Scattering Umklapp

Fig. 10. The temperature dependence of thermal conductivity for pyrolytic graphite in three different conditions [66]. The reduction of thermal conductivity with increasing temperature is attributed to increasing Umklapp scattering of phonons.

If dE/dk=/= 0, then Umklapp scattering will occur The crystal momentum phonon will and must "borrow" momentum and energy from the reciprocal lattice ... 

Finally, it may be remarked that the two salts MEM(TCNQ)2 and TEA(TCNQ)2 bear several points of resemblance in their physical properties with the most interesting salts of the (TMTTF)2X series (TMTTF = tetramethyltetrathiafulvalene) [14]. They are all organic conductors with a quasi-one-dimensional character, with p = 5 (or ), and with a dominant electron-electron interaction. They exhibit comparable modest values of the electrical conductivity at high temperature, which indicate that the electrons are not very delocalized in the materials, and in all of them an underlying 4kF dimerization is also present, due to the cations. These common features, which are thought to be at the origin of sizable Umklapp scattering effects in the salts of the (TMTTF)2 series [14], could also be able to produce the same kinds of effects in MEM(TCNQ)2 and TEA(TCNQ)2 (see also Chapter 2). 

We may incidentally remark that the same kinds of arguments could also be reasonably invoked for the electrical properties of some materials of more recent concern. For instance, a resistivity minimum that is not related to a phrase transition is observed at Tp in (DMDCNQI)2Ag (DMDCNQI = dimethylcicyanoquinonediimine) [59] as in (TMTTF)2PF6 [60]. This minimum is attributed to the opening of a gap in the electronic spectrum of the materials below Tp, under the effect of Umklapp scattering, with a corresponding Mott-Hubbard localisation [14]. It is, however, also... 

In addition to the gx and g2 terms in Eq. (3) there are g3 terms involving the transfer of two particles from one side of the Fermi surface to the other. Since the total momentum transfer is 4fcF, this process is allowed only if 4fcF is a reciprocal lattice vector (Umklapp scattering). 

Figure 3 (a) Ground states of the one-dimensional mean-field model. SM, LM, TS, and SS indicate spin modulation, lattice modulation, and triplet and singlet superconductivity, respectively, (b) The most divergent correlations of the onedimensional Hamiltonian in the presence of Umklapp scattering SDW correlations are more divergent than CDW correlations in the g, > 0 region. 

Umklapp scattering can occur from transverse as well as from longitudinal... 

To evaluate the sum over the first term in parentheses, wc write e e 9 and sum over Vj, holding tlj fixed. The sum is identically zero unless k + Q — k is zero or is a lattice wave number. In this representation, there is no distinction between normal and umklapp scattering. If k = k 3- Q, the sum is and the remaining sum has become a sum over (L. Similarly, the sum over the second term in parentheses leads to a factor Af., e" and Eq. (19-53) becomes... 

In the presence of Umklapp scattering, the following susceptibilities will be considered for the Peierls channel at 2k% one has the on-site p, = SDW for spin-density-wave and p = BOW for the bond-order-wave in the Cooper channel, one has singlet p = SS and triplet p=TS superconducting correlations. At two-loop level, the auxiliary responses are governed by the flow equations... 

Figure 23 Phase diagram of the electron gas model in the repulsive sector (g, >0) and in the presence of Umklapp scattering.

In the present alloys Qpt equals 2kF. Under this condition, the phonon-rotons can easily interact with electrons for T > T0 causing inelastic umklapp scattering of the electrons. Below T0, only elastic umklapp scattering and inelastic scattering with normal phonons occur. Above T0, phonon-rotons can be excited thermally as well as by electron scattering. Electronic transport properties versus temperature may therefore be strongly affected (5.5.4). 

The resistivity of the alloys considered here show similar T-dependences [5.80, 111, 112]. p(T) of Cu20A180 at low temperatures is given as an example strongly enlarged in Fig. 5.27a. The corresponding thermopower is shown in Fig. 5.27c and for completeness the Hall coefficient in Fig. 5.27d. Below T0 the absolute values of p as well as S (T) are dominated by elastic umklapp scattering as discussed above. The increase of p(T) with decreasing Thas been ascribed to weak localization [5.113] or e-e interaction [5.114] as well as to the phonon-... 

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