Quantum Cascade Detector Utilizing the Diagonal-transition Scheme for High Quality Cavities References and Links

A diagonal optically active transition in a quantum cascade detector is introduced as optimization parameter to obtain quality factor matching between a photodetector and a cavity. A more diagonal transition yields both higher extraction efficiency and lower noise, while the reduction of the absorption strength is compensated by the resonant cavity. The theoretical limits of such a scheme are obtained, and the impact of losses and cavity processing variations are evaluated. By optimizing the quantum design for a high quality cavity, a specific detectivity of 10 9 Jones can be calculated for λ = 8 µm and T = 300 K. Terahertz quantum cascade lasers with large wall-plug efficiency, " Appl. High power quantum cascade lasers, " New J. Quantum-cascade-laser structures as photodetectors, " Appl. High-speed infrared detection by uncooled photovoltaic quantum well infrared photodetectors, " Appl. InP-based quantum cascade detectors in the mid-infrared, " Appl. High resistance narrow band quantum cascade photode-tectors, " Appl. High detectivity short-wavelength II-VI quantum cascade detector, " Appl. Photonic crystal slab quantum cascade detector, " Appl. Plasmonic lens enhanced mid-infrared quantum cascade detector, " Appl. Antenna-coupled microcavities for enhanced infrared photo-detection, " Appl. Photonic crystal slab quantum well infrared photodetector, " Appl. Detectivity enhancement in quantum well infrared photodetectors utilizing a photonic crystal slab resonator, " Opt. Predictive circuit model for noise in quantum cascade detectors, " Appl. Temporal coupled-mode theory for the Fano resonance in optical resonators, " J. A flexible free-software package for electromagnetic simulations by the FDTD method, " Comp.


Introduction
Quantum cascade detectors (QCD) originated from the concept of quantum cascade lasers (QCL) [1][2][3][4].It was found, that, by not biasing a QCL, it can be operated as a photovoltaic detector [5].At this time, photovoltaic quantum well infrared photodetectors had already been demonstrated and showed very promising performance [6,7].By specifically designing a quantum cascade structure for detection, rather than lasing, a significant performance increase compared to unbiased QCLs could be achieved, which established the topic of QCDs [8][9][10][11].To date, detection has been shown in a broad spectral range, namely from the near-infrared to the THz, but only in the near-infrared and mid-infrared has room-temperature operation with decent device performance been demonstrated [12][13][14][15].Because of their versatility, QCDs are perfectly suited to use in combination with sophisticated cavities and have been demonstrated to yield performance increase with photonic crystals or plasmonic antennas [16][17][18].The highest possible performance is obtained when the quality factor (Q-factor) of the cavity and the detector material match.So far, this has been achieved by reducing the doping of the detector and thereby increasing its Q-factor [19,20].By that, the noise was reduced significantly, but the responsivity remained unchanged.
An alternative way to adjust Q-factors was introduced with a new QCD design concept, the diagonal-transition QCD [21], that showed a performance increase of almost an order of magnitude compared to other QCDs at the same wavelength.Here, the optically active transition occurs between two energy levels that are localized in adjacent wells.The dipole matrix element can be controlled, to match the cavity Q-factor, by changing the thickness of the separating barrier.A thicker barrier reduces the absorption strength , while the extraction efficiency and the resistance of the device is increased.In this way, both the responsivity and the detectivity can be increased.We will show that by fully utilizing both reduced doping and the diagonality of the optically active transition, the detector performance can be significantly enhanced.

Quantum design
The performance of photodetectors is commonly quantified through the responsivity or specific detectivity as figure of merit.The responsivity R is a measure of the unit of electrical output per optical input power.In the case of QCDs, it can be written as where λ is the wavelength, q the elementary charge, h the Planck constant, c the vacuum speed of light, η the absorption efficiency, p e the extraction efficiency and N the number of QCD cascades.The specific detectivity D * also contains a noise term and describes the signal to noise behavior.In the Johnson noise limit, the specific detectivity for QCDs near zero bias can be expressed as where R is the responsivity, R 0 is the device resistance, A the device area, k B the Boltzmann constant and T the temperature.By combining the QCD with a resonant cavity, the lifetime of photons in the QCD active region is increased, which increases the absorption efficiency η.This so-called resonant absorption enhancement is only efficient if the cavity and the photodetector material have similar Q-factors.The Q-factor is defined as and can be linked to the absorption coefficient of a material by where ñ = n + iκ is the refractive index, λ the wavelength and α the absorption coefficient related to the entire QCD period.It is assumed that the entire cavity is filled by the absorbing QCD material.Usually, both quantum-well infrared photodetectors and QCDs have an absorption coefficient that corresponds to a significantly lower Q-factor than that of resonant cavities.Typical values of α are 1000 -2000 cm −1 .Using above equation, this corresponds to a detector Q-factor of approximately 25 -12.5 in a InGaAs/InAlAs heterostructure, at λ = 8 µm.Such high absorption is necessary to get reasonable absorption efficiencies in typically used doublepass mesa structures.A higher detector Q-factor (or equivalently, lower detector absorption) can be obtained by the reduction of the doping.It was shown that the specific detectivity could be significantly increased by reducing the doping of a quantum well infrared photodetector and combining it with a photonic crystal slab resonant cavity [20].The benefit of lower doping is the reduction of the noise current density.In QCDs, a lower doping density increases the resistance, but according to Eq. 2, an c times higher resistance increases the specific detectivity by only √ c.With the diagonal-transition QCD, another means to control the absorption coefficient has been introduced, namely the diagonality of the optically active transition.The bandstructure of the diagonal-transition QCD that has been used for this analysis is depicted in figure 1 green region), the overlap integral and the dipole matrix element can be tuned.A thicker barrier increases the extraction efficiency and gives a higher electrical resistance [21].This yields a performance improvement despite the lower absorption coefficient of the diagonal-transition QCD, even without a resonant cavity.In contrast to the doping density, both the responsivity and the specific detectivity are directly proportional to the extraction efficiency.A number of simulations for varying doping density and barrier thickness was performed, using a single-particle Monte-Carlo transport simulator for quantum cascade devices [22].The following transport mechanisms were included: acoustic deformation potential, optical deformation potential, polar optical phonons, alloy scattering and interface roughness scattering.Using the obtained scattering rates, the resistance, the responsivity and the specific detectivity was calculated.The resistance was calculated based on the model from [23].A variation of the doping density generates a change of the bandstructure through band-bending, that can be compensated by a careful quantum design.This can be done in the range of sheet doping density used in this analysis.At even higher doping densities the relationship between extraction efficiency, resistance and sheet doping is strongly non-linear and the figures of merit deteriorate.That is why we limited the sheet doping density to n 2D = 10 12 cm −1 .We chose to perform the simulations without self-consistency for rapid prototyping of designs.Of course, for the design of growth structures, self-consistent simulations are necessary.In Fig. 2, the absorption coefficient α, the extraction efficiency p e and the resistance R 0 of the QCD are depicted as a function of the doping density n 2D and the barrier thickness d.The range of the barrier thicknesses was from 1-5 nm.With this we obtained dipole matrix elements of the optically active transition of 2.67-0.45nm.Arbitrary values of α can be reached by a variety of combinations of n 2D and d.Each different set of n 2D and d for a given α has a different extraction efficiency and resistance.Generally, the resistance and extraction efficiency are higher for designs with lower absorption.

Quality factor optimization
To find the optimal performance in terms of either responsivity or specific detectivity we have to formulate an equation that describes the dependence of the absorption efficiency η on the absorption coefficient α of the QCD, which is valid regardless of the used cavity.To obtain the results for a particular example, one needs to obtain the different Q-factors that describe the cavity and use it in the equation.The temporal coupled mode theory [24] is a powerful tool that can be used to evaluate such a problem.We obtain the following equation for the absorption efficiency: where Q c is the so-called cavity Q-factor that describes the coupling of the resonator mode to freespace radiation, Q d the detector Q-factor that describes the intersubband absorption strength of the QCD material, Q l the loss Q-factor describing the total loss of the device and ω 0 the resonance frequency of the cavity mode.The spectral width of a mode with a given Q-factor can be obtained using ω FW HM = ω res /Q.We assume in this analysis a constant frequency ω = ω 0 , such that the frequency dependent term in Eq. 5 vanishes.With this equation we evaluate the impact of variations of the different Q-factors on the performance of the device.To obtain the behavior of the resonant cavity photodetector under ideal circumstances, we initially neglect all losses and set 1 Q l = 0. We can calculate the responsivity and specific detectivity as a function of n 2D and d, by combining Eq. ( 5) with Eqs. ( 1) and ( 2) and applying it to the values from Fig. 2.
Figure 3 shows the responsivity and detectivity as a function of n 2D and d for a coupling strength of Q c = 200.The maximum of the responsivity for each fixed doping density occurs at the critical coupling condition, i.e. when η is maximum or, equivalently, Q c = Q d .If we look at the responsivity at a higher doping density, its maximum occurs at a higher barrier thickness, because a thicker barrier is necessary to maintain the same detector absorption.However, the maximum of the specific detectivity does not occur at the critical coupling condition.For a fixed doping density the maximum of the detectivity is at thicker barriers than the responsivity.The reason is the inclusion of the resistance in its definition, which is larger for thicker barriers.Generally formulated, the maximum of the specific detectivity occurs at a lower value of Q d than Q c .For different values of Q c , we will find the maxima at different values of n 2D and d, or equivalently at a different Q d .Also, the magnitude of the overall maxima will change depending on Q c .
The steep initial increase of the responsivity at low Q c comes from the increasing extraction efficiency.At higher values of Q c , it converges to the maximal achievable responsivity for η = 0.5 at λ = 8 µm and for the chosen number of 30 periods.The specific detectivity has an additional √ R 0 term in its definition.For higher detector Q-factors Q d , the resistance is also increasing.That means, without losses, there is no theoretical limit for the specific detectivity.
Q d , the specific detectivity gets arbitrarily large.For each value of Q c and given Q l , Q d was optimized for η.The optimum occurs at the critical coupling condition, which changes to 1 An increasing loss imposes an upper limit for reasonable detector Q-factors.
In reality, losses of different origins impose an upper limit to reasonable Q-factors.The most dominant loss mechanisms originate from the finite dimensions of the cavity (often referred to as the horizontal Q-factor) and free carrier absorption in the waveguide.The impact of loss in the system is evaluated by a combined loss Q-factor, that is obtained by using the following summation 1 where Q i are the loss Q-factors from N different sources.The horizontal Q-factor is strongly dependent on the device geometry.In the case of a photonic crystal cavity it is higher for a larger number of photonic crystal periods.Using an electromagnetic solver [25], we estimated the horizontal Q-factor by exciting a two-dimensional photonic crystal mode and obtain the evanescent decay of the mode.We calculated for a 100 × 100 µm 2 device a horizontal Q-factor of approximately 2000 at a wavelength of 8 µm, which can be increased further by making the resonator larger.Typical waveguide losses of dielectric slabs are in the range of 5 − 20 cm −1 , which corresponds to Q-factors in the range of Q l = 10 3 − 10 4 .Figure 5 shows the absorption efficiency as a function of Without loss, the maximum absorption efficiency is independent of Q c .An increasing loss gives a decreasing upper limit for reasonable detector Q-factors.
To further investigate the consequences of a mismatch between optimally designed Q-factors and a possible deviation, Fig. 6 shows the absorption efficiency of four devices, each with a different value of Q c , versus the detector Q-factor Q d .If no loss is included, the absorption efficiency of each curve has a maximum at Q c = Q d .On a logarithmic x-scale, all peaks have the same shape and are symmetric, centered around Q c .For a factor of two between optimal Q d and actual Q d , the absorption efficiency is still at approximately 90 % of its peak value, independent of Q c .The inclusion of loss reduces the maximum absorption efficiency.However, the curve retains its shape, thus giving the same impact of a Q-factor mismatch as without losses.
The maximum performance of a resonant cavity QCD is evaluated for a range of values of total loss Q-factor in Fig. 7.At zero loss, the responsivity converges to the maximum achievable responsivity of 107 mA/W, which is obtained from Eq. 1 with η = 0.5, p e = 1 at λ = 8 µm and 30 periods.By making 1/Q l larger, i.e. increasing the total losses, the maximum responsivity is obtained at lower values of 1/Q d and 1/Q c .When Q l has the same order of magnitude as Q d , Qc = 200 (b) Fig. 6.Absorption efficiency for different values of Q c versus the QCD Q-factor, without any losses.The absorption efficiency gets maximal for Q c = Q d .For a mismatch of a factor of two the absorption efficiency is still at approximately 90 % of its peak value.When loss is included the limit of the absorption efficiency decreases for higher losses, but the shape remains, giving the same impact of a Q-factor mismatch as without losses.
Fig. 7. Optimum responsivity and specific detectivity for this QCD design, versus the cavity Q-factor Q c for different Q l .For each value of Q c , Q d (n 2D , d) was optimized for R and then again for D * j .For increasing Q l , the responsivity converges against the maximum achievable responsivity of 107 mA/W for η = 0.5 at this wavelength and 30 periods at T = 300 K.For lower Q l , the optimal performance is only achieved with a QCD of equivalently lower absorption.the responsivity decreases significantly.However, the specific detectivity is proportional to the responsivity and the square root of the resistance.While the responsivity decreases for higher Q l , the specific detectivity compensates the reduction by an increase of the resistance.The maximum of the specific detectivity is found at higher values of Q d than for the responsivity.The most important consequence is, that a reduction of loss, increases the specific detectivity significantly.

Conclusion and outlook
We have presented a detailed analysis of a resonant cavity quantum cascade photodetector utilizing a diagonal-transition scheme.The diagonality of the optically active transition was introduced as new means to obtain quality factor matching between the photodetector and the resonant cavity.Neglecting losses and for high Q-factors, the maximum of the responsivity converges to its highest possible value for η = 0.5 and the chosen number of periods.The maximum of the specific detectivity gets arbitrarily large for increasing Q-factors.In reality such high values are inhibited by losses.We calculated the influence of varying losses on the device performance to obtain reasonable values for a detector Q-factor to design QCDs.The impact of a mismatch between designed Q-factors and possible deviations was analyzed.A deviation by a factor of two of Q c from its optimum value for given Q l and Q d , but keeping the correct resonance position, reduces the absorption efficiency by only 10%.The responsivity and specific detectivity was calculated for a range of loss values.The calculation predicts a possible responsivity of 95 mA/W and a specific detectivity of approximately 10 9 Jones at T = 300 K for QCDs with a diagonal optically active transition and optimized Q-factors.The optimum detectivity was obtained with a sheet carrier density of n 2D = 4.9x10 10 cm −2 and a barrier width of d = 4.8 nm.The resonance width is approximately 0.3cm −1 .

Fig. 1 .
Fig.1.Bandstructure of the diagonal-transition QCD that is used for this analysis.It is based on a diagonal-transition scheme, thus the resistance, the extraction efficiency and the absorption can be controlled by the thickness of the barrier between the active wells, indicated by the shaded green region.

Fig. 2 .
Fig. 2. Absorption coefficient α, extraction efficiency p e and resistance R 0 as twodimensional plots versus the thickness of the active barrier and the sheet doping density at T = 300 K. Standard QCDs have a typical absorption coefficient in the range of 1000 -2000 cm −1 .Such high absorption is obtained for either a low barrier thickness or a high doping density (or a combination of both), yielding low extraction efficiency and low resistance.High extraction efficiency and high resistance can only be obtained for devices with a low absorption coefficient.

Fig. 3 .Fig. 4 .
Fig. 3. Calculated responsivity and Detectivity at T = 300 K versus the barrier thickness and sheet doping density.All losses were neglected and 1Q l was set to zero.The maximum of the responsivity occurs for the so-called critical coupling condition, i.e. when the absorption efficiency is maximal or equivalently when Q d = Q c .The maximum of the specific detectivity occurs at a lower Q d .

Fig. 5 .
Fig. 5. Optimum absorption efficiency versus Q c for different values of waveguide losses.For each value of Q c and given Q l , Q d was optimized for η.The optimum occurs at the critical coupling condition, which changes to 1/Q d = 1/Q c + 1/Q l , if losses are taken into account.An increasing loss imposes an upper limit for reasonable detector Q-factors.