First Principles Explanation of the Positive Seebeck Coefficient of Lithium

Lithium is one of the simplest metals, with negative charge carriers and a close reproduction of free-electron dispersion. Experimentally, however, Li is one of a handful of elemental solids (along with Cu, Ag, and Au) where the sign of the Seebeck coefficient ($S$) is opposite to that of the carrier. This counterintuitive behavior still lacks a satisfactory interpretation. We calculate $S$ fully from first principles, within the framework of Allen’s formulation of Boltzmann transport theory. Here it is crucial to avoid the constant relaxation time approximation, which gives a sign for $S$ which is necessarily that of the carriers. Our calculated $S$ are in excellent agreement with experimental data, up to the melting point. In comparison with another alkali metal, Na, we demonstrate that within the simplest nontrivial model for the energy dependency of the electron lifetimes, the rapidly increasing density of states (DOS) across the Fermi energy is related to the sign of $S$ in Li. The exceptional energy dependence of the DOS is beyond the free-electron model, as the dispersion is distorted by the Brillouin zone edge; this has a stronger effect in Li than other alkali metals. The electron lifetime dependency on energy is central, but the details of the electron-phonon interaction are found to be less important, contrary to what has been believed for several decades. Band engineering combined with the mechanism exposed here may open the door to new “ambipolar” thermoelectric materials, with a tunable sign for the thermopower even if either $n$- or $p$-type doping is impossible.

Figure 1

Seebeck coefficient of Li, as a function of temperature. Black solid line is the calculated $S$ of bcc Li using the variational approach (extended to the temperature where 9R or liquid phase becomes stable experimentally), and the red dashed line denotes calculated result of bcc Li with constant $\tau$ from the boltztrap code. Discrete points are experimental data from LB (Landolt-Börnstein database) [11], Bidwell [8], Kendall [9], and Surla et al. [10]. The vertical dotted lines denote the temperatures of 9R-to-bcc phase transition and melting point.

Figure 2

Seebeck coefficient of Na, as a function of temperature. Black solid line is the calculated $S$ of bcc Na using the variational approach (extended beyond the experimental melting point), and the red dashed line denotes the calculated result of bcc Na with constant $\tau$. Discrete points are experimental data from Kendall [9] and LB (Landolt-Börnstein database) [11]. The vertical dotted line denotes the temperature of the melting point.

Figure 3

(a) Density of states, (b) square of the velocity (energy spectrum), (c) square of the velocity (energy spectrum) divided by the corresponding density of states of Li (solid black line) and Na (dashed red line). The vertical dotted line denotes the Fermi energy. The insets show a zoom around ${\epsilon }_{\mathit{F}}$. The orange shaded region covers ${\epsilon }_{\mathit{F}}\pm 8k_{\mathit{B}}T$ with $T=300\mathrm{K}$. The blue shaded region (darker) shows ${\epsilon }_{\mathit{F}}\pm {\omega }_m$, where ${\omega }_m$ is the maximum energy for phonons.

Figure 4

Calculated Seebeck coefficient of electron-doped Na using RTA and $1/\tau (\epsilon )\propto N(\epsilon )$, as a function of temperature at several doping concentrations.