Structural and electronic properties of Pb${}_{1-x}$Cd${}_x$Te and Pb${}_{1-x}$Mn${}_x$Te ternary alloys

A systematic theoretical study of two PbTe-based ternary alloys, Pb${}_{1-x}$Cd${}_x$Te and Pb${}_{1-x}$Mn${}_x$Te, is reported. First, using ab initio methods, we study the stability of the crystal structure of CdTe-PbTe solid solutions, to predict the composition for which the rock-salt structure of PbTe changes into the zinc-blende structure of CdTe. The dependence of the lattice parameter on Cd (Mn) content $x$ in the mixed crystals is studied by the same methods. The obtained decrease of the lattice constant with $x$ agrees with what is observed in both alloys. The band structures of PbTe-based ternary compounds are calculated within a tight-binding approach. To describe correctly the constituent materials, tight-binding parametrizations for PbTe and MnTe bulk crystals as well as a tight-binding description of rock-salt CdTe are proposed. For both studied ternary alloys, the calculated band gap in the L point increases with $x$, in qualitative agreement with photoluminescence measurements in the infrared. The results show also that in p-type Pb${}_{1-x}$Cd${}_x$Te and Pb${}_{1-x}$Mn${}_x$Te mixed crystals an enhancement of thermoelectrical power can be expected.

Figure 1

The total energy $E_{\mathrm{mix}}$ of Pb${}_{1-x}$Cd${}_x$Te mixed crystal, with RS (black stars) as well as with ZB (red dots) structure, vs Cd content. For each $x$, value zero at the energy scale denotes the total energy $E_{\mathrm{separate}}$ of separate, RS PbTe, and ZB CdTe, bulk phases. The total energies are normalized to the number of cation-anion pairs.

Figure 2

Lattice parameter of Pb${}_{1-x}$Cd${}_x$Te mixed crystal as a function of the Cd content $x$. Green and blue dots denote the calculated and experimental values, respectively.

Figure 3

Dependence of the lattice parameter of Pb${}_{1-x}$Mn${}_x$Te on the Mn content $x$. Black and red dots denote the calculated and experimental values, respectively.

Figure 4

Band structure of PbTe along the symmetry lines of the Brillouin zone calculated within our tight-binding model.

Figure 5

RS-CdTe band structure along the main symmetry lines of the Brillouin zone calculated with use of DFT-LDA method within the VASP code.

Figure 6

The band structure of MnTe in RS structure calculated within tight-binding model (with nearest- and next-nearest-neighbors interactions taken into account). The red dashed line is the Fermi level.

Figure 7

Dependence of the energy gaps in Pb${}_{1-x}$Cd${}_x$Te on the Cd content $x$. The blue squares, black stars, and red dots denote the energy gaps at $\Delta$, $\Sigma$, and L, respectively.

Figure 8

Dependence of the energy gaps in Pb${}_{1-x}$Mn${}_x$Te on the Mn content $x$. The blue squares, black stars, and red dots denote the energy gaps at $\Delta$, $\Sigma$, and L, respectively.

Figure 9

Side energy gaps $\Sigma$-L (black rhombuses) and $\Delta$-L (blue triangles) vs Cd concentration $x$ in Pb${}_{1-x}$Cd${}_x$Te.

Figure 10

Side energy gaps $\Sigma$-L (black rhombuses) and $\Delta$-L (blue triangles) vs Mn concentration $x$ in Pb${}_{1-x}$Mn${}_x$Te.

Figure 11

The values of the $D^{-1}dD\left(E\right)/dE$ expression at the Fermi level as a function of Cd content for $p=1\times {10}^{19}$ (black line), $3\times {10}^{19}$ (blue line), and $1\times {10}^{20}{\text{cm}}^{-3}$ (green line).

Figure 12

The dependence of the calculated thermoelectric power of $p$-type Pb${}_{1-x}$Cd${}_x$Te crystals on the Cd content. The black line corresponds to $p=1\times {10}^{19}{\text{cm}}^{-3}$, blue line to $p=3\times {10}^{19}{\text{cm}}^{-3}$, and green line to $p=1\times {10}^{20}{\text{cm}}^{-3}$.

Figure 13

The dependence of the calculated thermoelectric power of $p$-type Pb${}_{1-x}$Mn${}_x$Te crystals on the Mn content. The black line corresponds to $p=1\times {10}^{19}{\text{cm}}^{-3}$, blue line to $p=3\times {10}^{19}{\text{cm}}^{-3}$, and green line to $p=1\times {10}^{20}{\text{cm}}^{-3}$.