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

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. Finally, the developed simulation model was utilized to estimate the impact of varying the p-doping concentration on the design of breakdown voltage; while high p-doped structures are limited by the critical electric field at the interface, low p-doping designs need to contend with possible depletion of the entire p-GaN region and the consequent punch-through. A trade-off on the value of p-doping therefore exists to optimize the breakdown.

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, saturation velocity), making it the stronger choice for most applications. To further enhance the performance of GaN transistors, the focus of research is shifting from lateral to vertical architectures, which circumvent the breakdown limitations, surface trapping, and other challenges inherent to the lateral topology.
The adoption of GaN-based vertical diodes and transistors [1][2][3][4][5][6][7] is highly advantageous to high-speed and high-power electronics applications, presenting low R on and higher breakdown robustness in addition to improved thermal performance. Preliminary Fabricated on a 200 mm Si substrate, the diodes are based on an Mg-doped p + GaN body, where NA = 6 × 10 19 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 × 10 18 cm −3 , while the n − drift layer has a doping of ND = 4 × 10 16 cm −3 . The parameters have been sum- Fabricated on a 200 mm Si substrate, the diodes are based on an Mg-doped p + GaN body, where N A = 6 × 10 19 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 × 10 18 cm −3 , while the n − drift layer has a doping of N D = 4 × 10 16 cm −3 . The parameters have been summarized in Table 1. For the p + GaN, the value of N A = 6 × 10 19 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.

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. 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 region has been modelled separately, as illustrated in Figure 2b, and described as follows.
The observed rapid rise in leakage with temperature implies a strong thermally activated process, which is modelled in Figure 3a using the following equation: This equation describes the temperature dependence of the current conduction originated by thermionic emissions from Coulombic traps [6,19]. Here, A is a constant defining 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 region has been modelled separately, as illustrated in Figure 2b, and described as follows.
The observed rapid rise in leakage with temperature implies a strong thermally activated process, which is modelled in Figure 3a using the following equation: This equation describes the temperature dependence of the current conduction originated by thermionic emissions from Coulombic traps [6,19]. Here, A is a constant defining the almost vertical shift of the curve, E A is the thermal activation energy defining the slope of the curve, k B 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].
(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 region has been modelled separately, as illustrated in Figure 2b, and described as follows.
The observed rapid rise in leakage with temperature implies a strong thermally activated process, which is modelled in Figure 3a using the following equation: This equation describes the temperature dependence of the current conduction originated by thermionic emissions from Coulombic traps [6,19]. Here, A is a constant defining the almost vertical shift of the curve, 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].  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 E T 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 n [7,19,22,23] as follows: 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 Figure 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 simulations in the considered voltage range (described in detail in Section 3.2). 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 Figure 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 simulations in the considered voltage range (described in detail in Section 3.2).   [27][28][29]. Depending on the growth conditions in different GaN-based works, ≈ 10 −5 eV V −1/2 m 1/2 are generally reported [7,24,25,[27][28][29][30][31][32][33][34], as summarized in Figure 5.

CBM
Coulombic-well Theoretically, β can written as β = Zq 3 πε [7,24,25], where Z represents the charge on the Coulomb center (ionization state of the trap), and ε is the permittivity of GaN. The extracted β = 1.77 × 10 −5 eV V −1/2 m 1/2 in our measurements is close to the theoretical value (≈3.1 × 10 −5 eV V −1/2 m 1/2 ) considering a relative high-frequency GaN permittivity of 5.8, and Z = 1 for the carbon acceptor. Considering a simplified β = q πε [26], theoretical values are close to 3.2 × 10 −5 eV V −1/2 m 1/2 [27][28][29]. Depending on the growth conditions in different GaN-based works, β ≈ 10 −5 eV V −1/2 m 1/2 are generally reported [7,24,25,[27][28][29][30][31][32][33][34], as summarized in Figure 5. 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 evolution. The VRH mechanism, based on the theory developed by Mott and Hill, is illustrated in Figure 4b. The primary trap location is at A, with an energy ET and an exponentially distributed density of states (DOS). Electrons can hop from A to empty trap positions situated at B or C, within a distance of R from the primary trap, and within a range of energy surrounding ET.
The fit to measurements is presented in Figure 6 and modelled using the following relation with temperature T [7,35], valid for moderate to high electric fields, where represents the field contribution to strengthening the VRH conduction: 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 evolution. The VRH mechanism, based on the theory developed by Mott and Hill, is illustrated in Figure 4b. The primary trap location is at A, with an energy E T and an exponentially distributed density of states (DOS). Electrons can hop from A to empty trap positions situated at B or C, within a distance of R from the primary trap, and within a range of energy surrounding E T .
The fit to measurements is presented in Figure 6 and modelled using the following relation with temperature T [7,35], valid for moderate to high electric fields, where F 2 represents the field contribution to strengthening the VRH conduction: I 0 modifies the trap emission rate into current, C VRH represents a grouped constant, which can be written as C VRH = 4.626 × 10 −3 × qa U 2 [7], where q is the elementary 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 [7,27,30,36]. T 0 , the characteristic temperature, can be written as T 0 =  [7,24,25,[27][28][29][30]33,37,39] is provided in Figure 7 for reference, showing consistent results for the values extrapolated in this paper. Figure 5. Comparison of the Poole-Frenkel coefficient (β) ≈ 10 −5 eV V −1/2 m 1/2 obtained from different works on GaN-based systems [7,24,25,[27][28][29][30][31][32][33][34].
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 evolution. The VRH mechanism, based on the theory developed by Mott and Hill, is illustrated in Figure 4b. The primary trap location is at A, with an energy ET and an exponentially distributed density of states (DOS). Electrons can hop from A to empty trap positions situated at B or C, within a distance of R from the primary trap, and within a range of energy surrounding ET.
The fit to measurements is presented in Figure 6 and modelled using the following relation with temperature T [7,35], valid for moderate to high electric fields, where represents the field contribution to strengthening the VRH conduction:   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 defects within the drift region. Specific attention needs to be focused on the residual carbon concentration, considering its contribution to the low voltage regime.

TCAD Simulations of Diode Breakdown
To obtain an estimate of the breakdown voltage of the test structures, several samples 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 defects within the drift region. Specific attention needs to be focused on the residual carbon concentration, considering its contribution to the low voltage regime.

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).
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 defects within the drift region. Specific attention needs to be focused on the residual carbon concentration, considering its contribution to the low voltage regime.

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). The 2D-TCAD simulations, based on the drift-diffusion model for carrier transport, were employed to build a representative model of the measured devices, using the Sentaurus tool 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 defined as modified ohmic contacts to improve accuracy around the p-n junction; it does not impose the charge neutrality condition at vertices within the charged depletion regions [40]. In addition to suitable mobility and recombination models, the gate-dependent strain polarization model, especially suited to GaN devices, was activated. Since Mg has a relatively high ionization energy of 0.16 eV [41,42], Mg acceptors are not completely ionized at room temperature. Thus, to correctly model the p-doping levels and reproduce the breakdown voltage, the incomplete ionization model in Sentaurus was used, which is physically more accurate to model Mg doping. The effective doping concentration was computed internally based on ionization probability, derived from the ionization energy of the doping species. The 2D-TCAD simulations, based on the drift-diffusion model for carrier transport, were employed to build a representative model of the measured devices, using the Sentaurus tool 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 defined as modified ohmic contacts to improve accuracy around the p-n junction; it does not impose the charge neutrality condition at vertices within the charged depletion regions [40]. In addition to suitable mobility and recombination models, the gate-dependent strain polarization model, especially suited to GaN devices, was activated. Since Mg has a relatively high ionization energy of 0.16 eV [41,42], Mg acceptors are not completely ionized at room temperature. Thus, to correctly model the p-doping levels and reproduce the breakdown voltage, the incomplete ionization model in Sentaurus was used, which is physically more accurate to model Mg doping. The effective doping concentration was computed internally based on ionization probability, derived from the ionization energy of the doping species. Figure 9b 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. For high p-doping, such as for NA = 6 × 10 19 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  Figure 9b 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.
For high p-doping, such as for N A = 6 × 10 19 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 E Crit .
Micromachines 2021, 12, x FOR PEER REVIEW 9 of 12 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 which the peak E-field at the p + n junction reaches the critical field of GaN.
For the devices under test, the p-doping level was NA = 6 × 10 19 cm −3 , which indicates breakdown triggered by peak electric fields >3 MV/cm at the p + n junction, as can be seen from the evolution of the electric field with increasing voltage in Figure 11. The field values obtained in the range of 0 V-30 V were used in Figure 3b to verify the ∆ ∝ √ 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 NA = 1 × 10 18 cm −3 ), pushing VBR to higher voltages >200 V. Simulations indicated that reducing the p-doping significantly introduces a different constraint. For example, for NA = 6 × 10 17 cm −3 , we observed a much wider depletion of the p-GaN region. If the p-doping levels are too low (NA = 4 × 10 17 cm −3 in Figures 9b and 10a), complete depletion of the pGaN layer can 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 For the devices under test, the p-doping level was N A = 6 × 10 19 cm −3 , which indicates breakdown triggered by peak electric fields >3 MV/cm at the p + n junction, as can be seen from the evolution of the electric field with increasing voltage in Figure 11. For the devices under test, the p-doping level was NA = 6 × 10 19 cm −3 , which indicates breakdown triggered by peak electric fields >3 MV/cm at the p + n junction, as can be seen from the evolution of the electric field with increasing voltage in Figure 11. The field values obtained in the range of 0 V-30 V were used in Figure 3b to verify the ∆ ∝ √ 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 NA = 1 × 10 18 cm −3 ), pushing VBR to higher voltages >200 V. Simulations indicated that reducing the p-doping significantly introduces a different constraint. For example, for NA = 6 × 10 17 cm −3 , we observed a much wider depletion of the p-GaN region. If the p-doping levels are too low (NA = 4 × 10 17 cm −3 in Figures 9b and 10a), complete depletion of the pGaN layer can 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 process [44], which would reduce the effective concentration of Mg and accelerate the punch-through. The field values obtained in the range of 0 V-30 V were used in Figure 3b to verify 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 A = 1 × 10 18 cm −3 ), pushing V BR to higher voltages >200 V. Simulations indicated that reducing the p-doping significantly introduces a different constraint. For example, for N A = 6 × 10 17 cm −3 , we observed a much wider depletion of the p-GaN region. If the p-doping levels are too low (N A = 4 × 10 17 cm −3 in Figures 9b and 10a), complete depletion of the pGaN layer can occur, thus leading to punch-through even before the peak electric field reaches E Crit . This effect may be further 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. Figure 12 illustrates the evolution of electric field and ionized acceptor concentration with cathode voltage, for low (N A = 4 × 10 17 cm −3 ) and high p-doping (N A = 6 × 10 19 cm −3 ) cases.
Micromachines 2021, 12, x FOR PEER REVIEW Figure 12 illustrates the evolution of electric field and ionized acceptor concent with cathode voltage, for low (NA = 4 × 10 17 cm −3 ) and high p-doping (NA = 6 × 10 1 cases. Figure 12. Electric field profile and ionized Mg acceptor concentration for simulations at different voltages with the in complete ionization model: (a) low p-doping, NA = 4 × 10 17 cm −3 , (b) high p-doping, NA = 6 × 10 19 cm −3 . A trade-off exists o the value of p-doping: it must be sufficiently low to reduce the peak field, and sufficiently high to avoid punch-through First, from Figure 12b we note that the simulations accurately reproduced the down voltage of 180 V, for the samples under analysis, that had a doping level eq NA = 6 × 10 19 cm −3 . Second, we note that for the low NA case (Figure 12a), the ioniz concentration followed the progressive depletion of the p-GaN layer with higher vo with complete ionization at voltages higher than the punch-through voltage.
Finally, for high NA, the base ionization level was constant at 6% (=4 × 10 18 cm the defined NA for most of the p + GaN layer, except at the pn junctions, where th acceptors were almost fully ionized.
Our conclusion on this part is that a trade-off exists on the value of p-doping, must be sufficiently low to reduce the peak field, and sufficiently high to avoid p through.

Conclusions
In summary, we presented a detailed analysis of the leakage and breakdown of semi-vertical GaN-on-Si test structures. The results of the analysis indicated tha mionic carrier emissions from a trap state of 0.85 eV dominates leakage at low vo while variable range hopping is observable at high voltage.
TCAD simulation of the p + n diodes is employed to reproduce the breakdown v of semi vertical GaN on Si diodes. For the measured test structures with NA = 6 × 10 1  A trade-off exists on the value of p-doping: it must be sufficiently low to reduce the peak field, and sufficiently high to avoid punch-through.
First, from Figure 12b we note that the simulations accurately reproduced the breakdown voltage of 180 V, for the samples under analysis, that had a doping level equal to N A = 6 × 10 19 cm −3 . Second, we note that for the low N A case (Figure 12a), the ionized Mg concentration followed the progressive depletion of the p-GaN layer with higher voltages, with complete ionization at voltages higher than the punch-through voltage.
Finally, for high N A , the base ionization level was constant at 6% (=4 × 10 18 cm −3 ) of the defined N A for most of the p + GaN layer, except at the pn junctions, where the Mg acceptors were almost fully ionized.
Our conclusion on this part is that a trade-off exists on the value of p-doping, which must be sufficiently low to reduce the peak field, and sufficiently high to avoid punch-through.

Conclusions
In summary, 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 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 N A = 6 × 10 19 cm −3 , 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.