Determination of the Bulk and Local Diffusion-Length Values of Charge Carriers in MCT Films and in the Absorber Layers of MCT-Based Photovoltaic IR FPA Detectors

In the present chapter, we describe two new photoelectric-measurement-based methods that can be used for characterization of the diffusion process of charge carriers either in mercury-cadmium-tellurium (MCT) films intended for fabrication of infrared focal plane array (IR FPA) detectors or in the absorber layers of ready MCT-based photovoltaic IR FPA detectors. First-type measurements are photocurrent measurements to be performed on special test structures involving round photodio‐ des provided with coaxial light-shielding cap metal contacts. Second-type measure‐ ments are spot-scan measurements of MCT photovoltaic 2D IR FPA detectors traditionally used for measuring the crosstalk value of such detectors yet implement‐ ed at low and high levels of registered diode photocurrents. Both methods permit the determination of the bulk diffusion length of minority charge carriers in MCT material. The second method, in addition, permits the determination of the local effective diffusion length of minority charge carriers in the absorber-layer region under FPA diodes. The values of the bulk diffusion length of minority carriers obtained in MCT films and in the MCT absorber material of the examined middle-wave and long-wave IR FPA detectors were found to be in good agreement with previously reported relevant data. The estimated value of the local effective diffusion length of minority carriers in the film region under back-biased FPA diodes proved to be consistent with a theoretical estimate of this length.


Introduction
In infrared (IR) imaging and detection, cooled hybrid mercury-cadmium-tellurium (MCT) photovoltaic IR focal plane array (FPA) detectors have found widespread use [1,2]. Collection of photogenerated charge carriers by FPA diodes in the photodiode arrays of such detectors is a process essentially mediated by the diffusion of minority carriers in the MCT film. That is why the performance characteristics of MCT IR FPA detectors and their figures of merit depend on the distances the excess carriers diffuse over in the photosensitive film (PF) of such detectors [3,4]. Among the parameters that govern the diffusion process, of primary interest is the bulk diffusion length of minority charge carriers l d in the absorber material of the detector as this length characterizes the properties of the material irrespective of a particular FPA design, including the pixel architecture and array structure. Knowledge of l d is significant for quantitative analysis and numerical modeling of FPA detectors. With known mobility of minority carriers, this knowledge also permits evaluation of the lifetime of excess carriers in MCT, which is often hard to measure because of its small value. Finally, the length l d is generally regarded as a key parameter that defines the crosstalk value of FPA detectors [5][6][7][8][9].
In view of the aforesaid, of considerable interest are new experimental techniques and analysis methods that can be used for determining the length l d in the MCT material of MCT-based films and diode hetero-epitaxial structures.
In the present chapter, we describe two new photoelectric-measurement-based methods that can be used for characterization of the diffusion process of charge carriers either in MCT films intended for fabrication of IR FPA detectors or in the absorber layers of ready MCT-based photovoltaic IR FPA detectors. Both methods permit the determination of the bulk diffusion length of minority charge carriers l d in MCT material.
First-type photoelectric measurements can be performed on special test structures involving round photodiodes of radius r 0 provided with coaxial light-shielding cap metal contacts of different radii R i > r 0 . The contacts shield the vicinity of the photodiodes from the excitation radiation falling onto the front surface of the test structure. In the shielded region, the optical generation of excess charge carriers is nearly zero, and the diode photocurrents are therefore formed by those minority carriers that reach the diode via diffusion across the shielded annular region of width R i -r 0 . Numerically solving an appropriate 2D diffusion problem, we can preliminarily calculate the diode photocurrents I i as a function of l d for several experimentally implemented values of R i -r 0 . To avoid the necessity of performing absolute measurements and calculations, photocurrent ratios I 1 /I 2 , I 1 /I 3 , I 1 /I 4 , etc. for diode pairs with different contactto-diode size ratios can be calculated. Then, the calculated curves I 1 /I i (l d ) can be used as reference dependences allowing the determination of the length l d in the MCT film under study.
Second-type measurements are spot-scan measurements of MCT photovoltaic 2D IR FPA detectors traditionally used for measuring the crosstalk value of such detectors, yet implemented at low levels of registered diode photocurrents j ph . Low photocurrent levels can be achieved by lowering the gate potential V g of input field effect transistors (FETs) in the readout circuits of the photoelectric cells of the detector. The requirement for low values of the photocurrents j ph extracted by the readout circuits out of the photosensitive film of the detector arises as a condition for minimization of the influence of readout circuits on the lifetime and the effective (at j ph ≠ 0) diffusion length of photogenerated minority carriers in the photosensitive film l d eff . The effective diffusion lengths l d eff can be extracted from broadened spatial diode photoresponse profiles S(x) registered on successive reduction of the level of registered diode photocurrents. Then, the bulk diffusion length of minority charge carriers in the photosensitive film l d can be found as the extreme value of l d eff as j ph → 0. For revealing the asymptotic behavior of l d eff →l d for j ph →0, a special diffusion model with zero PF thickness was developed. This model was subsequently used to analyze the charge-carrier diffusion process at normal operational regime of the detector with large diode photocurrents j ph . The latter analysis has allowed us to evaluate the local diffusion length of minority charge carriers in the PF region under the back-biased FPA diodes l d eff . The obtained value of this length proved to be in good agreement with a theoretical estimate of this length. As a result, an experimental approach enabling a comprehensive spot-scan characterization of the charge-carrier diffusion process in the absorber layers of MCT IR FPA detectors at arbitrary levels of diode photocurrents has been developed.

A method for determining the diffusion length of minority charge carriers in MCT films with band-graded surface layers
The method for determining the diffusion length of minority carriers in MCT films l d (or L n , for electrons in p-type semiconductor) is based on the previously mentioned fact that this length presents an important parameter that defines, among other characteristics, the quantum efficiency of lateral diode structures prepared on such films [4].

Essence of the method
Methods for determining the lateral optical collection length of charge carriers l opt in MCT films based on quantum-efficiency measurements of lateral diode structures comprising variablearea diode arrays have previously been reported in the literature (see, for instance, [10,11]). The procedure for determining the length l opt in those methods is based on the fact that measured quantum efficiency of back-biased photodiodes in such structures is essentially defined, first, by the diffusion length of minority charge carriers l opt and, second, by the diode radius. However, measured length l opt presents only a rough estimate to the diffusion length of minority carriers in the film.
In [12], we have proposed another method for determining the length l d in photosensitive semiconductors. That method was subsequently implemented to perform measurements of the electron diffusion length L n in p-type MCT-film-based hetero-epitaxial structures [13]. The proposed method uses the fact that measured photocurrent produced by an illuminated p-n junction capped with a coaxial light-shielding metal contact varies in magnitude as we change the radius of the n-type region of the diode r 0 and/or the radius of the cap contact R. Of course, the radius r 0 is always smaller than R. A schematic of such a photodiode is shown in Fig. 1. The method works as follows. First, we measure the ratio between the photocurrents produced by two photodiodes having different values of R-r 0 . Preliminarily, the photocurrent ratio is theoretically calculated to be plotted versus the electron diffusion length L n . Then, with the measured photocurrent ratios at hand, we can use the theoretically calculated curves as reference dependences from which the length L n can be determined. Schematic of a photodiode provided with a coaxial light-shielding cap contact. 1 -р-type MCT film, 2 -ntype diode region of radius r 0 , 3 -substrate, 4 -shielding contact of radius R to the n-type region, 5 -peripheral contact to the р-type MCT film, 6 -dielectric, 7 -illumination.

Calculations
The photocurrent produced by a p-n photodiode was calculated by numerically solving the 2D stationary continuity equation for excess electron concentration in an axisymmetric quasineutral р-type region in the vicinity of the p-n diode illuminated, from the side of the cap contact, with an IR radiation flux that modeled the radiation flux used in the experiment. With the electron mobility and the electron lifetime assumed constant, the continuity equation has the form 2 2 In Eq. (1), the z axis is directed along the symmetry axis of the photodiode normally to the ptype absorber layer, and the r axis is directed along radius; n' = n -n p0 is the concentration of minority carriers in excess of their equilibrium concentration n p0 , D n is the electron diffusion constant in the р-type absorber material, and g(z,r) is a function that describes the rate of photogeneration of charge carriers in the illuminated film with allowance for the shielding action of the cap contact: is the Planck distribution function for the blackbody emissivity at temperature T b (see, for instance, [1]), λ 1 ≈ 20 μm is a wavelength far exceeding the long-wave cutoff wavelength λ с of the р-type MCT absorber layer, and F(r) is the shielding function (in a simplest model, F(r)=0 for r < R and F(r)=1 for r > R). Since, in what follows, ratios of diode-produced photocurrents will be calculated, then the knowledge of the absolute value of the radiation flux (which depends on the aperture angle and on the coefficient of radiation reflection from the surface of the test structure) is not necessary. The absorption coefficient α depends on the coordinate across the photosensitive film since, starting from a distance 0.4 μm from either the lower or upper surface of the film, the cadmium content of the MCT material and, hence, its bandgap energy in the test structure gradually increase in value toward the surface. The graded-band MCT surface layers were used to suppress the surface recombination of excess carriers at the surface boundaries of the p-type MCT film [14]. The computational domain in which Eq. (1) was solved is shown in Fig.  2. This domain is the region referring to the coordinate ranges r=0÷R ph and z=0÷H; the Cd x Hg 1-x Te material in this domain has fixed composition х.
. In solving the continuity equation, the following boundary conditions were adopted [15]: ∂ r = 0 for r = 0 and on the lateral cylindrical surface of the computational domain at r = R ph = r 0 +5L n, max , where L n, max is the maximum value of L n for which the computations were performed, and r 0 is the radius of the p-n junction.

non-illuminated
b. n ' = n p0 (exp(qV / kT ) − 1) -on the lateral and planar surfaces of the p-n junction; here, V is the voltage bias across the diode, q is the electron charge, k is the Boltzmann constant, and Т is temperature. In calculating the photocurrent, the bias voltage was assumed to be zero.
c. D n ∂ n ' ∂ z = S r n ' -on the upper and lower boundaries of the base layer at z = 0 and z = H, where S r is the effective rate of recombination of minority carriers at those surfaces.
With regard to the data of [16], the electron mobility value µ n =6×10 4 cm 2 /V×s for р-type Cd x Hg 1-x Te with х=0.223 was adopted in the calculations. The bandgap energy E g and the intrinsic concentration of charge carriers n i in the MCT material were calculated as functions of MCT-material stoichiometry and temperature by the formulas borrowed from the monograph [1]. The MCT absorption coefficient was calculated using the formulas reported in [17]. In the calculations, the length L n was varied by changing the electron lifetime τ n according to the formula L n =(D n τ n ) 1/2 . Equation (1) was transformed in a finite-difference equation to be solved, by the sweep method, on a grid with variable step over the p-type region. The total number of applied iterations reached 15000. The photocurrents across the planar and lateral surfaces of the p-n junction, j N = qD n ∂ n ' ∂ z and j L = qD n ∂ n ' ∂ r , were calculated via numerical differentiation of the obtained solution n'(r,z). The integral diode photocurrent I was calculated by numerical integration of photocurrent density over the pn junction surface.

Experimental
For experimental implementation of the method, a test photodiode structure whose fragment is shown in Fig. 3 was fabricated.
In the fabricated structure, the radius of the p-n junctions was fixed, equal to r 0 ≈6.5 μm, whereas the radius of the shielding contacts varied in the range from 10 to 100 μm (see Fig. 3 and Table 1). The distance between neighbor diodes was chosen to be sufficiently large, so that their contacts exerted no influence on the photocurrents produced by the neighbor diodes. At the periphery of the lateral diode structure, a second contact (to the р-type absorber layer) was provided. Diode photocurrents were determined from measured current-voltage characteristics of illuminated and nonilluminated diodes. The characteristics were measured with the help of a thin probe brought in contact to the diode caps. The structure of each individual lateral photodiode involved a peripheral metalinsulator-semiconductor (MIS) region formed by the cap metal contact, by the dielectric, and by the p-type MCT material (see Fig. 1). For avoiding the formation of surface inversion layers in the vicinity of the diodes, a sufficiently thick dielectric layer without a noticea-ble positive built-in charge and, also, a metal with a work function value close to that of the semiconductor need to be used, and measurements are to be performed on diodes biased with a sufficiently low bias voltage. In the test structures used in our experiments, a 0.15μm thick double-layer SiO 2 /Si 3 N 4 dielectric and indium metallization, allowing one to meet the above-formulated requirements, were used. In the experiments, a test structure fabricated on a variband MCT film grown on a GaAs substrate was placed on the surface of a corundum ceramic plate. The plate was cooled down to Т=78 K, via a copper cylinder, by liquid nitrogen into which the copper cylinder was immersed. The ceramic plate was necessary for preventing the reflection of IR radiation having passed through the thin MCT layer and through the transparent GaAs plate, from the metal surface of the cylinder. Such reflection could otherwise entail additional absorption of reflected radiation in the MCT material under the shielding contact, which was neglected in our calculations, and, hence, it could result in an increased value of measured photocurrent. The latter in turn could induce an additional inaccuracy in the found value of L n . The illumination of the test structure was performed from the side of the cap contacts by exposing the sample to 293-K thermal background radiation passing through an aperture angle of 32 • . The test structure was fabricated on sample 1-MCT130607 with the following characteristics of the р-type MCT film: thickness 8 μm, stoichiometric coefficient х=0.223, hole mobility µ p = 540 cm 2 /V×s, hole concentration р=6.7×10 15 cm -3 . As illustrated by Fig. 4, thin graded-band MCT layers were provided on both sides of the grown MCT film to suppress the surface recombination of charge carriers at its boundaries.

Experimental results
Current-voltage characteristics of the diode pairs involving neighbor diodes with different radii of cap contacts were measured, and the photocurrent values were determined from measured data. Results obtained for a diode pair with cap-contact radii R 1 = 10 μm and R 5 =70 μm are illustrated in Fig. 5.  Evidently, the photocurrents I 1 and I 5 remained roughly constant throughout the interval of low bias voltages till the bias voltage -150 mV. In addition, we have I 1 >>I 5 , which relation proves that the peripheral MIS regions of the photodiodes induced no substantial contribution to measured photocurrent values. Typically, photocurrents were calculated at a bias voltage of -50 mV. A bar chart of photocurrents I 1 and I 5 for seven diode pairs with cap-contact radii 10 and 70 μm is shown in Fig. 6. Here, the average photocurrent ratio I 1 /I 5 equals 9.84.  The theoretically calculated ratios of the photocurrents produced by a diode with the k-th shield (k = 2÷6) (see Table 1) and by a diode with shield 1 are shown in Fig. 7; as explained above, the photocurrent ratios were calculated by solving Eq. (1). For the dimensions of the pn junctions, the following values were adopted: radius r 0 =6.5 μm, inward extension of the pn junction in depth of the MCT film z 0 =2 μm. The thickness of the p-type absorber film was assumed to be H=8 μm. The effective rate of surface recombination of excess carriers on both surfaces of the MCT film was assumed to be zero due to the presence of band-graded MCT layers on those surfaces (see Fig. 4). The absorption of IR radiation in the band-graded MCT layers was neglected.  Each of the curves in Fig. 7 has an abruptly descending portion appropriate for determining diffusion-length values in a particular interval of L n .

Determination of the diffusion length L n
With reference to the properly chosen curve in Fig. 7, the experimentally measured photocurrent ratio I 1 /I 5 = 9.84 translates into a diffusion-length value 20.5 μm. Photocurrent ratios I 1 /I k for the diode pairs which comprised diodes with other values of cap-contact radii have also been measured. The values of L n obtained from the comparison of measured with calculated photocurrent ratios for those diodes are listed in Table 1. The inaccuracy in determining the length L n can be evaluated considering the scatter of measured ratios I 1 /I k for used photodiodes, whose total number for each shield radius R k typically amounted to 7. It is seen from Table 1 that very close data with average values of L n ranging from 20.2 μm to 20.5 μm were obtained for the diode pairs with cap-contact radii 10 and 24, 10 and 40, and 10 and 70 μm. Here, the data obtained from the photocurrent ratios I 1 /I 2 and I 1 /I 3 are the least accurate results since the curves in Fig. 4 for those diode pairs exhibit a gently sloping behavior in the range of L n -value around 20 μm.
Application of the largest shield with R 6 =100 μm yields somewhat overestimated values of L n due to the absorption, in the shielded region, of the radiation having passed through the MCT film and having been partially reflected from the lower surface of the GaAs substrate. Let us evaluate the effect due to the latter reflection. In the case of an 8-μm thick MCT film, one can readily obtain that, for α=2200 cm -1 , the radiation flux reaching the MCT film-substrate interface at wavelength λ=0.9λ c will amount to 17 % of the incident radiation flux. For reflection-coefficient value 0.27 at the air-GaAs interface, for the reflected radiation we obtain a value amounting to 3.8 % of the total flux initially incident onto the structure. Evaluation yields an additive to the photocurrent I k approximately equal to 0.05 I 1 . The larger value of I k will entail a smaller calculated value of I 1 /I k and, hence, a larger value of L n extracted from the graph of Fig. 7. Clearly, the contribution due to the mentioned additive to the photocurrent produced by the diode with the k-th shield can be neglected providing that 0.05 I 1 << I k , or I 1 /I k << 20. From Table 1, we see that the above condition was obviously violated for k=6.
To summarize, our measurements of the electron diffusion length in the fabricated test structure performed on the diodes with cap contacts considerably differing in size have yielded an average length value <L n >=20.2-22.8 μm. A similar analysis performed for the optimum diode pairs with the photocurrent ratios I 1 /I 3 and I 1 /I 4 , whose dependences on L n as revealed by calculations proved to be much more steeply sloping in the interval of L n -values around L n ≈20 μm, has yielded for the length L n values ranging in the interval from 19.3 to 21.3 μm (with allowance for the scatter of photocurrent-ratio values obtained for 14 measured diode pairs).

Determination of the bulk and local diffusion-length values of charge carriers in the photosensing film of ready MCT IR FPA detectors
Below, we present an analysis of the charge-carrier diffusion process in a photosensing film of MCT 2D IR FPA detectors based on spot-scan measurements of such detectors. The analysis was performed with due consideration given to the fact that the extraction of minority carriers by FPA diodes out of the absorber layer, generally speaking, reduces the lifetime of excess carriers in the photosensitive film of the detector in comparison with the "bare" (without FPA diodes) MCT film, making the diffusion length of those carriers in the film regions with photocurrent suction a shorter effective diffusion length l d eff . If the length l d eff , whose magnitude depends on the level of diode photocurrents, does not exceed the lateral sizes of the FPA diodes, we may speak of a local effective diffusion length of photogenerated charge carriers in the film region under the photodiodes. Below, we show how the bulk and local diffusionlength values of charge carriers in different parts of the FPA structure can be evaluated using the spot-scan data measured for FPA detectors at different levels of diode photocurrents.

Essence of the method
For evaluating the diffusion length of charge carriers in the photosensitive film of FPA detectors, the spot-scan technique was often used [5][6][7][8][9]. In a spot-scan procedure, the chargecarrier diffusion-length value is deduced from spatial diode photoresponses S(x) measured in the vicinity of a local illumination spot shined onto the FPA (see Fig. 8). However, the important fact that the spot-scan procedure itself largely modifies the distribution of minority carriers in the photosensitive film under measurement conditions with relatively high diode photocurrents has not been given due consideration in previous implementations of spot-scan measurements. It should be additionally mentioned here that not only the scanning diode but also its neighbor FPA diodes, which also produce their own photoresponses during scanning, seriously affect the distribution of photogenerated charge carriers in the scanned region. Like the scanning diode does, in performing scans, the neighbor diodes also continuously change their position with respect to the excitation spot. The latter circumstance hampers the treatment of spot-scan data if one attempts introducing corrections to gained data for photocurrent suction.
hnique was often used [5][6][7][8][9]. In a spot-scan procedure, the charge-carrier diffusion-length value is deduced from atial diode photoresponses S(x) measured in the vicinity of a local illumination spot shined onto the FPA (see Fig. 8). wever, the important fact that the spot-scan procedure itself largely modifies the distribution of minority carriers in the otosensitive film under measurement conditions with relatively high diode photocurrents has not been given due nsideration in previous implementations of spot-scan measurements. It should be additionally mentioned here that not ly the scanning diode but also its neighbor FPA diodes, which also produce their own photoresponses during anning, seriously affect the distribution of photogenerated charge carriers in the scanned region. Like the scanning de, in performing scans, the neighbor diodes also continuously change their position with respect to the excitation ot. The latter circumstance hampers the treatment of spot-scan data if one attempts introducing corrections to gained ta for photocurrent suction.
A good strategy toward diminishing the influence of measuring circuits on the distribution of charge carriers in absorber layer consists in decreasing the level of registered diode photocurrents j ph . By following this strategy, we ve recently proposed a new method for determining the bulk diffusion length of charge carriers l d in the absorber layer MCT IR FPA detectors [18]. In the new method, the spot-scan technique was implemented at low levels of diode otocurrents j ph , and the bulk diffusion length of charge carriers in the photosensitive film l d was proposed to be termined as the extreme value of their "spatially averaged" effective diffusion length eff d l for j ph → 0. The length eff d l low diode photocurrents j ph having been sufficiently large, the diffusion of photogenerated charge carriers in the otosensitive film of examined detectors was analyzed in [18] within a continuum approach. At a high level of diode otocurrents j ph , however, the latter approach becomes invalid because of a decreased value of eff d l . In the present per, we extend the analysis [18] of the diffusion process of charge carriers in the photosensitive film of FPA detectors the case of large diode photocurrents j ph . We show that, following the determination of the length l d by the method [18], atment of spot-scan profiles S(x) measured at large diode photocurrents within a simple 2D diffusion model makes it ssible to determine the local effective diffusion length l d eff of minority carriers in the film region under FPA diodes in rmal operational regime of the photodetector. In this way, we for the first time give a complete analysis of the chargerrier diffusion process in a photosensitive film of MCT IR FPA detectors at arbitrary diode photocurrents.

Experimental
In our experiments, two 320x256 Cd x Hg 1-x Te n-on-p photovoltaic detectors with long-wave cutoff wavelengths .4 (x ≈ 0.30) and ~10 μm (x ≈ 0.225), hereinafter to be referred to as detectors D-1 and D-2, were used. In both tectors, the diode pitch was p = 30 μm, the nominal size of FPA diodes (as defined by the implantation window size in dielectric) was 10x10 μm, and the thickness of the continuous p-type Hg vacancy-doped MCT photosensing layer s about 10 μm. The concentration of acceptors in the absorber layer was N a = (7÷10) x 10 15 cm -3 . The diode arrays in r detectors were fabricated using boron implantation. Both detectors contained graded-gap wideband MCT layers wn on both sides of the MCT film to ensure surface insulation of FPA diodes and suppress the recombination of arge carriers at the film boundaries [14]. The MCT film of the D-2 detector additionally contained a heavily doped n-  Scanning of a thin-line illumination spot with an FPA photodiode (V g is the gate potential of the photoelectric-cell input FETs with respect to the frame electrode of the diode array, S is the registered photosignal, С is the storage capacitor, and V cc is the readout integrated circuit (ROIC) supply voltage).
A good strategy toward diminishing the influence of measuring circuits on the distribution of charge carriers in the absorber layer consists in decreasing the level of registered diode photocurrents j ph . By following this strategy, we have recently proposed a new method for determining the bulk diffusion length of charge carriers l d in the absorber layer of MCT IR FPA detectors [18]. In the new method, the spot-scan technique was implemented at low levels of diode photocurrents j ph , and the bulk diffusion length of charge carriers in the photosensitive film l d was proposed to be determined as the extreme value of their "spatially averaged" effective diffusion length l d eff for j ph → 0. The length l d eff at low diode photocurrents j ph having been sufficiently large, the diffusion of photogenerated charge carriers in the photosensitive film of examined detectors was analyzed in [18] within a continuum approach. At a high level of diode photocurrents j ph , however, the latter approach becomes invalid because of a decreased value of l d eff . In the present paper, we extend the analysis [18] of the diffusion process of charge carriers in the photosensitive film of FPA detectors to the case of large diode photocurrents j ph . We show that, following the determination of the length l d by the method [18], treatment of spot-scan profiles S(x) measured at large diode photocurrents within a simple 2D diffusion model makes it possible to determine the local effective diffusion length l d eff of minority carriers in the film region under FPA diodes in normal operational regime of the photodetector. In this way, we for the first time give a complete analysis of the charge-carrier diffusion process in a photosensitive film of MCT IR FPA detectors at arbitrary diode photocurrents.

Experimental
In our experiments, two 320x256 Cd x Hg 1-x Te n-on-p photovoltaic detectors with long-wave cutoff wavelengths ~5.4 (x ≈ 0.30) and ~10 μm (x ≈ 0.225), hereinafter to be referred to as detectors D-1 and D-2, were used. In both detectors, the diode pitch was p = 30 μm, the nominal size of FPA diodes (as defined by the implantation window size in the dielectric) was 10x10 μm, and the thickness of the continuous p-type Hg vacancy-doped MCT photosensing layer was about 10 μm. The concentration of acceptors in the absorber layer was N a = (7÷10) x 10 15 cm -3 . The diode arrays in our detectors were fabricated using boron implantation. Both detectors contained graded-gap wideband MCT layers grown on both sides of the MCT film to ensure surface insulation of FPA diodes and suppress the recombination of charge carriers at the film boundaries [14]. The MCT film of the D-2 detector additionally contained a heavily doped n-type MCT layer with a larger bandgap energy provided to it for suppressing the debiasing effect in the diode array [19,20].
The scanning procedure and the experimental setup used in our experiments are schematically illustrated in Figs. 8 and 9. Different decreased levels of diode photocurrents were implemented by varying the gate-potential value V g of input field effect transistors (FETs) in the photoelectric cells of the detectors (see Fig. 8). The scans were performed at a 1-μm step by translational displacements of the objective lens installed on a high-precision PC-controlled motorized XYZ-stage. The photoresponse S was registered with the help of a shutter controlled by the same personal computer; usually, the photoresponse values were obtained by subtracting the average of typically 10-20 frames taken with closed shutter from the average of the same number of frames taken with open shutter. the dielectric) was 10x10 μm, and the thickness of the continuous p-type Hg vacancy-doped MCT photosensing layer was about 10 μm. The concentration of acceptors in the absorber layer was Na = (7÷10) x 10 15 cm -3 . The diode arrays in our detectors were fabricated using boron implantation. Both detectors contained graded-gap wideband MCT layers grown on both sides of the MCT film to ensure surface insulation of FPA diodes and suppress the recombination of charge carriers at the film boundaries [14]. The MCT film of the D-2 detector additionally contained a heavily doped ntype MCT layer with a larger bandgap energy provided to it for suppressing the debiasing effect in the diode array [19,20].
The scanning procedure and the experimental setup used in our experiments are schematically illustrated in Figs. 8 and 9. Different decreased levels of diode photocurrents were implemented by varying the gate-potential value Vg of input field effect transistors (FETs) in the photoelectric cells of the detectors (see Fig. 8). The scans were performed at a 1-μm step by translational displacements of the objective lens installed on a high-precision PC-controlled motorized XYZ-stage. The photoresponse S was registered with the help of a shutter controlled by the same personal computer; usually, the photoresponse values were obtained by subtracting the average of typically 10-20 frames taken with closed shutter from the average of the same number of frames taken with open shutter.
For diminishing the chromatic aberrations in the objective lens, as the secondary source of IR radiation, we used the exit slit of a monochromator through which IR radiation around wavelengths λ = 4.5 or 8 μm (respectively, in the case of the D-1 and D-2 detectors) cut off from the globar emission spectrum, passed. Those wavelengths were roughly at the maximum of the spectral sensitivity of the detectors. According to [21], the IR radiation at the above-mentioned wavelengths was absorbed in the absorber layer of the D-1 and D-2 detectors over the characteristic optical lengths labsorp ≈ 0.2 and 2.4 μm. In the chosen optical configuration, for typical slit-width values of 0.1-0.2 mm, the width of the focused image of the slit at the FPA location in the geometrical optics approximation was smaller than ~8 μm. The diffraction of IR radiation caused some smearing of the slit image on the FPA. The distributions of the illumination  For diminishing the chromatic aberrations in the objective lens, as the secondary source of IR radiation, we used the exit slit of a monochromator through which IR radiation around wavelengths λ = 4.5 or 8 μm (respectively, in the case of the D-1 and D-2 detectors) cut off from the globar emission spectrum, passed. Those wavelengths were roughly at the maximum of the spectral sensitivity of the detectors. According to [21], the IR radiation at the abovementioned wavelengths was absorbed in the absorber layer of the D-1 and D-2 detectors over the characteristic optical lengths l absorp ≈ 0.2 and 2.4 μm. In the chosen optical configuration, for typical slit-width values of 0.1-0.2 mm, the width of the focused image of the slit at the FPA location in the geometrical optics approximation was smaller than ~8 μm. The diffraction of IR radiation caused some smearing of the slit image on the FPA. The distributions of the illumination intensity I(x) across the excitation spot as implemented in the experiments with the D-1 and D-2 detectors (see Fig. 10, curves 1) were calculated under the assumption that the optical system used in our experiments was a diffraction-limited one, by summing the Airy disc distributions centered at each point of the geometric image of the slit [22]. The manner in which the radiation intensity I diffracted in the vicinity of a strip-like illumination spot of width b was calculated is illustrated by Fig. 11. The calculations of I in the geometrically illuminated region of FPA (x≤b) and outside this region (0≤x≤b) were performed by formulas 0 ( ) 2 ( , ) acos( ) acos( ) 2 ( , ) acos( ) , is the Airy function (ξ = 2π D λ θ, J 1 is the first-kind Bessel function, D is the lens diameter, λ is the radiation wavelength, θ=atan(r/R) is the diffraction angle, r is the radius-vector in the plane of FPA, and R is the lens-to-FPA separation).
The experimental conditions that were adopted in measuring the diode photoresponse profiles S(x) in the D-1 and D-2 detectors are summarized in Table 2. For extending, at low and high photocurrent values, the range of photocurrents j ph implemented in our experiments, we used such means as increasing the integration time t int (e.g., from 150 μs to 2 ms in the case of the D-2 photodetector while performing measurements at low currents j ph ) or attenuating the illumination intensity in the radiation spot via narrowing the entrance slit of the monochromator (in cases where off-scale swings of the analog-to-digital converter output signal at high photocurrents j ph had to be avoided). Figure 10 shows the wings of the profiles S(x) that were measured at various values of V g in experiments with the D-1 (curve 2 and symbols 3-5) and D-2 (curve 2 and symbols 3-6) detectors. As the right and left wings in the measured profiles were slightly asymmetrical, presumably because of some nonuniformity of the recombination properties of the absorber material and because of a slight difference in the electrical characteristics of the neighbor FPA diodes, Fig. 10 shows only the right wings of the profiles, from which data used in subsequent calculations were taken. For clarity, the curves S(x) measured for the D-1 detector in operation with V g ≤0.96 V and for the D-2 detector in operation with V g ≤0.90 V are shown in Fig. 10 in a smoothed form. The relation between the as-measured and smoothed profiles S(x) for the noisiest experimental curves (data for D-2 measured at V g =0.73 and 0.74 V) is illustrated in Fig.  12. It is seen that the case of V g =0.73 V was the noisiest experimental situation, while already at V g = 0.74 V the spot-scan profile S(x) was measured quite reliably. The profiles S(x) measured in both detectors at V g ≤0.9 V are noticeably widened in comparison with the profiles taken from the D-1 and D-2 detectors, respectively, at V g = 1.086 V and V g = 0.95 V, that is, at gate- potential values close to the optimum ones in normal operational regime of the detectors (see Fig. 10). On increasing the gate-potential value in excess of 0.95-1.00 V, the shape of the registered spatial photoresponses exhibited saturation, and the normalized profiles S(x), therefore, remained identical. Wishing to avoid dull data repetition, in Fig. 10 we do not illustrate this fact. Below, we give an analysis to measured profiles S(x) separately for the range of V g -values in which distinct widening of the bell-shaped profiles S(x) with decreasing the level of diode photocurrents j ph was observed (V g <0.95-1.00 V) and for the range of V g -values in which the shape of the profiles S(x) became independent of V g (V g >0.95-1.00 V) (conventionally, the cases of small and large diode photocurrents).  Notation: V g is the gate potential of photoelectric-cell input FETs with respect to the frame electrode of the diode array, t int is the integration time, I 0 is the photocurrent produced by a photodiode centered at the axis of the illumination spot, I is the IR radiation intensity in the illuminated spot (in arbitrary units different for D-1 and D-2), s is the width of the monochromator exit slit, l=l 1 +l 2 is the distance from the monochromator exit slit to the lens (see Fig. 9), D and F are the lens diameter and focal length, Δ is the width of the geometric image of the monochromator exit slit at the FPA location, l d eff is the "spatially averaged" effective diffusion length of minority carriers in the PF of the FPA detector in the model with uniform photocurrent suction, k is the value of the coefficient k found in the 1D diffusion model with uniform photocurrent suction, and k ph opt is the suction coefficient in the region under the photodiodes in the 2D diffusion model with structured photocurrent suction.

The case of small diode photocurrents j ph
As the IR radiation intensity reaching the n-regions of the FPA diodes was negligible, the measured diode photoresponses were primarily due to photoelectrons excited by IR radiation in the p-type absorber layer and sinking to the n-regions of the photodiodes. The spot-scan data obtained at small values of V g were analyzed within a continuum diffusion model that was developed to clarify the expected asymptotic behavior of l d eff (j ph ) for j ph → 0. This model could be used because, at the low levels of j ph , we had l d eff ≥ l SCR /2 and l d eff ≥ d (here l SCR is the separation between the depletion edges of adjacent FPA diodes and d is the PF thickness) [18].

Continuum diffusion model. Derivation of the asymptotic ld eff → ld for jph → 0
The model is based on the following assumptions: (i) the lateral diffusion of the photogenerated charge carriers proceeds in an infinitely thin film; (ii) the photoelectrons are extracted out of the photosensitive film, instead of a discrete diode array, by a large-area electrode continuously covering the entire surface of the film; (iii) the local photocurrent density jph varies over the diode area in proportion to the local sheet density of excess minority carriers ns with a coefficient whose magnitude depends on Vg; (iv) the total photocurrent across the diode can be calculated by integrating the local photocurrent density jph over the diode area, which has an efficient size t. For our detectors, a value t=14 μm was adopted in the calculations described below (the nominal window size plus 2 μm on either side of the photodiode) [23,24]. In the formulated model, the spot-scan profiles S(x) can be found by solving the 1D diffusion equation which can also be written as

Continuum diffusion model. Derivation of the asymptotic l d eff → l d for j ph → 0
The model is based on the following assumptions: (i) the lateral diffusion of the photogenerated charge carriers proceeds in an infinitely thin film; (ii) the photoelectrons are extracted out of the photosensitive film, instead of a discrete diode array, by a large-area electrode continuously covering the entire surface of the film; (iii) the local photocurrent density j ph varies over the diode area in proportion to the local sheet density of excess minority carriers n s with a coefficient whose magnitude depends on V g ; (iv) the total photocurrent across the diode can be calculated by integrating the local photocurrent density j ph over the diode area, which has an efficient size t. For our detectors, a value t=14 μm was adopted in the calculations described below (the nominal window size plus 2 μm on either side of the photodiode) [23,24]. In the formulated model, the spot-scan profiles S(x) can be found by solving the 1D diffusion equation which can also be written as Here, l d eff = l d / 1 + k and τ eff = τ / (1 + k) are the effective diffusion length and the effective lifetime of excess carriers in the photosensitive film (τ is the lifetime of excess carriers in the bulk absorber material, k is the coefficient that relates the local photocurrent density j ph to n s , j ph =k × n / τ), and g is the rate of sheet photogeneration of charge carriers in the photosensitive film. For the normalized profiles S(x), we then have [18]: The curves S(x) calculated by formula (3) Table   2; in Fig. 13, they are plotted (for fixed illumination intensity in the excitation spot) against the photocurrent I 0 that was produced by the scanning diode when the diode was centered at the spot axis (the current I 0 was evaluated from the maximum photosignal ΔV 0 in the registered scans by the formula I 0 = C × ΔV 0 / t int , where C=1.22 pF is the photoelectric-cell charge storage capacity and t int is the integration time).
The curves in Fig. 13 show the dependences l d eff (I 0 ) as predicted by the model; those dependences are defined by a nondimensional parameter α = ( j 0 × l d ) / G that varies in proportion to I 0

Optoelectronics -Materials and Devices
(here j 0 = I 0 / (q × p 2 ) is the average flux of photoelectrons through the spot-centered pixel area (q is the electron charge) and G is the rate of the sheet photogeneration of charge carriers in the illumination spot per unit length of the spot): ( ) Formula (4) can be derived as follows. Assuming that n(x) ∝ exp( ± x / l d eff ), putting g ≈ 0 outside the spot, and writing the coefficient k as j 0 / (n 0 × τ) (here, n 0 is the sheet density of photogenerated charge carriers in the spot), from Eq. (2) we obtain: Considering now the problem over the scale ~ l d eff and treating the distribution of illumination intensity in the spot as a delta-function, from the condition of matching the lateral photocurrents that spread from the spot to the left and to the right, for the photoelectron density n 0 we obtain: On substitution of expression (6) into relation (5), for x = l d eff l d we then derive a quadratic equation whose solution yields formula (4).
According to the asymptotic of (4), at low currents I 0 the effective diffusion length l d eff decreases quasi-linearly with increasing current I 0 in such a way that the slope α/I 0 defines simultaneously the weakly nonlinear behavior of l d eff (I 0 ). As it is seen from Fig. 13, both conclusions fairly well agree with the experimental data. Interestingly, from the found value of α/I 0 one can readily obtain an estimate of G. The 1D continuum model described above possesses the following advantageous features: (i) in treating the withdrawal of charge carriers out of the photosensitive film, this model permits leaving aside such things as boundary conditions for the minority carrier concentration n, which would be dealt with in the 3D diffusion model; (ii) the model explicitly involves a quantity l d eff whose asymptotic value at low photocurrents gives the bulk diffusion length of minority charge carriers l d in the photosensitive film; (ii) the model admits derivation of an analytical expression for the asymptotic behavior of l d eff (j ph ) as j ph → 0.

Evaluation of the length l d in the absorber layer of the examined IR FPA detectors
Approximation of the curve l d eff (I 0 ) to I 0 =0 according to asymptotic (4) yields for l d a value of 19.5 μm in the absorber material of the D-1 detector and a value of 24 μm in the absorber material of the D-2 detector. Those values are in good agreement with relevant data that were previously reported in the literature (see, e.g. [5,13,25], where the values of 19 μm [25], 20-23 μm [13], and 25-35 μm [5] were reported for the carrier diffusion length l d in p-type MCT at 78 K), and also with the diffusion-length values calculated from the lifetime and mobility data for p-type MCT of similar stoichiometry (see, e.g. [26], where the lifetime and mobility values τ=10-15 ns and µ e =6.8×10 4 cm 2 /V×s, translating into l d =21.4-26.2 μm, were obtained for a Cd x Hg 1-x Te material with x=0. 20-0.23).
It would also be instructive to correlate the diffusion-length data obtained in the present study with the available minority-carrier lifetime data for Hg-vacancy-doped MCT. According to [27], the lifetime of excess carriers in the p-type Cd x Hg 1-x Te material with x=0.225 and 0.30 doped with Hg vacancies to a concentration of (7-10)x10 15 cm -3 is approximately τ=2-4 and 5.5-10 ns. Evaluating the electron mobility in MCT by traditional formulas [21], we obtain the mobility values µ e =1.06x10 5 Fig. 10 is that for low gate voltages S(x)-data become less complicated and more prone to be fit, in an extended range of x, with a single exponential behavior. Fitting the S(x) data for our photodetectors with a single exponential decay constant yields values of l d eff that well coincide (within ~1-1.5 μm) with the values of l d eff obtained from the calculations based on formula (3).

The case of large diode photocurrents j ph (normal operational regime of FPA detectors)
At currents I o ≥0.5 nA/pixel in the D-1 detector and ≥1.5 nA/pixel in the D-2 detector, the length l d eff becomes comparable with the characteristic lateral size of the diode-array structure l SCR , making it necessary to take the nonuniformity of that structure into account in calculating the diode photoresponses S. Moreover, at large diode photocurrents, the local effective diffusion length of minority carriers in the region under the back-biased diodes l d eff becomes comparable with d, and the second applicability condition of the 1D diffusion model, l d eff > d, also becomes violated. Disregarding, for the time being, the second circumstance, in our analysis of spotscan data we tried to allow for the discrete structure of the FPA by performing calculations of 2D distributions of the excess carrier concentration n(x,y) in our locally illuminated photodetectors within a 2D diffusion model with d=0 presenting an extension of the previously used 1D diffusion model to the case of two lateral dimensions.

2D calculations
The 2D distribution of the sheet density of photogenerated charge carriers in the photosensitive film of the IR FPA detectors n s (x,y) with g(x) ∝ I (x) was calculated as the solution of the 2D diffusion equation In Eq. (7), the discrete structure of the diode array was taken into account via the function P(x, y), which was assumed to be unity in the region under the FPA diodes and zero in the region outside the FPA diodes. We thereby assumed that, over the area occupied by the FPA, the local diffusion length of charge carriers in the region outside the diodes was equal to l d . In the region under the diodes, a smaller value of the local (effective) diffusion length, l d eff , defined by a value of k ph to be found via the fit of calculated to experimental spot-scan data, was assumed. [Here, we would like to remind the reader that under the local diffusion length at a point of interest, we everywhere understand the average distance the charge carriers would move in the absorber material if they were spreading from that point in a spatially uniform photosensitive film with recombination properties of the absorber material being everywhere identical to the recombination properties of the material at the point under consideration. Under the effective diffusion length l d eff , we understand the diffusion length of charge carriers defined, in addition to their bulk recombination in PF (which, taken alone, defines the value of l d ), also by the disturbing action due to FPA diodes. The notion of local effective diffusion length of charge carriers, whose meaning can be comprehended by combining the above two definitions, is a notion pertinent to the 2D diffusion model with d=0 used in the present analysis].
The equation was solved by using the MathCad function relax in a 1024x1024-μm square domain covered by a square grid whose mesh size was 1 μm. The illumination spot with the distribution of illumination intensity I(x) stretched in the middle of the calculation domain along the y axis. At the calculation-domain boundaries parallel to the centerline of the spot, zero boundary conditions were adopted for the sheet density of excess carriers (n s =0). The starting boundary conditions at the domain boundaries normal to the illumination spot were initially taken from the solution of the related 1D diffusion problem, and then, at a second iteration made to refine the solution, from the condition of periodicity of the solution in the spot-parallel direction. Afterward, proper integration (over a 14x14-μm square) was used to calculate the relative values of S for various positions of the illumination spot with respect to the scanning diode. As an example, Fig. 14 shows the 2D plots and contour lines of the solutions n s (x,y) obtained in calculations with k ph =40 for the case in which the scanning diode in the D-2 detector was at the centerline of the illumination spot ( Fig. 14 (a)) and for the case in which its center was displaced from the spot axis by 10 μm (Fig. 14(b)). The purpose of our 2D calculations was to find such values of k ph at which the best coincidence between the normalized calculated and experimental diode photoresponse profiles S(x) could be achieved for the experimentally implemented values of V g , including those for the normal operational regime of the detectors. The data were obtained for the D-1 detector in the calculation with k ph =40.

Calculated data and their comparison with the measured spot-scan profile S(x)
The found values of k ph , k ph opt , are listed in the last column of Table 2. The values of k ph opt found for the D-1 detector in operation with V g ≤0.90 V and for the D-2 detector in operation with V g ≤0.95 V proved to be 4-5.5 times greater than the optimal values of the coefficient k, k , found in the continuum model of [18] with uniform current suction. With the factor 4-5.5 being close to the ratio between the pixel and diode areas, it can be concluded that both models, the model with continuous photocurrent suction and the model with structured photocurrent suction, yield adequate descriptions to the data measured at small values of V g providing that one and the same average photocurrent density on the area occupied by the FPA is assumed. The latter result can be regarded as verification of the applicability of the approach of [18] to the analysis of spot-scan data obtained for reduced potentials V g . For values of V g typical of the normal operational regime of the detectors (V g =1.086 V for D-1 and V g =0.95 V for D-2), the shape of the spatial diode photoresponses S(x) proves to be more complicated in comparison with the profiles S(x) measured at reduced values of V g , with the wings of the profiles S(x) being less prone to fitting with a single-exponential behavior (Fig. 10). A comparison between the spotscan profile measured at V g =0.95 V for the D-2 detector and the profiles S(x) calculated for the same detector by the 2D model with k ph =20, 40, and 60 is given in Fig. 15. Evidently, with k ph ≈ 20-40, a good agreement between the measured profile and the calculated spot-scan profiles was achieved, which shows that, at least in some practically important cases, the model with d=0 is also capable of providing a satisfactory approximation of the measured with calculated profiles S(x) for the values of V g typical of the normal operational regime of the detectors. then, we obtain: then, we obtain:

S(x)
x, m 1 2 3 4 5 Figure 15. Spot-scan profile S(x) in the D-2 detector measured at V g =0.95 V (curve 2) in comparison with the profiles S(x) calculated by the 2D diffusion model with k ph =20, 40, and 60 (curves 3, 4, and 5, respectively). Curve 1 is the distribution of the IR radiation intensity.
The above values of k ph , k ph opt ≈ 20-40, refer to the effective diffusion-length values of charge carriers in the region under the photodiodes l d eff = l d / 1 + k ph opt ≈ 3.7-5.2 μm. It should be noted here that the above-described fitting procedure yields an upper-type estimate for l d eff because, with increasing the magnitude of k ph over k ph opt ≈ 20-40, the calculated photoresponses S (and also the shape of the profiles S(x)) proved to be weakly dependent on k ph , exhibiting saturation in the vicinity of the measured S-values. Yet, it can be shown that the obtained values l d eff ≈ 3.7-5.2 μm agree well with a theoretical estimate of the length l d eff (see below). Such an a priori estimate of the length l d eff in the region under photodiodes can be obtained due to the fact that the values of the second derivatives of the excess carrier concentration n in the lateral direction normal to the spot axis and in the direction across the absorber layer are interrelated by the 3D diffusion equation, while some plausible assumptions about the shape of the cross-film and subsequent recombination of those carriers in this region) violates the previous tacitly adopted assumption that the photocurrent I o exp always presents a good approximation to the electron photocurrent being extracted by the photodiodes out of the photosensitive film. Indeed, due to a nonzero rate of "surface recombination" of excess charge carriers on the film surface covered by the photodiode, the electron flux across the p-n junction remains finite even as the net photocurrent through the forward-biased photodiode loaded to the high-resistance load tends to zero. This point can be comprehended by considering the diagram of the photocurrent components in the p-n junction shown in Fig. 18.  Thus, it is the photocurrent I o calc that had to be plotted instead of photocurrent I o calc along the abscissa axis in Fig. 13. Within the context of the present study, it is essential that the readily stemming modification of the determination procedure for the length l d (using I o calc -values instead of I o calc -values in plotting the l deff -vs-I 0 graphs) typically induces quite moderate or even negligible corrections to the initially obtained values of l d . Indeed, for the D-2 detector the modified procedure yields a value l d ≈ 25.5 μm, this length being rather close to the previously obtained value of 24 μm. For the D-1 detector, the modified procedure yields a value of l d being perfectly coincident with the previously obtained estimate l d ≈ 19.5 μm.

Conclusions
Two novel methods for determining the bulk diffusion length of minority charge carriers in photosensitive MCT films have been proposed.
The first method, suitable for determining the length l d in MCT films with suppressed surface recombination of excess carriers, uses tailored diode structures with photodiodes whose vicinity is shielded from incident radiation by a coaxial metal contact. Photocurrent measurements performed on the diodes with different radii of cap contacts can be used to determine the length l d via a comparison of measured photocurrents with the photocurrents numerically calculated for examined diode configurations. Experimentally, for an Hg-vacancy-doped Cd x Hg 1-x Te film with x=0.223 and hole concentration p=6.7x10 15 cm -3 l d -values ranging between 19 and 23 μm were obtained.
The second method, which can be used for determining the length l d in the continuous (thickness-uniform, without mesa-isolation of diodes) absorber layers of MCT 2D IR FPA detectors, is based on an analysis of spot-scan data obtained for such detectors at different levels of diode photocurrents. Experimental data gained for n-on-p MCT 2D MWIR and LWIR FPA detectors were analyzed using a 2D diffusion model taking the discrete structure of FPA into account. The properties of the used model were discussed to show that this model could indeed be applied to the analysis of the diffusion process under study. As a result, a general scheme for a comprehensive spot-scan analysis of MCT IR FPA detectors has been proposed. We believe that the newly proposed methods for determining the diffusion lengths of charge carriers in MCT films will add to the toolkit of characterization means for MCT-based IR FPA detectors.