An interpretation of the unipolar figure of merit is formulated for wide bandgap (WBG) semiconductors based on the on-state specific resistance ($RON,sp$) derived from the space charge limited current–voltage relationship (Mott–Gurney square law). The limitations of the traditional Ohmic $RON,sp$ for WBG semiconductors are discussed, particularly at low doping, while the accuracy of the Mott–Gurney based $RON,sp$ is confirmed by Silvaco ATLAS drift–diffusion simulations of diamond Schottky pin diodes. The effects of incomplete ionization are considered as well.

Wide-bandgap (WBG) semiconductors, such as SiC, GaN, Ga_{2}O_{3}, AlN, and diamond, have been developed for high power applications^{1–5} owing to their high breakdown electric fields (2.8–10 MV/cm), high thermal conductivities (11–2400 W/m K), high carrier mobilities (300–2100 cm^{2}/V s), and high saturation velocities (1.4–2.5 $\xd7$ 10^{7} cm/s).^{5} The impact of material properties on system level performance can be compared using figures of merit (FOMs) that incorporate parameters such as the breakdown voltage, on-resistance, and switching speed into a single metric encapsulating system performance for particular applications. Such FOMs are often used in comparing different material technologies for different applications. The unipolar FOM, formulated for WBG semiconductor devices,^{6,7} is given by

where $RON,sp$ is the specific on-state resistance and $VB$ is the avalanche breakdown voltage based on impact ionization.

Research in WBG device physics has revealed new on-state conduction and off-state breakdown mechanisms, which directly influence the parameters used in the FOM equations. One such on-state conduction mechanism that has been explored in the past decade is the space charge limited current (SCLC), given by the Mott–Gurney square law,^{7} which dominates in intrinsic or low-doped WBG semiconductors.^{8–11} We have recently shown experimentally and through numerical simulation^{12} that the current–voltage relation in diamond diodes transitions from thermionic emission (TE) directly into space charge limited Mott–Gurney square law dominant without an intermediate Ohmic or linear regime after the diode turns on. We also show that the ultimate theoretical limit to the forward current in WBG diodes is the Mott–Gurney square law, when contact and bulk resistance are negligible. This space charge limited conduction is an emergent phenomenon arising from the injection of a large concentration of free charge carriers into any WBG semiconductor region of low intrinsic free carrier concentration. The injection of free charge carriers may occur, for example, from the source of a FET into the drift region of a vertical transistor or over a Schottky barrier into the drift region of a diode. This current–voltage relation in the WBG semiconductor device on-state directly affects the calculation of $RON,sp$ and a different relationship for the FOM in (1), which we discuss below for diamond Schottky diode structure.

The unipolar FOM is based on a Schottky diode with a moderately doped drift region and a highly doped substrate with the Schottky contact to the drift region, as shown in Fig. 1.^{13} Traditionally, the $RON,sp$ for a Schottky diode is calculated assuming that the drift region is equal to the maximum depletion width obtained at the breakdown voltage and is given as

where $d$ is the thickness of the drift region, $\mu (NA\u2212)$ is the doping dependent carrier mobility, $NA\u2212$ is the ionized doping concentration in the drift region, and $q$ is the charge of an electron.^{13} This assumes that the on-state current conduction is Ohmic through the drift region. However, for low drift layer doping, the Ohmic conduction based on (2) is not consistent with $RON,sp$ values reported in the literature [0.05 m $\Omega $ cm^{2} (Ref. 12), 0.03 m $\Omega $ cm^{2} (Ref. 14)]. The current conduction in low to moderately doped diamond Schottky diodes is through holes injected from the heavily doped substrate into the lower doped drift region and over the junction barrier via thermionic emission (TE). As the injected charge increases with the increase in on-state voltage, the charge neutrality is disturbed as the injected charge carriers do not relax into the material quickly enough and, thereby, start accumulating in the drift region.^{15} This causes the current–voltage (*I–V*) relation to transition from exponential (due to TE) to space charge limited conduction (SCLC).

The ideal SCLC current due to injected charge is given by the Mott–Gurney square law^{16} as

where $\u03f5r$ is the material's relative dielectric permittivity, $\u03f50$ is the permittivity of free space, and $V$ is the voltage across the drift region of width $d$. The differential $RON,sp$ is calculated by taking the derivative of (3), which gives

Figure 2 shows a comparison of the simulated $RON,sp$ with the analytical $RON,sp,MG$(4) variation with drift layer thickness (black curves) for on-state voltages of 5 and 20 V as well as the variation with on-state voltage (red) for a constant drift layer thicknesses of 1, 10, and 50 $\mu m$, where the agreement between the numerical and analytical models is quite good, again emphasizing the dominant role of the Mott–Gurney behavior.

As can be seen from (4), the $RON,sp,MG$ reduces with on-state voltage across the drift region and has a cubic dependence on the drift layer thickness.

Traditionally, the FOM was evaluated for an abrupt junction in a parallel-plane configuration.^{13} The solution to Poisson's equation in the drift region with a uniform doping concentration leads to a triangular electric field. A maximum value, $EM$, at the Schottky end gradually decreases toward the abrupt junction, i.e., a non-punch-through (NPT) configuration, as shown in Fig. 1(a). The breakdown voltage and maximum depletion width ($Wd$) are then determined by the maximum electric field reaching the critical electric field $EC$ of the semiconductor material, i.e., $VB=WdEC/2$. In the original definition of the FOM, the drift layer width $d$ is made equal to $Wd$, which is what we use in the present paper, although this does not in fact correspond to the maximum FOM.^{17} Here, $Wd$ is dependent on the total doping concentration $NA$ because in reverse bias (and at breakdown voltage) the depletion charge constitutes the total dopants uncovered (or ionized) by the Fermi level going below (or above) the dopant level.

In diamond, P and B dopants have large activation energies [0.57 eV (Ref. 18) and 0.413 eV,^{19} respectively] and are incompletely ionized at room temperature. An incomplete ionization ratio $\eta $ can be obtained from the positive root of the quadratic equation of charge neutrality as shown by Sze *et al.*^{20} Also, the B activation energy in diamond is a function of its doping concentration $NA.$^{21} Therefore, the incomplete ionization $\eta $ becomes a nonlinear function of the doping concentration, i.e., $\eta NA$, which is given as follows:

where $\eta (NA)$ is the ionization ratio, $NA\u2212/NA$. Therefore, an incomplete ionization modified FOM similar to the one shown in Ref. 22, which gives a ratio of the losses in the on-state to the off-state power can then be written as

For WBG unipolar devices with low or zero doping, e.g., a Schottky diode, the depletion region extends across the drift layer and the electric field is constant throughout, leading to a punch-through (PT) electric field configuration shown in Fig. 1. The breakdown voltage for a drift layer width of $d$ can then be determined by $VB=d\xd7EC$. Therefore, for a critical electric field of $EC$ and for the same breakdown voltage, $VB$ only half as much of the drift layer width is required in a PT configuration in comparison with an NPT configuration.^{23} Consequently, to obtain a $RON,sp,MG$ vs $VB$ relationship, the drift layer thickness $d$ can be substituted in (4) to obtain

As evident from (7), $RON,sp,MG$ has a cubic relation with $VB$ compared to a square law in the traditional FOM case. In Fig. 3, $RON,sp$ vs $VB$ calculated by the traditional FOM from (5) assuming complete ionization (black curve) and including incomplete ionization (green curve) is compared to the SCLC $RON,sp,MG$ vs $VB$ calculated from (7) (blue curve) for a forward voltage of 20 V. Figure 3 (inset) shows the Boron doping concentration dependent activation energy $\Delta EA$ (eV) and incomplete ionization ratio [$\eta (NA)$]. Here, a constant mobility of 2100 cm^{2}/V s and a constant critical electric field, *E _{C}*, of 10 MV/cm were assumed for holes. As can be seen, the SCLC case has a significantly steeper slope than the traditional FOM due to the stronger $VB$ dependence, and the crossing point depends on the forward bias assumed. In the SCLC case, the FOM is a function of the breakdown voltage and is higher than the Ohmic FOM at low breakdown voltages where $RON,sp,MG$ is lower and smaller for high

*V*. Experimental values of $RON,sp$ vs breakdown voltage (BV) obtained from Refs. 12, 14, and 24–26 also indicate toward a steeper slope in comparison with the traditional FOM, as shown in Fig. 3.

_{B}Silvaco ATLAS simulations based on drift–diffusion charge transport are performed using the Schottky diode of Fig. 1 with an intrinsic drift layer. Physical models, such as a doping dependent effective conductivity due to hopping, incomplete ionization, Shockley–Read–Hall (SRH) recombination, and thermionic emission over a Schottky barrier, are included in the simulations as detailed in Ref. 12. Similar consideration of nonideal effects has been reported analytically by Umezawa *et al.*^{25} resulting in deviation from the straight-line behavior of the black curve in Fig. 3. The ATLAS simulations of $RON,sp$ vs $VB$ agree well with the analytical values calculated from (4) as shown in Fig. 3, with $VB=d\xd7EC$ assumed for both curves. This agreement strongly supports the supposition of Mott–Gurney behavior limiting the current in forward bias. The SCLC or Mott–Gurney behavior observed here in Silvaco ATLAS simulations is an emergent property from the drift–diffusion and accumulation of charge carriers (holes) in the applied electric field. The divergence of the two curves at lower *V _{B}* is due to the finite on-resistance of the 1 $\mu m$ thick p

^{++}substrate, which is accounted for in the ATLAS simulations.

The traditional FOM of (6) is optimized by varying the drift layer doping and fixing $d$ equal to the depletion width for a low Ohmic $RON,sp$. However, the assumption of Ohmic $RON,sp$ is not necessarily valid, even in the case of NPT diodes. An analysis of the slope $m=\u2202logJ/\u2202logV$ of the on-state characteristics reveals the dominant *I–V* relationship; a slope of ∼2 indicates space charge limited Mott–Gurney conduction while a slope of ∼1 indicates Ohmic conduction through the drift layer. For a given drift layer doping, $NA$, the slope *m* can be calculated from on-state simulations for different drift layer thicknesses, as shown in Fig. 4. An on-state voltage of 20 V is used in Fig. 4.

For the space charge limited current to dominate, the excess charge injected into the drift region must be swept out by the electric field before dielectric relaxation has time to establish neutrality. Assuming a carrier velocity $v$ and a uniform on-state potential *V* across a drift layer of thickness $d$, the carrier transit time across the drift layer can be given as $tt=\u222b0dv\u22121xdx=d2/\mu V$. The dielectric relaxation time, assuming a thermally generated free carrier concentration $p0$, can be given as $\tau d=\u03f5/\sigma =\u03f5/qp0\mu $. The transition from Ohmic conduction to SCLC occurs when $tt\u2248\u2009\tau d$. The value at which the transition occurs is, therefore, $d=\u03f5V/qp0$. For low to moderate $NA$ and $d$, $tt\u2264\tau d$, and, therefore, space charge limited current dominates with a slope *m* of 2. However, for large $d$, $tt$ becomes comparable or larger than $\tau d$ and the injected carriers have enough time to relax into the dielectric material, thereby reverting the dominant slope *m* back to 1, i.e., Ohmic conduction. For $d\u2009\u226a0.1\u2009\mu m,$ the forward current is dominated by the Ohmic resistance of the substrate, and the slope reverts to ∼1, see Fig. 4. Figure 4 confirms that over a wide range of doping and drift region thicknesses, the on-resistance in NPT Schottky diodes is dominated by Mott–Gurney behavior with a different $RON,sp$ than the Ohmic conduction used in the traditional FOM plot and requires different optimization than the Ohmic model used in, e.g., Ref. 17.

With regard to other materials besides diamond, the results shown in Figs. 2–4 should be equally applicable. The main differences in behavior between different materials relative to the onset of SCLC behavior are in terms of their mobility and device structure, specifically the drift region thickness. The same criteria hold in terms of the drift region transit time vs dielectric relaxation time for onset of SCLC behavior. Factors that make the observation of SCLC behavior more pronounced in diamond than other materials are the relatively high carrier mobility in diamond compared to other WBG materials, lower free carrier density due to incomplete ionization, and the high critical field, which allows the use of much narrower drift regions to achieve the same breakdown voltage as compared to, e.g., GaN.

The traditional unipolar FOM based on an Ohmic $RON,sp$ cannot be used for wide bandgap devices if the dominant *I–V* relationship is space charge limited. Indeed, in the presence of a nonlinear I–V relationship, such as (3), beyond the turn-on voltage of the diode, the basic unipolar FOM relationship (1) cannot be written, coming from a general definition of the overall power FOM in terms of the product of the forward on-current and the reverse breakdown voltage, $IonVB$, where an Ohmic relationship between $Ion$ and the forward voltage, *V*, is used.^{6} Starting with $IonVB$ for a power device as the metric for performance, and using the Mott–Gurney expression (3) for $Jon$, the unipolar FOM can be written simply as

where $V$ is the forward voltage. The dependence on the critical field is now linear rather than the usual cubic dependence given by (6) and depends explicitly on the ratio of the drift region thickness and the on-voltage (due to nonlinear dependence of the current on *V*), which is less general than (6), depending on where the diode is operated. Therefore, for the case of diamond Schottky pin diode operated at an on-state voltage of 20 V, the traditional constant unipolar FOM ($\u223c2.6\xd7105\u2009MW/cm2$ or $\u223c3\xd7104$ with respect to Si) under- and overestimates the actual diamond unipolar FOM below and above a breakdown voltage of ∼180 V, respectively.

In summary, an interpretation of the unipolar FOM based on the Mott–Gurney square law is discussed in comparison with the traditional Ohmic based FOM. Silvaco simulations are in excellent agreement with the analytically derived Mott–Gurney based $RON,sp,MG$ for low-doped PT diodes, validating the assertion that SCLC dominates. From the Mott–Gurney unipolar FOM, the optimization of WBG power devices in terms of on-state losses and off-state breakdown is based on the drift layer width and on-state operating voltage, rather than the simpler Ohmic unipolar FOM, which only depends on fundamental material parameters.

This work was supported as part of ULTRA, an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) Grant No. DE-SC0021230.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## References

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