Wannier state

The band gap of a semiconductor represents the energy needed to produce an electron and a hole in a stable configuration, which minimizes their Coulomb interaction. Theoretical considerations show that [4] when two of these charge carriers approach they create a so-called Wannier state, which can be represented by a hydrogen-like Hamiltonian, as follows ... 

D translational invariance of the system, we classify the excitons by their inplane wavevector k. Supposing that for some bands of Frenkel excitons in the OQW and Wannier-Mott excitons in the IQW the energy separation is much less than the distance to other exciton bands we take into account only the hybridization between these two bands. We choose as a basis set the pure Frenkel and Wannier states, i.e. the state (denoted by F,k)) when the OQW is excited, while the IQW is in its ground state, and vice versa (denoted by W, k)), their energies being Ep(k) and W(k). We seek the new hybrid states in the form... 

Double Rydberg states (double because both electrons are running electrons in this case) and the symmetrical double-ionisation problem were considered first by Wannier [320], and so the completely symmetric excited states with i = I2 and nj = ri2 as the double-ionisation threshold is approached are called Wannier states. These states have very interesting... 

To a rather good approximation, the valence and conduction band Wannier states are equivalent to the bonding and antibonding states, respectively, that is,... 

To base considerations on states with localized extra electrons or holes, the Wannier-type transformation is introduced, yielding localized states within each unit cell. The number of these states equals the number of bands included in the Wannier transformation. In the case of elemental semiconductors such as diamond or siUcon these are one-electron states localized at the middle of the nearest atoms bonds so that n=l,2,3,4 (see Chap. 3 for the discussion of the connection between Bloch and Wannier states). A local description arises through the decomposition of the bkn into a sum of local operators marked with a cell index R (translation vector of the direct lattice) ... 

The focus of interest in homogeneously mixed valence is the 4f electron de-localization. In normal lanthanide solids (which have by definition integral lanthanide valence), the number of inner 4f electrons is integral. Moreover the 4f electrons then remain completely localized on a given lanthanide atom as all other inner core electrons there is no 4f band formation. The outer 5d6s valence electrons are delocalized conduction electrons (Wannier states of the solid). 

In a homogeneous mixed valent material an electron must temporarily leave the 4f shell. After the valence transition from 4" to 4f it occupies one of the Wannier states of the outer 5d6s shell on the atom in question. It may get there in more than one intermediate step, but it must end up in the valence electron shell of the same atom because of local charge neutrality, at least in a metal. In the reverse transition... 

Band Wannier States and Order-N Electronic-Structure Calculations. 

The small and weakly time-dependent CPG that persisLs at longer delays can be explained by the slower diffusion of excitons approaching the localization edge [15]. An alternative and intriguing explanation is, however, field-induced on-chain dissociation, a process that does not depend on the local environment but on the nature of the intrachain state. The one-dimensional Wannier exciton model describes the excited state [44]. Dissociation occurs because the electric field reduces the Coulomb barrier, thus enhancing the escape probability. This picture is interesting, but so far we do not have any clear proof of its validity. 

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... 

These two types of exciton are schematically illustrated in Figure 4.13. The Mott-Wannier excitons have a large radius in comparison to the interatomic distances (Figure 4.13(a)) and so they correspond to delocalized states. These excitons can move freely throughout the crystal. On the other hand, the Frenkel excitons are localized in the vicinity of an atomic site, and have a much smaller radius than the Mott-Wannier excitons. We will now describe the main characteristics of these two types of exciton separately. 

Thus, Mott-Wannier excitons can give rise to a number of absorption peaks in the pre-edge spectral region according to the different states = 1, 2, 3,... As a relevant example. Figure 4.14 shows the low-temperature absorption spectrum of cuprous oxide, CU2O, where some of those hydrogen-like peaks of the excitons are clearly observed. These peaks correspond to different excitons states denoted by the quantum numbers = 2, 3, 4, and 5. 

Wannier, G. H. (1959). Solid State Theory . Cambridge University Press, Cambridge... 

Blossey DF (1970) Wannier exciton in an electric field. I. Optical absorption by bound and continuum states. Phys Rev B 2 3976... 

The systems of valent states and oxidation states introduced by chemists are not merely electron accounting systems. They are the systems which allow us to understand and predict which ratios of elements will form compounds and also suggests what are the likely structures and properties for these compounds (3). In the case of highly covalent compounds, the actual occupancy of the parent orbitals may seem to be very different than that implied from oxidation states if ionicity were high. Nonetheless, even some physicists have recognized the fundamental validity and usefulness of the chemist s oxidation state approach where the orbitals may now be described as symmetry or Wannier orbitals (6). 

While the electronic structure calculations addressed in the preceding Section could in principle be used to construct the potential surfaces that are a prerequisite for dynamical calculations, such a procedure is in practice out of reach for large, extended systems like polymer junctions. At most, semiempirical calculations can be carried out as a function of selected relevant coordinates, see, e.g., the recent analysis of Ref. [44]. To proceed, we therefore resort to a different strategy, by constructing a suitably parametrized electron-phonon Hamiltonian model. This electron-phonon Hamiltonian underlies the two- and three-state diabatic models that are employed below (Secs. 4 and 5). The key ingredients are a lattice model formulated in the basis of localized Wannier functions and localized phonon modes (Sec. 3.1) and the construction of an associated diabatic Hamiltonian in a normal-mode representation (Sec. 3.2) [61]. 

This approach was developed originally as an approximate method, if the wave functions of isolated atoms are taken as a basis wave functions Wannier functions. Only in the last case the expansion (1) and the Hamiltonian (2) are exact, but some extension to the arbitrary basis functions is possible. In principle, the TB model is reasonable only when local states can be orthogonalized. The method is useful to calculate the conductance of complex quantum systems in combination with ab initio methods. It is particular important to describe small molecules, when the atomic orbitals form the basis. 

In all cases the same Wannier exponent occurs, and for a selected photon energy the excess energies differ due to the state-dependent ionization energies Ef+. However, of importance in the present context are the state-dependent values for the constants of proportionality a0. Within the LS-coupling scheme the complete final state built from the ion core and the electron-pair wavefunction must have Sf = 0 and L = 1. Therefore one gets the following possibilities ... 

InSe and GaSe crystals are characterized with a weak interaction of 3D Wannier excitons with homopolar optical A -phonons [18, 19]. Therefore, when calculating the exciton absorption spectra, we took into consideration effects of broadening the exciton states using the standard convolution procedure (see in [18]) for theoretical values of a(7jco) the absorption coefficient in the Elliott s model [20] with y /io>) — 77 [n(E 2+/ 2)] the Lorentzian function in the Toyozawa s model [21], where r is the half-width at half-maximum which is usually associated with the lifetime tl/2r. 

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