The Joint Density of States

Under the assumption that the matrix elements can be treated as constants, they can be factored out of the integral. This is a good approximation for most crystals. By comparison with equation Al.3.84. it is possible to define a fiinction similar to the density of states. In this case, since both valence and conduction band states are included, the fiinction is called the joint density of states ... 

Within this approximation, the structure in 2 ( )vc related to structure in the joint density of states. The joint density of states can be written as a surface integral [1] ... 

CO = coq, has a discontinuity in tire first derivative. In figure A1.3.18 the characteristic structure of the joint density of states is presented for each type of critical point. 

Figure Al.3.18. Typical critical point structure in the joint density of states.

For a given pair of valence and conduction bands, there must be at least one and one critical points and at least tluee and tluee critical points. However, it is possible for the saddle critical points to be degenerate. In the simplest possible configuration of critical points, the joint density of states appears as m figure Al.3.19. 

It is possible to identify particular spectral features in the modulated reflectivity spectra to band structure features. For example, in a direct band gap the joint density of states must resemble that of critical point. One of the first applications of the empirical pseudopotential method was to calculate reflectivity spectra for a given energy band. Differences between the calculated and measured reflectivity spectra could be assigned to errors in the energy band... 

Amorphous materials exliibit speeial quantum properties with respeet to their eleetronie states. The loss of periodieify renders Bloeh s theorem invalid k is no longer a good quantum number. In erystals, stnietural features in the refleetivify ean be assoeiated with eritieal points in the joint density of states. Sinee amorphous materials eaimot be deseribed by k-states, seleetion niles assoeiated with k are no longer appropriate. Refleetivify speetra and assoeiated speetra are often featureless, or they may eonespond to highly smoothed versions of the erystalline speetra. 

This disorder decrease on alloying is in principle unexpected, because alloying introduces chemical disorder in addition to bond disorder. However, Silva et al. [14] reported results on the joint density of states as determined by EELS which showed a clear decrease in the density of tail states at 7-at.% N and showed a further increase for higher nitrogen content. 

The electron is excited from a filled initial state f below the Fermi level F to an empty final state f above F. Momentum conservation will be provided by a lattice vector or in some cases by a surface vector. The transition probability is mainly determined by the optical excitation matrix element containing the joint density of states. 

From the physicochemical point of view, the prevailing one in the present review, the question is how these coefficients relate to the characteristics of the constituant molecules and how this information can be extracted from (2,3). The answer relies on the fact that the essential contributions to these integrals come from only few nonoverlapping critical regions in the joint density of states (18,19) these are points, lines and surfaces depending on the spatial extension of the conjugated electron distribution. They are defined by the condition... 

However, UPS and XPS do not both image the density of states in entirely the same way. In XPS, the photoelectrons originating from the valence band leave the sample with kinetic energies over 1 keV. In UPS, the exciting energy is on the order of 21 eV, and the kinetic energy of the electrons is low, say between 5 and 16 eV. This means that the final state of the photoelectron is within the unoccupied part of the density of states of the metal. As a result, the UPS spectrum represents a convolution of the densities of occupied and unoccupied states, which is sometimes called the "Joint Density of States."... 

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 ... 

The WL procedure can be applied to any chosen macrovariable, M. But while a good estimate G(E) is sufficient to allow multicanonical sampling in E [and a definitive one is enough to determine Z(p), Eq. (A10)], the M-density of states does not itself deliver the desired analogues we need, instead, the joint density of states G(E, M) which determines the restricted, single-phase partition functions through... 

Defining the joint density of states, g ( ), as the number of pairs of states in bands j and j separated by energy, ha, one obtains ... 

The PPV spectra of Fig. 16 show all the signatures of exciton absorption and emission, such as in typical molecular crystals. The existence of well-defined structure in the absorption spectrum is not so easily accounted for in a band-to-band absorption model. In semiconductor theory, the main source of structure is in the joint density of states, and none is predicted in one-dimensional band structure calculations (see below). However, CPs have high-energy phonons (molecular vibrations) which are known (see, e.g., RRS spectra) to be coupled to the electron states. The influence of these vibrations has not been included in previous theories of band-to-band transition spectra in the case of such wide bands [176,183]. For excitons, the vibronic structure is washed out in the case of very intense transitions, corresponding to very wide exciton bands, the strong-coupling case [168,170]. Does a similar effect occur for one-electron bands Further theoretical work would be useful. 

The integral is the joint density of states for which the energy difference is constant. The shape of the absorption spectrum is the product of the energy dependence of this integral and the matrix element. 

The Urbach edge represents the joint density of states, but is dominated by the slope of the valence band, which has the wider band tail. Expression (3.37) for is therefore also an approximate description of the thermal broadening of the valence band tail. It is worth noting that the slope is quite strongly temperature-dependent above 200 K. This may have a significant impact on the analysis of dispersive hole transport, in which the temperature dependence of the slope is generally ignored. 

The shape of the defect absorption is given by the joint density of states and the matrix elements, and the position and width of the defect band can only be deduced by the appropriate deconvolution. The usual approach is to model the defect band, for example, by a gaussian, and to calculate the absorption from the known shape of the conduction band. 

Potential fluctuations have a different effect on the optical absorption. Long range Coulomb fluctuations cause a parallel shift of both band edges, so that the optical absorption is not spatially inhomogeneous. Therefore the joint density of states distribution derived from electrical measurements should not agree with the shape... 

The variation of matrix elements over the band is more subtle. The values of Xi(0) arc very large in covalent solids as compared to atoms, a fact that will be discussed in the next section, and their size would suggest that the interatomic matrix elements, such as those in Eq. (4-14), are dominant. The simplest approximation is the neglect of any dependence of the matrix elements upon initial and final states one should notice that this will give different answers depending upon whether one assumes that matrix elements of djdx are constant or that the i- arc constant. As usual, the assumption of equal probability on the basis of lack of information is not unique. A common approximation is that the matrix elements of X (or 1/m times the matrix elements of d/dx) are constant, which from Eq. (4-9) gives a X2( ) value directly proportional to the joint density of states ... 

A simple approximation to the optical absorption spectrum is the joint density of states, corresponding to the assumption that equal matrix elements of x couple occupied states of density n,(E) with empty states of density Hc(fc ). In the upper diagram, the joint density ofstates, JDOS = + ftw), is shown on the right for the triangular... 

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