Saturation of electrical resistivity

TL;DR

The paper addresses resistivity saturation in metals, focusing on why semiclassical Boltzmann theory (Ioffe-Regel limit) sometimes fails. It combines experimental trends with three theoretical frameworks: (i) a Boltzmann-based, weakly interacting picture predicting $l \gtrsim d$ saturation, (ii) a f-sum-rule–driven approach that bounds high-$T$ resistivity, and (iii) model analyses across weakly correlated metals, strongly correlated cuprates, and alkali-doped C$_{60}$ to explain when saturation occurs or is violated. Key findings show that saturation typically follows $l \sim d$, but notable exceptions (high-$T_c$ cuprates, C$_{60}$ compounds) arise due to strong correlations, interband participation, and coupling type (HI vs LE). The work emphasizes that a quantum-mechanical treatment beyond semiclassical pictures is essential to understand transport in complex and strongly correlated materials, with implications for interpreting high-resistivity behavior and localization phenomena. Overall, resistivity saturation emerges as a nuanced, material-specific phenomenon governed by the balance of kinetic energy, bandwidth, and scattering mechanisms.

Abstract

Resistivity saturation is observed in many metallic systems with a large resistivity, i.e., when the resistivity has reached a critical value, its further increase with temperature is substantially reduced. This typically happens when the apparent mean free path is comparable to the interatomic separations - the Ioffe-Regel condition. Recently, several exceptions to this rule have been found. Here, we review experimental results and early theories of resistivity saturation. We then describe more recent theoretical work, addressing cases both where the Ioffe-Regel condition is satisfied and where it is violated. In particular we show how the (semiclassical) Ioffe-Regel condition can be derived quantum-mechanically under certain assumptions about the system and why these assumptions are violated for high-Tc cuprates and alkali-doped fullerides.

Figures (14)

• Figure 1: Resistivity of Cu, Nb$_3$Sb (Fisk and Webb, 1976) and Nb (Abraham and Deviot, 1972). The figure also shows the the Ioffe-Regel (Ioffe and Regel, 1960) saturation resistivities of Nb$_3$Sb and Nb, setting the mean free path $l$ in Eq. (\ref{['eq:1']}) equal to the distance between the Nb atoms. The corresponding value for Cu, 260 $\mu\Omega$cm, falls outside the figure. The figure illustrates that for Nb$_3$Sb and Nb the resistivity saturates roughly as predicted by the Ioffe-Regel criterion, while $\rho(T) \sim T$ for Cu at large $T$.
• Figure 2: Resistivity of Bi$_2$Sr$_2$Ca$_{1-x}$Y$_x$Cu$_2$O$_{8+y}$ ($T_c=30$ K) (Wang et al., 1996ab), La$_{1.93}$Sr$_{0.07}$CuO$_4$ (Takagi et al., 1992), Nd$_{1.84}$Ce$_{0.16}$Cu$_{4-y}$ ($T_c=22.5$ K) (Hikada and Suzuki, 1989), YBa$_2$Cu$_3$O$_{6+x}$ ($T_c=60$ K) (Orenstein et al., 1990), Bi$_2$Sr$_2$Cu$_{6+y}$ ($T_c=6.5$ K) (Martin et al., 1990) and Nb$_3$Sb (Fisk and Webb, 1976). The arrow shows the Ioffe-Regel resistivity of La$_{1.93}$Sr$_{0.07}$CuO$_4$. The figure illustrates that there is no sign of saturation at the Ioffe-Regel resistivity, but in some cases perhaps at much larger resistivities. Observe the magnitude compared with Nb$_3$Sb.
• Figure 3: Resistivity of Rb$_3$C$_{60}$ (Hebard et al., 1993) La$_4$Ru$_6$O$_{19}$ (Khalifah et al., 2001), Sr$_2$RuO$_4$ (Tyler et al., 1998), Nb$_3$Sb (Fisk and Webb, 1976) and the Ioffe-Regel resistivity for Rb$_3$C$_{60}$. There is no sign of saturation at the Ioffe-Regel resistivity, but La$_4$Ru$_6$O$_{19}$ may saturate at a much larger resistivity.
• Figure 4: Resistivity of Ti$_{1-x}$Al$_x$ alloys. The figure suggests that the saturation resistivity is independent of the disorder (after Mooij, 1973).
• Figure 5: Resistivity of 3d (a) and 5d (b) transition metals (Bass, 1982).