Current relaxations in AlGaN/GaN high electron mobility transistors (HEMTs) often show a broad spread of relaxation times. These are commonly linked to the ionization energies of the traps in different regions of the devices and the relaxations are assumed to be exponential. To explain the observed spread of parameters, the presence of multiple centers is assumed. However, in actual spectra, only a few main peaks in the lifetimes distributions are observed, with considerable broadening of the peaks. In this paper, the authors examine the possible origin of the relaxation time broadening, including the presence of disorder giving rise to extended exponential decays and to physical broadening of discrete levels into bands. The latter is modeled by Gaussian broadening of the logarithm of relaxation time. The authors demonstrate the analysis of the peak positions and widths of the first derivative of the current transient by the logarithm in time, which is quite useful in deriving the relevant broadening parameters. They illustrate the approach for current relaxations in HEMTs for different pulsing modes.

## I. INTRODUCTION

High electron mobility transistors (HEMTs) based on AlGaN/GaN heterojunctions are of interest for high-power and high-frequency applications.^{1} These structures show breakdown voltages up to 1000 V, power output of up to 20 W/mm at 4 GHz, and are operable even beyond 400 °C.^{2,3} However, their performance is still handicapped by trapping by deep centers, which creates variations of ON-resistance, gate-lag, and drain-lag effects generally known as current collapse.^{3} The origin of these phenomena is often attributed to nonequilibrium capture of electrons by traps in the bulk or at surface of the AlGaN barrier between the gate edge and the drain, to capture of carriers by traps in the GaN buffer or at the AlGaN/GaN interface.^{4–7} Measuring the activation energy, capture cross section, and concentration of these traps is of crucial importance for optimizing HEMT growth and processing technology. The methods employed include measurements of individual drain and gate current relaxation curves at various temperatures,^{4,8} current and capacitance deep level transient spectroscopy (DLTS),^{9–13} and optical and thermal spectroscopies of drain current and photocurrent relaxations.^{14–16}

The relaxation curves of drain current are often not described by a single exponential decay.^{3,7,14,15} To deal with this, some have proposed to describe individual current build-up and relaxation curves by a sum of multiple exponents with decay or build-up times uniformly distributed over the entire probed time domain.^{8} Nonlinear least squares fitting procedure are then used to determine the characteristic times describing experimental curves. Temperature-dependent measurements allow a determination of specific activation energies for the processes involved and to infer the trapping/detrapping mechanisms. Drawbacks with this procedure include the presence of ripples in the relaxation time spectra and a strong dependence of the results on the choice of the base parameters if standard mathematical packages were used for fitting.^{17–19} Several modifications to the method have subsequently been proposed.^{17–19} However, the number of peaks in the spectra obtained is usually limited to 2–3, while the peaks come out more or less symmetrically broadened. However, the algorithms used do not allow assigning any physical meaning to such broadening.

An alternative approach widely used is the analysis of the derivative of the current relaxation by the logarithm of time.^{3,7,20} The values of characteristic time constants are determined from the peak positions and used to derive the emission energies. The number of peaks is also limited (usually two) and widely spaced in time. No attempt to derive information from the width of the peaks was made. Dealing with current transients described by two exponents considerably differing in time is easily treated in classical DLTS treatments designed exactly for automatically separating such contributions. However, there exist many reasons for the lifetimes at a given temperature to be spread over a certain range. If the electrons are captured by the surface states, the emission occurs from a band of states rather than discrete levels (for example, the density of surface states are known to form a wide U-like zone with spikes in the density of states [e.g., near E_{c}–0.3 eV (Ref. 21) or near E_{c}–(0.6–0.7) eV (Ref. 22)]. The presence of the Poole-Frenkel effect in HEMTs may strongly change the activation energy of the trap emission and lead to the zone formation for emission from traps located in the region of strong electric field near the gate edge.^{3,4,7} Similarly, nonuniform elastic strain in HEMTs can also lead to the formation of zones of trapping–detrapping states instead of discrete levels because of the presence of strong piezoelectric field in III-nitrides.^{3}

A further complication is that the threshold voltage of GaN-based HEMTs depends on the concentration of deep traps in the barrier, in the buffer, and at the interface.^{7,9,23,24} The concentration of these traps is often high,^{23} so that their local density variations can cause potential fluctuations in the channel and even lead to the appearance of percolation-type conduction,^{7,24,25} contributing to blurring of individual relaxation times. Furthermore, many traps in GaN are characterized by a strong electron-phonon interaction causing the appearance of capture barriers.^{7} Current relaxations in the presence of such traps often have stretched exponents rather than simple exponents.^{26,27} In GaN HEMTs, the stretched exponents are often invoked to describe the relaxation curves waveforms for current relaxations produced by illumination.^{14,15} Thus, there exist multiple reasons to expect the levels in GaN HEMTs to be either broadened into a band of states or the relaxation times to be spread over a long period due to the existence of potential fluctuations or to the existence of the barriers for capture of carriers. The band situation is most naturally described by the Gaussian broadening of respective peaks and the relaxation in the disordered system or a system with barriers for capture is traditionally described by invoking stretched exponents relaxation instead of pure exponents.^{26,27}

To our knowledge, no attempt has been made to deal with broadening effects and to treat the situation quantitatively by analyzing the individual relaxations at various temperatures or the waveform of the current transient derivatives by the logarithm of time. We compare the results of two models, one describing the relaxations by stretched exponents, the other taking into account in a very simplified manner the effects of Gaussian broadening of the trap level. We derive some criteria that allow to distinguish the cases of simple multiexponential relaxation from the effects of lifetime broadening and, in some cases, decide between the two broadening models. The approach is illustrated by analyzing the data obtained on AlGaN/GaN HEMTs under more or less typical pulsing conditions. We also compare the trap parameters determined from this advanced characterization with those observed in standard current DLTS (CDLTS) procedure. This latter is shown to yield trap parameters reasonably close to those calculated via the “exact” approach, albeit at the expense of losing the detailed information on broadening mechanisms.

## II. CALCULATION APPROACHES

### A. Pure exponents

The current relaxation is described, for a single exponent, as

where A1 is a constant representing the amplitude of relaxation, and τ is the relaxation time.

Differentiating it by ln(t) gives

By substituting δ = ln(t/τ), one gets a dimensionless equation

This derivative has the peak at δ = 0 (i.e., t = τ), and the amplitude of the peak is A1/e. The peak is asymmetric and is much broader to the left of the peak (half width FWHM approximately −1.5 in δ coordinates) than to the right of the peak (half width approximately 1 in δ coordinates).

### B. Extended exponents

For a single extended exponent, the relaxation has the form

where $0<\beta \u22641.$

The derivative of the relaxation by ln(t) then can again be represented as a function of dimensionless variable δ = ln(t/τ) as

The peak position in δ coordinates is still at δ = 0 (t/τ = 1), and the peak amplitude is −βA2/e (e is the base of the natural logarithm). The peak is also asymmetrical, but the halfwidths in δ are approximately 1.5/β (left) and 1/β (right).

### C. Gaussian broadened exponents

Consider a band of states emitting electrons into the conduction band. Suppose the density of states in this band has the Gaussian distribution peaked around energy E_{o} (in respect to the conduction band edge) with the standard deviation of energy σ_{E}

where N_{o} is the peak density of states, and E_{T} is the running energy value. Then, the current relaxation for the emission from such a band is a convolution of individual current relaxation contributions from each segment of the band given by the integral

Here, τ is the emission time corresponding to the running energy E_{T}

σ_{n} in Eq. (8) is the electron capture cross section, N_{c} is the density of states in the conduction band, v_{th} is the thermal velocity of electrons, and k is the Boltzmann constant.^{28} Here, we note that the integral in Eq. (7) can be expressed through ln(τ) using Eqs. (6)–(8) as

The expression has the form of the integral of the normal distribution of the density of states of ln(τ) multiplied by exp(−t/τ), where the peak in the density of states at ln(τ_{o}) corresponds to the peak in the energy distribution at E_{o}, the running τ value is given by Eq. (8), and the new normal deviation in ln(τ), σ_{ln(τ)}, is related to the normal deviation in energy σ_{E} by the expression

Then, we can substitute the integral in Eq. (9) by the sum by different states i of the form y_{o} + ∑N_{o}(i)exp(−t/τ_{i}) in which the amplitudes N_{0}(i) have the normal distribution on ln(τ) as in Eq. (9) [the term y_{o} takes into account the dark current not explicitly introduced in Eq. (9)]. The sum is taken over a large number of members equally spaced in ln(t) in the probed region (we tried running the sums with the number of points from 30 to 200).^{8} The huge advantage, however, is that there is no need to fit all 200 or so amplitudes and τ values over a very wide range of experimentally probed times, but do the summation only around the ln(τ_{o}) point over the interval of ±3σ_{ln(τ)}. The latter is chosen to provide a high level of accuracy (>99%) in the peak density of states. At that, since the amplitudes are given by the distribution N(ln(τ_{i})) = N_{o}exp[-(ln(τ_{o})-ln(τ_{i}))^{2}/(2σ_{ln(τ)}^{2})], the fitting requires only finding the right parameters N_{o}, ln(τ_{o}), σ_{ln(τ)}, and y_{o}. This process converges very much faster than the sum of independent exponents [we could as a rule obtain convergence of the fitting process by using 30 points around ln(τ_{o}), while for independent exponents, in our experience, even 200 exponents were not sufficient to suppress rippling and obtain consistency]. Moreover, numeric differentiation by ln(t) of simulated curves showed that, for not too high normal deviation values, the peak in the derivative is still fairly close to the t/τ = 1 point. Thus, one gets a good idea of the starting parameter to feed into the fitting process. Performing the fitting process for multiple temperature points yields the center of the peak position in respect to the conduction band and the electron capture cross section from Eq. (6). It also produces as a bonus the value of the normal deviation in ln(τ) that can be easily converted into the normal energy deviation using Eq. (10), thus providing the width of the band at the given temperature. Therefore, the procedure is in a way equivalent to the DLTS approaches treating the spectra of interface traps in MIS structures,^{22,28–31} although deriving the densities of states from the measured current relaxation curves requires a more detailed analysis of the relation between the fitted amplitudes and the density of states. Also, it is possible, in principle, to derive a similar procedure for the density of states distributions other than Gaussian, e.g., parabolic. The equations above are written for the band exchange with conduction band, but transitions involving hole emissions are naturally of the same form. However, the actual mechanisms giving rise to hole emission have to be constructed for HEMT structures.

### D. Discriminating between the three models

We have seen that, for pure exponents and for stretched exponents, the halfwidths of the peak in the derivative by ln(t) measurably differ and these halfwidths can be used to determine which of the model is applicable and to determine the value of β value in the stretched exponent. The problem, however, becomes less tractable for the case of several overlapping peaks. In principle, the peaks can be deconvoluted into separate contributions, e.g., by assuming the Gaussian distribution on δ or fitting individual relaxations and determining the β values from them. It is also a help to note that the first and second derivatives dependences on δ have a similar form in both cases

For two overlapping peaks, the β value can be estimated from the halfwidth in δ on the high δ side [FWHM (δ_{1/2}^{+})]. The ratio of the second and first derivatives at this point (the broadening on the low δ side is considerably stronger than on the high δ side) is described as

Eq. (13) can serve as an additional consistency check, when deconvoluting overlapping peaks.

Another consideration that can help to distinguish the cases of simple exponential relaxation and the stretched exponent case comes from looking at the physical origin of the latter mode. It is generally acknowledged that such a relaxation is the consequence of the presence of potential fluctuations in the system or the existence of the barrier for capture of charge carriers. In both cases, the relaxation times are expected to become shorter with the rising of the quasi-Fermi level and consequent decrease in the height of effective barrier. This change in the quasi-Fermi level can come as a result of increasing the intensity of illumination or the length of the pulse for light pulses.^{26,27,32} Similarly, one would anticipate decreasing of τ for increasing the width of the “injection” pulses in electrical pulsing. For simple emission from a discrete level, such a dependence on the quasi-Fermi level is not expected. The difference in behavior can serve to discriminate these two cases.

The situation with discriminating stretched exponents and the Gaussian-broadened relaxations is more complicated. The internal reasons for broadening are somewhat similar. At the same time, the complicated expressions for the Gaussian broadening case make it very difficult to obtain analytically the values of the halfwidths and their relation to the standard deviation. From what follows we will see that in many cases the stretched exponents and the Gaussian exponents yield similar quality of fitting of individual I(ln(t)) and I′(ln(t)) curves. Some guidance here can be obtained from noticing that in many cases the width of the energy band giving rise to the broadened ln(τ) distribution is not expected to vary significantly with temperature (although this is only the rule of thumb and the actual situations giving rise to the current transient have to be analyzed). Then, if Eqs. (9) and (10) hold, the values of the σ_{lnτ} obtained for different temperatures should decrease with rising temperature as ∼1/kT. We illustrate the above approach by the example of current relaxations analysis in AlGaN/GaN HEMTs.

## III. EXPERIMENT

### A. Samples

These experiments were carried out on epitaxial structures grown by metalorganic chemical vapor deposition on Si(111) substrates using thin AlN nucleation layer and several graded AlGaN strain relieving layers. The GaN buffer was carbon doped and semi-insulating, with thickness of 4 *μ*m. The barrier was Al_{0.25} Ga_{0.75} N with the thickness of 25 nm and a 0.5 nm GaN cap layer. These structures were processed into HEMTs with the distance between the source (S) and drain (D) of 5 *μ*m and the gate (G) that was 0.5 *μ*m long and 200 *μ*m wide. The Ohmic source/drain contacts were prepared by deposition of Ti/Al/Ni/Au and annealing at 850 °C for 60 s in N_{2}, and the Schottky gate was formed by deposition of Ni/Au. The gate-to-drain distance was 2 *μ*m. The fabrication process has been described in detail previously.^{23–25}

### B. Measurement techniques

Current–voltage (I–V) characteristics and pulsed current transients were measured with a current/voltage source/meter. Individual drain current relaxations following gate voltage pulses, drain voltage pulses, or optical injection pulses were measured in the temperature range 290–415 K with an accuracy of 0.1 K. On each relaxation curve, 15 000 points were measured and stored with time steps typically from 0.2 to 8 ms, so that the entire measurement time was up to ∼100 s. For optical pulses at fixed gate and drain voltages, light-emitting diodes (LEDs) emitting from 365 to 850 nm were used. The current changes during electrical or optical pulsing were studied in two modes: in the first, the length of pulse was varied, and in the second, the voltage was switched from the initial condition to the excited condition and the current change upon pulsing monitored. More details of the test setups have been given elsewhere.^{6,7}

### C. Signal processing

Individual relaxations measured at each temperature were smoothed to obtain a set of points equidistant in the log(t) space. The obtained set was then fitted with a low order polynome (usually second order) over the array of several consecutive points (5–25 depending on the actual noise) running from the start to the end of the entire relaxation curve. The first and second derivatives by log(t) were calculated by differentiating respective polynomes. The first derivative was plotted against the logarithm of time to determine the peak times further used for feeding into the fitting process for actual current relaxations at each temperature. The fitting with one of the relaxation models described in Secs. II A–II C was performed using the origin 9.0 software package and the nonlinear least squares minimization procedure. For logarithmic Gaussian fitting, a special script was written within the origin 9.0 package.

## IV. RESULTS AND DISCUSSION

I–V measurements showed that the threshold voltage of the HEMT was −4 V. This was determined by the gate-bias intercept of the linear extrapolation of the drain–current at the point of peak transconductance. In these structures, it did not vary more than a few tenths of volts with temperature because the carrier concentration is stable over our range of measurement temperatures.

Different pulsing regimes will be described. In the first, the drain voltage was set at 0.1 V, and the gate bias was fixed at −3.8 V and pulsed to 0 V for 2 s. In the second mode, the HEMT was kept at a drain voltage of 0.1 V with gate voltage of −3.9 V (almost closed) and illuminated with light pulses with different photon energies and different pulse lengths. In the third mode, the HEMT was kept at a drain voltage of 0.5 V (linear I–V region) and gate bias of −3.3 V (semi-on) and pulsed to V_{g} = −4 V (off-state). This sequence is more or less standard in assessing current collapse phenomena.^{3}

### A. Pulsing regime 1: V_{d} = 0.1 V, V_{g} = −3.5 V, pulsing to 0 V for 2 s

Several current relaxations taken above room temperature with time step of 0.2 ms are shown in Fig. 1(a). Figure 1(b) presents the first derivatives of the current relaxations by the logarithm of time. The derivatives suggest the presence of at least two relaxation processes. In Fig. 2, we show the result of fitting such a double peak by two simple exponential relaxations, two stretched exponents, and two Gaussian broadened exponents describing the current relaxation at respective temperature (397 K). The starting relaxation time values and the starting exponents amplitudes were taken as the peak position time and amplitude for the simple and stretched exponents. The β values for the stretched exponents were determined by analyzing the halfwidth of the derivative function at half maximum of respective peaks when plotting the derivative as a function of δ = ln(t/τ). The values were then fine-tuned by fitting. For Gaussian broadening, the current relaxation was differentiated numerically by the logarithm of time to decide how many peaks were present and then the I(t) or dI/dln(t) curve fitted. The relaxation times, the β values, and the σ_{lnτ} values obtained are shown in the caption of Fig. 2. The quality of fitting is the worst for simple exponents and similar for the stretched exponents and Gaussian exponents. For both latter cases, the slower relaxation process in Fig. 2 is closer to the simple exponential decay than the faster process [the β value 0.73 versus 0.68, the mean deviations the σ_{lnτ} 0.336 versus 1.435; from Eq. (10), we then get respective energy broadening of the energy bands as 0.011 and 0.049 eV]. If the procedure is repeated for the entire temperature range in which the relaxation curves were measured, one can get the Arrhenius plots of the 1/(τT^{2}) values for the fast and slow processes for the cases of stretched and Gaussian exponents as shown in Fig. 3.

### B. Pulsing regime 2: V_{d} = 0.1 V, V_{g} = −3.9 V, illumination with pulses of light of different duration

Figure 4(a) shows drain current relaxation curves at 415 K after illumination with 2-s-long pulse from LEDs with peak wavelength between 850 and 365 nm (the driving current in the pulse was in all cases 600 mA, with light output ∼100 mW). The current derivatives of the relaxations are shown in Fig. 4(b). For high photon energies (365, 385, 400 nm), the relaxation shows two distinct regions, fast and slow. For low photon energies, only the slow relaxation process is observed. Clearly, the relaxation times after optical excitation depend on the wavelength of light. The optical threshold for the fast process is ∼3.1 eV, and for the slow process it is ∼2.4 eV.

Figure 5(a) shows the waveforms of relaxations for the 365 nm wavelength excitation as a function of the pulse width, and Fig. 5(b) presents the respective derivatives. For short pulses, the current first rises and then falls with time. The relaxation time for both processes decreases with increasing pulse width and increased starting current from which the relaxation proceeds. Such a dependence is the hallmark of the relaxation processes described by stretched exponents and is commonly explained by the dependence of the barrier height for recombination on the quasi-Fermi level position.^{26,27,32} For the falling part of the I_{d}(t) curve in Figs. 5(a) and 5(b), one also observes two other peaks corresponding to longer relaxation times similar to the ones observed for lower photon energies of excitations.

A similar dependence of relaxation time on the pulse width is evident for the slow relaxation process predominant for photon energies with the optical threshold of 2.4 eV (530 nm). This is illustrated by the pulse width dependence of the relaxation waveforms for the 455 nm LED injection at high temperature [Figs. 6(a) and 6(b)]. The peaks in the derivatives in Fig. 6(b) are too broad even for stretched exponents and require the participation of at least two processes. The quality of fitting with Gaussian broadened exponents is as good as with stretched exponents. The mean deviation in ln(t) determined for various excitation wavelengths varied slightly. For 445 nm it was 1.6 at 388 K and 1.4 at 414 K, yielding the width of the energy band involved as ∼0.05 eV. For current relaxations involving a band of states, one might argue that the quasi-Fermi energy dependence of the relaxation time described above can also be observed in some cases whereas it is hard to justify it if the current decay proceeds via emission from a discrete level in the gap as assumed in current DLTS or photoinduced current transient approaches.^{4–6,13,33,34}

The Arrhenius plots of 1/(τT^{2}) for the two relaxation times observed after the 455 nm excitation with the long (2 s) injection pulse are shown in Fig. 7. The results were obtained by fitting the relaxation curves by two stretched exponents and yield the activation energies of 0.55 and 0.35 eV. For the fast relaxation time observed after the 365 nm excitation, the activation energy was 0.9 eV.

### C. Pulsing regime 3: V_{d} = 0.5 V, V_{g} = −3.3 V, pulsing to −4 V

A typical current relaxation at high temperature (409.6 K) is shown in Fig. 8(a) and could be fitted with two simple exponential decay curves. The derivative is also described by two exponential processes with very different amplitudes and relaxation time values [Fig. 8(b)]. The activation energy for the main process was 0.62 eV, and for the minor process, the activation energy could be deduced from the fits of derivatives and was 0.45 eV (Fig. 9).

## V. DISCUSSION

The current relaxations in HEMTs are often not adequately described by a sum of simple exponential decay processes and use of stretched exponents or some other broadening effects expressed in terms of Gaussian broadening of logarithm τ provide a more accurate representation. For stretched exponents, the analysis of the first derivative of the current relaxation by the logarithm of time allows determination of the lifetimes, β values, and amplitudes from assessing the width of the peaks. For Gaussian broadening, the procedure is more cumbersome, but the parameters of the model can still be obtained from fitting. The measurements and fitting procedures involved are time consuming, and, with nonlinear fitting, there is always a concern about the uniqueness of the calculated parameters. Therefore, it is interesting to compare the results of these analyses with much more standardized and automated CDLTS approaches.^{6,7,13} One would also like to better understand the physical phenomena underlying the relaxation processes.

For comparison with CDLTS results, we have performed these measurements on the same HEMTs using the same pulsing conditions. The experimental procedure has been described elsewhere.^{13,35} The number of centers and the activation energies derived from CDLTS were similar to the results yielded by the analysis of individual relaxations above. For example, CDLTS spectra measured above room temperature at V_{d} = 0.1 V, V_{g} = −3.5 V pulsed to 0 V reveal the presence of two major hole-trap-like peaks with activation energies of 0.88 eV (capture cross section of 5.2 × 10^{−14} cm^{2}) and 1.03 eV (cross section of 5.7 × 10^{−14} cm^{2}). The traps' signatures in Arrhenius plots are presented in Fig. 10 and are close to the two features discussed in Fig. 3. The apparent activation energies and capture cross sections differ for different sets of time windows (e.g., 0.9 eV, 1.2 × 10^{−13} cm^{2}, and 1.1 eV, 5.3 × 10^{−13} cm^{2} when the time windows ratio is taken as 1/10 rather than 1/3 as for the plots in Fig. 10). This is indicative of the relaxations not being strictly exponential, but still, the agreement with the advanced analysis based on treating individual relaxations is good.

For the second pulsing mode CDLTS measurements with long (2 s) injection pulses of the 365 nm excitation and 455 nm excitation, the activation energies obtained were 0.65 eV (455 nm LED excitation) and 0.85 eV (Fig. 10), close to the results of the exact analysis taking into account the actual nature of relaxation (Fig. 7), although the low energy process with the activation energy of 0.35 eV was not detected in CDLTS of this sample. In the third pulsing mode, with the gate pulsing from −3.3 to −4 V, the dominant feature in CDLTS spectra was the hole-trap-like peak with the activation energy of 0.66 eV (capture cross section 1.1 × 10^{−17} cm^{2}). This is close to that obtained from the exact current relaxations analysis for the main center.

Since the HEMT was grown on a C-doped semi-insulating buffer, then carbon is most likely present in the AlGaN barrier. Recent theoretical calculations predict for carbon on the nitrogen site C_{N} the existence of the deep acceptor state with the charge transition (0/-) level near E_{v} + 0.9 eV in GaN and E_{v} + 1.9 eV in AlN and a donor state of C_{N} with the charge transition level (0/+) near E_{v} + 0.35 eV in GaN and E_{v} + 1 eV in AlN.^{36} For the AlGaN barrier with the Al mole fraction of 0.25, the acceptor and donor levels will be located near E_{v} + (1–1.1) eV and E_{v} + (0.55–0.65) eV if the levels are approximately aligned with respect to the vacuum level.^{36} DLTS with optical injection performed on n-GaN reveals the presence of several acceptor levels near E_{v} + (0.9–1) eV and the level that can be reasonably assigned to C_{N} acceptors has an optical excitation energy between 2.1 and 2.4 eV.^{37,38} In our GaN buffer, the Fermi level should be pinned by the acceptor C_{N} state and the number of occupied acceptors is determined by the density of residual donors.^{39} In the AlGaN barrier, all acceptors near E_{v} + 1 eV are expected to be filled. Then, photons with energies above 2.4 eV will mainly deionize the filled C_{N}^{−} acceptors in the GaN buffer and transfer the optically excited electrons into the 2DEG region. For optical excitation of such acceptors in the AlGaN, a higher energy of ∼3.1 eV is required. Such photons will also contribute to the increase in the 2DEG density and leave behind nonequilibrium neutral deep acceptors in AlGaN. There will also be removal of electrons from the empty C_{N} acceptors in GaN and in AlGaN with the formation of empty C_{N}^{+} donor states for photon energies exceeding the optical threshold of ionization of the first electron from the filled acceptors (2.4 eV in GaN and ∼3.1 eV in AlGaN). Thus, one ends up with nonequilibrium empty acceptors and ionized C_{N} donors in GaN and AlGaN and photoexcited electrons in the 2DEG region. When the light pulse is terminated, the system has to return to its starting state by: (1) thermal emission of holes from neutral C_{N} acceptors (activation energies of 0.9–1 eV in GaN, 1–1.1 eV in AlGaN), (2) thermal emission of holes from the ionized C_{N} donors (activation energies of 0.35 eV in GaN and 0.55–0.65 eV in AlGaN), and (3) thermal emission of nonequilibrium 2DEG electrons over the barrier at the AlGaN/GaN interface (∼conduction band discontinuity ΔE_{c} minus the energy of the uppermost filled state in 2DEG) and capture by different C_{N} states in the AlGaN barrier. This set covers the experimentally observed range of activation energies. The fast process appears to be governed by the thermal emission of holes from the empty acceptors. All three types of processes should show a strong dependence of relaxation times on filling of the initial states.

The emergence of hole-trap-like relaxations in the V_{d} = 0.5 V, V_{g} = −3.3 V, pulsing to −4 V mode may result from hole injection from the Si/AlN/AlGaN interface or from leakage between the source and drain giving rise to hole injection into the buffer during the gate stress pulse and their capture by hole traps in the buffer or in the barrier. The traps observed in our case have signatures similar to those observed in backgating experiments.^{40–43}

## VI. SUMMARY AND CONCLUSIONS

Drain current relaxations in AlGaN/GaN HEMTs are often nonexponential, but can in some cases be described by a limited number of processes described by extended rather than simple exponents or by exponents with the logarithm of relaxation time determined by Gaussian distribution of amplitudes. The latter description naturally comes from assuming that the emission proceeds not from a discrete level, but from a band of states with Gaussian distribution of the density of states. Fitting using both the extended exponents and the Gaussian broadened exponents gives similarly good correspondence with experimental relaxation curves. The Gaussian broadened exponents treatment gives a physical interpretation of broadening and a useful estimate of the width of the energy band. This treatment can, in principle, be extended to other density of states distributions. The analysis of the first derivatives of relaxations on the logarithm of time can give a first approximation for the relaxation times for insertion into the fitting process and allow assessment of the amount of deviation from simple exponentials. For stretched exponents, the amplitude, the relaxation time, and the b value can be estimated from the broadening of the peak derivative. The activation energies determined from the relaxation curves analysis can originate from the ionization energy of the traps involved in the carrier emission, the heterojunction band offset, the magnitude of potential fluctuations in the buffer or the 2DEG channel or the temperature dependence of the buffer resistivity. The general type and the number of processes involved in current relaxations as well as the relevant activation energies can be reasonably estimated using the standard procedure in CDLTS measurements, even when the relaxations are not exponential. However, detailed analysis of individual relaxations allows to check the exponentiality of decays and to better understand the origin of the relaxation times broadening. A survey CDLTS characterization followed by the analysis of long individual current relaxations at selected temperature points is an effective method of assessing the electronic structure and defect states in HEMTs.

## ACKNOWLEDGMENTS

The work at NUST MISiS was supported in part by the Ministry of Education and Science of the Russian Federation in the framework of Increase Competitiveness Program of NUST «MISiS» (K2-2014-055). The work at Chonbuk National University was supported by National Research Foundation of Korea (NRF) funded by Ministry of Science, ICT and Future Planning (2015042417) and the Ministry of Trade, Industry and Energy (MOTIE), Korea Institute for Advancement of Technology (KIAT) through the Encouragement Program for The Industries of Economic Cooperation Region (R1603000041). The work at UF was supported by DTRA 11-1-0020.