Electronic structure and transport properties of doped PbSe

Understanding the electronic structure and transport properties of doped PbSe for its thermoelectric applications is an urgent need. Using a first-principles approach, we first explore the band structures of PbSe doped with a series of impurities, including cation-site substitutional impurities (Na, K, Rb; Mg, Ca, Sr; Cu, Ag, Au; Zn, Cd, Hg; Ga, In, Tl; Ge, Sn; As, Sb, Bi) and anion-site substitutional impurities (P, As, Sb; O, S, Te). Then we calculate the density of states (DOS) difference between the doped samples and pure host sample, which is a useful quantity to recognize the possibility of improving transport properties. The exhibited chemical trends and the nature of the impurity states are well explained with a simplified linear combination of atomic orbitals (LCAO) picture. Finally, we calculate the transport properties of these doped systems within the framework of Boltzmann theory and constant relaxation time approximation. Typical competing behavior between the electrical conductivity and Seebeck coefficient is exhibited, and a significant enhancement of thermoelectric power factor is found in the cation-site Au-doped $p$-type samples, and cation-site As-doped $n$-type samples.

Figure 1

(a) Band structure of 2-atom fcc unit cell PbSe calculated with or without SOC included. (b) $l$-decomposed site-projected partial density of states (PDOS) of PbSe. (c) Simplified LCAO description of PbSe band structure: (I) the initial atomic states; (II) bands formed by including only the Pb-$p$ and Se-$p$ interaction; (III) the same as (II) but also including the Pb-Se $s$-$p$ interaction. For clarity, we did not show the Pb-$s$ and Se-$s$ interaction. (d) Band structure of 64-atom cubic cell PbSe; SOC is not included. The energy zero is set at the valence band maximum.

Figure 2

Band structures and densities of states (DOSs) of PbSe doped with (a) Na, (b) K, and (c) Rb. The eigenvalues are aligned between the doped samples and the pure host sample as explained in Sec. 2. The horizontal dashed line at zero (green) denotes the VBM of pure PbSe, and the other (red) horizontal dashed line denotes the Fermi level in the doped sample. In the left panels, the band structures of doped samples and host PbSe are plotted using (red) solid lines and (green) double-dashed lines. In the right panels, the total DOSs of the doped samples and host PbSe are plotted with the same lines. Their difference is plotted with a (blue) thicker solid line: A positive value means an increase of DOS due to the doping.

Figure 3

(a) PDOS plot for PbSe:Rb. Energy zero is set at the host VBM, and the vertical line denotes the Fermi level in the doped sample. Se${}_1$ and Se${}_2$ denote the nearest and farthest Se atom away from Rb in the 64-atom supercell, respectively. We can say that Se${}_1$ is an impurity-related Se atom, while Se${}_2$ is a kind of hostlike Se atom. (b) Seebeck coefficient, (c) electrical conductivity, and (d) power factor of PbSe:Rb at 300 K compared with PbSe:Na, all relative to the carrier density, where $\tau$ is relaxation time.

Figure 4

The same as in Fig. 2 but for PbSe:I with I doped on the anion site.

Figure 5

The same as in Fig. 3 but for PbSe:I with I doped on the anion site.

Figure 6

The same as in Fig. 2 but for (a) PbSe:Mg, (b) PbSe:Ca, and (c) PbSe:Sr.

Figure 7

The same as in Fig. 2 but for (a) PbSe:Cu, (b) PbSe:Ag, and (c) PbSe:Au.

Figure 8

The same as in Fig. 3, but (a) for PbSe:Au, (b)–(d) for PbSe:X (X = Cu, Ag, Au).

Figure 9

The same as in Fig. 2 but for (a) PbSe:Zn, (b) PbSe:Cd, and (c) PbSe:Hg.

Figure 10

The same as in Fig. 3, but (a) for PbSe:Zn, (b)–(d) for PbSe:X (X = Zn, Cd).

Figure 11

The same as in Fig. 2 but for (a) PbSe:Ga, (b) PbSe:In, and (c) PbSe:Tl.

Figure 12

The same as in Fig. 3, but for PbSe:Tl.

Figure 13

The same as in Fig. 2 but for (a) PbSe:Ge and (b) PbSe:Sn.

Figure 14

The same as in Fig. 2 but for (a) PbSe:As${}_{\text{c}}$, (b) PbSe:Sb${}_{\text{c}}$, and (c) PbSe:Bi.

Figure 15

The same as in Fig. 3, but (a) for PbSe:As${}_{\text{c}}$, (b)–(d) for PbSe:As${}_{\text{c}}$, PbSe:Sb${}_{\text{c}}$, and the reference sample PbSe:I.

Figure 16

The same as in Fig. 2 but for (a) PbSe:P, (b) PbSe:As${}_{\text{a}}$, and (c) PbSe:Sb${}_{\text{a}}$.

Figure 17

The same as in Fig. 3, but (a) for PbSe:Sb${}_{\text{a}}$, (b)–(d) for PbSe:Sb${}_{\text{a}}$, and the reference sample PbSe:Na. In (a), Se denotes the hostlike Se atom which is farthest away from Sb. Pb${}_1$ denotes the impurity’s nearest neighboring Pb atom, and Pb${}_2$ denotes the hostlike Pb atom which is farthest away from Sb.

Figure 18

The same as in Fig. 2 but for (a) PbSe:O, (b) PbSe:S, and (c) PbSe:Te.

Figure 19

The same as in Fig. 17, but (a) for PbSe:O, (b)–(d) for PbSe:O, and the reference sample PbSe host.