Quasi-particle spectrum

It has been argued that, while the HOMO-LUMO energy difference suffers from systematic errors, a more reliable estimate of G can be obtained from the quasi-particle spectrum of the ionized system. For example, in NiO, the presence of an excess hole leads to the formation of a narrow band of unoccupied states in the VB region [249] (Fig. 11). It has been stressed [250] that the gap between this band and the CB edge approximates the optical/conductivity band gap. In Li NiO [251] and bulk NiO [249], this method yields a value of 4 eV, in agreement with optical absorption measurements [252]. It was also used to estimate G in an NiO(lOO) monolayer [85]. 

A first approximation to the total quasi-particle spectrum, is given by the DOS. As was the case for 7, A and G, a large part of the LDOS modifications on under-coordinated atoms is well accounted for by an effective one electron approach, because the shape of the LDOS reveals the pecu-... 

Quasi-particle spectrum has a gap over the entire Fermi surface (fig. 105a)... 

It is instructive to start with the excitation spectrum in the case of the ordinary 2SC phase when dfi = 0. With the conventional choice of the gap pointing in the anti-blue direction in color space, the blue quarks are not affected by the pairing dynamics, and the other four quasi-particle excitations are linear superpositions of ur>g and dr(J quarks and holes. The quasi-particle is nearly identical with a quark at large momenta and with a hole at small momenta. We represent the quasi-particle in the form of Q(quark, hole), then the four quasiparticles can be represented explicitly as Q(ur,dg), Q(ug, dr), Q(dr,ug) and Q(dg,ur). When S/i = 0, the four quasi-particles are degenerate, and have a common gap A. 

If there is a small mismatch (dp < A) between the Fermi surfaces of the pairing u and d quarks, the excitation spectrum will change. For example, we show the excitation spectrum of Q(ur, dg) and Q(dg,ur) in the left panel of Figure 3. We can see that 5p induces two different dispersion relations, the quasi-particle Q(dg,ur) has a smaller gap A — p, and the quasi-particle Q(ur,dg) has a larger gap A //. This is similar to the case when the mismatch is induced by the mass difference of the pairing quarks [16]. 

An immediate consequence of symmetry is the nature of the quasiparticle spectrum. The density of states p(co) can be calculated from the knowledge of the quasi-particle energies. For singlet states Ek,+ = Ek, with... 

The experimental 5s XPS spectrum in Xe7,8 (Fig. 32a) looks quite similar to the 4p spectrum in Ba, suggesting that the 5 s level lies just below the 5 1 level structure, forming a well-defined 5 s1/2 quasi-particle excitation and giving considerable strength to a prominent satellite spectrum, mainly 5 p25 d94, 95. The basic process is again giant Coster-Kronig fluctuation of the core hole... 

To cover the gap between them the Hubbard model Hamiltonian was quite generally accepted. This Hamiltonian apparently has the ability of mimicking the whole spectrum, from the free quasi-particle domain, at U=0, to the strongly correlated one, at U —> oo, where, for half-filled band systems, it renormalizes to the Heisenberg Hamiltonian, via Degenerate Perturbation Theory. Thence, the Heisenberg Hamiltonian was assumed to be acceptable only for rather small t/U values. 

Quantum field theory justifies the assumption that to a first approximation the structure of the energy level spectrum obeys the same principle as that of the energy levels of an ideal gas. In other words, any energy level can be obtained as the sum of energies of a certain number of quasi-particles or elementary excitations , with momenta p and... 

The calculation of a quasi-particle (QP) spectrum in a crystal requires the resolution of an equation of the type ... 

Table 4 lists the MBPT(2) band gaps of polyacetylene calculated with basis set 6-31G and DZP at three different geometries by us [36]. The cutoffs N and K are both 21. The geometries used in the calculations are listed in Table 5. The first two were given by Suhai [53,55] and the last one was an experimentally estimated geometry [97], The band gaps obtained are 4.033, 3.744, and 3.222 eV, respectively. There is no direct measurement of the band gap, defined as a quasi-particle energy difference of the lowest unoccupied and highest occupied orbitals. Instead, the absorption spectrum of polyacetylene crystalline films rises sharply at 1.4 eV and has a peak around 2.0 eV [97]. To explain this measured spectrum, one needs to calculate the density of the system s excited states and the absorption coefficients of the states.

Quasi-particle P3 calculations produce better accuracy than outer valence Green s function (OVGF). An application of the photoelectron spectrum of r-tetrazine illustrates the ability of the P3 method to predict correct final-stage orderings <1997IJQ291>. 

Fig. 10.9. (a) Calculated optical absorption spectrum of (rans-polyacetylene from a DFT-GWA-BSE calculation. The solid and dashed curves represent the exciton and quasi-particle spectra, respectively, (b) The electron-hole distribution function. Reprinted with permission from M. Rohlfing and S. G. Louie, Phys. Rev. Lett., 82, 1959, 1999. Cop5rright 1999 by the American Physical Society. 

As a result, quasi-particles in graphene exhibit the linear dispersion relation E = hvp, where vp is the Fermi velocity ( 10 m/s), as if they were massless relativistic particles. Thus, graphene s quasiparticles behave differently from those in conventional metals and semiconductors, where the energy spectrum can be approximated by a parabolic dispersion relation. Electron transport in all known condensed-matter systems is described by the (non-relativistic) Schrodinger equation and relativistic effects are usually negligible. In contrast, the electrons of graphene are described by the (relativistic) Dirac equation, i.e. they mimic relativistic charged particles with zero rest mass and constant velocity [10]. 

Correlation, or many-body effects, can be classified according to the many-body factor Xx- If Xx is close to 1, the MO picture, the aufbau principle, a Koopmans theorem and the quasi-particle picture hold. The analysis of the Auger spectrum can then be conducted solely in terms of MO theory. When more than one Xx enters in the wavefunction, we have hole-mixing effects and electronic interference in the transition cross sections, in analogy to the case of photoelectron spectra. When only one Xx is large, but this Xx is present in more than one state, one can then not associate a one-to-one correspondence between MOs (or MO factors in Eq. 3.39) and spectral bands (states). The states in question are thus associated with a breakdown of the MO picture. It could, finally, also be that no Xx is large, in which case we talk about a correlation-state satellite. 

Discretization in small partides is caused by their intermediate position between molecules, having a discrete spectrum of electron energy levels, and bulk sdids, which have the quasi continuous spectrum ctf electron energy levels. SmaU particles comprising thousands or tens of thousands of atoms have distances between energy levels of the order of 10 eV-i.e. rather small but detectable. 

---