Linear-T resistivity: strange-metal evidence and model limits

Scope

Linear-temperature resistivity describes a measured dependence over a specified range. The fit $\rho(T)=\rho_0+AT$ needs a sample, current direction, field, temperature interval and residuals. A straight-looking trace is not by itself evidence for a quantum critical point, a universal lifetime or a pairing interaction. This entry treats transport phenomenology; the linked Planckian entry discusses lifetime interpretations.

Separate a finite-field normal state, the region affected by superconducting fluctuations, and a higher-temperature regime that may have different scattering. Martin and colleagues' Bi-based crystal abstract reports strongly anisotropic transport; an in-plane trend is not an isotropic resistivity law. R1

Mathematical assumptions

A least-squares slope depends on both the model and the sampled window. For exact synthetic data $\rho=r_0+aT+bT^2$, equally weighted temperatures symmetric about $\bar T$ give

$A_{\mathrm{fit}}=a+2b\bar T.$

This follows by inserting the quadratic into $\operatorname{cov}(T,\rho)/\operatorname{var}(T)$. A narrower interval can hide curvature while changing the inferred slope. Independent synthetic fits confirm this counterexample; they are not fits to any material. Experimental model comparison also needs uncertainties, correlations and a justified treatment of residuals.

For a single isotropic Drude channel with temperature-independent carrier density $n$ and mass $m^*$, an inelastic pole relaxation rate $\gamma=\alpha k_BT/\hbar$ gives $A=\alpha m^*k_B/(ne^2\hbar)$. Thus a slope fixes $\alpha$ only after the carrier parameters and response model are specified. If those parameters change with temperature, differentiating resistivity produces additional terms. This conditional connection underlies the selected Legros analysis. R4

For a circular, parabolic two-dimensional band with total degeneracy $g$, use total sheet density $n_{2D}=gk_F^2/(4\pi)$ and $k_BT_F=\hbar^2k_F^2/(2m^*)$. The preceding Drude model then yields

$A_{\square}T_F=\frac{\alpha h}{g e^2}.$

The familiar factor of two corresponds to $g=2$, not an arbitrary multiband system. A bulk slope converted per conducting plane uses $A_{\square}=A/d$ with the stated effective plane spacing $d$. Independent occupied-disk and algebra checks cover $g=1,2,4$; neither degeneracy nor a mass should be transferred between materials without justification.

Parallel channels add conductivities. As an independent scalar, zero-field example, let $r(T)=r_0+aT>0$ describe one channel and let $\sigma_b\ge0$ be a constant background conductivity in matching units. Then

$\rho_{\mathrm{tot}}=\frac{r(T)}{1+\sigma_b r(T)},\qquad\frac{d\rho_{\mathrm{tot}}}{dT}=\frac{a}{[1+\sigma_b r(T)]^2}.$

The total slope is generally temperature dependent even though the first channel's resistance is exactly linear. Nine numerical derivative checks verify this model. It is not a multiband fit; in magnetic field, conductivity tensors must be added before inversion. Adding scattering mechanisms within one channel is a different assumption from adding parallel conductivities.

Experimental criteria

Legros and colleagues' earlier manuscript measures a Bi2212 film with nominal hole doping 0.23 in pulsed fields. Its 55 T trace and magnetoresistance-corrected, extrapolated zero-field curve are distinct datasets. The analysis estimates carrier parameters and geometric uncertainty; the checked passages do not turn its interpretation into a model-independent universal bound. R4

Cao and colleagues measure four-probe transport in gated twisted bilayer graphene. Their linear regime depends on filling and a crossover temperature. The operational scattering coefficient uses density and mass from low-temperature quantum oscillations; those assumptions matter when interpreting higher-temperature transport. The selected passages do not establish one shared microscopic mechanism across material families. R5

Romero and colleagues' earlier manuscript studies two 100 nm FeTe1-xSex films, x = 0.45 and 0.35, over 0.2–3 THz. Its narrow-channel fits use 20–40 K and report coefficients 3.02 and 2.42 in $\Gamma=\Gamma_0+\alpha k_BT/h$. These are channel-specific fit results, not coefficients of the total DC resistivity. R6

Record the raw voltage/current or conductivity data, geometry, composition, field direction, fit interval and uncertainty model. Identify excluded transition regions and test alternative fit windows and curvature. A successful fit does not establish that its parameters are uniquely identifiable. Normal-state measurement temperatures remain separate from superconducting transition temperatures in the records below.

Limits and disputes

Frequency conventions can create a factor-of-$2\pi$ error. For a Drude denominator $\Gamma-i\nu$ with ordinary frequency $\nu$, the corresponding pole rate is $\gamma=1/\tau=2\pi\Gamma$ in a denominator $\gamma-i\omega$, where $\omega=2\pi\nu$. Consequently $\Gamma=\alpha k_BT/h$ converts to $\gamma=\alpha k_BT/\hbar$. State the denominator and linewidth convention before comparing coefficients. This is a convention check, not a new measurement of a lifetime.

A shared temperature exponent does not identify a unique microscopic interaction. Drude transport, optical fits and single-particle lifetimes measure or infer different quantities. Extrapolating carrier parameters, suppressing superconductivity by field, or neglecting a parallel channel can affect an inferred coefficient. Report these choices alongside any claimed universality rather than treating the fit exponent as a proof.

Cooper's LSCO study and Bruin's cross-metal comparison remain bibliographic starting points here. Their complete datasets and numerical conclusions have not been reanalyzed in this entry. Selected manuscript checks above also leave final-version comparison and full uncertainty reproduction open. R2 R3

Discovery implications

Use linear-T transport to organize experiments, not to rank superconductors or predict a transition temperature by itself. A stronger comparison holds sample state, probe definition and fitting procedure fixed, then asks whether independent optical, thermodynamic or quantum-oscillation information supports the same parameter interpretation. Retain counterexamples and failed fits; correlation with superconductivity does not identify its cause.

Related topics

Continue with Planckian dissipation, zero resistance, critical fields, specific heat, cuprates, and multiband pairing.

Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.