Quantum-Dot Infrared Photodetector (QDIP)

1. Theory

• Definition: A quantum-dot infrared photodetector (QDIP) is a stack of $K$ layers of self-assembled quantum dots, typically InAs or InGaAs dots on GaAs with a base of 10 nm to 20 nm, a height of 4 nm to 8 nm and an areal density near $5\times 10^{10}\mathrm{c}{\mathrm{m}}^{-2}$, between heavily doped n-type contacts, in which the infrared photon excites an electron from a dot level to a higher dot state or to the continuum.
• Dots-in-a-well (DWELL): The dots are embedded in an InGaAs quantum well, so that the transitions between dot and well states are bound-to-bound or bound-to-continuum and depend on the bias.
• Normal incidence: The three-dimensional confinement removes the selection rule of the quantum well, so that n-type dots absorb normally incident light without a grating.
• Lifetime: The quantization in all three dimensions is expected to suppress the relaxation by LO phonons, and the carrier lifetime is quoted as 100 ps to 1 ns, with more than 1 ns expected, against about 10 μs for the minority carriers of HgCdTe.
• Linewidth: The dots differ in size, so that the ensemble transition has a width of about 50 meV and the absorption coefficient, which is inversely proportional to the linewidth, is small.
• Gain and efficiency: The current gain $g=1/(K{p}_{ok})$ is of order 1 to $10^6$, and the measured quantum efficiency is typically only 0.2% to 0.3%.
• Performance: Detectivities of $3.2\times 10^9$ to $1.0\times 10^{11}\mathrm{cm}\mathrm{H}{\mathrm{z}}^{1/2}/W$ at 77 K have been reported for different devices, and the detectors are aimed at the 3 μm to 5 μm band at 150 K.
• Doping: The optimum is about two electrons per dot, and the doping trades the dark current against the absorption.

2. Working Principle

• Absorption: A photon of energy $E=hc/\lambda$ excites the electron in a dot, and the absorption per layer is small, so that the stack has 10 to 70 layers with a spacing of 40 nm to 100 nm.
• Escape and collection: The excited electron leaves the dot by tunnelling or by thermal emission into the continuum and is swept to the contact by the bias, and the electrons injected from the emitter replace it.
• Capture and gain: An electron on its way is captured by a dot layer with the probability $p_{ok}$, and the gain $g=1/(K{p}_{ok})$ falls as the number of layers rises.
• Dark current: The thermionic emission over the barrier and the field-assisted tunnelling between the layers give the dark current, which rises by two orders of magnitude when the bias goes from 0.4 V to 1.2 V and falls by more than a factor of 10 when the spacing grows from 50 nm to 110 nm.
• Bias dependence: In a DWELL device the bias selects which transition dominates, so that the response shifts between bands such as 4.5 μm to 5.5 μm and 6.5 μm to 7.5 μm.
• Noise: The generation-recombination noise of the dark current, multiplied by the gain, sets the detectivity.

3. Applications

• Bias-tunable multicolor detection with normal incidence: A DWELL detector shifts its response between the mid and long wavelength bands with the voltage, in one structure and without a grating.
• High-operating-temperature mid-infrared arrays: Detectors for the 3 μm to 5 μm band at 150 K, where a thermoelectric cooler could replace the cryogenic cooler; this is a target stated for the technology.

Name	Formula	Related Formulae	Variables/Constants
Transition energy and spectral width	$E=\frac{hc}{\lambda },\frac{\Delta \lambda }{\lambda }=\frac{\Delta E}{E}$	$\lambda \left(\mu m\right)=\frac{1.2398}{E\left(\mathrm{eV}\right)}$ $\alpha \propto \frac{1}{{\Gamma }_{inh}}$	$\begin{array}{r}E:energybetweenthedotgroundstateand\\ thefinalstate\end{array}$ $\begin{array}{r}\Delta E:fullwidthathalfmaximumofthe\\ ensembletransition(about50meVfordots\\ ofnon-uniformsize)\end{array}$ ${\Gamma }_{inh}:inhomogeneouslinewidth$ $\alpha :absorptioncoefficient$ $\begin{array}{r}(E=155meVgives\lambda =8.0\mu \mathrm{m},and\Delta E=\\ 50meVgives\Delta \lambda /\lambda =32\%,computed;a\\ devicewith\Delta \lambda /\lambda =0.14at10.2\mu \mathrm{m}has\Delta E\\ =17meV,computed)\end{array}$
Absorption and quantum efficiency	$\alpha =\frac{\delta ⟨N⟩{\Sigma }_{QD}}{L},\eta \approx \alpha KL$	$\eta =1-e^{-\alpha KL}$ $⟨N⟩\approx 2,N_D\approx 2{\Sigma }_{QD}$	$\delta :crosssectionofthetransitionperelectron$ $⟨N⟩:meannumberofelectronsperdot$ ${\Sigma }_{QD}:arealdensityofthedots$ $L:periodofthedotlayers$ $K:numberofdotlayers(10to70)$ $\begin{array}{r}\eta :quantumefficiency(theapproximation\\ holdsfor\alpha KL\ll 1)\end{array}$ $\begin{array}{r}{N}_D:arealdopingdensity,theoptimum\\ occupancyisabouttwoelectronsperdot,\\ (for{\Sigma }_{QD}=5\times 10^{10}\mathrm{c}{\mathrm{m}}^{-2}thedopingis1\times 10^{11}\\ \mathrm{c}{\mathrm{m}}^{-2},computed)\end{array}$
Current gain and capture	$g=\frac{1}{K{p}_{ok}}$	$p_{ok}=\frac{1}{Kg}$ $g\sim 1\mathrm{to}10^6$	$g:currentgain$ $K:numberofdotlayers$ $\begin{array}{r}{p}_{ok}:probabilitythatanelectronis\\ capturedbyadotlayer(neutralcapture),\\ thegainfallsas1/K,thereportedvalues\\ ofdifferentdevicesspan1to10^6\end{array}$ $\begin{array}{r}(ariseofgby100timesbetween60Kand\\ 140Kmeansafallof{p}_{ok}bythesame\\ factor,derived)\end{array}$
Photocurrent and responsivity	$j_{photo}=q\eta {\Phi }_{s}g,\mathfrak{R}=\frac{j_{photo}}{h\nu {\Phi }_s}$	$\mathfrak{R}=\frac{q\lambda }{hc}\eta g$ $\eta g=\frac{\mathfrak{R}hc}{q\lambda }$	$j_{photo}:photocurrentdensity$ ${\Phi }_s:photonfluxdensity$ $\mathfrak{R}:responsivity$ $h\nu :photonenergy,\eta g:externalefficiency$ $\begin{array}{r}(2.16A/Wat10.2\mu \mathrm{m}gives\eta g=0.26,and\\ g=88for\eta =0.3\%(assumed),computed)\end{array}$
Dark current	$⟨j_{dark}⟩=j_{max}\frac{\Theta }{⟨N⟩}\exp \left[\frac{q\left(V+V_D-\frac{⟨N⟩}{N_{QD}}{V}_{QD}\right)}{\left(K+1\right)k_{B}T}\right]$	$j_{max}=A^{\ast }{T}^2$ $⟨j_{dark}⟩=\frac{q{\Sigma }_{QD}\left(G_{th}+G_{tun}\right)}{p_{ok}}$	$j_{max}:Richardson-Dushmancurrent$ $A^{\ast }:Richardsonconstant$ $\Theta :parameterofthebarrier$ $V:bias$ $V_D:built-involtage$ $V_{QD}:potentialofadotlayer$ $N_{QD}:densityofdotstates$ $k_B:Boltzmannconstant$ $\begin{array}{r}{G}_{th},G_{tun}:ratesofthermionicemissionand\\ field-assistedtunnellingfromadot,the\\ secondformisthebalanceoftheemission\\ andthecapture,(1.36\times 10^{-6}A/c{\mathrm{m}}^{2}at77K\\ and6.92\times 10^{-4}A/c{\mathrm{m}}^{2}at200Kgivean\\ effectiveactivationenergyof67meV,\\ computed)\end{array}$
Noise and detectivity	$D^{\ast }=\frac{\mathfrak{R}\sqrt{A\Delta f}}{i_n},i_n=\sqrt{4qg{I}_{dark}\Delta f}$	$D^{\ast }\propto \frac{\mathfrak{R}\sqrt{A}}{\sqrt{4q{I}_{dark}g}}$ $\mathrm{NEP}=\frac{i_n}{\mathfrak{R}}$	$D^{\ast }:specificdetectivity(\mathrm{cm}\mathrm{H}{\mathrm{z}}^{1/2}/W)$ $A:area$ $\Delta f:bandwidth$ $\begin{array}{r}{i}_n:generation-recombinationnoisecurrent\\ ofthedarkcurrent\end{array}$ $I_{dark}:darkcurrent$ $NEP:noise-equivalentpower$ $\begin{array}{r}(theratioofthedetectivities1.01\times 10^{11}\\ and6.4\times 10^{10}\mathrm{cm}\mathrm{H}{\mathrm{z}}^{1/2}/Wat77Kand100Kis\\ 1.6,computed)\end{array}$

Important Questions

Ques: Why can a QDIP absorb normally incident light, and is the predicted peak the measured one?

In a quantum well the intersubband transition couples only to the electric field along the growth direction, so n-type QWIPs need a grating. A dot confines the electron in all three dimensions, the selection rule differs, and n-type dots absorb normally incident light. For a DWELL device an 8-band $\mathbf{k}\cdot \mathbf{p}$ model with strain predicted a z-polarized absorption peak near 8.2 μm and a significant in-plane absorption near 20 μm, whereas the measured normal-incidence peak was at 6.5 μm, and the 20 μm response was not seen because those states lie too far below the continuum to give photocurrent. The measured value is the more likely to be representative, since the model depends on the assumed dot shape and composition; the source does not give the reason for the difference. Reference: Quantum dot infrared photodetectors: model after Ryzhii et al. (czasopisma.pan.pl, 11_paper.pdf; the fetch gave no title, the model and the numbers are as summarized); Jolley et al., Quantum Dots-in-a-Well infrared photodetectors (2006), openresearch-repository.anu.edu.au.

Ques: Why is the measured quantum efficiency only 0.2% to 0.3% when 40 to 60 layers are said to absorb 50%?

The two statements refer to different things. For uncoupled dot layers, 40 to 60 layers are quoted to absorb 50% of the light at the peak, which means $\alpha L=-\ln (0.5)/K=1.2\%$ to 1.7% per layer (computed). A typical measured efficiency of 0.2% to 0.3% for a stack of about ten layers (assumed) is $\eta /K\approx 0.02\%$ to 0.03% per layer (computed), i.e. 40 to 90 times lower. The large inhomogeneous width of about 50 meV spreads the oscillator strength over many energies, and the absorption is inversely proportional to the linewidth; non-uniform doping and strain defects reduce it further. The 50% figure is a best case at the peak, and the measured efficiency is the more likely for real devices. Reference: Laser Focus World, Research propels quantum dots forward; Quantum dot infrared photodetectors: model after Ryzhii et al. (czasopisma.pan.pl, 11_paper.pdf; the fetch gave no title, the model and the numbers are as summarized).

Ques: Why does the gain of a QDIP fall with the number of layers, and why does it rise with temperature?

An electron on its way to the collector can be captured by any of the $K$ dot layers, and the gain is the inverse of the total capture probability, $g=1/(K{p}_{ok})$. A stack with more layers absorbs more, but the gain falls as $1/K$, so that the responsivity is not raised in proportion. The gain of the reference device rose by more than 100 times between 60 K and 140 K; since $K$ is fixed, this means that $p_{ok}$ fell by the same factor (derived), as the capture into the neutral dots became less likely at higher temperature, the source not giving the mechanism. The values reported for different devices span 1 to $10^6$, which shows how sensitive the gain is to the capture. Reference: Quantum dot infrared photodetectors: model after Ryzhii et al. (czasopisma.pan.pl, 11_paper.pdf; the fetch gave no title, the model and the numbers are as summarized).

Ques: Why does the dark current of a QDIP depend so strongly on the layer spacing and the bias?

The dark current has a thermionic part, in which electrons are emitted over the barrier, and a part in which they tunnel from dot to dot, so that it is the sum $q{\Sigma }_{QD}(G_{th}+G_{tun})/p_{ok}$. Increasing the spacing from 50 nm to 110 nm cut it by more than a factor of 10, and raising the bias from 0.4 V to 1.2 V raised it by two orders of magnitude; the tunnelling between the layers falls with the spacing and rises with the field (a derived reading). A device with a 3 nm quaternary cap and 50 nm of GaAs had $1.36\times 10^{-6}A/c{\mathrm{m}}^2$ at 77 K and 1 V and $6.92\times 10^{-4}A/c{\mathrm{m}}^2$ at 200 K, a factor of 509 and an effective activation energy of 67 meV (computed), below the 121.5 meV of its 10.2 μm transition. This is consistent with a tunnelling component, as the authors attribute the low value to a suppression of tunnelling by the cap. Reference: Quantum dot infrared photodetectors: model after Ryzhii et al. (czasopisma.pan.pl, 11_paper.pdf; the fetch gave no title, the model and the numbers are as summarized); S. Chakrabarti et al., Appl. Phys. Lett. 99, 181102 (2011).

Ques: Is a QDIP better than a QWIP?

Theory predicted that QDIPs would be substantially better, because of the longer lifetime and the lower dark current, whereas the abstract of the Buffalo analysis states that most of the fabricated QDIPs had a worse performance than QWIPs. The measured detectivities are $3.2\times 10^9\mathrm{cm}\mathrm{H}{\mathrm{z}}^{1/2}/W$ for a 6.5 μm DWELL device at 77 K and $1.01\times 10^{11}$ for a 10.2 μm device with a quaternary cap at 77 K, to be compared with $1\times 10^{11}$ for a 8.5 μm QWIP array at 65 K and $3\times 10^{10}$ for a 13.5 μm array at 55 K. The best QDIP therefore equals the QWIP at 77 K, and the typical one is lower. The claim of a performance at 150 K comparable to HgCdTe at 77 K in the mid-infrared is a statement of the article, not a measurement given there. The superiority of QDIPs is therefore not supported by these data. Reference: Buffalo research record, Why QDIPs are still inferior to QWIPs: theoretical analysis (SPIE Proc. 4288, 396, 2001); Jolley et al., Quantum Dots-in-a-Well infrared photodetectors (2006), openresearch-repository.anu.edu.au; S. Chakrabarti et al., Appl. Phys. Lett. 99, 181102 (2011); NASA NTRS 20040068216 (QWIP arrays, for comparison); Laser Focus World, Research propels quantum dots forward.

Ques: Why does the ensemble linewidth matter, and what does vertical coupling of the dot layers do?

The dots grown by the Stranski-Krastanow method differ in size and composition, so that the transition energies are spread over about 50 meV, which is $\Delta \lambda /\lambda =32\%$ for 155 meV (computed), while the absorption coefficient is inversely proportional to the width. A narrower response was reached in a 35-layer stack of uncoupled dots: $\Delta \lambda /\lambda =0.14$ at 10.2 μm, i.e. 17 meV (computed). Coupling the dot layers vertically narrows the linewidth but raises the dark current, since the carriers tunnel between the layers more easily. A narrow line is thus bought with a higher dark current, and a thick barrier between the layers (at least 40 nm) is used to keep the layers uncoupled. Reference: Quantum dot infrared photodetectors: model after Ryzhii et al. (czasopisma.pan.pl, 11_paper.pdf; the fetch gave no title, the model and the numbers are as summarized); Laser Focus World, Research propels quantum dots forward; S. Chakrabarti et al., Appl. Phys. Lett. 99, 181102 (2011).

Ques: Why is the optimum occupancy about two electrons per dot?

The doping fixes how many electrons are in the dot ground state and available for absorption. With fewer carriers than states the absorption falls in proportion, whereas more carriers than the ground state holds must occupy higher levels and raise the dark current, so that the dopant density trades the dark current against the absorption. The ground state of a dot holds two electrons, with opposite spins (assumed), and this is the likely reason for the optimum of about two electrons per dot, i.e. a doping of twice the areal density of the dots, $1\times 10^{11}\mathrm{c}{\mathrm{m}}^{-2}$ for $5\times 10^{10}\mathrm{c}{\mathrm{m}}^{-2}$ (computed). The source states the optimum but not the reason. Precise control of the doping during the Stranski-Krastanow growth is difficult, which is a stated limitation. Reference: Quantum dot infrared photodetectors: model after Ryzhii et al. (czasopisma.pan.pl, 11_paper.pdf; the fetch gave no title, the model and the numbers are as summarized); Laser Focus World, Research propels quantum dots forward.

Ques: Why does the response of a dots-in-a-well detector shift with the bias but hardly with the width of the well?

The bias changes which transition dominates. A device with 30 periods showed photocurrent peaks near 225 meV (4.5 μm to 5.5 μm) and 165 meV (6.5 μm to 7.5 μm) at positive bias, 300 meV and 190 meV at negative bias, and a dominant 6.5 μm response at +6 V to +8 V, so the peak moved by tens of meV with the sign of the bias. Changing the width of the top well from 2 nm to 4 nm changed the spectra very little, and the calculation gave a tuning of at most about 5 meV even under favorable conditions, so that a shift could equally come from a change of the dot size. A likely reason, not stated by the sources, is that the dot states set the transition while the well only changes the escape path. Reference: N. H. Jabarullah, MSc dissertation, University of Hull (2010), DWELL photoluminescence and photocurrent; Jolley et al., Quantum Dots-in-a-Well infrared photodetectors (2006), openresearch-repository.anu.edu.au.

Ques: Why are QDIPs expected to work at higher temperatures, and what is measured?

The three-dimensional quantization should suppress the relaxation by LO phonons, which raises the carrier lifetime and lowers the thermal generation, so that the dark current is expected to be lower than in HgCdTe photodiodes and in QWIPs, and a 70-layer InAs/GaAs device had a lower dark current density than a commercial HgCdTe photodiode at 77 K. The measured evidence is limited: a 10.2 μm device had $D^{\ast }=1.01\times 10^{11}$ at 77 K and $6.4\times 10^{10}$ at 100 K, a drop of 1.6 times (computed) with a responsivity that decreases with temperature, and the DWELL measurements of the other study were made only at 77 K. The target of 150 K for the 3 μm to 5 μm band is therefore a goal, and the commercial use is expected only after further development. Reference: Laser Focus World, Research propels quantum dots forward; S. Chakrabarti et al., Appl. Phys. Lett. 99, 181102 (2011); Jolley et al., Quantum Dots-in-a-Well infrared photodetectors (2006), openresearch-repository.anu.edu.au.

Related pages in this vault: Quantum-Well Infrared Photodetector (QWIP), Colloidal Quantum-Dot Photodetector (CQD), Photoconductor, Type-II Superlattice and Barrier Photodetector (T2SL, nBn, XBn), Graphene and 2D-Material Photodetector, Quantum-Dot Laser, Quantum Dot LED (QLED), Quantum Cascade Laser (QCL), Nanocrystal Memory (Discrete Nanodot Storage).