G^{\prime }=\Delta p/\Delta n\gg 1$. Its simulations cover only 50 μm silicon and a 9 μm nanowire. Both views fit $G=({\tau }_{n,eff}/t_{tr})(1+G^{\prime }{\mu }_p/{\mu }_n)$ with a trap-extended lifetime ${\tau }_{n,eff}$, so $G=\tau /t_{tr}$ is the working formula once $\tau$ means that lifetime, and the claim of no intrinsic gain is more likely for trap-free material. Reference: A photoconductor intrinsically has no gain, arXiv:1511.03118; Graphene-based light sensing: fabrication, characterisation, physical properties and performance, arXiv:1809.04862.

Ques: Why do the gain and the bandwidth of a photoconductor trade off?

The response time of a lifetime-limited photoconductor is the lifetime itself, so $f_{3dB}=1/(2\pi \tau )$, whereas the gain is $G=\tau /t_{tr}$. The product $G{f}_{3dB}=1/(2\pi t_{tr})=\mu V/(2\pi L^2)$ (derived) does not depend on $\tau$: a longer lifetime buys gain and loses speed in the same proportion. For the black-phosphorus device, ${\tau }_0=0.13$ ms gives 1.2 kHz and a transit time of 1.3 ns gives a product of 120 MHz (computed); the measured bandwidth fell from 2.2 kHz to 1.1 kHz when the power fell from 5.6 μW to 1.6 nW, which agrees with a lifetime that grows when fewer traps are filled. The way to keep both is to shorten the transit time with a high mobility, a small electrode spacing and a high field. Reference: Black Phosphorus Mid-Infrared Photodetectors with High Gain, arXiv:1603.07346; Graphene-based light sensing: fabrication, characterisation, physical properties and performance, arXiv:1809.04862.

Ques: Is the exponent of the photoconductivity 1, 1/2, 3/5 or 3/4, and which is more likely?

Recombination on a fixed number of centres gives $b=1$ and bimolecular recombination $b=1/2$, so a mixture gives $0.5<b<1$. A CdS film gave $b=0.3$ to 0.5 at low and 0.6 to 0.7 at high illumination, depending on temperature, which a quadratic term in the recombination (1/3 to 1/2 at low intensity) explains. The hot-carrier model gives $\gamma =d/(d+1+{\beta }_v)$, which is 3/5 for ${\beta }_v=1$ and 3/4 for hot carriers in three dimensions, and four commercial photocells gave 3/4 in three samples and 2/3 in one, the author claiming a universal exponent that is independent of temperature. Because the CdS film data depend on temperature, a value set by the recombination kinetics is more likely, with 3/4 a typical value for cells. A cell with $R_{10}=20$ kΩ and $R_{100}=5$ kΩ (assumed) has $\gamma =0.60$ (computed). Reference: Large and Persistent Photoconductivity due to Hole-Hole Correlation in CdS, arXiv:1806.01894; Hot-Carrier Model for an Anomalous Exponent of Photoconduction, arXiv:1006.2496; Token Electronics, CdS Photoresistors (token.com.tw).

Ques: Why does the responsivity of a photoconductor fall at high optical power?

The traps that extend the lifetime are finite in number. The Hornbeck-Haynes form $I_{ph}=q\eta ({\tau }_0/t_{tr})F/(1+F/F_0)$ is linear for $F\ll F_0$ and saturates above it, so the responsivity has fallen to half at $F=F_0$ (computed). The black-phosphorus device gave 82 A/W at 1.6 nW, and its bandwidth rose from 1.1 kHz to 2.2 kHz at 5.6 μW which agrees with a lifetime that shortens as the traps fill. Reduced graphene-oxide films lost more than three orders of magnitude of responsivity with rising power, and a fluorographene device showed the same saturation. The linear dynamic range $10{\log }_{10}(P_{sat}/\mathrm{NEP})$ dB is therefore limited at the upper end by the trap density. Reference: Black Phosphorus Mid-Infrared Photodetectors with High Gain, arXiv:1603.07346; Graphene-based light sensing: fabrication, characterisation, physical properties and performance, arXiv:1809.04862.

Ques: Why does a larger gain not raise the background-limited detectivity, and when does it help?

In the background-limited regime both the signal and the generation-recombination noise carry the factor $G$, so that $\mathfrak{R}=\eta q\lambda G/(hc)$ and $i_{GR}=2qG\sqrt{\eta \Phi \Delta f}$ (with $\Phi ={\Phi }_{B}A$) give $D_{BLIP}^{\ast }=(\lambda /2hc)\sqrt{\eta /{\Phi }_B}$ (derived), independent of $G$. For $\eta =0.5$, $\lambda =5$ μm and ${\Phi }_B=1.23\times 10^{16}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$ this is $8.0\times 10^{10}\mathrm{cm}\mathrm{H}{\mathrm{z}}^{1/2}/W$ (computed), and a photodiode with the shot noise $2qI$ instead of $4qGI$ has a factor $\sqrt{2}$ more. The gain helps when the Johnson noise or the amplifier noise dominates, since the signal grows with $G$ and these noises do not; the two noises are equal at $G{I}_{ph}=k_{B}T/(q{R}_d)$, which is 25.9 nA for 1 MΩ at 300 K (computed). Reference: Leiden Observatory lecture notes Intrinsic Noise, DTL 02 2017 (home.strw.leidenuniv.nl/~brandl/DOL); OU lecture file 18 Detection (ou.edu/CoE/ECE/weng), the section on PbSe n+-p detectors; Graphene-based light sensing: fabrication, characterisation, physical properties and performance, arXiv:1809.04862.

Ques: Why does a CdS photoconductor respond slowly and remember its light history, and are the times 10 ms and 1 s?

One source gives about 10 ms for the turn-on and up to about 1 s for the recovery, another tens to hundreds of ms for both. In CdS films the decay has no single time constant: it is a stretched exponential with ${\beta }_s=0.43$ at short times, a single exponential with an activation energy of 0.55 eV at intermediate times, and a decay rate as low as 0.1 decade per hour, which the authors attribute to negative-U sulfur vacancies. One manufacturer’s specification pre-illuminates the cell for 2 h at 400 lux to 600 lux before the measurement and reads the dark resistance 10 s after the light is removed. The two time statements are consistent if 1 s is the upper figure of ordinary cells at moderate light, which is the more likely reading. Reference: Large and Persistent Photoconductivity due to Hole-Hole Correlation in CdS, arXiv:1806.01894; Wikipedia, Photoresistor; LCSC, The Photosensitive Resistor (LDR): In-Depth Principles, Testing Procedures, and Applications; Token Electronics, CdS Photoresistors (token.com.tw).

Ques: Why does cooling a PbS photoconductor raise its detectivity, extend its cutoff and slow it down?

The datasheet of thermoelectrically cooled PbS lists $D^{\ast }$ of $1.5\times 10^{11}$, $2.5\times 10^{11}$ and $2.8\times 10^{11}$ at $-20$, $-30$ and $-45{}^{\circ }$C, and a long-wavelength cutoff of 3.1 μm for the first and 3.2 μm for the others, i.e. 0.40 eV and 0.39 eV (computed); the supplier of a spectrophotometer states that cooling shifts the sensitivity to longer wavelengths and slows the response, and that PbS also works at room temperature. The mechanism is not stated in the sources; the likely one is that fewer thermally generated carriers raise the dark resistance and the lifetime, which lowers the thermal noise and lengthens the response. The three models are different elements, so the rise of $D^{\ast }$ is only partly a temperature effect. Reference: New England Photoconductor, PbS TE-cooled detectors datasheet (shop.amstechnologies.com); Shimadzu, How does a PbS photoconductive detector work? (ssi.shimadzu.com FAQ).

Ques: Why is the generation-recombination noise current $4qG{I}_{ph}\Delta f$ and not the shot-noise value $2qI\Delta f$?

In a photodiode only the generation of carriers is random, which gives the shot noise $2qI\Delta f$. In a photoconductor the number of free carriers fluctuates both by random generation and by random recombination, and each process adds an independent variance of the same size, so that the spectral density of the carrier number is twice that of the generation alone (derived). Each carrier also contributes $G$ electrons to the circuit, so that the noise current is $⟨i_{GR}^2⟩=4q^2{G}^2\eta \Phi \Delta f=4qG{I}_{ph}\Delta f$ with $I_{ph}=\eta q\Phi G$. This factor of two in the variance is the reason for the factor $\sqrt{2}$ in the limit of the detectivity against the photodiode. Reference: Leiden Observatory lecture notes Intrinsic Noise, DTL 02 2017 (home.strw.leidenuniv.nl/~brandl/DOL).

Related pages in this vault: Colloidal Quantum-Dot Photodetector (CQD), Ultraviolet Photodetector (AlGaN, SiC, Ga2O3, Diamond), Perovskite Photodetector (Photodiode and Photoconductor), Graphene and 2D-Material Photodetector, Phototransistor (HPT and Photo-FET), Quantum-Dot Solar Cell.