Wannier-Stark Ladders

The study of the effect of electric fields on the properties of solids dates back to Zener s (1934) investigation of electrical breakdown in solid dielectrics. Further pioneering work was carried out by Houston (1940) and Slater 

Having briefly noted the historical highlights of the WSL effect, we now examine the basic mathematical argument for its existence, namely, the idea that the energy spectrum of an infinite crystal is discretized by an applied held. For the present discussions, we assume that the applied field is linear, with its strength given by its gradient 7. 

We consider an atomic chain of spacing a, within the TBA, having the usual site energies 

Since the field-induced potential is assumed to be linear, and of the form 

Recalling that the anticommutation rules for the creation and annihilation operators are (App. C) 

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ANG AO ATA BF CB CF CNDO CPA DBA DOS FL GF HFA LDOS LMTO MO NN TBA VB VCA WSL Anderson-Newns-Grimley atomic orbital average t-matrix approximation Bessel function conduction band continued fraction complete neglect of differential overlap coherent-potential approximation disordered binary alloy density of states Fermi level Green function Flartree-Fock approximation local density of states linear muffin-tin orbital molecular orbital nearest neighbour tight-binding approximation valence band virtual crystal approximation Wannier-Stark ladder... 

We would like now to focus on initial conditions for the electron ensemble immediately after the optical excitation. The laser pulse may excite electron-hole pairs in the middle of the Wannier-Stark ladder, lower or above which within the miniband picture corresponds to excitation of electrons in the middle, bottom or top of the miniband, correspondingly. We assumed that all electrons were initially distributed in k-space in accordance with the normal distribution /(A J=(l/V2 o-i ) exp(-( -( j)2/2<7t2) where (k is the average k, and [Pg.201]

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