Density functional theory of the Seebeck coefficient in the Coulomb blockade regime

The Seebeck coefficient plays a fundamental role in identifying the efficiency of a thermoelectric device. Its theoretical evaluation for atomistic models is routinely based on density functional theory calculations combined with the Landauer-Büttiker approach to quantum transport. This combination, however, suffers from serious drawbacks for devices in the Coulomb blockade regime. We show how to cure the theory through a simple correction in terms of the temperature derivative of the exchange correlation potential. Our results compare well with both rate equations and experimental findings on carbon nanotubes.

Figure 1

Seebeck coefficient $S$ and density $N$ (inset) for the Anderson model vs gate $v$ for our corrected DFT (black), MB (blue), and RE (red). The $S_s$ (KS, green) and the xc correction $\partial v_{\mathrm{Hxc}}/\partial T$ (cyan) are also displayed. The parameters are $T=0.1$ and $\gamma =0.01$ (energies in units of $U$).

Figure 2

Density (left) and Seebeck coefficient (right) of CIM with two spin-degenerate levels computed from RE and DFT using the approximate functional of Eq. (8). The KS Seebeck coefficient is also shown.

Figure 3

Seeebeck coefficient for the Anderson model with nondegenerate single-particle levels (left) and for three spin-degenerate levels (right). The parameters are ${\varepsilon }_{\uparrow }^0=0$, ${\varepsilon }_{\downarrow }^0=0.3$ (left panel) and ${\varepsilon }_1^0=0$, ${\varepsilon }_2^0=0.3$, ${\varepsilon }_3^0=0.6$ (right panel). In both panels $T=0.03$ and $\gamma =0.001$ (all energies in units of $U$).

Figure 4

Conductance (upper panel) and Seebeck coefficient (lower panel) of a single-wall carbon nanotube from DFT (black) and experiment (red, data from Ref. [35]). Also shown is the KS Seebeck coefficient (dashed green). The single particle and charging energies are given in Table 1. The other parameters are $T=4.5\mathrm{K}$ and $\gamma =0.02\mathrm{meV}$.