American Institute of Mathematical Sciences

Advanced
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

Stefan Possanner 1, and  Claudia Negulescu 2,

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.
Keywords: kinetic matrix models, spin transport, Spintronics, spin-charge coupling..
Mathematics Subject Classification: Primary: 76P99, 76R50; Secondary: 81R2.
Citation: Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic & 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., (1998). Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

H.-P. Breuer and F. Petruccione, "The Theory of Open Quantum Systems,'', Oxford University Press, (2002). Google Scholar

[8]

I. A. Campbell, A. Fert and A. R. Pomeroy, Evidence for two current conduction in iron,, Phil. Mag., 15 (1967), 977. Google Scholar

[9]

M. I. Dyakonov, ed., "Spin Physics in Semiconductors,'', Springer-Verlag, (2008). Google Scholar

[10]

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

[11]

R. El Hajj, "Etude Mathématique et Numérique de Modèles de Transport: Application à la Spintronique,", Ph.D. thesis, (2008). Google Scholar

[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, (2007). Google Scholar

[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). Google Scholar

[14]

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

[15]

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

[16]

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

[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). doi: 10.1103/PhysRevB.46.8287. Google Scholar

[18]

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

[19]

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

[20]

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

[21]

O. Morandi and F. Schürrer, Wigner model for quantum transport in graphene,, J. Phys. A, (2011). Google Scholar

[22]

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

[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. doi: 10.1038/nature04233. Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[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. Google Scholar

[29]

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

[30]

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

[31]

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

[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). doi: 10.1103/PhysRevLett.80.4281. Google Scholar

[33]

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

[34]

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

[35]

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

[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). doi: 10.1103/PhysRevB.60.6710. Google Scholar

[37]

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

[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. doi: 10.1140/epjb/e2007-00004-0. Google Scholar

[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. doi: 10.1002/mma.1403. Google Scholar

[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). Google Scholar

[41]

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

[42]

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

show all references

References:
[1]

L. E. Ballentine, "Quantum Mechanics. A Modern Development,'' Revised edition,, World Scientific Publishing Co., (1998). Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

H.-P. Breuer and F. Petruccione, "The Theory of Open Quantum Systems,'', Oxford University Press, (2002). Google Scholar

[8]

I. A. Campbell, A. Fert and A. R. Pomeroy, Evidence for two current conduction in iron,, Phil. Mag., 15 (1967), 977. Google Scholar

[9]

M. I. Dyakonov, ed., "Spin Physics in Semiconductors,'', Springer-Verlag, (2008). Google Scholar

[10]

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

[11]

R. El Hajj, "Etude Mathématique et Numérique de Modèles de Transport: Application à la Spintronique,", Ph.D. thesis, (2008). Google Scholar

[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, (2007). Google Scholar

[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). Google Scholar

[14]

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

[15]

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

[16]

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

[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). doi: 10.1103/PhysRevB.46.8287. Google Scholar

[18]

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

[19]

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

[20]

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

[21]

O. Morandi and F. Schürrer, Wigner model for quantum transport in graphene,, J. Phys. A, (2011). Google Scholar

[22]

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

[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. doi: 10.1038/nature04233. Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[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. Google Scholar

[29]

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

[30]

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

[31]

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

[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). doi: 10.1103/PhysRevLett.80.4281. Google Scholar

[33]

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

[34]

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

[35]

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

[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). doi: 10.1103/PhysRevB.60.6710. Google Scholar

[37]

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

[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. doi: 10.1140/epjb/e2007-00004-0. Google Scholar

[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. doi: 10.1002/mma.1403. Google Scholar

[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). Google Scholar

[41]

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

[42]

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

[1]

Carlos J. Garcia-Cervera, Xiao-Ping Wang. Spin-polarized transport: Existence of weak solutions. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 87-100. doi: 10.3934/dcdsb.2007.7.87
[2]

Leif Arkeryd. A kinetic equation for spin polarized Fermi systems. Kinetic & Related Models, 2014, 7 (1) : 1-8. doi: 10.3934/krm.2014.7.1
[3]

Carlos J. García-Cervera, Xiao-Ping Wang. A note on 'Spin-polarized transport: Existence of weak solutions'. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2761-2763. doi: 10.3934/dcdsb.2015.20.2761
[4]

Liren Lin, I-Liang Chern. A kinetic energy reduction technique and characterizations of the ground states of spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1119-1128. doi: 10.3934/dcdsb.2014.19.1119
[5]

François Gay-Balma, Darryl D. Holm, Tudor S. Ratiu. Variational principles for spin systems and the Kirchhoff rod. Journal of Geometric Mechanics, 2009, 1 (4) : 417-444. doi: 10.3934/jgm.2009.1.417
[6]

Marco Cicalese, Matthias Ruf. Discrete spin systems on random lattices at the bulk scaling. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 101-117. doi: 10.3934/dcdss.2017006
[7]

M. I. Alomar, David Sánchez. Thermopower of a graphene monolayer with inhomogeneous spin-orbit interaction. Conference Publications, 2015, 2015 (special) : 1-9. doi: 10.3934/proc.2015.0001
[8]

Lei Yang, Xiao-Ping Wang. Dynamics of domain wall in thin film driven by spin current. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1251-1263. doi: 10.3934/dcdsb.2010.14.1251
[9]

Gong Chen. Strichartz estimates for charge transfer models. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1201-1226. doi: 10.3934/dcds.2017050
[10]

Dong Deng, Ruikuan Liu. Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3175-3193. doi: 10.3934/dcdsb.2018306
[11]

Per Danzl, Ali Nabi, Jeff Moehlis. Charge-balanced spike timing control for phase models of spiking neurons. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1413-1435. doi: 10.3934/dcds.2010.28.1413
[12]

Wolfgang Wagner. Some properties of the kinetic equation for electron transport in semiconductors. Kinetic & Related Models, 2013, 6 (4) : 955-967. doi: 10.3934/krm.2013.6.955
[13]

Mauro Garavello, Benedetto Piccoli. Coupling of microscopic and phase transition models at boundary. Networks & Heterogeneous Media, 2013, 8 (3) : 649-661. doi: 10.3934/nhm.2013.8.649
[14]

Sue Ann Campbell, Ilya Kobelevskiy. Phase models and oscillators with time delayed coupling. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2653-2673. doi: 10.3934/dcds.2012.32.2653
[15]

Pierre Degond, Hailiang Liu. Kinetic models for polymers with inertial effects. Networks & Heterogeneous Media, 2009, 4 (4) : 625-647. doi: 10.3934/nhm.2009.4.625
[16]

Seung-Yeal Ha, Doron Levy. Particle, kinetic and fluid models for phototaxis. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 77-108. doi: 10.3934/dcdsb.2009.12.77
[17]

Klas Modin. Geometry of matrix decompositions seen through optimal transport and information geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 335-390. doi: 10.3934/jgm.2017014
[18]

Charles Nguyen, Stephen Pankavich. A one-dimensional kinetic model of plasma dynamics with a transport field. Evolution Equations & Control Theory, 2014, 3 (4) : 681-698. doi: 10.3934/eect.2014.3.681
[19]

Tomasz Komorowski. Long time asymptotics of a degenerate linear kinetic transport equation. Kinetic & Related Models, 2014, 7 (1) : 79-108. doi: 10.3934/krm.2014.7.79
[20]

Michael Herty, Christian Ringhofer. Averaged kinetic models for flows on unstructured networks. Kinetic & Related Models, 2011, 4 (4) : 1081-1096. doi: 10.3934/krm.2011.4.1081

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences