Quantum-Well Infrared Photodetector (QWIP)

1. Theory

• Definition: A quantum-well infrared photodetector (QWIP) is a stack of 20 to 50 quantum wells, usually n-doped GaAs wells between AlGaAs barriers, in which the infrared photon excites an electron from the filled ground subband $E_1$ to the excited subband $E_2$, from where it escapes into the continuum and is collected as photocurrent under a bias.
• Wavelength: The photon energy $E_2-E_1=hc/\lambda$ is set by the well width and the barrier composition, with $\lambda \approx A/l_w$, so that the detection wavelength is tuned from the mid to the long wavelength infrared by design.
• Selection rule: In an n-type QWIP only the electric field component perpendicular to the layers is absorbed, so that normal-incidence light needs a grating, a 45° facet or a random reflector, whereas a p-type QWIP absorbs normal incidence by the mixing of light-hole and heavy-hole states.
• Gain and efficiency: The photoconductive gain $g=p_e/(N{p}_c)$ is of order one for n-type devices, and the quantum efficiency is low, 1.4% to 11% for the focal plane arrays quoted here.
• Dark current: The thermionic emission of electrons over the barrier, with an activation energy near the photon energy of the cutoff, dominates at high temperature, and below about 25 K a plateau appears that the thermionic and tunnelling models do not reproduce.
• Noise: The generation-recombination noise of the dark current, $i_n=\sqrt{4A_{d}qg\Delta f{J}_{dark}}$, sets the detectivity below the background-limited temperature.
• Figures of merit: The peak wavelength and the relative bandwidth $\Delta \lambda /\lambda$, the responsivity, the gain, the dark current, the detectivity $D^{\ast }$, the BLIP temperature and the noise-equivalent temperature difference.
• Arrays: The GaAs technology gives large, uniform arrays: 640 × 512 pixels of 25 μm with an operability of 99.9% for a broadband device.

2. Working Principle

• Doping and subbands: The wells are n-doped so that the Fermi level lies above $E_1$ and the ground subband is filled, and the barriers are wide enough to prevent tunnelling between the wells.
• Absorption: A photon of energy $E_2-E_1$ with a polarization component along the growth direction excites an electron, and the absorption per well is small, so the quantum efficiency follows $\eta =P(1-r)[1-e^{-B\alpha l_w}]$ with $P=0.5$ for n-type.
• Escape: The excited state lies in the continuum (bound-to-continuum) or just below the barrier top (bound-to-quasibound), and the field makes the electron leave the well.
• Gain: A carrier that leaves the well is replaced by one injected from the contact, so that the photocurrent exceeds the emission current by $g=p_e/(N{p}_c)$, which falls as the number of wells grows.
• Coupling: A two-dimensional grating of period $g_p$ diffracts the normal incidence light into waves with a perpendicular field component; the effective window is $2.3<\lambda /g_p<3.25$.
• Dark current: Electrons thermally excited over the barrier, and at low temperature tunnelling, give the dark current, and a longer cutoff wavelength lowers the barrier and raises it.
• Bias dependence: The bias raises the escape probability, so the responsivity rises and the cutoff shifts through field-assisted tunnelling.

3. Applications

• Multicolor and very-long-wavelength focal plane arrays: A single GaAs-based material system gave 640 × 512 arrays for the 8 μm to 9 μm band (noise-equivalent temperature difference 20 mK), a broadband array with a peak at 13.5 μm and a four-band array from 4 μm to 15.5 μm.

Name	Formula	Related Formulae	Variables/Constants
Intersubband transition and wavelength	$E_2-E_1=\frac{hc}{\lambda },\lambda =\frac{A}{l_w}$	$E_2-E_1=\frac{3{\pi }^2{\hslash }^2}{2m^{\ast }{l}_w^2}\left(\mathrm{infinite}\mathrm{well}\right)$ $\lambda \left(\mu m\right)=\frac{1.2398}{E_2-E_1\left(\mathrm{eV}\right)}$	$E_1,E_2:groundandexcitedsubbandenergies$ $l_w:wellwidth$ $\begin{array}{r}A:constantofthewellandbarrier\\ materials(4.83\times 10^{-10}\mathrm{c}{\mathrm{m}}^{2}for\\ GaAs/G{a}_{0.74}A{l}_{0.26}As)\end{array}$ $\begin{array}{r}{m}^{\ast }:effectivemass,inafinitewell{E}_2\\ liesclosetothetopofthebarrier\end{array}$ $\begin{array}{r}(l_w=4nmgives\lambda =12\mu \mathrm{m},and85.5meV\\ correspondsto14.5\mu \mathrm{m},computed)\end{array}$
Quantum efficiency	$\eta =P\left(1-r\right)\left[1-e^{-B\alpha l_w}\right]$	$\eta =\frac{2}{3}\left[1-e^{-\alpha N{l}_w}\right]$ $\alpha \propto {\left|M\right|}^2$	$\begin{array}{r}\eta :absorptionquantumefficiency\\ (probabilitythatanabsorbedphoton\\ excitesanelectron)\end{array}$ $\begin{array}{r}P:polarizationconstant(0.5forn-type\\ and1forp-type)\end{array}$ $r:reflectioncoefficient$ $B:numberofpasses$ $\alpha :intersubbandabsorptioncoefficient$ $l_w:totalwidthofthewells$ $\begin{array}{r}thesecondformistheworstcaser=1/3,\\ B=1forNwellsofwidth{l}_w\end{array}$ $\begin{array}{r}M:matrixelementoftheenvelope\\ function,(\alpha =25.4\times 10^3\mathrm{c}{\mathrm{m}}^{-1}withN=25\\ and{l}_w=4nm(assumed)gives\eta =15\%,\\ computed)\end{array}$
Grating coupling	$s=\frac{\lambda }{g},2.3<s<3.25$	$g=\frac{\lambda }{s}$ ${\lambda }_c\approx 3.25g$	$g:periodofthetwo-dimensionalgrating$ $\begin{array}{r}s:normalizedwavelength,onlytheTM\\ componentsofthediffractedwavesare\\ absorbed(theelectricfieldmusthavea\\ componentperpendiculartothewells),the\\ firstdiffractedorderspropagatewhenthe\\ wavelengthinthematerialisshorterthan\\ theperiod\end{array}$ $\begin{array}{r}(aperiodof3.3\mu \mathrm{m}givesacutoffof10.7\\ \mu \mathrm{m},and\lambda =9\mu \mathrm{m}needsg=2.8\mu \mathrm{m}to3.9\\ \mu \mathrm{m},computed)\end{array}$
Photoconductive gain and capture	$g=\frac{p_e}{N{p}_c}=\frac{1-P_c}{N{P}_c}$	$P_c=1-e^{-2l_w/L_c}$ $g=\frac{L}{t_c},L=\tau T_{qw}{\mu }_{eff}E$	$g:photoconductivegain$ $p_e:escapeprobability$ $p_c,P_c:captureprobabilityofawell$ $N:numberofwells$ $L_c:decaylengthofthecapture$ $L:meanfreepathofthecarrier$ $\begin{array}{r}{t}_c:totalthicknessofthewelland\\ barrierstack\end{array}$ $\tau :capturelifetime$ $T_{qw}:transmissionthroughawell$ ${\mu }_{eff}:effectivemobility$ $E:field$
Photocurrent and responsivity	$I_p=q{\Phi }_s\eta g,\mathfrak{R}=\frac{q\lambda }{hc}\eta g$	$\mathfrak{R}=\frac{q\eta \lambda }{hc}\cdot \frac{1-P_c}{N{P}_c}$ ${\eta }_{ext}=\eta g$	$I_p:photocurrent$ ${\Phi }_s:incidentphotonflux$ $\begin{array}{r}\mathfrak{R}:responsivity,thesecondformfollows\\ withaconstantcaptureprobability:the\\ responsivityisthenproportionaltothe\\ wavelength\end{array}$ ${\eta }_{ext}:externalquantumefficiency$ $\begin{array}{r}(\mathfrak{R}=83mA/Wat8.5\mu \mathrm{m}with{\eta }_{ext}=1.4\%\\ givesg=0.86,and250mA/Wat13.5\mu \mathrm{m}\\ with11\%givesg=0.21,computed)\end{array}$
Dark current	$J_{th}=e\frac{m^{\ast }}{\pi {\hslash }^2}\frac{v_d}{L_w}eF{L}_b\exp \left(\frac{-V_b-eF{L}_w-E_{fw}}{k_{B}T}\right)$	$J_{ST}=\frac{e{k}_{B}T}{\hslash L_w^2}P\left(E_1+eF{L}_w\right)\ln \left(\frac{1+e^{\left(E_{fw}-E_1\right)/k_{B}T}}{1+e^{\left(E_{fw}-eF{L}_b-E_1\right)/k_{B}T}}\right)$ $P\left(E\right)=\exp \left(-\frac{4\sqrt{2m_b^{\ast }}}{3eF\hslash }\left[{\left(V_b-E\right)}^{3/2}-{\left(V_b-eF{L}_b-E\right)}^{3/2}\right]\right)$	$\begin{array}{r}{J}_{th}:thermionicdarkcurrentdensity(the\\ Levineform)\end{array}$ $v_d:driftvelocity$ $F:field$ $L_w,L_b:wellandbarrierwidths$ $V_b:barrierheight$ $\begin{array}{r}{E}_{fw}:Fermilevelabove{E}_1,theactivation\\ energyisabout{E}_a=81meV\end{array}$ $J_{ST}:sequentialtunnellingcurrent$ $P\left(E\right):WKBtransmissionprobability$ $\begin{array}{r}{m}_b^{\ast }:effectivemassinthebarrier,(E_a=\\ 81meVgivesafactorof9.5between65K\\ and77K,prefactorneglected,computed)\end{array}$ $k_B:Boltzmannconstant$
Noise and detectivity	$i_n=\sqrt{4A_{d}qg\Delta f{J}_{dark}},D^{\ast }=\frac{\mathfrak{R}\sqrt{A_d\Delta f}}{i_n}$	$D_{BLIP}^{\ast }=\frac{\lambda }{2hc}\sqrt{\frac{\eta }{{\Phi }_B}}$ $\%\mathrm{BLIP}=\frac{i_p}{\sqrt{i_p^2+i_{nd}^2}}$	$i_n:generation-recombinationnoisecurrent$ $A_d:detectorarea$ $\Delta f:bandwidth$ $J_{dark}:darkcurrentdensity$ $D^{\ast }:specificdetectivity(\mathrm{cm}\mathrm{H}{\mathrm{z}}^{1/2}/W)$ ${\Phi }_B:backgroundphotonfluxdensity$ $BLIP:background-limitedperformance$ $i_p:photonnoisecurrent$ $\begin{array}{r}{i}_{nd}:darkcurrentnoise,thedetectivityat\\ BLIPdoesnotdependonthegain,(for\eta =\\ 1.5\%,\lambda =8.5\mu \mathrm{m}and{\Phi }_B=7.5\times 10^{15}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}\\ (300Kbackground,8\mu \mathrm{m}to9\mu \mathrm{m},f/2,\\ computed)D_{BLIP}^{\ast }=3.0\times 10^{10}\mathrm{cm}\mathrm{H}{\mathrm{z}}^{1/2}/W,\\ computed)\end{array}$

Important Questions

Ques: Why can an n-type QWIP not absorb normally incident light, and what does the grating do?

An intersubband transition in the conduction band couples only to the component of the electric field perpendicular to the wells, and light at normal incidence has none. A p-type QWIP avoids this, because the mixing of light-hole and heavy-hole states allows the absorption at normal incidence. For the n-type device a two-dimensional grating diffracts the light into waves with a perpendicular component, and only the TM components are absorbed. The first orders propagate if the wavelength in the material is shorter than the period, which gives the window $2.3<\lambda /g_p<3.25$: a period of 3.3 μm gave a cutoff of 10.725 μm, which is $3.25g_p$ (computed), and a calculated 2.7 times the responsivity of a 45° edge-coupled device. A broadband array uses a random reflector instead. Reference: DTIC report ADA272675 on p-type strained-layer QWIPs and grating coupling (the title is not given by the fetch); NASA Technical Reports Server 20040068216, large-format QWIP focal plane arrays (narrow-band, four-band and broadband; the title is not given by the fetch).

Ques: Why is the quantum efficiency of a QWIP low, and is the performance still useful?

The absorption per well is small and the n-type selection rule keeps only half of the polarization, so $\eta =P(1-r)[1-e^{-B\alpha l_w}]$ is small: $\alpha =25.4\times 10^3\mathrm{c}{\mathrm{m}}^{-1}$, 25 wells of 4 nm (assumed) and the worst case $r=1/3$ give 15% (computed), whereas the focal plane arrays reached 1.4% for the narrow-band, 1.5% for the array and 11% for the broadband device in a 45° double pass. The responsivity is $\mathfrak{R}=(q\lambda /hc)\eta g$, 83 mA/W for the narrow band. The performance is still useful because the noise at the operating temperature is low: the mean noise-equivalent temperature difference was 20 mK at 65 K (about 10 mK was estimated from test detectors), with a uniformity that suits large arrays. Reference: NASA Technical Reports Server 20040068216, large-format QWIP focal plane arrays (narrow-band, four-band and broadband; the title is not given by the fetch); Gadir, Harrison and Soref (2002), Modeling the capture probability and enhancing the photoconductive gain in quantum well infrared photodetectors (QWIPs); DTIC report ADA272675 on p-type strained-layer QWIPs and grating coupling (the title is not given by the fetch).

Ques: What sets the photoconductive gain of a QWIP, and why does it fall with the number of wells?

A photoexcited electron that leaves a well is replaced from the emitter contact, and if the replacement is not captured by the next well another electron is injected, so that the gain is $g=p_e/(N{p}_c)$, with the escape probability $p_e$ and the capture probability $p_c$ per well. It falls as $1/N$ because a longer stack gives more chances to capture; a model of the capture gives $P_c=1-e^{-2l_w/L_c}$, and the responsivity then rises faster than the wavelength, since a wider well captures more. The measured gain was 0.98 at 5 V for a narrow-band device, and about 0.015 for a strained p-type device, in which the heavy holes have a short lifetime. Adding absorbing wells therefore raises $\eta$ and lowers $g$, so that the responsivity cannot be raised by both. Reference: Wikipedia, Quantum well infrared photodetector (2016 version); Gadir, Harrison and Soref (2002), Modeling the capture probability and enhancing the photoconductive gain in quantum well infrared photodetectors (QWIPs); NASA Technical Reports Server 20040068216, large-format QWIP focal plane arrays (narrow-band, four-band and broadband; the title is not given by the fetch); DTIC report ADA272675 on p-type strained-layer QWIPs and grating coupling (the title is not given by the fetch).

Ques: Which thermionic model describes the dark current of a QWIP, and why does it follow an activation energy near the photon energy?

A 14.5 μm device with 40 wells followed the Levine thermionic expression best between 30 K and 60 K for bias below 2 V, with an activation energy of 81 meV, between $V_b-E_1=86$ meV and $V_b-E_{fw}=76$ meV; the Schneider-Liu and Kane forms with an exchange energy of $-13.6$ meV were also tested, and none of three tunnelling expressions reproduced the plateau below 25 K, for which a scattering-based model is called for. The activation energy is close to $hc/{\lambda }_c=85.5$ meV (computed) because the electron must reach the continuum, which is the same energy as the photon needs. Between 65 K and 77 K the dark current rises by a factor of 9.5 for 81 meV (computed). Reference: Ultimate performance of Quantum Well Infrared Photodetectors in the tunneling regime, arXiv:0901.1614.

Ques: Why is the background-limited detectivity independent of the gain, and is a detectivity of $10^{11}$ consistent with it?

Both the signal and the photon noise carry the gain, so $D_{BLIP}^{\ast }=(\lambda /2hc)\sqrt{\eta /{\Phi }_B}$ is independent of $g$ and depends on the efficiency and the background. For the 8 μm to 9 μm array, a 300 K background and f/2 optics the photon flux is $7.5\times 10^{15}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$ (computed), and $\eta =1.5\%$ gives $3.0\times 10^{10}\mathrm{cm}\mathrm{H}{\mathrm{z}}^{1/2}/W$ (computed), whereas the source gives $1\times 10^{11}$ at 65 K, 3.4 times higher; a value of 17% for $\eta$ or f/7 optics would be needed for agreement (computed). Neither is stated, so the computed limit is the more reliable bound, and the source value most likely refers to a smaller background or another spectral weighting than assumed here. Reference: NASA Technical Reports Server 20040068216, large-format QWIP focal plane arrays (narrow-band, four-band and broadband; the title is not given by the fetch); DTIC report ADA272675 on p-type strained-layer QWIPs and grating coupling (the title is not given by the fetch).

Ques: Why does a p-type QWIP need no grating and have a low dark current but a low responsivity?

In the valence band the light-hole and heavy-hole states mix, so that a transition between them is allowed for light at normal incidence, with the polarization constant $P=1$. In a strained In${}_{0.3}$Ga${}_{0.7}$As/In${}_{0.52}$Al${}_{0.48}$As structure the biaxial tension puts the light-hole level at the top and gives an effective barrier of about 155 meV with a miniband of about 10 meV. The device had a dark current density of $7\times 10^{-8}A/c{\mathrm{m}}^2$ at 2 V and 77 K and a responsivity of 34 mA/W to 51 mA/W, with $D_{BLIP}^{\ast }=5.9\times 10^{10}$ at 77 K. The gain was only about 0.015 because the heavy-hole mass is large and the lifetime short, so that the quantum efficiency of about 50% was converted into little current. Reference: DTIC report ADA272675 on p-type strained-layer QWIPs and grating coupling (the title is not given by the fetch).

Ques: Why does the responsivity of a QWIP rise so much with the bias?

The bias tilts the bands, so a larger part of the photoexcited electrons escapes before they are captured, and the escape probability $p_e$ in $g=p_e/(N{p}_c)$ rises, with the drift length $L=\tau T_{qw}{\mu }_{eff}E$ growing in proportion to the field. A 14.5 μm device gave 0.58 A/W at 1 V and 2.1 A/W at 2 V at 20 K, a factor of 3.6 for a doubling of the bias (computed), and an external efficiency of 5% and 18%. The cutoff wavelength also moves with the bias, which the authors attribute to photon- and field-assisted tunnelling. The narrow-band array had 83 mA/W at 1.1 V, with BLIP reached at 2 V and 72 K. Reference: Ultimate performance of Quantum Well Infrared Photodetectors in the tunneling regime, arXiv:0901.1614; NASA Technical Reports Server 20040068216, large-format QWIP focal plane arrays (narrow-band, four-band and broadband; the title is not given by the fetch).

Ques: Is the noise-equivalent temperature difference 10 mK or 20 mK?

The narrow-band array reached a mean of 20 mK at 65 K with 16 ms integration, a 300 K background and f/2 optics, whereas an estimate from test detectors gave about 10 mK. The estimate most likely contains only the noise of a single detector, while the array value also includes the nonuniformity of the pixels and the noise of the readout circuit, and the measured value covered 327,680 pixels. The measured 20 mK is therefore the figure that applies to the array. At higher temperature the dark current dominates the temporal noise above 72 K, where the BLIP condition is lost, and below 48 K the broadband array was limited by the nonuniformity and the readout noise. Reference: NASA Technical Reports Server 20040068216, large-format QWIP focal plane arrays (narrow-band, four-band and broadband; the title is not given by the fetch).

Ques: Why does a longer cutoff wavelength force a lower operating temperature?

The activation energy of the dark current follows the cutoff energy, $E_a\approx hc/{\lambda }_c$, so that $J_{dark}\propto \exp [-hc/({\lambda }_c{k}_{B}T)]$ (prefactors neglected). At 77 K, $hc/k_{B}T=186.8\mu \mathrm{m}$, and moving the cutoff from 9 μm to 14.5 μm multiplies the dark current by $\exp [186.8(1/9-1/14.5)]=2.6\times 10^3$ (computed). The temperature at which the dark current falls below the background current is therefore lower for the long-wavelength device. The four-band array showed this: its 14 μm bound-to-quasibound band dominated the dark current, while the other bands used bound-to-continuum transitions. Reference: Ultimate performance of Quantum Well Infrared Photodetectors in the tunneling regime, arXiv:0901.1614; NASA Technical Reports Server 20040068216, large-format QWIP focal plane arrays (narrow-band, four-band and broadband; the title is not given by the fetch).

Related pages in this vault: Quantum-Dot Infrared Photodetector (QDIP), Colloidal Quantum-Dot Photodetector (CQD), Photoconductor, Schottky-Barrier Photodiode, Type-II Superlattice and Barrier Photodetector (T2SL, nBn, XBn), Quantum Cascade Laser (QCL), Quantum-Well Laser (Single and Multiple Quantum Well), Germanium and Silicon-Germanium Electroabsorption Modulator, Quantum Well LED (InGaN-GaN and AlGaInP).