Temperature dependence of the energy gap

The PCS technique has demonstrated an ultra-sensitive ability to probe the conductive properties of the layered superconductor SnNbsSe We have shown that PCS is a unique tool for the detection of nanoclusters which in our case were likely formed by a small fraction of unreacted initial substances (A15 and dichalcogenide) used to synthesize the ternary compound. Finally, we have determined the value and the temperature dependence of the energy gap parameter in the SnNb5Se9 phase. 

We will also mention here the temperature dependencies of the energy gap Eg... 

Fig. 3 Lowest electronic states (a) for DyV04 crystal, temperature dependence of the energy gap between ground and excited doublets (b) (for comparison Raman scattering results are shown for the DyAs04 crystal with similar electronic and crystal structures), and the ultrasonic measurements (c) of the elastic constant Ci = l/2(Cn-Ci2) for DyV04 crystal...

When temperature is lowered, the band gaps usually increase [15]. There again, a few materials like lead sulphides or some copper halides are exceptions with a band gap increasing with temperature [96]. A quantitative analysis of the temperature dependence of the energy gaps must consider the electron-phonon interaction, which is the predominant contribution, and the thermal expansion effect. The effect of thermal expansion can be understood intuitively on the basis of the decrease of the interatomic distances when the temperature is decreased. A quantitative analysis of the electron-phonon contributions is more difficult, and most calculations have been performed for direct band-gap structures [75], Multi-parameter calculations of the temperature dependence of band gaps in semiconductors can be found in [81],... 

CuBr also shows an anomalous temperature dependence of the energy gap. The postulate that the vibration of Cu ions leads to an increase in the gap while that of the halide ion results in a decrease in the gap is thus reasonable. These observations have an important bearing on the temperature-dependent excitonic spectra of CuBr nanocrystals. 

This implies also that the temperature dependence of the real energy gap A(T) in highly one-dimensional systems taking account of the fluctuations is different from the mean-field theory prediction for the temperature dependence of the energy gap, Amf(T)- Figure 9.11 shows this in-principle result [11], which can be experimentally tested, since A(T=0) and Tp can be directly measured (see e.g. [12]). 

Figure 9.16a shows the temperature dependence of the energy gap, A(T) (solid curve), as obtained from a measurement of the magnetic susceptibility of the conduction electrons (see Sect 9.6.4 and [24]). With this information, the experimental temperature dependence a(1) was fitted to Eq. (9.14) (Fig. 9.17). The fit shows that the essential characteristics of a (7) from room temperature down to about 50 K, with a variation of more than eight orders of magnitude, are correctly described by this equation. It is thus justified to use Eq. (9.14) alone for the determination of the energy gap A(7) and the constant C. Figure 9.16a shows the energy gap A(T) (dashed curve), as determined directly from the conductivity a(T) (Fig. 1.13) using Eq. (9.14). From this fit, the constant C is also obtained (Table 9.3).

Using the value of xo. the temperature dependence of the energy gap, Aeff(T), was determined from the experimental values using this expression (Fig. 9.20). It shows qualitatively the same temperature dependence as was found from the temperature dependence of the specific electrical conductivity (cf Fig. 9.16a). In particular, it exhibits the effective band gap above the Peierls transition, which is caused by the fluctuations of the one-dimensional CDW conductor and has the result that even above the Peierls transition, no genuine metallic conductivity is present. 

Fig. 15. Schematic illustration of the temperature dependence of the Peierls transition in mean field theory for a one-dimensional metal. Above Tp the temperature dependence of the energy of the 2kp phonon is illustrated. Below Tp, the temperature dependence of the energy gap A is illustrated. The energy scale above and below Tp is not the same.

Table4.1-85 Temperature dependence of the energy gaps of indium compounds...

Fig. 12. Momentum and temperature dependence of the energy gap estimated ftom leading edge shifts of ARPES spectra for BSCCO-2212. (a) i-dependence of the gap in the rc = 87K, 83 K, and lOK samples, measured at 14 K. The inset shows the Brillouin zone with a large Fermi surface (FS) closing the (3t,3t) point, with the occupied region shaded, (h) Temperature dependence of the maximum gap in a near-optimal 7), = 87 K sample (circles), and two underdoped samples with rc = 83K (squares) and 7). = lOK (triangles). From Ding et...

Figure 4.10(b) shows the temperature dependence of the absorption spectrum expected for an indirect gap. It can be noted that the contribution due to becomes less important with decreasing temperature. This is due to the temperature dependence of the phonon density factor (see Equation (4.37)). Indeed, at 0 K there are no phonons to be absorbed and only one straight line, related to a phonon emission process, is observed. From Figure 4.10(b) we can also infer that cog shifts to higher values as the temperature decreases, which reflects the temperature dependence of the energy... 

The temperature and doping dependence of the energy gap can also be derived as follows. The energy to remove one electron pair from the condensate at absolute zero is... 

The second component, is from the temperature dependence of the energy levels with respect to each other. Optical absorption data find that the band gap, g, decreases with increasing temperature with a coefficient of 7 0 = 5 = 4.4 x 10 eV (Tsang and Street 1979). The Fermi energy is never further than half the band gap energy from the main conduction path and consequently Yg is at most Ygo- There is no detailed information about the temperature dependence of the defect levels and it is usually assumed that Yq is proportional to... 

The Fermi energy in doped samples is within the band tail, so there should be no significant contribution to the conductivity from the temperature dependence of the band gap. Thus, in the absence of any temperature dependence of E, is equal to the measured conductivity prefactor. 

Studies of radiationless transitions in matrix-isolated molecules represent a nice case of constructive interaction between theory and experiment. Most early experimental studies showed very poor agreement with the theoretical predictions. Thus in NH and OH, neither the expected temperature dependence, nor the energy-gap law predictions were fulfilled. Similarly, no steep temperature dependence of the relaxation rates was found in matrix-isolated CO. The experimental studies, however, permitted to identify the reasons for the failure of the simple theories. This in turn led to development of new models, describing more adequately the experimental results. " ... 

In sharp contrast to CuCl and CuBr, Cul exhibits a positive temperature dependence of the band gap mainly due to electron-phonon interaction. Gogolin et al. [10] studied the temperature dependence of exciton peak energies in Cul quantum dots (average radius = 6 nm) embedded in a glass matrix. The Zi,2 and Z3 exciton peaks of the zincblende type dots as well as Hi, H2 exciton peak in the hexagonal type dots both show a red shift upon heating. 

Energy gap values from tunneling experiments. Recently it has been suggested that the valence and conduction bands of SnTe are inverted from those of PbTef, on the basis of the temperature dependence of their energy gaps [24]. 

These features of the pseudogap are illustrated in figs. 12a,b which show the k-dependence of the energy gap and the dependence of the maximum gap on temperature from the ARPES measurements of Ding et al. (1996) on BSCCO-2212 samples with T s of 87 K, 83 K, and 10 K. The pseudogap and superconducting regions for BSCCO-2212... 

For comparison, in the BCS theory this ratio is 3,52. In relative values both the BCS and our dependence of the energy gap on the temperature are exactly the same (i.e. the dependences of A/Aq on T/Tc). The study of other physical properties, such as specific heat, is published in our previous paper [19]. Let us note that the Eq. 28.93 was derived without any specific requirements for the detailed mechanism of superconductivity in comparison with the BCS theory. It reflects the thermodynamical properties of non-adiabatic systems in a more general form, solely as a consequence of the solution of the extended Born-Handy formula. 

The Boltzmann equation (Equation 18.2) shows that, under equilibrium conditions, the ratio of the number (n) of ground-state molecules (A ) to those in an excited state (A ) depends on the energy gap E between the states, the Boltzmann constant k (1.38 x 10" J-K" ), and the absolute temperature T(K). 

A closely related test of the energy gap law for Ru complexes has come from temperature dependent lifetime and emission measurements for a series of complexes of the type RuO py I " (L py, substituted pyridines, pyrazine...). From the data, the variation in lnknr with Eem predicted by the energy gap law has been observed and it has been possible to observe the effect of changing the ligands L on the transition between the MLCT and dd states (20). 

Fig. 10 Theoretical curves [equation (12)] representing the temperature dependence of the signal intensities due to a triplet species that has a singlet manifold in equilibrium. The A -values represent the energy gap of the two states.

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