Densities of state

5 Density of States and Specific Heat 3.5.1 Density of States 

One frequently encounters lattice properties that are sums of the form S = I K[03.(q)]. (3.76) 

Examples are the mean energy E given by (2.131) or the specific heat 

Cy = dE/dT. Since the density of the values of q in reciprocal space is high 

An example of such a frequency integral is the specific heat given by (2.142). By comparing (3.79) with (3.77), we obtain 

The density of states per site, N(e), is calculated following the method used by Mele and Rice [16]  

In order to avoid divergences in the density of states we make the substitution e=e+ix (x=0.03 eV). This substitution introduces a Lorenzian shaped broadening of the true eigenstates corresponding to the Hamiltonian in Eq. 1, 

The electronic density of states (DOS) describes energy of electrons in solids. By definition, the density of states p(B) is a value, which being multiplied by a small interval of energies dE equals to a number of the electronic states with energies in an interval from Eto E dE. In solids, the density distributions are not discrete like a spectral density but continuous. A high density of states at a specific energy level means that there are many states available for occupation. A density of states of zero means that no states can be occupied at that energy level. Local variations, most often due to distortions of the system, are often called local density of states (LDOS). 

F ure 9.1 Calculated electronic density of states for silver (a) and platinum (b). A difference between the energy of electrons and the Fermi energy Ef is plotted onto abscissa axis zero is the Fermi energy, above which excited states are located (after [26]). 

Transitions from the initial state k to the final k are induced by the matrix elements (11.31). The wavevector k of the initial state can take values up to the 

TABLE 11.1 Matrix Elements for Electronic Transitions Obtained Using the 2D Bloch Model 

Here the vector a contains the expansion coefficients and the matrix elements of H are over the atomic orbitals Hij = i H j). The local charge density p(r) can be obtained as 

The density of states around r at a fixed energy E is defined through the equation 

If we assume that the electronic wavefunction is periodic in the (.x, y) direction of the lattice and has an infinite barrier at z = 0 and at in a metal slab with thickness L, the wave function is simply 

In a crystal, instead of this simple relationship between energy and momentum which applies to free electrons, we have to use the band-structure calculation for k. Then the expression for the density of states becomes 

The expansion is over as many principal axes as the dimensionality of our system in three dimensions (d = 3) there are three principal axes, characterized by the symbols oii,i = 1,2,3. Depending on the signs of these coefficients, the extremum can be a minimum (zero negative coefficients, referred to as type 0 critical point ), two types of saddle point ( type 1 and 2 critical points ) or a maximum ( type 3 critical point ). There is a useful theorem that tells us exactly how many critical points of each type we can expect. 

Theorem Given a function of d variables periodic in all of them, there are d /l d —1) critical points of type /, where I is the number of negative coefficients in the Taylor expansion of the energy, Eq. (5.5). 

With the help of this theorem, we can obtain the number of each type of critical point md — 3 dimensions, which are given in Table 5.1. 

Next we want to extract the behavior of the density of states (DOS) explicitly near each type of critical point. Let us first consider the critical point of type 0, Mo, in which case a 0, i = 1,2, 3. In order to perform the k-space integrals involved in the DOS we first make the following changes of variables in the neighborhood of the critical point at ko. 

This effect is also found for the bandwidth of the 0(2p) bands for the alkaline earth metal oxides which, at the LDA level of theory, decrease down the group from a calculated value of 4.44eV (MgO) to 1.83eV (BaO). This is partly due to the increase in lattice parameter, which spaces the O ions more widely in BaO than in MgO. However, in addition, it is found that the outermost valence electrons for the metal ions interact more strongly with the 0(2p) states in BaO than in MgO, giving more localization of the electron density at O and so a smaller anion in BaO [50]. 

Converging the self-consistent procedure in periodic calculations can be a difficult task, particularly if there are free states within any band. Movement of electrons between almost identical states has little effect on the energy of the system and so the search for the optimal distribution of electrons is hampered by many trivial alterations to the occupation numbers. To make the process more efficient, partial occupancies can be used for states near the highest filled level by introducing a smoothing function which defines the occupancy as a function of state energy, Tj,. The smoothing function used in the MgO calculation was  

This expression is based on the statistical mechanics of electrons which have an energy distribution given by Fermi-Dirac statistics [51]. The partial occupancy 

For a free electron in three dimensions the energy is given by 

The number of electrons that can be accommodated in states with energy E or less is  

Here the factor of two is because each state has two possible electron spins and the 1/8 is because we must take only positive values of nx, ny, and Hj. The density of these states g(E) per unit volume of reciprocal space in an energy interval dE is given by (l/V)(dN/dE), where V=L is the crystal volume (for a cube-shaped soHd), or  

This density of states will apply to any band extremum where the band can be approximated at least locally with a quadratic dependence of energy on momentum (free-electron-like). Since all band edges wiU have this general behavior. Equation 2.22 provides an approximate picture of the number of states per unit energy near a band edge. 

The importance of the density of states may be found in the calculation of the rate of any process in a solid, from scattering of an electron off a defect or another electron to absorption and emission of light. The rate of such a process is given in its most general form by Fermi s Golden Rule . Mathematically, the rate of a process (H) moving an electron from state vpi to state /f may be written in symbols as follows  

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Periodic boundary conditions force k to be a discrete variable with allowed values occurring at intervals of lull. For very large systems, one can describe the system as continuous in the limit of i qo. Electron states can be defined by a density of states defmed as follows ... 

The density of states for a one-dimensional system diverges as 0. This divergence of D E) is not a serious issue as the integral of the density of states remains finite. In tliree dimensions, it is straightforward to show that... 

One can detennine the total number of electrons in the system by integrating the density of states up to the highest occupied energy level. The energy of the highest occupied state is called the Eermi level or Eermi energy, E ... 

Another usefiil quantity is defining the electronic structure of a solid is the electronic density of states. In general the density of states can be defined as... 

Figure Al.3.15. Density of states for silieon (bottom panel) as ealeulated from empirieal pseudopotential [25], The top panel represents the photoemission speetra as measured by x-ray photoemission speetroseopy [30], The density of states is a measure of the photoemission speetra.

By examining the spatial eharaeter of the wavefiinetions, it is possible to attribute atomie eharaeteristies to the density of states speetnun. For example, the lowest states, 8 to 12 eV below the top of the valenee band, are s-like and arise from the atomie 3s states. From 4 to 6 eV below the top of the valenee band are states that are also s-like, but ehange eharaeter very rapidly toward the valenee band maximum. The states residing within 4 eV of the top of the valenee band are p and arise from the 3p 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] ... 

The types of critical points can be labelled by the number of less than zero. Specifically, the critical points are labelled by M. where is the number of which are negative i.e. a local minimum critical point would be labelled by Mq, a local maximum by and the saddle points by (M, M2). Each critical point has a characteristic line shape. For example, the critical point has a joint density of state which behaves as = constant x — ttiiifor co > coq and zero otherwise, where coq corresponds to thcAfQ critical point energy. At... 

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. 

Note that if we identify the sum over 8-fimctions with the density of states, then equation (A1.6.88) is just Femii s Golden Rule, which we employed in section A 1.6.1. This is consistent with the interpretation of the absorption spectmm as the transition rate from state to state n. 

The only modification of equation (Al.6.90) for spontaneous Raman scattering is the multiplication by the density of states of the cavity, equation (Al.6.24). leading to a prefactor of the fonn cojCOg. ... 

Photoelectron spectroscopy provides a direct measure of the filled density of states of a solid. The kinetic energy distribution of the electrons that are emitted via the photoelectric effect when a sample is exposed to a monocluomatic ultraviolet (UV) or x-ray beam yields a photoelectron spectrum. Photoelectron spectroscopy not only provides the atomic composition, but also infonnation conceming the chemical enviromnent of the atoms in the near-surface region. Thus, it is probably the most popular and usefiil surface analysis teclmique. There are a number of fonus of photoelectron spectroscopy in conuuon use. 

In general, the phonon density of states g(cn), doi is a complicated fimction which can be directly measured from experiments, or can be computed from the results from computer simulations of a crystal. The explicit analytic expression of g(oi) for the Debye model is a consequence of the two assumptions that were made above for the frequency and velocity of the elastic waves. An even simpler assumption about g(oi) leads to the Einstein model, which first showed how quantum effects lead to deviations from the classical equipartition result as seen experimentally. In the Einstein model, one assumes that only one level at frequency oig is appreciably populated by phonons so that g(oi) = 5(oi-cog) and, for each of the Einstein modes. is... 

Thus many aspects of statistical mechanics involve techniques appropriate to systems with large N. In this respect, even the non-interacting systems are instructive and lead to non-trivial calculations. The degeneracy fiinction that is considered in this subsection is an essential ingredient of the fonnal and general methods of statistical mechanics. The degeneracy fiinction is often referred to as the density of states. 

Then F( ) = S(/ -t d )- 2( ), and the density of states D E) = dS/d/ . A system containing a large number of particles N, or an indefinite number of particles but with a macroscopic size volume V, normally has the number of states S, which approaches asymptotically to... 

The complete thennodynainics of a system can now be obtained as follows. Let die isolated system withAi particles, which occupies a volume V and has an energy E within a small uncertainty E, be modelled by a microscopic Flamiltonian Ti. First, find the density of states p( ) from the Flamiltonian. Next, obtain the entropy as S(E, V, N) = log V E) or, alternatively, by either of the other two equivalent expressions... 

The canonical distribution corresponds to the probability density for the system to be in a specific microstate with energy E- H, from it one can also obtain the probability P( ) that the system has an energy between E and E + AE i the density of states D E) is known. This is because, classically. 

For a spin-zero particle in a cubic box, the density of states is... 

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

It may be iisefiil to mention here one currently widely applied approximation for barrierless reactions, which is now frequently called microcanonical and canonical variational transition state theory (equivalent to the minimum density of states and maximum free energy transition state theory in figure A3,4,7. This type of theory can be understood by considering the partition fiinctions Q r ) as fiinctions of r similar to equation (A3,4.108) but with F (r ) instead of V Obviously 2(r J > Q so that the best possible choice for a... 

Equation (A3.11.183) is simply a fommla for the number of states energetically accessible at the transition state and equation (A3.11.180) leads to the thenual average of this number. If we imagine that the states of the system fonu a continuum, then PJun, 1 Ican be expressed in tenus of a density of states p as in... 

The RRKM rate constant is often expressed as an average classical flux tlirough the transition state [18,19 and 20]. To show that this is the case, first recall that the density of states p( ) for the reactant may be expressed as... 

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