## 1. Introduction

It is of utmost technological importance to develop predictive tools that will provide us with information about design of materials' functional properties. In this context, density functional theory (DFT) first-principle calculations offer us such possibilities [1–4], allowing us calculation of structural, interfacial, vibrational, and electronic properties. Knowledge of these properties and how they depend on temperature and material’s composition are essential to assess total thermoelectric (TE) performance of thermoelectric device. Among recently investigated TE materials, silver-antimony-telluride (AgSbTe_{2})-based alloys have emerged as a promising class of materials for TE power generation in low- to mid-temperature range. These compounds are derivatives of lead-antimony-silver-telluride (LAST)-based alloys of AgPb_{m}SbTe_{2+m} form [5–9], which exhibit large TE figure-of-merit *ZT* values ranging from 1.3 to 1.7 [5, 10, 11], which are associated with the intrinsically good TE properties of AgSbTe_{2}-phase.

AgSbTe_{2}-based alloys serve not only for TE power conversion or cooling, but also for nonvolatile electronic memory, being classified as phase change materials, as demonstrated in thorough investigation by Wuttig and coworkers [12–15]. They have attracted scientific interest owing to their special nature of interatomic bonding and vibrational properties [16–18]. On TE aspect, their superior performance is associated mainly to glass-like, intrinsically low values of lattice thermal conductivities, as low as 0.6 W m^{−1} K^{−1} [19]. This anomaly is associated either to strong anharmonicity of interatomic forces [20, 21] or to relatively large variance of interatomic forces prevailing between Ag^{+} and Sb^{3+} cations, encouraging phonon scattering [22]. Additionally, resonant bonding yields high level of structural instability, that is accompanied, for instance, by spontaneous phase decomposition [23, 24]. This, intriguingly, what makes AgSbTe_{2}-based alloys good materials for both TE and phase change applications. Owing to these peculiarities, these alloys have recently been investigated extensively, either experimentally [19, 25–30] or computationally [18, 22, 31–33].

Despite of relatively high *ZT* values of AgSbTe_{2} phase, it is still challenging to increase them to the range of 2–3. Reaching at this limit will enable us employing this material for energy conversion at power levels >500 W [34]. Reduction in lattice thermal conductivity is a conventional way to enhance TE performance and is achieved by either doping with solute elements [35] or formation of second phases to stimulate phonon scattering [36–38]. These lattice defects affect, of course, electronic properties, mainly electrical conductivity and Seebeck coefficient. Attempts to improve TE properties of AgSbTe_{2}-based alloys by doping with different elements [19, 25, 39–47], as well as, by formation of second phase precipitates [46, 48–51], proved to be successful, as reported in the exhaustive studies of Zhang et al., Du et al., and Mohanraman et al. Alloying with second phase forming elements raises imperative question about material’s thermal stability, when employed in TE generators under service conditions, with engineering implications [51–53].

Among efforts to improve TE properties of AgSbTe_{2}-based alloys, Min et al. reported on improvement of electron transport properties due to La-doping [27]. Positive effects on PbTe compound due to La-doping were recorded, as well [54, 55]. In their recent study, Min et al. introduce a complete analysis of TE properties of AgSbTe_{2} doped with La of different concentrations [56].

Notwithstanding the aforementioned successful experimental and computational attempts, a set of experimental routines, that is initiated and directed by predictions from first principles for complete TE performance or any other computational procedure, is missing.

A significant step in this direction is introduced by our previous investigations of AgSbTe_{2}-based phase, involving both computational and experimental aspects [57, 58]. Vibrational properties of both AgSbTe_{2}-based and La-doped-AgSbTe_{2} alloys, including frequency-dependent vibrational density of states functions (v-DOS), temperature-dependent heat capacity, sound velocities, and Debye temperatures were evaluated employing lattice dynamics first-principles calculations. It was reported that La-doping reduces average sound velocity and varies v-DOS of AgSbTe_{2}-based phase [57, 58]. Quantitatively, lattice thermal conductivity of La_{0.125}Ag_{0.875}SbTe_{2} alloy was calculated to be lower by ca. 14%, than that of AgSbTe_{2}-based phase at 300 K [58]. Experimental validations of these effects of La-alloying on reducing lattice thermal conductivity were made, as well [58].

This chapter introduces a refined approach of evaluating temperature-dependent lattice thermal conductivity from data obtained *ab-initio*, as well as, calculations of electronic transport coefficients. Most importantly, this chapter presents experimental validations for the entire dataset obtained from first-principles, including thermal and electrical measurements.

## 2. Chapter outline

This chapter consists primarily of original computational and experimental data along with data, that were reported by us earlier [57, 58], and is aimed at drawing a complete picture depicting the role of lanthanum-alloying in silver-antimony-telluride-based alloys on a broad TE view. Herein, we demonstrate how alloying of AgSbTe_{2} (P4/mmm) alloy with lanthanum solute atoms brings about significant reduction in thermal conductivity with positive effects on TE power factor, as well; thus, achieving improved *ZT* values. This is achieved by DFT calculations of structural, interfacial, vibrational, and electronic properties performed for La-free and La-doped alloys, followed by experimental validation implemented by thermal and electronic transport measurements.

Computational procedures are divided into the following steps:

Total energy calculations for different polymorphs of AgSbTe

_{2}phase are implemented to evaluate their Helmholtz free energies, indicating which one is the most stable around and above room temperature.Vibrational calculations are performed for both La-free and La-doped lattices, including phonon dispersion and density of states, average sound velocity, and Debye temperature. These values enable us evaluating temperature-dependent lattice thermal conductivity values.

Electronic calculations of band structures of both La-free and La-doped lattices are performed, and the resulting transport coefficients are derived applying Boltzmann transport theory.

To consider the case in which Sb

_{2}Te_{3}- and Sb_{8}Te_{3}-phases precipitate inside AgSbTe_{2}-matrix, similar DFT transport coefficient calculations are performed for both phases, as well. In this context, the molar formation energies of both phases and the free energies of their interfaces with AgSbTe_{2}-matrix are simulated to predict their thermal stability and nucleation sequence.Additionally, to address the influence of deviations from AgSbTe

_{2}-stoichiometry on electron transport properties, the latter is simulated for two off-stoichiometric model alloys Ag_{3}SbTe_{4}and AgSb_{3}Te_{4}.

Experimental procedures are divided into the following steps:

Model ternary (AgSbTe

_{2}) and quaternary (3 at.% La-AgSbTe_{2}) alloys are synthesized by vacuum melting followed by quenching and hot-pressing. The appropriate conditions enabling formation of AgSbTe_{2}-matrix that dissolves La-atoms with no La-rich precipitates are found.Differential scanning calorimetry (DSC) tests are implemented for both La-free and La-doped alloys to address thermal stability issues and how they are influenced by La-additions.

Temperature-dependent thermal conductivity of both alloys is determined to realize effects of La-doping and to compare them with those predicted from first-principles.

Similarly, both temperature-dependent electrical conductivity and Seebeck coefficients are measured for ternary and quaternary alloys to realize effects of La-doping and to compare them with those predicted from first-principles.

Finally, to assess whether La-doping contributes to conversion efficiency, TE power factor and figure-of-merit are evaluated for La-free and La-doped materials.

## 3. Research methods

This section provides a brief description of computational and experimental methods applied in this research.

### 3.1. First-principles calculation

The primary calculations are performed for AgSbTe_{2} stoichiometric phase. To address, however, both optional cases of second-phase nucleation and deviations from stoichiometric composition, as described in Section 2, the following phases are simulated, as well: Sb_{2}Te_{3}, Sb_{8}Te_{3}, Ag_{3}SbTe_{4}, and AgSb_{3}Te_{4}.

#### 3.1.1. The base AgSbTe_{2} phase—structural and vibrational calculations

Silver-antimony-telluride of AgSbTe_{2} stoichiometry is commonly known to introduce a cubic lattice structure; however, it was suggested, that it may coexist with tetragonal and rhombohedral forms [59]. The following optional space group symmetries: cubic (Pm-3m, No. 221), tetragonal (P4/mmm, No. 123), and rhombohedral (R-3m, No. 166) have been simulated from first-principles [57]. Calculations of temperature-dependent Helmholtz free energy for these three polymorphs indicate that P4/mmm polymorph is the most stable one at temperatures above 400 K, whose energy exhibits close proximity to that of Pm-3m polymorph. Based on calculated v-DOS function for P4/mmm model alloy consisting of 4 atoms per simulation cell, **Figure 1a**, it was decided to test effects of doping with lanthanum atoms (to be discussed further below). To represent effects of La-doping with effective concentration of La atoms, that is close to realistic doping levels, a model compound of Ag_{7}LaSb_{8}Te_{16} stoichiometry was constructed having the same P4/mmm space group symmetry as of the original AgSbTe_{2} lattice. In this compound, consisting of 32 atoms per simulation cell, **Figure 1b**, La-atom substitutes for

#### 3.1.2. The base AgSbTe_{2} phase—electronic calculations

To simulate the effects of La-doping on electrical conductivity and Seebeck coefficient, electronic band structures are calculated for both lattices from first principles. A plane-wave basis set is implemented in Vienna *ab-initio* simulation package (VASP) [60–62] and *MedeA*^{®} software environment [63]. The exchange-correlation electronic energy is expressed by means of generalized gradient approximation (GGA) using PBEsol energy functional [64] and projector augmented wave (PAW) potentials, which are utilized to represent core electron density [65]. Sampling of Brillouin zone is carried out using a set of uniform Monkhorst-Pack *k*-point mesh with density ranges between 0.14 and 0.17 Å^{−1} and smearing method of linear-tetrahedron with Blöchl corrections [66]. To represent Kohn-Sham electronic wave functions, the plane waves are spanned with 400 eV energy cutoff for the structural relaxation or electronic calculations, respectively. Electronic optimization procedures are performed applying 10^{−5} eV energy convergence threshold.

The calculated 0 K band structures are used for evaluation of temperature-dependent electrical conductivity, electronic component of thermal conductivity, and Seebeck coefficient, applying near-equilibrium Boltzmann transport theory with constant relaxation time approximation, as implemented in BoltzTrap code [67].

The partial electrical conductivity tensor, *i*th energy band and a given **k**-point, is obtained from Cartesian component of electron group velocity by derivation of *i*th energy band,

where *e* is electron unit charge,

where

Finally, the explicit expression for Seebeck coefficient tensor,

#### 3.1.3. The Sb_{2}Te_{3} and Sb_{8}Te_{3} phases

Nonmagnetic DFT calculations are performed for Sb_{2}Te_{3} and Sb_{8}Te_{3} crystal structures having (R-3m) space group symmetry, which incorporate 15 and 33 atoms per simulation cell, respectively. Both lattice structures are rendered in **Figure 1****c** and **d**, respectively. A computational routine similar to aforementioned one is implemented with several differences. GGA approximation is applied for a set of uniform 9 × 9 × 9 Monkhorst-Pack *k*-point mesh, and plane waves are spanned with either 400 or 350 eV energy cutoff for structural relaxation or electronic calculations, respectively. Electronic optimization procedures are performed applying 10^{−6} eV energy convergence threshold.

Structural relaxation procedures are first performed, allowing variation of cell volume and atom positions at all degrees of freedom, setting a convergence threshold of 10^{−4} eV Å^{−1} for Hellman-Feynman forces. The resulting lattice parameters obtained for relaxed crystal structures are: a = b = 4.34 Å and c = 31.21 Å for Sb_{2}Te_{3}; and: a = b = 4.37 Å and c = 64.93 Å for Sb_{8}Te_{3}, which are in good agreement with data reported in the literature [72, 73]. Then, electronic band structure calculations are performed for both relaxed structures.

Band structures are calculated in the same manner as mentioned above, allowing calculations of temperature-dependent electrical conductivity, electronic component of thermal conductivity, and Seebeck coefficient values. These calculations yield p-type behavior for both structures, and we fine-tune the positions of electronic chemical potential to reside at the top of the valence bands. This yields Seebeck coefficient values that are very similar to those measured by us experimentally for pure Sb_{2}Te_{3} standard. Additionally, we set electron relaxation time to be 8 fs, so as to fit electrical conductivity values calculated for Sb_{2}Te_{3} with those measured for the same standards. We, then, apply the same relaxation time for Sb_{8}Te_{3}, as well.

To address bulk and interfacial energetic aspects related with nucleation of Sb_{2}Te_{3} and Sb_{8}Te_{3 }phases in AgSbTe_{2} phase, we have simulated formation energies of Sb_{2}Te_{3} and Sb_{8}Te_{3} phases and their interfaces with AgSbTe_{2} phase. Molar formation energy of model Sb_{p}Te_{q} cell,

where ^{−1}, respectively. The free energy of silver-antimony-telluride (AST)/antimony-telluride (SBT) interface is calculated constructing a slab model having AST/SBT generic form, and using the following expression [74]:

where *A* is AST/SBT interface cross-sectional area,

#### 3.1.4. Ag_{3}SbTe_{4} and AgSb_{3}Te_{4} model compounds

To simulate the effects of deviations from stoichiometric AgSbTe_{2} composition, we construct three model alloys based on P4/mmm space group symmetry, which is reduced to cubic P1 symmetry, by setting equal lattice parameter of a = 6.113 Å for all. The resulting structures simulated are: (AgSbTe_{2})_{2}, Ag_{3}SbTe_{4}, and AgSb_{3}Te_{4}, which appear in **Figure 1e, f,** and **g,** respectively. All three structures contain 8 atoms per unit cell and Sb/Ag ratios of 1, 1/3, and 3, respectively. To calculate band structures of these three model alloys, spin-orbit (SO) magnetic calculations were performed utilizing a similar GGA/PAW routine as described above for uniform 7 × 7 × 7 Monkhorst-Pack *k*-point mesh and 400 eV energy cutoff to represent Kohn-Sham electronic wave functions, applying 10^{−6} eV energy convergence threshold. SO coupling is often being considered in band structure calculations [32, 70, 75]. TE transport properties were calculated according to the procedure detailed by Eqs. (1)–(4).

### 3.2. Experimental procedure

#### 3.2.1. Materials synthesis

Experimental procedures implemented in this study are intended to validate the effects of La-alloying on TE performance, as predicted from first principles. They include synthesis of two model alloys, La-free and La-alloyed, having molar ratios (Ag:Sb:Te:La) of 18:29:53:0 and 15.75:29:53:2.25, respectively. Generally, synthesis procedures comprise vacuum melting and iced-water quenching, followed by uniaxial hot-pressing at two distinct temperatures, 540 and 500°C, yielding two series of 12.7 mm dia. pellets referenced below as Series A and Series B, respectively. The difference between these two series of alloys is manifested by their phase contents and average composition in matrix. These factors significantly affect TE performance, as will be discussed further below. A detailed description of the experimental procedures appears elsewhere [58].

#### 3.2.2. Materials and thermoelectric property characterization

Materials characterization procedures include microstructure, phase identification, and composition analysis employing scanning electron microscopy and X-ray diffraction [58]. Assessment of alloys’ thermal stability is investigated using SETARAM 1600 DSC with a scanning rate of 25 K min^{−1} at temperatures ranging from room temperature through 973 K.

Temperature-dependent electrical conductivity, σ(T), and Seebeck coefficient, S(T) (thermopower), of these pellets are measured in temperature range from 300 to ~700 K employing *Nemesis*^{®} SBA-458 apparatus (Netzsch GmbH), which is designed for simultaneous measurements of electrical conductivity and thermopower for planar geometry [76–78].

*MicroFlash*^{®} LFA-457 laser flash analyzer (LFA; Netzsch GmbH) is utilized to measure directly of thermal diffusivity, α(T), of pellets in the same temperature range applying pulse-corrected Cowan approximation to consider heat loss of the samples [79], yielding instrumental accuracy of 2%. Material’s density, _{p}(T), is simultaneously measured in LFA by comparative method using pure Al_{2}O_{3}—reference sample having similar geometry [76]. The resulting accuracy of evaluation of thermal conductivity values is equal to 10%. Pellets’ thermal conductivity values, κ, are then determined by measuring their temperature-dependent thermal diffusivity and heat capacity, as well as, density; κ is then expressed by [80]:

## 4. Effects of La-alloying on thermoelectric performance

In this section, we introduce the concept resting behind La-alloying: its origin and implications, predictions from first principles, and experimental validations. Comparative discussion of the results in view of TE performance is provided.

### 4.1. Predictions from first-principles

#### 4.1.1. The AgSbTe_{2}(P4/mmm) phase

### 4.1.1.1. Structural and vibrational properties

P4/mmm form of AgSbTe_{2} phase is found to be the most stable one compared to all three polymorphs at temperatures larger than 400 K and exhibits Helmholtz free energy values with close proximity to those of cubic polymorph [57]. Frequency-dependent v-DOS, *phonon engineering* has been coined [35]. Particularly, substitutions for Ag-sublattice sites by elements of different mass or atomic radius are expected to modify v-DOS with respect to that of pure AgSbTe_{2} phase by suppressing its major v-DOS peak. This, consequently, will reduce lattice thermal conductivity. La has been suggested as optional substitution atom due to its relatively large mass and atomic radius compared to average values of AgSbTe_{2}, that is, 138.91 a.m.u. and 187 pm vs. 121.21 a.m.u. and 143.98 pm, respectively, giving rise to enhanced phonon scattering by point defects [81–84]. Furthermore, La-alloying has commercial outcomes, since La is the most inexpensive element compared to constituents of AgSbTe_{2 }alloy and is one of the less inexpensive ones among *energy-critical elements* [85].

Three substitutional options were tested, in which La substitutes for Ag, Sb, or Te, and it was found that substitution at Ag-sublattice sites is the most energetically preferred state for P4/mmm symmetry [57]. Accordingly, La-doped structure was constructed, in which one La-atom substitutes for 1/8. of Ag-atoms, and is shown in **Figure 1b**. First, v-DOS was calculated for La-doped structure and 2.0 THz peak was suppressed, as expected. Second, phonon dispersion curves were calculated for both AgSbTe_{2} and LaAg_{7}Sb_{8}Te_{16} alloys close to Γ-point along c-crystallographic direction, indicating, that the slopes of the one longitudinal and two transverse acoustic modes of AgSbTe_{2}-lattice are greater, than those of La-alloyed one [57]. Quantitatively, average sound velocities derived for pure and La-alloyed materials are 1727 and 1046 m s^{−1}, respectively. Moreover, temperature-dependent heat capacity functions were determined for both structures, yielding slightly lower values for La-alloyed material. Both values of sound velocity and heat capacity that are found to decrease due to La-alloying imply, that La-alloying should reduce lattice thermal conductivity [57]. Additional calculations employing Debye approximation for low-temperature range of heat capacity yield Debye temperatures and sound velocities for both pure and La-alloyed materials, which are 112 K and 1684 m s^{−1} vs. 104 K and 1563 m s^{−1}, respectively [58]. It is noteworthy that evaluation of sound velocity in this manner is considered to be more physically reliable, since it represents the entire space of lattice directions, rather than individual one. It is, therefore, expected that this way of calculation should yield thermal conductivity values, that fit experimental data better than the former way does.

### 4.1.1.2. Effects of La-doping on thermal conductivity

Average sound velocity, _{2} alloy) [30, 57], only U-processes dominate. Second, Callaway model considers

where _{7}Sb_{8}Te_{16} andAgSbTe_{2} alloys, therefore, reflects the lower limit of relative reduction in thermal conductivity due to La-alloying. Applying dispersion curves close to Γ-point along c-crystallographic direction, it is predicted, that

A more thorough and accurate treatment of expression (8) considers the effects of La-alloying on *τ*, contributions of two major scattering mechanisms are taken into account. The first one is phonon-phonon inelastic interactions, i.e., U-processes, that prevail for these alloys above room temperature. Relaxation time for U-processes,

where *M* is average atomic mass of alloy. Second, to account for internal composition inhomogeneity or compositional modulations at unit-cell length scales, that are typical for such materials [5, 8, 16, 91], the boundary scattering mechanism is employed for characteristic period

To consider dependence of v-DOS on phonon frequency, frequency-averaged expression for

where

The resulting values of lattice thermal conductivity for LaAg_{7}Sb_{8}Te_{16} and AgSbTe_{2} alloys are obtained from Eq. (8) by substituting the respective physical magnitudes for both alloys in Eqs. (9)–(11) [19, 20, 30, 57, 58, 92] with _{7}Sb_{8}Te_{16} and AgSbTe_{2 }alloys calculated as function of temperature appear in **Figure 2**.

It is shown that thermal conductivity exhibits realistic values, that correspond with data documented in the literature [19, 20, 25, 98] with marked decrease due to La-doping, ranging between relative values of 11 and 19%, depending on temperature.

### 4.1.1.3. Effects of La-doping on electrical properties

It was shown that La-alloying reduces lattice thermal conductivity values, which affects TE performance positively. To address, however, the total effects of La-alloying on TE performance, evaluation of electrical conductivity and Seebeck coefficient is essential. This goal was achieved from first-principles applying Boltzmann transport theory as described above for LaAg_{7}Sb_{8}Te_{16} and AgSbTe_{2} alloys. The results are plotted in **Figure 3** in temperature range 50–1000 K.

It is found that La-doping results in reduction in electrical conductivity (e.g., from ca. 1800 down to 250 S cm^{−1} at room temperature) and, at the same time, increase in Seebeck coefficient, e.g., from ca. 4 up to 40 µV K^{−1} at room temperature. For the sake of comparison, Jovovic and Heremans reported on experimental measurements of electrical conductivity and Seebeck coefficients of stoichiometric and doped AgSbTe_{2} alloys at temperatures up to 400 K [19, 98]. For example, they report on electrical resistivity value of 5 × 10^{−5} Ohm m at 100 K for stoichiometric AgSbTe_{2 }alloy, which is equivalent to 200 S cm^{−1}.

They report also on electrical resistivity that increases with temperature, indicating charge carriers scattering. Additionally, electrical resistivity may either increase or decrease with doping, depending on dopant’s chemical identity. In the present case, electrical conductivity values are significantly larger, e.g., ca. 1900 S cm^{−1} at 100 K, and are decreasing with temperature, where La-doping reduces conductivity. Seebeck coefficient values reported by Jovovic et al. exhibit general trend of increase with temperature, which corresponds to trend calculated in the present case. Also, they report on general trend of increase in Seebeck coefficient values due to doping (except doping with AgTe), in agreement with the present study for La. Complementary trend is reported by Du et al. [25, 99]. Most interestingly, effects of La-doping on electrical properties of AgSbTe_{2} alloy are reported by Min et al. [27]. They report on trends that are qualitatively similar to those of the present study. First, La-doping was also reported to reduce electrical conductivity, e.g., from ca. 400 S cm^{−1} for undoped AgSbTe_{2} down to 66 S cm^{−1} for 3 at.% La-doping at room temperature. Second, La-doping increases Seebeck coefficients, e.g., from ca. 90 µV·K^{−1} for undoped AgSbTe_{2} up to ca. 220 µV K^{−1} for 3 at.% La-doping at room temperature. Quantitatively, values of electrical conductivity calculated in this study are considered to be large with respect to the above cited studies. Conversely, Seebeck coefficient values calculated in this study are considered to be smaller than those reported by the above studies. We note, however, that such calculations are most meaningful for comparative purposes, since they rest upon values, that should be calibrated against experimental data, such as electronic chemical potential and relaxation times.

It is indicated that the effects of La-doping on electrical conductivity and Seebeck coefficient are opposite to each other. Evaluation of TE power factor (PF; **Figure 4** displays PF calculated for LaAg_{7}Sb_{8}Te_{16} and AgSbTe_{2} alloys in temperature range 50–1000 K.

It is shown that La-doping has considerably positive effect on PF. This also corresponds with the data reported by Min et al. [27], specifically for low La-concentration regime. We note that, moreover, La-doping reduces lattice thermal conductivity, as shown in **Figure 2**. We conclude that La-alloying should improve energy conversion efficiency of AgSbTe_{2} (P4/mmm), as reflected by increased TE figure-of-merit.

#### 4.1.2. Formation of Sb_{2}Te_{3} and Sb_{8}Te_{3}(R-3m) phases

The single δ-phase is Sb-rich phase based on AgSbTe_{2} alloy. Since it has limited solubility to Sb with relatively moderate slope of Sb-solvus, it is likely to decompose to δ+Sb_{2}Te_{3} phase mixture [28, 29, 100–103], whereas Sb_{2}Te_{3} is equilibrium phase and may appear as different homologous forms [72, 73, 104–107]. Precipitation of antimony-telluride second phase in δ-matrix is expected to affect TE performance due to contributions from both matrix and precipitate phases or variation of the average matrix composition. In the following sections, we address both aspects. Section 4.1.2.1 introduces the issue of precipitation sequence based on bulk/interfacial energetic considerations, and Section 4.1.2.2 predicts the effects of phase formation on electronic properties. Then, Section 4.1.3 deals with compositional variations in the matrix and their effects on electronic properties.

### 4.1.2.1. The precipitation sequence: energetic aspects

Nucleation of Sb_{2}Te_{3} and Sb_{8}Te_{3} phases from Sb-saturated δ-AgSbTe_{2} matrix has been observed, which can be associated to different experimental conditions. To account for the sequence of phase formation, information on both bulk and interfacial energetics is required. The formation energies of Sb_{2}Te_{3} and Sb_{8}Te_{3} (R-3m) phases, calculated according to Eq. (5), are −62 and −56.6 kJ mol^{−1}, respectively. This implies that Sb_{2}Te_{3} is more energetically favorable, assuming, that Sb_{2}Te_{3} precipitates are adequately large, so that, interfaces do not play significant role. To address the role of interfaces, free energies of Sb_{2}Te_{3}/AgSbTe_{2} and Sb_{8}Te_{3}/AgSbTe_{2} interfaces are evaluated. To this end, two slab models of (Sb_{2}Te_{3})_{2}/(AgSbTe_{2})_{5}/(Sb_{2}Te_{3})_{2} (40 atoms) and (AgSbTe_{2})_{3}/(Sb_{8}Te_{3})_{6}/(AgSbTe_{2})_{3} (90 atoms) forms are constructed, respectively, consisting of two interfaces each, exhibiting **Figure 5**.

It is noted that two interfaces presented in both slabs shown in **Figure 5a** and **b** consist of different Sb- and Te-terminating planes, so that interfacial free energies calculated according to Eq. (6) represent an average value for both terminations. Correction factor is, therefore, applied to represent interfacial free energy of low-energy Sb-termination. The resulting values for Sb_{2}Te_{3}/AgSbTe_{2} and Sb_{8}Te_{3}/AgSbTe_{2} interfaces are^{−2}, respectively. These values are considered to be relatively low compared to those of intermetallic compounds and are comparable with those of pure metals [108]. This is, however, not surprising, considering the extremely small atomic misfit between the _{2}Te_{3} or Sb_{8}Te_{3} precipitates in the form of long lamellae along these planes [28, 29, 57, 58, 100–103]. These low values of interfacial free energy also initiate fast nucleation, thanks to low activation energy for nucleation, which is proportional to

Interestingly, Sb_{2}Te_{3} phase exhibits lower value of formation energy and higher value of interfacial free energy compared to those of Sb_{8}Te_{3} phase. This implies that Sb_{8}Te_{3} is metastable phase that may form prior to the nucleation of Sb_{2}Te_{3} equilibrium phase [28, 29, 100–103]. Suggested nucleation sequence is, therefore, supersaturated-δ → supersaturated-δ + Sb_{8}Te_{3} → equilibrium-δ + Sb_{2}Te_{3}.

### 4.1.2.2. Effects of Sb_{2}Te_{3} and Sb_{8}Te_{3} formation on electronic properties

In view of aforementioned prospect for the presence of either of Sb_{2}Te_{3}- or Sb_{8}Te_{3}-phases in AgSbTe_{2}-matrix, calculations of transport coefficients of these phases provide us with predictions of the effects such phase mixture on TE performance. **Figure 6** displays electrical conductivity and Seebeck coefficients calculated in temperature range 50–1000 K.

It is shown that both electrical conductivity and Seebeck coefficient values of Sb_{2}Te_{3} phase are larger than those of Sb_{8}Te_{3} phase in wide temperature range, e.g., ca. 2100 S cm^{−1} and 85 µV K^{−1} for Sb_{2}Te_{3} compared to 1380 S cm^{−1} and 29 µV K^{−1} for Sb_{8}Te_{3} at 300 K, respectively. Moreover, comparison of these results with the data shown in **Figure 3** for AgSbTe_{2}-matrix implies that precipitation of Sb_{2}Te_{3} phase yields positive influence on TE performance: both electrical conductivity and Seebeck coefficient increase. It is strikingly indicated that the effects of Sb_{2}Te_{3} precipitation are even greater, than those of La-doping. It is, therefore, concluded that the desirable material from TE viewpoint is La-doped, Sb-supersaturated δ-AgSbTe_{2}-matrix that is aged for a certain duration to form considerable amount of Sb_{2}Te_{3} phase. The effects of Sb_{8}Te_{3} phase on TE performance are, conversely, inferior to those of Sb_{2}Te_{3} phase. Sb_{8}Te_{3} is, however, metastable phase and is not expected to prevail for long durations at elevated temperatures (e.g., under service conditions of TE generator) due to low thermal stability.

#### 4.1.3. Effects of off-stoichiometry on electronic properties of the AgSbTe_{2} phase

An additional effect taking place during precipitation of any Sb_{p}Te_{q} phase from Sb-supersaturated δ-matrix is enrichment of δ-matrix with Ag-atoms and depletion of Sb. To simulate these compositional variations, two off-stoichiometric model alloys are constructed, namely Ag_{3}SbTe_{4} and AgSb_{3}Te_{4}, in addition to stoichiometric AgSbTe_{2 }phase. These model compounds, appearing in **Figure 1e, g,** and **f,** exhibit Sb/Ag ratios of 1/3, 3, and 1, respectively. **Figure 7** displays electrical conductivity and Seebeck coefficient values calculated in temperature range 50–1000 K.

It is shown, that increase in Sb/Ag ratio results in decrease in electrical conductivity simultaneously with increase in Seebeck coefficient. This trend corresponds well with study of Jovovic and Heremans [19], who reported on decrease in both Seebeck coefficient and electrical resistivity due to additions of 2% AgTe to stoichiometric AgSbTe_{2}-phase, i.e., reducing Sb/Ag ratio. It should be noted that comparison of this trend with data reported in the literature is not straightforward, since compositional changes involve in practice not only Sb/Ag ratio, but also ratio of Te to any of the other species. Additionally, deviations from given stoichiometry often involve formation of second phases, which is not directly simulated here. For instance, Zhang et al. reported on dependence of TE properties on composition for Ag_{2−y}Sb_{y}Te_{1+y }-based alloys and found that electrical conductivity increases, while Seebeck coefficient decreases with *y*-values increasing from 1.26 up to 1.38 [101].

Most importantly, this predicted effect of Sb/Ag ratio on electrical properties has major implications on the temporal evolution of TE performance of the material during aging heat treatments (below Sb-solvus), or of TE generator during service. Since Sb_{p}Te_{q} phases nucleate from Sb-supersaturated δ-matrix during heat treatments, Sb/Ag ratio in δ-matrix decreases. This should be accompanied by increase in electrical conductivity concurrently with decrease in Seebeck coefficient.

## 5. Experimental results

It was shown above how first-principles calculations provide us with information about all aspects concerning TE transport behavior, including thermal and electrical conductivity and Seebeck coefficient. Most importantly, this assists us in tailoring the material by introducing lattice defects to enhance its TE performance. In the following section, we introduce experimental procedures taken for validating the above predictions. Comparing between both aspects is, moreover, very instructive not only on engineering aspects, but also on universal aspect, by realizing how to implement computational tools to predict properties of other materials.

### 5.1. Microstructure and implications on thermoelectric behavior

As mentioned above, two classes of La-alloyed AgSbTe_{2}-based materials were prepared by uniaxial hot-pressing at 540 or 500°C, and are classified as Series A and Series B, respectively. The ideal case for testing the effects of La-doping is single δ-phase dissolving La homogeneously. This, however, is difficult to achieve. Hot-pressing at 540°C, that is, above Sb-solvus, expected to yield the desirable single δ-phase that does not contain Sb_{2}Te_{3} precipitates [101–103]. These Sb_{2}Te_{3} precipitates, indeed, were not observed in Series A samples; however, La-rich precipitates having stoichiometry close to LaTe_{2} were observed [110]. As a result, δ-matrix was found to be depleted of La [58], which does not allow us comparison between La-free and La-doped samples. Conversely, samples of Series B, that were hot-pressed at 500°C, exhibit considerable amount of Sb_{2}Te_{3} phase, which is expected; however, LaTe_{2} precipitates are dissolved, so that δ-matrix contains adequately large amount of La, close to its nominal concentration. Series B is, therefore, more suitable to exemplify the effects of La-doping. Moreover, as predicted from first-principles, the presence of Sb_{2}Te_{3} precipitates, in addition to La solute atoms, has positive effects on electronic transport.

### 5.2. Thermal analysis

Thermal conductivity measurements performed for both Series A and Series B indicate the expected trend. First, all thermal conductivity values lie in the range 0.6–0.8 W m^{−1} K^{−1} [58]. Second, samples of Series A did not exhibit any considerable difference between La-free and La-doped materials [58]. This is associated to depletion of δ-matrix from La solute atoms, so that, matrix composition of La-doped and La-free materials is practically the same. Third, and most importantly, it was found that thermal conductivity of La-doped materials is significantly lower than those of La-free materials of Series B, e.g., 0.8 W m^{−1} K^{−1} for La-free and 0.6 W m^{−1} K^{−1} for La-doped samples at 500 K. This is strikingly corroborated by predictions from first-principles, both qualitatively and quantitatively, as shown in **Figure 2**. It is also noteworthy that both values coincide at temperatures larger than 650 K, which can be associated with phase transition [58, 101–103]. Thorough discussion of thermal conductivity values measured for Series A and Series B materials and their relationship with microstructure appears elsewhere [58]. To address this issue of phase transition, DSC measurements were implemented for both La-free and La-doped samples, **Figure 8**.

Endothermic peak at around 630–650 K is observed for La-free material, which is associated with Ag_{2}Te + Sb_{2}Te_{3} → δ-AgSbTe_{2} phase transition at 360°C [101–103]. La-doped material, however, does not exhibit this transition. This corresponds well with thermal conductivity behavior reported by us earlier [58], in which temperature-dependent thermal conductivity of La-doped materials show up continuous trend, whereas La-free materials exhibit sharp drop of thermal conductivity around this temperature. This implies that La-additions help in stabilizing δ-phase against decomposition, which is expected to contribute to stability of TE device operation at service conditions. Sharp endothermic peak at ca. 860 K, which is common for both La-free and La-doped materials, is associated to melting.

### 5.3. Electrical property measurements

It was predicted from first-principles that La-doping reduces electrical conductivity and increases Seebeck coefficient, **Figure 3**. Measurements of electrical conductivity and Seebeck coefficients were carried out for both Series A and Series B materials. The samples of Series B are of our interest, since they dissolve La-atoms in δ-matrix; we will, therefore, introduce these results first. **Figure 9** displays experimentally collected electrical conductivity and Seebeck coefficient values of La-free and La-doped materials of Series B.

It is shown that electrical conductivity values decrease, e.g., from ca. 1400 down to 900 S cm^{−1} at room temperature, and Seebeck coefficient increase, e.g., from ca. 30 up to 70 µV K^{−1} at room temperature, due to La-doping. This behavior is, qualitatively, the same as that observed for calculated values shown in **Figure 3**. Moreover, temperature dependence, that is, electrical conductivity decreasing and Seebeck coefficient increasing with temperature for both La-free and La-doped materials, is identical to that indicated by calculated values shown in **Figure 3**. There are two major differences between experimental and calculated values appearing in **Figures 3** and **9**, respectively. First, the absolute values of measured Seebeck coefficient values are greater than calculated ones. Also, difference of electrical conductivity between La-doped and La-free materials is smaller for measured dataset than for calculated ones. This is probably due to difficulty to simulate low dopant concentrations in DFT [70]. Second, it is noteworthy that both values of electrical conductivity and Seebeck coefficients measured for La-free and La-doped materials converge at temperatures >650 K, **Figure 9**. Interestingly, these convergences occur due to sharp deviations of the values featured by La-free material, whereas the values of La-doped materials preserve their continuous trendline. This observation corresponds well with the behavior shown by DSC curves in **Figure 8**, where La-free compound decomposes at around 650 K, whereas La-doped compound seem to preserve its thermal stability. This also corresponds with converging thermal conductivities of the samples of Series B as discussed above [58]. Following our comparative discussion in Section 4.1.1.3, experimental values of electrical properties are found to be closer to experimental values reported in the literature than to calculated values [19, 25, 27, 98, 99].

To complement our understanding of the effects of La-doping on electronic properties, we measured temperature-dependent electrical conductivity and Seebeck coefficient values for the samples of Series A, as well. The results are plotted against temperature in **Figure 10**.

Comparison between the results attained for alloys of Series A and Series B is very instructive. As noted, the samples of Series A exhibited formation of LaTe_{2}-like precipitates, which “drain out” La atoms from δ-matrix, resulting in matrix compositions, that are nearly identical to each other for La-free and La-doped materials. For this reason, thermal conductivity values measured for La-free and La-doped materials seem to be practically identical in wide temperature range [58]. It is, therefore, not surprising to observe the same behavior for electrical properties, **Figure 10**.

It is indicated, that both electrical conductivity and Seebeck coefficient values measured for La-free and La-doped materials seem to be very close to each other in the entire temperature range, probably due to nearly identical matrix compositions for La-free and La-doped materials. Additionally, electrical conductivity and Seebeck coefficients featured by La-doped alloys exhibit relatively continuous temperature-dependent behavior, whereas values, measured for La-free alloys exhibit curled behavior. This, again, can be explained in terms of poor thermal stability of La-free materials, as discussed above.

### 5.4. Implications for thermoelectric power conversion

It has been shown that La-doping has unequivocally positive effect on reducing lattice thermal conductivity, both computationally and experimentally. The effects on electrical properties, particularly electrical conductivity and Seebeck coefficient, are opposing each other. To assess the effects of La-doping on device’s power capacity, TE PFs of La-free and La-doped materials of Series B are evaluated based on the data displayed in **Figure 9**. The results are shown in **Figure 11**.

It is clearly shown that La-doping affects positively PF for temperatures lower than 500 K, e.g., PF determined for room temperature increases from ca. 200 to 400 µW m^{−1} K^{−2} due to La-doping. At higher temperatures, PFs of La-free and La-doped materials are practically identical. The maximum PF values observed are around 1000 µW m^{−1} K^{−2}. This trend is similar to that reported by Min et al. [27], that is, PF increasing from 300 up to 1500 µW m^{−1} K^{−2} in respective temperature range from room temperature to 400°C for AgSbTe_{2} alloy.

La-doping was tested by them for different compositions, where composition yielding the greatest PF values is AgSb_{0.99}La_{0.01}Te_{2}, with PF values around 1000–1200 µW m^{−1} K^{−2} in the entire temperature range. Particularly, this La-doped material exhibits superior PF values up to ca. 325°C. This trend is similar to that reported by us in this study.

Finally, determination of TE figure-of-merit for both La-free and La-doped materials will provide us with the ultimate indication whether La-doping enhances TE power conversion efficiency. Based on thermal and electrical properties measured for Series A and Series B, temperature-dependent *ZT* values were determined and appear in **Figure 12**.

Most importantly, it is shown that La-doping increases *ZT* values markedly, **Figure 12b**, e.g., from ca. 0.3 to 0.45 at 473 K. This improvement is due to decrease in thermal conductivity in almost the entire temperature range and increase in PF at low-temperature regime due to La-doping. Above 600 K, again, both values of La-free and La-doped alloys converge due to poor thermal stability of La-free materials. *ZT* values of La-free and La-doped materials shown in **Figure 12b** correspond with those reported by Zhang et al. [101], where the effects of La-doping are comparable to those of stoichiometric variations about AgSbTe_{2} composition. Similar values are reported by Mohanraman et al. [43] and Jovovic and Heremans [19] for Bi-doping, as well as, for Pb-doping [19]. Chen et al. obtain similar *ZT* values for Ge-doping [111] and for Sn-doping [112], depending on concentration. *ZT* values reported in the present study are, however, lower than those reported by Du et al. [25, 99], probably owing to different processing conditions yielding higher electrical conductivity values [113].

The picture, revealed for alloys of Series A, is, however, different; it is shown in **Figure 12a** that La-doping has little or no effect on *ZT*. This is, again, not surprising and follows the trends featured by Series A alloys for electrical conductivity and Seebeck coefficient, **Figure 9**, and thermal conductivity [58] associated to depletion of La-atoms from δ-matrix in La-alloyed materials hot-presses at 540°C.

## 6. Summary and concluding remarks

This chapter introduced the following findings. Computationally, total energy calculations for different polymorphs of AgSbTe_{2} phase yield their Helmholtz free energies, implying, that P4/mmm space group symmetry is the most stable one at temperatures adequately higher than room temperatures. Predictions of the effects of doping on thermal conductivity are established on calculations of vibrational properties, such as phonon dispersion and density of states, from first-principles. Based on specific features in v-DOS curve, it is hypothesized that La-substitution for Ag-sites should result in reduced lattice thermal conductivity. These calculations predict reduction in average sound velocity from 1684 to 1563 m s^{−1} and of Debye temperature from 112 to 104 K due to La-doping. Applying Umklapp mechanism for phonon scattering with frequency-averaged inverse relaxation time, which is combined with boundary scattering, yields temperature-dependent functional forms for lattice thermal conductivity. Marked decrease due to La-doping, ranging between relative values of 11 and 19% depending on temperature, are observed. Then, calculations of electronic band structures of both La-free and La-doped lattices are performed, yielding TE transport coefficients applying Boltzmann transport theory. It is found that La-doping results in reduction in electrical conductivity (e.g., from ca. 1800 down to 250 S cm^{−1} at room temperature) at the same time with increase in Seebeck coefficient, e.g., from ca. 5 up to 40 µV K^{−1} at room temperature.

Attempts to infer conclusions with practical implications from DFT calculations must consider engineering aspects that extend further beyond single phase state having high symmetry unit cell, that maintains its physical properties with time. For example, considerations, such as long-term device operation under elevated service temperatures, should be taken into account. Such case requires original solution for simplified (or, sometimes, over-simplified) approach offered by DFT. Particularly, for the case of thermal stability, exposure of Sb-rich δ-phase to elevated temperatures results in precipitation of Sb_{p}Te_{q}-based phases at the same time with decrease in Sb/Ag ratio in δ-matrix. In this manner, thermal stability issues can be addressed by dividing the realistic conditions into a set of simplified problems, each can be handled by DFT. To this end, we first consider the case in which Sb_{2}Te_{3} and Sb_{8}Te_{3} phases precipitate inside AgSbTe_{2}-matrix. It is found that both electrical conductivity and Seebeck coefficient values of Sb_{2}Te_{3}-phase are larger, than those of Sb_{8}Te_{3}-phase in wide temperature range, e.g., ca. 2100 S cm^{−1} and 85 µV K^{−1} for Sb_{2}Te_{3} compared to1380 S cm^{−1} and 29 µV K^{−1} for Sb_{8}Te_{3} at 300 K, respectively. Moreover, it is estimated that precipitation of Sb_{2}Te_{3}-phase in AgSbTe_{2}-matrix is expected to improve the total values of both electrical conductivity and Seebeck coefficient. Concerning nucleation sequence of Sb_{2}Te_{3} and Sb_{8}Te_{3} phases in AgSbTe_{2}, their molar formation energies and interfacial free energies were calculated, suggesting that Sb_{8}Te_{3} nucleates first as metastable phase, prior to the formation of equilibrium Sb_{2}Te_{3} phase. Second, to address the influence of deviations from AgSbTe_{2} stoichiometry on electron transport properties, off-stoichiometric model alloys Ag_{3}SbTe_{4} and AgSb_{3}Te_{4} were simulated. It is found that increase in Sb/Ag ratio results in decrease in electrical conductivity simultaneously with increase in Seebeck coefficient. Considering both effects of Sb_{2}Te_{3} precipitation accompanied by simultaneous decrease in Sb/Ag ratio in δ-matrix taking place with aging time at temperatures below δ-solvus, it is expected that electrical conductivity of two-phase δ+Sb_{2}Te_{3} alloy should increase with aging time, disregarding effects, such as electron boundary scattering. Interestingly, these two effects have opposite consequences regarding Seebeck coefficient, so that, it is difficult to assess resulting Seebeck coefficient.

Experimentally, model ternary (AgSbTe_{2}) and quaternary (3 at.% La-AgSbTe_{2}) alloys were synthesized by vacuum melting followed by quenching and hot-pressing. The appropriate conditions enabling formation of AgSbTe_{2}-matrix that dissolves La-atoms with no La-rich precipitates were established. DSC tests enable observation of Ag_{2}Te+Sb_{2}Te_{3}→δ-AgSbTe_{2} phase transition at 360°C for La-free alloys only, indicating improvement of alloy’s thermal stability due to La-additions. Temperature-dependent thermal conductivity of both alloys indicate reduction in thermal conductivity as a result of La-alloying from 0.92 to 0.71 W m^{−1} K^{−1} at 573 K, which corresponds with the trend predicted from first-principles. Measurements of temperature-dependent electrical conductivity and Seebeck coefficients indicate that La-doping reduces electrical conductivity and increases Seebeck coefficients, as predicted from first-principles. Eventually, it is shown that La-doping has positive effects on TE figure-of-merit *ZT*, which is improved, e.g., from 0.35 up to 0.50 at 260°C.

We demonstrate how first-principles calculations serve as trustworthy tool for predicting TE performance of materials, screening the best candidates for application in TE devices. It is noteworthy that such DFT routines prove to be very efficient by prediction of TE properties in a way saving expensive and time-consuming experiments. The resulting materials that seem to possess improved performance are, eventually, processed in laboratory. We show how simple physical considerations can be implemented in DFT calculations and lead to improvement of power conversion efficiency. La-doping improves the alloys’ thermal stability and reduces their thermal conductivity, as well as enhances TE power factor in certain temperature range. As a result, the total TE figure-of-merit improves significantly. We, finally, emphasize the universal aspects of this approach that can be applied for other TE materials, as well.