Thermoelectricity near Anderson localization transitions

The electronic thermoelectric coefficients are analyzed in the vicinity of one and two Anderson localization thresholds in three dimensions. For a single mobility edge, we correct and extend previous studies and find universal approximants which allow us to deduce the critical exponent for the zero-temperature conductivity from thermoelectric measurements. In particular, we find that at nonzero low temperatures the Seebeck coefficient and the thermoelectric efficiency can be very large on the “insulating” side, for chemical potentials below the (zero-temperature) localization threshold. Corrections to the leading power-law singularity in the zero-temperature conductivity are shown to introduce nonuniversal temperature-dependent corrections to the otherwise universal functions which describe the Seebeck coefficient, the figure of merit, and the Wiedemann-Franz ratio. Next, the thermoelectric coefficients are shown to have interesting dependences on the system size. While the Seebeck coefficient decreases with decreasing size, the figure of merit first decreases but then increases, while the Wiedemann-Franz ratio first increases but then decreases as the size decreases. Small (but finite) samples may thus have larger thermoelectric efficiencies. In the last part we study thermoelectricity in systems with a pair of localization edges, the ubiquitous situation in random systems near the centers of electronic energy bands. As the disorder increases, the two thresholds approach each other, and then the Seebeck coefficient and the figure of merit increase significantly, as expected from the general arguments of Mahan and Sofo [J. D. Mahan and J. O. Sofo, Proc. Natl. Acad. Sci. USA 93, 7436 (1996)] for a narrow energy range of the zero-temperature metallic behavior.

Figure 1

Schematic pictures of the density of states, $\rho \left(E\right)$, as a function of the energy $E$ in three dimensions for (a) a nonrandom system, (b) with two mobility edges near the band edges, (c) with two mobility edges near the center of the band.

Figure 2

A schematic picture of the energy dependence of the zero-temperature conductivity in a finite system. The flat horizontal line is broader and higher for smaller samples.

Figure 3

The leading universal dependences of the Seebeck coefficient $S$ [in units of $k_{\text{B}}/\left|e\right|$, panel (a)], the figure of merit $ZT$ [panel (b)], and the Wiedemann-Franz ratio $\mathcal{L}$ [in units of $(k_{\text{B}}/{\left|e\right|)}^2$, panel (c)] versus $z=(\mu -E_c)/\left(k_{\text{B}}T\right)$, Eqs. (22), (23), and (24), with $x=0.5$ [dotted (magenta) curve], $x=1$ [small-dashed (red) curve], $x=1.5$ [medium-dashed (black) curve], and $x=2$ [large-dashed (blue) curve].

Figure 4

Comparisons of the full curves from Fig. 3, with $x=1.5$ [solid (black) curves], with the various approximants (dashed lines). Each panel shows two leading terms in Eqs. (A4), (A5), or (A6) for $z\gg 1$ [larger-dashed (magenta) curve] and in Eqs. (A12), (A14), or (A16) for $\left|z\right|\ll 1$ [dashed (red) line], and one leading term in Eqs. (A23), (A24), or (31) for $z\ll -1$ [dotted (blue) line].

Figure 5

The $x$ dependences of the coefficients $S_0$ (left) and $S_1$ (right) (in units of $k_{\text{B}}/\left|e\right|$) in Eq. (A13).

Figure 6

The $x$ dependences of the coefficients $Z_0$ and $Z_1$ in $ZT\approx Z_0+Z_{1}z$ for $\left|z\right|\ll 1$, Eq. (A15).

Figure 7

(a) The Seebeck coefficient in Eq. (19) and (b) the figure of merit, Eq. (20), versus $t=k_{\text{B}}T/\left|E_c\right|$ for $x=1.5$, and $y=1$, with $a=0$ [solid (black) curves], $a=0.05$ [dashed (red) curves], and $a=-0.05$ [dotted (blue) curves], as well as for $y=2$, with $a=0.01$ [largest-dashed (red) curves] and $a=-0.01$ [medium-dashed (blue) curves]: $(\mu -E_c)/\left|E_c\right|=-1$ (top), $(\mu -E_c)/\left|E_c\right|=0$ (middle), $(\mu -E_c)/\left|E_c\right|=1$ (bottom).

Figure 8

The Wiedemann-Franz ratio $\mathcal{L}$ in Eq. (21) versus $t=k_{\text{B}}T/\left|E_c\right|$ for $x=1.5, y=1$, with $a=0$ [solid (black) curve], $a=0.05$ [dashed (red) curve], and $a=-0.05$ [dotted (blue) curve], as well as for $y=2$, with $a=0.01$ [largest-dashed (red) curve] and $a=-0.01$ [medium-dashed (blue) curve]. (a) $(\mu -E_c)/\left|E_c\right|=0$, (b) $(\mu -E_c)/\left|E_c\right|=1$, (c) $(\mu -E_c)/\left|E_c\right|=-1$.

Figure 9

The Seebeck coefficient $S$ [in units of $k_{\text{B}}/\left|e\right|$, panel (a)], the figure of merit $ZT$ [panel (b)], and the Wiedemann-Franz ratio $\mathcal{L}$ [in units of $(k_{\text{B}}/{\left|e\right|)}^2$, panel (c)] as functions of the dimensionless energy ${\epsilon }_L^{}$, Eqs. (34), (35), and (36), for $x=1.5$ and $z=0$ [solid (black) curve], $z=-5$ [small-dashed (red) curve], and $z=-10$ [large-dashed (blue) curve]. Note that $\mathcal{L}$ is practically independent of $z$ for $z\ll -1$.

Figure 10

The Seebeck coefficient $S$ [in units of $k_{\text{B}}/\left|e\right|$, panel (a)], the figure of merit $ZT$ [panel (b)], and the Wiedemann-Franz ratio $\mathcal{L}$ [in units of $(k_{\text{B}}/{\left|e\right|)}^2$, panel (c)] versus $z=(\mu -E_c)/\left(k_{\text{B}}T\right)$ with $x=1.5$ for ${\epsilon }_L=0$ [solid (black) curve], ${\epsilon }_L=1$ [dotted (red) curve], ${\epsilon }_L=2$ [dashed (blue) curve], and ${\epsilon }_L=3$ [large-dashed (magenta) curve].

Figure 11

The Seebeck coefficient $S$ [in units of $k_{\text{B}}/\left|e\right|$, panel (a)], the figure of merit $ZT$ [panel (b)], and the Wiedemann-Franz ratio $\mathcal{L}$ [in units of $(k_{\text{B}}/{\left|e\right|)}^2$, panel (c)] versus $\overline{\mu }=\mu /\left(k_{\text{B}}T\right)$ for $x=1.5$ and for ${\overline{E}}_c=E_c/\left(k_{\text{B}}T\right)=0.2$ [solid (black) line], 1 [small-dashed (red) line], 2 [medium-dashed (blue) line], and 5 [large-dashed (magenta) line]. The larger-dashed four curves in the plot of the Seebeck coefficient [panel (a)] are calculated with the single-threshold expression.