Electron beam induced current (EBIC) measurements provide a powerful tool for characterizing semiconductor based materials and devices. By measuring the current generated by the electron beam of a scanning electron microscope (SEM), EBIC allows us to extract the minority carrier diffusion length (*L*) and the surface recombination velocity to diffusivity ratio (*S/D*) of a material. When combined with information on minority carrier lifetime (*τ*), for instance from time-resolved photoluminescence measurements, the minority carrier mobility of the material can be extracted. However, the EBIC technique relies heavily on the accurate modeling of the carrier generation and collection process. Traditionally, this was achieved using a combination of empirical analytical expressions (and later Monte Carlo simulations) for carrier generation and analytical diffusion/recombination expressions for carrier collection. This approach introduces significant uncertainties into the extracted material parameters. Here, we present a numerical approach to EBIC modeling which improves the spatial resolution of our model, while also retaining information regarding the relative EBIC signal as a function of incident beam energies and currents. We apply this technique to investigate the temperature dependent minority carrier mobility of InAs/InAsSb and InGaAs/InAsSb strained layer superlattice infrared detectors and compare our results to the values obtained using external quantum efficiency measurements of the same samples. Our approach not only allows for an improvement in the uncertainty of the extracted material parameters, but also offers insight into the material and device behavior as a function of nonequilibrium carrier concentration. The technique presented here offers potentially improved characterization of not only infrared detectors, but a range of semiconductor-based devices.

## I. INTRODUCTION

Electron beam induced current (EBIC) measurements have been used extensively to characterize the carrier dynamics of semiconductor materials and devices.^{1–8} In the EBIC technique, a beam of high energy electrons (typically in a scanning electron microscope, or SEM) is directed at the surface (or exposed cross-section) of a semiconductor device. The high energy electrons generate electron hole pairs (EHPs) which can then be collected at the device contacts, and the current generated by the electron beam is measured as a function of beam position. The EBIC measurement thus generates a “current image” of the device, as opposed to the secondary electron emission image typically obtained in an SEM. In principle, the extremely small electron beam spot size offers the potential for spatially resolved information of material parameters in active devices. This technique has been applied over the past few decades to characterize electronic and opto-electronic devices, providing valuable information on the electronic properties of the device material.^{6,8} In comparison to the closely related XBIC^{9} and LBIC^{10} measurements (x-ray- and laser- beam induced current, respectively), EBIC offers more accurate material parameter extraction for materials with short diffusion lengths^{11} as well as straightforward integration with scanning electron microscopes. In practice, however, the EBIC technique has limitations. Material parameters are most typically extracted by fitting the experimentally obtained EBIC data to a model which uses numerical techniques (Monte Carlo simulations) to determine the carrier generation profile,^{12} combined with an analytical model of carrier collection,^{4,7,8} which together give a predicted current vs. position plot for the device under test. Analytical integration of the product of the carrier generation profile and the carrier collection probability, however, requires the use of analytical fits to the carrier generation profile, fits which do not accurately reflect the numerical simulations, resulting in a loss of spatial resolution for the modeled EBIC, and poorer fits to the data. In addition, the most frequent examples of EBIC modeling fit normalized experimental data and modeled results (with fitting parameters of diffusion length *L* and surface recombination velocity to diffusivity ratio, *S/D*) for a range of electron beam energies.^{6–8} By normalizing both experimental and modeled data, these approaches look to fit only the shape of the EBIC signal, and omit valuable information obtained from the relative magnitude of the EBIC signal as a function of beam currents and energies.^{13} This results in uncertainty in the extracted parameters, with broad ranges of *L* and *S/D* offering similar fits to the experimental data, thus weakening the significance of the extracted data. For EBIC measurements on bulk materials and large areas or cross-sections with weak surface recombination and long diffusion lengths, these uncertainties are minimized, but this is not the case for more complicated devices, having shorter active regions, multiple material layers, and/or significant surface recombination. By retaining both the shape and the magnitude of the EBIC signal, improvements to the fit of the EBIC data, as well as improvements to the uncertainty in the extracted data, can be achieved. In addition, the comparison of excited EHP densities (which can be obtained from the beam energy and current) and the magnitude of the collected current can provide additional insight into carrier dynamics in devices, potentially offering the opportunity to observe transitions where carrier lifetimes are changing as a function of excess carrier concentration. Thus, it is conceivable that a new approach to EBIC modeling, which takes into account not only the shape of the EBIC data, but also its relative magnitude, would offer the potential to realize the full capability of EBIC.

One example of the more complex devices mentioned above is those leveraging strained layer superlattice (SLS) material systems, such as InAs/GaSb, InAs/InAsSb, and InGaAs/InAsSb, which have attracted significant interest over the past few decades due to their potentially superior performance in detecting mid-wave or long-wave infrared (MWIR or LWIR) light.^{14–27} Compared to the already commercialized state-of-the-art mercury cadmium telluride (MCT) detectors or quantum well infrared photodetectors (QWIPs), SLS detectors have competitive advantages such as a theoretically higher operating temperature, a suppression of Auger recombination, and an ability to control the detectors' effective bandgaps by engineering layer thicknesses in a binary system of ternary or quaternary alloys.^{14,17} However, the theoretically superior performance of SLS detectors is yet to be demonstrated experimentally or commercially, with material defects and growth imperfections considered as the major limiting factors for these infrared detector material systems.^{23,28,29} Improving the material quality of SLSs requires techniques to characterize and understand the carrier dynamics of this material. However, characterizing material quality, and understanding the effect of material quality on device operation, for the narrow effective bandgap SLSs, often consisting of hundreds of alternating layers of alternating group V materials, and the potential concerns regarding intermixing, strain, and defect formation, requires a multi-pronged approach. Invaluable information can be gleaned from electron microscopy^{28–30} and optical and electronic material and device characterization techniques.^{21,23,25,26} In addition, various techniques are already being utilized for measuring the minority carrier lifetime of SLSs.^{19,25} However, no viable technique has been reported so far to measure the vertical mobility, which is the other most important parameter for SLS detectors. EBIC offers a potential approach for characterizing both material quality and device operation, measuring the minority carrier diffusion characteristics of SLS materials, and has been utilized as a valuable supplemental characterization tool for understanding material quality, for instance in InAs/GaSb SLS detectors as a function of interfacial layers^{31} and for the promising Ga-free InAs/InAsSb T2LS material system.^{32} The latter SLS material system has attracted growing interest resulting from demonstrated reductions in dark currents and longer minority carrier lifetimes compared to the early versions of SLS detectors employing the InAs/GaSb material system, hypothesized to result from the absence of native defects associated with Ga-associated defects in the GaSb layers.^{19,22,24} Recently, the introduction of Gallium into the InAs layer of InAs/InAsSb SLSs has been proposed as a possible improvement to SLS detector active region design, and has been experimentally shown to improve the detector absorption coefficient due to increased overlap of electron and hole states in the superlattice.^{33} In addition, the InGaAs/InAsSb SLS material system, when compared to the Ga-free InAs/InAsSb material system, is theoretically expected to show higher vertical hole mobility due to reduced hole effective mass. However, the diffusion length and vertical carrier mobility in these SLS structures, key parameters for understanding carrier dynamics and potential device performance in the material system, are yet to be investigated.

In this work, we investigate the effect of the Ga content in In(Ga)As/InAsSb SLS detector devices using a combination of time-resolved photoluminescence (TRPL) and EBIC measurements, leveraging a new approach to EBIC parameter extraction. We demonstrate that the previous approaches to EBIC modeling leave uncertainty in the fitting parameters for our SLS materials, and introduce a new numerical approach to EBIC modeling which improves the spatial resolution of our model and reduces the uncertainty in our extracted fitting parameters, modeling not only the EBIC lineshape as a function of position, but the magnitude of the EBIC response. We use the latter to better understand the effects of nonequilibrium carrier concentration on the minority carrier transport properties, and discuss the limitations of EBIC associated with changes in minority carrier lifetime at higher nonequilibrium carrier concentrations. We apply our developed numerical EBIC modeling technique to understand the behavior of our In(Ga)As/InAsSb SLSs and discuss the agreement and discrepancies between our developed model and experimental results. The extracted fitting parameters are compared to those extracted using external quantum efficiency measurements performed on the same material with good agreement.

## II. SAMPLE GROWTH AND EXPERIMENTAL SETUP

Three In(Ga)As/InAsSb based nBn infrared photodetectors, grown by molecular beam epitaxy (MBE) on GaSb substrates, were investigated in this work. From the bottom to top, our devices consist of a GaSb buffer layer, a 2 *μ*m n-type SLS absorption layer (n-doped $2\xd71016\u2009cm\u22123$), an undoped 200 nm AlGaAsSb electron-blocking barrier layer, and a 200 nm n-type SLS top contact (n-doped $2\xd71016\u2009cm\u22123$). The three different SLS designs are: 17 ML InAs/5.5 ML InAs_{0.65}Sb_{0.35} (0% Gallium content), 13 ML In_{0.95}Ga_{0.05 }As/6.5 ML InAs_{0.65}Sb_{0.35} (5% Gallium content), and 8.5 ML In_{0.80}Ga_{0.20 }As/9ML InAs_{0.65}Sb_{0.35} (20% Gallium content), each designed to be strain balanced and also to have similar effective bandgaps. Figure 1(a) shows the schematic of the layer structure and device geometry of the SLS devices investigated in this work, while Figs. 1(c)–1(e) show the band diagram of two periods of the InAs/InAsSb, In_{0.95}Ga_{0.05 }As/InAsSb, and In_{0.80}Ga_{0.20 }As/InAsSb designs, respectively.

Samples were initially characterized with a Bruker v80V Fourier transform infrared (FTIR) spectrometer, using mid-IR photoluminescence (PL) spectroscopy in an amplitude modulation step-scan experiment. Figure 1(b) shows the low temperature (80 K) PL spectra for each sample, indicating similar cut-off wavelengths for the three SLSs detectors investigated in this work. The same samples were also characterized using temperature dependent time-resolved photoluminescence (TRPL). Here, the as-grown samples were pumped with a Q-switched diode-pumped laser emitting ∼1 ns pulses at λ=1064 nm, with 10 kHz repetition rate and varying pulse energies (controlled by neutral density filters at the laser output). The light emitted from the samples is collected with a parabolic mirror, and focused onto a high-speed MCT detector (Kolmar Technologies) using a Ge lens (which also serves, along with a low-pass 3.6 *μ*m filter, to block the scattered pump laser light). The output of the MCT detector is collected using a 14-bit LeCroy oscilloscope and the tail of the PL emission for all pulse energies is modeled using a single-exponential fit in order to extract the low-injection minority carrier lifetime.^{34} TRPL data are collected from all samples for temperatures from 80 K to 200 K.

The EBIC measurements were carried out on detector devices whose growth is described above, fabricated into mesa structures with solid, continuous metal contact pads, as shown in Fig. 1(a). For the EBIC measurements, the fabricated devices are cleaved through the top contact and mesa, and then mounted in the chamber of a JEOL 7000F scanning electron microscope (SEM) such that the cleaved surface is positioned normal to the SEM's electron beam [Fig. 2(a)]. The sample substrate is grounded to the SEM mount, and the top and bottom contacts to the detector are wire-bonded to ceramic stand-offs, connected to BNC cables. The current collected across top and bottom contacts of the device is amplified using a Stanford Research SR570 preamplifier connected via electrical feedthroughs in the SEM. The output of the pre-amplifier is fed into the SEM DigiScan control software and a “current image” of the sample is generated. It is important to note that this current image includes not only the EBIC signal, but also a DC dark current from the sample. This dark current (measured in the EBIC image as the current far from the collection junction) is removed from the final EBIC signal with a uniform background subtraction. The remaining EBIC image is averaged [parallel to the growth plane, along the y-direction of Fig. 2(a)] to produce an EBIC profile as a function of beam position in the growth direction. Each device is measured at temperatures from 80 K to 200 K with beam energies ranging from 10 keV to 30 keV in 5 keV increments. The experimental EBIC profile is compared to the modeled profile, which allows for an extraction of the minority carrier diffusion length, *L*, and the surface recombination velocity to diffusivity ratio, *S/D*.

Our EBIC data are compared to values for the vertical hole mobility extracted from external quantum efficiency (EQE) measurements. First, the EQE of fully reticulated single element detectors at different temperatures was measured^{33} following a standard radiometric characterization technique described in Ref. 35. The variation of the theoretically expected quantum efficiency with the diffusion length ($Lh$) for the same set of detectors was analytically calculated as described in Ref. 36, using the experimentally determined absorption coefficient from the detector materials, measured using the technique described in Refs. 37 and 38. The diffusion length values at different temperatures were then extracted by fitting the experimental EQE data points at corresponding temperatures.^{36}

## III. EBIC MODEL

The power of the EBIC technique is somewhat offset by the significant modeling required to extract meaningful values from the experimental data. The EBIC signal, for a given electron beam position on the cleaved surface, is proportional to the minority carrier collection efficiency of the device junction. In the idealized picture of EBIC, an extremely narrow electron beam acts as a point source generator of electron hole pairs (EHPs), and thus for each beam position one can assume a singular collection efficiency. In reality, the small spot size, high energy beam of electrons scatters upon reaching the material surface, and thus results in an EHP generation volume extending both laterally from the beam position (x, along the growth direction and y, along the growth plane) and into the sample (z, depth from cleaved surface). Figure 2(a) shows the orientation of the electron beam on the SLS infrared detector. We define the probability of collection ($\phi (x,y,\u2009z)$), as the probability that a hole generated at point $(x,z)$ will diffuse to the detector junction and be collected as a photocurrent. The collection probability for EHPs generated at different x positions (for the same position of the electron beam, $xo$) will vary, depending on their relative proximity to the junction. The EHPs' probability of collection will also vary as a function of depth into the sample ($z$), due to surface recombination effects. Finally, $\phi (x,y,\u2009z)$ can also depend on *y*, for EBIC measurements close to detector mesa sidewalls, on samples with metal aperture contacts, or for samples with inhomogeneous defect densities (on a length scale on the order of the width of the carrier generation volume). The efficiency of collection, $\eta xo$, is effectively a measure of the EBIC current generated by the EHP generation distribution associated with a beam position $xo$, and can be calculated by taking the volume integral of the product of the probability of collection, $\phi (x,z)$, and the electron hole pair (EHP) generation function, $gEbx\u2212xo,y,zEHPs\u2212cm3$, where the “$Eb$” subscript indicates the fact that our EHP distribution depends on the electron beam energy. The solid metal contact pads, the location of our EBIC measurement (far from the mesa sidewalls), and the fact that we see consistent EBIC signals across the lateral range (y-direction) of the sample cross-sections enable us to approximate the probability of collection in our measurement as having no y-dependence, such that we can integrate our three-dimensional (3D) generation function to give a two-dimensional (2D) generation function $hEbx\u2212xo,z=\u222b\u2212\u221e\u221egEbx\u2212xo,y,zdy$. For a sample using metal contacts with apertures for light transmission, or with defect spacing on the order of the width of the generation function, the full 3D generation function would need to be used. At each beam position ($xo$), we solve the integral

where $x=0$ denotes the position of the collection junction (at the barrier/absorber interface) and $z=0$ the cleaved surface of the device. The 2D generation function $hEbx\u2212xo,z$ is determined numerically using CASINO Monte Carlo software^{12} simulations for each position ($xo$) across the growth direction of the device, while the probability of collection, $\phi x,z$, can be derived from the diffusion equation.^{8}

In previous approaches to EBIC modeling, the normalized 2D carrier generation function $hEbx\u2212xo,z$, is analytically expressed as shown in Eq. (2) (Ref. 8)

where *H* is the normalization factor, σ_{1} represents the spread of the generation profile in the $x$ direction, and $\sigma 2$ is the spread of the profile in the $z$ direction (both of which have empirical dependence on beam energy). Equation (2) is used to fit the normalized Monte-Carlo simulated generation distribution, with the fitting performed in the $x$ and $z\u2009$ directions separately. Figures 2(b) and 2(c) show the Monte Carlo-simulated EHP generation profiles in the *x* and *z* directions, respectively, for incident electron beam energies of 10 and 15 keV, for the electron beam impinging at $xo=1\u2009\mu m$ (with $x=0$ again being the barrier/absorber interface). The analytical fitting functions ($hEbx\u2212xo,z$) for each beam energy are plotted as dashed lines in Figs. 2(b) and 2(c). Comparison between the numerically simulated generation distribution and the analytical fitting functions shows good agreement in the $z$ direction, into the sample. However, the analytical expression for the *x*-dependence (along the growth direction) of the generation profile does not accurately capture the strength of EHP generation near the position of the incident electron beam ($xo$). To quantify this, we compare the percentage of EHPs generated within the range $xo\xb140\u2009nm$ for both the analytical and numerical generation distributions as a function of beam energy. At low beam energies (10 keV), the two approaches are similar (62% vs. 61%), but as beam energy increases, the analytical approach diverges from our numerical results: 37% vs. 30%, 27% vs. 19%, 21% vs. 13%, and 17% vs. 10% (numerical vs. analytical) for 15, 20, 25, and 30 keV beam energies, respectively. In addition, the normalization of the beam profiles removes the ability to model the relative amplitude of the EBIC signal as a function of beam energy and current.

Once the generation distribution is calculated, the diffusion equation, describing the transport of beam-generated minority carriers, is used to determine the probability of collection, $\phi (x,z)$.^{7,8} Originally, this expression was derived for EBIC modeling of bulk p-n junction diode devices^{2,3,5} and later also used to model SLS p-n junction detectors.^{8} The probability of collection expression for SLS nBn detectors must be adjusted somewhat, and is given below in Eqs. (3a)–3(d)

*k*, depends entirely on the minority (hole) carrier diffusion length, $Lh$, and the surface recombination velocity to diffusivity ratio, $Sh/Dh$. Here the

*k'*s denote discrete solutions to the transcendental equation governing the z-dependence of the probability of collection, which for thick materials ($z\u226bL$) becomes continuous, resulting in the integrals of Eq. (3), as detailed in Ref. 8. In Figs. 2(d)–2(f), we show contour plots of the probability of collection in our SLS structures for various $Lh$ and $Sh/Dh$ combinations: $Lh=0.5\u2009\mu m$ and $Sh/Dh=1\u2009\mu m\u22121$, $Lh=0.5\u2009\mu m$ and $Sh/Dh=10\u2009\mu m\u22121$, and $Lh=1\u2009\mu m$ and $Sh/Dh=1\u2009\mu m\u22121$, respectively. From these plots, one can clearly see the effect of higher $Sh/Dh$ values, significantly reducing the probability of collection at the cleaved surface ($z=0\u2009\mu m$), as more of the generated EHPs recombine via surface states before they can be collected. As expected, increases in $Lh$ broaden the probability of collection towards the substrate.

As described in Eq. (1) above, the integral of the product of the 2D EHP generation function, $hEb(x\u2212xo,z)$, and the probability of collection, $\phi (x,z)$, over the *x-z* plane, returns the efficiency of collection, $\eta (xo)$. The use of the analytical expression for the EHP generation function given in Eq. (2) and the probability of collection expressions derived from the diffusion equation shown in Eq. 3 allows for an analytical expression to be derived for the collection efficiency $\eta Eb(xo)$,^{8} significantly simplifying the EBIC modeling process. However, because both the EHP generation function and the probability of collection are typically normalized, the resulting $\eta Eb(xo)$ does not reflect the relative changes in $\eta Eb(xo)$ with changing beam energy and current. In addition, as mentioned earlier, while the analytical expressions for generation distribution allow for a final analytical expression for $\eta Eb(xo)$, they do not offer an entirely accurate picture of the actual EHP generation distribution, especially in the lateral ($x$ and $y)$ directions. This can lead to weaker fits to the data, and thus slightly more uncertainty in the extracted values for $Lh$ and $Sh/Dh$.

With the rapid recent increase in computational speed and power, an analytical approach to EBIC modeling may no longer be the only suitable technique for parameter extraction. To demonstrate this, we developed a technique for our EBIC modeling which leverages a numerical approach to solve Eq. (1). Our Monte Carlo simulations effectively return a volumetric distribution of EHP generation, $hEbxi\u2212xo,\u2009zj$, for some number of incident high energy electrons (*N*, typically in the 10's of thousands). Using the recorded beam current ($Ibeam$, measured by a Faraday cup) for each beam energy, we can determine the EHP generation rate as a function of position, $GEHPxi\u2212xo,\u2009zj=hEbxi\u2212xo,zjIbeamNe[EHPs/s]$, where *e* is the charge of an electron. Using this Monte Carlo-simulated generation distribution, as opposed to the analytical fit of Eq. (2), we can perform a numerical integration by dividing our device into discrete differential volumes and summing the current contributed to the total EBIC signal from each of these differential volumes [Eq. (4a)], which we can use to determine the EBIC current ($iEb(xo)$) and normalized EBIC current ($IEbxo$) as a function of position [Eq. (4b)]

The fitting process then begins by looking at the lowest energy electron beam data. For our initial fitting, we normalize both our modeled and experimental data such that $I10keVxo=0=1$, using scaling factors $Amod$ and $A\u2009exp(10\u2009keV)$, respectively. We evaluate the numerical sum in Eq. (4a), and the scaling of Eq. (4b), for a range of $Lh$ and $Sh/Dh$ values, and then measure the fit error by summing the square of the deviation of our modeled current to the experimental current. Figure 3 shows the contour plot generated from this fitting process for both the analytical [Fig. 3(a)] integration and the numerical [Fig. 3(b)] summation approaches, in this case using our data from the InAs/InAsSb SLS device at 120 K as representative data.

A number of features can be observed from the comparison offered by Fig. 3. First, we see that the effect of the $Sh/Dh$ parameter on our fit quality is minimal, particularly for low $Sh/Dh$ values ($<0.1\u2009\mu m\u22121$). This is to be expected, as once the surface recombination is slower than the diffusion of carriers toward the junction (effectively the carrier lifetime $\tau p$), one would not expect surface recombination to have a significant quantitative effect on the model. In addition, we do observe a narrower range of our fit quality when we utilize the numerical integration technique, suggesting at least a slight reduction in the uncertainty of the extracted $Lh$ term, presumably a result of the improved spatial resolution offered by employing the raw output of our Monte Carlo simulations, as opposed to the analytical fit to this simulation.

Once the best fit for the $Lh$ and $Sh/Dh$ parameters is obtained for the low energy electron beams, we model the higher beam energies by inserting the $Lh$ and $Sh/Dh$, extracted for low beam energies, into the expression for $\phi xi,\u2009zj$, and now using generation distribution, $hEb>15keVxi\u2212xo,\u2009zj$ obtained for the higher beam energy, and the beam current measured for the higher beam energy conditions, we calculate the new EBIC curve. For a beam energy of 20 keV, this would appear as

The result of this approach is an EBIC curve scaled in amplitude (due to the changes in $Ibeam$ and $\eta Eb>15keV$), with only slight changes in the EBIC profile shape resulting from the change in the carrier generation distribution at higher beam energies. The immediate and clear benefit of this approach is the ability to model not only the change in the EBIC profile, but the amplitude as well. Figure 4(a) shows, for 10 and 15 keV beam energies, the comparison of our normalized experimental data to the normalized EBIC model using the analytical fit to the carrier generation profile. Thus for the data of Fig. 4(a), we scale both our modeled and experimental $IEbx$ such that, once again, $I10keVxo=0=I15keVxo=0=1$, and in doing so lose any information regarding the relative strengths of the EBIC signals. In Fig. 4(b), however, we show the comparison of modeled and experimental data using our numerical approach [Eq. (5)]. Here, the experimental data are scaled by the very same factor as our 10 keV data, $A\u2009exp\u2009(10\u2009keV)$, while the scaling of our modeled data uses both the same factor as our 10 keV model, $Amod$, and the additional scaling coming from the change in the beam current. Thus, the change in the magnitude of the modeled EBIC profile results from measurable experimental parameters (beam energy and current), more accurately reflecting the change in experimental parameters with increasing beam energy. The advantage of this approach can be clearly seen in Fig. 4(b), where we observe excellent fits to both the shape and magnitude of the EBIC profile. As indicated in Fig. 4, our averaged sum of squared error (SSE) for the 10 and 15 keV data improves from 0.0577 to 0.0346 using our numerical EBIC approach. At higher beam energies, the normalized EBIC model will give better fits, for reasons discussed later.

Our approach to EBIC modeling not only offers improved fitting to the experimental data with decreased uncertainty in the extracted values of $L$ and $S/D$, but also allows us to predict the amplitude of the EBIC signal for increasing beam energies, as we can see in Fig. 4(b). This will provide us with additional valuable data points for understanding EBIC measurements of our samples. Below we discuss the results from the samples investigated, and benefits and challenges of the EBIC modeling technique described above.

## IV. RESULTS AND DISCUSSION

While the results from the EBIC measurement provide diffusion lengths for our devices as a function of temperature, a more holistic understanding of minority carrier transport is achieved when the extracted diffusion lengths are combined with minority carrier lifetimes ($\tau p$), allowing for the calculation of the hole diffusivity ($Dh$) and, using the Einstein relationship, minority carrier vertical mobility ($\mu h$)

Time-dependent photoluminescence (TRPL) spectroscopy allows for the measurement of the minority carrier lifetime in the low injection regime, where typical IR detectors operate. The full expression for carrier lifetime in a SLS is most accurately expressed as^{22}

where $\delta n$ is the excess carrier concentration, $\tau po$ and $\tau no$ are the minority and majority carrier Shockley-Read-Hall (SRH) lifetimes, $B$ is the bulk radiative coefficient, and $Cn$ is the Auger recombination coefficient. For low injection, $\delta n\u226ano$, and lightly doped material ($no<2.5\xd71015\u2009cm\u22123$), the minority carrier lifetime is dominated by the contributions from SRH and radiative recombination, and can be described by a single value for lifetime (independent of excess carrier concentration). Recent results have demonstrated that for a more highly doped material ($no>2.5\xd71015\u2009cm\u22123$), the low injection lifetime is dominated by Auger recombination, although the resulting TRPL data can still be fitted with a single exponential.^{39} Figure 5 shows the TRPL results from all of our samples for temperatures of 80, 120, 160, and 200 K, using a single-exponential fit to the tail of the TRPL data, from which we can extract the temperature dependent carrier lifetime. From these data we observe decreasing carrier lifetimes with increasing Ga content of our samples, with the 0% Ga sample showing a factor of 2 or greater lifetime than the 20% Ga sample across the entire temperature range investigated. All samples display similar decreases in minority carrier lifetimes as a function of temperature, an indication that the extracted carrier lifetime is an Auger-limited lifetime, as opposed to resulting from SRH recombination, as would be expected at our intended doping concentrations of $no=2\xd71016\u2009cm\u22123$.^{39} The observed decrease in carrier lifetime with increased Ga concentration, in a more lightly doped SLS, could indicate the presence of additional defects associated with Ga in our SLSs. Alternatively, the decrease in lifetime with increasing Ga could result from the increased overlap between electron and hole wavefunctions in the InGaAs/InAsSb SLS material system,^{33,36} and thus shorter radiative recombination times. However, low-injection lifetimes in our material system, as discussed above, are most likely Auger-limited. Thus, from the lifetime data alone, it cannot be said that the presence of Ga either introduces non-radiative recombination centers or improves radiative lifetimes in the SLS structure. The decrease in lifetime as a function of Ga content is more likely a result of changes in the Auger lifetime of our highly doped samples, an effect we will discuss below.

Extracting our temperature dependent lifetime from the TRPL measurements [Fig. 6(a)], and our temperature-dependent diffusion length from the numerical model of the EBIC measurement [Fig. 6(b)], we are able to measure the vertical hole mobility for each of the SLS samples as a function of temperature. Figure 6(c) shows the resulting temperature dependent vertical hole mobility for all three of our samples, where the uncertainty shown in our data is determined using the $SSE<0.02$ metric depicted in Fig. 3. In a separate analysis, the hole diffusion lengths for the same set of samples were determined by fitting the experimental external quantum efficiency (EQE) of the detectors to the theoretically expected EQE at temperatures in the same range (80–200 K). The EQE experimental approach is detailed in Ref. 37, and requires careful absorption and reflection measurements with a (spatially and spectrally) well-calibrated IR light source, and accurate current measurements, which when combined with an analytical model, allow for the extraction of the minority carrier diffusion length. Using those values of $Lh$ and the TRPL lifetime values, another set of values for the vertical hole mobility was obtained. We compare these results in Fig. 6 and both approaches show vertical hole mobilities increasing as a function of temperature. In addition, both the EQE and EBIC techniques show increasing mobility, at all temperatures, for increasing Ga content in the SLS samples. Finally, we also observe a significantly stronger temperature dependence for the 0% Ga SLS sample (a factor of ∼20) than the 5% and 20% Ga samples (factors of 4 and 5, respectively) across the T = 80 K to T = 200 K range of temperatures investigated.

The increase in vertical carrier mobility with increasing Ga content can be understood by recalling that a primary benefit of the addition of Ga to the InAs layers of an InAs/InAsSb SLS is the increase in the wavefunction overlap between electron and hole states in the SLS.^{36} This overlap is caused not only by the weaker quantization of the states in the conduction band (due to the decrease in the conduction band offset between the InAsSb and the In(Ga)As), but also from the decrease in thickness of the In(Ga)As barriers between the hole states in the InAsSb, which allows increased extension of hole states into the In(Ga)As hole barriers (which for the 20% Ga sample are only 8.5 ML thick), and thus improved vertical transport. Band structure calculations of the vertical hole effective mass indicated that the effective mass decreased from 2.97 m_{o} to 1.49 m_{o} as the gallium composition was increased from 0 to 20%. These calculations do not take into account intersubband scattering effects,^{40} which could explain the discrepancy between the expected change in mobility and the actual change.

Previous approaches to EBIC modeling search for the optimized fitting parameters ($L$ and $S/D$) which most accurately fit the EBIC profiles (lineshapes) for all beam energies. The argument for this approach is that at higher beam energies, the carrier generation volume probes deeper into the device, essentially providing a variation in the effective depth of the average EHP generated. Finding the optimized fitting parameters for all beam energies is thus argued to offer the ability to extract a more accurate $S/D$ value. However, for many material systems, large variations in the $S/D$ value have little to no effect on the accuracy of the fit to EBIC data, as can clearly be observed in Fig. 3. In fact, for the normalized EBIC fittings, many different combinations of $L$ and $S/D$ values can produce very similar EBIC profiles, with a greater uncertainty in the extracted diffusion length. At the same time, by focusing solely on the EBIC profile, it is possible that this technique discards valuable information which could be extracted from the relative magnitudes of the EBIC profiles. Using our scaled EBIC fittings, however, not only do we decrease the uncertainty in extracted $Lh$ and $Sh/Dh$ for a given experimental condition, we are able to use the relative magnitude of our modeled EBIC signal to obtain improved fits and qualitative information regarding the performance of our devices as a function of excess carrier concentration.

Figure 7(a) shows the scaled fits to our EBIC data for the 0% Ga SLS sample at 120 K for all of the beam energies investigated in this work, in addition to the 10 and 15 keV data already presented in Fig. 4(b). The fits to our experimental data become progressively poorer as we move to higher beam energies. We can understand this effect by returning to our expression for carrier-dependent lifetime in Eq. (7), which indicates that for higher carrier concentrations, we would expect a decrease in the average lifetime of excited carriers due to increased Auger recombination, regardless of whether our SLS minority carrier lifetime is Auger- or SRH-limited at low excess carrier concentrations. A higher beam energy results in not only a broader carrier generation distribution, but also significantly higher carrier concentration (which is also affected by beam current). Thus, it would be expected that as we increase beam energy (and/or current), we would observe an increased deviation from our EBIC model (which we fit to the low beam energy data).

In fact, the deviation from our fit offers qualitative information regarding the carriers effectively “lost” to increased recombination rates (Auger) in our experiment. Figures 7(b)–7(d) show the difference between our scaled experimental and modeled EBIC signal at the SLS junction ($x=0$) for the samples studied in this work. A largely monotonic increase in the deviation is observed for all samples at all temperatures (with the exception of one outlier data point: the 160 K, 20 keV data for the InAs/InAsSb SLS). In addition, we observe a weaker increase in the difference between our model and our data for the 0% Ga SLS than for the 5% Ga SLS, and significantly weaker than the 20% Ga SLS, whose experimental EBIC signal is far smaller than that predicted by our model. These results suggest that the effects of additional, carrier concentration-dependent, recombination mechanisms are correlated with increasing Ga content in our SLS structures. As Auger recombination is known to be quenched by increased quantization of charge carriers, the stronger signature of Auger recombination observed with increasing Ga content could be attributed to the decreasing conduction band offset and thus weaker quantization of conduction band electrons with increasing Ga content. Measuring the change in the difference between the modeled and experimental EBIC signal provides only a qualitative measure of the change in carrier lifetime. Future efforts will attempt to develop a quantitative understanding of this measure using samples with a clear transition between SRH and Auger limited lifetimes.

Previous approaches to EBIC modeling thus not only miss valuable information obtained from the relative magnitudes of the EBIC signal and model, but potentially could result in inaccurate parameter extraction. These approaches attempt to fit EBIC data from all beam energies simultaneously, including the higher beam energy data, where carrier lifetimes can be very different than for lower beam energy excitation. While this approach may be sufficiently accurate for materials with long diffusion lengths and relatively constant carrier lifetimes (small Auger coefficients), for materials with shorter diffusion lengths and larger Auger coefficients (such as narrow bandgap semiconductors), the fit to the data then may not accurately reflect the device parameters for typical operating conditions (low excess carrier concentration). Although our improved EBIC modeling technique is thus far only able to extract qualitative information regarding the behavior of our devices as a function of carrier concentration, future efforts will look to develop a more quantitative approach to understand the effects of beam energy (and/or current) on the EBIC profiles of our narrow bandgap materials. In particular, we will look to investigate beam energy dependence of devices as a function of background doping. In doing so, we will look to observe the transition between SRH- and Auger-limited lifetimes, either by control of beam energy or doping, and use these data to develop quantitative modeling techniques to extract device parameters as a function of excess carrier concentration.

## V. CONCLUSIONS

In conclusion, we have presented a technique for modeling electron beam induced current measurements which offers improved fitting to experimental data, lower uncertainty in parameter extraction, and qualitative information on carrier dynamics as a function of carrier concentration. Our approach utilizes Monte Carlo-simulated carrier generation distributions combined with an expression for carrier diffusion modified for the devices investigated, with a numerical integration to obtain a modeled EBIC profile which more accurately fits our experimental data. We use the EBIC model presented to extract the minority carrier diffusion length and the surface recombination velocity to diffusivity ratio for In(Ga)As/InAsSb strained-layer superlattice detectors with 0, 5, and 20% Ga content. Although we use the presented technique to measure vertical hole mobility in narrow bandgap SLS materials with nBn detector architectures and diffusion dominated transport, our approach could well be adapted for lateral mobility studies (plan view) or for studying alternative material systems and/or detector architectures, with adjustments to the probability of collection expression [Eq. (3)] and our Monte Carlo simulation parameters. Together with time-resolved photoluminescence (TRPL) measurements, we use our EBIC technique to extract the temperature dependent mobility of our samples. We observe increasing hole mobility as a function of the temperature, and higher mobilities for the InGaAs/InAsSb devices than the InAs/InAsSb device at all temperatures. In addition, we compare the deviation of our modeled EBIC response from the experimental data, and use this discrepancy to qualitatively understand the effect of additional recombination mechanisms, or changes in the existing recombination rates, in our samples. The In(Ga)As/InAsSb SLS material system provides the opportunity to investigate the effects of electron/hole wavefunction overlap in narrow bandgap materials by control of Ga content in the In(Ga)As layers. The extracted temperature dependent mobility and beam-energy dependent current amplitudes for our samples are discussed using the framework of wavefunction overlap and carrier quantization, and offer an understanding of the effects of bandstructure design on carrier dynamics in the SLS material system. The presented work offers an approach to electron beam induced current measurements and parameter extraction with improved fitting of the experimental data, lower uncertainty and the potential for measuring device properties as a function of injection regime. While in this work we investigate narrow bandgap SLS materials, the approach presented is applicable to the investigation of a wide range of semiconductor-based electronic and optoelectronic devices.

## ACKNOWLEDGMENTS

The authors (N.Y. and D.W.) gratefully acknowledge support from the Army Research Office Multi-Disciplinary Research Initiative (Grant No. W911NF-10-1-0524). N.Y. also acknowledges funding from the Semiconductor Research Corporation (SRC) company-named Intel Fellowship. The authors (C.R., G.A., J.S., and J.D.) acknowledge funding from the Air Force Research Laboratory, Sensors Directorate under project “III-V Focal Plane Array Development Using Novel Superlattices.”