Electron Mobility in Bulk n-Doped SiC-Polytypes 3C-SiC, 4H-SiC, and 6H-SiC: A Comparison

Abstract

This communication presents a comparative study on the charge transport (in transient and steady state) in bulk n-type doped SiC-polytypes: 3C-SiC, 4H-SiC and 6H-SiC. The time evolution of the basic macrovariables: the “electron drift velocity” and the “non-equilibrium temperature” are obtained theoretically by using a Non-Equilibrium Quantum Kinetic Theory, derived from the method of Nonequilibrium Statistical Operator (NSO). The dependence on the intensity and orientation of the applied electric field of this macrovariables and mobility are derived and analyzed. From the results obtained in this paper, the most attractive of these semiconductors for applications requiring greater electronic mobility is the polytype 4H-SiC with the electric field applied perpendicular to the c-axis.

Appendices

APPENDIX A

COLLISION TERMS IN EQS. (6)–(9)

The detailed expressions for the collision operators in Eqs. (6)–(9) are:

$$\begin{gathered} {{J}_{{{{{\mathbf{P}}}_{{{\text{ph}}}}}}}}(t) = \sum\limits_{{\mathbf{k}},{\mathbf{q}},\ell ,\eta } {\frac{{2\pi }}{\hbar }\hbar {\mathbf{q}}{\text{|}}M_{\eta }^{\ell }({\mathbf{q}}){{{\text{|}}}^{2}}[{{\nu }_{{{\mathbf{q}},\eta }}}(t){{f}_{{\mathbf{k}}}}(t)} \\ \, \times (1 - {{f}_{{{\mathbf{k}} + {\mathbf{q}}}}}(t)) - {{f}_{{{\mathbf{k}} + {\mathbf{q}}}}}(t)(1 + {{\nu }_{{{\mathbf{q}},\eta }}}(t)) \\ \, \times (1 - {{f}_{{\mathbf{k}}}}(t))]\delta ({{\epsilon }_{{{\mathbf{k}} + {\mathbf{q}}}}} - {{\epsilon }_{{\mathbf{k}}}} - \hbar {{\omega }_{{{\mathbf{q}},\eta }}}), \\ \end{gathered} $$

$${{{\mathbf{J}}}_{{{{{\mathbf{P}}}_{{{\text{imp}}}}}}}}(t) = - \frac{{128\sqrt {2\pi m_{e}^{*}} {{{({{k}_{{\text{B}}}}T_{e}^{*}(t))}}^{{3/2}}}}}{{{{n}_{I}}{{{(\mathcal{Z}{{e}^{2}}{\text{/}}{{\epsilon }_{0}})}}^{2}}G(t)}}{{{\mathbf{P}}}_{e}}(t),$$

$$\begin{gathered} {{J}_{{E,{\text{ph}}}}}(t) = \sum\limits_{{\mathbf{k}},{\mathbf{q}},\ell ,\eta } {\frac{{2\pi }}{\hbar }{\text{|}}M_{\eta }^{\ell }({\mathbf{q}}){{{\text{|}}}^{2}}({{\epsilon }_{{{\mathbf{k}} + {\mathbf{q}}}}} - {{\epsilon }_{{\mathbf{k}}}})[{{\nu }_{{{\mathbf{q}},\eta }}}(t)} \\ \, \times {{f}_{{\mathbf{k}}}}(t)(1 - {{f}_{{{\mathbf{k}} + {\mathbf{q}}}}}(t)) - {{f}_{{{\mathbf{k}} + {\mathbf{q}}}}}(t) \\ \, \times (1 + {{\nu }_{{{\mathbf{q}},\eta }}}(t))(1 - {{f}_{{\mathbf{k}}}}(t))] \\ \, \times \delta ({{\epsilon }_{{{\mathbf{k}} + {\mathbf{q}}}}} - {{\epsilon }_{{\mathbf{k}}}} - \hbar {{\omega }_{{{\mathbf{q}},\eta }}}), \\ \end{gathered} $$

$${{J}_{{{\text{LO}},an}}}(t) = - \sum\limits_{\mathbf{q}} {\hbar {{\omega }_{{{\mathbf{q}},{\text{LO}}}}}\frac{{{{\nu }_{{{\mathbf{q}},{\text{LO}}}}}(t) - {{\nu }_{{{\mathbf{q}},{\text{AC}}}}}(t)}}{{{{\tau }_{{{\text{LO}},an}}}}}} ,$$

$${{J}_{{{\text{AC}},dif}}}(t) = - \sum\limits_{\mathbf{q}} {\hbar {{\omega }_{{{\mathbf{q}},{\text{AC}}}}}\frac{{{{\nu }_{{{\mathbf{q}},{\text{AC}}}}}(t) - {{\nu }_{0}}}}{{{{\tau }_{{ac,dif}}}}}} .$$

The quantities $M_{\eta }^{\ell }$(q) are the matrix elements of the interaction between carriers and branch η-type phonons (η = LO, AC for longitudinal optical and acoustical phonons, respectively), with supraindex ℓ indicating the kind of interaction (polar, deformation potential, piezoelectric, etc.); ν_q,η(t) and f_k(t) are, respectively, the phonons and electrons distribution; δ is delta function and ${{\epsilon }_{{\mathbf{k}}}}$ = ℏ²k²/2$m_{e}^{*}$; n_I is the density of impurities, ℒ the units of charge of the impurity, and

$$G(t) = \ln (1 + b(t)) - \frac{{b(t)}}{{1 + b(t)}},$$

where b(t) = 24${{\epsilon }_{0}}m_{e}^{*}{{[{{k}_{{\text{B}}}}T_{e}^{*}(t)]}^{2}}{\text{/}}{{n}_{I}}{{e}^{2}}{{\hbar }^{2}}$. Finishing, τ_LO,an is a relaxation time which is obtained from the band width in Raman scattering experiments [58], and τ_AC,dif is a characteristic time for heat diffusion, which depends on the particularities of the contact of sample and reservoir [59]. We notice that more details for the collision operators are given in [32].