Negative Hall Factor of Acceptor Impurity Hopping Conduction in p-Type 4H-SiC

Abstract

Experimental data obtained from temperature-dependent Hall-effect measurements on Al-doped p-type 4H-SiC samples, which exhibit an anomalous sign reversal of the Hall coefficient to negative at low temperatures, are analyzed on the basis of a previously proposed impurity hopping conduction model. According to the small-polaron theory for the nonadiabatic case, the activation energy E₃ for the drift mobility of nearest-neighbor hopping is deduced, taking into account the temperature dependence of the preexponential factor. Existing models for the sign of the Hall coefficient are critically examined. It is shown that the anomalous sign reversal of the Hall coefficient can be well explained by assuming a hopping Hall factor in the form $A_{{\rm H}3} = \left( {k_{\rm B} T/J_{3} } \right)\exp \left( {K_{{\rm H}} E_{3} /k_{\rm B} T} \right)$ with a negative sign of J₃.

Appendix: Calculation Methods for Concentration, Mobility, and Hall Factor of Free Holes

The concentration n_v of free holes is calculated by $n_{\rm v} = N_{\rm v} {\fancyscript{F}}_{1/2} \left( \eta \right)$, where $N_{\rm v} = 2\left( {2\pi m_{\rm d} k_{\rm B} T/h^{2} } \right)^{3/2}$ is the effective DOS of the valence band, η is the reduced Fermi level with respect to the edge of the valence band, and ${\fancyscript{F}}_j \left( \eta \right)$ is the normalized Fermi–Dirac integral of order j. In the present study, we use the temperature-dependent DOS effective masses, rather than the temperature-independent DOS effective masses, for free holes in p-type 4H-SiC, as in the study by Koizumi et al.71 The temperature-dependent DOS effective mass m_d(T) is defined as

$$ n_{\rm v} = 4\pi \left( {\frac{{2m_{d} (T)}}{{h^{2} }}} \right)^{3/2} \int_{0}^{\infty } {f(E,T)E^{1/2} {\text{d}}E} , $$

(4)

where f(E, T) is the Fermi–Dirac distribution function, and has been obtained by Wellenhofer and Rössler91 for 4H-SiC. We find that the temperature-dependent DOS effective mass obtained by Tanaka et al.57 can be approximated by $m_{\rm d} \left( T \right) = 2.5m_{0} /\left[ {1 + \left( {24/T} \right)^{3/4} } \right]$ for p-type 4H-SiC in the temperature range between 100 K and 900 K.

Tanaka et al.57 proposed a model to calculate the Hall mobility μ_H and the drift mobility μ_d of p-type 4H-SiC and fit the calculated results to the experimental data on their almost noncompensated p-type 4H-SiC samples. Their model includes the effects of the anisotropic valence band structure and the anisotropic relaxation times. They showed that the drift mobility μ_d of almost noncompensated p-type 4H-SiC is dominated by acoustic and nonpolar optical phonon scattering at high temperatures while ionized impurity scattering has a negligible impact. Furthermore, through the fits to the temperature and acceptor density dependence of the mobility, they obtained empirical expressions for the drift mobility and Hall factor as

$$ \mu_{\rm d} (T,N_{\rm A} ) = \frac{{95\,{\text{cm}}^{2} /{\text{Vs}} \times \left( {\frac{T}{300\,\text{K}}} \right)^{ - 2.1} }}{{1 + \left( {\frac{T}{300\,\text{K}}} \right)^{ - 1.5} \left( {\frac{{N_{\rm A} }}{{1 \times 10^{19} \,\text{cm}^{ - 3} }}} \right)^{0.71} }} $$

(5)

and

$$ A_{{\rm Hv}} (T,N_{\rm A} ) = 1.16 \times \left( {\frac{T}{{300\,{\text{K}}}}} \right)^{ - 0.9} \times \frac{{1 + \left( {\frac{T}{{300\,{\text{K}}}}} \right)^{ - 1.5} \left( {\frac{{N_{\rm A} }}{{1 \times 10^{19} \,{\text{cm}}^{ - 3} }}} \right)^{0.7} }}{{1 + \left( {\frac{T}{{300\,{\text{K}}}}} \right)^{ - 1.8} \left( {\frac{{N_{\rm A} }}{{3 \times 10^{18} \,{\text{cm}}^{ - 3} }}} \right)^{0.6} }}, $$

(6)

respectively. The former expression agrees with both μ_H/A_H and μ_Hn_H/n_v calculated using their model within about $\pm$10% error in the range of 200 K to 700 K. At lower temperature, however, this expression slightly overestimates the mobility. Furthermore, they suggested that larger errors may arise for higher acceptor concentrations. They also noted that, in the case of highly compensated samples, this expression will require some modification to take account of the compensating donor concentration. Such errors at lower temperatures for higher acceptor concentrations as well as the modification required for the highly compensated case can arise from the effect of ionized impurity scattering.

In the present study, therefore, the effect of impurity scattering is explicitly included in order to calculate the drift mobility μ_v of free holes in moderately compensated heavily doped p-type 4H-SiC at low temperatures. Namely, the drift mobility μ_v is calculated as $1/\mu_{\rm v} = 1/\mu_{\text{phonon}} + 1/\mu_{\rm imp}$, where μ_phonon and μ_imp denote the lattice- and impurity-limited mobility, respectively. We regard μ_d(T, 0) as μ_phonon. The impurity-limited mobility is calculated as $1/\mu_{\rm imp} = 1/\mu_{\rm ii} + 1/\mu_{\rm ni}$, where μ_ii and μ_ni denote the mobility due to ionized- and neutral-impurity scattering, respectively.

Tanaka et al.57 calculated conductivity tensors and the drift mobility on the basis of the relaxation-time approximation. In Ref. 57, the relaxation time τ_ii due to ionized-impurity scattering was calculated using the Brooks–Herring formula. Then, Tanaka et al.57 multiplied the relaxation time τ_ii by a factor of 2 to take into account the p-type symmetry of wavefunctions in place of the s-type wavefunctions, while Pernot et al.36 as well as Koizumi et al.71 multiplied it by a factor of 3/2. However, Poklonski et al. showed that the Brooks–Herring formula overestimates the mobility (and thus the relaxation time τ_ii) for p-type Si92 as well as for n-type InSb,93 even when the value is not multiplied. When using the Brooks–Herring formula to calculate the hole mobility, Lowney and Bennett94 preferred not to multiply τ_ii by the overlap factor because of the relative strength of small-angle scattering and the fact that the overlap factor due to the p-wave nature of holes goes to unity for small angles. In the present study, therefore, the relaxation time τ_ii due to ionized-impurity scattering is calculated using the Brooks–Herring formula without any multiplication, as in previous studies by the author on p-type materials.35,47

On the other hand, for the calculation of τ_ii, we take into account the effect of the increase of the static dielectric constant at the insulator side of the MI transition. It has been shown both experimentally and theoretically that the static dielectric constant at low temperature increases with the impurity concentration to diverge at the critical net acceptor concentration N_NAcr for the onset of the MI transition.95,96 According to Poklonski et al.,96 we assume the form

$$ \varepsilon_{\rm eff} (N_{\rm A}^{0} ) = \left( {\varepsilon_{\rm s} + 2{{N_{\rm A}^{0} } \mathord{\left/ {\vphantom {{N_{\rm A}^{0} } {N_{\rm NAcr} }}} \right. \kern-0pt} {N_{\rm NAcr} }}} \right)\left( {1 - {{N_{\rm A}^{0} } \mathord{\left/ {\vphantom {{N_{\rm A}^{0} } {N_{\rm NAcr} }}} \right. \kern-0pt} {N_{\rm NAcr} }}} \right)^{ - 1} . $$

(7)

Regarding the critical concentration for the MI transition in Al-doped p-type 4H-SiC, we adopt the value N_NAcr = 2.1 × 10²⁰ cm⁻³ according to Persson et al.,97 and calculate ε_eff according to Eq. 7.

In the calculation of the energy-dependent relaxation time τ_ii(E) due to ionized-impurity scattering, Tanaka et al.57 as well as Koizumi et al.71 and Parisini et al.83 took into account not only screening due to free holes but also that due to hopping carriers98 when calculating the inverse screening length β_s. (Note that there are nontrivial errors related to the Fermi–Dirac integral in Eq. A2 for β_s in Ref. 83.) In the present study, however, only screening due to free holes is taken into account when calculating τ_ii(E) of free holes without the effect of hopping carriers. To calculate the drift mobility, it is necessary to average τ_ii(E) by the integral of E. The Brooks–Herring formula for τ_ii(E) contains a factor $B_{\rm ii} (b) = \ln (b + 1) - {b \mathord{\left/ {\vphantom {b {(b + 1)}}} \right. \kern-0pt} {(b + 1)}}$, where $b = {{8m_{\rm d} E} \mathord{\left/ {\vphantom {{8m_{\rm d} E} {\left( {\hbar \beta_{\rm s} } \right)^{2} }}} \right. \kern-0pt} {\left( {\hbar \beta_{\rm s} } \right)^{2} }}$. Although the factor B_ii(b) is a function of E, its dependence is slow. Owing to this, B_ii(b) can be replaced before the integral of EdE by a constant value of B_ii(b_max), where b_max represents the value of b at which the integral becomes maximum. One can calculate b_max as 99

$$ b_{\hbox{max} } = \frac{{4\pi \varepsilon_{\rm eff} (k_{\rm B} T)^{1/2} h}}{{e^{2} (2m_{\rm d} )^{1/2} }}\frac{5}{\sqrt \pi }{\fancyscript{F}}_{3/2} \left( \eta \right)/\left[ {{\fancyscript{F}}_{ - 1/2} \left( \eta \right){\fancyscript{F}}_{1/2} \left( \eta \right)} \right]. $$

(8)

Then, the drift mobility due to ionized-impurity scattering can be calculated as

$$ \mu_{\rm ii} = \frac{{\left( {4\pi \varepsilon_{\rm eff} } \right)^{2} (k_{\rm B} T)^{3/2} }}{{N_{\rm ii} \pi e^{3} (2m_{\rm d} )^{1/2} B_{\rm ii} (b_{\hbox{max} } )}}2{\fancyscript{F}}_{2} \left( \eta \right)/{\fancyscript{F}}_{1/2} \left( \eta \right). $$

(9)

The calculation of μ_ni is performed according to Erginsoy’s model; Namely, it is calculated as

$$ \mu_{\rm ni} = \frac{e}{{20a_{\rm B} \hbar }}\frac{{{{m^{*} } \mathord{\left/ {\vphantom {{m^{*} } {m_{0} }}} \right. \kern-0pt} {m_{0} }}}}{{4\pi \varepsilon_{\rm eff} \varepsilon_{0} N_{\rm A}^{0} }}, $$

(10)

where $a_{\rm B} = 4\pi \varepsilon_{\rm eff} \varepsilon_{0} \hbar^{2} /m^{*} e^{2}$ is the Bohr radius describing the bound hole at the neutral acceptor. Tanaka et al.57 treated the hole effective mass in Eq. 10 as an adjustable parameter which is independent of the DOS and conductivity effective masses, assuming it to be 1.0 × m₀ to reproduce the experimental mobility at low temperatures and high concentrations of acceptors in p-type 4H-SiC. In the present study, the hole effective mass for the calculation using Eq. 10 is also assumed to be 1.0 × m₀.