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## Creator

[Hideki T. Miyazaki](https://orcid.org/0000-0003-4152-1171), [Takaaki Mano](https://orcid.org/0000-0002-6955-260X), [Takeshi Noda](https://orcid.org/0000-0002-6705-8552), [Takeshi Kasaya](https://orcid.org/0000-0002-1976-8760), [Yusuf B. Habibullah](https://orcid.org/0000-0002-8129-1545)

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VC 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://
creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0208399[Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/)

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[Antenna-enhanced high-resistance photovoltaic infrared detectors based on quantum ratchet architecture](https://mdr.nims.go.jp/datasets/967aaf41-543a-4af8-9e69-01535b08b661)

## Fulltext

Microsoft Word - Miyazaki_QRD_SuppMaterR_Clean_240526.docx1  Antenna-enhanced high-resistance photovoltaic infrared detectors based on quantum ratchet architecture: Supplementary Material  Hideki T. Miyazaki,a) Takaaki Mano, Takeshi Noda, Takeshi Kasaya, and Yusuf B. Habibullahb) Affiliations National Institute for Materials Science, Tsukuba 305-0047, Japan a)Author to whom correspondence should be addressed: miyazaki.hideki@nims.go.jp b)Present address: Rakuten Mobile, Inc., Tokyo 158-0094, Japan.   S1. Current noise spectral density for the background state The current noise spectral density for the background state insd,BG is complicated compared with that for the dark state; therefore, the derivation of Eq. (4a) in the main text is presented here. The insd,BG value is the shot noise by the detected background radiation from a 300 K environment [1,2]:  𝑖 , 𝑅 𝜆 𝑃 𝜆 𝐴𝑑𝜆, (S1) where e is the electron charge, Nw the number of periods of the unit structure, Resp the responsivity, PBG the background Planck radiation intensity, A the detector area, and the wavelength. As shown in Eq. (2) in the main text, Resp is proportional to the absorption efficiency abs and the escape probability pe. In Eq. (S1), as the noise gain, 1/Nw is considered. While some authors include pe in the noise gain [1], our experimental results support the absence of pe [2-4]. In the photovoltaic quantum-well infrared photodetectors (PV-QWIPs) discussed in this paper, the integration in Eq. (S1) can be simply described with values at the peak responsivity wavelength p because the bandwidth is narrow:  𝑅 𝜆 𝑃 𝜆 𝑑𝜆 𝑅 , 𝑃 , ∆𝜆, (S2) where Resp,p is the peak responsivity, PBG,p the background Planck radiation intensity at p, and  the effective bandwidth determined so that Eq. (S2) holds. Moreover,  𝑅 ,, , (S3) where h is the Planck constant, c the speed of light, and abs,p the absorption efficiency at p. In reality, the background state is continuously connected with the dark state. Delga showed that insd2 is generally expressed as the sum of the shot noise and the Johnson noise [2]. This permits a discussion on the dependence of insd on both the temperature of the detector T and the bias voltage Vb. In considering the Johnson noise, there can be two types of resistance R0: a static one, Rs = dV/dI at Vb = 0 (V : voltage, I : current), and a dynamic one, Rd = dV/dI. Delga pointed out that those resistances should be properly selected depending on the polarity of Vb. According to our experimental results, however, neither exhibited a remarkable difference, although in practice Rs agreed well with the experiment throughout the entire Vb range. While T is relatively high, Johnson noise is dominant; thus, the detector performance is described by the dark-state properties. However, as T decreases, both noises become comparable with each other [background-limited infrared photodetector (BLIP) temperature, TBLIP]. At a sufficiently low T, the shot noise by the background radiation becomes dominant (BLIP region). Even in the T range of the BLIP region, the dark current drastically increases for elevated Vb, and the performance is described by the dark-state properties in the high Vb region. The BLIP operation holds within a finite Vb range near zero bias at a T range below TBLIP (more details given in supplementary material S5).  S2. Calculation of the global transition rate and device resistance In this work, global transition rates Gij between the states i (Si) and j (Sj) at equilibrium were calculated based on Koeniguer et al. [5] and Ferreira and Bastard [6]. Gij gives the resistance-area product R0A by Eq. (5) in the main text. In this study, we applied several approximations in determining Gij. In the calculation of transition rates Sij based on Ferreira and Bastard [6], depending on the relative magnitude of the energetic distance E and longitudinal optical (LO) phonon energy ℏLO, the initial or final state was fixed at their minimum energy point by ignoring the finite distribution of electron momentum. It was confirmed that the influence of this approximation is not essential. At E ~ ℏLO, Sij exhibits singular behavior due to the resonant LO phonon scattering. To avoid unrealistic results, the damping coefficient of LO phonons of 0.25 meV was incorporated [7]. 2  In the calculation of R0A based on Koeniguer et al. [5], all possible intersubband paths have to be accounted for. In our work, the device layer is sandwiched by 48-nm-thick heavily doped contact layers. These relatively thin contact layers also work as quantum wells, and discrete subbands are formed in the contact layers as well. We conducted the calculation of the resistance R0A considering all possible combinations of these states. However, it was found that R0A at around 77 K, mainly discussed in this work, is well described by the simple summation between S1 and Sj in the devices as shown in Eq. (5) in the main text. This means that the ground state S1 works as a bottleneck, and this is a general feature in a wide temperature range. The R0A obtained by Eq. (5) offers fairly good agreement with the experimental results, as presented in the inset of Fig. 5(a) in the main text. However, at a relatively high temperature (T > 140 K), careful consideration of the electron transport between the contact layer and the states in the device becomes important. In a low temperature range (T < 140 K), among the transports between the left contact layer and Sj in the device region, the transport to S1 is dominant. In contrast, in a high temperature range (T > 140 K), other direct paths from the left contact layer to individual states Sj in the device layer other than S1 gradually become remarkable; i.e., currents irrelevant to the photoexcitation become outstanding. This leads to a discrepancy of R0A between the calculation and experiment [inset of Fig. 5(a)] in a high temperature range. In addition, this reasonably explains the sudden decrease in responsivity in a high temperature range beyond 140 K, which is discussed in supplementary material S5. However, in this work, further details are not discussed. Figure 1 quantitatively displays the significance of the spatial and energetic distances between S1 and Sj on R0A for a virtual, representative model case. As a typical wavefunction, we assume the ground state of a 5-nm-wide GaAs quantum well (QW) between AlGaAs barriers with a conduction band offset of Ec = 0.3 eV. The color indicates the G1j between the ground state S1 at the origin and a state with the identical wavefunction shape virtually placed at (z, E) at 77 K. For generality, we assume that the energy level of S1 is located at the Fermi level. Here, we would like to make three remarks for interpreting Fig. 1. First, overall, Gij values are underestimated by a few orders of magnitude compared with the actual values, since the energy level of S1 is set to the Fermi level. Usually, S1 is set below the Fermi level to fill S1 with sufficient electrons. The actual electron densities of the other states are similarly higher, and they should exhibit greater conductance than the estimation in Fig. 1. Second, the G1j values at the positions of j = 8 and 7, shown by color in Fig. 1, are overestimated by 1‒2 orders of magnitude. This is because the actual wavefunctions for S8 and S7 have a very different form than the assumed shape, and their actual overlap with the wavefunction of S1 should be much smaller. Finally, the G1j values for the extraction region (j ≥ 5) are underestimated. In reality, the wavefunctions other than that of S1, particularly j ≥ 5, extend over a wide space covering multiple QWs. Therefore, the overlap of the wavefunctions remains high for a large z, and it should give higher G1j values than in Fig. 1. The actual values of G1j based on the band diagrams in Fig. 2 are shown by solid lines in Fig. S1. For reference, the values based on the model calculation at the gravity center positions plotted in Fig. 1 are also shown by dotted lines. From a comparison between the dotted and solid lines, we can confirm the above three features: The dotted lines are a few orders of magnitude lower as a whole, S8 and S7 are outstandingly high, and the reducing trend of the dotted lines associated with the state progress (from S8 to S1) is sharper. The most important point understood from the actual G1j values (solid lines) is that S6 and S5 exhibit similar conductance as S7 for the quantum cascade detector (QCD). To be precise, G17 = 2.16×1016 m-2s-1, G16  FIG. S1. Actual value of G1j of each state Sj based on the band diagrams in Fig. 2 in the main text for QRD (red) and QCD (blue): solid lines and closed circles. Values of G1j at the actual (z, E) positions denoted by color in Fig. 1 in the main text are also shown by dotted lines and open circles. 3  = 2.36×1016 m-2s-1, and G15 = 2.39×1016 m-2s-1; all three of these are nearly equivalent, and S5 gives the highest value. This is why the R0A of the QCD was mainly determined by S5 and S6. In contrast, for the quantum ratchet detector (QRD), G16 is three orders of magnitude smaller that of the QCD, and G15 is smaller by more than five orders of magnitude, reflecting the exponential decay of the form factor against the spatial distance. Therefore, in the R0A of the QRD, the contribution after S6 is negligible, and R0A is simply determined by S8 and S7. In the solid lines at S8 and S7 in Fig. S1, other important information can also be found. First, QRD shows lower values than QCD. This is because a large part of the wavefunction of the QRD flows out to the step QW W2, and this leads to a smaller overlap with S1. Next, S7 is higher than S8 for both detectors, since S7 is energetically closer to S1. The estimation in Fig. 1 includes various inaccuracies as discussed above. Nevertheless, the diagram in Fig. 1 suggests some fundamental strategies as a semi-quantitative guideline expressing the overall trend. We believe the concepts presented in Fig. 1 offer a clear vision for the rational design of QW structures.  S3. Calculation of the conduction band diagram, wavefunction, and transition time The conduction band profiles and the wavefunctions were numerically obtained by self-consistently solving the Schrödinger and Poisson equations (nextnano GmbH, nextnano3/++). Material parameters were based on Vurgaftman et al. [8]. However, a small correction was applied so that the calculation would give consistent results with the numbers of experiments; we used the conduction band offset Ec between GaAs and AlxGa1-xAs as an adjustment parameter. Vurgaftman's original relationship Ec = 0.66 Eg (Eg: bandgap difference) was slightly modified to Ec = 0.61 Eg. In the actual crystal growth by molecular beam epitaxy (MBE), errors in the thickness and composition arose. On the basis of X-ray diffraction and responsivity spectrum evaluated by a Brewster-angle detector, errors of +7% and -7% in the thickness and composition x, respectively, were assumed for the diagrams in Fig. 2. The LO phonon scattering time ij from Si to Sj was calculated as ij = ni/Gij, where ni is the two-dimensional electron density in Si, based on the Gij obtained according to supplementary material S2. For obtaining the total transition time through parallel paths S8 and S7, the oscillator strengths from S1 to the respective states were taken into account. The tunneling time from W1 to W2 was evaluated as half the duration of a Rabi oscillation cycle [9]. Nevertheless, the important dynamics was determined by the LO phonon scattering, since the tunneling (0.19‒0.39 ps) is sufficiently faster than the LO phonon scattering in this study. As shown in Eq. (3b) in the main text, the specific detectivity for the dark state D *DK of a PV-QWIP is determined by abs, pe, R0A, and Nw. In this study, abs was enhanced by optical antennas, and Nw was minimized (Nw = 1). Other parameters, pe and R0A are related to the overlap of the wavefunctions. As shown in Eq. (5) in the main text, R0A is given by G18, G17,,, G12. On the other hand, pe is determined by the relative magnitudes between the transition rate from the excited states S7 and S8 in the forward direction (for) and that in the backward direction (back): for = 1/76 + 1/86 + 1/for,others, and back = 1/71 + 1/81 + 1/back,others, where 1/for,others is a minor transition rate from S7 and S8 in the right direction except for S6, and 1/back,others is that in the left direction. For maximizing R0A, minimization of G16 by suppressing the overlap of the wavefunctions of S6 and S1 is of particular importance. On the other hand, for maximizing pe, minimization of 76 and 86 by increasing the overlap of the wavefunctions of S7/S8 and S6 is necessary. These requirements conflict with each other. In the present study, for for QRD (8.7×1011 s-1) was smaller than that for QCD (1.22×1012 s-1). This is because in the QRD, the wavefunction of S6 overlaps only with the outer edges of those of S7 and S8. However, quantitatively, there was no great difference between the estimated pe values: 0.69 for QRD and 0.72 for QCD, only 4% difference. Therefore, we expect that QRDs can support both high R0A and high pe. We attribute this studyʼs moderate pe value for QRD to inappropriate material parameters, particularly the conduction band offset, in the QW design. We also observed a change in the optical/electronic properties due to wafer bonding. Further refinement of the QW design and fabrication process is necessary. However, in the previous studies on QCDs, correction of band alignment by biasing has been used as well [10,11]. Although the development of precisely designed quantum cascade lasers has been intensively promoted, further studies are still 4  required for QW engineering based on very precise band alignment.  S4. Details on the fabrication, electromagnetic calculation, and characterization Refer to the earlier work [12] for basic descriptions of the fabrication, electromagnetic calculation, and characterization. Only the details unique to the present paper are described in the following text. The intrinsic properties of the QW structures grown using MBE were examined by fabricating test devices for dark current measurement and Brewster-angle detectors for responsivity, and then the results were fed back to the QW structure design. A few wafers judged to be promising were selected for wafer transfer and patch antenna fabrication. Electromagnetic properties in this paper were obtained by finite element analysis (COMSOL, COMSOL Multiphysics). The semiconductor layer was divided into five regions: left contact, the first barrier, the first active QW, other device layers (called cascade layer), and the right contact layer. The dielectric constant of the cascade layer was obtained from the volume average of GaAs and AlGaAs. In the vertical component of the dielectric function of the first QW, a Lorentzian term -p2/(2-2+i) was applied. For our quantum ratchet detector (QRD) and the reference quantum cascade detector (QCD), we set (, , p) = (189, 10.5, 130) and (181, 13.0, 140) in meV, respectively, so that the maximum responsivity of the biased Brewster-angle detector (assumed as unity escape probability) and the calculated absorption would be consistent. The oscillator strengths f and the dipole matrix elements <z> estimated from these values were f = 0.65 and <z> = 1.39 nm for QRD and f = 0.71 and <z> = 1.48 nm for QCD. For the measurement of responsivity and noise, a Gifford-McMahon cryostat (NIKI Glass, LTS-101DL-IR-OPT-LV) was used. The current noise spectrum density insd was obtained from a plateau region, as a peak in the histogram. The position of the plateau region typically falls within the frequency range of 10‒1000 Hz. The plateau feature for the dark state was sensitive to the temperature; the position would be determined by the relationship of the impedance of the device and the current amplifier. For the dark state, the value to be measured (~10-15 A/Hz1/2) was close to the limit of the amplifier.  S5. Other properties This section presents important results that could not be shown in the main text. Figure S2 exhibits the measured absorption spectrum of the QRD (red curve), obtained from the microscopic reflection by the Fourier transform infrared spectrometer. Based on the choice of optimum size of the antenna arrays, nearly perfect absorption of 0.96 was obtained. Calculated absorption by each layer is also displayed. The total absorption is well reproduced by the calculation. As discussed for Fig. 4(b) in the main text, the absorption by the QW layer is about 0.2 at most. Other absorption takes place in the Au and contact layers, and then simply dissipates as heat rather than generating an optical signal. It has been suggested that the absorption in the QW layer abs could be improved with the proper choice of Nw [13]. Consequently, future improvement is expected. The results for the reference QCD are very similar, so they are not presented. As shown in Figs. 4(b) and (c), Resp spectra for QRD and QCD exhibit drastic changes by Vb. Here, we address the change in p. Figures S3(a) and (b) show Resp spectra at representative Vb values for QRD and QCD, respectively. The change in p is summarized in Fig. S3(c). The stepwise shift is due to the resolution of the Fourier transform infrared spectrometer (16 cm-1). p generally showed a red shift up to 2 meV depending on the increase in Vb. However, an interesting behavior was found for the QCD. Within the general trend of the red shift, we observed a blue shift of 3 meV at an intermediate range of Vb = 0.12‒0.20 V. This is due to the switching of the dominant excited states from S7 to S8. In Fig. S3(b), a widening of the spectrum due to the coexistence of two states was observed in the transition range (Vb = 0.16 and 0.20 V). In Fig.  FIG. S2. Experimental absorption spectrum of antenna-enhanced QRD (red solid line). Calculated absorption ratio of each layer is also shown. 5   FIG. S3. Resp spectra for (a) QRD and (b) QCD at representative Vb values. (c) Vb dependence of p obtained from these spectra (solid lines) for QRD (red) and QCD (blue). Dotted lines show p for wafers without antennas.  S3(c), the p change in the wafers themselves without antennas, evaluated by Brewster-angle detectors, is also shown by dotted lines. It was confirmed that the blue-shift feature of the QCD originates from the QW structures (wafers). In an antenna-enhanced device, part of the Resp spectrum of the wafer is enhanced by the resonance of the antennas. Figure S4(a) shows the temperature dependence of the photovoltaic responsivity at zero bias. At low temperatures, the responsivity is nearly flat. However, at around 140 K, there is a sudden drop for both detectors. Although many QCDs have demonstrated room-temperature responsivity, in this work, the QCD also exhibited similar responsivity drop in the high temperature range. Therefore, the unfavorable high-temperature properties in this work would be due to a common reason for both detectors, rather than due to the QW design in the QRD. While QRD offers inferior zero-bias responsivity to QCD at 77 K, it exhibits greater response at T > 160 K. This suggests that QRD has greater potential for high-temperature operation than QCD.  As displayed in Fig. 4 in the main text, when biased, both the QRD and the QCD represent similar responsivity values. This is also denoted in Fig. S4(a) by the marks at 77 K. Conduction band diagrams of this high Vb situation are shown in Fig. S5. The excited states of the active well W1 are confined at the tip of the barrier, and their right side is open. This is nearly the same situation as the conventional bound-to-bound photoconductive QWIPs. The photoelectrons escape from W1 by tunneling. The tunneling times are estimated as 0.14 ps and 0.34 ps for QRD and QCD, respectively [3]. On the other hand, the backward transition times (1/back) are 2.58 ps (QCD) and 2.12 ps (QCD) for zero bias. In our thin active wells (~ 5 nm), the influence of the electric field on the backward transition times would be small. From these values, we can estimate the escape probability  6   FIG. S4. (a) Temperature dependence of peak responsivity Resp,p at zero bias for QRD (red) and QCD (blue). Maximum Resp,p values at 77 K are shown by marks; cross: QRD, plus: QCD. (b) Temperature dependence of specific detectivity D * at zero bias for QRD (red) and QCD (blue). Filled circles: dark state; open circles: background state. Maximum D *BG values at 77 K are denoted by marks; cross: QRD, plus: QCD. Gray dotted lines indicate predicted D *DK and D *BG for the QRD under the assumption of solving the band alignment issue.  as pe = 0.95 (QRD) and 0.86 (QCD); thus, pe ~ 1. By using this result, pe for zero bias can be evaluated. In conventional QWIPs at high Vb, Resp generally saturates at a constant value [3]. Nevertheless, in Fig. 4(c), Resp quickly decreased after showing these peaks. We attribute this sudden decrease to the same mechanism as the Resp drop at T > 140 K. As can be expected from Fig. S5, at these high Vb values, direct transport from the left contact layer to individual states Sj, other than S1, would become remarkable. Since currents irrelevant to the photoexcitation are outstanding, Resp could suddenly decrease. Figure S4(b) shows the temperature   FIG. S5. Conduction band diagrams with squared wavefunction profiles for (a) QRD at Vb = 0.32 V and (b) QCD at Vb = 0.20 V.  dependence of the D * at zero bias. As for the D *DK, QRD is consistently superior to QCD, demonstrating the effect of high resistance. Despite the inferior zero-bias D *BG of QRD to QCD at 77 K, as seen in Fig. 5(b) in the main text, QRD demonstrates a higher performance at a finite bias voltage [denoted by marks in Fig. S4(b)]. Here, we would like to discuss the predicted performance of the QRD under the assumption of solving the band alignment issue. As shown in the calculation results (circles) in Fig. 4(b), abs values for the QRD and the QCD are equivalent. Improvement of pe by a factor of ~3 is expected. In that case, Eqs. (3b) and (4b) predict the enhancement of D *DK by 3 times and D *BG by √3 times. These estimated properties are added to Fig. S4(b) using gray dotted lines. A zero-bias D *BG similar to that of the QCD would be possible. The most striking improvement would appear in TBLIP, i.e., the movement of the crossing point of D *DK and D *BG as a result of the increase in D *DK. The present TBLIP of 98 K could be increased to ~110 K, which would contribute to relaxing the required cooling power. Figures S6(a) and (b) show the Vb dependence 7  of insd and D *, respectively, for the background state. In the flat insd,BG regions in Fig. S6(a) (QRD: -0.01‒0.20 V, QCD: 0‒0.08 V), the background current overwhelms the dark current. As a result, the detectors operate, in practice, as PV detectors even at Vb ≠ 0 and can support a great D *BG value. In fact, as Fig. S6(b) shows, both detectors recorded the maximum D *BG at finite Vb values within this range. Moreover, this Vb range is wider for a detector with a lower dark current, i.e., with a higher          resistance (R0A). Despite the inferior zero-bias Resp, the QRD offered a higher D *BG than the QCD at a finite Vb due to its high resistance. By improving the band alignment, the QRD would present much higher D *BG at a lower finite Vb.  References [1] F. R. Giorgetta, E. Baumann, M. Graf, Q. Yang, C. Manz, K. Kohler, H. E. Beere, D. A. Ritchie, E. Linfield, A. G. Davies, Y. Fedoryshyn, H. Jackel, M. Fischer, J. Faist, and D. Hofstetter, IEEE J. Quantum Electron. 45, 1039‒1052 (2009). [2] A. Delga, “Quantum cascade detectors: A review,” in Mid-infrared Optoelectronics, (Elsevier, 2020), pp. 337‒377. [3] B. F. Levine, J. Appl. Phys. 74, R1‒R81 (1993). [4] H. Schneider and H. C. Liu, Quantum Well Infrared Photodetectors: Physics and Applications (Springer, Berlin, 2007). [5] C. Koeniguer, G. Dubois, A. Gomez, and V. Berger, Phys. Rev. B 74, 235325 (2006). [6] R. Ferreira and G. Bastard, Phys. Rev. B 40, 1074‒1086 (1989). [7] J. S. Blakemore, J. Appl. Phys. 53, R123‒R181 (1982). [8] I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, J. Appl. Phys. 89, 5815‒5875 (2001) [9] C. Jirauschek and T. Kubis, Appl. Phys. Rev. 1, 011307 (2014). [10] A. Bigioli, G. Armaroli, A. Vasanelli, D. Gacemi, Y. Todorov, D. Palaferri, L. Li, A. G. Davies, E. H. Linfield, and C. Sirtori, Appl. Phys. Lett. 116, 161101 (2020). [11] G. Quinchard, C. Mismer, M. Hakl, J. Pereira, Q. Lin, S. Lepillet, V. Trinité, A. Evirgen, E. Peytavit, J. L. Reverchon, J. F. Lampin, S. Barbieri, and A. Delga, Appl. Phys. Lett. 120, 091108 (2022). [12] H. T. Miyazaki, T. Mano, T. Kasaya, H. Osato, K. Watanabe, Y. Sugimoto, T. Kawazu, Y. Arai, A. Shigetou, T. Ochiai, Y. Jimba, and H. Miyazaki, Nat. Commun. 11, 565 (2020). [13] M. F. Hainey, Jr., T. Mano, T. Kasaya, Y. Jimba, H. Miyazaki, T. Ochiai, H. Osato, Y. Sugimoto, T. Kawazu, A. Shigetou, and H. T. Miyazaki, Opt. Express 29, 43598‒43611 (2021).   FIG. S6. (a) Bias dependence of current noise spectral density for background state insd,BG at 77 K for QRD (red) and QCD (blue). (b) Bias dependence of the specific detectivity for the background state D *BG at 77 K for QRD (red) and QCD (blue). Filled circles: zero bias; open circles: peak bias.