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April  2022, 15(2): 257-282. doi: 10.3934/krm.2022007

Formal derivation of quantum drift-diffusion equations with spin-orbit interaction

1. 

Dipartimento di Matematica e Informatica "Ulisse Dini", Università degli Studi di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italia

2. 

Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8–10, 1040 Wien, Austria

*Corresponding author: Ansgar Jüngel

Received  September 2021 Revised  January 2022 Published  April 2022 Early access  March 2022

Fund Project: The first author acknowledges support by Italian National Group for Mathematical Physics (INdAM-GNFM). The last two authors have been partially supported by the Austrian Science Fund (FWF), grants P30000, P33010, F65, and W1245. This work received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme, ERC Advanced Grant NEUROMORPH, no. 101018153

Quantum drift-diffusion equations for a two-dimensional electron gas with spin-orbit interactions of Rashba type are formally derived from a collisional Wigner equation. The collisions are modeled by a Bhatnagar–Gross–Krook-type operator describing the relaxation of the electron gas to a local equilibrium that is given by the quantum maximum entropy principle. Because of non-commutativity properties of the operators, the standard diffusion scaling cannot be used in this context, and a hydrodynamic time scaling is required. A Chapman–Enskog procedure leads, up to first order in the relaxation time, to a system of nonlocal quantum drift-diffusion equations for the charge density and spin vector densities. Local equations including the Bohm potential are obtained in the semiclassical expansion up to second order in the scaled Planck constant. The main novelty of this work is that all spin components are considered, while previous models only consider special spin directions.

Citation: Luigi Barletti, Philipp Holzinger, Ansgar Jüngel. Formal derivation of quantum drift-diffusion equations with spin-orbit interaction. Kinetic and Related Models, 2022, 15 (2) : 257-282. doi: 10.3934/krm.2022007
References:
[1]

M. Ancona and G. Iafrate, Quantum correction to the equation of state of an electron gas in a semiconductor, Phys. Rev. B, 39 (1989), 9536-9540.  doi: 10.1103/PhysRevB.39.9536.

[2]

L. Barletti, P. Holzinger and A. Jüngel, Quantum drift-diffusion equations for a two-dimensional electron gas with spin-orbit interaction, In Recent Advances in Kinetic Equations and Applications, (ed. F. Salvarani), Springer, 48 (2021), 51–67. doi: 10.1007/978-3-030-82946-9_2.

[3]

L. Barletti and F. Méhats, Quantum drift-diffusion modeling of spin transport in nanostructures, J. Math. Phys., 51 (2010), 053304, 20 pp. doi: 10.1063/1.3380530.

[4]

Y. Bychkov and E. Rashba, Properties of a 2D gas with lifted spectral degeneracy, J. Exper. Theor. Phys. Lett., 39 (1984), 78-81. 

[5]

X. Q. Chen and L. Chen, The bipolar quantum drift-diffusion model, Acta Math. Sin. Engl. Ser., 25 (2009), 617-638.  doi: 10.1007/s10114-009-7171-2.

[6]

P. DegondS. Gallego and F. Méhats, An entropic quantum drift-diffusion model for electron transport in resonant tunneling diodes, J. Comput. Phys., 221 (2007), 226-249.  doi: 10.1016/j.jcp.2006.06.027.

[7]

P. DegondF. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118 (2005), 625-667.  doi: 10.1007/s10955-004-8823-3.

[8]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), 587-628.  doi: 10.1023/A:1023824008525.

[9]

R. Duboscq and O. Pinaud, On the minimization of quantum entropies under local constraints, J. Math. Pure Appl., 128 (2019), 87-118.  doi: 10.1016/j.matpur.2019.05.001.

[10]

R. E. Hajj, Diffusion models for spin transport derived from the spinor Boltzmann equation, Commun. Math. Sci., 12 (2014), 565-592.  doi: 10.4310/CMS.2014.v12.n3.a9.

[11] G. Folland, Harmonic Analysis in Phase Space, Princeton University Press, Princeton, 1989.  doi: 10.1515/9781400882427.
[12]

A. Glitzky, Analysis of a spin-polarized drift-diffusion model, Adv. Math. Sci. Appl., 18 (2008), 401-427. 

[13]

A. Glitzky and K. Gärtner, Existence of bounded steady state solutions to spin-polarized drift-diffusion systems, SIAM J. Math. Anal., 41 (2009/10), 2489-2513.  doi: 10.1137/080736454.

[14]

A. Jüngel, Transport Equations for Semiconductors, Springer, Berlin, 2009. doi: 10.1007/978-3-540-89526-8.

[15]

J. L. López and J. Montejo-Gámez, On the derivation and mathematical analysis of some quantum-mechanical models accounting for Fokker–Planck type dissipation: Phase space, Schrödinger and hydrodynamic descriptions, Nanoscale Sys., 2 (2013), 49-80.  doi: 10.2478/nsmmt-2013-0004.

[16]

F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics, J. Stat. Phys., 140 (2010), 565-602.  doi: 10.1007/s10955-010-0003-z.

[17]

F. Méhats and O. Pinaud, The quantum Liouville–BGK equation and the moment problem, J. Differ. Eqs., 263 (2017), 3737-3787.  doi: 10.1016/j.jde.2017.05.004.

[18]

S. Possanner and C. Negulescu, Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport, Kinet. Relat. Models, 4 (2011), 1159-1191.  doi: 10.3934/krm.2011.4.1159.

[19] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I: Functional Analysis, Academic Press, New York, 1972. 
[20]

N. Zamponi and A. Jüngel, Two spinorial drift-diffusion models for quantum electron transport in graphene, Commun. Math. Sci., 11 (2013), 807-830.  doi: 10.4310/CMS.2013.v11.n3.a7.

[21]

I. Žutić, J. Fabian and S. Das Sarma, Spin-polarized transport in inhomogeneous magnetic semiconductors: Theory of magnetic/nonmagnetic $p$-$n$ junctions, Phys. Rev. Lett., 88 (2002), 066603, 4 pp.

[22]

I. ŽutićJ. Fabian and S. Das Sarma, Spintronics: Fundamentals and applications, Rev. Modern Phys., 76 (2004), 323-410.  doi: 10.1103/RevModPhys.76.323.

show all references

References:
[1]

M. Ancona and G. Iafrate, Quantum correction to the equation of state of an electron gas in a semiconductor, Phys. Rev. B, 39 (1989), 9536-9540.  doi: 10.1103/PhysRevB.39.9536.

[2]

L. Barletti, P. Holzinger and A. Jüngel, Quantum drift-diffusion equations for a two-dimensional electron gas with spin-orbit interaction, In Recent Advances in Kinetic Equations and Applications, (ed. F. Salvarani), Springer, 48 (2021), 51–67. doi: 10.1007/978-3-030-82946-9_2.

[3]

L. Barletti and F. Méhats, Quantum drift-diffusion modeling of spin transport in nanostructures, J. Math. Phys., 51 (2010), 053304, 20 pp. doi: 10.1063/1.3380530.

[4]

Y. Bychkov and E. Rashba, Properties of a 2D gas with lifted spectral degeneracy, J. Exper. Theor. Phys. Lett., 39 (1984), 78-81. 

[5]

X. Q. Chen and L. Chen, The bipolar quantum drift-diffusion model, Acta Math. Sin. Engl. Ser., 25 (2009), 617-638.  doi: 10.1007/s10114-009-7171-2.

[6]

P. DegondS. Gallego and F. Méhats, An entropic quantum drift-diffusion model for electron transport in resonant tunneling diodes, J. Comput. Phys., 221 (2007), 226-249.  doi: 10.1016/j.jcp.2006.06.027.

[7]

P. DegondF. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118 (2005), 625-667.  doi: 10.1007/s10955-004-8823-3.

[8]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), 587-628.  doi: 10.1023/A:1023824008525.

[9]

R. Duboscq and O. Pinaud, On the minimization of quantum entropies under local constraints, J. Math. Pure Appl., 128 (2019), 87-118.  doi: 10.1016/j.matpur.2019.05.001.

[10]

R. E. Hajj, Diffusion models for spin transport derived from the spinor Boltzmann equation, Commun. Math. Sci., 12 (2014), 565-592.  doi: 10.4310/CMS.2014.v12.n3.a9.

[11] G. Folland, Harmonic Analysis in Phase Space, Princeton University Press, Princeton, 1989.  doi: 10.1515/9781400882427.
[12]

A. Glitzky, Analysis of a spin-polarized drift-diffusion model, Adv. Math. Sci. Appl., 18 (2008), 401-427. 

[13]

A. Glitzky and K. Gärtner, Existence of bounded steady state solutions to spin-polarized drift-diffusion systems, SIAM J. Math. Anal., 41 (2009/10), 2489-2513.  doi: 10.1137/080736454.

[14]

A. Jüngel, Transport Equations for Semiconductors, Springer, Berlin, 2009. doi: 10.1007/978-3-540-89526-8.

[15]

J. L. López and J. Montejo-Gámez, On the derivation and mathematical analysis of some quantum-mechanical models accounting for Fokker–Planck type dissipation: Phase space, Schrödinger and hydrodynamic descriptions, Nanoscale Sys., 2 (2013), 49-80.  doi: 10.2478/nsmmt-2013-0004.

[16]

F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics, J. Stat. Phys., 140 (2010), 565-602.  doi: 10.1007/s10955-010-0003-z.

[17]

F. Méhats and O. Pinaud, The quantum Liouville–BGK equation and the moment problem, J. Differ. Eqs., 263 (2017), 3737-3787.  doi: 10.1016/j.jde.2017.05.004.

[18]

S. Possanner and C. Negulescu, Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport, Kinet. Relat. Models, 4 (2011), 1159-1191.  doi: 10.3934/krm.2011.4.1159.

[19] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I: Functional Analysis, Academic Press, New York, 1972. 
[20]

N. Zamponi and A. Jüngel, Two spinorial drift-diffusion models for quantum electron transport in graphene, Commun. Math. Sci., 11 (2013), 807-830.  doi: 10.4310/CMS.2013.v11.n3.a7.

[21]

I. Žutić, J. Fabian and S. Das Sarma, Spin-polarized transport in inhomogeneous magnetic semiconductors: Theory of magnetic/nonmagnetic $p$-$n$ junctions, Phys. Rev. Lett., 88 (2002), 066603, 4 pp.

[22]

I. ŽutićJ. Fabian and S. Das Sarma, Spintronics: Fundamentals and applications, Rev. Modern Phys., 76 (2004), 323-410.  doi: 10.1103/RevModPhys.76.323.

Figure 1.  Left: A two-dimensional electron gas (2DEG) is confined between two different semiconductor materials A and B (for instance, InAlAs and InGaAs). Right: The electrons of the 2DEG experience an effective magnetic field $ \alpha_R(p\times \mathit{\boldsymbol{e}}_3) $ orthogonal to both the electron momentum $ p $ and the confinement direction $ \mathit{\boldsymbol{e}}_3 $, where $ \alpha_R>0 $ and $ \mathit{\boldsymbol{e}}_3 = (0, 0, 1)^T $
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