Direct observation of the capture cross section is challenging due to the need for extremely short filling pulses in the two-gate Deep-Level Transient Spectroscopy (DLTS). Simple estimation of the cross section can be done from DLTS and admittance spectroscopy data but it is not feasible to distinguish temperature dependence of pre-exponential and exponential parts of the emission rate equation with sufficient precision conducting a single experiment. This paper presents experimental data of deep levels in β-Ga2O3 that has been gathered by our group since 2017. Based on the gathered data, we propose a derivation of apparent activation energy and capture cross section assuming the temperature dependent capture via the multiphonon emission model, which resulted in a strong correlation between and according to the Meyer–Neldel rule, which allowed us to estimate low- and high-temperature capture coefficients and as well as capture barrier . It also has been shown that without considering the temperature dependence of capture cross section, the experimental values of are overestimated by 1–3 orders of magnitude. A careful consideration of the data also allows to be more certain identifying deep levels by their “fingerprints” ( and ) considering two additional parameters ( and ) and to verify the density functional theory computation of deep-level recombination properties.
I. INTRODUCTION
In recent years, Ga2O3 has been actively investigated by numerous research groups due to its promising potential for power electronics and solar-blind UV detectors.1,2 To advance the application of gallium oxide devices, detailed studies have been conducted on crystal growth,1,3,4 epitaxial film growth,1 intentional doping, and electrically active defects.3 Our emphasis has been focused on the deep-level defects investigation by capacitance spectroscopy, mainly Deep-Level Transient Spectroscopy (DLTS)5 and Admittance Spectroscopy (AS).6,7 These two techniques are widely used for characterizing electrically active defects by evaluating their concentration , thermal activation energy , and electron capture cross section .
These parameters are obtained from electron detrapping-kinetics characterized by the emission rate dependence on temperature T, as given by5
The kinetics of many thermally activated processes in chemistry and physics, including the emission rate of carriers from deep levels, are usually calculated from so called Arrhenius plots, which allow us to extract the pre-exponential factor and activation energy as the temperature independent parameters for the sake of mathematical treatment simplification. This approach works well for simple estimation kinetics phenomena; however, if one is lacking knowledge of exact temperature dependence of pre-exponential factor and activation energy, it is not possible to reduce it to Arrhenius equation term by term, and, therefore, the kinetics cannot be characterized with high precision. Furthermore, a capture cross section and activation energy cannot be determined accurately.
A. Meyer–Neldel rule in semiconductors
Various research groups confirmed this rule for carrier emission kinetics in A3B5 compounds,9,10 ZnO, and other semiconductors.11,12 The most reasonable physical explanation of the MNR in terms of emission rate is based on considering the total change in Gibbs energy with entropy part as attributed to vibrational entropy . It has been concluded that vibrational entropy (estimated as rearrangement entropy of n interacting phonons of N total phonons in interaction volume) could not explain extremely small values of which are ∼10−23 cm2, and that without accumulation of a significant amount of data, it is not clear whether new ideas on this issue will be proposed.11
From the theoretical side, Alkauskas et al. in 2014 developed13 a theory on computing nonradiative capture cross section for deep-level transitions occurring via multiphonon emission, which gives insights on defect recombinational properties at different temperatures and allows us to identify defects from experimental data.
The current paper reveals new aspects of experimental deep-level parameters determination, provides results to verify such Density Functional Theory (DFT) defects computations, and demonstrates new data on capture coefficients for the main electron traps in β-Ga2O3.
II. SAMPLING
This analysis is based on DLTS and admittance spectroscopy data that has been gathered since 2017 for deep levels in a wide range of β-Ga2O3 samples. The studied samples of β-Ga2O3 were cut from various types of wafers purchased from Tamura/Novel Crystals, Inc., Tokyo, Japan: β-Ga2O3 (−201) and (010) oriented edge defined film-fed grown (EFG) wafers doped with Sn, unintentionally doped EFG (−201) wafer, (010) oriented EFG wafers doped with Fe and (001) orientated unintentionally doped halide vapor phase epitaxy (HVPE) grown layers on bulk n+-EFG substrates doped with Sn.14 Different sets of treatments were employed to understand the presence and origin of electrically active deep levels in β-Ga2O3. A detailed description of the experiments and results along with the depiction of the deep-level spectra could be found in our previous works.15–21
The sampling contains 1242 uncategorized data entities each corresponding to a single measurement of activation energy and capture cross section from the Arrhenius plot in vs axes. The peak temperature is taken at the smallest window in the measurement and used only to improve quality of deep-level data clustering.
Up to now, a large number of groups have already done a significant work in the field of theoretical and experimental identification of electrically active defects in gallium oxide. Let us briefly describe the results of deep-level characterization that are taken as reference data in this analysis.
Center E1 (with found in the range of 0.45–0.65 eV and found in the range of 0.3–7 × 10−13 cm2) has been introduced by H-plasma treatment,19 proton irradiation,18,21 and ampoule annealing in H2.15,22 It has been observed that the E1 is a donor, and according to the theoretical models, a possible configuration is the complex of H with shallow donors Si or Sn.15
Center E2 [Ea = (0.74−0.82) eV, σn = (0.6−23) × 10−15 cm2] is often detected in EFG, HVPE grown samples, and assigned to Fe acceptors.23–32 The E2* centers [Ea = (0.75−0.78) eV, σn = (2−7) × 10−14 cm2] typically have been observed after radiation or implantation and demonstrate a linear increase with irradiation exposure.21 This implies the possible origin of E2, which is a complex of intrinsic point defects of gallium and oxygen vacancies.32
For the E3 (Ea = 1.05 eV, σn = 4.1 × 10−13 cm2) level detected in unintentionally doped EFG-grown β-Ga2O3, it has been suggested that the possible nature of the center is a deep donor related to Ti.33 However, a defect with a similar and tends to increase in concentration after irradiation with high-energy particles (neutrons and protons) and Ar plasma treatment.21 So, the issue with these two interpretations could be the same as for E2 and E2* at early stages of β-Ga2O3 research.32
The E8 (Ea = 0.28 eV, σn = 6 × 10−18 cm2) center is an intrinsic point defect or complex detected after irradiations and treatment with Ar and H plasma.21
A. Deep-level clustering
Data clustering was performed using a Gaussian Mixture model with variational inference algorithm.34 This method assumes all data can be represented by a finite mixture of Gaussian distributions with unknown parameters which are determined from a variational lower boundary. The above procedure reduced the total data entities from 1242 to 1033, excluding dropouts, and produced 6 clusters assigned as main deep levels E2*, E4, E8, E1, E2, and E3. Results of data clustering can be seen in the pairwise plot in Fig. 1, where normalized distributions of , , and for each trap are presented on the main diagonal plots and pairs of parameters plotted pairwise on off-diagonal plots.
Pairwise plot of clustered data for main deep levels in β-Ga2O3. (a), (e), and (i) Distributions of experimental parameters for trap clusters. (b), (c), (f) and (d), (g), (h) represent the same data and show the correlation within measured parameters. These scatterplots demonstrate no correlation of or with , in which variance is determined with technique limitations. However, strongly depends on and this phenomenon will be studied more thoroughly in Sec. III C below. (j) One of the DLTS measurements from the gathered data, which provides 3 of 1242 data entities represented with Arrhenius plot (k) and attributed to E1, E2, and E4 levels.
Pairwise plot of clustered data for main deep levels in β-Ga2O3. (a), (e), and (i) Distributions of experimental parameters for trap clusters. (b), (c), (f) and (d), (g), (h) represent the same data and show the correlation within measured parameters. These scatterplots demonstrate no correlation of or with , in which variance is determined with technique limitations. However, strongly depends on and this phenomenon will be studied more thoroughly in Sec. III C below. (j) One of the DLTS measurements from the gathered data, which provides 3 of 1242 data entities represented with Arrhenius plot (k) and attributed to E1, E2, and E4 levels.
Variances and mean values of , , and for each deep-level can be determined from the data presented in Figs. 1(a), 1(e), and 1(i). The vertical alignment of clusters on Figs. 1(c) and 1(f) demonstrates no correlation of and with (since and are computed from Arrhenius plots in temperature ranges higher than variance) but there is strong correlation of with on Fig. 1(b). This relationship will be used further for Eq. (2) fit and results analysis.
III. DERIVATIONS
Typically, the most straightforward and accurate way to determine capture cross section is through direct observation of capture kinetics, but this method necessitates the usage of extremely short pulsing times even for materials with low doping levels in the two-gate DLTS35 . More simple ways are to calculate the capture cross section from standard DLTS and AS approaches,5–7 but these techniques are not suitable for precise capture cross-section measurements, especially with the assumption of its strong temperature dependence. Emission rates can be measured by implementing long and short windows for DLTS to compute the low- and high-temperature cross section from Arrhenius plots, but in this case, it is not feasible to separate the temperature dependence of pre-exponential and exponential parts with sufficient precision conducting a single experiment.
A. Vibrational entropy
The entropy term, as mentioned in the introduction, was proposed to explain the observed correlation of and for other semiconductor materials. In addition, its appearance in Gibbs free energy fits well with temperature independent behavior described by Eq. (2), but the vibrational entropy is relatively small 36,37 when atomic rearrangements around defect are neglected.
B. Carrier capture by multiphonon emission
Configurational diagram. Ed—energy of electron on a defect with Q = Qd (non-equal to 0 with the presence of electron–lattice interaction) and Ek—excited state that is delocalized and since then Q = 0 (small electron–lattice coupling to local mode Q).
Configurational diagram. Ed—energy of electron on a defect with Q = Qd (non-equal to 0 with the presence of electron–lattice interaction) and Ek—excited state that is delocalized and since then Q = 0 (small electron–lattice coupling to local mode Q).
Arrhenius plot and extracted parameters as a function of temperature. Fully analytical model (gradient line) with the accounting of temperature dependence of capture cross section as Eq. (5) and simplified model equation (15) (red line) of temperature dependence of measured and . (a) Fully analytical Arrhenius plot showing different slope values at low and high temperatures, (b) and (c) and calculated from Arrhenius plot showing step-like function, and (d) parametric plot of and with full and simplified models based on our E2 data (violet crosses) and other groups’ data (blue symbols).
Arrhenius plot and extracted parameters as a function of temperature. Fully analytical model (gradient line) with the accounting of temperature dependence of capture cross section as Eq. (5) and simplified model equation (15) (red line) of temperature dependence of measured and . (a) Fully analytical Arrhenius plot showing different slope values at low and high temperatures, (b) and (c) and calculated from Arrhenius plot showing step-like function, and (d) parametric plot of and with full and simplified models based on our E2 data (violet crosses) and other groups’ data (blue symbols).
It can be clearly seen from (8) and (10) that appears to be step-like function [Fig. 3(b)] of T, with low- and high-temperature plateaus that are roughly and , with shift due to the presence of in . More precise values for lower and upper boundaries are and , where shift value is . The same temperature behavior of is presented on Fig. 3(c) with the lower and upper boundaries at the same temperatures as .
Summing up the intermediate results, (10) and (11) imply that and correlates through the temperature at which DLTS/AS can be performed, but experimentally we are limited in the analysis for the same reason elaborated in Sec. II A—data points from DLTS/AS data taken in broad temperature interval (around for E2 trap) when noticeable changes in or are happening in a narrow temperature range. This narrow temperature range appears from data linearity in Fig. 1(b) implying by (12) [ should be constant or vary slowly to give linearity in Fig. 1(b)].
Since and data show no correlation to due to experiment limitations, and the biggest changes in measured parameters appear around , we can expand and near , excluding the temperature from equations and establishing functional dependence in form .
For at ,
C. Model fitting
To find all four ( , , , ) model parameters from experimental data, it is needed to impose two more restrictions besides (13) and (18).
IV. RESULTS
Computed and for deep levels E2, E2*, E1, E8, and E3 (E4 is omitted here due to the lack of extensive collected data) in β-Ga2O3 are presented in Fig. 4. These data are then used for solving Eq. (21) to provide the right-hand side for Eqs. (iii) and (iv) of (19).
Collected data on main deep levels in β-Ga2O3 [magnified plot from Fig. 1(b)] and fit with parameters from Table I.
Results of model fitting for main deep levels in β-Ga2O3.
Trap . | Ea (eV) . | Eb (eV) . | C0 (cm3/s) . | C1 (cm3/s) . |
---|---|---|---|---|
E2 | 0.68 | 0.21 | 7.0 × 10−10 | 1.4 × 10−6 |
E2* | 0.61 | 0.21 | 1.3 × 10−9 | 2.6 × 10−6 |
E1 | 0.52 | 0.17 | 1.3 × 10−8 | 3.1 × 10−5 |
E8 | 0.23 | 0.12 | 2.0 × 10−11 | 5.3 × 10−8 |
E3 | 0.89 | 0.19 | 2.0 × 10−8 | 2.0 × 10−5 |
Trap . | Ea (eV) . | Eb (eV) . | C0 (cm3/s) . | C1 (cm3/s) . |
---|---|---|---|---|
E2 | 0.68 | 0.21 | 7.0 × 10−10 | 1.4 × 10−6 |
E2* | 0.61 | 0.21 | 1.3 × 10−9 | 2.6 × 10−6 |
E1 | 0.52 | 0.17 | 1.3 × 10−8 | 3.1 × 10−5 |
E8 | 0.23 | 0.12 | 2.0 × 10−11 | 5.3 × 10−8 |
E3 | 0.89 | 0.19 | 2.0 × 10−8 | 2.0 × 10−5 |
The distributions of gathered and are represented in box-plots on Fig. 5. Boxes on Fig. 5 represent and at temperature and used as right-hand side of Eqs. (i) and (ii) of (19). Solutions of (21) for each trap are presented in Table I and plotted in Fig. 6.
and distributions of main deep levels in β-Ga2O3, based on collected data.
In comparison with Fig. 5, activation energies and capture cross sections obtained in experiments are overestimated due to the interplay of the temperature dependence of the capture cross section and the thermal emission terms, and, in this case, the previously measured values of the capture cross section (around ) are 1–3 orders of magnitude higher than obtained with the suggested model.
Approaching the same problem from the theoretical side of the issue, similar results were obtained by Wickramaratne et al.39 In their paper, the computed temperature dependent capture cross section via multiphonon emission mechanism was inserted into DLTS formalism, and it appeared to shift apparent activation energy obtained from the Arrhenius plot to higher values at higher temperatures. Nevertheless, the study39 was not supported by any experimental data, unlike the present paper, which is possibly the first experimental observation on this matter.
V. SUMMARY AND CONCLUSIONS
In this paper, we have considered that the carrier emission rate from main deep levels in β-Ga2O3 follow the MN-rule and it has been shown that the capture cross section in the multiphonon emission model explains the observed and shift.
We have applied the theory to the main deep-level centers in β-Ga2O3 and accurately calculated all parameters (Table I), including activation energy , barrier height for carrier capture , as well as low- and high-temperature capture coefficients and .
This suggests using of and as two additional parameters to identify defects, employing , , and to estimate capture coefficient with more detailed and advanced approach, and verifying DFT results on recombinational properties of deep levels.
ACKNOWLEDGMENTS
This research at NUST MISIS was funded by the Ministry of Science and Higher Education of the Russian Federation (Grant No. 075-15-2022-1113). This work at UF was performed as part of Interaction of Ionizing Radiation with Matter University Research Alliance (IIRM-URA), sponsored by the Department of the Defence, Defence Threat Reduction Agency under Award No. HDTRA1-20-2-0002. The content of the information does not necessarily reflect the position or the policy of the federal government, and no official endorsement should be inferred.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
A. A. Vasilev: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). A. I. Kochkova: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). A. Y. Polyakov: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). A. A. Romanov: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). N. R. Matros: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). L. A. Alexanyan: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). I. V. Shchemerov: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). S. J. Pearton: Conceptualization (equal); Writing – original draft (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.