Critical Point Analysis

    Optical and electron-energy-loss spectroscopies are well established methods of probing the electronic structure of materials. Comparison of experimental spectroscopic results with theory is complicated by the fact that the experiments extract information about the interband transition strength of electrons, while theoretical calculations provide information about individual valence and conduction bands. Based on the observation that prominent features in the optical response arise from critical points in the joint density of states, critical point modeling was developed to gain an understanding of these spectral features in terms of specific critical points in the band structure. These models were usually applied to derivative spectra and restricted to the consideration of isolated critical points. The authors present a new approach to critical point modeling of the undifferentiated spectra and interpret the model in terms of balanced sets of critical points which describe the interband transition strength arising from individual pairings of valance and conduction bands. This approach is then applied to achieve a direct, quantitative comparison of theoretical and experimental data on aluminum nitride.

    Major contributions to the interband transition strength Jcv(E) occur when (equation 10). Points in k-space where this condition is fulfilled are called critical points or van-Hove-singularities. In the vicinity of a critical point the interband energy can be approximated with a quadratic form:

The number of non-vanishing coefficients determines the dimensionality nj of jth critical point. The type of critical points (maximum, saddle point, minimum) corresponds to the possible choices of signs for the coefficients b1, b2 and b3. In the three dimensional case the critical points are labeled Mo, M1, M2 and M3. For the two and one dimensional band system they are denoted Do, D1 and D2 and Po and P1, respectively, while a zero dimensional band system is denoted S_o. A model of a real electronic structure may be constructed by summing together the contributions from individual critical points. Each feature in an experimental Jcv(E) spectrum has to be described by a balanced set of critical points (Loughin et al., 1996, Loughin, 1992). A balanced set consists of a number of critical points such that outside the energy range over which transitions can occur the model predicts a vanishing optical response. Therefore the line shape of the critical points used for a set have to cancel outside the interesting range. A computer program (Loughin et al., 1996) was used for optimizing the parameters of the critical point model.

Fig 6. Sums of critical point functionals are used to create balanced sets, shown here for the 3D case (a), 2D case (b), 1D case (c), and the 0D case (d).

Fig. 10. The critical point model for Al2O3, consisting of 3D balanced sets, is fitted to experimental data collected on single crystal sapphire at room temperature.

Figure 5. Critical point models consisting of 1D balanced sets, used to model the interband transitions of a linear polymer, poly(di-n-hexylsilane). This model consists of an exciton peak associated with a 1D set for the Si backbone and a second 1D set for the hexyl side chain transitions.

Comment: (c) 2003 Roger H. French , frenchrh@lrsm.upenn.edu
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