Electrical transport near quantum criticality in low-dimensional organic superconductors

We propose a theory of longitudinal resistivity in the normal phase of quasi-one-dimensional organic superconductors near their quantum critical point where antiferromagnetism borders with superconductivity under pressure. The linearized semiclassical Boltzmann equation is solved numerically, fed in by the half-filling electronic umklapp scattering vertex as derived from one-loop renormalization-group calculations for the quasi-one-dimensional electron-gas model. The momentum and temperature dependence of umklapp scattering has an important impact on spin fluctuations and on the behavior of longitudinal resistivity in the the normal phase. Resistivity is found to be linear in temperature around the quantum critical point at which spin-density-wave order joins superconductivity along the antinesting axis, to gradually evolve towards the Fermi-liquid behavior in the limit of weak superconductivity. A critical analysis of the predictions is made from a comparison with experiments performed on the ${\left(\mathrm{TMTSF}\right)}_2{\mathrm{PF}}_6$ member of the Bechgaard salt series under pressure. Fair agreement between theory and experiment is then found in the low-temperature range linked to quantum criticality while deviations from predictions become apparent at high temperature.

Figure 1

Calculated phase diagram of the quasi-one-dimensional electron-gas model showing the SDW-$\mathrm{SC}d$ sequence of instabilities as a function of antinesting $t_{\perp }^{\prime }$ and initial $g_3$ (for the model parameters used in the calculations, see Sec. 3a).

Figure 2

Renormalized umklapp scattering amplitude projected in the $(k_{\perp 1},k_{\perp 2})$ plane in the normal phase at different temperatures for the initial $g_3=0.03$ ($t_{\perp }^{\prime *}=38\mathrm{K}$). (a) $t_{\perp }^{\prime }=33\mathrm{K}(<t_{\perp }^{\prime *})$; (b) $t_{\perp }^{\prime }=44\mathrm{K}(>t_{\perp }^{\prime *})$ (for other model parameters used in the calculations, see Sec. 3a).

Figure 3

Renormalized umklapp scattering amplitude projected in the $(k_{\perp 1},k_{\perp 2})$ plane in the normal phase of the $\mathrm{SC}d$ sector of the phase diagram,as obtained at different temperatures and initial values of $g_3$ (for other model parameters used in the calculations, see Sec. 3a).

Figure 4

Calculated longitudinal resistivity over the whole temperature interval for different $t_{\perp }^{\prime }$ at $g_3=0.03$ ($t_{\perp }^{\prime *}=38\mathrm{K}$). The continuous line refers to the Fermi-liquid limit at constant scale-independent $g_3(=0.03)$ yielding linear $T$ resistivity in the high-temperature 1D domain $(T>t_{\perp }=200\mathrm{K}$) followed by a crossover toward a 2D $T^{2}lnT$ regime at low temperature (for other model parameters used in the calculations, see Sec. 3a).

Figure 5

Calculated resistivity at low temperature in the SDW domain of the phase diagram for different $t_{\perp }^{\prime }$ and $g_3$ (for other model parameters used in the calculations, see Sec. 3a).

Figure 6

Calculated resistivity at low temperature in the $\mathrm{SC}d$ domain of the phase diagram for different $t_{\perp }^{\prime }$ and $g_3$ (for other model parameters used in the calculations, see Sec. 3a).

Figure 7

Variation of the power-law exponent $\alpha$ of resistivity in the phase diagram for $g_3=0.026$ (top) and $g_3=0.033$ (bottom). The arrows indicate the location of $t_{\perp }^{\prime *}$ (for other model parameters used in the calculations, see Sec. 3a).

Figure 8

Variation of resistivity components along the Fermi surface at different temperatures: SDW (top) and $\mathrm{SC}d$ (bottom) domains (for other model parameters used in the calculations, see Sec. 3a).

Figure 9

(a) Longitudinal resistivity of Ref. [11] observed at different pressures or $T_c$ for ${\left(\mathrm{TMTSF}\right)}_2{\mathrm{PF}}_6$. (b) Inelastic part of the data ($\mathrm{\Delta }{\rho }_a={\rho }_a-{\rho }_a^0$) traced in a log-log plot showing the growth of the power-law exponent $\alpha$ for resistivity with pressure (for other model parameters used in the calculations, see Sec. 3a).