Colloquium: Planckian dissipation in metals

The appearance of the Planckian time ${\tau }_{\mathrm{Pl}}=\hslash /k_{B}T$ is reviewed in both conventional and unconventional metals. A pedagogical discussion of the various different timescales (quasiparticle, transport, many-body) that characterize metals is given, with an emphasis on conditions under which these times are the same or different. The possibility of a Planckian bound on dissipation is discussed from both a quasiparticle and a many-body perspective. Planckian quasiparticles can arise naturally from a combination of inelastic scattering and mass renormalization. Many-body dynamics, on the other hand, is constrained by the basic timescales and length scales of local thermalization.

Figure 1

$T$-linear resistivity from $T_{\mathrm{c}}\approx 7$ up to 700 K in ${\mathrm{Bi}}_2{\mathrm{S}\mathrm{r}}_2{\mathrm{C}\mathrm{u}\mathrm{O}}_{6+\delta }$. ARPES measurements of the nodal electrons, which are believed to control transport, show a feature in their dispersion due to coupling to a known optical phonon in this material at around 63 meV ([71]). We indicate the corresponding temperature scale as $(1/3)T_{\mathrm{ph}}$, where $T$-linear scattering by this mode is expected to set in. Below this temperature the resistivity is not due to conventional electron-phonon scattering by this mode. At temperatures above $T_{\mathrm{ph}}$, scattering by this phonon mode becomes increasingly elastic, as further discussed in Sec. 3c. This plot schematically illustrates that different mechanisms for $T$-linear resistivity with the same slope may plausibly be at work in a given unconventional metal. From [130].

Figure 2

Imaginary part of the self-energy (14) due to scattering by an Einstein phonon, normalized at large frequencies. From bottom to top, temperatures are $T/T_{\mathrm{ph}}=0.1$, 0.3, 0.4, 0.5, 0.7, 1, 3. At low temperatures no scattering occurs below the frequency ${\omega }_{\mathrm{ph}}$, while at high temperatures all frequencies are scattered.

Figure 3

Inverse quasiparticle lifetime as a function of temperature due to scattering by an Einstein phonon, following from Eq. (14). The electron-phonon coupling is $2\pi {\lambda }_{e\text{−}\mathrm{ph}}=5$ (top curve) and $2\pi {\lambda }_{e\text{−}\mathrm{ph}}=1$ (bottom curve). The dashed line shows the Planckian rate. Inset: corresponding resistivity. The resistivity is $T$ linear for $T\gtrsim (1/3)T_{\mathrm{ph}}$ with a slope proportional to the coupling. Even with the larger coupling, however, the inverse lifetime becomes sub-Planckian for $T\lesssim T_{\mathrm{ph}}$ due to a temperature-dependent mass renormalization.

Figure 4

$T$-linear resistivity and logarithmically enhanced specific heat upon tuning the ferromagnetic heavy fermion ${\mathrm{CeRh}}_6{\mathrm{Ge}}_4$ to a quantum critical point ([192]). Left panel: $n$ represented as the logarithmic derivative of the resistivity (i.e., $n=1$ is $T$ linear), given as a function of pressure and temperature. Right panel: the resistivity and specific heat as a function of temperature at the critical pressure.

Figure 5

Left panel: MFL decay rate (20) as a function of temperature for $\lambda =0.3$, 1, 3 (bottom to top). The dashed line shows the Planckian rate. At $T\ll T_{\star }$ the decay rate is sub-Planckian, but for a large coupling $\lambda$ the Planckian limit can be surpassed as temperature is increased. Inset: corresponding $T$-linear resistivity with a slope proportional to $\lambda$. Right panel: temperature evolution of $1/\tau$ (dotted line), $m_{\star }\propto 1/Z$ (dashed line), and $\rho \propto m_{\star }/\tau$ (solid line) for $\lambda =3$.

Figure 6

Schematic diagram showing the evolution of the effective mass as a function of the temperature and tuning parameter. Quantum oscillations measure the low-temperature mass, while Planckian scattering is claimed at the lower end of the $T$-linear transport regime. Note that observed mass divergences as the critical point is approached at low temperatures do not necessarily correspond to the quasiparticles that dominate transport.

Figure 7

Left panel: a low-temperature $T^2$ scattering rate crosses over to a high-temperature $T$-linear scattering rate in ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ at a field $B=7.4\mathrm{T}$ ([28]). The solid line is the data and the dashed line is the extrapolation of the low-temperature behavior. Here the strong low-temperature growth of the scattering rate is cut off by Planckian scattering. Right panel: a low-temperature $T$-linear scattering rate ([113]) crosses over to a high-temperature $T^2$ scattering rate ([171]) in LCCO at $x=0.16$. Here the high-temperature scattering rate rises above the low-temperature Planckian scattering. The $T=0$ scattering rate has been subtracted in both cases. For ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ the masses and densities used to extract a scattering rate from transport have been obtained from quantum oscillations ([28]). In the absence of quantum oscillation data on LCCO, the masses were estimated from quantum oscillations in ${\mathrm{Nd}}_{2-x}{\mathrm{C}\mathrm{e}}_x{\mathrm{C}\mathrm{u}\mathrm{O}}_4$ ([113]). This procedure introduces some uncertainty in the density.