Seebeck coefficient in correlated low-dimensional organic metals

We study the influence of inelastic electron-electron scattering on the temperature variation of the Seebeck coefficient in the normal phase of quasi-one-dimensional organic superconductors. The theory is based on the numerical solution of the semiclassical Boltzmann equation for which the collision integral equation is solved with the aid of the renormalization-group method for the electronic umklapp scattering vertex. We show that the one-loop renormalization-group flow of momentum and temperature-dependent umklapp scattering, in the presence of nesting alterations of the Fermi surface, introduce electron-hole asymmetry in the energy dependence of the anisotropic scattering time. This is responsible for the enhancement of the Seebeck coefficient with respect to the band $T$-linear prediction and even its sign reversal around the quantum critical point of the phase diagram, namely, where the interplay between antiferromagnetism and superconductivity and also the strength of spin fluctuations are the strongest. A comparison of the results with available data on low-dimensional organic superconductors is presented and critically discussed.

Figure 1

Renormalization-group results for the phase diagram of the quasi-1D electron-gas model as a function of the antinesting tuning parameter $t_{\perp }^{\prime }$ and for the model parameters specified in Sec. 3a.

Figure 2

Renormalized umklapp scattering amplitude $g_3(k_{\perp 1},-k_{\perp 1};k_{\perp 3},-k_{\perp 3})$, projected in the $(k_{\perp 1},k_{\perp 3})$ plane at different temperatures for the metallic phase model parameters specified in Sec. 3a. (a) SDW, $t_{\perp }^{\prime }/k_B=25\mathrm{K}(<t_{\perp }^{\prime *}/k_B)$, (b) SCd, $t_{\perp }^{\prime }/k_B=35\mathrm{K}(>t_{\perp }^{\prime *}/k_B)$, and (c) SCd, $t_{\perp }^{\prime }/k_B=42$ and 46 K at $T=1$ K.

Figure 3

The longitudinal Seebeck coefficient as a function of temperature for (a) different values of antinesting $t_{\perp }^{\prime }$ in the metallic phase and (b) model parameters in the more correlated case with stronger umklapp scattering and lower amplitudes of hopping integrals $[t_{\perp }^{\prime }/k_B=15$ K, $t_{\perp }/k_B=100$ K; see Sec.5a on (TMTTF)${}_{2}X$ salts]. The dashed lines gives the band contribution $Q_a^0$ of Eq. (13) for constant relaxation time in energy.

Figure 4

Open Fermi surface of the quasi-1D electron-gas model (top) and typical variations of the scattering time along the Fermi surface for different high temperatures ($t_{\perp }^{\prime }=t_{\perp }^{\prime *}$; bottom).

Figure 5

(a) The longitudinal Seebeck coefficient $Q_a$ and (b) the ratio $Q_a/T$ as a function of $T$ at low temperature and different values of antinesting $t_{\perp }^{\prime }$. The value with an asterisk stands for the critical $t_{\perp }^{\prime *}$. The dashed line corresponds to the Fermi liquid limit using a momentum- and temperature-independent $g_3$ ($=0.025$) at $t_{\perp }^{\prime *}$. The inset in (b) displays the enhancement on a logarithmic temperature scale. The solid line refers to $Q_a/T\sim lnT.$

Figure 6

(a) Variation of the normalized scattering time as a function of energy near the Fermi level at low temperature. Here $t_{\perp }^{\prime }/k_B=25$ and 35 K for the blue and red curves, respectively. (b) The anisotropy of the normalized energy derivative of the scattering time along the Fermi surface at low temperature for different $t_{\perp }^{\prime }$.

Figure 7

Amplitude of the Seebeck coefficient at low temperature as a function of antinesting.

Figure 8

Temperature dependence of Seebeck coefficients in (a) a few (TMTTF)${}_{2}X$ salts at ambient pressure (after Ref. [37]) and (b) (TMTSF)${}_{2}X$ in the metallic state above $T_{\mathrm{SDW}}, X={\mathrm{PF}}_6$, and $T_c, X={\mathrm{ClO}}_4$. The inset shows the sign reversal of the ratio $Q_a/T$ in the low-temperature domain (after Ref. [15]).