Fermi surface and electron dispersion of PbTe doped with resonant Tl impurity from KKR-CPA calculations

We present results of a detailed study on the electron dispersions and Fermi surface of lead telluride doped with 2% of thallium, which is a resonant impurity in PbTe. Using the Korringa-Kohn-Rostoker method with the coherent potential approximation (KKR-CPA), Bloch spectral functions (BSFs), which replace the dispersion relations in alloys, are calculated, and BSF intensity maps over the Brillouin zone (alloy Fermi surface cross sections) are presented. It is shown that, close to the valence band edge, Tl does not create an isolated impurity band, but due to its resonant character, strongly disturbs the host electronic bands, leading to the disappearance of sharp and well-defined electronic energy bands. The consequences of this effect on the transport properties are discussed and a qualitative explanation for the improvement in the thermoelectric properties of PbTe:Tl is suggested.

Figure 1

Top panel: Valence band DOS of PbTe and Tl${}_{0.02}$Pb${}_{0.98}$Te, with the two resonant peaks, around $-5.5$ and 0 eV, visible. $E_v$ is the energy of the valence band (VB) edge. Left bottom panel: Total DOS close to the VB edge with atomic-decomposed contributions. Vertical lines mark the position of the chemical potential (at $T=300$ K) for the carrier concentrations: (1) $3\times {10}^{20}$ (nominal carrier concentration), (2) $1\times {10}^{20}$, (3) $5\times {10}^{19}$, all in cm${}^{-3}$. Right bottom panel: Partial DOS of the thallium atom (per one Tl atom, i.e., not multiplied by Tl concentration).

Figure 2

The two-dimensional projections of Bloch spectral functions of Tl${}_{0.02}$Pb${}_{0.98}$Te in high-symmetry directions. Here and in the following figures, the black color corresponds to a BSF value greater than 300 a.u., i.e., 22 eV${}^{-1}$. The right panel, with the $\Sigma -L$ direction, has the same energy scale as the left one.

Figure 3

Bloch spectral functions of Tl${}_{0.02}$Pb${}_{0.98}$Te plotted at k = $(0.4,0.4,0.4)$, $(0.45,0.45,0.45)$ (close to the $L$ point), and $(0.375,0.375,0.0)$ ($\Sigma$ point), all in units $2\pi /a$. Contributions to BSF from Tl+Pb and Te sites are plotted in colors.

Figure 4

Cuts through the BZ with Bloch spectral function values marked by colors, plotted in a logarithmic scale (Fermi surface cross sections) for the carrier concentration $p=5\times {10}^{19}$ cm${}^{-3}$.

Figure 5

Cuts through the BZ with Bloch spectral function values marked by colors, plotted in a logarithmic scale (Fermi surface cross sections) for the carrier concentrations $p=5\times {10}^{19}$ and $1.0\times {10}^{20}$ cm${}^{-3}$. The top panels show the rectangle between the $K-K-U-U$ points in BZ, marked in magenta in the BZ picture, and the bottom panels show the rectangle lying between the $X_3$ and $K$ points, marked in navy in the BZ picture.

Figure 6

Bloch spectral functions of Tl${}_{0.02}$Pb${}_{0.98}$Te plotted in a broader energy range. The BSF of the “hyperdeep” state around $-5.5$ eV is visible, and resembles the isolated impurity band, in contrast to the RL located close to the VB edge.