Influence of passivation, doping and geometrical
parameters on the avalanche breakdown of GaN SBDs

B. Orfao, R A Peña, B G Vasallo, S Pérez, J Mateos, T González

► To cite this version:

B. Orfao, R A Peña, B G Vasallo, S Pérez, J Mateos, et al.. Influence of passivation, doping and geometrical parameters on the avalanche breakdown of GaN SBDs. Journal of Computational Electronics, 2025, 24, pp.189. 10.1007/s10825-025-02430-2 . hal-05376884

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<figure><img src="image_1.png" /></figure>
# Influence of passivation, doping and geometrical parameters on the avalanche breakdown of GaN SBDs

B. Orfao<fnref n="1" /> · R. A. Peña<fnref n="2" /> · B. G. Vasallo<fnref n="2" /> · S. Pérez<fnref n="2" /> · J. Mateos<fnref n="2" /> · T. González<fnref n="2" />

Received: 30 July 2025 / Accepted: 11 September 2025 / Published online: 27 September 2025
© The Author(s) 2025

## Abstract

The breakdown of GaN-based Schottky barrier diodes associated with impact ionization events initiated by electrons injected by tunneling is physically analyzed by means of a Monte Carlo simulator self-consistently coupled with a two-dimensional solution of the Poisson equation. Simulations of a realistic topology where different geometrical parameters are modified allow to identify their influence on the breakdown voltage. The correct physical modeling of two-dimensional effects is essential for a proper prediction of the breakdown. Epilayer doping and thickness, dielectric used for the passivation and lateral extension of the epilayer are analyzed. As expected, the lower the doping and the thicker the epilayer, the higher the value found for the breakdown voltage, but, interestingly, the results also indicate that the peak electric field present at the edge of the Schottky contact, which may be reduced by means of high-k dielectric passivation and a short lateral extension of the epilayer, plays a key role in the breakdown.

**Keywords** Schottky barrier diode · Impact ionization · Tunnel current · Breakdown

## 1 Introduction

In the last years, GaN has generated an increased interest due to its properties such as wide bandgap, high mobility, high breakdown field and high electron saturation velocity [1], which makes it suitable for high-frequency and high-power applications [2]. Presently, THz and sub-THz electronic sources rely on frequency multipliers based on the classic GaAs Schottky barrier diode (SBD) technology[3, 4], but the output power at frequencies above 500 GHz is limited by the low breakdown voltage and the small size of the diodes (necessary to be used at high frequencies) [5]. Apart from complex designs including multiple diodes, exploiting the “power-combining” circuit topology [6, 7], the use of GaN SBDs can potentially increase the RF power output level at the intermediate stages of frequency-multiplied sources. Indeed, the key parameter for weighting the power performance of GaN SBDs is their breakdown voltage, which is

theoretically much higher than that of GaAs-based diodes [8, 9]. However, the diode breakdown mechanism is not only related to the semiconductor properties, but it is also intimately linked to the presence of a reverse leakage current, problem widely studied in GaN SBDs and partially mitigated by recent technological advances. The presence of trap-related mechanisms such as trap-assisted tunneling, Poole-Frenkel emission or variable range hopping [8, 10, 11] leads to a higher leakage current in comparison with the expected ideal one, which includes thermionic emission and tunneling contributions [12, 13]. This problem can be very significant in non-mature technologies such as GaN SBDs, but the use of native GaN substrates instead of foreign ones (SiC, Si or sapphire) allows mitigating the excess leakage current by presenting a lower density of defects in the semiconductor, which improves the Schottky contact quality [14].

The electric field crowding effect at the edge of the junction also provokes an increase of the leakage current that could lead to premature breakdown [15]. Indeed, the high values of the electric field appearing at the metal-semiconductor interface enhance the electron injection by tunneling [16]. Thus, a breakdown voltage improvement could be achieved by using dielectric passivation, which smooths the electric field at the contact edges [17]. A correct physical modeling of all the involved processes is essential in the

<figure><img src="image_2.png" /></figure>

<footnote>B. Orfao
beatriz.orfao-e-vale-tabernero@univ-lille.fr</footnote>

<footnote n="1">¹ CNRS-IEMN, UMR8520, Université de Lille, Av. Poincaré, 59650 Villeneuve d'Ascq, France</footnote>

<footnote n="2">² Applied Physics Department and USAL-NANOLAB, Universidad de Salamanca, 37008 Salamanca, Spain</footnote>
analysis of the breakdown. In particular, electron and hole impact ionization (II) processes produced by tunneled electrons should be considered when analyzing the reverse leakage current because they are, together with the high electric fields at the contact edges, at the origin of the avalanche breakdown in SBDs [18]. Additionally, the barrier profile should be self-consistently calculated with the carrier concentration in the depletion region, since the full-depletion assumption may be no longer valid [19].

In this work, the role played by II in the breakdown voltage of SBDs has been physically studied through a home-made semiclassical ensemble Monte Carlo (MC) simulator coupled with a two-dimensional (2D) Poisson's equation solver [19–21]. The influence of different technological parameters on the breakdown voltage will be qualitatively analyzed, as the dielectric passivation material, the epilayer doping level, and the epilayer thickness and lateral extension from the Schottky contact. The paper is organized as follows. The simulated structure of the diodes and the MC model are described in Sect. 2. The results and their discussion are reported in Sect. 3. Finally, the main conclusions are drawn in Sect. 4.

## 2 Physical model and simulated structure

The ideal current mechanisms in SBDs are thermionic emission and tunnel injection. The tunnel injection becomes more relevant for reverse bias. To determine these current contributions, it is necessary to know the barrier profile along the structure. To calculate the tunneling current at a given energy $\epsilon$, we use the general expression:

$$J(\epsilon)d\epsilon = -eN(\epsilon)T_c(\epsilon)d\epsilon, \quad (1)$$

where $N(\epsilon)$ is the number of electrons per unit area incident on the barrier with an energy $\epsilon$ per unit time and per unit energy and $T_c(\epsilon)$ is the transmission coefficient. This expression assumes that the occupation of the final states is negligible. For solving the Schrodinger equation, we have used the Wentzel–Kramers–Brillouin (WKB) approximation, obtaining that $T_c$ is equal to [22]:

$$T_c(\epsilon) = \exp \left( -\frac{2}{\hbar} \int_{x_1}^{x_2} \sqrt{2m^*(\epsilon_c(x) - \epsilon)} dx \right), \quad (2)$$

being $\hbar$ the Planck constant, $m^*$ the electron effective mass, and $x_1$ and $x_2$ the two classical turning points for the potential barrier. In the MC simulator, a region where the tunnel injection takes place is defined. $T_c$ is calculated for each energy interval in which the conduction band is discretized. At each of these intervals, the details of injected charge are determined: number of injected particles, and time and position at which they are injected [23]. As in [12, 19], we have

considered the exact expression of $N(\epsilon)$, including transversal states [24]:

$$N(\epsilon) = \frac{A^*T}{ek_B} \ln \left[ 1 + \exp \left( -\frac{\epsilon - \epsilon_F}{k_B T} \right) \right], \quad (3)$$

where $A^*$ is the Richardson constant, $T$ the temperature, $k_B$ the Boltzmann constant, $e$ the electron charge and $\epsilon_F$ is the Fermi energy.

The GaN conduction band model used for electrons in our MC simulator consists of three non-parabolic spherical valleys, $\Gamma_1$, (6 equivalent) U and $\Gamma_3$ [25]. Carriers are subjected to scattering mechanisms, which are randomly selected according to their respective probabilities. The mechanisms considered in the simulations include inter-valley scattering, phonon scattering (acoustic and polar, both optical and non-optical), ionized impurity scattering and piezoelectric scattering. The probabilities of these mechanisms, as well as the final carrier states, have been previously studied in detail [25, 26]. The parameters and scattering mechanisms of electrons in the conduction band are the same as in [26], summarized in Table 1. The MC model for holes in the valence band consists of only one valley globally representing heavy, light and split-off bands. The hole parameters have been adjusted by comparison with the three-valley model used in [19].

<table><caption>Table 1 Simulation parameters for GaN and electron transport</caption><thead><tr><th>Parameter</th><th colspan="3">GaN</th></tr></thead><tbody><tr><td>Density (kg/m<sup>3</sup>)</td><td colspan="3">6150</td></tr><tr><td>Sound velocity (m/s)</td><td colspan="3">6560</td></tr><tr><td>Optic dielectric constant</td><td colspan="3">5.35</td></tr><tr><td>Static dielectric constant</td><td colspan="3">8.9</td></tr><tr><td>Optical phonon energy (eV)</td><td colspan="3">0.09120</td></tr><tr><td>Band gap (eV)</td><td colspan="3">3.44</td></tr><tr><td>Lattice parameter (Å)</td><td colspan="3">5.185</td></tr><tr><td></td><td>Γ<sub>1</sub></td><td>U</td><td>Γ<sub>3</sub></td></tr><tr><td>Effective mass (m*/m<sub>0</sub>)</td><td>0.22</td><td>0.39</td><td>0.28</td></tr><tr><td>No parabolicity coef. (eV<sup>-1</sup>)</td><td>0.37</td><td>0.50</td><td>0.22</td></tr><tr><td>Energy from Γ valley</td><td>0.0</td><td>2.2</td><td>2.4</td></tr><tr><td>Acoustic def. pot. (eV)</td><td>8.3</td><td>8.3</td><td>8.3</td></tr><tr><td>Optic def. pot. (eV)</td><td>0.0</td><td>0.0</td><td>0.0</td></tr><tr><td>Intervalley def. pot. (10<sup>10</sup> eV/m)</td><td></td><td></td><td></td></tr><tr><td>From Γ<sub>1</sub> to</td><td>0</td><td>10</td><td>10</td></tr><tr><td>From U to</td><td>10</td><td>10</td><td>10</td></tr><tr><td>From Γ<sub>3</sub> to</td><td>10</td><td>10</td><td>0</td></tr><tr><td>Intervalley phonon energy (eV)</td><td></td><td></td><td></td></tr><tr><td>From Γ<sub>1</sub> to</td><td>0</td><td>0.09120</td><td>0.09120</td></tr><tr><td>From U to</td><td>0.09120</td><td>0.09120</td><td>0.09120</td></tr><tr><td>From Γ<sub>3</sub> to</td><td>0.09120</td><td>0.09120</td><td>0</td></tr></tbody></table>
For high-reverse-bias conditions, in addition to tunnel injection, II events should be considered for both electrons and holes. II may occur when electrons in the conduction band have an energy exceeding a threshold value $\epsilon_{th}$, resulting in the generation of an electron-hole pair. This phenomenon has been introduced in the simulator as an additional scattering mechanism using the Keldysh approach, under which the probability per unit time of having an II event is given by the following expression [27, 28]:

$$ 
\begin{aligned}
P(\epsilon) &= S\left(\frac{\epsilon - \epsilon_{th}}{\epsilon_{th}}\right)^2 & \text{if } \epsilon > \epsilon_{th} \\
P(\epsilon) &= 0 & \text{if } \epsilon < \epsilon_{th}
\end{aligned} 
\qquad (4) 
$$

being $\epsilon$ the carrier kinetic energy, $\epsilon_{th}$ the ionization threshold energy (3 eV for both electrons and holes) and $S$ the softness or hardness of the threshold ($5 \times 10^{12} \text{ s}^{-1}$ for electrons and $25 \times 10^{12} \text{ s}^{-1}$ for holes). The generated carriers appear in the lower-energy valleys, since these are the most probable to be occupied. The electron that produces the II event remains in the valley where it is located, losing an energy equivalent to the sum of the bandgap and the thermal energy transferred to the electron-hole pair. For high reverse bias, II phenomena may provoke a sharp increase of the current, eventually leading to the diode breakdown. In our simulations, the breakdown is identified by the inability to reach a steady-state condition due to the indefinite increase in the number of carriers, which is usually accompanied by a previous sharp growth of the current.

Surface charges present at the dielectric-semiconductor interface, inducing carrier depletion through Coulomb repulsion, are also included in the simulation. Local values of surface charge density $\sigma$ are self-consistently calculated with the number of carriers present near the interface [17]. In this way, $\sigma$ is consistent with the electron distribution for each bias condition and adapted to the geometry of every structure.

The MC models used in this work (carrier transport, II processes, surface charge at the interfaces, tunnel injection, etc.) have already been validated in previous studies on other devices [19–21, 23, 26, 27].

The simulated device, based on real planar GaN SBDs, consists of a highly doped n+ substrate with doping $N_s = 5 \times 10^{18} \text{ cm}^{-3}$ and a n- epilayer with doping $N_D$, whose influence on the breakdown conditions is analyzed. The Schottky contact length, $L_{Sch}$, is 200 nm. Figure 1 shows a scheme of the complete 2D topology. The red dashed rectangle delimits the one-dimensional (1D) diode corresponding to the vertical region under the Schottky contact, which will be also simulated for comparison. The epilayer thickness $W_{EP}$ and lateral extension from the Schottky contact $L_{EP}$ are also magnitudes under study. The surface charges $\sigma$ located at the dielectric-semiconductor interface are represented by a violet line. As mentioned, besides $N_D$, $W_{EP}$ and $L_{EP}$, we

<figure><img src="image_3.png" /><figcaption>Fig. 1 Scheme of the simulated SBDs. The red dashed rectangle indicates the 1D diode structure. A study of the device breakdown is carried out with respect to the parameters: $N_D$, $W_{EP}$, $L_{EP}$ and $\kappa_{pas}$</figcaption></figure>

will study the influence on the breakdown performance of the passivation material, characterized by its dielectric constant $\kappa_{pas}$. The breakdown voltage can be improved by the presence of a passivation dielectric, as it smooths the sharp potential drop at the edges of the Schottky contact, an effect previously reported in [17]. To ensure the accuracy of the simulations, a time step $\Delta t=0.2$ fs is considered for a total time of 40 ps. The device mesh discretization has been done by using rectangular cells of 2–5 nm sides, smaller than the Debye length in the considered semiconductor.

## 3 Results and discussion

In this section, the influence of the different geometrical parameters on the breakdown voltage ($\kappa_{pas}$, $N_D$, $W_{EP}$ and $L_{EP}$) will be analyzed in detail.

### 3.1 Passivation

In order to study the influence on the breakdown voltage of the dielectric passivation material, simulations for a 2D structure with $N_D = 3 \times 10^{17} \text{ cm}^{-3}$, $W_{EP} = 350$ nm, $L_{EP} = 200$ nm, and without ($\kappa_{pas} = 1$) and with high-k dielectric passivation ($\kappa_{pas} = 25$) were carried out. The resulting I-V curves in reverse bias, considering and not II, are shown in Fig. 2. In the absence of II, the current smoothly increases following the ideal tunneling behavior. However, II events lead to a sharp increase of the current appearing for high reverse bias and thus to the avalanche breakdown of both diodes, which is identified by the fact that a stationary situation is
<figure><img src="image_4.png" /><figcaption>**Fig. 2 I-V** curves of the 2D structure of Fig. 1 (with $N_D=3 \times 10^{17}$ cm⁻³, $W_{EP}=350$ nm, $L_{EP}=200$ nm) calculated with (close circles) and without (open circles) including II events in the simulation, with air (black) or a high-k dielectric (cyan) considered as dielectric. Inset: Electron (triangles) and hole (squares) current contributions. The case of a 1D structure (consisting of just the vertical region under the Schottky contact) is also shown in red for comparison.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 3** Color maps of **a**, **c** electron and **b**, **d** hole concentrations for $N_D=3 \times 10^{17}$ cm⁻³, $W_{EP}=350$ nm, $L_{EP}=200$ nm and an applied voltage of –35 V with **a**, **b** air and **c**, **d** high k as dielectric material</figcaption></figure>

not reached in the simulation due to a progressive increase of the carrier number and, as consequence, of the current. Thus, the inclusion of II events and hole transport in the simulations is essential to understand and optimize this physical behavior.

To further evidence the influence of the passivation layer (air or a high-k material) and the associated 2D effects on the avalanche breakdown, simulations of a 1D structure, consisting of the vertical diode delimited by the red dashed line in Fig. 1, were carried out. The increase of the current when increasing the reverse bias is more pronounced in the 2D structures due to the enhanced electric field at the edge of the Schottky contact [17], not present in the 1D case, which leads to intense II events in the neighboring region.

Besides, the higher the permittivity of the passivation material, the lower the value of the electric field peak found at the edge of the Schottky contact [17] and thus the lower the value of the current achieved when considering II. This is evident when the reverse bias is lower than 40 V, above that voltage the current is similar for both non-passivated and passivated SBDs, which indicates that the influence of 2D effects is reduced. This occurs because, even if the peak electric field at the edge is still slightly different, the contribution along the whole Schottky contact becomes more relevant than just the high value at the edge.

The MC simulator allows us to distinguish between two current contributions, considering the number of carriers that reach the Schottky contact: one associated with electrons (tunnel injection) and the other one associated with

holes (coming from II events). These two contributions are presented in the inset of Fig. 2. For the lower voltages, the hole current contribution is negligible. As the bias is increased, it becomes comparable to the electron contribution, and finally, for the higher voltages (just before the appearance of the avalanche breakdown), it becomes dominant. The breakdown is explained as a consequence of II as follows. For high enough voltages, II leads to an increasing presence of carriers at the vicinity of the Schottky contact, which in turn provokes an increase of the electric field (mainly at the contact edge), with the subsequent enhancement of electron injection by tunneling. Consequently, more II events take place, ultimately leading to the breakdown of the diode.

Figure 3 presents color maps of electron and hole concentration considering air and high k as dielectric passivation materials for an applied voltage of –35 V, when II is evident but far enough from the avalanche breakdown. For the bias presented in the figure, the epilayer is almost fully depleted. As observed, the tunnel injection is located at the edge of the Schottky contact, being more pronounced when the diode is not passivated. As mentioned previously, the electric field is stronger at the contact edge, leading to the appearance of II events and therefore to the presence of a non-negligible hole concentration. As the electric field peak is reduced when the dielectric permittivity increases, also the hole current contribution (inset of Fig. 2) is lower due to both the lower electron injection and the less frequent II events.
## 3.2 Epilayer doping and thickness

The second technological parameter we have studied is the epilayer doping level $N_D$. Simulations of the 2D structure with $W_{EP}=350$ nm, $L_{EP}=200$ nm and without passivation ($\kappa_{pas}=1$) were performed for three different values of $N_D$: $10^{17}$, $3 \times 10^{17}$ and $6 \times 10^{17}$ cm$^{-3}$. Note that the values of $W_{EP}$, $L_{EP}$ and $\kappa_{pas}$ are not the optimal ones in terms of the breakdown behavior. However, these values allow to more clearly identify the influence of $N_D$. In particular, such small value of $W_{EP}$, while useful for high-frequency applications, leads to a reduction of the breakdown voltage as will be shown later. In Fig. 4, the corresponding I–V characteristics are represented. As observed, the breakdown voltage lies in the range 60–70 V, being lower when the doping is increased due to the higher electric field present at the contact for the same bias, which leads to a higher tunnel injection.

The breakdown for diodes with $W_{EP}=350$ nm is strongly linked to the full depletion of the epilayer. Indeed, the epilayer thickness $W_{EP}$ must be analyzed in relation to $N_D$, since the depletion depends on both parameters. In the inset of Fig. 4, we have represented the current as a function of the value of the peak electric field at the edge of the Schottky contact. Note that for a higher $N_D$, at a given applied voltage a higher electric field is achieved (as example, see circles in the inset of Fig. 4, corresponding to an applied voltage of –5 V). The vertical lines indicate the field at which the full depletion of the epilayer is observed in the simulations (not represented for $N_D=6 \times 10^{17}$ cm$^{-3}$ since the breakdown appears when biasing below such condition). As expected,

we have obtained similar values of the current as long as the epilayer is not fully depleted. However, once the epilayer is fully depleted, an increase of the current is observed. As a consequence, it is to note that for $N_D=1$ and $3 \times 10^{17}$ cm$^{-3}$ the diodes are not correctly designed ($W_{EP}=350$ nm is too small), since the current increase due to the epilayer depletion leads to a lowering of their breakdown voltage.

Figure 5 presents the I–V curves obtained from the simulations of diodes with the same doping level ($N_D=6 \times 10^{17}$ cm$^{-3}$) and without passivation ($\kappa_{pas}=1$), but with different epilayer thickness, together with the resulting current in a diode with $N_D=10^{17}$ cm$^{-3}$ and $W_{EP}=2$ µm. Note that for all of them the breakdown appears for voltages below the theoretical full depletion of the epilayer.

By comparison of the total current obtained for the diodes with $W_{EP}=2$ µm and different $N_D$, we confirm that, as expected, when reducing the doping, the breakdown voltage is significantly increased due to the lower electric field present at the contact for a given applied voltage and the associated smaller number of II processes taking place. The ratio between the current values obtained in the presence and in the absence of II processes in the simulation is presented in the inset of Fig. 5. For the highest doping ($N_D=6 \times 10^{17}$ cm$^{-3}$), there is an abrupt increase of this ratio near the breakdown voltage due to the stronger presence of II processes, while for the lowest doping the increase is smoother.

Ideally, within the full-depletion approximation, for a given doping level, the breakdown voltage should be the same regardless of the value of $W_{EP}$ (as long as not all the epilayer is depleted). However, MC simulations show that

<figure><img src="image_6.png" /><figcaption>**Fig. 4** Simulated I–V curves obtained for a 2D structure with $W_{EP}=350$ nm, $L_{EP}=200$ nm and different epilayer doping levels: $N_D=10^{17}$ (cyan), $3 \times 10^{17}$ (red) and $6 \times 10^{17}$ cm$^{-3}$ (green). Inset: Current as a function of the electric field at the edge of the Schottky contact (the symbols surrounded by circles correspond to an applied voltage of –5 V). The vertical lines indicate the condition for which a total epilayer depletion is found in the simulations (Color figure online)</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 5** Simulated I–V curves obtained for diodes with $N_D=6 \times 10^{17}$ cm$^{-3}$, $L_{EP}=200$ nm and different epilayer thickness: $W_{EP}=350$ nm (green), 1 µm (gray) and 2 µm (cyan); and a diode with $N_D=10^{17}$ cm$^{-3}$, $L_{EP}=200$ nm and $W_{EP}=2$ µm (yellow). The inset shows the ratio of the current values obtained with and without considering II processes in the simulation for the diodes with an epilayer thickness of 2 µm (Color figure online)</figcaption></figure>
2D effects and the strong tunnel injection at the contact edge invalidate the full-depletion approximation, finally resulting in higher breakdown voltages for thicker epilayers. Thus, the largest diode ($W_{EP}=2$ µm) presents a higher breakdown voltage, and interestingly, it exhibits an increase of the current near the breakdown range which is less abrupt than for the smallest one ($W_{EP}=350$ nm).

Figure 6 shows the breakdown voltage as a function of the epilayer (a) doping and (b) thickness. It is found to decrease with increasing epilayer doping, Fig. 6a, following a linear dependence according to the reported fitting, while it increases for thicker epilayer thickness, Fig. 6b, in this case with a quadratic dependence. These analytical expressions provide a simple way to quantify the observed trends and can serve as practical guidelines for device optimization.

Experimental values of breakdown voltage in GaN diodes with epilayer thickness and doping in the order of those considered here can be found in the literature. For example, Song et al. have employed a gradient doping technique in a

200 nm GaN drift layer to analyze the breakdown in different cases, obtaining a breakdown around 35 V for $N_D = 10^{17}$ cm$^{-3}$ and 20 V for $N_D = 4 \times 10^{17}$ cm$^{-3}$ [29]. Liang et al. experimentally measured the breakdown voltage in GaN Schottky barrier diodes ($W_{EP}=200$ nm and $N_D = 5 \times 10^{17}$ cm$^{-3}$), reporting values around 15 V [30]. Chen et al. demonstrated an improvement of the breakdown voltage (from 78 to 145 V) by applying a selective fluorine treatment in SBDs with $W_{EP}=1$ µm and $N_D = 7 \times 10^{16}$ cm$^{-3}$ [31]. While the trends of the experimental values with $N_D$ and $W_{EP}$ are the same as those found in our simulations, the breakdown voltages obtained in this work are slightly higher than the experimental ones, as expected according to the quasi-ideal conditions considered here, since we are neglecting non-ideal leakage current mechanisms, like those related to traps, or the influence of dislocations.

To further analyze the behavior of the breakdown voltage, Fig. 7 shows the vertical profiles of electron (solid lines) and hole concentration (dashed lines) at the edge of the Schottky contact calculated for the largest diode ($W_{EP}=2$ µm) with two different doping levels and an applied voltage of -100 V. As observed in Fig. 5, this is practically the breakdown voltage for the diode with $N_D=6 \times 10^{17}$ cm$^{-3}$, while the avalanche is far for the diode with $N_D = 10^{17}$ cm$^{-3}$. For $N_D = 6 \times 10^{17}$ cm$^{-3}$, the electron concentration (solid blue line) at the epilayer is even higher than the doping concentration. The electric field shows a sharp increase at the interface between the epilayer and the metal (taking values even above of the critical electric field, but along a very short length), originating a strong electron injection and intense II processes. The hole concentration is also extremely high throughout the epilayer, which indicates that the breakdown is originated by II events taking place

<figure><img src="image_8.png" /><figcaption>**Fig. 6** Breakdown voltage as a function of *a* the epilayer doping $N_D$ for an epilayer thickness $W_{EP}=350$ nm, *b* the epilayer thickness $W_{EP}$ for an epilayer doping of $N_D=6 \times 10^{17}$ cm$^{-3}$. The lines correspond to the indicated analytical fittings</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 7** Vertical profiles of the electron (solid lines) and hole (dashed lines) concentration represented together with the electric field (thick lines) at the edge of the Schottky contact for $N_D = 10^{17}$ cm$^{-3}$ (red) and $6 \times 10^{17}$ cm$^{-3}$ (blue) at $V = -100$ V (Color figure online)</figcaption></figure>

<figure><img src="image_10.png" /></figure>
all along the epilayer. On the other hand, the electron con-
centration corresponding to the diode with N<sub>D</sub> = 10<sup>17</sup> cm<sup>-3</sup>
(solid red line) for this bias (still far from the breakdown)
shows a partial depletion of the epilayer. Furthermore, as
expected, the peak of the electric field at the interface is
smaller than for N<sub>D</sub> = 6 × 10<sup>17</sup> cm<sup>-3</sup> and takes high values
only in the depleted region. Thus, holes are present only near
the Schottky contact, where II processes occur, and the hole
concentration is lower. In summary, the key aspect which
makes the difference when increasing the value of N<sub>D</sub> is the
enhanced tunnel injection (as a consequence of a narrower
barrier) which leads to more frequent II events; the associ-
ated generation of holes further increases the electron injec-
tion through the Schottky contact, thus leading the avalanche
to take place at lower voltages.

The electric field as a function of the applied voltage V
at three different positions of the Schottky contact (x=0,
x=L<sub>Sch</sub>/2 and x=L<sub>Sch</sub>) is shown in Fig. 8 for the two diodes
with N<sub>D</sub>=6×10<sup>17</sup> cm<sup>-3</sup> studied above (W<sub>EP</sub>=350 nm and 2
µm). For x=0 and x=L<sub>Sch</sub>/2, the values of the electric field
are similar. However, as expected, at the edge of the Schottky
contact a higher electric field is obtained. For the largest
diode (W<sub>EP</sub>=2 µm), the electric field remains practically
constant for |V|> 40 V, while for the small diode (W<sub>EP</sub>=350
nm) there is a sharp increase of the electric field near the
breakdown voltage associated with onset of the increased
hole concentration in the vicinity of the Schottky contact.
Thus, the physical origin of the avalanche breakdown signifi-
cantly depends on the 2D topology of the diode, particularly

<figure><img src="image_11.png" /><figcaption>**Fig. 8** Electric field at the metal–semiconductor interface as a function of the voltage for two diodes with N<sub>D</sub>=6×10<sup>17</sup> cm<sup>-3</sup>, L<sub>EP</sub>=200 nm and different epilayer thickness: W<sub>EP</sub>=350 nm (close circles) and 2 µm (open circles), calculated at three positions: x=0; x=L<sub>Sch</sub>/2 and x=L<sub>Sch</sub>. L<sub>Sch</sub>=200 nm is the size of the Schottky contact. A scheme of the diode, where the positions are indicated, is included</figcaption></figure>

around the Schottky contact and its edge. In view of these
results, the influence of L<sub>EP</sub> is analyzed in the next section.

**3.3 Epilayer lateral extension**

Since the edge effects are extremely significant in the
appearance of the avalanche breakdown, the epilayer lateral
extension L<sub>EP</sub> must be optimized. To this aim, diodes with
N<sub>D</sub>=6×10<sup>17</sup> cm<sup>-3</sup>, W<sub>EP</sub>=2 µm and three different values of
L<sub>EP</sub> have been simulated. No passivation (κ<sub>pas</sub>=1) is consid-
ered in these diodes. Note that this topology is not the opti-
mal one, but it was selected because II effects are relevant
enough to easily identify the influence of L<sub>EP</sub>.

The I–V curves are presented in Fig. 9. Lower values of L<sub>EP</sub> involve lower values of the current. In order to further evidence the importance of the 2D topology on the break- down voltage, the I–V curve for the equivalent 1D structure (red rectangle of Fig. 1) has been included in the figure. As expected, the current is reduced with respect to the diodes with L<sub>EP</sub>=100 nm and 200 nm, which can be explained by the absence of 2D effects (peak electric field at the contact edge). However, the current is higher than that obtained for a diode with L<sub>EP</sub>=0 nm.

To illustrate this behavior, the MC profiles of electron and
hole concentrations for V=-50 V are presented in Fig. 10 for
the four considered diodes. The electron concentration near
the Schottky contact is smaller when reducing L<sub>EP</sub> due to the
influence of the surface charge located at the vertical side-
wall of the epilayer [17]. When the epilayer and the Schottky
contact are aligned (L<sub>EP</sub>=0 nm), the electron concentration
dramatically decreases due to the depletion provoked by the

<figure><img src="image_12.png" /><figcaption>**Fig. 9** I–V curves obtained from the simulation of diodes with N<sub>D</sub>=6×10<sup>17</sup> cm<sup>-3</sup>, W<sub>EP</sub>=2 µm and different L<sub>EP</sub>: 200 nm (blue), 100 nm (red) and 0 nm (gray). The I–V curve of a 1D vertical diode (green) with the same W<sub>EP</sub>, N<sub>D</sub> and L<sub>Sch</sub> is also shown for comparison (Color figure online)</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Fig. 10** Color maps for an epilayer with $W_{EP}=2$ µm and $N_D=6 \times 10^{17}$ cm$^{-3}$ and an applied voltage of -50 V for **a**, **e** $L_{EP}=200$ nm, **b**, **f** $L_{EP}=100$ nm, **c**, **g** $L_{EP}=0$ nm and **d**, **h** a 1D structure with the same contact length. **a–d** Electron and **e–h** hole concentration</figcaption></figure>

surface charge. On the other hand, the hole concentration is higher for longer $L_{EP}$ (200 nm), which is associated with a stronger tunnel injection due to the higher value of the vertical electric field at the contact edge. For $L_{EP}=0$ nm, the electric field is smaller due to the depletion, which is coherent with the lower injection.

In the 1D structure, quantities are uniform along the hor-
izontal direction. As mentioned, interestingly, the current
is higher than for the diode with $L_{EP}=0$ nm. The strong
influence of the surface charge when $L_{EP}=0$ nm in the 2D
structure reduces the current with respect to the 1D diode
due to the lower electric field and consequently the lower
tunnel injection and II events. Despite the depletion induced
by the surface charges, the 2D cases with $L_{EP}=100$ and 200
nm provide a higher current than the 1D structure due to the
higher electric field at the contact edge and the lateral exten-
sion beyond the contact where injected carriers can move
along the vertical direction. Therefore, in order to optimize
the design of these devices 2D simulations are essential,
indicating that higher breakdown voltages are reached in
diodes with shorter $L_{EP}$.

In all the previous results, self-heating effects have been
neglected. In small diodes like those analyzed here, the
influence of the associated temperature rise is not found
to be critical in the device breakdown, even if it could be
slightly shifted to lower voltages. The temperature rise can
be estimated by means of a thermal resistance ($R_{th}$) model.

Considering a representative value of $R_{th}$ of 300 K/W [32, 33], a circular diode with a radius of 200 nm (the value of $L_{Sch}$ considered here), a current level of 2500 A/m (around which the avalanche process starts in most of the simulated diodes) and an applied voltage of 60 V, the temperature rise is about 28 K, far from values leading to a thermal runaway or a significant current increase.

**4 Conclusions**

By means of 2D MC simulations, we have physically ana-
lyzed the influence of several technological parameters on
the breakdown voltage of GaN SBDs originated by II. Under
high-reverse-bias conditions, it is necessary to carefully
consider tunnel injection and II events, since the positive
feedback between these mechanisms leads to a significant
increase of the carrier number ultimately provoking the
breakdown of the diodes. A 2D modeling of the physical
processes leading to the diode breakdown is essential for
their correct description. In particular, the strong carrier
injection taking place at the contact edge and the lateral
depletion induced by the surface charges present at the semi-
conductor/dielectric interface require a 2D model to be cor-
rectly accounted for.

The results indicate that the passivation of the diodes
is beneficial because lower values of electric field are
<table><caption>**Table 2** Summary of design parameters influencing breakdown voltage</caption><thead><tr><td>Parameter</td><td>Guideline</td><td>Trade-off</td></tr></thead><tbody><tr><td>Passivation (κ<sub>pas</sub>)</td><td>High κ<sub>pas</sub></td><td>Increase of the edge effect capacitance</td></tr><tr><td>Doping level (N<sub>D</sub>)</td><td>Low N<sub>D</sub></td><td>Higher series resistance</td></tr><tr><td>Epilayer thickness (W<sub>EP</sub>)</td><td>Thick epilayer</td><td>Higher series resistance</td></tr><tr><td>Epilayer lateral extension (L<sub>EP</sub>)</td><td>Self-aligned epilayer (L<sub>EP</sub> = 0)</td><td>Fabrication limit</td></tr></tbody></table>

reached at the contact edge the higher is the dielectric constant of the passivation material, leading to a lower tunnel injection, which may improve the breakdown voltage. The doping level and the thickness of the epilayer are key parameters for the improvement of the breakdown voltage. The thicker the epilayer and the lower the doping level, the higher the breakdown voltage, at the expense of a higher series resistance that may deteriorate the high-frequency performance of the SBDs. The optimum design should be the result of a trade-off between a good frequency performance and high breakdown voltage. Finally, a short lateral extension of the epilayer also helps to increase the breakdown voltage thanks to the reduction of the electric field at the contact edge induced by the surface charges at the semiconductor/dielectric interface: a self-aligned epilayer being the optimum design (Table 2).

We remark that our analysis is qualitative, and some effects not considered in our model, like self-heating of the devices or trap-assisted tunneling, could modify the specific voltage values at which the breakdown takes place. But the observed tendencies and the conclusions reached about the influence of the different technological parameters remain valid as long as the breakdown is originated by II processes and tunneling. Moreover, other leakage current mechanisms related to the presence of traps, dislocations or defects may lead to breakdown voltages much lower than those predicted here.

**Acknowledgements** This work has been partially supported through grants PID2023-1475550B-I00 and PDC2023-145896-I00 funded by MCIN/AEI/https://doi.org/10.13039/501100011033, and the Junta de Castilla y León and FEDER through project SA136P23. The work of R. A. Peña was supported by the Junta de Castilla y León.

**Author contributions** All authors contributed to the study conception and design. Simulations, data collection and analysis were performed by B. O., J. M. and T. G. The first draft of the manuscript was written by B. O. and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

**Funding** Open access funding provided by FEDER European Funds and the Junta de Castilla y León under the Research and Innovation Strategy for Smart Specialization (RIS3) of Castilla y León 2021-2027.

**Data availability** No datasets were generated or analysed during the current study.

## Declarations

**Competing interests** The authors declare that they have no competing interests.

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