The intersubband quantum-well infrared photodetectors (QWIPs) [1–4] based on the A₃B₅ heterostructures have been developed since 1960s [5–8]. These photodetectors are still the subject of intensive theoretical and experimental studies [9–15]. one devices and systems is much less than of those based on CdHgTe and InSb. The intersubband QWIP disadvantages include the large thermal dark current at elevated temperatures, which prevents the room operation [1–3] and the need for the radiation coupling structures (for the n-type QWIPs). Therefore, the standard A₃B₅QWIPs cannot compete with the interband CdHgTe photodiodes (PDs) [3] in performance. The recent progress in the fabrication of CdHgTe QW heterostructures [16–19] provides an opportunity for a further enhancement of the CdHgTe photodetector technology. In this paper, we propose and evaluate the QWIPs using the interband transitions in the CdHgTe heterostructures, in particular, with the HgTe QWs. The interband HgTe-CdHgTe QWIP operation is associated with the photoexcitation of the electrons in the QWs followed by their escape. These processes result in the redistribution of the device potential, varying the electric field at the emitter and the electron injection current from the emitter. The electric potential distribution in question is governed by the balance of the photoexcitation from the QWs and the electron capture into the QWs similar to what takes place in the standard QWIPs [20–24]. Due to the features of the energy spectrum in the QWs, these interband QWIPs can operate at the normal IR radiation incidence. In this regard, the interband HgTe-CdHgTe QWIP operation is akin to the operation of the vertical graphene-layer infrared photodetectors (GLIPs) [25–27].

The interband electron transitions in the HgTe-CdHgTe QWIPs can provide a rather strong absorption. A weak capture of the electrons into the QWs leads to a high photoconductive gain (phototransistor effect). The required spectral characteristics of the interband QWIPs can be obtained by a proper choice of the CdHgTe composition and the QW width. The possibility of the QW engineering enables the fabrication of the interband HgTe-CdHgTe QWIPs with desirable spectral characteristics (by using, for example, the HgTe QWs with different width). We compare the QWIPs under consideration with the standard p-i-n CdHgTe photodiodes (PDs) and discuss the advantages of the former.

1. Interband QWIP device model

The QWIP structure under consideration consists of a number of the undoped HgTe QWs (N = 1, 2, 3, …) with the energy gap, Δ_QW, between the top of the highest hole subband and the bottom of the lowest electron subband. The QWs are separated by a material with the energy gap Δ_G> Δ_QW(Cd_xHg_1−xTe with 0 < x < 1), so that the barrier of the height Δ_Bis formed for electrons (see Fig. 1). The structure is sandwiched between the emitter and collector n-doped layers. For the definiteness we assume that both these layers are the same QWs (as the inner QWs) but highly doped by donors. Figures 1(a) and 1(b) show the QWIP device composition and the band diagram under sufficiently strong bias voltage V ≪ V_bi, k_BT/e (where V_biis the built-in voltage between the n-doped contact and the undoped inner QWs, T is the temperature, k_B is the Boltzmann constant, and e is the electron charge).

 

Fig. 1 Schematic views of (a) interband HgTe-CdHgTe QWIP structure, (b) its band diagram at the bias voltage V. The inset in Fig. 1(b) shows a fragment of the band diagram in more detail. Wavy arrows correspond to the incident photons and to the processes of the electron transitions from the subband of the QW valence band to the electron subband (above its bottom). Solid arrows indicate the propagation of the electrons (both injected from the emitter and the preceding QWs) above the barriers, capture of these electrons into the QW, and tunneling of the photoexcited electrons from the QW.

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The electron photoexcitation in the QWs in question under the normally incident radiation can be associated with both the intersubband transitions within the QW conduction band and with the interband transitions. Considering that the electron density in the undoped QWs is relatively small and that the pertinent matrix elements are small, we focus on the contribution of the interband transitions.

To provide an effective escape of the photoexcited electrons from the QWs into the states above the barriers, the following conditions are assumed:

Using a simplified model for the characteristics of the vertical photodetectors using the photoexcitation from the localized states in the structure and the electron injection from the emitter (used previously in the papers on the standard QWIPs as well as in the GLIPs [19–27]), one can obtain for the photocurrent density in the QWIP j_photoand its photodetector responsivity R = j_photo/I_ωħω

The factor ξ_N≲ 1 describes a nonideality of the emitter (see Ref. [26] and the references therein): ξ_N≃ N/(γ^3/2 + N) with γ = (Δ_B− ε_F)/Δ_B, where N is the number of the inner QWs, ε_F is the electron Fermi energy in the emitter QW counted from bottom of the lowest electron subband in the QW and Δ_Bis the energy spacing between the barrier top and the bottom of this subband [see Fig. 1(b)]. Equation (2) corresponds to the net rate of the photoescape β_ωθ_ωξ_NN and the phoconductive gain g = 1/(Np_c).

At ħω ≤ ħω_thand ħω ≥ ħω_th,

2. Energy spectra, spectral characteristics of the interband absorption and capture probabilities

The functions β_ωand θ_ωdepend on the QW energy spectra, which, in turn, depend on the compositions of the QWs and barrier layers materials and on the QW width, d. We calculated the energy spectra and the spectral characteristics of the absorption coefficients for the heterostructures grown on [013] CdTe surface with different content of Cd in the barrier layers at temperatures T = 77 and 200 K. The choice of the ranges of the composition and the QW widths variations corresponds to the the energy gaps Δ_QW∼ 0.025 − 0.20 eV, i.e., to the pertinent photon energies (mainly to ħω ≃ 0.08 − 0.18 eV). The refractive index of the barrier is set to be $n=\sqrt{15.2}$.

Figure 2 shows examples of the energy spectra of the HgTe QWs surrounded by the Cd_0.27Hg_0.73Te and Cd_0.3Hg_0.7Te barriers calculated for different values of the QW width and T = 77 K and T = 200 K. The spectra shown in Fig. 2 correspond to two lowest electron subbands marked as e1 and e2 (virtually undistinguished due to a weak interface inversion asymmetry splitting) and two sets of split hole subbands marked as h1, h2 and h3, h4.

 

Fig. 2 Energy spectra of the HgTe QWs (a) with the width d = 2.2 nm surrounded by the Cd_0.27Hg_0.73Te barriers at T = 77 K and (b) with the width d = 3.2 nm surrounded by Cd_0.3Hg_0.7Te barriers at T = 200 K: e1 and e2 lines correspond so slightly split lowest electron subband, h1, h2 and h3, h4 correspond to two split hole subbands. Horizontal dotted lines show the bottom of the barrier conduction band E c (for the Cd contents x = 0.27 and x = 0.3) and the conduction band bottom energy minus the optical phonon energy ħω₀, respectively. The energy is counted from the top of the CdTe valence band.

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Figure 3 shows the energy gap, Δ_QW, between the hole and electron subbands and the energy separation, Δ_B, between the barrier conduction band and the bottom of the lowest electron subband (the barrier height) as functions of the QW width d calculated for different content of Cd “x” in the barriers at different temperatures.

 

Fig. 3 The energy gap Δ_QWand the barrier height Δ_Bversus the QW width d for different Cd content x at (a) T = 77 K and (b) T = 200 K.

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The calculations of β_ωare based on the following equation:

In the calculations of the capture probability, p_c, we assumed that the capture of the electrons propagating over the inter-QW barriers into the QWs is primarily associated with the emission of optical phonons (with the energy ħω₀= 0.015 eV).

Figure 4 shows the spectral characteristics of the absorption coefficient (absorption probability) associated with the interband transitions in the HgTe–Cd_0.27Hg_0.73Te (at T = 77 K) and in the HgTe–Cd_0.3Hg_0.7Te (at T = 200 K) heterostructures with the HgTe QWs of different width d. The two-step increase in the absorption probability as a function of the photon energy is associated with a noticeable split of the h1, h2 and h3, h4 hole subband pairs (see Fig. 2).

 

Fig. 4 The interband absorption coefficient β_ωversus photon energy ħω (a) in the HgTe–Cd_0.23Hg_0.73Te heterostructures with the QW widths d = 2.2 nm and d = 1.7 nm (at T = 77 K) and at (b) in the HgTe–Cd_0.3Hg_0.7Te heterostructures with the QW thicknesses d = 3.2 nm and d = 2.5 nm (at T = 200 K).

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3. Interband QWIP characteristics

Figures 5 and 6 show the spectral characteristics of the responsivity R for the interband HgTe-CdHgTe QWIPs with the same parameters as those in Fig. 4 and ξ_N= 0.738 (γ = 0.5 and N = 1) calculated using Eqs. (2) – (4) for different relative electric fields U = E/E_tunn. It is assumed that τ_esc/τ_relax= 0.1.

 

Fig. 5 The spectral dependences of the responsivity R for the interband HgTe–Cd_0.3Hg_0.7Te QWIPs with γ = 0.5, N = 1, and different normalized electric fields U = E/E_tunn: (a) d = 3.2 nm (p_c = 0.36%) and (b) d = 2.5 nm (p_c = 0.58%) at T = 200 K.

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Fig. 6 The spectral dependences of the responsivity R for the interband HgTe–Cd_0.3Hg_0.7Te QWIPs with different number of the QWs N (N = 1, 3, and 10) at T = 200 K, γ = 0.5, and U = E/E^tunn = 0.5: (a) d = 3.2 nm, p_c = 0.36% and (b) d = 2.5 nm, p_c = 0.58%,.

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The plots in Fig. 5(a) correspond to the capture probabilities p_c = 2.5% and p_c = 1.1% of the electrons with the average energy in the barrier layers ε = k_BT ≃ 6 meV. The plots in Fig. 5(b) correspond to p_c = 0.36% and p_c = 0.58% (of the average electron energy ε = k_BT ≃ 17 meV).

The obtained values of the capture probability p_c ≃ (0.36 − 0.58)% are generally of the same order of magnitude (or somewhat larger) as the p_c values for the GLs [30]. The fact that p_c in the QWIP under consideration can be larger at some QW parameters than in GLs, can be explained by large energies of the optical phonons in GLs in comparison with HgTe or CdHgTe. As a consequence, the emission of the optical phonons accompanying the electron capture into GLs requires larger variations of the electron momentum.

As follows from Eq. (2), the dependence of the interband QWIP responsivity on the number of the inner QWs, N, is determined by the factor ${\xi }_N=N/({\gamma }_N^{3/2}+N)$ with γ = (Δ_B− ε_F)/Δ_B), where ε_Fis the Fermi energy in the emitter QW counted from the electron subband bottom. Since γ varies from unity (ε_F≃ 0) to zero (ε_F= Δ for a heavily doped emitter QW), the factor 0.5 < γ_N< 1 for arbitrary N. This implies that the interband QWIP responsivity is a weak function of the number of inner QWs. A similar situation occurs in the intersubband QWIPs and interband GLIPs.

Some increase (although a relatively slow) in the interband QWIP responsivity with increasing number of the QWs N is confirmed by Fig. 6 (N = 1, 3, and 10). At N ≳ 10, an increase in R with N becomes insignificant.

Larger values of the responsivity corresponding to the plots in Fig. 5(a) [and in Fig. 6(a)] in comparison to those shown in Fig. 5(b) [and in Fig. 6(b)] are attributed to a smaller value of the capture probability p_c for the former figure.

As follows from the above data, the interband QWIP responsivity can be fairly large.

4. Comparison with other photodetectors

The interband HgTe-CdHgTe QWIPs surpass the QWIPs using the intersubband (intraband) both based on similar HgTe-CdHgTe heterostructures and on the A₃B₅QW heterostructures. This is due to higher probability of the interband electron photoexcitation in the former compared to the probability of the intersubband photoexcitation. This is confirmed by the calculations of the intersubband absorption coefficient using an equation similar to Eq. (5) (as for the calculations of the interband absorption coefficient) accounting for the population of the QW subbands. The smallness of ${\beta }_{\omega }^{intersub}$ in the case of normal photon incidence under consideration is due to the relative smallness of the intersubband matrix elements and can not be compensated by an increase in the doping. Indeed, for the case of the Cd0.3Hg−0.7Te QW with the widths d = 3.5 nm at T = 200 K, our calculations for the matrix elements square yield ${\left|{\nu }_{i,j}^x\right|}^2+{\left|{\nu }_{i,j}^y\right|}^2\simeq 3\times {10}^{312}/{\text{cm}}^2$ for the interband transitions and ${\left|{\nu }_{i,j}^x\right|}^2+{\left|{\nu }_{i,j}^y\right|}^2\simeq 1.2\times {10}^{282}/{\text{cm}}^2$ for the intersubband transitions. One needs to stress that the intersubband matrix elements are not equal to zero (at the normal in incidence) only because of the nonparabolicity of the electron spectra in the QWs which is rather modest even in the materials considered above. The use of the electron intersubband transitions (both in Hg-CdHgTe and A₃B₅QWIPs) requires a substantial donor doping of the QWs. However the latter leads to a decrease in the thermogeneration activation energy and negatively affects the detector detectivity.

Below we compare the responsivity and dark-current-limited detectivity of the interband QWIPs under consideration with the traditional interband p-i-n PDs. The operation of the latter is associated with the interband photogeneration of the electrons and holes in the depleted bulk i-layer CdHgTe and their propagating in the vertical direction.

4.1. Responsivity

Using the above results and considering Eq. (A1) at α_ωW = β_ω,PD < 1, where α_ωand W are the interband absorption coefficient in the PD depletion layer and the thickness of the latter, for the ratio of the pertinent responsivities R and R_PD we find

4.2. Dark-current-limited detectivity

One of the most important figure-of-merit of the interband QWIP is the dark-current-limited detectivity, which depends on the detector noise at moderate and elevated temperatures. The detectivity can be expressed via the dark current density and the photoconductive gain.

Compare the interband QWIP detectivity D^* and the p-i-n PD detectivity $D_{PD}^{*}$ considering that $D^{*}\propto {\beta }_{\omega }N/\sqrt{NG}$ and $D_{PD}^{*}\propto {\beta }_{\omega ,PD}/\sqrt{2G_{PD}}$, where G and G_PDare the rates of generation in the dark conditions. We assume that the barrier material in the QWIP and the material of the p-i-n PD are chosen to provide the equal values of ħω_th= Δ_QW+ Δ_B= Δ_G[see Eq. (1)], where Δ_Gis the energy gap in the depletion region of the p-i-n PDs under comparison. Since the energy gap in the interband QWIP barrier layers exceeds Δ_QW+ Δ_B, the former is larger than the energy gap, Δ_G, in the PDs under comparison. Due to this, one can neglect the thermogeneration in the barrier layers in comparison to the thermogeneration in the QWs. As a result,

Assuming m = 0.025m₀ (m₀ is the mass of bare electron), β_ω= 3 × 10⁻³, α_ω = 2 × 10³ cm⁻¹, W = 10⁻⁴ cm (i.e., β_ω,PD= 0.2), and T = 200 K, we obtain from Eq. (8) $D^{*}/D_{PD}\simeq 0.119\sqrt{N}$. Even at N ∼ 10 − 20, the latter ratio is somewhat smaller than unity. The Auger generation in the QWs can further decrease D*. However, in reality, the Auger generation in CdHgTe p-i-n PDs operating in long wavelength radiation range also can result in elevated dark currents and, hence, in a lower detectivity [32,33] (see also, Refs. [12,13]. A detailed comparison of the Auger generation role in both type of the devices as well as the the consideration of the trap-assisted tunneling from the QWs require a substantial generalization of the model that is beyond the scope of this paper.

In conclusion, we proposed the interband HgTe-CdHgTe QWIPs and analyzed their characteristics. The analysis of these photodetectors demonstrates their substantial advantages over the intersubband (intraband) HgTe-CdHgTe QWIPs, the conventional p-i-n PDs, and the intersubband A₃B₅QWIPs.