September  2013, 6(3): 505-532. doi: 10.3934/krm.2013.6.505

Semi-classical models for the Schrödinger equation with periodic potentials and band crossings

1. 

Department of Mathematical Science, Tsinghua University, Bejing 100084, China

2. 

Department of Mathematics, Institute of Natural Sciences, and MOE Key Lab in Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai 200240, China

3. 

Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, United States

Received  February 2013 Revised  March 2013 Published  May 2013

The Bloch decomposition plays a fundamental role in the study of quantum mechanics and wave propagation in periodic media. Most of the homogenization theory developed for the study of high frequency or semi-classical limit for these problems assumes no crossing of the Bloch bands, resulting in a classical Liouville equation in the limit along each Bloch band.
    In this article, we derive semi-classical models for the Schrödinger equation in periodic media that take into account band crossings, which is important to describe quantum transitions between Bloch bands. Our idea is still based on the Wigner transform (on the Bloch eigenfunctions), but in taking the semi-classical approximation, we retain the off-diagonal entries of the Wigner matrix, which cannot be ignored near the points of band crossings. This results in coupled inhomogeneous Liouville systems that can suitably describe quantum tunneling between bands that are not well-separated. We also develop a domain decomposition method that couples these semi-classical models with the classical Liouville equations (valid away from zones of band crossings) for a multiscale computation. Solutions of these models are numerically compared with those of the Schrödinger equation to justify the validity of these new models for band-crossings.
Citation: Lihui Chai, Shi Jin, Qin Li. Semi-classical models for the Schrödinger equation with periodic potentials and band crossings. Kinetic & Related Models, 2013, 6 (3) : 505-532. doi: 10.3934/krm.2013.6.505
References:
[1]

N. W. Ashcroft and N. D. Mermin., "Solid State Physics,", Saunders College, (1976).

[2]

G. Bal, A. Fannjiang, G. Papanicolaou and L. Ryzhik, Radiative transport in a periodic structure,, Journal of Statistical Physics, 95 (1999), 479. doi: 10.1023/A:1004598015978.

[3]

P. Bechouche, Semi-classical limits in a crystal with a Coulombian self-consistent potential: Effective mass theorems,, Asymptotic Analysis, 19 (1999), 95.

[4]

P. Bechouche, N. J. Mauser and F. Poupaud, Semiclassical limit for the Schrödinger-Poisson equation in a crystal,, Communications on Pure and Applied Mathematics, 54 (2001), 851. doi: 10.1002/cpa.3004.

[5]

F. Bloch, Über die quantenmechanik der elektronen in kristallgittern,, Zeitschrift für Physik A Hadrons and Nuclei, 52 (1929), 555.

[6]

A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu and J. Zwanziger, "The Geometric Phase in Quantum Systems. Foundations, Mathematical Concepts, and Applications in Molecular and Condensed Matter Physics,", Texts and Monographs in Physics, (2003).

[7]

R. Carles, P. A. Markowich and C. Sparber, Semiclassical asymptotics for weakly nonlinear Bloch waves,, Journal of Statistical Physics, 117 (2004), 343. doi: 10.1023/B:JOSS.0000044070.34410.17.

[8]

M. S. Child, "Atom-Molecule Collision Theory,", Plenum, (1979).

[9]

Richard Courant and David Hilbert, "Methods of Mathematical Physics, Differential Equations,", Wiley-VCH, (2008).

[10]

K. Drukker, Basics of surface hopping in mixed quantum/classical simulations,, Journal of Computational Physics, 153 (1999), 225. doi: 10.1006/jcph.1999.6287.

[11]

W. E, J. Lu and X. Yang, Asymptotic analysis of quantum dynamics in crystals: The Bloch-Wigner transform, Bloch dynamics and berry phase,, Acta Mathematicae Applicatae Sinica, (2011), 1. doi: 10.1007/s10255-011-0095-5.

[12]

Clotilde Fermanian Kammerer and Caroline Lasser, Wigner measures and codimension two crossings,, J. Math. Phys., 44 (2003), 507. doi: 10.1063/1.1527221.

[13]

P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms,, Communications on Pure and Applied Mathematics, 50 (1997), 323. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.

[14]

L. Gosse and N. J. Mauser, Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice. III. From ab initio models to WKB for Schrödinger-Poisson,, Journal of Computational Physics, 211 (2006), 326. doi: 10.1016/j.jcp.2005.05.020.

[15]

George A. Hagedorn, "Molecular Propagation through Electron Energy Level Crossings,", Mem. Amer. Math. Soc., 111 (1994).

[16]

I. Horenko, C. Salzmann, B. Schmidt and C. Schütte, Quantum-classical Liouville approach to molecular dynamics: Surface hopping Gaussian phase-space packets,, The Journal of Chemical Physics, 117 (2002), 11075. doi: 10.1063/1.1522712.

[17]

Z. Huang, S. Jin, P. A. Markowich and C. Sparber, A Bloch decomposition-based split-step pseudospectral method for quantum dynamics with periodic potentials,, SIAM Journal on Scientific Computing, 29 (2008), 515. doi: 10.1137/060652026.

[18]

S. Jin, H. Liu, S. Osher and Y.-H. R. Tsai, Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation,, Journal of Computational Physics, 205 (2005), 222. doi: 10.1016/j.jcp.2004.11.008.

[19]

S. Jin, P. Qi and Z. Zhang, An Eulerian surface hopping method for the Schrödinger equation with conical crossings,, SIAM Multiscale Modeling and Simulation, 9 (2011), 258. doi: 10.1137/090774185.

[20]

S. Jin and D. Yin, Computational high frequency waves through curved interfaces via the Liouville equation and geometric theory of diffraction,, Journal of Computational Physics, 227 (2008), 6106. doi: 10.1016/j.jcp.2008.02.029.

[21]

L. Landau, Zur theorie der energieubertragung II. Physik,, Physics of the Soviet Union, 2 (1932), 46.

[22]

C. Lasser, T. Swart and S. Teufel, Construction and validation of a rigorous surface hopping algorithm for conical crossings,, Communications in Mathematical Sciences, 5 (2007), 789.

[23]

C. Lasser and S. Teufel, Propagation through conical crossings: An asymptotic semigroup,, Comm. Pure Appl. Math., 58 (2005), 1188. doi: 10.1002/cpa.20087.

[24]

P.-L. Lions and T. Paul, Sur les mesures de Wigner,, Rev. Mat. Iberoamericana, 9 (1993), 553. doi: 10.4171/RMI/143.

[25]

P. A. Markowich, N. J. Mauser and F. Poupaud, A Wigner-function approach to (semi) classical limits: Electrons in a periodic potential,, Journal of Mathematical Physics, 35 (1994), 1066. doi: 10.1063/1.530629.

[26]

C. C. Martens and J.-Y. Fang, Semiclassical-limit molecular dynamics on multiple electronic surfaces,, Journal of Chemical Physics, 106 (1997), 4918. doi: 10.1063/1.473541.

[27]

O. Morandi and F. Schürrer, Wigner model for quantum transport in graphene,, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 5301.

[28]

G. Panati, H. Spohn and S. Teufel, Effective dynamics for Bloch electrons: Peierls substitution and beyond,, Communications in Mathematical Physics, 242 (2003), 547. doi: 10.1007/s00220-003-0950-1.

[29]

G. Panati, H. Spohn and S. Teufel, Motion of electrons in adiabatically perturbed periodic structures,, in, (2006), 595. doi: 10.1007/3-540-35657-6_22.

[30]

D. S. Sholl and J. C. Tully, A generalized surface hopping method,, The Journal of Chemical Physics, 109 (1998), 7702. doi: 10.1063/1.477416.

[31]

M. Sillanpää, T. Lehtinen, A. Paila, Y. Makhlin and P. J. Hakonen, Landau-Zener interferometry in a Cooper-pair box,, Journal of Low Temperature Physics, 146 (2007), 253. doi: 10.1007/s10909-006-9262-0.

[32]

G. Sundaram and Q. Niu, Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects,, Phys. Rev. B, 59 (1999), 14915. doi: 10.1103/PhysRevB.59.14915.

[33]

J. C. Tully, Molecular dynamics with electronic transitions,, The Journal of Chemical Physics, 93 (1990), 1061. doi: 10.1063/1.459170.

[34]

J. C. Tully and R. K. Preston, Trajectory surface hopping approach to nonadiabatic molecular collisions: The reaction of H with D,, The Journal of Chemical Physics, 55 (1971), 562.

[35]

E. Wigner, On the quantum correction for thermodynamic equilibrium,, Physical Review, 40 (1932), 749.

[36]

D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties,, Reviews of Modern Physics, 82 (2010), 1959. doi: 10.1103/RevModPhys.82.1959.

[37]

C. Zener, Non-adiabatic crossing of energy levels,, Proceedings of the Royal Society of London, 137 (1932), 696. doi: 10.1098/rspa.1932.0165.

show all references

References:
[1]

N. W. Ashcroft and N. D. Mermin., "Solid State Physics,", Saunders College, (1976).

[2]

G. Bal, A. Fannjiang, G. Papanicolaou and L. Ryzhik, Radiative transport in a periodic structure,, Journal of Statistical Physics, 95 (1999), 479. doi: 10.1023/A:1004598015978.

[3]

P. Bechouche, Semi-classical limits in a crystal with a Coulombian self-consistent potential: Effective mass theorems,, Asymptotic Analysis, 19 (1999), 95.

[4]

P. Bechouche, N. J. Mauser and F. Poupaud, Semiclassical limit for the Schrödinger-Poisson equation in a crystal,, Communications on Pure and Applied Mathematics, 54 (2001), 851. doi: 10.1002/cpa.3004.

[5]

F. Bloch, Über die quantenmechanik der elektronen in kristallgittern,, Zeitschrift für Physik A Hadrons and Nuclei, 52 (1929), 555.

[6]

A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu and J. Zwanziger, "The Geometric Phase in Quantum Systems. Foundations, Mathematical Concepts, and Applications in Molecular and Condensed Matter Physics,", Texts and Monographs in Physics, (2003).

[7]

R. Carles, P. A. Markowich and C. Sparber, Semiclassical asymptotics for weakly nonlinear Bloch waves,, Journal of Statistical Physics, 117 (2004), 343. doi: 10.1023/B:JOSS.0000044070.34410.17.

[8]

M. S. Child, "Atom-Molecule Collision Theory,", Plenum, (1979).

[9]

Richard Courant and David Hilbert, "Methods of Mathematical Physics, Differential Equations,", Wiley-VCH, (2008).

[10]

K. Drukker, Basics of surface hopping in mixed quantum/classical simulations,, Journal of Computational Physics, 153 (1999), 225. doi: 10.1006/jcph.1999.6287.

[11]

W. E, J. Lu and X. Yang, Asymptotic analysis of quantum dynamics in crystals: The Bloch-Wigner transform, Bloch dynamics and berry phase,, Acta Mathematicae Applicatae Sinica, (2011), 1. doi: 10.1007/s10255-011-0095-5.

[12]

Clotilde Fermanian Kammerer and Caroline Lasser, Wigner measures and codimension two crossings,, J. Math. Phys., 44 (2003), 507. doi: 10.1063/1.1527221.

[13]

P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms,, Communications on Pure and Applied Mathematics, 50 (1997), 323. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.

[14]

L. Gosse and N. J. Mauser, Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice. III. From ab initio models to WKB for Schrödinger-Poisson,, Journal of Computational Physics, 211 (2006), 326. doi: 10.1016/j.jcp.2005.05.020.

[15]

George A. Hagedorn, "Molecular Propagation through Electron Energy Level Crossings,", Mem. Amer. Math. Soc., 111 (1994).

[16]

I. Horenko, C. Salzmann, B. Schmidt and C. Schütte, Quantum-classical Liouville approach to molecular dynamics: Surface hopping Gaussian phase-space packets,, The Journal of Chemical Physics, 117 (2002), 11075. doi: 10.1063/1.1522712.

[17]

Z. Huang, S. Jin, P. A. Markowich and C. Sparber, A Bloch decomposition-based split-step pseudospectral method for quantum dynamics with periodic potentials,, SIAM Journal on Scientific Computing, 29 (2008), 515. doi: 10.1137/060652026.

[18]

S. Jin, H. Liu, S. Osher and Y.-H. R. Tsai, Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation,, Journal of Computational Physics, 205 (2005), 222. doi: 10.1016/j.jcp.2004.11.008.

[19]

S. Jin, P. Qi and Z. Zhang, An Eulerian surface hopping method for the Schrödinger equation with conical crossings,, SIAM Multiscale Modeling and Simulation, 9 (2011), 258. doi: 10.1137/090774185.

[20]

S. Jin and D. Yin, Computational high frequency waves through curved interfaces via the Liouville equation and geometric theory of diffraction,, Journal of Computational Physics, 227 (2008), 6106. doi: 10.1016/j.jcp.2008.02.029.

[21]

L. Landau, Zur theorie der energieubertragung II. Physik,, Physics of the Soviet Union, 2 (1932), 46.

[22]

C. Lasser, T. Swart and S. Teufel, Construction and validation of a rigorous surface hopping algorithm for conical crossings,, Communications in Mathematical Sciences, 5 (2007), 789.

[23]

C. Lasser and S. Teufel, Propagation through conical crossings: An asymptotic semigroup,, Comm. Pure Appl. Math., 58 (2005), 1188. doi: 10.1002/cpa.20087.

[24]

P.-L. Lions and T. Paul, Sur les mesures de Wigner,, Rev. Mat. Iberoamericana, 9 (1993), 553. doi: 10.4171/RMI/143.

[25]

P. A. Markowich, N. J. Mauser and F. Poupaud, A Wigner-function approach to (semi) classical limits: Electrons in a periodic potential,, Journal of Mathematical Physics, 35 (1994), 1066. doi: 10.1063/1.530629.

[26]

C. C. Martens and J.-Y. Fang, Semiclassical-limit molecular dynamics on multiple electronic surfaces,, Journal of Chemical Physics, 106 (1997), 4918. doi: 10.1063/1.473541.

[27]

O. Morandi and F. Schürrer, Wigner model for quantum transport in graphene,, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 5301.

[28]

G. Panati, H. Spohn and S. Teufel, Effective dynamics for Bloch electrons: Peierls substitution and beyond,, Communications in Mathematical Physics, 242 (2003), 547. doi: 10.1007/s00220-003-0950-1.

[29]

G. Panati, H. Spohn and S. Teufel, Motion of electrons in adiabatically perturbed periodic structures,, in, (2006), 595. doi: 10.1007/3-540-35657-6_22.

[30]

D. S. Sholl and J. C. Tully, A generalized surface hopping method,, The Journal of Chemical Physics, 109 (1998), 7702. doi: 10.1063/1.477416.

[31]

M. Sillanpää, T. Lehtinen, A. Paila, Y. Makhlin and P. J. Hakonen, Landau-Zener interferometry in a Cooper-pair box,, Journal of Low Temperature Physics, 146 (2007), 253. doi: 10.1007/s10909-006-9262-0.

[32]

G. Sundaram and Q. Niu, Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects,, Phys. Rev. B, 59 (1999), 14915. doi: 10.1103/PhysRevB.59.14915.

[33]

J. C. Tully, Molecular dynamics with electronic transitions,, The Journal of Chemical Physics, 93 (1990), 1061. doi: 10.1063/1.459170.

[34]

J. C. Tully and R. K. Preston, Trajectory surface hopping approach to nonadiabatic molecular collisions: The reaction of H with D,, The Journal of Chemical Physics, 55 (1971), 562.

[35]

E. Wigner, On the quantum correction for thermodynamic equilibrium,, Physical Review, 40 (1932), 749.

[36]

D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties,, Reviews of Modern Physics, 82 (2010), 1959. doi: 10.1103/RevModPhys.82.1959.

[37]

C. Zener, Non-adiabatic crossing of energy levels,, Proceedings of the Royal Society of London, 137 (1932), 696. doi: 10.1098/rspa.1932.0165.

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