Abstract
This chapter is devoted to the derivation of k⋅p multi-band quantum transport models, in both the pure-state and mixed-state cases. The first part of the chapter deals with pure-states. Transport models are derived from the crystal periodic Hamiltonian by assuming that the lattice constant is small, so that an effective multi-band Schrödinger equation can be written for the envelopes of the wave functions of the charge carriers. Two principal approaches are presented here: one is based on the Wannier-Slater envelope functions and the other on the Luttinger-Kohn envelope functions. The concept of Wannier functions is then generalized, in order to study the dynamics of carriers in crystals with varying composition (heterostructures). Some of the most common approximations, like the single band, mini-bands and semi-classical transport, are derived as a limit of multi-band models. In the second part of the chapter, the mixed-state (i.e. statistical) case is considered. In particular, the phase-space point of view, based on Wigner function, is adopted, which provides a quasi-classical description of the quantum dynamics. After a theoretical introduction to the Wigner-Weyl theory, a two-band phase-space transport model is developed, as an example of application of the Wigner formalism to the k⋅p framework. The third part of the chapter is devoted to quantum-fluid models, which are formulated in terms of a finite number of macroscopic moments of the Wigner function. For mixed-states, the maximum-entropy closure of the moment equations is discussed in general terms. Then, details are given on the multi-band case, where “multi-band” is to be understood in the wider sense of “multi-component wave function”, including therefore the case of particles with spin or spin-like degrees of freedom. Three instances of such systems, namely the two-band k⋅p model, the Rashba spin-orbit system and the graphene sheet, are examined.
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Notes
- 1.
Equation (1.36) has been rewritten here with a slightly different notation. Moreover the mass is set equal to 1
- 2.
- 3.
- 4.
In dimensionless variables, all the identities involving Weyl quantization are obtained from the original ones by the formal substitution ℏ↦ε.
- 5.
Of course, there are also examples of multi-band QFD equations with an arbitrary number of bands, see e.g. [96].
- 6.
For notational convenience we adopt “+” and “−” to denote “up” and “down” instead of the more common “ ↑” and “ ↓”.
- 7.
This shows clearly that, indeed, p is not a momentum but a “pseudomomentum” (or “crystal momentum”): the latter has rather different properties.
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Acknowledgements
This work was partially supported by the Italian Ministry of University (MIUR National Project “Kinetic and hydrodynamic equations of complex collisional systems”, PRIN 2009, Prot. n. 2009NAPTJF_003).
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Barletti, L., Frosali, G., Morandi, O. (2014). Kinetic and Hydrodynamic Models for Multi-Band Quantum Transport in Crystals. In: Ehrhardt, M., Koprucki, T. (eds) Multi-Band Effective Mass Approximations. Lecture Notes in Computational Science and Engineering, vol 94. Springer, Cham. https://doi.org/10.1007/978-3-319-01427-2_1
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