Section 10_Metals-Electron_dynamics_and_Fermi_surfaces

Section 10_Metals-Electron_dynamics_and_Fermi_surfaces -...

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Physics 927 E.Y.Tsymbal 1 Section 10 Metals: Electron Dynamics and Fermi Surfaces Electron dynamics The next important subject we address is electron dynamics in metals. Our consideration will be based on a semiclassical model. The term “semiclassical” comes from the fact that within this model the electronic structure is described quantum-mechanically but electron dynamics itself is considered in a classical way, i.e. using classical equations of motion. Within the semiclassical model we assume that we know the electronic structure of metal, which determines the energy band as a function of the wave vector. The aim of the model is to relate the band structure to the transport properties as a response to the applied electric field. Given the functions E n ( k ) the semiclassical model associates with each electron a position, a wave vector and a band index n . In the presence of applied fields the position, the wave vector, and the index are taken to evolve according to the following rules: (1) The band index is a constant of the motion. The semiclassical model ignores the possibility of interband transitions. This implies that within this model it assumed that the applied electric field is small. (2) The time evolution of the position and the wave vector of an electron with band index n are determined by the equations of motion: ( ) 1 ( ) n n dE d dt d = = k r v k k (1) ( , ) ( , ) d t e t dt = = k F r E r (2) Strictly speaking Eq.(2) has to be proved. It is identical to the Newton’s second law if we assume that the electron momentum is equal to k . The fact that electrons belong to particular bands makes their movement in the applied electric field different from that of free electrons. For example, if the applied electric field is independent of time, according to Eq.(2) the wave vector of the electron increases uniformly with time. ( ) (0) e t t = E k k (3) Since velocity and energy are periodic in the reciprocal lattice, the velocity and the energy will be oscillatory. This is in striking contrast to the free electron case, where v is proportional to k and grows linearly in time. The k dependence (and, to within a scale factor, the t dependence) of the velocity is illustrated in Fig.2, where both E ( k ) and v( k ) are plotted in one dimension. Although the velocity is linear in k near the band minimum, it reaches a maximum as the zone boundary is approached, and then drops back down, going to zero at the zone edge. In the region between the maximum of v and the zone edge the velocity actually decreases with increasing k , so that the acceleration of the electron is opposite to the externally applied electric force! This extraordinary behavior is a consequence of the additional force exerted by the periodic
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Section 10_Metals-Electron_dynamics_and_Fermi_surfaces -...

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