M. Calandra and O. Gunnarsson In a metal, the electrical resistivity ρ, grows
with the temperature T, due to the increased scattering of the electrons by phonons. Typi-cally, ρTT for large temperatures. For some metals with a very largeρ, however, the resistivity essentially saturates for large T. The resistivity is often described in a semiclassical picture, where an electron, on the average, trav-els a mean free path l before it is scattered.
Typically, ld, where d is the atomic separa-tion. For the systems with resistivity saturation, however, l becomes comparable to d. Work in the 1970’s suggested that saturation occurs universally when ld. Later work has, how-ever, found apparent exceptions, such as alkali-doped fullerenes and high-Tc cuprates. Intu-itively, resistivity saturation seems natural. One might expect that at worst, an electron could be scattered at each atom, leading to ld. Such a semiclassical picture, however, breaks down when ld, and it is contradicted by the lack of saturation for the fullerenes.
Saturation is clearly observed for, e.g., A15 compounds, such as Nb3Sb, where the resis-tivity saturates at about 0.10 to 0.15 mΩcm.
For the high-Tc cuprates and the fullerenes, the Ioffe-Regel condition predicts a larger satura-tion resistivity, about 0.7 and 1 mΩcm, respec-tively. Experimentally, however, the resistivi-ties reach much larger values; several mΩcm
for the cuprates and almost 20 mΩcm for hole-doped fullerenes, suggesting a lack of saturation in these systems.
To discuss resistivity saturation, we use the f-sum rule the separation of the scattering centers, Ω is the volume of a unit cell and TK is the the ki-netic energy per unit cell. We assume that T is large enough that the Drude peak has been smeared out and that σω is a smooth func-tion. The removal of the Drude peak may be due to any scattering mechanism, electron-phonon, electron-electron or disorder scatter-ing. We furthermore assume that σω= 0 for ωW, where W is the one-particle band width.
This should be a good approximation, since ωW would involve multiple electron-hole ex-citations and have a small weight. We can then write ex-pect γto be a bit larger than unity, and explicit calculations for models with electron-phonon or electron-electron scattering give values of the order two to four. Localization could lead to aγ
much smaller than unity and a resistivity much larger than predicted below (byρ= 1σ(ω= 0)).
Since, however, we consider large tempera-tures, localization should not occur.
We first consider non-interacting electrons in a band with the orbital degeneracy n at T = 0.
Then we can write TK2n
µ
W2εNεdε2nαW where Nε is the density of states per orbital and spin, µ is the chemical potential andα de-pends on the shape Nε and the filling. For a half-filled semielliptical Nε α0.1. This result is relatively independent of the specific shape of Nε. We then obtain
For a weakly correlated transition metal com-pound like Nb3Sn, the Nb 4d orbital plays the main role, and we use n = 5. Consider-ing the A15 lattice of Nb3Sn, the resistivity should not exceed a value of about 0.14 mΩcm [M. Calandra et al., Physical Review Letters 26, 266601-1 (2001)]. This value is shown by the horizontal line in Fig. 21 and it compares well with the experimental saturation resistivity of about 0.15 mΩcm for Nb3Sn. This resistivity corresponds to a mean free path of the order d.
We next consider the low T behavior, assum-ing an electron-phonon scatterassum-ing mechanism (strengthλ)
ρT8π2λT kB
¯
hΩ2pl (8)
where kBis the Boltzmann constant. Ωpl is the plasma frequency, which depends on the aver-age Fermi velocity. Figure 21 shows the straight line corresponding to Eq. (8) as well as a cal-culation of the resistivity. The resistivity curve initially follows Eq. (8), describing how the am-plitude of the Drude peak is gradually reduced, reducingσω0and increasingρT. When the two lines intersect, the Drude peak has been entirely lost. At roughly this T the rapid growth
ofρTstops and saturation occurs. Nb3Sn has a rather largeλand a very smallΩpl, leading to very steep slope.
Figure 21: Resistivity for a modelNb3Sn. The hori-zontal line shows the saturation resistivity (Eq. (7)) and the linear in T resistivity (Eq. (8)). Saturation happens before these two lines intersect.
As a result the curves intersect at a value of T which is experimentally accessible. For most metals, however, the slope of the low T curve is such that the intersection would happen well above T’s that can be reached experimentally.
Then no saturation is seen. Nb3Sn has a rather large unit cell, leading to a complicated band structure with many forbidden crossings. This leads to rather flat bands, a small Ωpl and a rapid increase ofρT. This is in contrast to Nb, which has only one atom per unit cell and a sub-stantially largerΩpl. For Nb signs of saturation are seen, but at a larger T and less pronounced than for Nb3Sn.
We now consider a model of a strongly corre-lated system, appropriate for the cuprates. In these systems the transport properties should mainly be determined by the Cu–O antibonding band of Cu x2– y2and O 2p character. Thus we consider one orbital per site and the orbital de-generacy n = 1. We furthermore assume that the Coulomb interaction U between two electrons on the same site is so large that double occu-pancy of a site can be neglected. We therefore use a two-dimensional tJ model for describ-ing the system.
We first notice that if the model has one elec-tron per site, hopping is completely suppressed by the Coulomb energy, and TK= 0. Introduc-ing dopIntroduc-ing (x holes per site), however, makes hopping possible. The probability that a given site is occupied is (1 – x). An electron can hop to a neighboring site only if this site is empty, the probability for which is about x. From this we obtain a crude estimate
TK4 t x1x
where four is the number of nearest neighbor and t is the hopping integral. Explicit calcula-tions for the tJ model show this result over-estimatesTKby about 15%. Insertion of this result (with the prefactor 3.4) in Eqs. (5, 6) gives
ρT 035
x1x mΩcm (9) For small x, this result is much larger than the Ioffe-Regel resistivity (0.7 mΩcm). It is also much larger than the saturation resistiv-ity 0.14 mΩcm derived above for a model of weakly correlated transition metal compounds.
This is due to the strong reduction of the kinetic energy in the cuprates, due to strong correlation effects, in particular for small dopings.
Figure 22 compares the experimental resistivity for La2xSrxCuO4 with the saturation resistiv-ity deduced from the kinetic energy calculated in the tJ model. For all values of x, the re-sistivity is smaller than the predicted saturation resistivity. Therefore, the experimental results do not demonstrate absence of saturation. Ac-tually, for x = 0.04 and x = 0.07 the experimental data show signs of saturation, while for x = 0.15 and x = 0.34 one would have to study much higher (and experimentally unaccessible) T to demonstrate lack of saturation. We therefore conclude that the results for La2xSrxCuO4are consistent with resistivity saturation, although the saturation happens at much larger resistivi-ties than predicted by the Ioffe-Regel condition.
Figure 22: The resistivity as a function of T for La2xSrxCuO4and the saturation resistivity accord-ing to the f-sum rule for x = 0.04 (full curve) x = 0.07 (broken curve) x = 0.15 (dotted curve) and x = 0.34 (dash-dotted curve). The horizontal arrow shows the saturation resistivity expected from the Ioffe-Regel condition. The figure illustrates that there are signs of saturation for small x at roughly the resistivities where saturation is expected. For larger x, much larger T would have to be considered to test whether there is saturation.
We finally consider the alkali-doped fullerenes, A3C60 (A = K, Rb). These systems have orien-tational disorder. Therefore, already at T = 0 the resistivity is comparable to the saturation resis-tivity estimated from Eq. (8). In these system the main scattering mechanism is believed to be due to intramolecular phonons, which cou-ple to the level energies and create fluctuations in these energies. Due to the small band width, the fluctuations become comparable to the band width at temperatures that can be reached ex-perimentally. The band width then grows with T, roughly as
ab T, where a and b are ma-terial constants. The fluctuations reduce the ki-netic energy, since two neighboring levels may have very different energies, which strongly suppresses hopping. A crude estimate shows that TKc
ab T, where c is a material con-stant. Inserting this in Eqs. (5, 6) gives
ρTρT0dT
where d is a material constant. Indeed, quan-tum Monte-Carlo calculations for a model of these systems show a lack of saturation even for extremely large values of T [O. Gunnarsson et al., Nature 405, 1027 (2000)], as predicted above. We therefore expect a lack of saturation for the fullerenes, putting these systems in a dif-ferent class than the high-Tccuprates.
The fullerenes differ from systems of the type of Nb3Sn in two respects. Firstly, in Nb3Sn the phonons couple primarily to the hopping ma-trix elements. As a result, both TKand W grow with T. These effects therefore tend to cancel to some extent (see Eqs. (5, 6)). Secondly, the tem-perature scale for these effects are much large for Nb3Sn due to the larger band width, and the effects are therefore rather small at realistic T’s, leading to a weak T dependence after saturation has occurred.