FACTA UNIVERSITATIS Series: Electronics and Energetics Vol. 34, No 3, September 2021, pp. 323-332 https://doi.org/10.2298/FUEE2103323S © 2021 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper ANALYTICAL STUDY OF EFFECT OF ENERGY BAND PARAMETERS AND LATTICE TEMPERATURE ON CONDUCTION BAND OFFSET IN AlN/Ga2O3 HEMT * Rajan Singh1, Trupti Ranjan Lenka1, Hieu Pham Trung Nguyen2 1Department of Electronics and Communication Engineering, National Institute of Technology Silchar, Assam, India 2Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark, New Jersey, 07102, USA Abstract. Apart from other factors, band alignment led conduction band offset (CBO) largely affects the two dimensional electron gas (2DEG) density ns in wide bandgap semiconductor based high electron mobility transistors (HEMTs). In the context of assessing various performance metrics of HEMTs, rational estimation of CBO and maximum achievable 2DEG density is critical. Here, we present an analytical study on the effect of different energy band parameters—energy bandgap and electron affinity of heterostructure constituents, and lattice temperature on CBO and estimated 2DEG density in quantum triangular-well. It is found that at thermal equilibrium, ns increases linearly with ΔEC at a fixed Schottky barrier potential, but decreases linearly with increasing gate-metal work function even at fixed ΔEC, due to increased Schottky barrier heights. Furthermore, it is also observed that poor thermal conductivity led to higher lattice temperature which results in lower energy bandgap, and hence affects ΔEC and ns at higher output currents. Key words: 2DEG density, CBO Conduction Band Offset, Heterojunction, HEMT, Lattice Temperature, Barrier, Buffer, Ga2O3 1. INTRODUCTION Due to its suitable material properties and availability of high quality and cost-effective native substrates, gallium oxide (Ga2O3) is being exhaustively investigated for power electronics applications [1]. Currently, this domain of high-power and high-frequency devices are dominated by wide-bandgap semiconductors like silicon carbide (SiC) and gallium nitride (GaN). Some of the unique features—excellent carrier confinement in the form of 2DEG, high Received February 10, 2021; received in revised form May 06, 2021 Corresponding author: Rajan Singh Department of Electronics and Communication Engineering, National Institute of Technology Sichar, Assam, India E-mail: rajan_rs@ece.nits.ac.in * An earlier version of this paper was presented at the International Conference on Micro/Nanoelectronics Devices, Circuits, and Systems (MNDCS-2021), January 29-31, 2021, India [1]. 324 R. SINGH, T. R. LENKA, H. P. T. NGUYEN carrier mobility, and large breakdown voltage of GaN-based HEMTs have made it one of the most useful devices for high-power and high-frequency applications [2-4]. Despite some key challenges on the substrate side, GaN technology however survived beyond the expected life cycle of a typical semiconductor technology [5]. Recently, researchers across the globe have started to look towards ultra-wide bandgap (UWB) semiconductors—Ga2O3, AlN, and Diamond for high voltage applications [6]. Among these UWB semiconductors, Ga2O3 has emerged as an ultimate choice for future power electronics devices on the back of preliminary results that are encouraging enough to prove its capabilities to supplement existing SiC/GaN technologies. It is worth noting here that, apart from lower bulk electron mobility, 150 - 200 cm2/Vs [5-6], Ga2O3 has very low thermal conductivity, 0.13 – 0.27 W/cm K [5]. The following equation relates the electron affinity of a semiconductor with lattice temperature (TL), as given in [7]: 𝜒(𝑇𝐿 ) = 𝜒(300) − 𝐶𝐻𝐼. 𝐸𝐺. 𝑇𝐷𝐸𝑃 × (𝐸𝑔 (𝑇𝐿 ) − 𝐸𝑔 (300)) (1) where, parameter CHI.EG.TDEP is a ratio, in range (0 – 1) with a default value of 0.5 (used here), which specifies a fraction of the change in the bandgap due to the temperature change. For β-Ga2O3, the energy bandgap at TL is given by Varshni equation [8]: 𝐸𝑔 (𝑇𝐿 ) = 𝐸𝑔 (300) − 𝛼𝑇𝐿 2 𝑇𝐿+𝛽 (2) where fitting parameters α, and β for AlN, and β-Ga2O3 are taken from [9], [10] respectively and extrapolated at higher temperatures. The different bandgaps of two materials of heterojunction create these band discontinuities, and the band offset parameters—conduction and valence band offset (ΔEC and ΔEV) have a large impact on the charge transport in the heterostructure [7]. Higher values of 2DEG density are anticipated for large ΔEC values [11], while ΔEC is also dependent on different electron affinity values of heterojunction materials besides their bandgaps as stated earlier. Considering the importance of 2DEG density in the operation of HEMTs, various physics-based analytical models for ns [12-18] are available mostly for AlGaN/GaN HEMTs. In this work, we primarily investigated the estimation of ns based on different values of alignment—a fraction of bandgap difference to ΔEC, at fixed Schottky barrier and at higher metal work function led increasing Schottky potential at fixed ΔEC through TCAD simulations. The study is also done considering the heat flow in the device as the poor thermal conductivity of Ga2O3 and high currents in power devices resulting in high lattice temperature. Table 1 Symbols Used and Meaning [1] Symbol Physical meaning Symbol Physical meaning χ Electron affinity ε Static dielectric permittivity Eg Energy bandgap D Density of states ϕB Schottky barrier height Ef Position of Fermi level ϕM Metal work function d Thickness of barrier layer q Electron charge qV0 Built-in potential ns Electron density in the 2DEG Vth Thermal voltage ΔEC Conduction band offset (CBO) NC Conduction band density ΔEV Valence band offset (VBO) NV Valence band density Analytical Study of Effect of Energy Band Parameters and Lattice Temperature on Conduction Band... 325 Fig. 1 Energy band diagram of a typical heterojunction showing band offset [1]. The energy band diagram of AlN/β-Ga2O3 abrupt heterojunction, having β-Ga2O3 buffer layer (Eg1, and χ1) and AlN barrier layer (Eg2, and χ2), where Eg2 > Eg1 and χ2 < χ1 is shown in Figure 1. The developed model is used to optimize ns considering band parameters of barrier and buffer layer materials reported in Ga2O3 experimental HEMTs. The 2DEG charge density ns relating conduction band offset ΔEC, using charge control equation [12], can be written as: 𝑛𝑠 = 𝜀 𝑞𝑑 [𝑉𝑔 − 𝜙𝑏 + 𝑉𝑝𝑏 − 𝐸𝑓 + 𝛥𝐸𝑐 ] (3) where Vpb is barrier layer pinch-off voltage and Vg is the applied gate voltage. The other symbols used along with their physical meaning are listed in Table 1. The device under study here is AlN/β-Ga2O3 HEMTs, as the experimental measurements of band offset parameters—ΔEV and ΔEC at the III-nitride (GaN, and AlN)/β-Ga2O3 heterostructure are readily available [19, 20]. Additionally, as the in-plane lattice mismatch between [-201] AlN and [0002] AlN planes is as small as 2.4 % [20], AlN/β-Ga2O3 is anticipated as a potential candidate for future high-power applications. 2. 2DEG CHARGE DENSITY AND DEVICE MODEL DESCRIPTION In the triangular quantum well, 2DEG charge density ns is related with Fermi level Ef and two sub-bands E0 and E1, using Fermi-Dirac statistics, as given by [12] 𝑛𝑠 = 𝐷𝑉𝑡ℎ [𝑙𝑛 {𝑒 (𝐸𝑓−𝐸0) 𝑉𝑡ℎ⁄ + 1} + 𝑙𝑛 {𝑒 (𝐸𝑓−𝐸1) 𝑉𝑡ℎ⁄ + 1}] (4) where first energy level E0 = γ0ns 2/3, and second E1= γ1ns 2/3. In case of complete ionization of barrier layer, equation (1) can be written as: 𝑛𝑠 = 𝜀 𝑞𝑑 {𝑉𝑔𝑜 − 𝐸𝑓 } (5) where Vgo = Vg – Voff. The 2DEG density model, for AlGaN/GaN HEMT, developed so far explained ns behavior concerning Vg. It was assumed that only the first sub-band E0 lies 326 R. SINGH, T. R. LENKA, H. P. T. NGUYEN below Ef for Vg > Voff. Here, a more simplified expression of Fermi level (in volts) in terms of ns is obtained under steady-state conditions. 𝐸𝑓 = 𝑛𝑠 2𝐷 + 𝐸0+𝐸1 2 − 𝑉𝑡ℎ 2 (6) The above equation is obtained using the approximation ln(1 + 𝑥) ≈ 𝑥, 𝑓𝑜𝑟 𝑥 ≪ 1 and after some mathematical manipulations. Next, E0 and E1 can be replaced to get the following explicit relation between Ef and ns: 𝐸𝑓 = 𝑛𝑠 2𝐷 + ( 𝛾0+𝛾1 2 ) 𝑛𝑠 2/3 − 𝑉𝑡ℎ 2 (7) Furthermore, higher charge confinement in the triangular well is anticipated based on higher energy difference between Ef and E0 [21]. It is worth mentioning that, some fraction, say 60 or 80 % and very rarely up to 100% [22], of the heterostructures’ material bandgap difference appears as CBO. Therefore, while estimating ns, careful measurement of CBO (ΔEC) is important. In this work, for the estimation of confined charge density in the quantum well, the relative position of E0 to Ef is analyzed under varying band alignment and varying Schottky barrier height under thermal equilibrium. This is illustrated in Figure 2. The device model analyzed here is comprised of an AlN barrier on β-Ga2O3 buffer layer having a thickness of 10 and 50 nm respectively. The layer sequence cum device cross-section is shown in Figure 3. Source and drain contacts are considered to be ohmic, while gate contact is Schottky type. Silicon nitride (Si3N4) is used for surface passivation and to limit the parasitic capacitances as mentioned in [23]. The various material parameters for β-Ga2O3 are taken from [24, 25] and are shown in Table 2 along with for AlN taken from [7]. Different material parameters used in the simulation of AlN/β-Ga2O3 HEMT constituents are listed in Table 2. Fig. 2 The relative position of E0 and Ef to CBO for fixed Schottky potential [1]. Fig. 3 Schematic diagram showing layer sequence of AlN/β-Ga2O3 HEMT; dashed line below AlN barrier represents 2DEG charges [1]. Analytical Study of Effect of Energy Band Parameters and Lattice Temperature on Conduction Band... 327 Table 2 Material parameters of β-Ga2O3 and AlN used in different calculations of TCAD simulations, taken from [7, 24-25]. Symbol β-Ga2O3 AlN χ (eV) 3.15 1.4 Eg (eV) 4.9 6.1 NC (cm-3) 3.6 × 1018 4.42 × 1018 NV (cm-3) 2.86 × 1020 6.76 × 1018 𝒏𝒊 (cm -3) 2.23 ×10-22 1.51 × 10-33 ε 10.2 8.5 3. RESULTS AND DISCUSSION The device under the test (Figure 3) is simulated to estimate CBO and 2DEG density using Atlas TCAD under steady-state conditions and at different bias voltages enabling heat-flow in the device. The duo investigations are performed using the alignment-based rule—ΔEC as a fraction of ΔEg, due to the significant difference between band offsets estimated using standard values of electron affinity and experimental measurements. 3.1. At Steady State Condition 3.1.1. Fixed Schottky Barrier Height Earlier, various high-performance AlN Schottky barrier diodes were demonstrated [26- 28] and barrier heights ranging from 1.6 – 2.3 eV between AlN and different metals were measured [26]. Here, Ti and Au AlN Schottky contacts with a barrier height of 1.6 eV are used to estimate CBO and analyze 2DEG density under three different degrees of alignments— 60, 80, and 100%. As conduction band offset ΔEC increased from 0.65 to 1.15 eV, 2DEG density increases as higher conduction band alignment boost carrier confinement as illustrated in Figure 4. Fig. 4 Estimation of CBO keeping fixed barrier height of 1.6 eV under 60, 80, and 100% alignment of bandgap difference. 328 R. SINGH, T. R. LENKA, H. P. T. NGUYEN 3.1.2. Fixed alignment of 60% Considering a moderate value of alignment, say 60% of the bandgap difference (ΔEg = 1.24 eV) between AlN and β-Ga2O3 is assigned here as CBO. The three different metals—Ti, Ni, and Au on AlN with barrier heights of 1.6, 1.8, and 2.3 eV are considered to analyze to estimate ΔEC and consequently 2DEG density and are shown in Figure 5. Although, a fixed fraction of bandgap difference (0.6 of 1.24 = 0.744 eV) is assigned to conduction band discontinuity, the estimated value of ΔEC is slightly less than the assigned value. This may be attributed to surface and or interface states at the AlN/β-Ga2O3 boundary [26]. Fig. 5 Estimation of ΔEC, and ns under three increasing Schottky barrier heights for fixed alignment of 60 %; decreasing values of ns (5.0, 4.9, and 4.7) × 10 13 cm-2 are estimated. 3.2. Heat-flow Simulation Poor thermal conductivity of Ga2O3 and higher output currents in Ga2O3 based power devices commonly result in high lattice temperature in absence of device-level thermal management. Here, after enabling the relevant model in simulation, maximum lattice temperature under the gate area is gauged under different bias voltages. The subsequent effects on electron affinity and energy bandgap are also estimated. The increased affinity values led to reduced energy bandgap results in higher bandgap difference at the heterointerface. The maximum lattice temperature at elevated currents is shown in Figure 6, and resulting energy bandgap and electron affinity at different lattice temperature is shown in Figure 7. Analytical Study of Effect of Energy Band Parameters and Lattice Temperature on Conduction Band... 329 Fig. 6 Maximum lattice temperature; extracted from ATLAS (left) at higher drain current, and plotted versus corresponding drain voltage at zero gate voltage (right). Fig. 7 Energy band gap and electron affinity as a function of maximum lattice temperature. 3.2.1. Effect on Conduction band offset As the lattice temperature increases, the energy bandgap of β-Ga2O3 shrinks as per equation (2) and is shown in Figure 7. Now the maximum bandgap difference available between AlN and β-Ga2O3, corresponding to the maximum lattice temperature of 1063 K at VDS = 15 V (VGS = 0 V), is given as ∆𝐸𝑔 = (𝐸𝑔 𝐴𝑙𝑁 − 𝐸𝑔 𝐺𝑎2𝑂3 ) ≈ 5.8 − 3.2 ≈ 2.6 𝑒𝑉 Here, it is evident that a larger change in Ga2O3 energy bandgap led to a 46 % higher bandgap difference between AlN and Ga2O3 compared to its value, 1.24 eV, at 300 K and consequently higher values of CBO result. Further, corresponding to 80 % alignment of 330 R. SINGH, T. R. LENKA, H. P. T. NGUYEN ΔEg (ΔEC = 0.8 × 2.6 = 2.08 eV), 2DEG density ns is estimated for trio barrier heights and is shown in Figure 8. Fig. 8 2DEG estimation at the fixed alignment of 80 % for different Schottky barrier heights. 2DEG density decreases with higher Schottky barrier heights. The important inferences from the results exhibited above are summarized in Table 3. It is found that a higher degree of CBO results in increased 2DEG density, both at steady-state and at higher bias voltages. However, 2DEG density in the latter scenario is relatively low as compared to the previous case. This is attributed to the enhanced electron-phonon interaction with increasing lattice temperature. Additionally, confined carrier density decreases with increasing Schottky barrier heights in both cases. This can be due to the presence of interface charges and defect states at the AlN-β-Ga2O3 boundary. Table 3 Estimated values of ΔEC, and ns under fixed Schottky barrier and fixed alignment Alignment (%) / Schottky height (eV) Under steady state (TL = 300 K) 𝐸𝑔 𝛽−𝐺𝑎2𝑂3 = 4.9 𝑒𝑉, 𝐸𝑔 𝐴𝑙𝑁 = 6.1 𝑒𝑉 At VDS /VGS = 15 / 0V (TL = 1063 K) 𝐸𝑔 𝛽−𝐺𝑎2𝑂3 = 3.2 𝑒𝑉, 𝐸𝑔 𝐴𝑙𝑁 = 5.8 𝑒𝑉 ΔEC (eV) ns ( × 1013 cm-2) ΔEC (eV) ns ( × 1013 cm-2) 60 / 1.6 0.65 5.0 1.47 4.6 80 / 1.6 0.9 5.12 2.0 4.8 100 / 1.6 1.15 5.23 2.5 5.0 80 / 1.6 0.9 5.12 2.0 4.8 80 / 1.8 0.9 5.0 2.0 4.7 80 / 2.3 0.9 4.8 2.0 4.5 4. CONCLUSION To summarize, the effect of energy bandgap difference enabled conduction band offset on 2DEG density in AlN/β-Ga2O3 HEMT is studied analytically. The analytical expression of Fermi level is derived to conclude that the relative position of Ef and E0 largely affects 2DEG density. Alignment-based rule—CBO as a fraction of ΔEg is found in more agreement with its value measured in experimental devices. By varying band alignment Analytical Study of Effect of Energy Band Parameters and Lattice Temperature on Conduction Band... 331 and Schottky barrier heights, the resultant effect on ΔEC and 2DEG density are estimated. It is found that apart from conduction band offset dependency, ns is also affected by Schottky barrier height. It is also shown that poor thermal conductivity led to higher lattice temperature which results in large ΔEg and CBO, but yielded relatively lower 2DEG density as compared to steady-state condition. In steady-state, for fixed Schottky barrier height of ϕB = 1.6 eV (Ti/AlN) , 2DEG density increases from 5.0 × 10 13 to 5.23 × 1013 cm-2 when ΔEC changes from 60 to 100 %, and from 4.6 × 10 13 to 5.0 × 1013 cm-2 at VDS/VGS =15/0 V. On the other hand, even at fixed ΔEC, 2DEG density decreases from 5.12 × 10 13 to 4.8 × 1013 cm-2 when ϕB increases from 1.6 to 2.3 eV in steady-state, and 4.8 × 10 13 to 4.5 × 1013 cm-2 at lattice temperature of 1063 K corresponding to VDS/VGS =15/0 V. These conclusions can be beneficial to access the limitations in β-Ga2O3 HEMT performance, which critically depends on the careful estimation of 2DEG density. REFERENCES [1] R. Singh, T. R. Lenka, S. A. Ahsan, and H. P. T. Nguyen, “Analytical Study of Conduction Band Discontinuity supported 2DEG Density in AlN/Ga2O3 HEMT,” In Proceedings of the International Conference on Micro/Nanoelectronics Devices, Circuits, and Systems (MNDCS-2021), 29-31 Jan 2021. http://mndcs.nits.ac.in/ [2] U. K. Mishra, L. Shen, T. E. Kazior, Y. F. Wu, “GaN-based RF power devices and amplifiers,” In Proceedings of the IEEE. 16, Jan 2008, vol. 96, no. 2, pp. 287–305. [3] P. Parikh, Y. Wu, M. Moore, P. Chavarkar, U. Mishra, R. Neidhard, et al., “High linearity, robust, AlGaN-GaN HEMTs for LNA and receiver ICs,” In Proceedings of the IEEE Lester Eastman Conf. High Perform. Devices, Aug. 2002, pp. 415–421. [4] P. Kordos, A. Alam, J. Betko, P. P. Chow, M. Heuken, P. Javorka, et al., “Material and device issues of GaN-based HEMTs,” In Proceedings of the 8th IEEE Int. Symp. High Perform. Electron Devices Microw. Optoelectron. Appl., Nov. 2002, pp. 61–66. [5] S. J. Pearton, F. Ren, M. Tadjer, and J. Kim, “Perspective: Ga2O3 for ultra-high power rectifiers and MOSFETs,” Journal of Applied Physics, vol. 124, no. 22, p. 220901, Dec 2018. [6] E. Ahmadi, and Y. Oshima, “Materials issues and devices of α – and β – Ga2O3,” Journal of Applied Physics, vol. 126, p. 160901, Oct. 2019. [7] Atlas, Device Simulator. “Atlas user’s manual.” Silvaco International Software, Santa Clara, CA, USA (2016). [8] Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica 34, vol. 1, pp. 149–154, 1967. [9] S. Rafique, L. Han, S. Mou, and H. Zhao, “Temperature and doping concentration dependence of the energy band gap in β-Ga2O3 thin films grown on sapphire,” Optical Material Express 3561, vol. 7, no. 10, Oct. 2017. [10] K. B. Nam, J. Li, J. Y. Lin, and H. X. Jiang, “Optical properties of AlN and GaN in elevated temperatures,” Applied Physics Letters, vol. 85, no. 16, Oct 2004. [11] Y. Zhang, et al., “Demonstration of high mobility and quantum transport in modulation-doped β- (AlxGa1-x)2O3/Ga2O3 heterostructures,” Applied Physics Letters, vol. 112, no. 17, p. 173502, Apr. 2018. [12] S. Kola, J. M. Golio, and G. N. Maracas, “An analytical expression for Fermi level versus sheet carrier concentration for HEMT modeling,” IEEE Electron Device Lett., vol. 9, no. 3, pp. 136–138, Mar. 1988. [13] X. Cheng, M. Li, and Y. Wang, “An analytical model for current voltage characteristics of AlGaN/GaN HEMTs in presence of self-heating effect,” Solid State Electron., vol. 54, no. 1, pp. 42–47, Jan. 2010. [14] S. Khandelwal, N. Goyal, and T. A. Fjeldly, “A physics based analytical model for 2DEG charge density in AlGaN/GaN HEMT devices,” IEEE Trans. Electron Devices, vol. 58, no. 10, pp. 3622–3625, Oct. 2011. [15] X. Cheng and Y. Wang, “A surface-potential-based compact model for AlGaN/GaN MODFETs,” IEEE Trans. Electron Devices, vol. 58, no. 2, pp. 448–454, Feb. 2011. [16] T. R. Lenka, and A. K. Panda, “Effect of structural parameters on 2DEG density and C ~ V characteristics of AlxGa1-xN/AlN/GaN-based HEMT,” Indian Journal of Pure & Applied Physics, vol. 49, pp. 416-422, Jun 2011. [17] S. Khandelwal and T. A. Fjeldly, “A physics based compact model for I–V and C–V characteristics in AlGaN/GaN HEMT devices,” Solid State Electron., vol. 76, pp. 60–66, Oct. 2012. 332 R. SINGH, T. R. LENKA, H. P. T. NGUYEN [18] F. M. Yigletu, S. Khandelwal, T. A. Fjeldly, and B. Iniguez, “Compact Charge-Based Physical Models for Current and Capacitances in AlGaN/GaN HEMTs,” IEEE Trans. Electron Devices, vol. 60, no. 11, pp. 3746–3752, Nov. 2013. [19] W. Wei, et al., “Valence band offset of β-Ga2O3/wurtzite GaN heterostructure measured by X-ray photoelectron spectroscopy,” Nanoscale Research Letters; vol. 7, p. 562, Dec. 2012. [20] H. Sun, et al., “Valence and conduction band offsets of β-Ga2O3/AlN heterojunction,” Applied Physics Letters, vol. 111, no. 16, p. 162105, Oct. 2017. [21] Y. K. Verma, V. Mishra, S. K. Gupta, “A Physics Based Analytical Model for MgZnO/ZnO HEMT,” Journal of Circuits, Systems, and Computers, Jan 2019. [22] H. Sun et al., “Nearly-zero valence band and large conduction band offset at BAlN/GaN heterointerface for optical and power device application,” Applied Surface Science, vol. 458, pp. 949–953, Jul. 2018. [23] R. Singh, T R Lenka, R T Velpula, B Jain, H Q T Bui, H P T Nguyen, “A novel β-Ga2O3 HEMT with fT of 166 GHz and X-band POUT of 2.91 W/mm,” Int. J. Numer Model El., e2794, 2020. [24] A. Mock, R. Korlacki, C. Briley, V. Darakchieva, B. Monemar, Y. Kumagai, K. Goto, M. Higashiwaki, M. Schubert, Phys. Rev. B Condens. Matter, vol. 96, no. 24, p. 245205, 2017. [25] Z. Zhang, E. Farzana, A.R. Arehart, and S.A. Ringel, “Deep level defects throughout the bandgap of (010) β-Ga2O3 detected by optically and thermally stimulated defect spectroscopy,” Applied Physics Letters, vol. 108, p. 052105, Feb. 2016. [26] P. Reddy, I. Bryan, Z. Bryan, J. Tweedie, R. Kirste, R. Collazo, and Z. Sitar, “Schottky contact formation on polar and nonpolar AlN,” Journal Applied Physics, vol. 116, no. 19, p. 194503, Nov. 2014. [27] Y. Irokawa, E. Villora, and K. Shimamura, “Schottky Barrier Diodes on AlN Free-Standing Substrates,” Japanese Jornal of Applied Physics, vol. 51, no. 4R, p. 040206, Mar. 2012. [28] T. Kinoshita, T. Nagashima, T. Obata, S. Takashima, R. Yamamoto, R. Togashi, Y. Kumagai, R. Schlesser, R. Collazo, A. Koukitu, and Z. Sitar, “Fabrication of vertical Schottky barrier diodes on n-type freestanding AlN substrates grown by hydride vapor phase epitaxy,” Appl. Phys. Exp., vol. 8, no. 6, p. 061003, May 2015.