Enhancement of the thermoelectric power by electronic correlations in bad metals: A study of the Kelvin formula

In many strongly correlated electron metals the thermoelectric power has a nonmonotonic temperature dependence and values that are orders of magnitude larger than for elemental metals. Inspired by Kelvin, Peterson and Shastry derived a particularly simple expression for the thermopower in terms of the temperature dependence of the chemical potential, now known as the Kelvin formula. We consider a Hubbard model on an anisotropic triangular lattice at half filling, a minimal effective Hamiltonian for several classes of organic charge transfer salts. The finite temperature Lanczos method is used to calculate the temperature dependence of the thermopower using the Kelvin formula. We find that electronic correlations significantly enhance the magnitude of the thermopower and lead to a nonmonotonic temperature dependence. The latter reflects a crossover with increasing temperature from a Fermi liquid to a bad metal. Although, the Kelvin formula gives a semiquantitative description of some experimental results it cannot describe the directional dependence of the sign of the thermopower in some materials.

Figure 1

Temperature dependence of the thermoelectric power in the organic metal $\kappa$-(BEDT-${\mathrm{TTF})}_2\mathrm{Cu}\left[N{\left(\mathrm{CN}\right)}_2\right]\mathrm{Br}$. The two different curves correspond to two different intralayer directions in the crystal. Experimental data is taken from Ref. [2]. Note the nonmonotonic temperature dependence and that the thermopower is comparable to $k_B/e=86\mu \mathrm{V}/\mathrm{K}$. For temperatures below about 50 K the thermopower is approximately linear in temperature, as expected in a Fermi liquid. The material becomes a superconductor below $T_c=12$ K. The inset shows a schematic of the electron (blue) and hole (red) Fermi surfaces deduced from a tight-binding band structure [20]. Transport in the $a$ and $c$ directions will be dominated by holes and electrons, respectively.

Figure 2

Enhancement of the thermopower by strong correlations. The temperature dependence of the Kelvin thermopower $S_K$ is shown for several different values of the Hubbard $U$. All results are for the isotropic triangular lattice, $t^{\prime }=t$. As the Mott metal-insulator transition $\left(U_c\simeq 7.5t\right)$ [26] is approached the magnitude of the thermopower increases to values that are an order of magnitude larger than for noninteracting electrons $\left(U=0\right)$ for temperatures of about $T\sim t/10$. The maximum in $|S_K|$ at low temperatures corresponds to the crossover from a Fermi liquid at low temperatures to a bad metal at higher temperatures. This maximum is also seen in the specific heat [26] and the spin susceptibility. The curves have been linearly extrapolated from their value at $T=0.06t$ to zero at zero-temperature. All results shown are for a 16-site cluster. Also shown is the linear temperature dependence obtained by a Sommerfeld expansion for noninteracting electrons [27].

Figure 3

Limited finite-size effects. The same quantities as shown in Fig. 2 are plotted here for smaller 12- and 14-site clusters. All the main features, namely, that $S_K$ increases with $U$ at low temperatures, is nonmonotonic in $T$, and is of the order of $k_B/e$ close to the MIT, are present. However, at lowest $T$ there are still some finite size effect and $S_K$ still seems to slightly increase with system size.

Figure 4

The dependence of $S_K$ on $U$ for several different fixed temperatures $T$ is shown. It is seen how correlations increase $S_K$ at low temperatures, with possibly a signature of the MIT near the critical $U_c\sim 7.5$ [26].

Figure 5

Energy dependence of the density of states $g\left(\epsilon \right)$ for the tight-binding band structure associated with noninteracting electrons with $t^{\prime }=t$. For half filling $\mu =E_F=0.82t$ at zero temperature, and $g\left(E_F\right)=0.14/t$ and $g^{\prime }\left(E_F\right)=0.056/t^2$. The latter determines the slope of the Kelvin thermopower versus temperature for noninteracting electrons. Reversing the sign of $t^{\prime }$, or both $t$ and $t^{\prime }$, corresponds to a particle-hole transformation and reverses the sign of this derivative $\left[g^{\prime }\left(E_F\right)\right]$.