ANHARMONICITY AS AN EXPLANATION FOR ANOMALOUS RESISTANCE OF HIGH-TC SUPERCONDUCTORS

Abstract Experimental evidence about the temperature dependence of phonon frequencies in V3Si is used to demonstrate that anharmonicity is a major factor in causing the resistance anomalies recently observed in many high-Tc superconductors.

factor in causing the resistance anomalies recently observed in many high-Tsuperconductors. [p(300) -Po] can be estimated from the data to be s-p and d-band constituents) deviates strongly from the about 0.5. The data in Fig. 1 is taken from a published Bloch-Gruneisen 2 behavior generally found in other paper3 and the drawing of slopes is hazardous. We can metals. A typical example is V 3Si for which the data 3 be in error in estimating~(300) by as much as 50% in are shown in Fig. 1. Usually the Bloch-Gruneisen theory either direction. is interpreted to mean that the difference between the Many possible explanations for this discrepancy resistivity p(T) for T>>°D from the limiting (impurity have been given. 6 In this note we present evidence for a dominated) resistivity for T -÷ 0 is linear in temperature, particularly simple explanation, namely that Xe,. in i.e. equation (1)  (2) mass, and pep(T) is a complicated integral over the phonon distribution which simplifies at high tempera-where i~tris a purely electronic factor involving the electures to a linear power of T. The strength of the elec-tron density of states and matrix elements averaged over tron-phonon coupling is measured by the dimensionless the Fermi surface, Mis an averaged atomic mass, and number4 Xe,. which is closely related to the coupling (w2) is a mean square phonon frequency calculated with constant X of superconductivity theory.5 It is immedi-a weight factor F(w)/w where F(w) is the phonon denately clear that the data of Fig. 1 cannot satisfy equation sity of states. (This involves an assumption that the (1) because the high temperature resistivity is not really coupling factor~r(~') is roughly independent of w, an linear. If we denote by p*(T) the extrapolated resis-approximation which probably only fails at small w.) If tivity at zero temperature from the slope at we assume the validity of equation (1) at high temperatures, then we can write * Supported in part by the National Science Foundation 2)T (5) Fig. 1. Experimental values of electrical resistance p as a function of temperature T for V 3Si (from reference 3). where V34 schematically denotes the dependence oil The value p*(296) -Po = 35 p~2-cm found by linear the cubic and quartic anharmonicity and (u 2)T is the extrapolation deviates significantly from the Bloch-(;runeisen value of 0. mean square displacement at a temperature T. Notice that to O((u2>) the contribution of the cubic anharmonicity would be such as to decrease the phonon frequencies and of the quartic anharmonicity to increase them.
The experimental results lead one to conclude that the 6.7 quartic anharmonicity is dominant in V 3S1 (and we suspect also in other transition metal compounds). The physical origin of the strong quartic anharmonicity probably lies in the strong short range repulsion of tightly bound d shells.
We may put for~i where we determine c by fitting to the data of Fig. 2.

LOG I
Choosing B (w2(0))''2 296K, we find that a value Fig. 2. Experimental values of log (w2)(calculated from c = 0.84 fits the available data. The resulting curve is the phonon densities of states of reference 7) plotted vs shown as the solid line in Fig. 2, and yields a value log T. The solid line is an empirical fit using equation (6). (d log~w2)/d log T) = 0.31. Inserting in (3) this gives us only about 60% of the effect we are looking for. How-Our mechanism is based on the experimental results ever, as already mentioned our deduction of~(30O) on (w 2(T)) which can only be understood if the anhar. from the data is quite uncertain. So is the deduction of monic forces in V 3SI are anomalously large. Indepen-(d log (w 2)/d log 7) at room temperature from the data dent evidence that this is so are provided by the extrawhich extends only to room temperature. More precise ordinary harmonic generation observed by Testardi11 in knowledge of resistivity as well as measurements of (w2) V 3Si. The confirmation of strong anharmonicity in at higher temperatures are required to see whether other high temperature superconductors discussed in mechanisms other than the one discussed here contri-reference 1 must await determination of (w 2(T)) for bute significantly to the resistivity anomaly.
them. The reason for the relatively strong anharmoni-Reichardt and collaborators at Karisruhe have also city of high temperature superconductors is that the sent us unpublished data for F(w) in V 3Ga, V3Ge, and harmonic force constants in these materials as measured, Nb3Sn at two different temperature each. The moments say, by the low temperature sound velocities, are relaare shown in Table 1. For the varadium compounds, tively small. This leads to a larger mean square displacean incoherent scattering method was used, while for ment and effectively a larger anharmonic contribution Nb3Sn, an average of several scattering angles was taken to the phonon frequencies. with a polycrystalline target. The interpolated values of In the A-is structure extraordinary anharmonicity d log (~2)/ d log Tare smaller than in V3Si, but still has earlier been inferred by Testardi and Batemari, 12 fairly large, and tend to scale with T~, in agreement with who also speculated on its effect on the resistivity. Fisk's observations1 about resistance anomalies. As can Testardi's13 heuristic model for the interatomic potenbe seen from Table 1, anharmonic effects can account tial in A-l 5 compounds also leads to temperature depenfor 30% of the anomaly in Nb 3Sn. dence of phonon frequencies of the form given by Previous attempts 81°to explain the resistivity equation (6). anomaly have mainly invoked a temperature dependence of (m/n)i~trwhich arises from having the electronic density of states vary rapidly with energy over a range kT. It is hard to estimate the magnitude of this effect