In this work, first principles calculations of defect energetics are used to understand the thermo- dynamic stability of different types of point defects in PbTe. For Pb-rich chemical potentials and crystal growth conditions, we find that the most stable defects with the lowest formation energy are Te vacanciesVT e+2 and Pb vacancies VP b−2, whereas those for Te-rich growth conditions are Te
300 600 900 1200 1500
T(K)
0 0.2 0.4 0.6 0.8 1.0
FaysandsGillson MonirisandsPetot GravemannsandsWallbaum Kharifsetsal.
Kimura
Pb Molesfraction,sTe Te
Liquid
Liquid + PbTe PbTe + Liquid
Pb + PbTe PbTe + Te
Figure 3.4: (Color online) Pb-Te phase diagram calculated with the CALPHAD method and com- pared with experimental data.
anti-sitesT e+2P b and Pb vacanciesVP b−2. The Fermi level is determined at various temperatures by numerically solving the charge neutrality equation, which tells us the equilibrium defect and carrier concentrations for various defects in each charged state for both growth conditions. It is found that the electron donating Te vacanciesVT e+2are the highest concentration defects for Pb-rich conditions, which are the cause of itsn-type conductivity, whereas it is the electron accepting Pb vacanciesVP b−2 that are the cause ofp-type conductivity in PbTe for Te-rich growth conditions. This result, along with calculated electron and hole concentrations, agrees well with data from previous experimental works.
This information is then used to modify and re-develop the thermodynamic model of the Pb-Te system using the CALPHAD method. A previous thermodynamic model of this system assumed an anti-site defect model for the PbTe phase, i.e., (Pb, Te)1:(Te, Pb)1. However, from our calculations we found that it is actually the vacancy defects that predominate in this phase over anti-site or interstitial defects. Accordingly, the defect model for this phase was changed to (Pb, Va)1:(Te, Va)1. In addition to this, the defect formation energies calculated from DFT were used in the parameters of the Gibbs free energy expression of this phase and the resulting non-stoichiometric
5,, 6,, 7,, 8,, 9,,
*,,,
**,,
*2,,
*3,,
TAGKu
499, 4995 5,,, 5,,5 5,*,
BrebrickAandAAllgaier HewesAetAalM BrebrickAandAGubner SealyAandACrocker ChouAetAalM
Pb MoleAfractionCATeAGS*,o4u Te
Liquid
Liquid + PbTe PbTe + Liquid
PbTe + Te
DFT:APborich DFT:ATeorich
Figure 3.5: (Color online) Solidus lines of the PbTe phase calculated with the CALPHAD method compared with solubility lines calculated from DFT, and with experimental data.
range was in very good agreement with experimental data. The entire system was then re-assessed based on experimental information on the phase diagram and thermodynamic data on the mixing enthalpy of the liquid phase and chemical potential of Te in the liquid phase. Thus, in this work, we have demonstrated the use of DFT in conjunction with the CALPHAD method by calculating defect formation energies of various intrinsic point defects in PbTe using DFT, and then using those numbers as input to the thermodynamic model of the technologically important PbTe phase, which resulted in solubilities in excellent agreement with experiments. Such a methodology is important for the computation of solubility/stability ranges of dopants in TE materials in different regions of the phase diagrams, which is critical to the fine tuning of TE properties.
-30 -25 -20 -15 -10 -5 0
Enthalpy4of4mixing,4KJ/mol
0 0.2 0.4 0.6 0.8 1.0
Blachnik4and4Gather Moniri4and4Petot Castanet4et4al.
Maekawa4et4al.
Mole4fraction,4Te
Figure 3.6: (Color online) Enthalpy of mixing of the liquid phase calculated at T = 1200 K with reference to the liquid phase states of Pb and Te, and compared with experimental data.
-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0
ChemicalPpotentialPofPTe,PKJ/mol
0 0.2 0.4 0.6 0.8 1.0
PredelPetPal.
BrebrickPandPStrauss
MolePfraction,PTe
Figure 3.7: (Color online) Chemical potential of Te in the liquid phase calculated at T = 1200 K with reference to the fcc phase of Pb and hexagonal phase of Te, and compared with experimental data.
Chapter 4
Calculation of dopant solubilities and phase diagrams of X-Pb-Se (X = Br, Na) limited to defects with localized charge
4.1 Introduction
Impurity dopants are key to unlocking the potential of semiconductors for a variety of applications.
Impurities act as extrinsic dopants that allow for precise control over charge carrier sign and densi- ties. In a semiconductor with an ideal dopant, the excess charge supplied is directly related to the dopant concentration. In PbSe, for example, the substitution of a Br atom for a Se atom should make an ideal n-type dopant because each Br brings one extra electron even though the electronic states of Br are essentially the same as those of Se. Similarly the substitution of Na for Pb should be ideal for p-type PbSe. In many applications, the performance of a semiconductor is limited by the dopant solubility, which is the amount of dopant that can be incorporated [79] before a dopant-rich secondary phase precipitates from the semiconductor (known as dopant precipitation).
Such formation and evolution of unwanted secondary phases often harms lifetime performance, or even the stability and integrity of the material. A phase diagram provides essential information for material design to address these challenges. However, experimental phase diagrams are not always available, or available without enough detail in the region of interest. In such cases, calculated phase diagrams could provide alternative guidelines and help in the development new materials.
Density functional theory (DFT) [14, 15] based defect energy calculations provide powerful insights for understanding properties of lightly doped semiconductors used as photovoltaics or op- toelectronics (see, for example Ref. [80]) that have band gaps of nearly 1 eV or greater. Such studies have been particularly helpful in identifying charge state transitions of deep defects, states that form well inside the forbidden energy gap.
In this study, we use DFT methods to investigate the defect thermodynamics of ideal dopants in the narrow band gap (≈ 0.3 eV) semiconductor PbSe. Experiments performed for this work show that Br is indeed an ideal donor dopant and Na is an ideal acceptor dopant with essentially 100% doping effectiveness, indicating that one charge carrier is measured for each impurity atom.
Doping effectiveness is defined as the ratio of carrier concentration of the sample to the amount of dopant added to it and is given by,
η = n−p X
d,q
c0d,q
or p−n X
d,q
c0d,q
, (4.1)
where the numerator is electron concentration (n−p) in the case of Br-Pb-Se and (p−n) in the case of Na-Pb-Se, andX
d,q
c0d,q is the total atom concentration of the dopant in the PbSe phase found by summing over all the dopant containing defects. PbSe is chosen because the lead chalcogenides (PbQ, Q = Te, Se, S) with the rock-salt structure are excellent thermoelectric materials [13, 81–88]
for applications between 600 K and 900 K [89], where its zT exceeds 1 [88] for both p-type [82]
and n-type [84] materials. Br is chosen due to its comparable size and electronic structure to Se, and reportedzT = 1.1±0.1 at 850 K [84]. Similarly, Na provides fine control over hole carrier concentration [81, 90] that leads tozT close to 1 at 850 K in PbSe [82, 91].
Despite the frequent use of these impurity dopants, no experimental literature is available (to the authors0 knowledge) on the phase diagrams of Br-Pb-Se or Na-Pb-Se, which is unsurprising due to the complexity of the experiments required to determine a ternary phase diagram. Accu- rate calculations of dopant solubilities could supplant tiresome experiments in the search for new semiconductors.