December  2011, 4(4): 1159-1191. doi: 10.3934/krm.2011.4.1159

Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport

1. 

Institute of Theoretical and Computational Physics, Graz University of Technology, Petersgasse 16, 8010 Graz

2. 

Laboratoire CMI/LATP, Université de Provence, 39, rue Joliot Curie, 13453 Marseille Cedex 13

Received  May 2011 Revised  September 2011 Published  November 2011

The aim of the present paper is the mathematical study of a linear Boltzmann equation with different matrix collision operators, modelling the spin-polarized, semi-classical electron transport in non-homogeneous ferromagnetic structures. In the collision kernel, the scattering rate is generalized to a hermitian, positive-definite $2\times2$ matrix whose eigenvalues stand for the different scattering rates of, for example, spin-up and spin-down electrons in spintronic applications. We identify four possible structures of linear matrix collision operators that yield existence and uniqueness of a weak solution of the Boltzmann equation for a general Hamilton function. We are able to prove positive-(semi)definiteness of a solution for an operator that features an anti-symmetric structure of the gain respectively the loss term with respect to the occurring matrix products. Furthermore, in order to obtain matrix drift-diffusion equations, we perform the diffusion limit with one of the symmetric operators assuming parabolic spin bands with uniform band gap and in the case that the precession frequency of the spin distribution vector around the exchange field of the Hamiltonian scales with order $\epsilon^2$. Numerical simulations of the here obtained macroscopic model were carried out in non-magnetic/ferromagnetic multilayer structures and for a magnetic Bloch domain wall. The results show that our model can be used to improve the understanding of spin-polarized transport in spintronics applications.
Citation: Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic and Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159
References:
[1]

L. E. Ballentine, "Quantum Mechanics. A Modern Development,'' Revised edition, World Scientific Publishing Co., Inc., River Edge, NJ, 1998.

[2]

L. Barletti and G. Frosali, Diffusive limit of the two-band k.p model for semiconductors, J. Stat. Phys., 139 (2010), 280-306. doi: 10.1007/s10955-010-9940-9.

[3]

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

[4]

G. S. D. Beach, M. Tsoi and J. L. Erskine, Current-induced domain wall motion, J. Magn. Magn. Mater., 320 (2008), 1272-1281. doi: 10.1016/j.jmmm.2007.12.021.

[5]

L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current, Phys. Rev. B, 54 (1996), 9353-9358. doi: 10.1103/PhysRevB.54.9353.

[6]

D. V. Berkov and J. Miltat, Spin-torque driven magnetization dynamics: micromagnetic modeling, J. Magn. Magn. Mater., 320 (2008), 1238-1259. doi: 10.1016/j.jmmm.2007.12.023.

[7]

H.-P. Breuer and F. Petruccione, "The Theory of Open Quantum Systems,'' Oxford University Press, New York, 2002.

[8]

I. A. Campbell, A. Fert and A. R. Pomeroy, Evidence for two current conduction in iron, Phil. Mag., 15 (1967), 977-981, arXiv:0711.4478.

[9]

M. I. Dyakonov, ed., "Spin Physics in Semiconductors,'' Springer-Verlag, Berlin Heidelberg, 2008.

[10]

R. El Hajj, Diffusion models for spin transport derived from the spinor Boltzmann equation,, Comm. Math. Sci., (). 

[11]

R. El Hajj, "Etude Mathématique et Numérique de Modèles de Transport: Application à la Spintronique," Ph.D. thesis, Institut de Mathámatiques de Toulouse (IMT), Université Paul Sabatier, 2008.

[12]

H. A. Engel, E. I. Rashba and B. I. Halperin, Theory of spin Hall effects in semiconductors, Handbook of Magnetism and Advanced Magnetic Materials, John Wiley & Sons, Ltd., 2007, arXiv:cond-mat/0603306v3.

[13]

C. Ertler, A. Matos-Abiague, M. Gmitra, M. Turek and J. Fabian, Perspectives in spintronics: Magnetic resonant tunneling, spin-orbit coupling, and GaMnAs, J. Phys. Conf. Ser., 129 (2008), 012021, arXiv:0811.0500.

[14]

A. Fert, Nobel lecture: Origin, development, and future of spintronics, Rev. Mod. Phys., 80 (2008), 1517-1530. doi: 10.1103/RevModPhys.80.1517.

[15]

A. Fert and I. A. Campbell, Electrical resistivity of ferromagnetic nickel and iron based alloys, J. Phys. F: Metal Phys., 6 (1976), 849-871. doi: 10.1088/0305-4608/6/5/025.

[16]

O. Gunnarsson, Band model for magnetism of transition metals in the spin-density-functional formalism, J. Phys. F: Metal Phys., 6 (1976), 587. doi: 10.1088/0305-4608/6/4/018.

[17]

R. Q. Hood and R. M. Falicov, Boltzmann-equation approach to the negative magnetoresistance of ferromagnetic-normal-metal multilayers, Phys. Rev. B, 46 (2007), 8287. doi: 10.1103/PhysRevB.46.8287.

[18]

M. Johnson and R. H. Silsbee, Interfacial charge-spin coupling: Injection and detection of spin magnetization in metals, Phys. Rev. Lett., 55 (1985), 1790. doi: 10.1103/PhysRevLett.55.1790.

[19]

J. A. Katine and E. E. Fullerton, Device implications of spin-transfer torques, J. Magn. Magn. Mater., 320 (2008), 1217-1226. doi: 10.1016/j.jmmm.2007.12.013.

[20]

D. Loss and D. P. DiVincenzo, Quantum computation with quantum dots, Phys. Rev. A, 57 (1998), 120-126. doi: 10.1103/PhysRevA.57.120.

[21]

O. Morandi and F. Schürrer, Wigner model for quantum transport in graphene, J. Phys. A, accepted, 2011, arXiv:1102.2416.

[22]

W. Nolting and A. Ramakanth, "Quantum Theory of Magnetism,'' Springer-Verlag, Berlin-Heidelberg, 2009. doi: 10.1007/978-3-540-85416-6.

[23]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos and A. A. Firsov, Two-dimensional gas of massless Dirac fermions in graphene, Nature, 438 (2005), 197-200. doi: 10.1038/nature04233.

[24]

F. Piéchon and A. Thiaville, Spin transfer torque in continuous textures: semiclassical Boltzmann approach, Phys. Rev. B, 75 (2007), 174414. doi: 10.1103/PhysRevB.75.174414.

[25]

F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers, Asympt. Anal., 4 (1991), 293-317.

[26]

Y. Qi and S. Zhang, Spin diffusion at finite electric and magnetic fields, Phys. Rev. B, 67 (2003), 052407. doi: 10.1103/PhysRevB.67.052407.

[27]

D. C. Ralph and M. D. Stiles, Spin transfer torques, J. Magn. Magn. Mater., 320 (2008), 1190-1216.

[28]

S. Saikin, A drift-diffusion model for spin-polarized transport in a two-dimensional non-degenerate electron gas controlled by spin-orbit interaction, J. Phys.: Condens. Matter, 16 (2010), 5071-5081, arXiv:cond-mat/0311221v3.

[29]

E. Simanek, Spin accumulation and resistance due to a domain wall, Phys. Rev. B, 63 (2001), 224412. doi: 10.1103/PhysRevB.63.224412.

[30]

J. C. Slonczewski, Current-driven excitation of magnetic multilayers, J. Magn. Magn. Mater., 159 (1996), L1-L7. doi: 10.1016/0304-8853(96)00062-5.

[31]

M. D. Stiles and J. Miltat, Spin-transfer torque and dynamics, Top. Appl. Phys., 101 (2006), 225. doi: 10.1007/10938171_7.

[32]

M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi and P. Wyder, Excitation of a magnetic multilayer by an electric current, Phys. Rev. Lett., 80 (1998), 4281. doi: 10.1103/PhysRevLett.80.4281.

[33]

T. Valet and A. Fert, Theory of the perpendicular magnetoresistance in magnetic multilayers, Phys. Rev. B, 48 (1993), 7099. doi: 10.1103/PhysRevB.48.7099.

[34]

P. C. van Son, H. van Kempen and P. Wyder, Boundary resistance of the ferromagnetic-nonferromagnetic metal interface, Phys. Rev. Lett., 58 (1987), 2271. doi: 10.1103/PhysRevLett.58.2271.

[35]

L. Villegas-Lelovsky, Hydrodynamic model for spin-polarized electron transport in semiconductors, J. Appl. Phys., 101 (2007), 053707. doi: 10.1063/1.2437570.

[36]

C. Vouille, A. Barthélémy, F. Elokan Mpondo, A. Fert, P. A. Schroeder, S. Y. Hsu, A. Reilly and R. Loloee, Microscopic mechanisms of giant magnetoresistance, Phys. Rev. B, 60 (1999), 6710. doi: 10.1103/PhysRevB.60.6710.

[37]

M. Wenin and W. Pötz, Optimal control of a single qubit by direct inversion, Phys. Rev. A, 74 (2006), 022319. doi: 10.1103/PhysRevA.74.022319.

[38]

J. Xiao, A. Zangwill, and M. D. Stiles, A numerical method to solve the Boltzmann equation for a spin valve, Eur. Phys. J. B, 59 (2007), 415-427. doi: 10.1140/epjb/e2007-00004-0.

[39]

N. Zamponi and L. Barletti, Quantum electronic transport in graphene: A kinetic and fluid-dynamic approach, Mathematical Methods in Applied Sciences, 34 (2011), 807-818. doi: 10.1002/mma.1403.

[40]

J. Zhang, P. M. Levy, S. Zhang and V. Antropov, Identification of transverse spin currents in noncollinear magnetic structures, Phys. Rev. Lett., 93 (2004), 256602, arXiv:cond-mat/0405610v1.

[41]

S. Zhang, P. M. Levy and A. Fert, Mechanisms of spin-polarized current-driven magnetization switching, Phys. Rev. Lett., 88 (2002), 236601. doi: 10.1103/PhysRevLett.88.236601.

[42]

I. Zutic, J. Fabian and S. Das Sarma, Spintronics: Fundamentals and applications, Rev. Mod. Phys., 76 (2004), 323-410.

show all references

References:
[1]

L. E. Ballentine, "Quantum Mechanics. A Modern Development,'' Revised edition, World Scientific Publishing Co., Inc., River Edge, NJ, 1998.

[2]

L. Barletti and G. Frosali, Diffusive limit of the two-band k.p model for semiconductors, J. Stat. Phys., 139 (2010), 280-306. doi: 10.1007/s10955-010-9940-9.

[3]

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

[4]

G. S. D. Beach, M. Tsoi and J. L. Erskine, Current-induced domain wall motion, J. Magn. Magn. Mater., 320 (2008), 1272-1281. doi: 10.1016/j.jmmm.2007.12.021.

[5]

L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current, Phys. Rev. B, 54 (1996), 9353-9358. doi: 10.1103/PhysRevB.54.9353.

[6]

D. V. Berkov and J. Miltat, Spin-torque driven magnetization dynamics: micromagnetic modeling, J. Magn. Magn. Mater., 320 (2008), 1238-1259. doi: 10.1016/j.jmmm.2007.12.023.

[7]

H.-P. Breuer and F. Petruccione, "The Theory of Open Quantum Systems,'' Oxford University Press, New York, 2002.

[8]

I. A. Campbell, A. Fert and A. R. Pomeroy, Evidence for two current conduction in iron, Phil. Mag., 15 (1967), 977-981, arXiv:0711.4478.

[9]

M. I. Dyakonov, ed., "Spin Physics in Semiconductors,'' Springer-Verlag, Berlin Heidelberg, 2008.

[10]

R. El Hajj, Diffusion models for spin transport derived from the spinor Boltzmann equation,, Comm. Math. Sci., (). 

[11]

R. El Hajj, "Etude Mathématique et Numérique de Modèles de Transport: Application à la Spintronique," Ph.D. thesis, Institut de Mathámatiques de Toulouse (IMT), Université Paul Sabatier, 2008.

[12]

H. A. Engel, E. I. Rashba and B. I. Halperin, Theory of spin Hall effects in semiconductors, Handbook of Magnetism and Advanced Magnetic Materials, John Wiley & Sons, Ltd., 2007, arXiv:cond-mat/0603306v3.

[13]

C. Ertler, A. Matos-Abiague, M. Gmitra, M. Turek and J. Fabian, Perspectives in spintronics: Magnetic resonant tunneling, spin-orbit coupling, and GaMnAs, J. Phys. Conf. Ser., 129 (2008), 012021, arXiv:0811.0500.

[14]

A. Fert, Nobel lecture: Origin, development, and future of spintronics, Rev. Mod. Phys., 80 (2008), 1517-1530. doi: 10.1103/RevModPhys.80.1517.

[15]

A. Fert and I. A. Campbell, Electrical resistivity of ferromagnetic nickel and iron based alloys, J. Phys. F: Metal Phys., 6 (1976), 849-871. doi: 10.1088/0305-4608/6/5/025.

[16]

O. Gunnarsson, Band model for magnetism of transition metals in the spin-density-functional formalism, J. Phys. F: Metal Phys., 6 (1976), 587. doi: 10.1088/0305-4608/6/4/018.

[17]

R. Q. Hood and R. M. Falicov, Boltzmann-equation approach to the negative magnetoresistance of ferromagnetic-normal-metal multilayers, Phys. Rev. B, 46 (2007), 8287. doi: 10.1103/PhysRevB.46.8287.

[18]

M. Johnson and R. H. Silsbee, Interfacial charge-spin coupling: Injection and detection of spin magnetization in metals, Phys. Rev. Lett., 55 (1985), 1790. doi: 10.1103/PhysRevLett.55.1790.

[19]

J. A. Katine and E. E. Fullerton, Device implications of spin-transfer torques, J. Magn. Magn. Mater., 320 (2008), 1217-1226. doi: 10.1016/j.jmmm.2007.12.013.

[20]

D. Loss and D. P. DiVincenzo, Quantum computation with quantum dots, Phys. Rev. A, 57 (1998), 120-126. doi: 10.1103/PhysRevA.57.120.

[21]

O. Morandi and F. Schürrer, Wigner model for quantum transport in graphene, J. Phys. A, accepted, 2011, arXiv:1102.2416.

[22]

W. Nolting and A. Ramakanth, "Quantum Theory of Magnetism,'' Springer-Verlag, Berlin-Heidelberg, 2009. doi: 10.1007/978-3-540-85416-6.

[23]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos and A. A. Firsov, Two-dimensional gas of massless Dirac fermions in graphene, Nature, 438 (2005), 197-200. doi: 10.1038/nature04233.

[24]

F. Piéchon and A. Thiaville, Spin transfer torque in continuous textures: semiclassical Boltzmann approach, Phys. Rev. B, 75 (2007), 174414. doi: 10.1103/PhysRevB.75.174414.

[25]

F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers, Asympt. Anal., 4 (1991), 293-317.

[26]

Y. Qi and S. Zhang, Spin diffusion at finite electric and magnetic fields, Phys. Rev. B, 67 (2003), 052407. doi: 10.1103/PhysRevB.67.052407.

[27]

D. C. Ralph and M. D. Stiles, Spin transfer torques, J. Magn. Magn. Mater., 320 (2008), 1190-1216.

[28]

S. Saikin, A drift-diffusion model for spin-polarized transport in a two-dimensional non-degenerate electron gas controlled by spin-orbit interaction, J. Phys.: Condens. Matter, 16 (2010), 5071-5081, arXiv:cond-mat/0311221v3.

[29]

E. Simanek, Spin accumulation and resistance due to a domain wall, Phys. Rev. B, 63 (2001), 224412. doi: 10.1103/PhysRevB.63.224412.

[30]

J. C. Slonczewski, Current-driven excitation of magnetic multilayers, J. Magn. Magn. Mater., 159 (1996), L1-L7. doi: 10.1016/0304-8853(96)00062-5.

[31]

M. D. Stiles and J. Miltat, Spin-transfer torque and dynamics, Top. Appl. Phys., 101 (2006), 225. doi: 10.1007/10938171_7.

[32]

M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi and P. Wyder, Excitation of a magnetic multilayer by an electric current, Phys. Rev. Lett., 80 (1998), 4281. doi: 10.1103/PhysRevLett.80.4281.

[33]

T. Valet and A. Fert, Theory of the perpendicular magnetoresistance in magnetic multilayers, Phys. Rev. B, 48 (1993), 7099. doi: 10.1103/PhysRevB.48.7099.

[34]

P. C. van Son, H. van Kempen and P. Wyder, Boundary resistance of the ferromagnetic-nonferromagnetic metal interface, Phys. Rev. Lett., 58 (1987), 2271. doi: 10.1103/PhysRevLett.58.2271.

[35]

L. Villegas-Lelovsky, Hydrodynamic model for spin-polarized electron transport in semiconductors, J. Appl. Phys., 101 (2007), 053707. doi: 10.1063/1.2437570.

[36]

C. Vouille, A. Barthélémy, F. Elokan Mpondo, A. Fert, P. A. Schroeder, S. Y. Hsu, A. Reilly and R. Loloee, Microscopic mechanisms of giant magnetoresistance, Phys. Rev. B, 60 (1999), 6710. doi: 10.1103/PhysRevB.60.6710.

[37]

M. Wenin and W. Pötz, Optimal control of a single qubit by direct inversion, Phys. Rev. A, 74 (2006), 022319. doi: 10.1103/PhysRevA.74.022319.

[38]

J. Xiao, A. Zangwill, and M. D. Stiles, A numerical method to solve the Boltzmann equation for a spin valve, Eur. Phys. J. B, 59 (2007), 415-427. doi: 10.1140/epjb/e2007-00004-0.

[39]

N. Zamponi and L. Barletti, Quantum electronic transport in graphene: A kinetic and fluid-dynamic approach, Mathematical Methods in Applied Sciences, 34 (2011), 807-818. doi: 10.1002/mma.1403.

[40]

J. Zhang, P. M. Levy, S. Zhang and V. Antropov, Identification of transverse spin currents in noncollinear magnetic structures, Phys. Rev. Lett., 93 (2004), 256602, arXiv:cond-mat/0405610v1.

[41]

S. Zhang, P. M. Levy and A. Fert, Mechanisms of spin-polarized current-driven magnetization switching, Phys. Rev. Lett., 88 (2002), 236601. doi: 10.1103/PhysRevLett.88.236601.

[42]

I. Zutic, J. Fabian and S. Das Sarma, Spintronics: Fundamentals and applications, Rev. Mod. Phys., 76 (2004), 323-410.

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