Calculation of the electron Hall mobility and Hall scattering factor in 6 H-SiC

The calculated electron Hall Mobility and Hall scattering factor in n-type 6H-SiC based on numerical solutions to the Boltzmann transport equation are presented. These results were obtained by solving the Boltzmann equation exactly for the electron Hall mobility using the contraction mapping principle and the electron drift mobility with Rode’s iterative method. The relative importance of the various scattering mechanisms for these calculations is discussed, and a new set of values for the acoustic deformation potential, intervalley deformation potential, and intervalley phonon energy is found, which simultaneously fit experimental Hall mobility and Hall scattering factor data. The calculated Hall mobility and Hall scattering factor are in good agreement with experimental results. In addition, predicted values of the Hall scattering factor for various temperatures and doping concentrations are given. © 2009 American Institute of Physics. doi:10.1063/1.3212532


I. INTRODUCTION
During the past few decades, the various polytypes of silicon carbide ͑SiC͒ have been intensively studied in the hope that it will become a suitable material for high power and high temperature electronic applications.3][4] However, for any practical realization of electronic devices on 6H-SiC to occur, certain transport properties need to be known.One such quantity is the electron mobility, and, to this end, experimental studies have been carried out to determine its value. 5,6][9][10] Although these computational studies had adequate success in explaining the experimental results, all of these calculations have either ͑i͒ focused on calculating the electron drift mobility, while the experimental work has usually measured the electron Hall mobility; or ͑ii͒ been inexact solutions based on the relaxation time approximation.Comparisons between the Hall mobility and drift mobility have been justified by either assuming a Hall scattering factor of unity, or through a calculation of the Hall scattering factor, which also relied on the relaxation time approximation.Experiments have shown that the former assumption can be off by as much as 19% depending on temperature and doping concentration. 11Meanwhile, Hall scattering factors calculated with the relaxation time approximation differ significantly from experimental data. 10n this paper a more accurate calculation of the electron Hall mobility and Hall scattering factor for 6H-SiC is presented by calculating the electron Hall mobility exactly using the contraction mapping principle. 12At the same time, the low-field electron drift mobility is solved exactly using the iterative method of Rode, [13][14][15] and the Hall scattering factor is then arrived at by taking the ratio of the two values multiplied by a mass factor.The various scattering mechanisms are discussed, and a single set of values for the acoustic deformation potential, the intervalley deformation potential, and the intervalley phonon energy, which are adjusted to simultaneously match the experimental Hall scattering factor and Hall mobility data, are given.The calculated electron Hall mobilities and Hall scattering factors are compared with the experimental data of Karmann et al. 5 and Rutsch et al., 11 respectively.First, however, a concise overview of the techniques used to solve the Boltzmann transport equation is given.

II. TECHNIQUES FOR SOLVING THE BOLTZMANN TRANSPORT EQUATION
Under steady-state, spatially homogeneous conditions with Fermi-Dirac statistics, the Boltzmann transport equation is given by [13][14][15] − qF

͑1͒
where q is the electron charge, F is the electric field, f k is the distribution function as a function of wave vector k, and S k,k Ј and S k Ј ,k are the transition rates between states k and kЈ.By writing the distribution function as [13][14][15] where f 0 is the Fermi-Dirac distribution, g k is the perturbed distribution function, and x is the direction cosine between F and k, and inserting into Eq.͑1͒, an iterative form for g k can be found.This is given by [13][14][15] g k,i+1 = and can be solved iteratively for the unknown distribution function by using an initial guess for g k .The operators S o and S i ͑g k,i ͒ are the out-scattering and in-scattering rates, respectively.The out-scattering rate is [13][14][15] S o = v el + 1/ in , ͑4͒ where v el is the sum of the momentum relaxation rates of elastic processes, and 1 / in is the sum of the relaxation rates of inelastic processes.The in-scattering rate is 13-15 where X is the angle between the initial and final wave vectors, and S k Ј ,k inel and S k Ј ,k inel are the transition rates of the inelastic processes.With the perturbation distribution function g k solved to arbitrary accuracy, the drift mobility is [13][14][15] = − ប where is the electron drift mobility, ប is the reduced Planck constant, and m ‫ء‬ is the effective mass of the electron.This iterative technique to solve for the mobility in the presence of a low electric field was first discussed by Rode in 1973. 13 similar technique, also introduced by Rode, 12 solves the Boltzmann transport equation in the presence of an arbitrary magnetic field.In this case, the distribution function is given by 12 where h k is the perturbation distribution function due to the magnetic field, and y is the direction cosine from B ϫ F to k. Insertion of Eq. ͑7͒ into Eq.͑1͒ yields a coupled pair of equations that can be solved iteratively as 12 g k,i+1 = where ␤ = qvB / បkS o , v is the electron group velocity, and B is the magnetic field.Using these two perturbed distribution functions, the electron Hall mobility H is 12 The iterative sequences of Eqs.͑8͒ and ͑9͒, as well as Eq.͑3͒, have been shown to be contraction mappings, which guarantee converging solutions. 12ith the drift mobility and Hall mobility determined, the Hall scattering factor r H is This solution is a more accurate calculation of the Hall scattering factor compared to the more commonly used expression where ͗͗ 2 ͘͘ and ͗͗͘͘ are the specially averaged scattering times.The latter expression for the Hall scattering factor is derived from the relaxation time approximation, which assumes that scattering times are independent of the distribution function.More importantly, the relaxation time approximation is only valid when the scattering processes are either elastic or isotropic, which is not true for 6H-SiC where polar optical phonon scattering dominates the low-field mobility ͑this scattering mechanism is neither elastic nor isotropic͒.

III. SCATTERING MECHANISMS IN 6H-SIC
In order to calculate S i and S o and properly determine the Hall and drift mobilities, the relevant scattering mechanisms must be taken into account.For this study, six scattering mechanisms are considered: ͑i͒ acoustic deformation potential scattering, ͑ii͒ ionized impurity scattering, ͑iii͒ neutral impurity scattering, ͑iv͒ piezoelectric scattering, ͑v͒ intervalley phonon deformation potential scattering, and ͑vi͒ polar optical phonon scattering.The values of the acoustic deformation potential, intervalley phonon deformation potential, and intervalley phonon energy were taken as adjustable parameters to simultaneously fit the Hall scattering factor and Hall mobility data.In addition, parabolic conduction bands are assumed, with the values of the effective masses being m M⌫ ‫ء‬ ͑m 1 ‫ء‬ ͒ = 0.75m 0 , m MK ‫ء‬ ͑m 2 ‫ء‬ ͒ = 0.24m 0 , and m ML ‫ء‬ ͑m 3 ‫ء‬ ͒ = 1.83m 0 , and the density of states effective mass is 16 Interaction of electrons with nonpolar acoustic phonons results in the acoustic deformation potential scattering.The momentum relaxation rate is given by [13][14][15]17 where D ac is the acoustic deformation potential, E k is the carrier energy, v S is the longitudinal acoustic velocity, and is the mass density.For 6H-SiC, v S = 13 730 m / s 2 and = 3211 kg/ m 3 . 18As previously stated, D ac is treated as an adjustable parameter to fit experimental data ͑discussed later͒ and found to be 5.5 eV.Coulomb scattering by ionized impurities, important at low temperatures and high doping concentrations, is handled through the Brooks and Herring approach.The momentum relaxation rate is given by [13][14][15]17 where N I is the ionized impurity concentration, 0 is the low frequency dielectric constant, 0 is the permittivity of free space, and where k B is the Boltzmann constant and T is the temperature.For 6H-SiC, the value of the relative dielectric constant is 0 = 9.7. 18ecause of the large ionization energies of nitrogen donors in 6H-SiC, neutral impurity scattering makes a considerable contribution to the total scattering rate, especially at low temperatures and high doping concentrations.For neutral impurity scattering, the form proposed by Sclar was used, with the momentum relaxation rate given by 17 where the binding energy is E T = 0.75͑m ‫ء‬ / m͒͑1 / 0 ͒ 2 eV.Silicon carbide crystals lack inversion symmetry, and are therefore piezoelectric. 19,20Thus polar acoustic interactions play a role, especially at lower temperatures.The momentum relaxation rate for this process is given by 13,15,17

͑17͒
For the mobility with the electric field perpendicular to the c axis of the crystal, the coupling coefficient K for wurzite structures is The last two scattering mechanisms, intervalley phonon scattering and polar optical phonon scattering, are inelastic processes.Therefore, in addition to the scattering rates, the integral of Eq. ͑5͒ must be evaluated for each of the transition rates.The relaxation rate for intervalley phonon scattering is given by [13][14][15]17 where Z is the number of final valleys, D if is the intervalley deformation potential, ប if is the intervalley phonon energy, N i is the intervalley phonon occupation number determined by Bose-Einstein statistics, and ⌬E fi is the energy separation between the bottom of the conduction bands of the final and initial valleys.Here we only consider equivalent intervalley scattering, so Z = 4 and ⌬E fi =0. 22As will be discussed later, fits of experimental data yield D if = 1.25ϫ 10 11 eV/ m and ប if = 65 meV.For the in-scattering operator, the integral of Eq. ͑5͒ vanishes for intervalley phonon scattering. 13inally, for polar optical phonon scattering, the momentum relaxation rate is given by [13][14][15]17

͑22͒
where ϱ is the high frequency dielectric constant, ប o is the polar optical phonon energy, and N o is the polar optical phonon occupation number.The polar optical phonon energy is taken to be 120 meV.For the in-scattering operator, the result of Eq. ͑5͒ for the polar optical phonon transition rate is 13

͑25͒
To see the relative importance of each of the scattering mechanisms in n-type 6H-SiC, the scattering rates for doping concentrations of N D =10 16 cm −3 and N D =10 19 cm −3 are shown in Figs. 1 and 2, respectively, as a function of energy for two different temperatures.Previous studies have indicated that the ratio of impurities at cubic sites to impurities at hexagonal sites is about 2:1. 5,6,11Therefore, for both of these plots, it is assumed that there are twice as many impurities at cubic sites than there are at cubic sites, and that there is no compensation.The free electron concentration is found by solving the charge neutrality equation where n is the electron concentration, N A is the acceptor compensation level, E C is the conduction band minimum, E F is the energy of the Fermi level, N DH and N DK are the concentrations of impurities at hexagonal and cubic sites, re-spectively, g H and g K are the donor degeneracy levels of the hexagonal and cubic levels, respectively, and E DH and E DK are the ionization energies of the impurities at hexagonal and cubic sites, respectively.Both the donor degeneracy levels are assumed to be 2.For this example, the donor ionization energies were set at 94 and 118 meV for the hexagonal and cubic sites, respectively.The lattice scattering in these plots includes acoustic deformation potential, intervalley phonon, and polar optical phonon scattering.At lower temperatures for both doping concentrations, neutral impurity scattering is significant for lower energy carriers, necessitating its inclusion as a scattering mechanism.In addition, at lower temperatures, piezoelectric scattering is the strongest scattering mechanism at lower doping concentrations while still being significant at higher doping concentrations for low energy carriers.][9][10] At higher temperatures, ionized impurity scattering generally dominates.For lower doping concentrations, lattice scattering makes a significant contribution, while at higher doping concentrations, neutral impurity scattering is again important.

IV. ELECTRON HALL MOBILITY AND HALL SCATTERING FACTOR
Using Eq. ͑11͒ and the techniques discussed in Sec.II, and including the scattering mechanisms discussed above, the Hall mobility and the Hall scattering factor were calculated and were fitted to the experimental data measured by Karmann et al. 5 and Rutsch et al. 11 These were calculated assuming a measurement configuration of ͓B ʈ c , j Ќ c͔ and with magnetic fields of B = 0.4 T and B = 0.741 T, as in the experiments.Because of the measurement configuration, a mass factor is included in the Hall scattering factor. 10s previously mentioned, the values of the acoustic deformation potential, intervalley deformation potential, and intervalley phonon energy were adjusted to obtain a best fit for the Hall mobility and Hall scattering factor data simultaneously.In a manner similar to previous studies, 10,23 the parameters were first adjusted to the Hall mobility data, and then subsequently used in the calculation of the Hall scattering factor data.For the most part, adjustment of the acoustic deformation potential had the effect of shifting the Hall mobility curve higher or lower.Meanwhile, adjustment of the intervalley deformation potential and the intervalley phonon energy primarily affected the slope of the data at temperatures above 100 K. Figure 3 shows the comparison of the calculated and the experimental Hall mobility for three different samples, while Fig. 4 shows the comparison of the calculated and the experimental Hall scattering factor.The calculated results fit the experimental data very well.This is particularly true for the calculated Hall scattering factor, which matches experimental data more so than calculations with a relaxation time approximation by Iwata and Itoh. 10   The deviation of the calculated Hall scattering factor in this study from experimental data never exceeds 8.8%.
However, the aforementioned fitting results in values of D ac = 5.5 eV, D if = 17.5 eV/ m, and ប if = 65 meV, which diverge drastically from previously reported values of D ac = 17.5 eV, D if = 0.6ϫ 10 11 eV/ m, and ប if = 85.4 meV. 7,9his warrants some discussion.The first study to attempt to extract the deformation potentials and intervalley phonon energy ͑from which the latter set of parameters originated͒ by Mickevicius and Zhao 7 used a Monte Carlo method to calculate the electron drift mobility.This calculated mobility was then fitted to experimental Hall data without accounting for the Hall scattering factor.A subsequent study by Dhar and Ghosh 9 used the same set of parameters in a calculation of the drift mobility using Rode's iterative method.The Hall scattering factor in this case was calculated using Eq.͑12͒, which is an inexact solution, and without using a mass factor to account for the anisotropy of the Hall measurement configuration.It is no surprise that the same set of parameters yields similar results for both the Monte Carlo and Rode's iterative method, as both are recognized as exact methods of obtaining the electron drift mobility.Neither method, however, solves for the Hall mobility.It can therefore be argued that the more accurate parameters are the ones obtained in this study.
Because of the large deviations of the Hall scattering factor from unity and its importance, it is clear that in order to deduce the electron drift mobility from experimental Hall mobility data the Hall scattering factor must be accurately known.Figure 5 shows predicted values of the Hall scattering factor against temperature at four different doping concentrations for B = 0.5 T. These calculated values assume a 2:1 ratio of impurities at cubic sites to impurities at hexagonal sites, and a compensation ratio of N A / N D = 0.01.These curves show that the doping concentration and the temperature both greatly impact the Hall scattering factor for the ͓B ʈ c , j Ќ c͔ Hall measurement configuration, so that extreme care must be taken when trying to extract the electron drift mobility from Hall measurements in 6H-SiC.

V. SUMMARY
As previous experimental works have shown, the differences between the Hall mobility and drift mobility in 6H-SiC are not trivial.Nevertheless, the computational studies in this area have usually not completely addressed all the factors pertaining to these differences.This paper studied the Hall mobility and Hall scattering factor of 6H-SiC in detail and to a greater degree of accuracy than previous works.To this end, an examination of the importance of the various scattering mechanisms in 6H-SiC was carried out.Using the contraction mapping principle, the Hall mobility and Hall scattering factor were solved for exactly, and calculated to fit experimental data.From this fitting, a new set of values for the acoustic deformation potential, intervalley deformation potential, and intervalley phonon energy was determined.These values differed significantly from those of previous works, and this is attributed to the more accurate method of calculating the Hall mobility.Finally, the Hall scattering  factor at different temperatures for various impurity concentrations was calculated in order to aid the extraction of the electron drift mobility from experimental Hall measurements.

FIG. 5 .
FIG.5.Predicted values of the Hall scattering factor plotted against temperature with doping concentration as a parameter for n-type 6H-SiC.
21,15,17= e 33 − e 31 −2e15, and e 15 , e 31 , and e 33 are the piezoelectric constants.The averaged transverse and longitudinal elastic constants for wurzite structures, c t and c l , are13,15,17   11, c 13 , c 33 , and c 44 are the elastic constants.In this study, the piezoelectric constants calculated by Mirgorodsky et al.20are used, while the elastic constants measured by Kamitani et al.21are used.