1. Introduction

Quantum cascade photodetectors (QCDs), which are based on intersubband (ISB) transitions of multiple quantum well (QW) structures, have been attracted great interests in the last ten years. A standard QCD consists of tens of periodic multiple QWs. Each period contains a doped QW used for photon absorption and a cascade QW structure to extract the photo-excited electrons. Such periodic excitation-extraction structures can produce a pure photocurrent without any external bias. As a result, the detected optical signal is converted to voltage signal of the device. So QCDs are photovoltaic detectors with extremely weak dark current. As ISB devices, QCDs are also high-speed detectors whose fundamental speed limit is the ISB scattering time of electrons (within picosecond magnitude). These advantages of QCD make it a promising candidate for applications in wide spectral detection ranging from near-infrared (NIR) to terahertz (THz) region.

In practice, QCDs have been demonstrated in sensitive detection and high-speed operation in the mid-infrared (MIR) [1–7] and NIR [8–12] regions. One of the important factors for the achievement of QCDs in MIR and NIR regions is the carefully designed extraction cascades in those devices, which are adapted to the LO-phonon energy. In such an extraction structure, LO-phonon scattering is the dominant process which leads to high detectivity and high-speed operation [13]. However, for very long wave infrared (12~20 μm) or even THz wave detection, the LO-phonon stair extractor would fail since the photon energy is comparable or lower than the LO-phonon energy. A chirped miniband structure was used to replace the LO-phonon extraction stair in a QCD for 16.5 μm field [5]. And a THz QCD using energy mini-steps for electron transport was also presented [14, 15]. Although these structures keep a high device resistance and low Johnson noise level, the escape probability of an excited electron is reduced which results in a low photodetector gain. Considering the importance of LO-phonon stair in high-performance of QCDs, here, we present a scheme to extend the detection wavelength into THz region without trading off high extraction efficiency (LO-phonon stair). We design a THz absorption QW within the LO-phonon cascade QWs, then demonstrate that the structure can response THz wave with the MIR pumping light. We study the dynamical behavior of electrons within such structure in detail and show that the performance of the device for the THz detection can reach the same level of MIR QCDs with appropriate pumping and doping.

2. Quantum design

The schematic pumping-detection structure in one period of our design is shown in Fig. 1(a). It can be seen that the pumping of the MIR field is prerequisite for the absorption of THz photons. Under illumination of a MIR field, electrons in the ground state can be pumped to the first excited state in a MIR active well. Through resonant tunneling, the excited electrons transport unidirectionally into the following phonon stair with m steps, and then accumulate in the ground state of the THz absorption well, in preparation for resonant excitation by the THz field. After absorbing a THz photon successfully, the excited electrons are extracted by another LO-phonon stair with n steps to the ground state of the next period, resulting in a measurable photocurrent. According to the law of conservation of energy, the band structure should be designed following the relationship of hν₂ = (m+n)E_LO −hν₁, where hν₁ is the MIR pumping photon energy, hν₂ is the detected THz photon energy, and E_LO is the LO-phonon energy. We notice that when hν₂ is close to E_LO, the extraction efficiency will be largely reduced due to the LO-phonon scattering from the upper state back into the lower state in the detection active well. [16] However, for hν₂ < E_LO, in the absence of LO-phonon scattering in the THz absorption well, the extraction efficiency is greatly promoted, which leads to a more robust performance for application in the region for ν₂ < 8 THz.

 

Fig. 1 (a) Scheme of pumping-detection structure in one period of the active region of the MIR-pumped THz QCD. (b) Calculated conduction band profile of a THz QCD designed according to the above concept.

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We apply the above concept to a specific design of THz QCD sensitive at wavelength of 77.1 μm (16.1 meV). Based on GaAs QWs and Al_0.45Ga_0.55As barriers system, we calculate the conduction band profile of one period and plot it in Fig. 1(b). Here, we establish two extractors totally with 6 LO-phonon steps for rapid electron transport from one period to the next period. Thus, a MIR field with wavelength of 5.6 μm (222.5 meV) is appropriate for pumping electrons. The absorption of the MIR field occurs in QW A. And QWs F-I constitute the THz detection region. QWs F and G have identical potential, forming a double QW with resonant tunnel coupling. Similarly, QWs H and I form another pair of coupled QWs. The THz detection transition takes place between hybrid states in these adjacent double QWs. The design of double well potentials gives strong confinements of wave functions, and enlarges the space between state |7〉(|2〉) and the upper hybrid states |5〉 and |6〉 (the lower hybrid states |3〉 and |4〉), which dramatically decreases the undesired couplings of hybrid states to extractor states. The overlap integrals of the participating energy levels can be reduced to zero by tailoring the thicknesses of QWs and barriers in the detection region. This ensures minimizing the leakage current. Using the series expansion method [18], we calculate the layer sequence as follows: 55/65/13/60/18/43/23/36/28/34/35/56/35/40/32/56/32/35/41/33 Å, where values in bold indicate the barrier layers. The active QW A is doped with a density of n_s = 2.5 × 10¹¹ cm⁻². The transition energies for THz wave detection from states |3〉 or |4〉 to states |5〉 or |6〉 are E_5,3 =16 meV, E_{6 3} =21.1 meV, E_5,4 =12.4 meV, and E_6,4 =17.5 meV, respectively. And the energy intervals of the phonon stairs are all about 40 meV. The calculated dipole moment matrix elements are d_1,10 = 1.21 nm, d_1,11 = 1.00 nm, d_3,5 = 1.22 nm, d_3,6 = 0.94 nm, d_4,5 = 2.41 nm, and d_4,6 = 1.63 nm. The large dipole moments for the diagonal transitions indicate strong absorption of THz waves.

It should be noted that the pumping scheme may provide a little probability for excited carriers to absorb another MIR photons to get into the continuum states. On one hand, in the absence of external bias, carriers in the continuum states have equal probability to relax to each bound state, which makes a little change of the steady state populations, but obviously has no contribution to dark current. On the other hand, carriers, who, with help of the pumping, get across the barriers in the detection region to excited states |5〉 or |6〉 by relaxing from the continuum states, may become an origin of dark current. However, the possibility of this process is dramatically reduced when it goes through the cascade structure, which leads to a low dark current level. Therefore, when we investigate the dynamics of the system, we ignore the effects of pumping to continuum states discussed above.

Obviously, as demonstrated in [14], thermal activation of electrons in the THz detection region can give rise to serious dark current, especially in high operation temperature. This is a general problem in THz detection. Therefore, THz QCDs need to work under low temperature condition.

Of course, the implementation of the pumping-detection structure would require a relatively complex architecture, for example, the integration of a MIR source with the structure. The state of the art in MIR quantum cascade lasers (QCLs) gives a broad frequency range and a watt level power [17], which is sufficient for the MIR pumping as shown in the following calculation. In addition, its compact volume (e.g. 3 mm× 0.15 mm× 0.015 mm) and analogous cascade structure make the possibly needed integration technics viable.

3. Dynamics

To investigate the dynamics of the system, we presume that periods of the THz QCD precisely have the same structure which results in the same performance. Therefore, when we describe the interaction of the whole system with the optical fields, it can be restricted in one period whose Hamiltonian can be written as

In our study, we mainly consider the acoustic scattering in all QWs and LO phonon scattering in QW A and the extraction stairs. The two scattering processes are temperature dependent. However, in low temperature region, the scattering rates change with temperature within one order of magnitude [19]. For simplicity, we take the scattering rates at zero temperature according to [16] as γ₁ = 1 THz in QW A, γ₂ = 10 GHz between QWs in the THz absorption wells, and Γ = 1.5 THz in LO-phonon stairs. Actually, scattering processes in our structure also include interaction between electron and ionized impurity, electrons, alloy disorder, and interface roughness, which are not taken into account in calculations. In fact, for a QCD system, the electron-electron scattering effect is relatively weak due to low populations in the excited states. In our structure, to reduce ionized impurity scattering effect on THz detection region, we design a remote doping in QW D as shown in Fig. 1(b). Alloy disorder and interface roughness scattering in structures such as QCLs and QCDs is usually determined by specific samples, and then the corresponding scattering rates are difficult to give typical values. For example, alloy disorder and interface roughness scattering lead to low R₀A (where R₀ is the resistance per period and A the area of the detector) in [15]: 10⁻³ Ωcm² and 10⁻⁵ Ωcm², respectively. However, a R₀A around 10 Ωcm² per period is reported in THz QCD [14] and 10² Ωcm² per period are often reported for 10 μm QCDs.

By numerically solving the master equation, we obtain the populations P_i of |i〉, and P_E of the empty state |E〉. Here, |E〉 is introduced to describe the periodic dynamic behavior of electrons tunneling out one period into the next. And then the current responsivity R of the QCD can be calculated by [21]

4. Results and discussions

4.1. Responsivity

The responsivity shows the capability of a QCD converting specific photons into electric signal. In Fig. 2, we plot the calculated responsivity and photocurrent density spectra of our QCD at the MIR field resonant with transition between states |1〉 and |10〉. The parameters for fields are hν₁ = 222.5 meV, I₁ = 10.5 W/cm² and I₂ = 0.02 W/cm². Since the THz absorptions occur between two ground states (|3〉 and |4〉) and two excited states (|5〉 and |6〉), the two ground states are no longer isolated but correlated by optical dynamic processes. Only one prominent absorption peak would be at $\frac{1}{2}(E_5+E_6-E_3-E_4)=16.75\text{meV}(134.6{\text{cm}}^{-1})$, if coupling strengths of the four pairs of transitions are all the same. However, in our designed band structure, the coupling strengths differ from each other, which leads to a shift of the main peak and splitting peaks in the spectrum curve. Here, the intensively responded wavenumber is shifted to 129.8 cm⁻¹ (16.1 meV) with responsivity of 9.1 mA/W. And a small peak splitted from the main one is found at 166 cm⁻¹ (20.6 meV). The band structure also determines a large full width at half maximum (FWHM) of the responsivity spectrum. The FWHM is 79.0 cm⁻¹, giving a relative linewidth of 61%. This design of detection region may provide a possible way to accomplish broadband THz QCD.

 

Fig. 2 Responsivity spectrum of the THz QCD at the MIR field resonant with transition |1〉 ↔ |10〉.

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We also notice that the peak photocurrent density is about 1.5×10⁻⁴ A/cm². According to the measured dark current of 10⁻⁵ ~ 10⁻⁴ A/cm² under temperature 6~30 K in [14], the THz detection using this structure needs low temperature condition.

4.2. MIR field modulation

In this pumping-detection structure, the photocurrent responsivity for THz signal is modulated by the MIR field, since the dynamics of two optical processes are correlated by LO-phonon scattering. In Fig. 3, we show variation of the peak responsivity versus I₁ and I₂. It is found that the responsivity with specific I₂ reaches its maximum when I₁ ≫ I₂. Actually, when detecting weak THz signal, the responsivity is proportional to the absorption coefficient α of the detection well. Therefore, the MIR field basically modulates the absorption efficiency of the THz field. This can be understood by reducing the system of correlated dynamics to a four-level configuration, as depicted in the inset of Fig. 3. After optical excitations, electron transport through LO-phonon stairs, i.e. |10〉 → |3〉 and |5〉 → |1〉 is modeled as an incoherent process with scattering rate Γ′. For the sake of simplicity, Rabi frequencies Ω₁ and Ω₂ are used to describe couplings of the MIR and THz fields, respectively, and γ is used to denote decay rates from the higher levels to the lower levels. Under the condition of Γ′, γ ≫ Ω₁,Ω₂,we solve the steady state rate equations of the four-level configuration, and give populations of the four states as $P_1\approx \frac{{\mathrm{\Omega }}_2^2}{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}$, $P_3\approx \frac{{\mathrm{\Omega }}_1^2}{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}$, and P₁₀ = P₅ ≈ 0. In addition, the absorption coefficients of the MIR field and the THz field can be expressed as ${\alpha }_1=C_1\frac{{\mathrm{\Omega }}_2^2}{({\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2)(\gamma +{\mathrm{\Gamma }}^{\prime })}$ and ${\alpha }_2=C_2\frac{{\mathrm{\Omega }}_1^2}{({\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2)(\gamma +{\mathrm{\Gamma }}^{\prime })}$, respectively, with constants $C_1=\frac{4\pi n_s{e}^2{d}_{1,10}^2{v}_1}{{\epsilon }_{0}cn\overline{h}}$ and $C_2=\frac{4\pi n_s{e}^2{d}_{3,5}^2{v}_2}{{\epsilon }_{0}cn\overline{h}}$, where ε₀ is the vacuum di-electric constant, c is the speed of light, and n is the refractive index. In the limit of I₁ ≪ I₂, i.e. Ω₁ ≪ Ω₂ (because d_1,10 ≈ d_3,5), P₁ → 1, P₃ → 0, ${\alpha }_1\approx \frac{C_1}{\gamma +{\mathrm{\Gamma }}^{\prime }}$, and ${\alpha }_2\ll \frac{C_2}{\gamma +{\mathrm{\Gamma }}^{\prime }}$. Obviously, electrons almost accumulate in the ground state |1〉, and the absorption coefficient of the MIR field is analogous to that of a two-level system. This indicates that there exists almost no absorption of the THz field, leading to an extremely low response to the input signal. However, in the limit of I₁ ≫ I₂, P₁ → 0, P₃ → 1, ${\alpha }_1\ll \frac{C_1}{\gamma +{\mathrm{\Gamma }}^{\prime }}$ and ${\alpha }_2\approx \frac{C_2}{\gamma +{\mathrm{\Gamma }}^{\prime }}$. This implies that since the oscillation induced by the MIR field is much faster than that induced by the THz field, electrons are much easier to pump into and accumulate in the lower states for the THz field absorption. And then a linear absorption, as a standard QCD does in the two-level system in its absorption QW, is achieved in the detection region. Generally speaking, under the condition of I₁ ≫ I₂, our THz QCD can obtain a peak absorption efficiency having the same order of magnitude as standard MIR QCDs.

 

Fig. 3 The peak responsivity of the THz QCD versus I₁ and I₂. Inset: a four-level configuration modeling the dynamics of two optical processes correlated by LO-phonon scattering.

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It should be noted that, under a high I₁, electron accumulation in the THz active wells F and G may distort the subband lineup. According to the calculation in [20], we estimate that an electron density of 2.5 × 10¹¹ cm⁻² in the QWs can produce a potential of up to 0.4 meV, which can induce a change of about 0.1 THz in the detection frequency.

4.3. Photodetector gain

From Eq. (3–5), we calculate the escape probability of an excited electron in the detection QWs by p_e = hν₂Rp_c/(α₂e) ≈ 0.99, here p_c = 1 is used [21]. For comparison, we perform similar calculations of a standard MIR QCD [1] and a traditional THz QCD [14] operated under the same condition. The results show that p_e = 0.6 and 0.5 for peak detection in the MIR and THz QCDs, respectively, which are close to the values estimated in [1] and [14]. Actually, in these available QCDs, the phonon scattering times from the upper resonant state towards the ground state (relaxation time τ_rel in the active QWs) and towards the extractor state (escape time τ_esc) are in the same order of magnitude, i.e. τ_rel ≈ τ_esc is about the magnitude of picosecond in the MIR QCD, and hundreds of picoseconds in the THz QCD. This means that only one-half of the excited electrons escape the active QWs to form photocurrent. However, in our design, the bandstructure gives τ_rel ≫ τ_esc, indicating that almost all the excited electrons escape. Therefore, according to Eq. (5), the promoted p_e leads to a higher photodetector gain in our structure.

4.4. Response speed

Time to reach the steady state is related to the frequency response of the THz QCD. In a standard QCD, each absorption QW is identically doped, thus it is prepared with plenteous carriers for the optical transition. Therefore, the main factor limiting the speed of the device is the transit time of electrons through the extractor between two consecutive active QWs. In our design, the LO-phonon scattering ensures the high-speed transport. However, distinct from the standard QCD, it requires two groups of absorption QWs in each period. We notice that different ways of doping these absorption QWs can change the response speed of the device greatly. Due to the weak driving force of the detected THz field, if each period is solo doped in QW A (QW D), it will take a long time for electrons accumulating in the ground state of the THz absorption QWs (MIR absorption QW A). Here, to reduce impurity scattering, we dope QW D to provide carriers for the THz absorption wells in stead of directly doping QWs F/G/H/I. Figure 4(a) shows that the system with solo doping needs more than 1 μs to reach its steady state. However, when QWs A and D are both doped, it only needs 1.6 ps to reach the steady state, as shown in Fig. 4(b). This means that the response of THz detection remains in picosecond magnitude as that of a standard MIR QCD. For a THz QCD with energy mini-step extraction structure as in [14], it would be difficult to reach such a high speed response.

 

Fig. 4 Time evolution of the electronic density of the empty state. (a) Only QW A or QW D doped. (b) Both QW A and QW D doped. The field parameters are hν₁ = 222.5 meV, hν₂ = 16.1 meV, I₁ = 42 W/cm², I₂ = 1.7 W/cm².

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5. Conclusion

In summary, we propose a scheme to extend the detection wavelength of a QCD into THz region. Aiming at high-efficient photocurrent responsibility, we design a pumping-detection structure with the adoption of LO-phonon extractor. A MIR field is in charge of the pumping work in one absorption well to provide the energy space for detection of THz wave within LO-phonon extraction stairs. We apply this concept to a design of THz QCD sensitive at wavelength of 77.1 μm. A diagonal transition for the THz field detection occurs between hybrid states in adjacent double QWs. This design not only reduces undesired couplings between extractor states and hybrid states, increases the device resistance, but also gives a broad detection range with a relative linewidth of 61%. Numerical simulation indicates that the photoexcited electron escape probability is high (p_e ≈ 0.99). Modulated by F₁, the absorption process can be analogous to that in a two-level system, with the absorption efficiency at maximum. Therefore, the total quantum efficiency of the THz QCD is higher than that of a standard QCD. The design of such a pumping-detection quantum cascade structure will extend the QCD to the high efficient and high speed THz detection, and may be also used in development of all-optical THz modulators.