Understanding the Leakage Mechanisms and Breakdown Limits of Vertical Gan-On-Si P+Nn Diodes

micromachines

Article Understanding the Leakage Mechanisms and Breakdown Limits of Vertical GaN-on-Si p+n−n Diodes: The Road to Reliable Vertical MOSFETs

Kalparupa Mukherjee 1,*, Carlo De Santi 1 , Matteo Buffolo 1 , Matteo Borga 2, Shuzhen You 2, Karen Geens 2, Benoit Bakeroot 3 , Stefaan Decoutere 2, Andrea Gerosa 1, Gaudenzio Meneghesso 1 , Enrico Zanoni 1 and Matteo Meneghini 1

1 Department of Information Engineering, University of Padua, 35131 Padova, Italy; [email protected] (C.D.S.); [email protected] (M.B.); [email protected] (A.G.); [email protected] (G.M.); [email protected] (E.Z.); [email protected] (M.M.) 2 Imec, Kapeldreef 75, 3001 Leuven, Belgium; [email protected] (M.B.); [email protected] (S.Y.); [email protected] (K.G.); [email protected] (S.D.) 3 CMST IMEC/UGent, 9052 Ghent, Belgium; [email protected] * Correspondence: [email protected]

Abstract: This work investigates p+n−n GaN-on-Si vertical structures, through dedicated measurements and TCAD simulations, with the ultimate goal of identifying possible strategies for leakage and breakdown optimization. First, the dominant leakage processes were identiﬁed through temperature- dependent current–voltage characterization. Second, the breakdown voltage of the diodes was modelled through TCAD simulations based on the incomplete ionization of Mg in the p+ GaN layer. Citation: Mukherjee, K.; De Santi, C.; Finally, the developed simulation model was utilized to estimate the impact of varying the p-doping Buffolo, M.; Borga, M.; You, S.; Geens, K.; Bakeroot, B.; Decoutere, S.; Gerosa, A.; concentration on the design of breakdown voltage; while high p-doped structures are limited by the Meneghesso, G.; et al. Understanding critical electric ﬁeld at the interface, low p-doping designs need to contend with possible depletion of the Leakage Mechanisms and the entire p-GaN region and the consequent punch-through. A trade-off on the value of p-doping Breakdown Limits of Vertical GaN-on-Si therefore exists to optimize the breakdown. p+n−n Diodes: The Road to Reliable Vertical MOSFETs. Micromachines Keywords: semi-vertical; vertical; GaN; pn diodes; leakage modeling; device modeling; TCAD 2021, 12, 445. https://doi.org/ 10.3390/mi12040445

Academic Editor: Giovanni Verzellesi 1. Introduction Wide bandgap semiconductors (WBGs) are being widely advocated to meet the Received: 26 February 2021 demands for higher efficiency, robustness, and power handling capabilities [1–16], as Si is Accepted: 13 April 2021 Published: 16 April 2021 approaching its limits in market sectors where high efficiency and high power are required, such as transportation, aerospace, and power conversion systems. The contenders include β Publisher’s Note: MDPI stays neutral SiC [13,14], GaN, -Ga2O3 [15], diamond [11,12], and AlN [16] in order of increasing with regard to jurisdictional claims in bandgap and thus breakdown field. So far, only GaN and SiC have progressed from published maps and institutional affil- research-level devices to commercially available technologies, whereas β-Ga2O3, diamond, iations. and AlN prototypes, while highly promising, have not yet reached the maturity for high level production. Currently, SiC-based devices are dominating the WBG market for power applications requiring >1200 V, voltages that are not yet reached by commercial lateral GaN devices. However, GaN is superior to SiC in overall material properties (breakdown field, mobility, Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. saturation velocity), making it the stronger choice for most applications. To further enhance This article is an open access article the performance of GaN transistors, the focus of research is shifting from lateral to vertical distributed under the terms and architectures, which circumvent the breakdown limitations, surface trapping, and other conditions of the Creative Commons challenges inherent to the lateral topology. Attribution (CC BY) license (https:// The adoption of GaN-based vertical diodes and transistors [1–7] is highly advanta- creativecommons.org/licenses/by/ geous to high-speed and high-power electronics applications, presenting low Ron and 4.0/). higher breakdown robustness in addition to improved thermal performance. Preliminary

Micromachines 2021, 12, 445. https://doi.org/10.3390/mi12040445 https://www.mdpi.com/journal/micromachines Micromachines 2021, 12, 445 2 of 11

demonstrators of quasi-vertical or vertical diode structures [1–10] have already presented good performance metrics, handling the associated epitaxial challenges in various ways. Within the GaN power field, emerging GaN-on-Si vertical technologies have the competitive advantage. However, owing to the lattice mismatch between GaN and Si, the epitaxial thicknesses and doping of the vertical stack require careful optimization to achieve high breakdown voltages. For well-designed structures, the drift region becomes especially relevant because it dominantly controls the reverse breakdown capability. Currently, the wide use of GaN-on-Si vertical devices is being impeded by the lack of understanding of the leakage mechanisms and breakdown of the vertical stack. Leakage paths along the passivation, drift, and other transition layers, and through the Si substrate can critically affect the reliability of GaN-on-Si diodes. Reverse leakage conduction can arise from electrode limited, surface limited, or bulk limited conduction mechanisms [5,7,17]. Hence, the study and identification of vertical leakage conduction and doping con- straints to optimize breakdown voltage are major research concerns in the design of vertical pn GaN diodes and FETs [5–7]. This analysis is different from studies of leakage in lateral GaN devices. In lateral GaN transistors, the buffer is typically isolating and compensated with carbon; in vertical GaN devices, most of the stack is constituted by a drift region that is not intentionally doped and which needs to be highly conductive. To investigate the leakage and breakdown, technology computer aided design (TCAD) simulations provide cost-effective and versatile perspectives towards the optimization and design of these devices. In this work, we use a combination of TCAD simulations and experimental measurements to understand how to overcome the breakdown and leakage limits of GaN-on Si vertical devices, and to provide general design rules. Within this work, Section2 presents the structural details of the investigated quasi- vertical GaN-on-Si diodes. Section3 is divided into two parts: Section 3.1 presents the modeling of the reverse leakage current of the tested devices, to identify the dominant mechanism for low and high voltage ranges, while Section 3.2 uses a simplified TCAD model to understand how the nature of breakdown changes as a function of the chosen doping level within the p+ body. Section4 reviews the main results and concludes the work.

2. Materials and Methods The p+n diode is the core of the epitaxy of semi-vertical/vertical trench-MOSFETs [8–10]; hence, understanding its breakdown limits is fundamental for the development of robust vertical GaN transistors. The samples in this study are test vehicles for investigation into Micromachines 2021, 12, x FOR PEER REVIEW+ − 3 of 12 the p n n GaN-on-Si semi-vertical diode conﬁguration (Figure1), enabling the dedicated characterization of diode properties and doping effects.

− FigureFigure 1. SchematicSchematic of of the the p+n p−+ dioden diode test vehicle test vehicle structures, structures, the fundamental the fundamental unit of a complete unit of a complete semi-verticalsemi-vertical MOSFET. MOSFET.

Fabricated on a 200 mm Si substrate, the diodes are based on an Mg-doped p+ GaN body, where NA = 6 × 1019 cm−3, and a lightly doped n− drift layer. The cathode is defined at the buried n+ layer below the n-drift region. The n+ layers have a doping of 5 × 1018 cm−3, while the n−drift layer has a doping of ND = 4 × 1016 cm−3. The parameters have been summarized in Table 1. For the p+ GaN, the value of NA = 6 × 1019 cm−3 represents the test structures under test. However, the simulated p-doping levels have been varied to discuss the impacts of choosing low versus high p-doping values.

Table 1. Structural properties of the vertical GaN-on-Si diodes under test.

Layer (GaN) Doping (cm−3) Thickness (nm) n+ 5 × 1018 250 p+ body 6 × 1019 400 n− drift 4 × 1016 750

3. Results and Discussion 3.1. Physical Origin of Leakage Current The conduction in the depletion region of a reverse biased p+n junction can be considered analogous to leakage through a dielectric subjected to high fields [5,7,18]. The off- state leakage mechanisms can then be categorized with respect to (a) properties of the metal-dielectric (semiconductor) contact—referred to as electrode-limited conduction mechanisms, or (b) the properties of the dielectric, and thus the existing trap levels—referred to as bulk-limited conduction mechanisms. Both kinds of processes might be simultaneously applicable; however, electrode-limited mechanisms such as thermionic emission, Schottky emission, and direct or F–N tunneling should not be the limiting factors in well-designed vertical diodes with the peak field located deeper at the p+n junction. As such, the dominant mechanisms are usually bulk-limited, such as Poole–Frenkel emission, space charge limited conduction (SCLC), and variable range hopping (VRH), among others. To identify the vertical leakage mechanisms of the studied p+n−n diodes, the temperature (T) dependence of the reverse biased diode characteristic is measured, as illustrated in Figure 2a.

Micromachines 2021, 12, 445 3 of 11

Fabricated on a 200 mm Si substrate, the diodes are based on an Mg-doped p+ GaN 19 −3 − body, where NA = 6 × 10 cm , and a lightly doped n drift layer. The cathode is deﬁned at the buried n+ layer below the n-drift region. The n+ layers have a doping of 18 −3 − 16 −3 5 × 10 cm , while the n drift layer has a doping of ND = 4 × 10 cm . The parameters + 19 −3 have been summarized in Table1. For the p GaN, the value of NA = 6 × 10 cm represents the test structures under test. However, the simulated p-doping levels have been varied to discuss the impacts of choosing low versus high p-doping values.

Table 1. Structural properties of the vertical GaN-on-Si diodes under test.

Layer (GaN) Doping (cm−3) Thickness (nm) n+ 5 × 1018 250 p+ body 6 × 1019 400 n− drift 4 × 1016 750

3. Results and Discussion 3.1. Physical Origin of Leakage Current The conduction in the depletion region of a reverse biased p+n junction can be considered analogous to leakage through a dielectric subjected to high ﬁelds [5,7,18]. The off-state leakage mechanisms can then be categorized with respect to (a) properties of the metal-dielectric (semiconductor) contact—referred to as electrode-limited conduction mechanisms, or (b) the properties of the dielectric, and thus the existing trap levels—referred to as bulk-limited conduction mechanisms. Both kinds of processes might be simultaneously applicable; however, electrode-limited mechanisms such as thermionic emission, Schottky emission, and direct or F–N tunneling should not be the limiting factors in well-designed vertical diodes with the peak ﬁeld located deeper at the p+n junction. As such, the dominant mechanisms are usually bulk-limited, such as Poole–Frenkel Micromachines 2021, 12, x FOR PEER REVIEW 4 of 12 emission, space charge limited conduction (SCLC), and variable range hopping (VRH), among others. To identify the vertical leakage mechanisms of the studied p+n−n diodes, the temperature (T) dependence of the reverse biased diode characteristic is measured, as illustrated in Figure2a.

-4 + − -4 10 p n diode, T (°C) 10 p+ n− diode, T (°C) 10-5 50 60 70 80 90 10-5 50 60 70 80 90 10-6 100 110 120 130 10-6 100 110 120 130 10-7 10-7 10-8 10-8 10-9 10-9 -10 -10 (A) Current 10 Current (A) Current 10 10-11 10-11 Model in Region 1 10-12 Model in Region 2 10-12 Region 1 Region 2 10-13 10-13 0 20406080 0 20406080 Cathode Voltage (V) Cathode Voltage (V)

(a) (b) FigureFigure 2. 2. ((aa)) Reverse Reverse leakageleakage of of the the p +pn+−nn−n diodes diodes measured measured for for T = T 50 = ◦50C to°C 130 to 130◦C. Two°C. Two distinct distinct regions regions can be can identiﬁed; be identified; (b(b) )illustrates illustrates the ﬁtfit toto the the models models used used to to describe describe the the leakage leakage evolutions evolutions in the in two the regions. two regions. The curve exhibits two distinct regions (Region 1 and Region 2), with a notable second The curve exhibits two distinct regions (Region 1 and Region 2), with a notable sec- rise in slope for voltages higher than 40–60 V (depending on temperature). Each region has beenond modelledrise in slope separately, for voltages as illustrated higher inthan Figure 40–602b, andV (depending described as on follows. temperature). Each re- gionThe has observed been modelled rapid rise separa in leakagetely, as withillustrated temperature in Figure implies 2b, and a strong described thermally as follows. activatedThe process, observed which rapid is modelledrise in leakage in Figure with3a usingtemperature the following implies equation: a strong thermally activated process, which is modelled in Figure 3a using the following equation: −𝐸 𝐼 = 𝐴𝑇 𝑒𝑥𝑝 , (1) 𝑘𝑇 This equation describes the temperature dependence of the current conduction orig- inated by thermionic emissions from Coulombic traps [6,19]. Here, A is a constant defining the almost vertical shift of the curve, 𝐸 is the thermal activation energy defining the slope of the curve, kB is the Boltzmann constant, and T is the temperature. An activation energy of ≈0.85 eV is extracted from the slope in Figure 3b, indicating a possible role of carbon acceptors in the leakage process [20,21].

1.0 1.5

Voltage from 0.5V to 30V, step 0.5V 10n 0.8 2 1n 1.0 0.6 880.0m 870.0m (eV) A 860.0m 100p 850.0m 0.4 840.0m 830.0m 820.0m 0.5 Adjusted R Current (A) 10p 810.0m 0.2 800.0m β = 1.77x10-5 eV.V-1/2 .m1/2 790.0m

Activation energy (eV) 6000 7000 8000 9000 10000

1p Activation energy E Square root of electric field (V/m)1/2 0.0 0.0 320 340 360 380 400 0 5 10 15 20 25 30 Voltage (V) Temperature (K) (a) (b) Figure 3. (a) Modeling reverse leakage diode behavior using Coulombic potential well model from 0 V to −30 V (Region 1) for T = 50 °C (323 K) to 130 °C (403 K); (b) extrapolated trap activation energy of 0.85 eV. The R2 of the fit is near one in the entire analyzed voltage range, confirming the good quality of the fit. (inset) Lowering in EA showing F−1/2 dependence.

Micromachines 2021, 12, x FOR PEER REVIEW 4 of 12

(a) (b) Figure 2. (a) Reverse leakage of the p+n−n diodes measured for T = 50 °C to 130 °C. Two distinct regions can be identified; (b) illustrates the fit to the models used to describe the leakage evolutions in the two regions.

The curve exhibits two distinct regions (Region 1 and Region 2), with a notable second rise in slope for voltages higher than 40–60 V (depending on temperature). Each re-

Micromachines 2021, 12, 445 gion has been modelled separately, as illustrated in Figure 2b, and described as follows.4 of 11 The observed rapid rise in leakage with temperature implies a strong thermally activated process, which is modelled in Figure 3a using the following equation: −𝐸 𝐼 = 𝐴𝑇 𝑒𝑥𝑝 𝑘 𝑇, (1) 2 −EA This equation describes the temperatureITE = AT exp dependence, of the current conduction orig- (1) kBT inated by thermionic emissions from Coulombic traps [6,19]. Here, A is a constant defining This equation describes the temperature dependence of the current conduction origi- the almost vertical shift of the curve, 𝐸 is the thermal activation energy defining the nated by thermionic emissions from Coulombic traps [6,19]. Here, A is a constant deﬁning slope of the curve, kB is the Boltzmann constant, and T is the temperature. An activation energythe almost of ≈0.85 vertical eV shiftis extracted of the curve, from EtheA isslope the thermalin Figure activation 3b, indicating energy a deﬁning possible the role slope of carbonof the curve,acceptorskB is in the the Boltzmann leakage process constant, [20,21]. and T is the temperature. An activation energy of ≈0.85 eV is extracted from the slope in Figure3b, indicating a possible role of carbon acceptors in the leakage process [20,21].

1.0 1.5

Activation energy (eV) 6000 7000 8000 9000 10000

1p Activation energy E Square root of electric field (V/m)1/2 0.0 0.0 320 340 360 380 400 0 5 10 15 20 25 30 Voltage (V) Temperature (K) (a) (b)

Figure 3. ((a)) Modeling Modeling reverse leakage diodediode behaviorbehavior usingusing CoulombicCoulombic potentialpotential well well model model from from 0 0 V V to to− −3030 VV (Region(Region 1) 1)for for T =T 50= 50◦C °C (323 (323 K) K) to to 130 130◦C °C (403 (403 K); K); (b ()b extrapolated) extrapolated trap trap activation activation energy energy of of 0.85 0.85 eV. eV. The The R2 Rof2 of the the fit fit is nearis near one one in in the the entire analyzed voltage range, confirming the good quality of the fit. (inset) Lowering in EA showing −F1/2−1/2 dependence. entire analyzed voltage range, confirming the good quality of the fit. (inset) Lowering in EA showing F dependence. The corresponding conduction mechanism is described in Figure4a. At low field F, the potential near traps can be assumed to be Coulombic, while at higher fields, the potential is

deformed. Depending on the nature of the deformation, charge emission from an occupied primary trap state located at an energy of ET from the conduction band minimum (CBM) (labelled A in Figure4) can be strengthened through different conduction mechanisms. A higher temperature can lead to phonon assisted tunneling processes (contribution labelled PhaT in Figure4a). With increasing temperature, the overall thermal energy of the trapped electron is higher, leading to a thinner barrier for carrier tunneling, as illustrated in the transition from A to B in blue in Figure4a. However, under high ﬁelds, Poole–Frenkel lowering of the barrier height becomes relevant. A lower effective barrier can be directly overcome by the carriers by thermionic emission (contribution labelled PF in Figure4a), as illustrated in the transition from A to B in red in Figure4a. The Poole–Frenkel effect thus results in a change in the emission rate en [7,19,22,23] as follows: 1 ! ET − βF 2 en ∝ exp − . (2) kBT

This process facilitates emissions from trap centers√ at high fields (the Poole–Frenkel coefficient β quantifies the lowering in the barrier = β F). As presented in the inset of Figure4b, the field dependence of the extracted trap level at 0.85 eV is found to follow this behavior. The peak electric field values are taken from corresponding numerical simulations in the considered voltage range (described in detail in Section 3.2). Micromachines 2021, 12, x FOR PEER REVIEW 5 of 12

The corresponding conduction mechanism is described in Figure 4a. At low field 𝐹, the potential near traps can be assumed to be Coulombic, while at higher fields, the potential is deformed. Depending on the nature of the deformation, charge emission from an occupied primary trap state located at an energy of ET from the conduction band minimum (CBM) (labelled A in Figure 4) can be strengthened through different conduction mechanisms. A higher temperature can lead to phonon assisted tunneling processes (contribution labelled PhaT in Figure 4a). With increasing temperature, the overall thermal energy of the trapped electron is higher, leading to a thinner barrier for carrier tunneling, as illustrated in the transition from A to B in blue in Figure 4a. However, under high fields, Poole–Frenkel lowering of the barrier height becomes relevant. A lower effective barrier can be directly overcome by the carriers by thermionic emission (contribution labelled PF in Figure 4a), as illustrated in the transition from A to B in red in Figure 4a. The Poole– Frenkel effect thus results in a change in the emission rate 𝑒 [7,19,22,23] as follows:

𝑒 ∝𝑒𝑥𝑝− . (2) This process facilitates emissions from trap centers at high fields (the Poole–Frenkel coefficient β quantifies the lowering in the barrier = β√F). As presented in the inset of Fig- ure 4b, the field dependence of the extracted trap level at 0.85 eV is found to follow this behavior. The peak electric field values are taken from corresponding numerical simula- Micromachines 2021, 12, 445 tions in the considered voltage range (described in detail in Section 3.2). 5 of 11

CBM Coulombic-well CBM

1/2 βF B PF E T VRH ET B PhaT DOS A C R

(a) (b)

FigureFigure 4. 4.Schematic Schematic representation representation of of the the leakage leakage processes: processes: (a ()a Poole–Frenkel) Poole–Frenkel (PF) (PF) emission emission (used (used to to model model Region Region 1 1 in in FigureFigure2); 2); ( b ()b variable) variable range range hopping hopping (used (used to to model model Region Region 2 in2 in Figure Figure2). 2).

q 3 Theoretically, β can written as β = Zq𝑍𝑞[7,24,25], where Z represents the charge on Theoretically, β can written as 𝛽= πε 𝜋𝜀 [7,24,25], where Z represents the charge the Coulomb center (ionization state of the trap), and ε is the permittivity of GaN. The extractedon the Coulombβ = 1.77 center× 10− (ionization5 eV V−1/2 statem1/2 ofin the our trap), measurements and 𝜀 is the is permittivity close to the theoreticalof GaN. The −5 −1/2 1/2 valueextracted (≈3.1 β× = 101.77−5 ×eV 10 V− eV1/2 Vm1/2m) considering in our measurements a relative high-frequency is close to the GaN theoretical permittivity value (≈3.1 × 10−5 eV V−1/2 m1/2) considering a relative high-frequency GaNq permittivity of 5.8, of 5.8, and Z = 1 for the carbon acceptor. Considering a simpliﬁed β = q [26], theoretical 𝑞 πε −5 −1/2 1/2 Micromachines 2021, 12, x FOR PEER REVIEWvaluesand Z are = 1 close for the to 3.2 carbon× 10 acceptor.eV V Consideringm [27–29 a]. simplified Depending 𝛽= on the growth𝜋𝜀 [26], conditions theoretical6 of 12

invalues different are GaN-based close to 3.2 works, × 10−5 βeV≈ V10−1/2−5 meV1/2 V[27–29].−1/2 m 1/2Dependingare generally on the reported growth[7, 24conditions,25,27–34 ]in, asdifferent summarized GaN-based in Figure works,5. 𝛽 ≈ 10−5 eV V−1/2 m1/2 are generally reported [7,24,25,27–34], as summarized in Figure 5.

Figure 5. Comparison of the Poole–Frenkel coefficient (β) ≈ 10−5 eV V− 1/2 m1/2 obtained from Figuredifferent 5. Comparison works on GaN-basedof the Poole–Frenkel systems [ 7coefficient,24,25,27– 34(β].) ≈ 10−5 eV V−1/2 m1/2 obtained from different works on GaN-based systems [7,24,25,27–34]. For the second region of the leakage curves, the variable range hopping [6,7,35] model,For the which second describes region theof the current leakage associated curves, with the thevariable hopping range of hopping electrons [6,7,35] from one model,trap which state to describes another the distributed current associated across different with the energies, hopping is of found electrons to best from represent one trap the stateleakage to another evolution. distributed The VRH across mechanism, different en basedergies, on is the found theory to best developed represent by the Mott leakage and Hill, E evolution.is illustrated The VRH in Figure mechanism,4b. The based primary on the trap theory location developed is at A, by with Mott an and energy Hill, isT illus-and an exponentially distributed density of states (DOS). Electrons can hop from A to empty trap trated in Figure 4b. The primary trap location is at A, with an energy ET and an exponen- positions situated at B or C, within a distance of R from the primary trap, and within a tially distributed density of states (DOS). Electrons can hop from A to empty trap positions range of energy surrounding E . situated at B or C, within a distanceT of R from the primary trap, and within a range of The fit to measurements is presented in Figure6 and modelled using the following energy surrounding ET. relation with temperature T [7,35], valid for moderate to high electric fields, where F2 The fit to measurements is presented in Figure 6 and modelled using the following represents the field contribution to strengthening the VRH conduction: relation with temperature T [7,35], valid for moderate to high electric fields, where 𝐹 represents the field contribution to strengthening the VRH conduction: 𝑇 𝑇 𝐼 =𝐼𝑒𝑥𝑝 −1.76 𝑇 𝐶 𝑇 𝐹 , (3)

Voltage from 70V to 75V, step 0.1V

100n Current (A) Current

320 340 360 380 400 Temperature (K)

Figure 6. Modeling reverse leakage diode behavior using variable range hopping (VRH) from V= −70 V to −75 V (Region 2) for T = 50 °C (323 K) to 130 °C (403 K).

I0 modifies the trap emission rate into current, CVRH represents a grouped constant, 𝑞𝑎 which can be written as 𝐶 = 4.626 10 𝑈 [7], where q is the elementary

Micromachines 2021, 12, x FOR PEER REVIEW 6 of 12

Figure 5. Comparison of the Poole–Frenkel coefficient (β) ≈ 10−5 eV V−1/2 m1/2 obtained from different works on GaN-based systems [7,24,25,27–34].

Micromachines 2021, 12, 445 6 of 11 For the second region of the leakage curves, the variable range hopping [6,7,35] model, which describes the current associated with the hopping of electrons from one trap state to another distributed across different energies, is found to best represent the leakage " 1 3 # 4 4 evolution. The VRH mechanism, based onT 0the theory developedT0 2 by Mott and Hill, is illus- IVRH = I0exp −1.76 + CVRH F , (3) trated in Figure 4b. The primary trap locationT is at A, withT an energy ET and an exponentially distributed density of states (DOS). Electrons can hop from A to empty trap positions I0 modiﬁes the trap emission rate into current, CVRH represents a grouped constant, situated at B or C, within a distance of −R3 from qa the2 primary trap, and within a range of which can be written as CVRH = 4.626 × 10 × U [7], where q is the elementary charge, energywhile a surroundingand U are constants ET. related to the physical properties of the trap states. U indicates the characteristicThe fit to measurements energy of the is DOS, presented and a represents in Figure the 6 and localization modelled radius using of thethe wave following relationfunction with corresponding temperature to theT [7,35], trapped valid electron. for moderate It can be estimated to high electric to the effective fields, Bohrwhere 𝐹 representsradius of the the bound field contribution electron and liesto strengthening within the range the of VRH 1 nm conduction: to 10 nm [7, 27,30,36]. T0, = 18 the characteristic temperature, can be written as T0 (k D a3) [7,35,37 ,38]. It is inversely B T −1 𝑇 −1 𝑇 proportional to the trap𝐼 DOS=𝐼D𝑒𝑥𝑝T (volume −1.76energy 𝑇 )𝐶 at the primary 𝑇 (see𝐹 A, in Figure4b) (3) trap energy and can vary depending on a. A summary of different T0 values reported in the literature [7,24,25,27–30,33,37,39] is provided in Figure7 for reference, showing consistent results for the values extrapolated in this paper.

Voltage from 70V to 75V, step 0.1V

Micromachines 2021, 12, x FOR PEER REVIEW 7 of 12

100n charge, while a and U are constants related to the physical properties of the trap states. 𝑈

indicates (A) Current the characteristic energy of the DOS, and 𝑎 represents the localization radius of the wave function corresponding to the trapped electron. It can be estimated to the effective Bohr radius of the bound electron and lies within the range of 1 nm to 10 nm 18 [7,27,30,36]. T0, the characteristic temperature, can be written as 𝑇 = 𝑘𝐷𝑎 320 340 360 380 400 [7,35,37,38]. It is inversely proportional to the trap DOS DT (volume−1 energy−1) at the primary (see A in Figure 4b)Temperature trap energy and (K) can vary depending on 𝑎. A summary of different T0 values reported in the literature [7,24,25,27–30,33,37,39] is provided in Figure 7 forFigureFigure reference, 6. 6. ModelingModeling showing reverse reverse consistent leakage leakage results diode diode for behavior the values usingusin extrapolatedg variable rangerange in this hoppinghopping paper. (VRH)(VRH) from from V= ◦ ◦ −70V= V− to70 − V75 to V− (Region75 V (Region 2) for 2) T for = T50 = °C 50 (323C (323 K) K)to 130 to 130 °C (403C (403 K). K).

I0 modifies the trap emission rate into current, CVRH represents a grouped constant, 𝑞𝑎 which can be written as 𝐶 = 4.626 10 𝑈 [7], where q is the elementary

Figure 7. Comparison of the characteristic temperature (T0) obtained from different works on

FigureGaN-based 7. Comparison systems of [7 the,24 ,characteristic25,27–30,33,37 temperature,39]. (T0) obtained from different works on GaN-based systems [7,24,25,27–30,33,37,39]. Based on these results, we conclude that a substantial reduction in leakage current and in its temperature sensitivity can be obtained through the reduction of the density Based on these results, we conclude that a substantial reduction in leakage current of defects within the drift region. Speciﬁc attention needs to be focused on the residual and in its temperature sensitivity can be obtained through the reduction of the density of carbon concentration, considering its contribution to the low voltage regime. defects within the drift region. Specific attention needs to be focused on the residual carbon concentration, considering its contribution to the low voltage regime.

3.2. TCAD Simulations of Diode Breakdown To obtain an estimate of the breakdown voltage of the test structures, several samples were subjected to reverse bias sweeps until failure at room temperature, as illustrated in Figure 8. Very little dispersion was observed within the diode characteristics, and the mean reverse breakdown voltage was found to be around 170 V (inset of Figure 8).

100 -1 10 + − + 10-2 p n n GaN-on-Si diodes -3 10 At RT 10-4 10-5 10-6 10-7 175 10-8 -9 10 170 -10

Current (A) Current 10 -11 10 165 10-12 10-13 Breakdown Voltage (V) 0 50 100 150 200 Cathode Voltage (V)

Micromachines 2021, 12, x FOR PEER REVIEW 7 of 12

charge, while a and U are constants related to the physical properties of the trap states. 𝑈 indicates the characteristic energy of the DOS, and 𝑎 represents the localization radius of the wave function corresponding to the trapped electron. It can be estimated to the effective Bohr radius of the bound electron and lies within the range of 1 nm to 10 nm 18 [7,27,30,36]. T0, the characteristic temperature, can be written as 𝑇 = 𝑘𝐷𝑎 [7,35,37,38]. It is inversely proportional to the trap DOS DT (volume−1 energy−1) at the primary (see A in Figure 4b) trap energy and can vary depending on 𝑎. A summary of different T0 values reported in the literature [7,24,25,27–30,33,37,39] is provided in Figure 7 for reference, showing consistent results for the values extrapolated in this paper.

Figure 7. Comparison of the characteristic temperature (T0) obtained from different works on GaN-based systems [7,24,25,27–30,33,37,39].

Based on these results, we conclude that a substantial reduction in leakage current and in its temperature sensitivity can be obtained through the reduction of the density of

Micromachines 2021, 12, 445 defects within the drift region. Specific attention needs to be focused on the residual car-7 of 11 bon concentration, considering its contribution to the low voltage regime.

3.2. TCAD Simulations of Diode Breakdown 3.2.To TCADobtain Simulationsan estimate ofof Diode the breakdown Breakdown voltage of the test structures, several samples were subjectedTo obtain to anreverse estimate bias of sweeps the breakdown until fail voltageure at room of the temperature, test structures, as severalillustrated samples in Figurewere 8. subjectedVery little to dispersion reverse bias was sweeps observ untiled failurewithin atthe room diode temperature, characteristics, as illustrated and the in meanFigure reverse8. Very breakdown little dispersion voltage was was observed found to within be around the diode 170 V characteristics, (inset of Figure and 8). the mean reverse breakdown voltage was found to be around 170 V (inset of Figure8).

100 -1 10 + − + 10-2 p n n GaN-on-Si diodes -3 10 At RT 10-4 10-5 10-6 10-7 175 Micromachines 2021, 12, x FOR PEER REVIEW10-8 8 of 12 -9 10 170 -10

Current (A) Current 10 -11 10 165 Figure10-12 8. Reverse bias I–V diode characteristics until failure on four diodes. (inset) Mean break- 10-13 Breakdown Voltage (V) down voltage0 is 170 V. 50 100 150 200 Cathode Voltage (V) The 2D-TCAD simulations, based on the drift–diffusion model for carrier transport, Figurewere 8. Reverseemployed bias to I–V build diode a representative characteristics until model failure of the on fourmeasured diodes. devices, (inset) Mean using breakdown the Sen- voltagetaurus is 170 tool V. from Synopsys [40]. To investigate the nature of breakdown, a simplified fully vertical (n+-p+-n−-n+) diode structure (see Figure 9a) was used. The anode and cathode are The 2D-TCAD simulations, based on the drift–diffusion model for carrier transport, defined as modified ohmic contacts to improve accuracy around the p–n junction; it does werenot employed impose the to charge build a neutrali representativety condition model at vertices of the measured within the devices, charged using depletion the Sen- re- taurusgions tool [40]. from In addition Synopsys to [suitable40]. To investigatemobility and the recombination nature of breakdown, models, the a simplifiedgate-dependent fully verticalstrain (n polarization+-p+-n−-n+ )model, diode especially structure (seesuited Figure to GaN9a) devices, was used. was The activated. anode andSince cathode Mg has area defined relatively as modifiedhigh ionization ohmic en contactsergy of to0.16 improve eV [41,42], accuracy Mg acceptors around theare p–nnot junction;completely it doesionized not impose at room the temperature. charge neutrality Thus, conditionto correctly at model vertices the within p-doping the chargedlevels and depletion reproduce re- gionsthe [ 40breakdown]. In addition voltage, to suitable the incomplete mobility ionization and recombination model in Sentaurus models, the was gate-dependent used, which is strainphysically polarization more model,accurate especially to model suitedMg doping. to GaN The devices, effective was doping activated. concentration Since Mg was has a relativelycomputed high internally ionization based energy on ionization of 0.16 probability, eV [41,42], derived Mg acceptors from the are ionization not completely energy ionizedof the at doping room temperature.species. Thus, to correctly model the p-doping levels and reproduce the breakdownFigure 9b voltage, presents the the incomplete electric field ionization evolution along model the in simulated Sentaurus structure was used, as a which func- is physicallytion of the more chosen accurate p-doping to modellevel, for Mg a cathod doping.e voltage The effective of 160 V, doping i.e., just concentration below the meas- was computedured breakdown internally voltage. based on ionization probability, derived from the ionization energy of the doping species.

V Cathode = 160 V + + 3 n Doping Conc. n 250 nm + ANODE + N (cm-3) 2.5 n p A (5×1018) 4x1017 17 + 400 nm 2 6x10 p GaN 1x1018 1x1019 1.5 6x1019 n− GaN n− 750 nm 1 n+ × 16 (4 10 ) × 18 0.5 (5 10 ) Electric Field (MV/cm)Electric Field n+ GaN 0 CATHODE -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Y Dimension (μm)

(a) (b)

FigureFigure 9. (a )9. Schematic (a) Schematic of the of the simulated simulated structure. structure. ( b(b)) Electric Electric field profile profile along along the the simulated simulated vertical vertical p+n pdiode+n diode for N forA = 4 × 1017 17cm−3, 6− 3× 1017 cm17–3, 1 × –31018 cm−3, 181 × 10−193 cm−3, and19 6 × 10−319 cm−3. 19 −3 NA = 4 × 10 cm , 6 × 10 cm , 1 × 10 cm , 1 × 10 cm , and 6 × 10 cm .

For high p-doping, such as for NA = 6 × 1019 cm−3 (see Figure 10b), we observed the peak electric field at the p+ to n− interface approaching the critical field value for GaN (3.3 MV/cm [43]). The applied potential dropped almost entirely across the n-drift region, with little to negligible depletion observed within the p-GaN layer. As such, we can expect breakdown to occur when the peak electric field crosses ECrit.

Micromachines 2021, 12, 445 8 of 11

Figure9b presents the electric ﬁeld evolution along the simulated structure as a function of the chosen p-doping level, for a cathode voltage of 160 V, i.e., just below the measured breakdown voltage. 19 −3 Micromachines 2021, 12, x FOR PEER REVIEW For high p-doping, such as for NA = 6 × 10 cm (see Figure 10b), we observed9 of 12 the

peak electric ﬁeld at the p+ to n− interface approaching the critical ﬁeld value for GaN Micromachines 2021, 12, x FOR PEER REVIEW(3.3 MV/cm [43]). The applied potential dropped almost entirely across the n-drift9 region, of 12

with little to negligible depletion observed within the p-GaN layer. As such, we can expect breakdown to occur when the peak electric ﬁeld crosses ECrit.

Figure 10. Two failure processes are identified from electric field profiles: (a) For lowly doped samples, punch-through occurs, due to the full depletion of the p-GaN, and (b) for high p-GaN, doping breakdown corresponds to the voltage for Figure 10. Two failure processes are identified from electric field profiles: (a) For lowly doped samples, punch-through whichFigure the 10. peak Two E-field failure at processes the p+n junction are identified reaches from the crelectricitical field field of profiles: GaN. (a) For lowly doped samples, punch-through occurs,occurs, due due to tothe the full full depletion depletion of ofthe the p-GaN, p-GaN, and and (b) ( bfor) for high high p-GaN, p-GaN, doping doping breakdow breakdownn corresponds corresponds to tothe the voltage voltage for for + whichwhich the the peak peak E-field E-field at atthe the p+ pnFor junctionn junction the devices reaches reaches underthe the critical critical test, field the field ofp-doping ofGaN. GaN. level was NA = 6 × 1019 cm−3, which indicates breakdown triggered by peak electric fields >3 MV/cm at the p+n junction,19 − 3as can be seen For the devices under test, the p-doping level was NA = 6 × 10 cm , which indicates 19 −3 frombreakdown theFor evolutionthe devices triggered of under the by electric peaktest, the electricfield p-doping with fields increasing level >3 MV/cm was voltage NA at = the6 in× p10 Figure+n cm junction, 11., which as canindicates be seen + breakdownfrom the evolutiontriggered ofby the peak electric electric field fields with >3 increasing MV/cm at voltage the p n in junction, Figure 11 as. can be seen from the evolution of the electric field with increasing voltage in Figure 11. 3.5

3.53

2.53

2.52

1.52

1.51 0.5

Electric Field (MV/cm) 1

0.50

Electric Field (MV/cm) 0 50 100 150 200 0 Cathode Voltage (V) 0 50 100 150 200 19 19−3 −3 FigureFigure 11. 11.ElectricElectric field fieldCathode profile profile versus versusVoltage voltage voltage for(V) for NA N =A 6 =× 610× 10cm cm, corresponding, corresponding to the to devices the devices underunder test, test, illustrating illustrating a field a field value value ≈3 MV/cm≈3 MV/cm at 170 at V. 170 V. Figure 11. Electric field profile versus voltage for NA = 6 × 1019 cm−3, corresponding to the devices underThe test,The field illustrating field √values values aobtained field obtained value in ≈ 3the in MV/cm the range range at of 170 0 of V.V–30 0 V–30 V were V were used used in Figure in Figure 3b3 tob toverify verify the ∆E ∝ F dependence. It can be expected that by decreasing the p-doping within a the ∆𝐸 ∝A √𝐹 dependence. It can be expected that by decreasing the p-doping within a certain range, we can reduce the peak electric field (see Figure 10b for N = 1 × 1018 cm−3), certainThe range, field we values can reduceobtained the in peak the rangeelectric of field 0 V–30 (see V Figure were used10b for in NFigureAA = 1 ×3b 10 to18 cmverify−3), pushing V to higher voltages >200 V. Simulations indicated that reducing the p-doping pushingthe ∆𝐸 V∝BR√ toBR𝐹 higher dependence. voltages It can>200 be V. expected Simulation thats indicated by decreasing that reducing the p-doping the p-doping within a 17 −3 significantly introduces a different constraint. For example, for N = 6 × 10 18cm −,3 we significantlycertain range, introduces we can reduce a different the peak constraint. electric Forfield example, (see Figure for 10bNA =for 6A ×N 10A =17 1cm × −103, we cm ob-), observed a much wider depletion of the p-GaN region. If the p-doping levels are too low pushing VBR to higher voltages >200 V. Simulations indicated that reducing the p-doping served a much wider17 −depletion3 of the p-GaN region. If the p-doping levels are too low (NA = 4 × 10 cm in Figures9b and 10a), complete depletion of the pGaN17 −3 layer can (NsignificantlyA = 4 × 1017 cmintroduces−3 in Figures a different 9b and 10a),constraint. complete For depletionexample, offor the NA pGaN = 6 × 10layer cm can, weoccur, ob- thusserved leading a much to punch-throughwider depletion even of the before p-GaN the peakregion. electric If the field p-doping reaches levels ECrit .are This too effect low may(NA =be 4 ×further 1017 cm worsened−3 in Figures under 9b and real 10a), conditions complete by depletionthe presence of the of pGaNhydrogen layer during can occur, the epi-growththus leading process to punch-through [44], which would even before reduce the the peak effective electric concentration field reaches of E MgCrit. andThis accel- effect eratemay thebe further punch-through. worsened under real conditions by the presence of hydrogen during the epi-growth process [44], which would reduce the effective concentration of Mg and accelerate the punch-through.

Micromachines 2021, 12, 445 9 of 11

Micromachines 2021, 12, x FOR PEER REVIEW 10 of 12

occur, thus leading to punch-through even before the peak electric ﬁeld reaches ECrit. This effect may be further worsened under real conditions by the presence of hydrogen during the epi-growth processFigure [44 12], illustrates which would the evolution reduce the of effective electric field concentration and ionized of acceptor Mg and concentration accelerate thewith punch-through. cathode voltage, for low (NA = 4 × 1017 cm−3) and high p-doping (NA = 6 × 1019 cm−3) Figure 12 cases.illustrates the evolution of electric field and ionized acceptor concentration with 17 −3 19 −3 cathode voltage, for low (NA = 4 × 10 cm ) and high p-doping (NA = 6 × 10 cm ) cases.

17 17 17 4×10 ) 5x10 4×10 -3 2.5 VCathode VCathode + n+ p 100 V 4x1017 n (a) 100 V 2 (5×1018) 120 V (5×1018) 120 V 140 V 3x1017 140 V 1.5 160 V 160 V 180 V 180 V 2x1017 1 200 V 200 V − − n n + 17 + n 1x10 16 n 0.5 (4×1016) (4×10 ) (5×1018) (5×1018)

Electric Field (MV/cm) Electric Field p 0 0 Ionized Acceptor (Mg) Conc. (cm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 μ Y Dimension (μm) Y Dimension ( m)

19 19 6×10 ) 6×10 3.5 -3 19 6x10 VCathode VCathode 3 + 100 V 100 V 5x1019 n (b) × 18 120 V 2.5 n+ 120 V (5 10 ) 19 140 V (5×1018) p+ 140 V 4x10 − 2 n 160 V 160 V × 16 3x1019 (4 10 ) 180 V 1.5 180 V + p 200 V 200 V 19 1 − 2x10 n n+ 16 + 0.5 (4×10 ) n 19 × 18 × 18 1x10 (5 10 )

Electric (MV/cm) Field (5 10 ) 0 4×1018 0

-0.5 Ionized Acceptor (Mg) Conc. (cm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Y Dimension (μm) Y Dimension (μm)

Figure 12. Electric field profile and ionized Mg acceptor concentration for simulations at different 17 −3 Figure 12. Electricvoltages field with profile the incomplete and ionized ionization Mg acceptor model: concentration (a) low p-doping, for simulations NA = 4 ×at 10differentcm voltages,(b) high with the incomplete ionization model: (a) low 19p-doping,−3 NA = 4 × 1017 cm−3, (b) high p-doping, NA = 6 × 1019 cm−3. A trade-off exists on p-doping, NA = 6 × 10 cm . A trade-off exists on the value of p-doping: it must be sufficiently the value of p-doping:low to reduce it must the peakbe sufficiently field, and low sufficiently to reduce high the to peak avoid field, punch-through. and sufficiently high to avoid punch-through.

First, from FigureFirst, 12 fromb we Figure note that 12b the we simulationsnote that the accurately simulations reproduced accurately the reproduced break- the breakdown voltagedown of 180 voltage V, for the of 180 samples V, for under the samples analysis, under that hadanalysis, a doping that levelhad a equal doping to level equal to 19 −3 NA = 6 × 10 NcmA = 6. × Second, 1019 cm we−3. Second, note that we for note the low that N forA case the (Figurelow NA 12 casea), the(Figure ionized 12a), Mg the ionized Mg concentrationconcentration followed the progressive followed the depletion progressive of the depletion p-GaN layer of the with p-GaN higher layer voltages, with higher voltages, with completewith ionization complete at voltages ionization higher at voltages than the higher punch-through than the punch-through voltage. voltage. × 18 −3 Finally, for highFinally, NA, thefor basehigh ionizationNA, the base level ionization was constant level atwas 6% constant (=4 10 at 6%cm (=4) × 1018 cm−3) of of the deﬁned N for most of the p+ GaN layer, except at the pn junctions, where the Mg theA defined NA for most of the p+ GaN layer, except at the pn junctions, where the Mg acceptors wereacceptors almost fully were ionized. almost fully ionized. Our conclusionOur on thisconclusion part is that on athis trade-off part is exists that ona trade-off the value exists of p-doping, on the whichvalue mustof p-doping, which be sufficientlymust low to be reduce sufficiently the peak low field, to reduce and sufficiently the peak high field, to and avoid sufficiently punch-through. high to avoid punch- 4. Conclusionsthrough.

In summary,4. Conclusions we presented a detailed analysis of the leakage and breakdown limits of semi-vertical GaN-on-Si test structures. The results of the analysis indicated that thermionic carrier emissions fromIn summary, a trap state we presented of 0.85 eV a dominates detailed analysis leakage of at the low leakage voltage, and while breakdown limits variable rangeof hopping semi-vertical is observable GaN-on-Si at high test voltage.structures. The results of the analysis indicated that thermionic carrier emissions from a trap state of 0.85 eV dominates leakage at low voltage, while variable range hopping is observable at high voltage. TCAD simulation of the p+n diodes is employed to reproduce the breakdown voltage of semi vertical GaN on Si diodes. For the measured test structures with NA = 6 × 1019 cm−3,

Micromachines 2021, 12, 445 10 of 11

TCAD simulation of the p+n diodes is employed to reproduce the breakdown voltage of 19 −3 semi vertical GaN on Si diodes. For the measured test structures with NA = 6 × 10 cm , breakdown is estimated to correspond to the voltage for which the peak E-ﬁeld at the p+n junction reaches the critical ﬁeld of GaN, and simulation can effectively reproduce the experimental results. For lowly doped samples, punch-through occurs, due to the full depletion of the p-GaN, as demonstrated by simulations. In conclusion, the doping of the p-GaN layer can strongly impact the breakdown voltage of the analyzed structures, and a trade-off between the occurrence of punch-through and junction breakdown needs to be considered to optimize device robustness.

Author Contributions: Conceptualization, K.M., C.D.S., M.B. (Matteo Buffolo), M.B. (Matteo Borga), A.G., S.Y., K.G., B.B., S.D., G.M., E.Z. and M.M.; investigation, K.M., C.D.S., M.B. (Matteo Buf- folo), M.B. (Matteo Borga), S.Y., K.G., B.B., S.D., G.M., E.Z. and M.M.; data curation, K.M., C.D.S., B.B., S.Y. and M.M.; writing—original draft preparation, K.M., C.D.S., M.M., K.G., S.Y. and B.B.; writing—review and editing, K.M., C.D.S., M.M., K.G., S.Y. and B.B.; supervision, A.G., G.M., E.Z., M.M. and S.D.; funding acquisition, M.M. and S.D. All authors have read and agreed to the published version of the manuscript. Funding: This project has received funding from the ECSEL Joint Undertaking (JU) under grant agreement No. 826392. The JU receives support from the European Union’s Horizon 2020 research and innovation program and Austria, Belgium, Germany, Italy, Norway, Slovakia, Spain, Sweden, and Switzerland. This research activity was partly funded by project “Novel vertical GaN-devices for next generation power conversion”, NoveGaN (University of Padova), through the STARS CoG Grants call. Part of this work was supported by MIUR (Italian Minister for Education) under the initiative “Departments of Excellence” (Law 232/2016). Conﬂicts of Interest: The authors declare no conﬂict of interest.

References 1. Kizilyalli, I.C.; Edwards, A.P.; Nie, H.; Disney, D.; Bour, D. High Voltage Vertical GaN p-n Diodes with Avalanche Capability. IEEE Trans. Electron Devices 2013, 60, 3067–3070. [CrossRef] 2. Hu, J.; Zhang, Y.; Sun, M.; Piedra, D.; Chowdhury, N.; Palacios, T. Materials and processing issues in vertical GaN power electronics. Mater. Sci. Semicond. Process. 2018, 78, 75. [CrossRef] 3. Sun, Y.; Kang, X.; Zheng, Y.; Lu, J.; Tian, X.; Wei, K.; Wu, H.; Wang, W.; Liu, X.; Zhang, G. Review of the Recent Progress on GaN-Based Vertical Power Schottky Barrier Diodes (SBDs). Electronics 2019, 8, 575. 4. Zhang, Y.; Sun, M.; Piedra, D.; Azize, M.; Zhang, X.; Fujishima, T.; Palacios, T. GaN-on-Si Vertical Schottky and p-n Diodes. IEEE Electron Device Lett. 2014, 35, 618–620. 5. Zhang, Y.; Wong, H.Y.; Sun, M.; Joglekar, S.; Yu, L.; Braga, N.A.; Palacios, T. Design space and origin of off-state leakage in GaN vertical power diodes. In Proceedings of the 2015 IEEE International Electron Devices Meeting (IEDM), Washington, DC, USA, 7–9 December 2015. 6. Rackauskas, B.; Dalcanale, S.; Uren, M.J.; Kachi, T.; Kuball, M. Leakage mechanisms in GaN-on-GaN vertical pn diodes. Appl. Phys. Lett. 2018, 112, 233501. [CrossRef] 7. Musolino, M.; Van Treeck, D.; Tahraoui, A.; Scarparo, L.; De Santi, C.; Meneghini, M.; Riechert, H. A physical model for the reverse leakage current in (In,Ga)N/GaN light-emitting diodes based on nanowires. J. Appl. Phys. 2016, 119, 044502. [CrossRef] 8. Liu, C.; Khadar, R.A.; Matioli, E. GaN-on-Si Quasi-Vertical Power MOSFETs. IEEE Electron Device Lett. 2017, 39, 71–74. [CrossRef] 9. Otake, H.; Chikamatsu, K.; Yamaguchi, A.; Fujishima, T.; Ohta, H. Vertical GaN-Based Trench Gate Metal Oxide Semiconductor Field-Effect Transistors on GaN Bulk Substrates. Appl. Phys. Express 2018, 1, 011105. [CrossRef] 10. Mukherjee, K.; Borga, M.; Ruzzarin, M.; De Santi, C.; Stoffels, S.; You, S.; Meneghini, M. Analysis of threshold voltage instabilities in semi-vertical GaN-on-Si FETs. Appl. Phys. Express 2020, 13, 024004. [CrossRef] 11. Saremi, M. Modeling and Simulation of the Programmable Metallization Cells (PMCs) and Diamond-Based Power Devices. Ph.D. Thesis, Arizona State University, Tempe, AZ, USA, 2017. 12. Saremi, M.; Hathwar, R.; Dutta, M.; Koeck, F.A.; Nemanich, R.J.; Chowdhury, S.; Goodnick, S.M. Analysis of the reverse I-V characteristics of diamond-based PIN diodes. Appl. Phys. Lett. 2017, 111, 043507. [CrossRef] 13. Mahabadi, S.E.J.; Moghadam, H.A. Comprehensive study of a 4H–SiC MES–MOSFET. Phys. E 2015, 74, 5–29. [CrossRef] 14. Moghadam, H.A.; Dimitrijev, S.; Han, J.; Haasmann, D.; Aminbeidokhti, A. Transient-current method for measurement of active near-interface oxide traps in 4H-SiC MOS capacitors and MOSFETs. IEEE Trans. Electron Devices 2015, 62, 2670–2674. [CrossRef] 15. Liu, Z.; Li, P.G.; Zhi, Y.S.; Wang, X.L.; Chu, X.L.; Tang, W.H. Review of gallium oxide based ﬁeld-effect transistors and Schottky barrier diodes. Chin. Phys. B 2019, 28, 017105. [CrossRef] 16. Palacios, T.A.; Fujishima, T. Aluminum Nitride Based Semiconductor Devices. U.S. Patent 9337301B2, 10 May 2016. Micromachines 2021, 12, 445 11 of 11

17. Han, D.P.; Oh, C.H.; Kim, H.; Shim, J.I.; Kim, K.S.; Shin, D.S. Conduction Mechanisms of Leakage Currents in InGaN/GaN-Based Light-Emitting Diodes. IEEE Trans. Electron Devices 2015, 62, 587–592. 18. Chiu, F.-C. A Review on Conduction Mechanisms in Dielectric Films. Adv. Mater. Sci. Eng. 2014, 2014, 1–18. [CrossRef] 19. Mazzola, M.S.; Saddow, S.E.; Neudeck, P.G.; Lakdawala, V.K.; We, S. Observation of the D-center in 6H-SiC p-n diodes grown by chemical vapor deposition. Appl. Phys. Lett. 1994, 64, 2730. [CrossRef] 20. Demchenko, D.O.; Diallo, I.C.; Reshchikov, M.A. Yellow Luminescence of Gallium Nitride Generated by Carbon Defect Complexes. Phys. Rev. Lett. 2013, 110, 087404. [CrossRef] 21. Huber, M.; Silvestri, M.; Knuuttila, L.; Pozzovivo, G.; Andreev, A.; Kadashchuk, A.; Lundskog, A. Impact of residual carbon impurities and gallium vacancies on trapping effects in AlGaN/GaN metal insulator semiconductor high electron mobility transistors. Appl. Phys. Lett. 2015, 107, 032106. [CrossRef] 22. Hill, R.M. Poole Frenkel conduction in amorphous solids. Philos. Mag. 1971, 23, 59. [CrossRef] 23. Frenkel, J. On Pre-Breakdown Phenomena in Insulators and Electronic Semi-Conductors. Phys. Rev. 1938, 54, 647. [CrossRef] 24. Shan, Q.; Meyaard, D.S.; Dai, Q.; Cho, J.; Fred Schubert, E.; Kon Son, J.; Sone, C. Transport-mechanism analysis of the reverse leakage current in GaInN light-emitting diodes. Appl. Phys. Lett. 2011, 99, 253506. [CrossRef] 25. Zhou, S.; Lv, J.; Yini, W.; Zhang, Y.; Zheng, C.; Liu, S. Reverse leakage current characteristics of InGaN/GaN multiple quantum well ultraviolet/blue/green light-emitting diodes. Jpn. J. Appl. Phys. 2018, 57, 051003. [CrossRef] 26. Simmons, J.G. Conduction in thin dielectric films. J. Phys. D Appl. Phys. 1971, 4, 613. [CrossRef] 27. Kuksenkov, D.; Temkin, H.; Osinsky, A.; Gaska, R.; Khan, M. Origin of conductivity and low-frequency noise in reverse-biased GaN pn junction. Appl. Phys. Lett. 1998, 72, 1365–1367. [CrossRef] 28. Kim, J.; Kim, J.Y.; Tak, Y.; Kim, J.; Hong, H.G.; Yang, M.; Chung, U.I. Investigation of reverse leakage characteristics of InGaN/GaN light-emitting diodes on silicon. IEEE Electron Device Lett. 2012, 33, 1741–1743. [CrossRef] 29. Jung, E.; Lee, J.K.; Kim, M.S.; Kim, H. Leakage Current Analysis of GaN-Based Light-Emitting Diodes Using a Parasitic Diode Model. IEEE Trans. Electron Devices 2015, 62, 3322–3325. [CrossRef] 30. Look, D.C.; Reynolds, D.C.; Kim, W.; Aktas, Ö.; Botchkarev, A.; Salvador, A.; Morkoç, A. Deep-center hopping conduction in GaN. J. Appl. Phys. 1996, 80, 2960–2963. [CrossRef] 31. Lee, M.; Lee, H.U.; Song, K.M.; Kim, J. Significant improvement of reverse leakage current characteristics of Si-based homoepitaxial InGaN/GaN blue light emitting diodes. Sci. Rep. 2019, 9, 970. [CrossRef] 32. Hirsch, L.; Barrière, A.S. Electrical characterization of InGaN/GaN light emitting diodes grown by molecular beam epitaxy. J. Appl. Phys. 2003, 94, 5014. [CrossRef] 33. Ferdous, M.S.; Wang, X.; Fairchild, M.N.; Hersee, S.D. Effect of threading defects on InGaN/GaNInGaN/GaN multiple quantum well light emitting diodes. Appl. Phys. Lett. 2007, 91, 231107. [CrossRef] 34. Kim, J.; Kim, J.; Tak, Y.; Chae, S.; Kim, J.; Park, Y. Effect of V-Shaped Pit Size on the Reverse Leakage Current of InGaN/GaN Light-Emitting Diodes. IEEE Electron Device Lett. 2013, 34, 1409–1411. [CrossRef] 35. Hill, R.M. Hopping conduction in amorphous solids. Philos. Mag. 1971, 24, 1307. [CrossRef] 36. Pollak, M.; Riess, I.J. A percolation treatment of high-field hopping transport. Phys. C Solid State Phys. 1976, 9, 2339. [CrossRef] 37. Zhao, L.; Chen, L.; Yu, G.; Yan, D.; Yang, G.; Gu, X.; Lu, H. Tunneling-Hopping Transport Model for Reverse Leakage Current in InGaN/GaN Blue Light-Emitting Diodes. IEEE Photonics Technol. Lett. 2017, 29, 1447–1450. [CrossRef] 38. Yu, D.; Wang, C.; Wehrenberg, B.L.; Guyot-Sionnest, P. Variable Range Hopping Conduction in Semiconductor Nanocrystal Solids. Phys. Rev. Lett. 2004, 92, 216802. [CrossRef] 39. Tsou, C.; Ji, M.; Bakhtiary-Noodeh, M.; Detchprohm, T.; Dupuis, R.D.; Shen, S. Temperature-Dependent Leakage Current Characteristics of Homojunction GaN p-i-n Rectifiers Using Ion-Implantation Isolation. IEEE Trans. Electron Devices 2019, 66, 4273–4278. [CrossRef] 40. Sentaurus Device User Guide; Synopsys: Mountain View, CA, USA, 2015. 41. Podor, B. Thermal ionization energy of Mg acceptors in GaN: Effects of doping level and compensation. Int. Conf. Solid State Cryst. 2000, 4412, 299–303. 42. Sabui, G.; Parbrook, P.J.; Arredondo-Arechavala, M.; Shen, Z.J. Modeling and simulation of bulk gallium nitride power semiconductor devices. Aip Adv. 2016, 6, 055006. [CrossRef] 43. Cheng, K.; Liang, H.; Van Hove, M.; Geens, K.; De Jaeger, B.; Srivastava, P.; Chung, H. AlGaN/GaN/AlGaN Double Het- erostructures Grown on 200 mm Silicon (111) Substrates with High Electron Mobility. Appl. Phys. Express 2012, 5, 011002. [CrossRef] 44. Yang, J.; Zhao, D.; Jiang, D.; Chen, P.; Zhu, J.; Liu, Z.; Du, G.T. Influence of hydrogen impurities on p-type resistivity in Mg-doped GaN films. J. Vac. Sci. Technol. A 2015, 33, 021505. [CrossRef]