Frequency behavior of AlInAsSb nBn photodetectors and the development of an equivalent circuit model

: We report the frequency response of Al 0.3 InAsSb/Al 0.7 InAsSb nBn photodetectors. The 3-dB bandwidth of the devices varies from ∼ 150 MHz to ∼ 700 MHz with different device diameters and saturates with bias voltage immediately after the device turn on. A new equivalent circuit model is developed to explain the frequency behavior of nBn photodetectors. The simulated bandwidth based on the new equivalent circuit model agrees well with the bandwidth and the microwave scattering parameter measurements. The analysis reveals that the limiting factor of the bandwidth of the nBn photodetector is the large diffusion capacitance caused by the minority carrier lifetime and the device area. Additionally, the bandwidth of the nBn photodetector is barely affected by the photocurrent, which is found to be caused by the barrier structure in the nBn photodetector.


Introduction
Mid-wave infrared (MWIR) to long-wave infrared (LWIR) photonics possesses the potential for realizing next generation breakthroughs in a wide range of areas including medical diagnostics [1,2], spectroscopy [3,4], environmental monitoring [5], communication [6] and defense. Consequently, photodetectors that operate in this spectral range are attracting growing interest [7]. However, due to the narrow bandgap material used for this spectral range, the devices need to be cooled to cryogenic temperatures in order to avoid excessive dark current. The n-barrier-n (nBn) photodetector has been demonstrated as one of the most effective approaches for addressing this challenge, owing to its low dark current achieved by suppressing Shockley-Read-Hall (SRH) recombination and generation [8], which enables it to work at much higher operating temperatures compared to conventional P-I-N photodiodes [9,10]. To date, nBn photodetectors have been substantially improved based on various materials systems, such as InAs/GaSb type-II superlattices (T2SL) [11,8] and InAs/InAsSb type-II strained layer superlattices (T2SLS) [12][13][14][15]. They have also been successfully integrated onto silicon wafers to facilitate MWIR silicon photonics [16,17]. Recently, we demonstrated nBn photodetectors based on AlInAsSb digital alloy materials system [18][19][20]. This material system has the benefit of a nearly-zero valence band discontinuity [21], which is highly favorable for the hole transport required to achieve high detectivity and low turn on voltage.
Although material systems are different, much of the research on nBn photodetectors has focused on achieving lower dark current, higher quantum efficiency and longer detection wavelength. However, there have not been studies of the frequency behavior and bandwidth of nBn photodetectors except for a few pulse response reports [22,23]. With rapidly expanding applications in the MWIR and LWIR range, device speed is attracting more and more attention [24], such as for high-speed imaging [25,26], MWIR frequency combs [3,27,28] and LIDAR systems [29]. Therefore, a thorough understanding of the frequency behavior of the nBn photodetector is important.
In this paper, we investigate the frequency characteristics of the AlInAsSb nBn photodetectors reported in Ref. [18], by bandwidth, capacitance-voltage (C-V) and microwave scattering parameters (S-parameters) measurements. Subsequently, a new equivalent circuit model is developed for the nBn photodetector to further understand its frequency behavior and the limiting factors of its bandwidth. To our knowledge, this is the first detailed analysis of the frequency behavior of nBn photodetectors. This work provides a framework for understanding the bandwidth performance of nBn photodetectors, which can be used for further optimization of nBn photodetectors for high-frequency applications in the MWIR and LWIR spectral range.

Device structure and fabrication
The epitaxial structure of the Al 0.3 InAsSb/Al 0.7 InAsSb nBn photodetector is shown in Fig. 1. The structure was grown by molecular beam epitaxy (MBE) as a digital alloy of four binary constituents: AlAs, AlSb, InAs, and InSb lattice matched to an n-type GaSb substrate [31]. Details regarding the fabrication of the device are included in Ref. [18]. It is worth noting that no coplanar waveguide (CPW) pad was designed for these devices. The frequency response was measured by directly probing the top and bottom metal contacts using a ground-signal (G-S) probe.  Figure 2 shows a schematic of the experimental setup for the frequency response measurement. A CW optical signal from a temperature-stabilized 2-µm semiconductor laser was directed through a polarization controller and then modulated by a LiNbO 3 Mach-Zehnder modulator (MZM). The modulator was biased at the quadrature point and driven by a vector network analyzer (VNA). The modulated optical signal was then focused onto the device with a lensed fiber, and the photoresponse was collected through a G-S probe. The RF photoresponse was measured by the VNA using a bias-tee. The S 11 of the device was measured by the same setup without laser illumination to further study the equivalent circuit model of the nBn photodetector. Circuit and probe loss have been calibrated for the measurement.  Figure 3 shows the normalized response of the nBn photodetectors measured from 20 MHz to 3 GHz. The 3-dB bandwidth was extracted and plotted in Fig. 4. Devices with varying device diameters show a similar bandwidth saturation phenomenon, where the bandwidth initially increases with bias voltage but then quickly saturates beyond ∼ -0.5 V. It is worth noting that the bandwidth saturation voltage is exactly equal to the turn-on voltage of the nBn photodetectors as discussed in [18], which reflects the voltage needed to overcome carrier trapping [30,32]. Therefore, it appears that the bandwidth of the nBn photodetector saturates immediately after the device turns on.  Theoretically, the bandwidth of a photodiode is limited by the carrier transit time and the resistance-capacitance (RC) time constant, as expressed by the equation

Results
where f 3dB is the total 3-dB bandwidth of the device, f TR is the transit time limit bandwidth, and f RC is the RC time limit bandwidth [33]. One of the factors that can limit the bandwidth of an nBn photodetector is the long transit time due to slow carrier diffusion through the thick absorber. However, as shown by the band diagram and electric field simulation in Fig. 5, more than half of the absorber remains undepleted when the device is turned on at ∼ -0.5 V. As the depletion width and electric field extend further into the absorber, the transit time bandwidth should increase since more photogenerated carriers begin to drift rather than diffuse. To verify this, the transit time bandwidth of the nBn photodetector was simulated using the Lumerical Charge solver [34], and the result is shown in Fig. 6. As expected, the transit time bandwidth continues to increase with bias voltage even beyond the turn-on voltage. It is also worth noting that the value of the simulated transit time bandwidth is far above the measured 3-dB bandwidth. Therefore, it appears that the bandwidth of the nBn photodetector is not limited by transit time.  Another possible limiting factor is the RC-limited bandwidth. The equivalent circuit of a typical P-I-N photodiode is as shown in Fig. 7, and the RC-limited bandwidth is expressed by the equation where C j is the junction capacitance formed in the depleted intrinsic region, R j is the junction resistance which is hundreds of megaohms and can be regarded as an open circuit, and R s and R L are the series resistance and the load resistance, respectively. To study the RC-limited bandwidth of the nBn photodetector, capacitance-voltage (C-V) curves were measured using an LCR meter at a frequency of 10 kHz. Theoretical junction capacitance was also calculated by assuming a parallel plate capacitor model as expressed by equation where C is the theoretical junction capacitance, ε is the permittivity, d is the device diameter, and w d is the depletion width [35]. As shown in in Fig. 8, the measured total capacitance and the theoretical junction capacitance show good agreement. Then, by assuming a load resistance of 50 Ω and negligible series resistance, the theoretical RC bandwidth of the nBn photodetectors was calculated based on the photodiode circuit model (Eq. (2)) and the measured capacitance ( Fig. 8). However, as shown in Fig. 8, the theoretical RC bandwidth calculated using this model is significantly higher than the measured 3-dB bandwidth. The discrepancy indicates that the nBn photodetector is likely to have a different equivalent circuit model compared to a P-I-N photodiode. A new equivalent circuit model is needed to understand the bandwidth of the nBn photodetector.

Discussion
Recall in Fig. 1 that the nBn photodetector is an N-I-N structure composed of an unintentionally doped (UID) region (buffer, barrier, and absorber layers) sandwiched between two n-type contact layers. As shown in Fig. 5(a), this structure is similar to a phototransistor where one forward-biased junction is connected to a reverse-biased junction in series. Therefore, it is likely that the equivalent circuit model of an nBn photodetector is similar that of a phototransistor. As studied in Ref. [36], the equivalent circuit model of a phototransistor is shown in Fig. 9(a), and the RC-limited bandwidth is expressed by the equation where C e and C c are emitter diffusion capacitance and collector junction capacitance, respectively; R e is the diffusion resistance between the emitter and the base; and R L is the load resistance. Since phototransistors have internal gain, the left part of the equation expresses the gain-bandwidth product where G is the photogain. Although the band structures are similar, nBn photodetectors are intrinsically different from phototransistors due to the barrier layer that blocks the majority carriers from diffusing into the absorber. As a result, the multiplied photocurrent Gi p that results from these majority carriers in a phototransistor does not contribute to the total current in an nBn photodetector and therefore should be left open in the circuit model. Correspondingly, the gain-bandwidth product f RC · (G + 1) in Eq. (4) should be reduced to the RC bandwidth f RC . Based on these modifications, the equivalent circuit model of an nBn photodetector is shown in Fig. 9 . 9. Equivalent circuit model of (a) an N-P-N phototransistor [36] and (b) an nBn photodetector the RC bandwidth expressed as The lumped elements in the equivalent circuit model of the nBn photodetector serve similar roles as in a phototransistor, but physically they originate from different sources and need to be discussed and re-defined as follows. Correspondence between the band structure and the equivalent circuit model is schematically plotted in Fig. 10(a) to make the following discussion more understandable.
C diff : diffusion capacitance formed by the stored charge near the bottom edge of the barrier layer. In a typical N-P-N phototransistor, the emitter diffusion capacitance exists due to the fact that the injected minority carriers accumulate under forward bias voltage. This leads to charge stored near both sides of the depletion region and forms a diffusion capacitance [36]. In an nBn photodetector, there is no p-type base region but a UID barrier layer instead. Nevertheless, the barrier layer has a similar effect on creating a diffusion capacitance. Since the barrier layer has a large conduction band offset, electrons that diffuse from the bottom contact layer to the absorber will be blocked. As shown in Fig. 10(b), blocked electrons accumulate near the bottom edge of the barrier layer. Since it takes time for the stored electrons to recombine with holes, these carriers create a diffusion capacitance. It is worth noting that the knik around 0.8 µm is indicating the edge of the depletion region.
R diff : diffusion resistance of the forward biased junction between the bottom contact later and the UID region. In a typical N-P-N phototransistor, the diffusion capacitance is associated with the slope of current-voltage (I-V) curve for the forward-biased emitter-base junction [36]. As shown in Fig. 10(a), the junction between the bottom contact layer and the UID region in an nBn photodetector is forward biased and therefore the diffusion resistance in an nBn photodetector is similar to that of an N-P-N phototransistor.
C j : junction capacitance in the depleted absorber under reverse bias. This is similar to the junction capacitance in a P-I-N photodiode. According to the equivalent circuit model in Fig. 9(b), the total capacitance C measured with an LCR meter is a series connection of C j and C diff if the measuring frequency is low: where C is the total measured capacitance. We found that the total capacitance C measured with an LCR meter stabilizes when the frequency is lower than ∼ 100 kHz. As shown in Fig. 8, the measured total capacitance at 10 kHz fits well with the theoretical junction capacitance. This agreement indicates that C diff is likely to be much larger than C j . Accurate characterization of C j and C diff will be addressed by the S-parameter fitting and will be discussed in the next section. R s : series resistance of the device. R L : load resistance in the measurement setup, set as 50 Ω in this work.
To accurately determine the value of each lumped element in the new equivalent circuit model, S 11 parameters of the devices were measured. Figure 11 shows the measured S 11 parameters of the nBn photodetectors with different device diameters at various bias voltages, plotted on Smith Charts from 20 MHz to 5 GHz.
The measured S 11 parameters show a similar saturation phenomenon as the bandwidth measurement, where S 11 curves converge quickly after the turn-on voltage (∼ -0.5 V). At the same time, a strong device area dependence also appears when comparing Fig. 11(a)-(d), where larger devices are clearly more capacitive. To fit each lumped element, S-parameter fitting was performed using Keysight Advanced Design System (ADS) [37] based on the equivalent circuit model shown in Fig. 9(b). As shown in Fig. 12, simulated S 11 curves show good agreement with the measured results. The fitted parameters at -2 V are summarized in Table 1 for different device diameters. Fitted parameters at other voltages are not listed since the measured S 11 parameters converge beyond the turn-on voltage. To further verify the circuit model, the theoretical 3-dB bandwidth of the nBn photodetectors was calculated using Eq. (5) and the fitted parameters in Table 1. As shown in Table 2, the theoretical 3-dB bandwidth also agrees well with the measured results.  So far, the equivalent circuit model developed for the nBn photodetectors fits well with both the bandwidth and S 11 measurements. However, it is worth noting that the fitted parameters in Table 1 show a different trend compared to a phototransistor, i.e., the diffusion capacitance of an nBn photodetector is proportional to the device area. For a typical N-P-N phototransistor, diffusion capacitance is proportional to photocurrent and carrier lifetime as expressed by equation where I ph is the photocurrent and τ is the minority carrier lifetime [38]. It is evident that C diff is independent of device area because the diffusion capacitance is determined by the total current flow through the forward bias junction, which is continuous with the photocurrent in a phototransistor [39]. However, nBn photodetectors have a barrier layer that blocks the diffusion of electrons into the absorber. As a result, the electron current is no longer continuous across the whole device; the electron current between the bottom contact layer and the barrier layer is the diffusion current of the majority carriers and should be much larger than the minority electron current in the absorber. On the contrary, hole current is still continuous since no barrier exists on the valence band. Based on this new current continuity condition, the diffusion capacitance and resistance of the nBn photodetectors can be qualitatively explained as follows. As shown in Fig. 10 (a), assume voltage applied on the reversed biased absorber (junction 1) is V 1 , and the electron and hole currents flowing through the junction are I n 1 and I p 1 , respectively. Likewise, voltage applied across the forward biased junction between the bottom contact layer and the UID region (junction 2) is V 2 , and the electron and hole currents flowing through the junction are I n 2 and I p 2 , respectively; photocurrent is I ph . The hole current through the entire device is continuous. As a result, kT − 1) = I p 1 = I ph (8) where I p 0 is the dark saturation hole current, q is absolute value of electron charge, k is Boltzmann's constant and T is absolute temperature. Therefore, V 2 can be expressed as Diffusion resistance of a forward biased junction is defined as the derivative of the voltage to current [38], therefore R diff is expressed as which shows that R diff is inversely proportional to photocurrent I ph . Diffusion capacitance in a forward biased junction is expressed by Eq. (7). However, since electron current in an nBn photodetector is not continuous, current blocked at the barrier layer is no longer equal to photocurrent but is a majority current from the forward biased junction 2, and the electron current density J n 2 is where J n 0 is the dark saturation electron current density. It is worth noting that V 2 is still determined by the photocurrent, as shown in Eq. (9), to make the hole current continuous. Therefore, by substituting Eq. (9) into Eq. (11), we get J n 2 ≅ J n 0 I ph I p 0 ∝ I ph (12) which shows that electron current density at junction 2 is proportional to photocurrent. The total electron current at junction 2 should be electron current density multiplied by the device area. Therefore, the expression of the diffusion capacitance in Eq. (7) should be modified to where A is the area of the device. Since electron current density J n 2 is proportional to I ph , C diff can be further modified by substituting Eq. (12) to be C diff ∝ I ph Aτ (14) which shows that the diffusion capacitance in an nBn photodetector is proportional to photocurrent, carrier lifetime, and device area. In summary, the area dependence of the diffusion capacitance in an nBn photodetector originates from its barrier structure in the conduction band. As a result, the electron current is no longer continuous through the entire device, and therefore the electron current blocked at the barrier layer is not equal to the photocurrent (the minority current in the absorber). Instead, it is a majority current from the forward biased junction 2 that is proportional to the junction area and therefore results in the area-dependent diffusion capacitance. It is worth noting that photocurrent still affects the diffusion capacitance. However, it is not due to the photocurrent limiting the majority electron current blocked at the barrier layer but results from the photocurrent determining the voltage applied on the forward biased junction 2 to make the hole current continuous. Therefore, the photocurrent affects the majority electron current blocked at the barrier layer indirectly. This process is similar to the origin of the photogain in a phototransistor, but the difference in an nBn photodetector is that the majority electron current from the forward biased junction no longer results in photogain. Instead, it is blocked by the barrier layer, stores charges there, and ultimately increases the diffusion capacitance.
After deriving the expression of the diffusion resistance and diffusion capacitance, the limiting factor of the bandwidth of the nBn photodetector can be deduced. As shown in Table 1, the diffusion capacitance in an nBn photodetector is significantly larger than the junction capacitance. Therefore, the limiting factor of the bandwidth, as shown in Eq. (5), should be the product of R diff and C diff . By substituting Eqs. (10) and (14) into Eq. (5), the bandwidth expression of nBn photodetector can be simplified to It is worth noting that the influence of the photocurrent is canceled in Eq. (15), indicating that the bandwidth of an nBn photodetector should be unaffected by photocurrent level. This conclusion is clearly different from a phototransistor whose bandwidth is inversely proportional to the total photocurrent-described as the gain-bandwidth product-since the total photocurrent in a phototransistor is multiplied by the photogain [36]. To verify this conclusion, the frequency response of the AlInAsSb nBn photodetectors with diameters of 80 µm, 100 µm, 150 µm, and 200 µm are plotted in Fig. 13(a) to 13(d), respectively at different photocurrent levels. As shown in Fig. 13, with a photocurrent that is two times higher, the frequency response curves of the AlInAsSb nBn photodetectors are almost the same, which agrees well with the prediction from Eq. (15). Equation (15) also suggests possible means for improving the bandwidth of nBn photodetectors. The most straightforward method is to reduce the device area, which can reduce the majority electron current that diffuses to the barrier layer and therefore reduce the diffusion capacitance. The other possible method is reducing the minority carrier lifetime, which results in fast recombination of the accumulated carriers and therefore reduces diffusion capacitance. However, there can also be a negative aspect to this since long carrier lifetimes are preferred in an nBn photodetector in order to maintain enough diffusion length [9,18]. Consequently, reducing carrier lifetime may result in lower quantum efficiency if the diffusion length drops below the absorption length. Therefore, although the principles behind the bandwidth characteristic of an nBn photodetector and a normal-incidence photodiode are different, they have a similar intrinsic problem, i.e., the trade-off between the bandwidth and the quantum efficiency. Therefore, approaches such as waveguides [40][41][42] and surface photon trapping structures [43][44][45] that can improve the quantum efficiency without requiring a longer absorption length may be promising methods for achieving higher efficiency-bandwidth products for nBn photodetectors.

Conclusion
We report the frequency response of Al 0.3 InAsSb/Al 0.7 InAsSb nBn photodetectors. The 3-dB bandwidth shows a strong RC time limit and saturates with bias voltage right after the device turn on. A new equivalent circuit model is developed for nBn photodetectors due to the discrepancy between their frequency behavior and the conventional photodiode equivalent circuit model. The theoretical bandwidth predicted by the new equivalent circuit model and the S-parameter fitting agree well with the measured results. Subsequent analysis further reveals that the limiting bandwidth factor of nBn photodetectors are carrier recombination lifetime and device area. Additionally, the bandwidth of nBn photodetectors is barely affected by photocurrent level, which is shown to result from the barrier structure in the conduction band. This work provides valuable insights into optimizing the frequency response of nBn photodetectors, which enable the expansion of its applications in high-speed scenarios in the MWIR and LWIR spectral range. Disclosures. The authors declare that there are no conflicts of interest related to this article. Data availability. Data underlying the results presented in this paper are available from the corresponding author upon request.