Thermionic emission or tunneling? The universal transition electric field for ideal Schottky reverse leakage current: A case study in β-Ga₂O₃

With a unique combination of an ultra-high breakdown electric field of ∼8 MV/cm,^1,2 a decent electron mobility of ∼200 cm²/V·s,³ an availability of melt-grown substrates,⁴ and controllable n-type doping,⁵ β-Ga₂O₃ is an attractive ultra-wide bandgap semiconductor material for applications demanding high power handling capability.⁶ To date, promising performance in Ga₂O₃ devices has been demonstrated, including Schottky barrier diodes with a breakdown voltage (BV) over 2 kV,^7–9 high-voltage power transistors,^10–12 and RF transistors.^13,14

Ga₂O₃ Schottky barrier diodes (SBDs) are highly versatile. They can be used as high-speed rectifiers for efficient power regulation^15–17 and also function as UV photodetectors.^18,19 In addition, a Schottky contact is a key device building block, offering gate control for metal-semiconductor field-effect transistors (MESFETs),¹ as well as serving potentially as a p–n junction replacement for high field management.^20,21

Central among all the aforementioned functionalities is the reverse blocking capability of the Schottky barrier, which is critically dependent on the reverse leakage current. In general, the ideal total reverse leakage current (J_R,tot) through a Schottky barrier consists of the transport of electrons both above and below the top of the barrier. The former mechanism is thermionic emission (TE), while the latter is barrier tunneling (BT), which comprises thermionic-field emission (TFE) and field emission (FE).^22,23 As a result, J_R,tot can be expressed as

where J_TE is the thermionic emission current and J_BT the barrier tunneling current.

It has been widely recognized that, at a certain temperature, there should exist a transition voltage (V_T) or transition electric field (E_T), below which TE dominates and above which BT dominates.^22,23 E_T is defined as the surface electric field where $JTE=JBT$⁠, while V_T is defined as the corresponding reverse-bias voltage. Knowledge about V_T or E_T is highly valuable since it determines the appropriate bias or electric field ranges for TE and BT models. However, due to the difficulty in calculating the tunneling current analytically near this transition region, there has not been a simple closed-form expression for V_T or E_T. In previous studies, V_T in β-Ga₂O₃ has been calculated numerically,²⁴ but the dependence on the doping concentration, barrier height, and temperature is very complicated. Also, the non-monotonic temperature dependence of V_T is questionable. In this study, we first show from the numerical calculation that, unlike V_T, E_T is nearly independent of the doping concentration and the barrier height; furthermore, there exists a universal monotonic temperature dependence of E_T.

Experimentally, most of the previous analyses on the reverse leakage current in Ga₂O₃ SBDs use either the TE model^19,25 or the TFE model,^26,27 without considering their respective appropriate ranges. We have previous observed and verified near-ideal bulk reverse leakage current in Ga₂O₃ SBDs fabricated on bulk substrates.²⁰ However, due to the high doping concentration employed, the transition electric field could not be accessed in the study in Ref. 20. In this work, we fabricate SBDs on an n^--Ga₂O₃ epitaxial layer, which allows us to access the surface electric field around E_T with the edge leakage current sufficiently suppressed with a field-plate structure.

For the calculation of the reverse leakage current, two effects on the shape of the Schottky barrier potential should be considered: image-force lowering (IFL) and doping effects, as illustrated schematically in Figs. 1(a) and 1(b), respectively. With both effects included, the potential energy distribution of the Schottky barrier under a surface electric field E is given by

where $ϕB$ is the barrier height, $ND−NA$ is the net doping concentration, and $εs$ = 10 $ε0$ is the dielectric constant of β-Ga₂O_3.²⁸ Here, the Fermi-level energy in metal (⁠$EFm$⁠) is taken as the zero-energy level, as shown in Fig. 1(a). Due to the presence of IFL, the barrier is lowered by $Δϕ=eE/(4πεs)$⁠, i.e., $Ec,max=eϕB−Δϕ$⁠. Assuming a transmission probability of unity for electrons with an energy $E$ higher than $Ec,max$⁠, $JTE$ is given by the familiar expression,

where $A*=4πm*kB2e/h3$ is the Richardson constant.

On the other hand, including both IFL and doping effects in the calculation of the barrier tunneling current is analytically intractable, and thus, we developed a numerical approach, as presented in our previous work.²⁰ To obtain solely the barrier tunneling current, the upper limit of the integration in Ref. 20 should be changed to $Ec,max$⁠, i.e.,

where $TE$ is the transmission probability, whose expression is given in Ref. 20. A single effective mass $m*=0.31 m0$ is adopted for both the Richardson constant and the tunneling effective mass in $TE$ due to the single-valley and near-isotropic nature of the conduction band in β-Ga₂O₃.^29,30

Figure 1 shows the calculated J_R,tot using the numerical model and its constituent components (J_TE, J_BT). Here, we have temporality neglected the doping effect to compare with the analytical TE and TFE models derived by Murphy and Good,²² which consider only the IFL. It can be seen that the calculated J_R,tot from our numerical model agrees well with Murphy and Good's models within their respective applicable ranges, indicating that our numerical method is valid. It is worth noting that Murphy and Good's TE model actually includes both the TE and BT currents despite what the name suggests, and thus, it comes as no surprise that a match with J_R,tot is observed, rather than with J_TE.

With the numerical model established, the transition electric field E_T can be calculated by equating J_TE with J_BT. Figure 2(a) shows the calculated E_T as a function N_D-N_A at different temperatures, at a barrier height of 1.2 eV. E_T is primarily a function of temperature and has a very weak dependence on N_D-N_A. It is nearly constant when N_D-N_A < 10¹⁷ cm⁻³ and only increases slightly (<0.08 MV/cm) when N_D-N_A approaches 2 × 10¹⁸ cm⁻³, indicating that the influence of the doping effect is near negligible. The surface electric field at zero bias (E₀) is also calculated by the familiar expression: $E0=2(ND−NA)(eVbi,0−kBT)/εs$⁠, where V_bi,0 is the built-in potential at zero bias. If $E0$ is larger than E_T, J_R,tot would be dominated by barrier tunneling, as in the case of the SBD we reported in Ref. 20. This illustrates the superiority of using the surface electric field instead of the reverse bias as the variable to characterize the transition region, as V_T would have a large dependence on N_D-N_A even with a constant E_T.

Knowing the weak dependence on the doping concentration, we calculate E_T as a function of temperature without considering the doping effect, as shown in Fig. 2(b). Here, we examine the influence of the barrier height ranging from 0.5–2.0 eV. Interestingly, there is a negligible dependence on the barrier height from 100 K to 800 K. This is because near the transition electric field, the majority of contribution to J_BT arises from the tunneling electrons with energy $E$ close to $Ec,max$⁠, such that the natural log term in Eq. (4) can be simply replaced by $exp−E−EFmkBT$⁠, as long as the barrier height is not too small. Consequently, it can be shown that $JBT∝exp(−eϕB/kBT)$⁠, and thus, the condition of $JBT=JTE$ does not depend on the barrier height. Below 100 K, however, there exists a sharp transition of E_T to zero at some “transition” temperature, depending on the barrier height. This comes from the fact that the contribution of tunneling electrons with energy $E$ close to $EFm$ can no longer be neglected, leading to more complicated dependence on the barrier height in $JBT$⁠. Because of the exponential decrease in J_TE at cryogenic temperatures, to match J_BT with J_TE, the required E_T is very low, <0.06 MV/cm. Also, with N_D-N_A$≥$ 1 × 10¹⁶ cm⁻³ in practice, the value of E_T around this transition region would already be lower than E₀, rendering it unobservable.

Due to the negligible dependence on the barrier height and very weak dependence on the doping concentration, we can describe E_T in β-Ga₂O₃ with a near-universal empirical temperature dependence as obtained from a quadratic fitting to the numerical calculation in Fig. 2(b),

where T is in units of K. Eq. (5) is valid within the temperature range of 100–800 K and a barrier-height range of 0.5–2.5 eV, with maximum errors of 0.01 MV/cm for N_D-N_A$≤$ 1 × 10¹⁷ cm⁻³ and 0.08 MV/cm for N_D-N_A$≤$ 2 × 10¹⁸ cm⁻³.

To verify the existence of the transition region experimentally, we fabricated field-plated SBDs on a (001) Ga₂O₃ epitaxial wafer grown by halide vapor phase epitaxy (HVPE), as schematically shown in Fig. 3(a). The field plate length is designed to be 30 μm for the purpose of suppressing the electric field crowding at the anode edge. The cathode Ohmic contact is based on Ti/Au (50/100 nm), while the anode Schottky contact is based on Ni/Au (40/150 nm). The fabrication processes for the formation of cathode and anode contacts were the same as described in Ref. 31. After the anode formation, a 31 nm Al₂O₃ was deposited by atomic layer deposition (ALD) at 300 °C, acting as the dielectric for the field plate. Finally, a contact hole was etched, followed by the deposition of the field plate, which comprises a stack of Ni/Ti/Al/Pt (30/10/80/20 nm).

Temperature-dependent capacitance–voltage (C–V) measurements were performed on co-fabricated SBDs without the field plate. Figure 3(b) shows the extracted doping profile, which shows an average N_D-N_A value of ∼7 × 10¹⁵ cm⁻³ in the Si-doped n^- drift layer. The 1/C²-V plot is shown in the inset, from which the values of V_bi,0 are extracted to be 1.10 V at 25 °C and 0.92 V at 200 °C, corresponding to barrier heights of 1.27 eV and 1.21 eV, respectively (also see Figs. 4 and 7).

Figure 4(a) shows the measured temperature-dependent forward current–voltage (I–V) characteristics. Since the doping concentration is low, the depletion width at zero bias is ∼0.4 μm. In this case, the TE model is inappropriate for the analysis of the forward I–V characteristics since the electron transport through the depletion region cannot be assumed ballistic, especially considering that the mobility of Ga₂O₃ is not very high.³ Therefore, we analyze the data using the thermionic emission-diffusion (TED) model,^32,33 which considers the drift-diffusion transport in the depletion region. In the calculations, the temperature-dependent drift mobility model in Ref. 3 is used with the Hall factor considered. The constant of proportionality is adjusted to match the Hall mobility of 145 cm²/V·s measured at 25 °C on a similar wafer.⁵ 

The extracted barrier heights and ideality factors (n) from the TED model at each temperature are plotted in Fig. 4(b). The ideality factor is 1.02 at 25 °C and decreases to below 1.01 beyond 100 °C, indicating a very good Schottky contact quality. The image-force controlled ideality factor limit (n_IF) is calculated to be 1.007 using the standard method.³⁴ It can be seen that the extracted ideality factor approaches n_IF beyond 75 °C, further suggesting a near-ideal interface. Both the apparent barrier height and the barrier height after image-force (IF) correction (∼0.026 eV)³⁴ were plotted. The IF-corrected barrier height is around 1.20 eV.

To verify the effect of the field plate in suppressing the edge leakage current, temperature-dependent reverse I–V measurements were performed on both non-field-plated and field-plated SBDs on the same wafer, as shown in Figs. 5(a) and 5(b), respectively. Without the field plate, the SBDs exhibit a large, near temperature-independent reverse leakage current below 100 °C. As have been pointed out in our previous report,³¹ such a leakage behavior is characteristic of field-emission dominated edge leakage current due to electric-field crowding at the anode edge. On the other hand, the leakage current in SBDs with FP is much reduced, suggesting that the FP structure is effective in suppressing the edge electric-field crowding. Note that between −200 V and −100 V, there still exists a very low level of leakage current at 25–75 °C, which does not show a much temperature dependence, suggesting that there is still some edge leakage current not completely eliminated. However, considering the very low magnitude (<10⁻⁸ A/cm²) and the weak temperature dependence, it will not significantly “pollute” the uniform bulk leakage current from 100 °C to 200 °C, which we will model using our numerical models.

Figure 6 shows the temperature-dependent reverse leakage current as a function of the surface electric field, i.e., J–E characteristics. The reverse leakage characteristics from 100 °C to 200 °C can be well-fitted with the calculated total reverse leakage current (J_R,tot) using our numerical model [Eqs. (1)–(4)], with the barrier height as the only fitting parameter at each temperature. The individual components of J_R,tot, i.e., J_TE and J_BT, are also plotted in Fig. 6. While J_TE matches with the measured data at the lower end of E and J_BT at the higher end of E, neither J_TE nor J_BT alone can capture well the field dependence throughout the electric-field range (0.07–0.69 MV/cm). These results strongly suggest the presence of the transition regime, where both J_TE and J_BT are important.

The validity of the fitting depends critically on the fact that the measured bulk reverse leakage current is near ideal. A good check would be comparing the barrier height values extracted from the reverse leakage characteristics with those extracted from other methods. Figure 7 shows such comparisons. The barrier height values extracted from forward I–V, reverse I–V, and C–V measurements exhibit good agreement. These provide further evidence that the measured reverse leakage current is near ideal, which, in turn, corroborates the identifications of the transition region from the data fitting in Fig. 6. The barrier heights extracted from C–V measurements are slightly larger than those from the I–V methods, especially at lower temperatures. This behavior is commonly observed (see, e.g., Ref. 19) and could be due to the presence of barrier height inhomogeneity³⁵ and/or uncertainty of the doping concentration close to the Schottky contact interface.

In conclusion, the transition electric field (E_T) separating the thermionic emission- and barrier tunneling-dominated reverse leakage regimes is calculated in β-Ga₂O₃ Schottky barrier diodes, by using a numerical reverse leakage model. E_T is shown to have a very weak dependence on the doping concentration and the barrier height. A near-universal empirical expression for E_T is obtained, which is valid for wide temperature, doping, and barrier-height ranges in β-Ga₂O₃ SBDs. Experimentally, we confirmed the presence of the transition region in field-plated Ga₂O₃ SBDs, which shows near-ideal bulk reverse leakage current well-matched with our numerical model. With the knowledge about the transition electric field, the long-standing confusion about whether the thermionic emission model or the tunneling model should be used is lifted: if the surface electric field is much lower than E_T, the thermionic emission model can be used; conversely, the barrier tunneling model should be employed. Near E_T, it is important to consider both models. These results and methodologies are highly valuable for the design of functional Schottky barriers in nearly all semiconductors that rely on the precise knowledge about the reverse leakage current.

This work was supported in part by ACCESS, an AFOSR Center of Excellence (No. FA9550-18-1-0529), and AFOSR (No. FA9550-20-1-0148) and was carried out at the Cornell Nanoscale Facility and CCMR Shared Facilities sponsored by the NSF NNCI program (No. ECCS-1542081), MRSEC program (No. DMR-1719875), and MRI No. DMR-1631282.