High Field Transport in (Ultra) Wide Bandgap Semiconductors: Diamond Versus Cubic GaN

We provide an analysis of nonlinear transport in diamond and cubic GaN (c-GaN) with emphasis on the different types of optical phonon scattering [i.e., optical deformation potential (ODP) scattering versus polar optic phonon (POP) scattering] limiting the carrier velocity. Both types of carrier mobilities and carrier saturation (peak) velocities in diamond and c-GaN as functions of different doping types and concentrations are obtained by directly solving the Boltzmann equation. Our model indicates that the nonrandomizing nature of POP scattering causes carrier temperature cooling, resulting in higher carrier drift velocity than with ODP scattering. This effect, in addition to the small carrier effective masses and large optical phonon energies, is responsible for the higher peak velocities in c-GaN, compared to carrier drift velocity in diamond.

Although there have been numerical methods such as Monte Carlo simulation to address these issues in both materials by calculating their carrier drift velocity [16], [18], [19], these techniques are tedious and require long computation times.In contrast, solving the Boltzmann equation analytically [20], [21], [22] is especially powerful and convenient for (U)WBG materials, because the optical phonon energy is large (≫k B T ), the conduction band is nearly parabolic, and the Maxwell-Boltzmann (M-B) approximation holds for relatively high doping concentrations.
Here, we adopt a fast, accurate semi-analytic method to solve the Boltzmann equation to compare the transport properties of diamond and c-GaN by calculating the drift velocity for both electrons and holes as a function of the electric field at various doping types and concentrations.We place special emphasis on the influence of the respective types of optical phonon scattering mechanisms [i.e., polar optic phonon (POP) for c-GaN and optical deformation potential (ODP) for diamond] on the peak and saturation velocities of the two materials.Asides from shining new light on the (U)WBG carrier distribution in high fields, our analysis indicates that POP scattering is responsible for the higher peak or saturation velocity in c-GaN over the effect of ODP scattering in diamond.
where e is the electron charge, ⃗ F 0 is the electric field, f ( ⃗ k) is the distribution function (DF), ℏ is the Planck constant, and is the scattering rate from the state ⃗ k ′ to ⃗ k, where i denotes the specific scattering mechanism.
Fig. 1 sketches the physical model that was used.In our approach, we partition the k-space in concentric regions limited by energy surfaces E(k) = nℏω op , where n = 1, 2, . . .and ω op is the optical phonon frequency.In isotropic materials, these surfaces are concentric Debye spheres, for which the carrier momentum is defined as where k c is a critical momentum and m * c is the conductivity effective mass.Because of the large value of the optical phonon energy in both materials (i.e., 92 meV for c-GaN and 163 meV for diamond, see Table I), both much larger than k B T , we can restrict our analysis to the lowest two energy regions of the k-space [i.e., 0 < E(k) ≤ ℏω op and ℏω op < E(k) ≤ 2ℏω op divided by the first Debye sphere].In the former, optic phonon emission is prohibited, and optic phonon absorption can be ignored since the phonon occupation number N q = {exp(ℏω op /k B T ) − 1} −1 is relatively small at room temperature (i.e., 0.002 for diamond and 0.03 for c-GaN), while low-energy scattering mechanisms by ionized and neutral impurities (NI), acoustic deformation potential (ADP), and piezoelectric (PZ) phonons can be treated as elastic within a relaxation time approximation.Outside the Debye sphere [i.e., ℏω op < E(k)], optical phonon emission dominates, which scatters the high-energy carriers back into the low-energy region 0 < E(k) ≤ ℏω op .We apply this model to charge carriers in the central isotropic (energy lowest) -valley of the lowest c-GaN conduction band as well as the highest valence bands in c-GaN and diamond.For the multivalley conduction band of diamond that exhibits a sixfold X-like degeneracy for lowest electron energy around the Brillouin zone edge, we adopt an isotropic scattering model with an average density of states consisting of a geometric average of two transverse and one longitudinal effective masses [24], rather than considering individual interand intravalley scattering.In this single valley approximation, the relationship between the energy E and the wave vector k is therefore parabolic with a conductivity effective mass obtained from harmonic average [23].
Assuming the electric field is in the positive z-direction, (1) can be split into two equations in each of the energy regions [20], [21], [22] eF where f < and f > denotes the DF inside and outside of the first Debye sphere, respectively, while f 0 denotes the even function part of f < that satisfies the detailed balance for elastic scattering mechanisms and that can be approximated by an M-B distribution with the carrier temperature T c (≥T ): Here, F is the magnitude of the electric field, τ is the total momentum relaxation time for lowenergy scattering, and S op ( ⃗ k ′ , ⃗ k) is the optical phonon emission scattering rate.
The relaxation time for low-energy ionized impurity (ii) scattering reads [23] 1 as derived from the momentum-relaxation time approximation, where n ii is the ionized impurity concentration, ε 0 and ε s are the vacuum permittivity and the dielectric constant, respectively, k is the magnitude of the carrier momentum, and q s is the screening wave vector determined as the inverse of the Debye length.Phosphorus (P) donors and boron (B) acceptors are considered in diamond, while silicon (Si) donors and magnesium (Mg) acceptors are considered in c-GaN.For ADP scattering, we use [25] 1 where E ac is the ADP, T is the lattice temperature, ρ is the crystal density, and v s is the longitudinal sound velocity.
Here, it is assumed that carrier energies are much smaller than E ac (≥6.2 eV, see Table I) and equipartition approximation holds, so that the scattering is nearly elastic [25].NI plays an important role in diamond, as most dopants are neutral even at room temperature due to the high activation energy (≥300 meV) of dopants.Neutral impurity (NI) scattering is empirically given by the following equation [26]: where N N denotes the NI density, w = E Inc /E B , where E Inc denotes the energy of the incident carriers and E B denotes the binding energy of dopants.Finally, PZ scattering limits the low-field drift velocity in c-GaN because the Ga-N bond polarity is high compared to other III-V materials.The momentum relaxation rate for PZ scattering reads [25] 1 where e 14 is the PZ coefficient.The total momentum relaxation time is calculated by averaging total relaxation rate over the distribution: Carrier-carrier scattering is neglected in this analysis.
In the energy region outside the Debye sphere, elastic scattering mechanisms are much weaker than optical phonon emission and are neglected.In diamond, scattering occurs through intervalley (IV) phonons [ODP and zone-edge ADP, where phonon energy is assumed constant to the weighted average of f and g scattering [24] (see Table I)] for electrons and intravalley ODP for holes.The scattering rate where E and E ′ denote the energy of the state ⃗ k and ⃗ k ′ , D 0 is the effective ODP, and V ol is the volume of the material.In c-GaN, the intravalley POP scattering rate S POP ( ⃗ k ′ , ⃗ k) for electrons and holes reads [25]  where ε ∞ and ε s are the dynamic and static dielectric constants, respectively.IV ODP scattering for electrons in c-GaN is neglected for the energy range of interest of this study as we limit the validity of our study to electric field up to 100 kV/cm, for which nearly all the electrons occupy the -valley [16].Similarly, intravalley ODP scattering for holes in c-GaN is neglected because the POP scattering rate is about two orders higher than the ODP scattering rate in this field range.
Fig. 2 plots the scattering rate as a function of the carrier momentum magnitude for n-type and p-type diamond and c-GaN.For all materials, impurity scattering and PZ scattering peaks at low k to decrease as a function of k, while ADP scattering rate monotonically increases with k.For k > k c , high energy optical phonon scattering dominates other mechanisms in all four materials, validating our approach that ignores elastic scattering outside the Debye sphere.
As (2b) is homogenous in f > (k), it can be integrated directly that gives where denotes the DF at the Debye sphere boundary, whereas ϒ(k z , k ⊥ ) is a decay function caused by phonon emission outside the Debye sphere.For diamond ODP where a ODP = D 2 0 m * /(2ρω op π eFℏ) and is the k z value on the Debye sphere.For POP in c-GaN Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.[11], [15], [17].For holes in c-GaN, there are no data reported up to the saturation. where Here, expansion at k = k c is used to obtain an analytic expression.Now, substituting the explicit solution for f > (k) of (9a) into the Boltzmann equation inside the Debye sphere (2a) yields the following equation: Again, for diamond and for c-GaN (12b-2), as shown at the bottom of the next page.
As 10) yields an equation implicit in f b (k 0 z ) that is solved numerically by iterations.Then, f b (k 0 z ) generates the DF over the entire defined area via (9a) and (10).The carrier temperature T c = 2 ⟨E⟩/3k B and the various momentum relaxation times are iterated self-consistently over the distribution.
The parameter sets used in this calculation for both diamond and c-GaN are displayed in Table I.Optical phonon energy and ODP of diamond electrons are effective values, considering IV scattering at different energies in a similar way as ODP scattering [24].ADP values for diamond electrons are the effective values from uniaxial and dilatation ADPs [25], [33].

III. RESULTS AND DISCUSSION
Fig. 3(a) displays the calculated diamond electron and hole drift velocity as a function of electric field compared with the experimental results for natural diamond [11] and CVD-grown diamond [15].Our model uses nitrogen donors with 1.62-eV activation energy [36] in natural diamond without phosphorus and boron.The agreement between the theoretical calculations and the experiments is excellent.Fig. 3(b) shows our results for the c-GaN electron velocity compared with calculations in [17], which also shows a good agreement between our model and the literature.The holes in c-GaN could not be compared with literature as there is no data regarding the field versus drift velocity reported until the saturation.
Fig. 4 shows the carrier drift velocity curves obtained from our model for P-doped diamond, B-doped diamond, Si-doped c-GaN, and Mg-doped c-GaN at different doping densities, where it can be seen that our results fit the Canali empirical formula where v is the drift velocity, µ is the carrier mobility, F is the electric field, and v sat is the saturation velocity or peak velocity (in c-GaN electrons).The saturation (or peak) velocities are 1.9, 1.6, 2.7, and 1.6 × 10 7 cm/s, for electrons in diamond, holes in diamond, electrons in c-GaN, and holes in c-GaN, respectively.Table II summarizes the extracted carrier mobilities as a function of doping density as well as the saturation velocity of each material.The high mobility of electrons in diamond at low impurity concentration higher than the conventional value of 1000-2000 cm 2 ] agrees well with the highest mobility of 4500 cm 2  •V −1 •s −1 reported in [14].Still, the highest mobility of holes in diamond is limited by ADP scattering to the value of 1820 cm 2

TABLE II EXTRACTED MOBILITIES AT DIFFERENT DOPING CONCENTRATIONS
AND THE SATURATION VELOCITY in [14], which suggests lower experimental ADP value in real diamond.The electron mobility in c-GaN with doping concentration lower than 10 16 cm −3 obtained in this work is higher than reported for hexagonal GaN (h-GaN) because of presence of additional longitudinal optical (LO) branch near 26 meV in h-GaN [19].In addition, our model does not include low-energy dislocation and defect scattering mechanisms that are abundant in experimental GaN.
All our mobility values fit into the following equation [13]: where µ min and µ max are the minimum and maximum mobility, respectively, N is the doping concentration, N c is the critical doping concentration, and γ is the fitting parameter.Here, µ min = 0 is assumed as µ min cannot be extracted from theoretical calculations.Fig. 5 compares the electron transport in diamond and c-GaN, for which the impurity concentration is fixed as 1.0 × 10 17 cm −3 in both materials.In Fig. 5(a), one can see that the drift velocity of c-GaN achieves higher values than those of diamond at high field, even though the diamond optical phonon energy is larger than in c-GaN.This is due to the different nature of the optical phonon scattering between the two materials, i.e., ODP in diamond versus POP in c-GaN, which in the former randomizes the carrier momenta during collisions, while it favors forward momentum scattering in the latter.Indeed, if one normalizes the drift velocities with respect to k c , thereby normalizing their effective masses and optical phonon energies, the electron peak velocity in c-GaN reaches 0.584 k c , whereas their saturation velocity in diamond achieves a lower value to 0.526 k c .In order to quantify this optical phonon scattering effect, Fig. 5(b) displays the effective electron temperatures (i.e., distribution broadenings) in the direction perpendicular T c⊥ and parallel T cz to the electric field that are defined by the following equation: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
resulting from a DF that follows an anisotropic drifted M-B distribution The factor of 1/2 in T c⊥ accounts for the fact that ( 16) is considering the average energy in two dimensions perpendicular to the field.The data show that both effective T c s in c-GaN are lower than in diamond, indicating less distribution spreading in the perpendicular direction as well as in the parallel direction with more carriers characterized by a large momentum toward the field direction for the former than the later material.As mentioned above, this is caused by the preferably forward POP scattering in c-GaN that overall lowers the carrier temperature compared to randomizing ODP scattering.This "cooling" effect due to the POP scattering was already reported earlier [37].Fig. 5(c) displays the contour plot of the electron distribution at 20 kV/cm for diamond and c-GaN that resemble ellipsoids or ovals elongated toward the field direction.It can be seen that the distributions resemble gaussian distributions in the k ⊥ -direction with rotational symmetry around the k z -axis, where they peak near the average carrier momentum, as a result of carrier acceleration along the field balanced by optical phonon scattering.As mentioned earlier, the distributions in both diamond and c-GaN can then be approximated by drifted M-B distributions instead of semi Fermi circles as previously described [38].At the same electric field, one observes a striking difference between the two M-B distributions due to the abovementioned difference between nonrandomizing POP scattering in c-GaN and randomizing ODP scattering in diamond.In the latter, the larger carrier spreading in both the k z -and k ⊥ -directions results in lower drift velocity because the balance between the drifted and scattered carriers is achieved at lower energy than c-GaN.From a general standpoint, in addition to the small carrier effective mass and the large optical phonon energy, the polar nature of the optical phonon in III-N semiconductors is one of the fundamental reasons why they exhibit high peak drift velocity.
IV. CONCLUSION In conclusion, we compare the transport properties of diamond and c-GaN with an accurate, fast semi-analytic method to calculate the carrier drift velocity as a function of the electric field in (U)WBG semiconductors.In this process, mobilities and saturation velocities of diamond and c-GaN at various doping concentrations are reported.Our analysis shows that the POP scattering mechanism in c-GaN decreases the carrier temperature making the carrier drift velocity higher at the steady state, contrary to diamond where randomizing ODP scattering governs the transport properties.
It is worth noting that this method, despite ignoring carriercarrier scattering, shows realistic results for (U)WBG materials due to their large activation energies and can be generally utilized for emerging cubic (U)WBG materials, such as cubic AlN or cubic BN with the appropriate material parameters.The findings in this work highlight cubic AlN and cubic BN as potentially high-saturation-velocity materials.

Fig. 1 .
Fig. 1.Schematic evolution of the carrier distribution under electric field in the Debye sphere.1) equilibrium distribution.2) Drift of the distribution in the field and optical phonon emission outside the Debye sphere.3) Carrier repopulation into the Debye sphere from phonon emission.

Fig. 2 .
Fig. 2. Scattering rates for electrons and holes in diamond and c-GaN as a function of carrier momentum k for ii, ADP, NI, PZ (in c-GaN), IV (in P-doped diamond), ODP (in B-doped diamond), and POP (in c-GaN) scattering.(a) Electron scattering for P-doped diamond, (b) hole scattering for B-doped diamond, (c) electron scattering for Si-doped c-GaN, and (d) hole scattering for Mg-doped c-GaN.Doping concentration is 1.0 × 10 17 cm −3 for all cases.

Fig. 3 .
Fig. 3. (a) Electron and hole drift velocities in diamond and (b) c-GaN as a function of electric field.The calculated results are compared with the experiments and other existing theoretical data[11],[15],[17].For holes in c-GaN, there are no data reported up to the saturation.

Fig. 4 .
Fig. 4. (a) Electron drift velocity in phosphorus-doped diamond, (b) hole drift velocity in boron-doped diamond, (c) electron drift velocity in silicon-doped c-GaN, and (d) hole drift velocity in magnesium-doped c-GaN.Different doping levels are indicated above.

Fig. 5 .
Fig. 5. Comparison of electronic transport in diamond and c-GaN.(a) Electron drift velocities as a function of electric field, (b) effective k B T c (i.e., broadening of the DF) in the direction parallel and perpendicular to the electric field, and (c) contour plots of the electron DF as a function of ⃗ k/k c .Both materials are doped with the doping density of 1.0 × 10 17 cm −3 .

TABLE I PARAMETER
SET FOR THE CALCULATION [23]PHYSICAL MODELThe steady-state Boltzmann equation for a nondegenerate, homogeneous semiconductor reads[23] Table III summarizes the results.