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We report on the device model for the infrared photodetectors based on the van der Waals (vdW) heterostructures with the radiation absorbing graphene layers (GLs). These devices rely on the electron interband photoexcitation from the valence band of the GLs to the continuum states in the conduction band of the inter-GL barrier layers. We calculate the photocurrent and the GL infrared photodetector (GLIP) responsivity at weak and strong intensities of the incident radiation and conclude that the GLIPs can surpass or compete with the existing infrared and terahertz photodetectors. The obtained results can be useful for the GLIP design and optimization.
© 2017 Optical Society of America
1. Introduction
Unique properties of graphene [1] have enabled its use in different detectors of terahertz (THz) and infrared (IR) radiation [2–10]. Progress in the fabrication of the van der Waals (vdW) heterostructures [11] incorporating the graphene layers (GLs) stimulated the development of new devices, which provide the advantages of combining GLs with such materials as hBN, WS2, InSe, and similar materials [12–21]. Recently (see preprint [22]), we proposed and evaluated GL-vdW IR photodetectors using the interband transitions of the electrons in the GLs with their subsequent escape to the continuum states in the barrier layers. The features of these photodetectors (in the following referred to as GLIPs) are associated with the specifics of the interband absorption and slow capture of the photoexcited electrons propagating across the heterostructure in the continuum states above the barriers. This promises high values of the GLIP responsivity combined with a high speed due to the short electron transit times. The device model presented in [22] is limited to the GLIP operation at low and moderate IR radiation intensities. However, the applications of the GLIPs for the IR detection at the background limited conditions and for the generation of the THz radiation using the photomixing or the ultra-short pulsed excitation require the model accounting for the GLIP response to high power optical signals. In this paper, we evaluated the GLIP response at elevated IR radiation intensities. This, in particular, can provide the insight on the GLIP operation in the background limited infrared performance (BLIP) regime and evaluate the dynamic range of the GLIP operation. Using the developed device model, we calculate the dependences of the photocurrent on the incident IR radiation intensity in GLIPs with different structural parameters (different conduction band offsets, numbers of the GLs, and electron capture efficiency).
2. Device structure and model
The GLIP under consideration consists of the GL-vdW heterostructure, which comprises the N ≥ 1 undoped (inner) GLs clad by the barrier layers and two the doped contact GLs [so that the heterostructure includes (N + 2) GLs in total and (N + 1) barrier layers]. The the inner GLs have the indices n = 1, 2, 3, ..., N. The index n marks the GL betweem the n-th and (n + 1)-th barrier layer. The barrier layers have the indices n = 1 (near-emitter barrier) and n = 2, 3, ..., N + 1 (other barriers, including that at the collector GL). It is assumed that the conduction band offset, Δ, between the GLs and barrier layers is smaller than the pertinent valence band offset. A sufficiently strong dc bias voltage V is applied between the contact GLs. The emitter and collector GLs are assumed to be doped either by donors or acceptors to provide their sufficiently high lateral conductivity.
The GLIP operation is associated with the following processes [22]: (1) the photogeneration of the electron-hole pairs in the GLs due to the interband radiative transitions; (2) the tunneling injection of the thermalized electrons from the ground states in the GLs and the escape of the photogenerated electrons from their excited states followed by the propagation across the barrier layers; (3) the electron capture from the continuum states above the inter-GL barriers into the inner GLs.
Figures 1(a) and 1(b) schematically show the band diagrams of a GLIP with the n-type emitter and collector GLs [22] without the IR irradiation at relatively low bias voltages (with the electric field, EE, in the near-emitter barrier layer small in comparison with the electric field, EB, in the heterostructure bulk, i.e., in other barrier layers) and under a strong intensity of the incident IR radiation (when EE is larger than EB), respectively. The band profile at a weak irradiation can also correspond to Fig. 1(a).
Fig. 1 Band diagrams of a GLIP [22] (a) in dark at relatively low voltages when EE < EB (weaker inclination of the band profile in the near-emitter barrier layer than in the barrier layers in the structure bulk) and (b) under strong irradiation when EE > EB (the band profile in the near-barrier is steeper than in others). Solid and wavy arrows indicate generation of the electron-hole pair (electron photoexcitation due to the absorption of normally incident radiation) and different electron paths, respectively.
The consideration the GLIP operation at the elevated radiation intensities requires to account for the self-consistent potential redistribution across the heterostructure due to the competition between the processes of the electron escape from and capture into the GLs, the variation of the electron (hole) density in the emitter GL caused by the variation of the self-consistent electric field at the emitter GL, and, in principle, the saturation of the radiative interband transitions and the pertinent drop in the IR radiation absorption with the increasing intensity. Our calculations are based on the concept of the capture probability used in the theory of single and multiple quantum-well photodetectors (QWIPs) [23–27] and, more recently, in the theory of the electron transport in GL-vdW heterostructures [28]. The main difference between GLIPs and QWIPs is using the interband radiative transitions and the intraband (intersubband) transitions in GLIPs and QWIPs, respectively. Due to this, the GLIPs are sensitive to the normally incident IR radiation and their response is virtually independent of the GL doping. A small efficiency of the electron capture into GLs [29] promotes elevated GLIP responsivity due to the effect of the photoelectric gain.
3. Main equations
To simplify the model, we neglect the thermionic emission of electrons and holes from the GLs in comparison with the tunneling and radiative processes due to relatively large ratios of the work functions under consideration and the thermal energy (see Appendix A) and disregard the local electric-field dependence of the capture efficiency [28, 34]. The balance between the electrons leaving the GLs due to their tunneling and the photoexcitation (direct or followed by tunneling) and the electrons captured into the GLs is governed by the following equations:
Here j is the density of the electron current across GLs, e is the electron charge, pn is the capture efficiency [22–27] for the electrons crossing the GL with the index n = 1, 2, ..., N, GE, RE = β θE I, Gn, and Rn = β θnIn are the rates of the injection and photoemission from the emitter GL and the inner GLs (with the index n = 1, 2, ..., N), respectively, is the probability of the interband photon absorption in GLs [1], α = e2/ħc ≃ 1/137 is the fine structure constant, c is the speed of light in vacuum, and is the barrier material refractive index, θn is the probability of the photoexcited electrons tunneling escape from the n−th GL to the continuum states above the inter-GL barriers, I is the IR radiation photon flux crossing the GLs. The quantity θn characterizes the escape of the photoexcited electrons from the GLs. The above value of β corresponds to the linear energy spectrum of carriers in GLs: ε = vW |p|, where vW ≃ 108 cm/s and p is the carrier momentum.At the sufficiently strong IR irradiation, the electric field, EE = E1, at the emitter GL can markedly exceed the electric fields, En (with n = 2, 3, ..., N + 1), in other inter-GL barriers. At such conditions, the injection of the thermalized electrons from the emitter GL to the heterostructure bulk is determined by the thermally activated tunneling, while such an injection from the inner GLs is small in comparison with the photoemission from these GLs. Hence, we set
Here jm is the maximum current density which can be extracted from an undoped GL and , where Δ is the conduction band offset between the GLs and the barrier layer material, is the field characterizing the tunneling through the triangular barrier top [30], εF is the Fermi energy of the thermalized electrons in the n-type emitter GL in the absence of the electric field at the emitter GL (determined by the emitter GL donor density Σ0), , m and κ are the electron effective mass and the dielectric constant in the barrier layers. In the GLIPs with the emitter GL of p-type, one needs to replace εF and F by −εF and −F, respectively, although in the following we focus on the GLIPs with donor doped emitter GL. In contrast to our previous work [22], the quantity γE accounts for the variation of the Fermy energy in the emitter GL by the electric field. This effect can be pronounced in the case of elevated bias voltages considered below.The quantities θE and θn are given by the following expressions:
where ηΩ = [(Δ − ħΩ/2)/Δ] if ħΩ/2 < Δ and ηΩ = 0 if ħΩ/2 ≥ Δ. τesc is the try-to-escape time, τrelax is the characteristic time of the photoexcited electrons energy relaxation, and n = 1, 2, ..., N.Using Eq. (1) together with Eqs. (2) and (3), we arrive at the following set of (N+1) equations governing the electric fields, EE and En, in the barriers:
This set of equations should be supplemented by the following condition corresponding to the voltage applied between the emitter and collector equal to , where d is the thickness of the barrier layers.In the GLIPs with not too large number of the GLs N < β−1, taking into account that (1 − β)n ≃ exp(−nβ) ≃ 1 − nβ ≃ 1 (if Nβ ≪ 1), the electric fields in the barriers in the heterostructure bulk are very close to each other, i.e., En ≃ EB (for n = 2, 3, ..., N + 1) with EB = (V/d − EE)/N. In this case, considering Eq. (4), we arrive at the following equations for EE and EB as functions of the IR radiation intensity I:
4. Dark characteristics
In the absence of irradiation at relatively low voltages (V ≪ dEtunn), Eq. (5) yields the following analytical expressions for the electric fields and :
where γ0 = (Δ − εF)/Δ. One can see that at low voltages, the ratio of the electric fiels at the emitter and in the bulk . This case corresponds to the band diagram schematically shown in Fig. 1(a). At elevated voltages V and sufficiently small capture efficiency p when V > NdEtunn/ ln(1/p), instead of Eqs. (7) and (8) we obtain In the latter case, as follows from Eqs. (9), EE > EB. The difference between the electric fields in the emitter barrier and in the bulk change the sign at the voltage V0 when , which satisfies the following equation: This implies that V0 < dEtunn/(N +1) ln(1/p) ≪ dEtunn. At V < V0 and at V > V0, one obtains and , respectively.5. Photoresponse at low IR radiation intensities
At low intensities, the processes of the electron photoexcitation from the GLs can be considered as a perturbation in comparison with the tunneling processes of the thermalized electrons, so that the electric fields EE and EB are close to their values in the dark given by Eqs. (7) –(9). In this situation, for the range of low voltages, from Eqs. (1) –(3) one can find an analytic expression for the photocurrent density jphoto = j − jdark (where jdark is the current density in the absence of irradiation, i.e., the dark current density):
As follows from Eq. (11) the photocurrent density can be markedly larger than the current density of the electrons photoexcited from all the GLs if the photoelectric gain . This is associated with the amplification of the photoexcited electrons current by the extra injection from the emitter GL. A similar effect takes place in single and multiple-QW QWIPs [27].Equation (11) leads to the expression for the GLIP responsivity ℛ = jphoto/ħΩIin, where Iin is the intensity of the incoming radiation, which is weakly dependent on the number of GLs in the GLIP at the voltage V ∝ (N + 1)dEtunn and markedly increases with a decrease in the capture efficiency p. Since the latter in the GLIPs can be very small [29], their responsivity should be fairly large. Using Eq. (11) and taking into account the IR radiation reflection from the device surface [that yields ], for the characteristic GLIP responsivity (at low intensities) R = ℛ|ħΩ=Δ/2 we obtain
For κ∞ = 5, p = 0.01 − 0.02, τesc/τrelax = 0.1, N = 1, εF = 0.1 eV (γ0 = 0.75), ħΩ = 0.8 eV, Δ = 0.4 eV, and, Eq. (12) yields ℛ ≃ 0.135 − 0.270 A/W.Figure 2 shows the low intensity responsivity ℛ as a function of the photon energy ħΩ calculated for different GL parameters (different values of Δ, N, p, and τesc/τrelax) at different normalized voltages U = V/dEtunn. Naturally, the responsivity ℛ is somewhat smaller than the characteristic responsivity R [22] calculated without accounting for the reflection effect.
Fig. 2 Spectral dependences of the responsivity ℛ of GLIPs (a) with different conduction band offsets Δ, at a given normalized voltage U = 0.2 and (b) with Δ = 0.4 eV at different normalized voltages U. Dashed line in the left panel corresponds to Δ = 0.4 eV and U = 0.25.
It is worth noting that the GLIPs can exhibit good values of the responsivity even at ħΩ < 2Δ if the bias voltage V is sufficiently high, see Fig. 2(a). As seen from Fig. 2(b), the GLIP responsivity in the range ħΩ < 2Δ is very sensitive to the applied voltage. The latter is because an increase in the voltage leads to a substantial rise of the escape rate of the photoexcited electrons from the GLs. This can be used for an effective voltage control of the GLIP spectral characteristics.
An increase in V leads to a higher responsivity but also to a stronger dark current. The latter decreases the dark current limited detectivity.
The responsivity of the GLIPs intended for far-IR range of the spectrum with a smaller Δ can exhibit high values than those in the above estimate. Smaller values of the capture efficiency p [29] also promote an increase in the GLIP responsivity. Comparison of the GLIP responsivity ℛ with the QWIP responsivity ℛQWIP yields the following rough estimate for their ratio: ℛ/ℛQWIP ∼ (pQWIP/p)(β/βQWIP). Since, pQWIP > p) and β > βQWIP, the responsivity ration can be large (about the order of magnitude or even more).
6. Nonlinear response - high IR radiation intensities
At a strong IR irradiation, the photoexitation pronouncedly affects the electric fields in the emitter barrier, EE, and in other barriers, EB. When I varies from zero to a certain value IV, the electric fields EE and EB vary from their values in dark to EE ≃ V/d and EB ≃ 0, respectively.
Figure 3 shows the variation of EE and EB with increasing normalized intensity S = eβI/jm at different normalized voltages U = V/dEtunn, found from the numerical solution of Eqs. (5) and (6), for a GLIP with N = 1, ħΩ = 0.6 eV, and different values of the normalized voltage U = V/dEtunn. Here and in the consequent figures we assume that Δ = 0.4 eV, εF = 0.1 eV, f = Etunn/F = 30, τesc/τrelax = 0.1, and p = 0.01. One can see from Fig. 3 that at the chosen voltages, EE < EB at relatively low S (and, hence, ). However, at certain values of S (different for different U), the quantity EE −EB changes its sign. This is in line with Eqs. (9) and (10).
Fig. 3 Normalized electric fields EE/Etunn(solid lines) and EB/Etunn (dashed lines) versus the normalized intensity S for a GL with N = 1 and ħΩ = 0.60 eV at different normalized voltages U.
Figures 4 and 5 show the dependences of the electric fields EE and EB and the dependences of the normalized current J = jphoto/jm on the normalized intensity S calculated numerically using Eqs. (5) and (6) and Eq. (1), respectively, for GLIPs with Δ = 0.4 eV, different number of the GLs N and different photon energies ħΩ. As seen from Fig. 4, EE tends to V/d (i.e., EE/Etunn tends to UV = 0.5) and EB tends to zero when the intensity approaches a certain value SV = eβ IV/jm. In particular, as follows from Eq. (5) at ħΩ ≥ 2Δ (when ηΩ = 0), we can obtain
At this intensity, j ≃ jphoto = jV, whereFig. 4 Normalized electric fields EE/Etunn and EB/Etunn versus the normalized intensity S for GLIPs with different number of GLs N and photon energies ħΩ at U = 0.5: solid lines correspond to N = 1, dashed lines - N = 2, and dashed-doted lines - N = 8.
Fig. 5 Normalized current density J versus normalized intensity S in GLIPs with different N and ħΩ at U = 0.5: as in Fig. 4, solid lines correspond to N = 1, dashed lines - N = 2, and dashed-doted lines - N = 8.
As seen from Fig. 5, the normalized current J increases with the normalized intensity S (and, hence, jphoto increases with I). Thus, the current density jphoto increases with the increasing voltage V and intensity I. However, the jphoto − I dependence, being a linear one at low intensities, slows down at moderate and elevated intensities and tends to a saturation. This is attributed to the saturation of EE as a function of I (as pointed above EE tends to V/d), so that the exponents in the left parts of Eqs. (2) and (3) also approach to constants. Such a behavior is in line with the results on numerical calculations of EE and EB shown in Figs. 3 and 4.
Slowing down of the J − S and, therefore, jphoto − I dependences results in a moderate rise of J with increasing S seen in the spectral characteristics shown in Fig. 6 for a GLIP with N = 1 at different S and U. Indeed, when S increases by two orders of magnitude, J increases only by a factor smaller than ten.
Fig. 6 Spectral dependences of the normalized current density J at different normalized intensities S at (a) U = 0.5 and (b) U = 0.25.
Figure 7 shows the spectral characteristics of the responsivity ℛ for a GLIP with N = 1 at different S and U. The relatively slow jphoto − I dependence implies that the GLIP responsivity ℛ is a decreasing function of the intensity I at elevated values of the latter. The spectral characteristics of the photocurrent density and responsivity, jphoto versus ħΩ and R versus ħΩ at high radiation intensity and voltage are mainly determined not by a decrease in the effective activation energy of the photoexcited electrons with increasing ħΩ (i.e., the factor ηω) and but by a decreasing factor 1/ħΩ in formula ℛ = jphoto/ħΩI.
Fig. 7 Spectral dependences of the responsivity, ℛ, of a GLIP with N = 1 at different normalized intensities S and voltages U.
One can see from Fig. 7 that the GLIP responsivity in the photon energy range ħΩ < 2Δ at relatively high voltages can markedly exceed that at relatively low voltages shown in Figs. 2(a) and 2(b). The dashed line in Fig. 2(a), obtained considering the weak irradiation as a perturbation, and the dependence for S = 0.001 in Fig. 7(b) are similar. Relatively high values of the GLIP responsivity in the range ħΩ < 2Δ are attributed to a substantial increase in the barrier tunneling transparency at elevated voltages. However, the elevated values of the responsivity at high voltages are accompanied by rather high dark currents (corresponding to the values J at S = 0 in Fig. 5).
7. Comments
In our model, we neglected the dependence of the efficiency, p, of the electron capture to the n-GL on the electric fields in the barriers surrounding this GL, i.e., on En and En+1. The inclusion of this effect, can account for charging not only the GL adjacent to the emitter GL but also a few following GLs, although their charges would be relatively small. (as was demonstrated considering the dark currents in the multiple-QW and -GL heterostructures [25, 26, 28, 32, 33]). Since the capture efficiency is much more sensitive to the average electric field Ē = V/d(N + 1) in the heterostructure [34], our simplification appears to be justified. Nevertheless, the dependence of p versus Ē can somewhat affect the GLIP current-voltage characteristics both in dark conditions and under irradiation. When I exceeds IV, Eq. (5) becomes invalid, because in this case it formally yields EB < 0. At I > IV, the electron transport in the GLIP bulk becomes diffusive, and one might expect that a further increase in I does not lead to a marked variation of the current. Thus, the quantity IV is actually the IR radiation intensity corresponding to the photocurrent saturation. It defines the dynamic range of the GLIP normal operation (I < IV). Estimating jm = eΣT/τesc, where ΣT = (π/6)(T/ħvW)2 is the electron and hole density in the undoped GL [1], from Eqs. (11) and (12), we arrive at
Assuming that T = 100−300 K, τesc = 10−13 s, τesc/τrelax = 0.1, Δ = 0.4 eV, εF = 0.1 eV, p = 0.01, κ = κ∞ = 5, m = 0.28m0 (m0 is the electron mass in vacuum), V/dEtunn ∼ 0.1, we obtain IV ≃ 4 × (1022 − 4 × 1023) s−1cm−2. For ħΩ = 0.80 eV, this photon flux corresponds to PV = ħΩ IV ≃ (5 − 50) kW/cm2. This implies that at I ∼ IV, the power density absorbed by a GLIP with the N inner GLs is about of . Hence, the GLIP operation at such an intensity can be realized in the pulsed regime.
An increase in V leads to an increase in EE and, consequently in the decrease of the factor γV/d in Eqs. (15) and (16). At sufficiently large V, this factor can be close to zero. This implies that the device model under consideration is limited by the voltage range U = V/dEtunn ≃ Umax = Vmax/dEtunn ∼ (Δ/εF)2(F/Etunn). At the above parameters, this inequality yields U ≃ Umax ∼ 0.5.
Above we also disregarded the following processes which, in principle, might affect the GLIP characteristics: the thermionic emission of electrons from the GLs, particularly, the hot electrons in the GLs heated by the absorbed IR radiation and the saturation of the radiation absorption. The former mechanism leads to an increase in the current emitted by the GLs due the thermionic emission contribution. However, as shown by our estimates, this contribution is relatively small in a wide range of the GLIP parameters, applied voltages, and radiation intensities corresponding to the normal device operation (except, possibly, the range of relatively low photon energies (see Appendix A).
The saturation of the radiation absorption in the GL due to the Pauli principle associated with the accumulation of the photoexcited electrons with the energy ħΩ/2 and in the vicinity of this energy, should result in the saturation of the photocurrent as a function of the intensity. However, this can occur at sufficiently high intensities beyond real practical GLIP applications (see Appendix B).
8. Conclusions
We calculated the current in the GL-vdW heterostructures for GLIPs caused by the IR irradiation in a wide dynamic range - from low to high intensities. A strong IR radiation results in nonuniform electric-field distributions with a high-electric field domain near the emitter. The spatial redistribution of the electric-field leads to the photocurrent amplification (the effect of photoelectric gain) and to a slowing down of the photocurrent rise with increasing intensity. Since the capture efficiency in the GLIPs can be much smaller than in the standard QW-heterostructures, the GLIP responsivity can markedly exceed that of other QW-photodetectors, in particular, QWIPs [27] as well as the unitravelling-carrier photodiodes [35]. A substantial advantage of the GLIPs is associated with their sensitivity to the normally incident IR radiation. The saturation of the IR radiation absorption due to the Pauli principle can be an additional mechanism of the current-intensity characteristic nonlinearity but at relatively high IR radiation powers. The obtained results can be useful for the development and optimization of the GLIPs in starring arrays, optical communication devices, and terahertz photomixers.
Appendix A. Thermionic electron emission
The thermionic emission of the electrons thermalized in the GLs can contribute to the net current in the GLIPs [18]. If the effective temperature of these electrons is close to the lattice temperature T, the thermionic contribution is relatively weak if Δ/T > edEtunn N/V. The latter takes place at not too low bias voltages V.
The heating of the thermalized electrons due to the radiation absorption leads to the variation of the net current j, which increases with the increasing I. Such a mechanism can be essential in the GLIPs with relatively small Δ at elevated temperatures. This might limit the applicability of the results obtained above.
The heating of the thermalized electron-hole system in the GLs is associated with a fraction of the kinetic energy of the photoexcited electrons and all of the energy of the photoexcited holes going to this system, particularly, in the emitter GL. This implies that the electron-hole system in the emitter GL approximately gets the power of S ≃ β IħΩ(1/2 + w). Here w is the probability that a photoexcited electronn does not escape from the GL: . Depending on τesc/τrelax and ħΩ, w varies in the range 0 < w < 1/2. Neglecting the energy contribution of the Drude absorption (due to high photon energy in the spectral range under consideration) and considering that the specific heat of the degenerate electron system in the emitter GL is given by the standard formula , and equalizing the energy received from the IR radiation and the energy transferring to phonons, one can obtain the following estimate for the effective temperature TE:
where τε is the electron energy relaxation time. Assuming the same parameters as for the estimate for IV in Sec. 5 and setting τε = (10−11 − 10−12) s, we obtain IT ≃ 3 × (1023 − 1024 s−1cm−2. These values are of the same order of magnitude or larger than IV.At low intensities, TE/T ≃ 1 + I/2IT. In this case, the variation of the thermionic current density from the emitter GL due to the electron heating by the absorbed IR radiation is estimated as . Comparing this value with the photocurrent density, associated with the photoemitted electrons (at ħΩ ≃ 2Δ when ηΩ ≃ 0), we find that the latter substantially exceeds the thermionic component if
Inequality (18) is satisfied in a wide range of parameters and temperatures. Indeed, setting Δ = 0.4 eV, Δ/εF = 4, and τesc/τε = 0.1, for T = 0.025 eV (≃ 300 K) the term in the right-hand side of this inequality is approximately equal to 430. At Δ = 0.2 eV and T = 0.017 eV (≃ 200 K), this term is about of 30. When ħΩ is markedly smaller that 2Δ, the rate of the photoexcited electrons escape from the GLs can be decreased. At such photon energies, the heating mechanism can be crucial. However, the device model for the GLIPs operating in the range of relatively small photon energies should be generalized accordingly, that is out of the scope of the present work.At elevated radiation intensities, considering Eq. (A1), the density of the thermionic current created by the hot electrons can be estimated as . The comparison of JT and JV given be Eq. (11) shows that JT ≪ JV when
The term in the right-hand side of inequality (A3) is large except the cases of impractically small the band offcets and applied voltages. Since the validity of the formulas obtained in Secs. 4 and 5 is limited by I ≤ IV, taking into account that IV ≤ IT and considering inequality (19), one might expect that the nonlinearity mechanism associated with the electric field distribution is more crucial than the heating mechanism.Appendix B. Saturation of absorption
At sufficiently strong irradiation, the distribution function of the photoexcited electrons at the energies close ε ≃ ħΩ/2 can become close to 1/2. This can lead to the interband transitions saturation [31] and, hence, to the drop of the GLIP responsivity.
Taking into account the linearity of the GL energy spectrum and considering the Pauli principle, the interband photoexcitation rate with the estimated values of the electron distribution function at the energy εΩ = ħΩ/2 can be presented as
Here θ is given by Eqs. (3), Σ is the density of the photoexcited electrons in the GL, and ΔΩ is the incident IR radiation spectral width. Estimating Σ considering the balance between the electron photoexcitation and the escape and substituting the obtained value to Eq. (10), we arrive at where for the most interesting situations when τesc ≪ τrelax and ħΩ ∼ 2Δ one obtainsAs follows from Eq. (B1), the effect of the absorption saturation can be accounted for by replacing the factor jm/eβ I in Eq. (5) by jm(1 + I/IS)/eβ I. This leads to a saturation of j as a function of I and to decrease in the GLIP responsivity ℛ ∝ (ħΩI)−1 with increasing I. As a result, at large intensities, the modulation parameter M = δj/δI should drop and tend to zero at very strong irradiation.
Setting ħΩ = 0.8 eV, ħΔΩ ∼ ħ/τesc ∼ 0.02 eV, we find IS ∼ 2.6 × 1027 s−1 cm−2 (or SS = ħΩ IS ∼ 330 MW/cm2). The above results are in line with those predicted theoretically [31] and obtained experimentally [36]. Comparing the characteristic intensities IV and IS, one can conclude that the nonlinearity mechanism invoked in our model is more crucial at much lower intensities than the saturation mechanism.
Due to short tunneling delay times, the bandwidth of the GLIP response to the modulated IR radiation is primarily limited by the electron transit time ttrans ∝ (N + 1)d across the barrier layers. This time in the GLIPs with sufficiently thin inter-GL layers can be fairly short. In this regard, the GLIPs can surpass or, at least, compete with such IR detectors as unitravelling-carrier PDs [35]. The GLIP operation using the heating mechanism is much slow, because the pertinent speed is limited by the electron energy relaxation time τε ≫ τesc, ttrans.
Funding
The work at RIEC and UoA was supported by the Japan Society for Promotion of Science, KAKENHI Grant No. 16H06361. The work at RIEC with UB was supported by the RIEC Nation-Wide Cooperative Research Project. VR also acknowledges the support by the Russian Scientific Foundation, Grant No.14-29-00277. The work by VL and DS was supported by the Russian Foundation for Basic Research, Grant No. 16-37-60110. The work at RPI were supported by the US ARL Cooperative Research Agreement.
Acknowledgments
The authors are thankful to A. Satou, V. Y. Aleshkin, and A. A. Dubinov for useful discussions.
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