### Abstract

We study the semi-classical limit of the dynamics of electrons in a crystal under the influence of an external electric field. Two small parameters e and h are introduced. They are respectively related to the scaled lattice length and the scaled Planck constant. When ε/h and h go to zero, we prove that in the limit process the dynamics of electrons is described by a Vlasov equation. When h is fixed and ε tends to zero, the dynamics is governed by an effective Schrödinger equation. In both cases, the mass of electrons is replaced by a matrix, the so-called effective mass.

Original language | English (US) |
---|---|

Pages (from-to) | 1897-1918 |

Number of pages | 22 |

Journal | Communications in Partial Differential Equations |

Volume | 21 |

Issue number | 11-12 |

State | Published - 1996 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics

### Cite this

*Communications in Partial Differential Equations*,

*21*(11-12), 1897-1918.

**Semi-classical limits in a crystal with exterior potentials and effective mass theorems.** / Poupaud, F.; Ringhofer, Christian.

Research output: Contribution to journal › Article

*Communications in Partial Differential Equations*, vol. 21, no. 11-12, pp. 1897-1918.

}

TY - JOUR

T1 - Semi-classical limits in a crystal with exterior potentials and effective mass theorems

AU - Poupaud, F.

AU - Ringhofer, Christian

PY - 1996

Y1 - 1996

N2 - We study the semi-classical limit of the dynamics of electrons in a crystal under the influence of an external electric field. Two small parameters e and h are introduced. They are respectively related to the scaled lattice length and the scaled Planck constant. When ε/h and h go to zero, we prove that in the limit process the dynamics of electrons is described by a Vlasov equation. When h is fixed and ε tends to zero, the dynamics is governed by an effective Schrödinger equation. In both cases, the mass of electrons is replaced by a matrix, the so-called effective mass.

AB - We study the semi-classical limit of the dynamics of electrons in a crystal under the influence of an external electric field. Two small parameters e and h are introduced. They are respectively related to the scaled lattice length and the scaled Planck constant. When ε/h and h go to zero, we prove that in the limit process the dynamics of electrons is described by a Vlasov equation. When h is fixed and ε tends to zero, the dynamics is governed by an effective Schrödinger equation. In both cases, the mass of electrons is replaced by a matrix, the so-called effective mass.

UR - http://www.scopus.com/inward/record.url?scp=0040062089&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040062089&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0040062089

VL - 21

SP - 1897

EP - 1918

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 11-12

ER -