Quasi-particle gap

Table 9 DFT-LDA electronic gaps in Ge-NWs and Si-NWs are reported respectively in the third and fifth column, quasi-particle gaps are reported for Ge-NWs in the fourth column. All values are in eV...

Nevertheless, the calculation described above for the quasi-particle gap in alternating fra j-polyacetylene proves that one can start from a high-quality HF wave function for infinite systems as well (in contrast to... 

Correlation Energy and Quasi-Particle Gap in a Cytosine Stack... 

Therefore the quasi particle gap is always smaller than the HF one. Further considerations /12,22/ indicate also that the band widths of the conduction and valence bands, respectively, are smaller in the QP (correlated) description, than in the HF case. 

On the other hand we have seen in the case of (CH) that by the application of a good basis the quasi particle gap becomes 3.0 eV instead of the minimal basis value of 8.3 eV /22, and point 3.1... 

Our results fit also with a previous investigation (9) on polyenes based on a version of the 2h-lp Cl scheme restricted to the virtual one-electron states generated by a minimal basis. In our case, however, the fragmentation of lines into satellites is much more pronounced. The reason lies in the size-consistency of the ADC[3] approach (as contrasted with the size-inconsistency of any truncated form of Cl (27d), in the full handling of the virtual space, and (10) in the inclusion of correlation corrections to the reference ground state, leading to (37) a net reduction of the quasi-particle band gap of conjugated polymers. 

It is important to realize that each of the electronic-structure methods discussed above displays certain shortcomings in reproducing the correct band structure of the host crystal and consequently the positions of defect levels. Hartree-Fock methods severely overestimate the semiconductor band gap, sometimes by several electron volts (Estreicher, 1988). In semi-empirical methods, the situation is usually even worse, and the band structure may not be reliably represented (Deak and Snyder, 1987 Besson et al., 1988). Density-functional theory, on the other hand, provides a quite accurate description of the band structure, except for an underestimation of the band gap (by up to 50%). Indeed, density-functional theory predicts conduction bands and hence conduction-band-derived energy levels to be too low. This problem has been studied in great detail, and its origins are well understood (see, e.g., Hybertsen and Louie, 1986). To solve it, however, requires techniques of many-body theory and carrying out a quasi-particle calculation. Such calculational schemes are presently prohibitively complex and too computationally demanding to apply to defect calculations. 

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]. 

As a characteristic feature, both the gap functions have nodes at poles (9 = 0,7r) and take the maximal values at the vicinity of equator (9 = 7t/2), keeping the relation, A > A+. This feature is very similar to 3P pairing in liquid 3He or nuclear matter [17, 18] actually we can see our pairing function Eq. (39) to exhibit an effective P wave nature by a genuine relativistic effect by the Dirac spinors. Accordingly the quasi-particle distribution is diffused (see Fig. 3)... 

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. 

When two trans-polyacetylene chains with different phases are put together, an obvious disturbance occurs in the standard conjugation pattern. The bond alternation defect that appears is known as a neutral soliton (Fig. 1.7). This kind of quasi-particle has an unpaired electron but is electrically neutral and is isoenergetically mobile along the polymer chain in both directions. This soliton gives rise to a state in the middle of the otherwise empty energy gap that can be occupied by zero, one or two electrons (Fig. 1.8). 

Key words superconductivity quasi-particles pseudo-gap bound states... 

One way of measuring the gap G, in insulators, consists in superposing direct and inverse photoemission spectra (quasi-particle spectra of the compound), and recording the smallest energy difference between them. G is thus related to the ionization potential and electron affinity by ... 

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]. 

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.

---