Interband Cascade Infrared Photodetector (ICIP)

1. Theory

• Definition: An interband cascade infrared photodetector (ICIP) is a series connection of $N$ identical-type stages, each consisting of a thin absorber, an electron barrier and a relaxation region, grown as one epitaxial stack on InAs or GaSb; the stage is the active stage of an interband cascade laser, so the two devices share one material system.
• Absorber: The absorber is an InAs/GaSb or Ga-free InAs/InAsSb type-II superlattice, or a bulk GaInAsSb layer, with a cutoff wavelength from 2.9 μm (short-wave) over 4.3 μm to 5 μm (mid-wave) to about 10.5 μm (long-wave); an effective gap of 248.3 meV corresponds to 4.99 μm (computed).
• Electron barrier: A few GaSb/AlSb quantum wells block the photoelectrons of the absorber and pass the holes by interband tunnelling, which is possible because the conduction band of InAs lies below the valence band of GaSb (a broken-gap type-II alignment).
• Relaxation region: A digitally graded InAs/AlSb region carries the photoelectrons down to the electron barrier of the next stage, where they recombine with holes tunnelling out of that barrier, so that the electrons and holes of different stages annihilate and the same current flows in all stages.
• Current continuity: The photocurrent of the stack is the photocurrent of the weakest stage, $I_{ph}=q\Phi {\min }_m{\eta }_m$; the absorbers are therefore made thicker with depth, so that the stages absorb equally (current matching).
• Short absorbers: Each absorber is shorter than the diffusion length of the minority electrons, which was 450 nm at 300 K in short-wave devices and falls to 350 nm at 320 K and to 170 nm or less at 340 K, so most photocarriers are collected where a single thick absorber would collect only those generated within $L$ of the barrier.
• Noise and resistance: The zero-bias resistance is the sum of the resistances of the stages, $R_{0}A={\sum }_m{R}_0{A}_m$, while the photocurrent stays that of one stage, so that the noise per unit signal falls with the number of stages.
• Gain: A photoconductor-like gain $g=\tau /t_{tr}$ is proposed for some devices when the transit time through a stage is shorter than the carrier lifetime; it is not established.
• Figures of merit: The responsivity $\mathfrak{R}$, the resistance-area product $R_{0}A$, the Johnson-noise-limited detectivity $D^{\ast }$, the 3-dB bandwidth and the saturation photocurrent.

2. Working Principle

• Absorption and diffusion: A photon absorbed in a stage creates an electron-hole pair; the minority electron diffuses to the barrier on its collecting side over a distance shorter than $L$, and the hole stays in the absorber.
• Cascade transport: The electron relaxes through the graded region, recombines at the next electron barrier with a hole that has tunnelled from the valence band of the GaSb, and the process repeats in each stage, so the sum of the voltages of the stages appears across the contacts.
• Zero-bias operation: The detector works as a photovoltaic device at zero bias with a Johnson-noise-limited detectivity; a small reverse bias raises the blackbody responsivity from 0.018 A/W at 0 V to 0.024 A/W at $-0.3$ V and then saturates, in a mid-wave device with five stages.
• Dark current: Diffusion dominates at small bias, with an activation energy of 236.2 meV against a gap of 248.3 meV, and tunnelling at large reverse bias, with 160.3 meV at $-4$ V.
• Speed: The thin absorbers shorten the transit time, but the undepleted absorbers limit the bandwidth at low bias; the bandwidth of a 20 μm device rose from 0.91 GHz at $-1$ V to 7.04 GHz at $-5$ V as more of the stack was depleted.
• Direction of the electrons: In the regular configuration the electrons move in the direction of the photons and in the reverse configuration against them, closer to the collecting barrier, which collects more efficiently when the absorber is thicker than the diffusion length.
• Temperature: The responsivity of a short-wave device at zero bias was nearly constant between 280 K and 340 K, while $R_{0}A$ fell with temperature by a factor of 8.9 from 300 K to 340 K for the two-stage device (computed).

3. Applications

• Room-temperature mid-wave detection with a bandwidth of several GHz: Five-stage InAs/GaSb devices reached 7.04 GHz at $-5$ V and a saturation current of 290 μA at 5 GHz for 40 μm devices, and three-stage devices 1.3 GHz without cooling.
• Monolithic integration with interband cascade lasers: Because the stage of the detector and of the laser is the same, an edge-illuminated detector and a laser can be grown and processed on one chip, as in the monolithic frequency comb platform cited by the Roadmap on mid-infrared photonics.
• Thermoelectrically cooled long-wave detection: Three-stage Ga-free InAs/InAsSb devices with a GaAs immersion lens had a detectivity above $10^{10}$ $\mathrm{cm}\mathrm{H}{\mathrm{z}}^{1/2}/W$ at about 9 μm and 210 K.

Name	Formula	Related Formulae	Variables/Constants
Effective gap, cutoff and activation energy	$E_g^{eff}=E_{C1}-E_{V1},{\lambda }_c\left(\mu \mathrm{m}\right)=\frac{1.2398}{E_g^{eff}\left(\mathrm{eV}\right)}$	$J_{dark}=J_0\exp \left(-\frac{E_a}{k_{B}T}\right)$ $E_a\approx E_g^{eff}\left(\mathrm{diffusion}\mathrm{limit}\right)$	$\begin{array}{r}{E}_{C1},E_{V1}:lowestconductionandhighest\\ valenceminibandedgesofthetype-II\\ superlatticeabsorber\end{array}$ ${\lambda }_c:cutoffwavelength$ $E_a:activationenergyofthedarkcurrent$ $J_0:prefactor$ $k_B:Boltzmannconstant$ $\begin{array}{r}(E_g^{eff}=248.3meVgives4.99\mu \mathrm{m},computed,\\ againstameasuredcutoffofabout5\mu \mathrm{m};\\ E_a=236.2meVat-0.1Vand160.3meVat\\ -4V)\end{array}$
Stage photocurrent and current matching	$I_{ph}=q\Phi {\min }_m{\eta }_m,{\eta }_m=\left(1-R\right)e^{-\alpha D_{m-1}}\left(1-e^{-\alpha d_m}\right){\eta }_{c,m}$	$D_{m-1}={\sum }_{j<m}{d}_j$ $d_m=-\frac{1}{\alpha }\ln \left(1-{\eta }_1{e}^{\alpha D_{m-1}}\right),{\eta }_1=1-e^{-\alpha d_1}$	$I_{ph}:photocurrentofthestack$ $\Phi :incidentphotonflux$ ${\eta }_m:efficiencyofstagem$ $R:reflectanceofthefrontsurface$ $\alpha :absorptioncoefficient$ $d_m:absorberthicknessofstagem$ $\begin{array}{r}{\eta }_{c,m}:collectionefficiencyofthestage,\\ thesamecurrentflowsinallstages,so\\ theweakeststagelimitsit\end{array}$ $\begin{array}{r}(thematchingexpressionisderivedfor\\ {\eta }_{c,m}=1andR=0;d_1=312nmand\alpha =\\ 3.0\times 10^3\mathrm{c}{\mathrm{m}}^{-1}give344.5nmand384.6nm,\\ computed,against344.5nmand383.5nmin\\ thethree-stagemid-wavedevice)\end{array}$
Collection efficiency of a stage	${\eta }_c=\frac{1}{1-e^{-\alpha d}}{\int }_0^d\alpha e^{-\alpha x}P\left(x\right)dx$	$P_{rev}\left(x\right)=\frac{\cosh \left[\left(d-x\right)/L\right]}{\cosh \left(d/L\right)}$ $P_{reg}\left(x\right)=\frac{\cosh \left(x/L\right)}{\cosh \left(d/L\right)}$	$\begin{array}{r}P:probabilitythataminorityelectron\\ generatedatdepthxreachesthe\\ collectingbarrier(derivedfora\\ collectingbarrierandareflecting\\ barrier)\end{array}$ $L:electrondiffusionlength$ $\begin{array}{r}x:depthfromtheilluminatedside,rev:\\ electronsmoveagainstthephotons,reg:\\ electronsmovewiththephotons\end{array}$ $\begin{array}{r}(forL=450nmand\alpha =3.3\times 10^3\mathrm{c}{\mathrm{m}}^{-1}the\\ efficiencyis0.803against0.796ford=\\ 0.4\mu \mathrm{m}and0.397against0.346ford=1.2\\ \mu \mathrm{m},computed)\end{array}$
Resistance-area product of the stack and diffusion length	$R_{0}A={\sum }_{m=1}^N\frac{k_{B}T}{q^2{g}_{th}L\tanh \left(d_m/L\right)}$	$\frac{{\left(R_{0}A\right)}_3}{{\left(R_{0}A\right)}_2}=1+\frac{\coth \left(d_3/L\right)}{\coth \left(d_1/L\right)+\coth \left(d_2/L\right)}$ ${\lim }_{L\to 0}\frac{{\left(R_{0}A\right)}_3}{{\left(R_{0}A\right)}_2}=\frac{3}{2}$	$g_{th}:thermalgenerationrateoftheabsorber$ $\begin{array}{r}{d}_m:absorberthicknessofstagem,the\\ firstformisthesumofthe\\ diffusion-limitedresistancesofthe\\ stages(derivedfromJ=q{g}_{th}L\tanh (d/L)),the\\ secondformcomparesathree-stagewitha\\ two-stagestackthatsharesthe\\ thicknesses{d}_{1}and{d}_2,inwhich{g}_{th}cancels\end{array}$ $k_B:Boltzmannconstant$ $\begin{array}{r}(d_1,d_2,d_3=570,644,741nmandL=450\\ nmgive1.4696andL=350nmgives\\ 1.4829,computed,againstthemeasured\\ 1.470and1.483;themeasured1.500at340\\ Kequalsthelimit,soitgivesonlyan\\ upperboundofL)\end{array}$
Noise current and detectivity of the stack	$S_I=\frac{2qI}{N}\coth \left(\frac{qV}{2N{k}_{B}T}\right)$	$S_I\left(V\to 0\right)=\frac{4k_{B}T}{R_0},S_I\left(qV\gg N{k}_{B}T\right)=\frac{2qI}{N}$ $D^{\ast }=\mathfrak{R}\sqrt{\frac{R_{0}A}{4k_{B}T}}$	$S_I:spectraldensityofthecurrentnoise$ $I:darkcurrent$ $V:appliedbias$ $\begin{array}{r}N:numberofstages,R_0=N{R}_{0,1}forN\\ identicalstages\end{array}$ $\begin{array}{r}\mathfrak{R}:currentresponsivity,thefirstform\\ isderivedforNidenticalstagesin\\ serieswithindependentnoiseandthe\\ Nyquist-Schottkyformforeachstage\end{array}$ $k_B:Boltzmannconstant$ $\begin{array}{r}(theformreducestotheJohnsonnoiseat\\ zerobiasandtotheshotnoiseatlarge\\ bias;addingstagesofthesamethickness\\ leavesthephotocurrentunchangedand\\ lowersthenoisecurrentby\sqrt{N})\end{array}$
Temperature dependence of the dark current	$\frac{J\left(T_2\right)}{J\left(T_1\right)}={\left(\frac{T_2}{T_1}\right)}^3\exp \left[\frac{E_g}{k_B}\left(\frac{1}{T_1}-\frac{1}{T_2}\right)\right]$	$J_{diff}\propto T^3\exp \left(-\frac{E_g}{k_{B}T}\right)$ $R_{0}A\left(T\right)\propto \frac{1}{J_{diff}\left(T\right)}$	$J:darkcurrentdensityinthediffusionlimit$ $E_g:gapoftheabsorber$ $\begin{array}{r}{T}_1,T_2:temperatures,theexponent3\\ followsfromthestandardideal-diodeform\end{array}$ $k_B:Boltzmannconstant$ $\begin{array}{r}(along-wavestackwith{E}_g=0.118eV\\ (10.5\mu \mathrm{m})from210Kto300Kgivesa\\ factor21,computed,againstameasured\\ falloftheresistanceat-100mVfrom540\\ \Omega to23\Omega ,afactor23)\end{array}$
Bandwidth	$\frac{1}{f_{3dB}^2}=\frac{1}{f_{RC}^2}+\frac{1}{f_{tr}^2},f_{RC}=\frac{1}{2\pi \left(R_s+R_L\right)\left(C_j+C_p\right)}$	${\tau }_{diff}=\frac{4d^2}{{\pi }^{2}D},{\tau }_{diff}\left(N\right)=\frac{{\tau }_{diff}\left(1\right)}{N^2}$ $C_j=\frac{{ϵ}_0{ϵ}_{r}A}{d_{dep}}$	$f_{RC}:cutoffofthecircuit$ $R_s:seriesresistance$ $R_L:load$ $C_j:capacitanceofthedepletedstages$ $C_p:capacitanceofthepadandtheairbridge$ $\begin{array}{r}{\tau }_{diff}:lowest-modediffusiontimeofan\\ absorberofthicknessd(derived)\end{array}$ $D:diffusivity$ $N:numberofstagesatafixedtotalthickness$ $\begin{array}{r}(a20\mu \mathrm{m}devicehad0.91GHzat-1Vand\\ 7.04GHzat-5Vwithadepletedthickness\\ of1.24\mu \mathrm{m};stagesof347nminsteadof\\ 1040nmshortenthediffusiontimebya\\ factorof9,computed)\end{array}$
Photoconductive-like gain	$g=\frac{\tau }{t_{tr}}$	$\mathfrak{R}=g\frac{\eta q\lambda }{hc}$ $g_{app}=\frac{{\mathfrak{R}}_{meas}}{{\mathfrak{R}}_{calc}}$	$g:photoconductivegain$ $\tau :carrierlifetime$ $t_{tr}:transittime$ $\eta :quantumefficiency$ $\lambda :wavelength$ $\begin{array}{r}{g}_{app}:ratioofthemeasuredtothe\\ calculatedresponsivity(againisclaimed\\ onlyifitislargerthantheuncertainty\\ of\alpha ,Rand{\eta }_c)\end{array}$ $\begin{array}{r}(0.25A/Wmeasuredat3\mu \mathrm{m}against0.234\\ A/Wcalculatedfor\alpha =3.3\times 10^3\mathrm{c}{\mathrm{m}}^{-1}andR=\\ 0gives1.07,andagainst0.164A/WforR\\ =0.3(assumed)gives1.5,computed)\end{array}$

Important Questions

Ques: Why can a cascade with a short absorber in each stage reach a higher detectivity than a single absorber of the same total thickness?

A single absorber thicker than the diffusion length collects only the carriers generated within about $L$ of the barrier, so the efficiency saturates while the zero-bias resistance, $k_{B}T/(q^2{g}_{th}L)$, stops growing. In a current-matched stack each absorber is thinner than $L$, so the efficiency of the limiting stage and the summed resistance both grow with the number of stages. A model with $L=450$ nm, $\alpha =3.3\times 10^3\mathrm{c}{\mathrm{m}}^{-1}$ and an equal thermal generation rate (assumed) gave a ratio $\eta \sqrt{R_{0}A}$, which is proportional to $D^{\ast }$, of 1.30 for three matched stages of 1.05 μm total thickness and of 1.6 for eight stages of 2.2 μm, against a single absorber of the same thickness (computed). The gain of the cascade thus comes from the collection and the resistance, not from the absorption. Reference: Recent developments in interband cascade infrared photodetectors, Proc. SPIE 9819, 98190Q (2016) (OSTI record); H. Lotfi, Interband Cascade Structures for Infrared Photodetectors and Thermophotovoltaic Devices, University of Oklahoma dissertation (2016) (abstract).

Ques: Why must the absorbers become thicker with depth, and which absorption coefficient does the 312, 344.5, 383.5 nm sequence assume?

The light is attenuated by the stages before it, so a stage at depth $D_{m-1}$ absorbs the fraction $e^{-\alpha D_{m-1}}(1-e^{-\alpha d_m})$, and equal currents require $d_m=-{\alpha }^{-1}\ln (1-{\eta }_1{e}^{\alpha D_{m-1}})$ (derived). With $d_1=312$ nm, $\alpha =3.0\times 10^3\mathrm{c}{\mathrm{m}}^{-1}$ reproduces 344.5 nm exactly and gives 384.6 nm for the third stage (computed), against 383.5 nm printed, whereas the absorption of $3.3\times 10^3\mathrm{c}{\mathrm{m}}^{-1}$ measured at 3 μm on a one-stage 1040 nm device gives 347.8 nm and 393.0 nm. The two values differ by 1% to 2%, and the design value is the more likely to describe the grown stack, since the thicknesses were fitted to the wavelength of the response. Reference: Recent developments in interband cascade infrared photodetectors, Proc. SPIE 9819, 98190Q (2016) (OSTI record).

Ques: Is there a photoconductive gain in the mid-wave ICIP, or is the high responsivity explained by the absorption?

The three-stage mid-wave device had 0.25 A/W at 3 μm. The source estimates that this response needs an absorption coefficient above $5.4\times 10^3\mathrm{c}{\mathrm{m}}^{-1}$, against $3.3\times 10^3\mathrm{c}{\mathrm{m}}^{-1}$ measured on a one-stage absorber, and attributes the difference to a possible gain from a transit time shorter than the lifetime. For matched stages with no front reflection the measured $3.3\times 10^3\mathrm{c}{\mathrm{m}}^{-1}$ gives 0.234 A/W, i.e. $g_{app}=1.07$, and a reflectance of 0.3 (assumed) raises the required coefficient to $5.6\times 10^3\mathrm{c}{\mathrm{m}}^{-1}$ and $g_{app}$ to 1.5 (computed). The inferred coefficient therefore follows from the reflectance, and a gain is not proved: a value of 1 to 1.5 lies within the uncertainty of $\alpha$, $R$ and ${\eta }_c$, which makes the absence of gain the more likely reading. Reference: Recent developments in interband cascade infrared photodetectors, Proc. SPIE 9819, 98190Q (2016) (OSTI record).

Ques: How is the diffusion length extracted from the resistance of the two-stage and the three-stage device, and why is the 340 K value only an upper bound?

The resistance of each stage is $k_{B}T/(q^2{g}_{th}L\tanh (d/L))$, and the two devices share the absorbers of 570 nm and 644 nm, so that the ratio of the two $R_{0}A$ values depends on $L$ alone, $1+\coth (d_3/L)/[\coth (d_1/L)+\coth (d_2/L)]$ (derived). Solving for $L=450$ nm gives 1.4696 and $L=350$ nm gives 1.4829 (computed), which match the measured 1.470 and 1.483. When $L\ll d$ every $\coth$ tends to 1 and the ratio tends to 3/2, so the measured 1.500 at 340 K lies at the limit and fixes only $L\le 170$ nm, which is the form given in the source. The method needs a diffusion-limited dark current and the bulk $R_{0}A$ without surface leakage. Reference: Recent developments in interband cascade infrared photodetectors, Proc. SPIE 9819, 98190Q (2016) (OSTI record).

Ques: Why does the bandwidth of the five-stage device rise with reverse bias, and is a dark current of $1.04A/c{\mathrm{m}}^2$ at $-5$ V compatible with the description ‘relatively low’?

At low bias the stages are only partly depleted, and the photocarriers of the undepleted absorbers move by diffusion, so the bandwidth of a 20 μm device was 0.91 GHz at $-1$ V; at $-5$ V the depleted thickness was 1.24 μm and the bandwidth 7.04 GHz at 60 μA, falling to 3.50 GHz at 80 μA by space-charge screening. The capacitance of the depleted stack of a 20 μm device is about 35 fF for ${ϵ}_r=15.5$ (assumed), which gives an $RC$ cutoff of the junction alone near 80 GHz for 55 Ω (assumed, computed), so the transit and the parasitic elements, not the junction, set the speed. The dark current of $1.04A/c{\mathrm{m}}^2$ at 300 K and $-5$ V is high for a photodiode, and the description holds only for the small bias, $9.39\times 10^{-2}A/c{\mathrm{m}}^2$ at $-0.1$ V. Reference: High-speed InAs/GaSb mid-wave infrared interband cascade photodetector at room temperature, arXiv:2008.13386.

Ques: Why does the regular or reverse direction of the electrons matter for a thick absorber but hardly for a thin one?

In the reverse configuration the collecting barrier lies on the illuminated side, where the absorption is strongest, so the electrons generated near the surface need a short diffusion path, whereas in the regular configuration they must cross the whole absorber. The collection probability is $\cosh [(d-x)/L]/\cosh (d/L)$ against $\cosh (x/L)/\cosh (d/L)$ (derived). For $L=450$ nm and $\alpha =3.3\times 10^3\mathrm{c}{\mathrm{m}}^{-1}$ the efficiency of a 0.4 μm absorber is 0.803 against 0.796 and of a 1.2 μm absorber 0.397 against 0.346 (computed). Since the stages of the mid-wave devices (312 nm to 384 nm) are shorter than $L$, the difference is under 1% there, and the direction matters only for thicker stages. Reference: Recent developments in interband cascade infrared photodetectors, Proc. SPIE 9819, 98190Q (2016) (OSTI record).

Ques: Why does the detectivity of the long-wave ICIP fall by more than an order of magnitude between 210 K and 300 K, and which of the two values at 300 K is right?

The cutoff of 10.5 μm corresponds to a gap of 0.118 eV (computed), and the diffusion current grows as $T^3\exp (-E_g/k_{B}T)$, i.e. by a factor 21 from 210 K to 300 K (computed), in line with the resistance at $-100$ mV falling from 540 Ω to 23 Ω (23). The responsivity fell from 0.6 A/W to 0.056 A/W as well, so that $D^{\ast }\propto \mathfrak{R}\sqrt{R_{0}A}$ drops by a factor of about 50 (computed with the bias resistances as a proxy), from above $10^{10}$ to a few $10^8\mathrm{cm}\mathrm{H}{\mathrm{z}}^{1/2}/W$. The abstract gives about $3\times 10^8$ and the text $4.2\times 10^8$ for 300 K; the text value, tied to a stated peak at 7.5 μm, is the more specific, and both lie in the same decade. Reference: Gawron et al., Demonstration of T2SLs InAs/InAsSb Based Interband Cascade Detector Supported by Immersion Lens for LWIR, J. Electron. Mater. 52, 7068 (2023); Martyniuk et al., Ultimate Performance of IB CID T2SLs InAs/GaSb and InAs/InAsSb Longwave Photodetectors for High Operating Temperature, J. Electron. Mater. 48, 6093 (2019) (abstract).

Ques: Why does the noise of an ICIP not simply add the shot noise and the Johnson noise of the stack?

For one diode the current noise is $2qI\coth (qV/2k_{B}T)$, which tends to the Johnson noise $4k_{B}T/R_0$ at zero bias and to the shot noise $2qI$ at large bias, so the two are limits of one expression and not separate sources. For $N$ identical stages in series with independent noise the conductance falls to $G_1/N$ and the noise current to $S_{I,1}/N$ (derived), which gives $S_I=(2qI/N)\coth (qV/2N{k}_{B}T)$ with a limit $4k_{B}T/R_0$ and $R_0=N{R}_{0,1}$. Adding a stage of the same thickness leaves the photocurrent unchanged and lowers the noise by $\sqrt{N}$. The published unified treatment of the shot and Johnson noise argues that separate treatment distorts the detectivity of ICIPs; its expressions were not available, so the form above is a derivation. Reference: R. Q. Yang, Shot and Johnson noises in interband cascade infrared photodetectors, Appl. Phys. Lett. 121, 051105 (2022) (abstract); Mid-wavelength and long-wavelength interband cascade infrared photodetectors, University of Oklahoma (shareok.org record, abstract).

Related pages in this vault: Type-II Superlattice and Barrier Photodetector (T2SL, nBn, XBn), HgCdTe Photodiode, PN Photodiode, Uni-Traveling-Carrier Photodiode (UTC-PD), Colloidal Quantum-Dot Photodetector (CQD), Inter-band Cascade Laser (ICL), Quantum Cascade Laser (QCL).