Ab initio studies of the electronic structure of defects in PbTe

Understanding the detailed electronic structure of deep defect states in narrow band-gap semiconductors has been a challenging problem. Recently, self-consistent ab initio calculations within density functional theory using supercell models have been successful in tackling this problem. In this paper, we carry out such calculations in PbTe, a well-known narrow band-gap semiconductor, for a large class of defects: cationic and anionic substitutional impurities of different valence, and cationic and anionic vacancies. For the cationic defects, we study the chemical trends in the position of defect levels by looking at series of compounds $R{\mathrm{Pb}}_{2n-1}{\mathrm{Te}}_{2n}$, where $R$ is vacancy or monovalent, divalent, or trivalent atom. Similarly, for anionic defects, we study compounds $M{\mathrm{Pb}}_{2n}{\mathrm{Te}}_{2n-1}$, where $M$ is vacancy, S, Se or I. We find that the density of states near the top of the valence band and the bottom of the conduction band get significantly modified for most of these defects. This suggests that the transport properties of PbTe in the presence of impurities may not always be interpreted by simple carrier doping (from bound impurity states in the gap) concepts, confirming such ideas developed from qualitative and semiquantitative arguments.

Figure 1

(Color online) Supercell model of $R{\mathrm{Pb}}_{31}{\mathrm{Te}}_{32}$ where $R$ is either a vacancy or an impurity atom. Small balls are for Pb and Te, large balls are for $R$.

Figure 2

(a) The total DOS of PbTe with and without Pb vacancy, (b) the difference in the partial DOS between PbTe with and without Pb vacancy where the six nearest-neighbor Te atoms of the Pb vacancy are denoted as Te2 and there are three other inequivalent Te atoms (denoted as Te1, Te3, and Te4), and (c) the difference in the total DOS $\left(\Delta {\rho }_{\mathrm{tot}}\right)$, in the DOS associated with the charges inside the atomic spheres $\left(\Delta {\rho }_{\mathrm{at}}\right)$, and in the DOS associated with interstitial regions $(\Delta {\rho }_{\mathrm{i}}=\Delta {\rho }_{\mathrm{tot}}\text{−}\Delta {\rho }_{\mathrm{at}})$. The energy origin in (b) and (c) is chosen to be the top of the valence band of PbTe; whereas the energy origin in (a) is the highest occupied Kohn-Sham single-particle state (which will be denoted as the Fermi energy) of the system with defect.

Figure 3

The difference in the total DOS $\left(\Delta {\rho }_{\mathrm{tot}}\right)$ between PbTe with and without monovalent impurities: Na, K, Rb, Cs, Cu, and Ag. The energy origin $\left(0\mathrm{eV}\right)$ is chosen to be at the top of the valence band of PbTe.

Figure 4

The difference in the total DOS $\left(\Delta {\rho }_{\mathrm{tot}}\right)$ between PbTe with and without $s$-type divalent atom: Zn, Cd and Hg. The energy origin $\left(0\mathrm{eV}\right)$ is chosen to be at the top of the valence band of PbTe.

Figure 5

(a) The total DOS of PbTe with and without Cd impurity, (b) their difference $\Delta {\rho }_{\mathrm{tot}}$, (c) partial DOS per Te associated with nearest-neighbor Te atoms, and (d) Cd partial DOS of $s$-character. The Fermi energy of PbTe with Cd impurity is at the energy origin $\left(0\mathrm{eV}\right)$.

Figure 6

The difference in the total DOS $\left(\Delta {\rho }_{\mathrm{tot}}\right)$ between PbTe with and without $p$-type divalent atom: Sn and Ge. The energy origin $\left(0\mathrm{eV}\right)$ is chosen to be at the top of the valence band of PbTe.

Figure 7

The difference in the total DOS $\left(\Delta {\rho }_{\mathrm{tot}}\right)$ between PbTe with and without trivalent group III impurities: Ga, In, and Tl. The energy origin $\left(0\mathrm{eV}\right)$ is chosen to be at the top of the valence band of PbTe.

Figure 8

(a) The total DOS of PbTe with and without Ga impurity, (b) their difference $\Delta {\rho }_{\mathrm{tot}}$, (c) partial DOS per Te associated with nearest-neighbor Te atoms, and (d) Ga partial DOS of $s$ character. The Fermi energy of PbTe with Ga impurity is at the energy origin $\left(0\mathrm{eV}\right)$.

Figure 9

The difference in the total DOS $\left(\Delta {\rho }_{\mathrm{tot}}\right)$ between PbTe with and without trivalent group V impurities: As, Sb, and Bi. The energy origin $\left(0\mathrm{eV}\right)$ is chosen to be at the top of the valence band of PbTe.

Figure 10

(a) The total DOS of PbTe with and without As impurity, (b) their difference $\Delta {\rho }_{\mathrm{tot}}$, (c) partial DOS per Te associated with nearest-neighbor Te atoms, and (d) As partial DOS of $p$ character. The Fermi energy of PbTe with As impurity is at the energy origin $\left(0\mathrm{eV}\right)$.

Figure 11

(a) The total DOS of PbTe with and without Te vacancy, (b) their difference $\Delta {\rho }_{\mathrm{tot}}$, and (c) partial DOS per Pb associated with nearest-neighbor Pb atoms. The Fermi energy of PbTe with Te vacancy is at the energy origin $\left(0\mathrm{eV}\right)$.

Figure 12

(a) The total DOS of PbTe with and without S impurity, (b) their difference $\Delta {\rho }_{\mathrm{tot}}$, (c) partial DOS per Pb associated with nearest-neighbor Pb atoms, and (d) $S$ partial DOS of $p$ character. The Fermi energy of PbTe with S impurity is at the energy origin $\left(0\mathrm{eV}\right)$.

Figure 13

(a) The total DOS of PbTe with and without Se impurity, (b) their difference $\Delta {\rho }_{\mathrm{tot}}$, (c) partial DOS per Pb associated with nearest-neighbor Pb atoms, and (d) Se partial DOS of $p$ character. The Fermi energy of PbTe with Se impurity is at the energy origin $\left(0\mathrm{eV}\right)$.

Figure 14

(a) The total DOS of PbTe with and without I impurity, (b) their difference $\Delta {\rho }_{\mathrm{tot}}$, (c) partial DOS per Pb associated with nearest-neighbor Pb atoms, (d) I partial DOS of $p$ character. The Fermi energy of PbTe with I impurity is at the energy origin $\left(0\mathrm{eV}\right)$.

Figure 15

The total DOS of PbTe with In impurity obtained in PAW calculations using two different models: $\mathrm{In}{\mathrm{Pb}}_{31}{\mathrm{Te}}_{32}$ with one In impurity at the corner, and ${\mathrm{In}}_2{\mathrm{Pb}}_{30}{\mathrm{Te}}_{32}$ with one In at the corner and the other one at the center of a $(2\times 2\times 2)$ supercell; the distance between the two In impurities being $13.1\mathrm{\AA }$ and $11.345\mathrm{\AA }$, respectively. The two lines are almost identical except at$\sim -0.5\mathrm{eV}$ and $\sim 4.5\mathrm{eV}$ where the integrated DOS of the deep and hyperdeep defect states of ${\mathrm{In}}_2{\mathrm{Pb}}_{30}{\mathrm{Te}}_{32}$ are larger than those of $\mathrm{In}{\mathrm{Pb}}_{31}{\mathrm{Te}}_{32}$ by a factor of 2. The Fermi energy is at the top of the upper localized band formed by the deep defect state.

Figure 16

The total DOS of PbTe with In impurity obtained in PAW calculations using two different supercell sizes, $(2\times 2\times 2)$ supercell $\left(\mathrm{In}{\mathrm{Pb}}_{31}{\mathrm{Te}}_{32}\right)$ and $(3\times 3\times 3)$ supercell $\left(\mathrm{In}{\mathrm{Pb}}_{107}{\mathrm{Te}}_{108}\right)$; the distance between the two In impurities being $13.1\mathrm{\AA }$ and $19.65\mathrm{\AA }$, respectively. The Fermi energy is at the top of the upper localized band formed by the deep defect state.