Parabolic band

The intercept provides the reference for zero Knight shift. In a parabolic band the shift should be linear with N J3, as it is for except for p-type PbTe, whose VB is strongly non-parabolic. Reprinted with permission from [213], Copyright 1973 by the American Physical Society... 

In this approach to quantitatively analyzing the distribution of carrier concentrations, it was noted that the spatial length scale of dopant concentration fluctuations was an area for future exploration [207]. It is certainly clear that if each crystallite possessed a different dopant and thus carrier concentration, then this approach would be valid to the extent that the Knight shift followed an nj3 functional dependence as expected for a parabolic band. 

A very crude model to calculate the increase in bandgap energy is the effective-mass particle-in-a-box approximation. Assuming parabolic bands and infinitely high barriers the lowest conduction band (CB) level of a quantum wire with a square cross-section of side length w is shifted by AEC compared to the value Ec of the bulk crystal [Lei, Ho3] ... 

In Appendix A1 we have determined that for the general case of parabolic bands, p ci>) is given by... 

For some direct-gap materials, the quantum electronic selection rules lead to = 0. However, this is only strictly true at / = 0. For 0, it can be assumed, in a first order approximation, that the matrix element involving the top valence and the bottom conduction states is proportional to k that is, Pif k. Within the simplified model of parabolic bands (see Appendix Al), it is obtained that Tuo = Tuog + flp., and therefore Pif k co — cog). Thns, according to Equations (4.31) and (4.32), the absorption coefficient for these transitions (called forbidden direct transitions) has the following spectral dependence ... 

In order to obtain the freqnency dependence for the joint density of states p co) (Eqnation (4.32)), we assume the parabolic band structure given in Figure 4.8(a). For simplicity, we suppose that the bottom of the conduction band (E f = Eg) and the top of the valence band ( , = 0) are both at I = 0, as shown in Figure ALL Then, the E-k relationships are given by ... 

Figure Al.l E-k curves for a parabolic band structure, showing the frequency range (o (o + A(o between two k values, k and k2-...

It must be emphasized that these cross sections are only valid for an electron excitation into free-electron like final states (conduction band states with parabolic band shape) and not for resonance transitions as f — d or p - d excitations. If too low excitation energies (< 10 eV, see Table 1) are used in UPS, the final states are not free-electron like. Thus the photoemission process is not simply determined by cross-sections as discussed above but by cross-sections for optical transitions as well as a joint density of states, i.e. a combination of occupied initial and empty final states. 

In the metallic regime the electronic specific heat (Vest et al 1958, Zumsteg 1976) and the magnetic susceptibility (Zumsteg 1976, Graener et al 1969) provide evidence that the effective mass should be given by mcff/m 1.6, but the measured quantities vary as x rather than x1/3, as would be the case for a parabolic band... 

The photoconductivity and absorption spectra of the multilayer polydiacetylene are shown in Fig. 22 [150]. The continuous and dotted line relate to the blue and red polymer forms respectively. Interpretation is given in terms of a valence to conduction band transition which is buried under the vibronic sidebands of the dominant exciton transition. The associated absorption coefficient follows a law which indicates either an indirect transition or a direct transition between non-parabolic bands. The gap energies are 2.5 eV and 2.6 eV for the two different forms. The transition is three dimensional indicating finite valence and conduction band dispersion in the direction perpendicular to the polymer chain. 

The effect of critical scattering on the resistance was evaluated assuming that the holes reside in a simple parabolic band (Omiya et al. 2000). The corresponding contribution to the resistivity has to take into account the presence of correlation between neighboring spins, (Si Sj) (Sf)Sij (Dietl 1994),... 

To account for quantum mechanical effects, an approximate quantum model that reproduces the findings of the two classical spin-based approaches was constructed in a next step.37 One foundation of this model was the finding that several (nonfmstrated) molecular antiferromagnets of N spin centers 5 (which can be decomposed into two sublattices) have as their lowest excitations the rotation of the Neel vector, that is, a series of states characterized by a total spin quantum number S that runs from 0 to N x 5. In plots of these magnetic levels as a function of S, these lowest S states form rotational (parabolic) bands with eigenvalues proportional to S(S +1). While this feature is most evident for nonfmstrated systems, the idea of rotational bands can be... 

In the analyses of conventional ZB semiconductors, we frequently assume a symmetric parabolic band for the conduction band state, and the Luttinger-Kohn Hamiltonian is used to describe the valence band states. In general, the effective Hamiltonian is derived from a k.p perturbation theory or from the theory of invariants developed by Pikus and Bir. In the latter theory, the operator form of the effective Hamiltonian can easily be constructed from symmetry consideration alone. Within this framework, the lowest two conduction bands and the upper six valence bands are described to the second order of k. The invariant forms of the Hamiltonians are written as follows [26,27] ... 

For WZ compounds, we must consider hexagonal symmetry in the effective Hamiltonian. The Luttinger-Kohn Hamiltonian is constructed under the condition of cubic symmetry and the form reflects cubic crystal symmetry. Thus, in the analysis of WZ nitrides, we must use a k-dependent parabolic band for the conduction band state and Bir-Pikus Hamiltonians for the valence band states. The Hamiltonians for the upper six valence bands and the lowest two conduction bands are given by [28]... 

In the analyses of conventional zincblende (ZB) semiconductors, we frequently assume a parabolic band for the conduction bands, and the 6 x 6 Luttinger-Kohn Hamiltonians are used to describe the upper valence bands [1,2], In treating the valence bands together with the conduction bands on an equal footing, as when estimating the momentum matrix elements, we often make use of the 8 x 8 Kane Hamiltonian [3], However, the form of the Hamiltonians reflects the crystal symmetry, and Kane Hamiltonians are constructed under the condition of cubic symmetry. For wurtzite (WZ) materials, therefore, we must consider hexagonal symmetry in the effective Hamiltonian. Let us consider the 8 x 8 k.p Hamiltonian for WZ structure [4,5],... 

In the range of 1.55-6.35 eV, the absorption coefficient of SI was represented by one CL model. Based on the assumptions that the Hf-N bonds and Hf-O bonds independently form the parabolic band structure, and that total absorption coefficient is the sum of the absorption coefficients related to the Hf-N bonds and the Hf-O bonds two CL models were used as models for S2 and S3 films. Due to the degradation of Hf-O-N thin films, the simple grade model was applied after the fitting of films. 

Schematic parabolic band structure for CdSe, which has a band gap of 1.75 eV. The conduction band is labeled C, and several valence bands (V,) are shown. The filled and open circle symbols indicate the position of quantized k values mr/ai allowed for the / = 1 and n = 2 states of an NC with radius a. The solid arrow shows the / = 1 transition in which an electron is excited and a hole is created (open circle). The dashed arrow shows how the position of this n = i transition would change for a nanocrystal of smaller radius 32- (Adapted from Ref. 7.) This simple diagram is for the cubic zinc blend structure the hexagonal wurtzite structure has a small gap k= 0 between the and V2 bands.

Eg between the valence band and the conduction band. The band structure of a direct II-VI intrinsic semiconductor like CdSe can be represented reasonably well by a parabolic band model like that shown schematically in Fig. 2. Here, k = 7r/ris the wave vector and r is the radial distance from an arbitrary origin in the center of the crystal. The kinetic energy of the electron is proportional to E- and the energy minimum of the conduction band and the maxima of the valence bands occur at k = 0 (corresponding to r = co in a bulk sample). 

A more exact treatment of the electronic states of NCs reqnires nonparabolic bands that can be conpled to each other, a treatment of the energy dependence of the effective masses, consideration of both the nonsphericity of the NCs and the leakage of the wavefnnction ont of the confines of the NC, and inclnsion of electron-hole exchange." Althongh these are important refinements, the remarkable thing is the fact that many essential features of NC spectroscopy can be captnred by models as simple as the effective-mass parabolic band and Brus strong confinement descriptions. 

Fig. 7.17 estimates the parameter g for the actual conduction band density of states distribution of a-Si H in Fig. 3.16. The integral of the density of states up to energy E is plotted against N E). The equivalent ordered state is taken to be a parabolic band with the density of states of crystalline silicon. The parameter g decreases from the middle of the band to the band edge as expected and the results indicate that the mobility edge should occur near N E = 10 cm" eV", which is quite close to the value indicated by experiment. Unfortimately, this does not provide an accurate procedure for measuring E(, because there is not an exactly equivalent crystal with which to compare the density of states. Nevertheless it illustrates the principle.

It should be noted here that the overall shape of the K-phase FS can also be obtained by a simple free-electron treatment. With the usual parabolic bands and the known electron density one obtains a circular FS which cuts the Brillouin zone at approximately the point where the calculated gaps in Fig, 2.19 occur. Folding back these FS parts into the first Brillouin zone results in an only slightly modified topology compared to the calculated tight-binding FS of K-(ET)2l3. The effective masses estimated from the predicted band-structures are close to the free-electron mass, rrie. These values, however, are in contradiction to the experimentally extracted masses from optical [165, 166] and also dHvA or SdH measurements (see Sect. 4.2). 

The explicit form up to second order of the oscillatory part of M B) for a parabolic band was first given by Landau [250]. Taking into account phase smearing effects due to finite temperature, electron scattering, and electron spin finally resulted in the so-called Lifshitz-Kosevich formula [251] which is given here for one extremal area of the FS ... 

Figure 1 shows the ratio of the contribution between electrons and holes to thermopower as a function of reduced chemical potential. We have also shown the results in the case of parabolic bandCfS set to be zero while kept fixed value conesponds to the parabolic band). 

Acoustic Phonon Optical Phonon Ionised Impurity o Parabolic Band... 

For k-vectors away from the centre of the BZ, the whole EM Hamiltonian combines with the Bloch functions of the r + and p7+ VB states, and this breaks the parabolicity of the T7+ VB. Normally, for a parabolic band, the energy levels are independent from i and given by En (a.u.) = 1/n2. When non-parabolicity is taken into account, it has been shown by [14] that the modification of the energy levels from band mixing leads to a perturbation term yielding ... 

The analysis of the shape of the absorption edge of the high-pressure phase (Fig. 13) shows the existence of two spectral ranges with different types of energy dependence on the absorption coefficient. At high values of absorption it follows the empirical Tauc relation [57] in the case of parabolic band edges (Fig. 13(b)), while at smaller absorption a so-called Urbach or exponential absorption tail [58, 59] is observed (Fig. 13(c)). The existence of this kind of absorption edge is normally related to amorphous semiconductors. The optical absorption gap determined from our experiment is 0.6-0.7 eV and it decreases with pressure (see below). The slope of the Urbach tail, which can be considered as a measure of a random microfield [59] is found to be T=2.6 eV at 160 GPa. This is very close to what one would expect for an amorphous phase with a coordination of 2.5 [59]. 

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