P4/mmm form of AgSbTe2 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, *gp* (*ω*), calculated for this compound applying Debye approximation exhibits two major peaks at ca. 2.0 and 2.7 THz, and discloses interesting feature. Whereas, 2.7 THz peak comprises equal contributions from lattice vibrations of all sublattice sites, 2.0 THz one is primarily ascribed to vibrations of Ag-sublattice site atoms [57, 58]. This opens up the option of tuning v-DOS pattern by introducing point defects, a discipline for which the term *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 AgSbTe2 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 AgSbTe2 , 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 AgSbTe2 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 AgSbTe2 and LaAg7 Sb8 Te16 alloys close to Γ-point along c-crystallographic direction, indicating, that the slopes of the one longitudinal and two transverse acoustic modes of AgSbTe2 -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.

#### *<IP_ADDRESS>. Effects of La-doping on thermal conductivity*

Average sound velocity, *vs* , and Debye temperature, *θD* , evaluated from first-principles serve as input, that is required to evaluate lattice thermal conductivity, *κp* . To this end, one possibility is to employ Callaway model for lattice thermal conductivity [86, 87], which has become conventional, particularly in the field of TE materials [36–38, 88–93]. In present case, however, there is no need to employ Callaway model for several reasons. First, Callaway model is specified for low temperatures, where contributions of either Normal (N)- or Umklapp (U)-processes are at the same order of magnitude. For temperatures adequately higher than Debye temperature (e.g., *θD* ≈ 112 K for AgSbTe2 alloy) [30, 57], only U-processes dominate. Second, Callaway model considers *gp* (*ω*) and *Cp* (*T*) functions that are simplistically approximated based on Debye model [94]. In present case, however, the explicit *gp* (*ω*) and *Cp* (*T*) functions have already been calculated for both pure and La-alloyed materials. Alternatively, the following expression for lattice thermal conductivity is employed [94–96]:

$$
\kappa\_p = \frac{1}{3} C\_v v\_s^2 \,\tau \,\,\,\,\,\tag{8}
$$

where *τ* is phonon relaxation time. To first approximation, it has been assumed that La-doping influences mostly sound velocity and heat capacity and has negligible effect on *τ*. The ratio of *Cv vs* 2 -products obtained for LaAg7 Sb8 Te16 andAgSbTe2 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 *κp* should decrease by factor of ca. 2.7 due to La-doping. Alternatively, applying sound velocity values derived from Debye approximation, *κp* is expected to decrease by ca. 14% at room temperature due to La-doping [58].

A more thorough and accurate treatment of expression (8) considers the effects of La-alloying on *τ*, as well. To evaluate *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, *τU*, is represented by [96, 97]:

$$
\pi\_{\mathcal{U}}^{-1} \approx \frac{\hbar \,\gamma^2}{M \,\mathrm{v}\_\*^{-2} \theta\_D} \,\omega^2 \, T \, e^{\left(\frac{\theta\_d}{M}\right)} \,\tag{9}
$$

where *γ* is Grüneisen parameter that reflects the degree of lattice anharmonicity [69], and *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 *l*, represented by relaxation time *τB* , so that [96]:

$$
\pi\_{\rm B}^{-1} \approx \frac{\nu\_s}{I} \,. \tag{10}
$$

To consider dependence of v-DOS on phonon frequency, frequency-averaged expression for *τU* is introduced, so that *gp* (*ω*) serves as weighting function:

$$
\left< \tau\_{\mathcal{U}}^{-1} \right>\_{\omega} = \frac{\hbar \gamma^2}{M \nu\_\* \,^2 \theta\_0 \,\omega\_0} T \, e^{\left(\frac{\theta\_0}{\mathcal{N}}\right)} \Big|\_{0}^{\omega\_p} \omega^2 \, \mathcal{g}\_p(\omega) d\omega \,\, \,\tag{11}
$$

where *ωD* is Debye frequency. Equivalent relaxation time is then expressed as:

$$
\tau^{-1} = \left< \tau\_{\ll}^{-1} \right>\_{\omega} + \tau\_{\gg}^{-1} \,. \tag{12}
$$

The resulting values of lattice thermal conductivity for LaAg7 Sb8 Te16 and AgSbTe2 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 *l* ≈ 1 nm [5, 8, 16, 91]. Lattice thermal conductivity for LaAg7 Sb8 Te16 and AgSbTe2 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.

#### *<IP_ADDRESS>. Effects of La-doping on electrical properties*

Γ-point along c-crystallographic direction, indicating, that the slopes of the one longitudinal

[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 experi-

ity is to employ Callaway model for lattice thermal conductivity [86, 87], which has become conventional, particularly in the field of TE materials [36–38, 88–93]. In present case, however, there is no need to employ Callaway model for several reasons. First, Callaway model is specified for low temperatures, where contributions of either Normal (N)- or Umklapp (U)-processes are at the same order of magnitude. For temperatures adequately higher than

(*ω*) and *Cp*

tions have already been calculated for both pure and La-alloyed materials. Alternatively, the

3 *Cv vs*

Te16 andAgSbTe2

where *τ* is phonon relaxation time. To first approximation, it has been assumed that La-doping influences mostly sound velocity and heat capacity and has negligible effect on *τ*. The ratio of

of relative reduction in thermal conductivity due to La-alloying. Applying dispersion curves

by factor of ca. 2.7 due to La-doping. Alternatively, applying sound velocity values derived

A more thorough and accurate treatment of expression (8) considers the effects of La-alloying on *τ*, as well. To evaluate *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

mated based on Debye model [94]. In present case, however, the explicit *gp*

following expression for lattice thermal conductivity is employed [94–96]:

Sb8

close to Γ-point along c-crystallographic direction, it is predicted, that *κp*

, and Debye temperature, *θD* , evaluated from first-principles serve

. To this end, one possibil-

(*ω*) and *Cp*

(*T*) func-

should decrease

alloy) [30, 57], only U-processes dominate.

2 *τ* , (8)

alloys, therefore, reflects the lower limit

is expected to decrease by ca. 14% at room temperature due to

(*T*) functions that are simplistically approxi-

and two transverse acoustic modes of AgSbTe2

156 Thermoelectrics for Power Generation - A Look at Trends in the Technology

mental data better than the former way does.

Average sound velocity, *vs*

*Cv vs* 2

*<IP_ADDRESS>. Effects of La-doping on thermal conductivity*

Debye temperature (e.g., *θD* ≈ 112 K for AgSbTe2

*κp* = \_\_1

Second, Callaway model considers *gp*

from Debye approximation, *κp*

La-doping [58].

as input, that is required to evaluate lattice thermal conductivity, *κp*

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 LaAg7 Sb8 Te16 and AgSbTe2 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 AgSbTe2 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 AgSbTe2 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 AgSbTe2 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 AgSbTe2 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 AgSbTe2 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.

**Figure 2.** The lattice thermal conductivity values calculated from first-principles for AgSbTe2 (pure AST; filled red circles) and LaAg7 Sb8 Te16 (La-doped AST; empty red circles) alloys in temperature range 300–1000 K.

**Figure 3.** Electrical conductivity and Seebeck coefficient values calculated for AgSbTe2 (pure AST; filled black squares and blue circles, respectively) and LaAg7 Sb8 Te16 (La-doped AST; empty black squares and blue circles, respectively) alloys in temperature range 50–1000 K from first-principles applying Boltzmann transport theory.

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; *S* 2 *σ*) is, therefore, necessary in order to realize how La affects TE power conversion. **Figure 4** displays PF calculated for LaAg7 Sb8 Te16 and AgSbTe2 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 AgSbTe2 (P4/mmm), as reflected by increased TE figure-of-merit.

#### *4.1.2. Formation of Sb2 Te3 and Sb8 Te3 (R-3m) phases*

and Heremans reported on experimental measurements of electrical conductivity and Seebeck

For example, they report on electrical resistivity value of 5 × 10−5 Ohm m at 100 K for stoichio-

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

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

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.

increases Seebeck coefficients, e.g., from ca. 90 µV·K−1 for undoped AgSbTe2

**Figure 2.** The lattice thermal conductivity values calculated from first-principles for AgSbTe2

Te16 (La-doped AST; empty red circles) alloys in temperature range 300–1000 K.

down to 66 S cm−1 for 3 at.% La-doping at room temperature. Second, La-doping

alloys at temperatures up to 400 K [19, 98].

alloy are reported by Min et al. [27]. They

up to ca. 220 µV

(pure AST; filled red circles)

coefficients of stoichiometric and doped AgSbTe2

La-doping on electrical properties of AgSbTe2

AgSbTe2

and LaAg7

Sb8

metric AgSbTe2 alloy, which is equivalent to 200 S cm−1.

158 Thermoelectrics for Power Generation - A Look at Trends in the Technology

The single δ-phase is Sb-rich phase based on AgSbTe2 alloy. Since it has limited solubility to Sb with relatively moderate slope of Sb-solvus, it is likely to decompose to δ+Sb2 Te3 phase mixture [28, 29, 100–103], whereas Sb2 Te3 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 <IP_ADDRESS> introduces the issue of precipitation sequence based on bulk/interfacial energetic considerations, and Section <IP_ADDRESS> 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.

**Figure 4.** Thermoelectric power factor (PF) calculated for AgSbTe2 (pure AST; filled black squares) and LaAg7 Sb8 Te16 (La-doped AST; empty black squares) alloys in temperature range 50–1000 K from first-principles applying Boltzmann transport theory.

#### *<IP_ADDRESS>. The precipitation sequence: energetic aspects*

Nucleation of Sb2 Te3 and Sb8 Te3 phases from Sb-saturated δ-AgSbTe2 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 Sb2 Te3 and Sb8 Te3 (R-3m) phases, calculated according to Eq. (5), are −62 and −56.6 kJ mol−1, respectively. This implies that Sb2 Te3 is more energetically favorable, assuming, that Sb2 Te3 precipitates are adequately large, so that, interfaces do not play significant role. To address the role of interfaces, free energies of Sb2 Te3 /AgSbTe2 and Sb8 Te3 /AgSbTe2 interfaces are evaluated. To this end, two slab models of (Sb2 Te3 ) 2 /(AgSbTe2 ) 5 /(Sb2 Te3 ) 2 (40 atoms) and (AgSbTe2 ) 3 / (Sb8 Te3 ) 6 /(AgSbTe2 ) 3 (90 atoms) forms are constructed, respectively, consisting of two interfaces each, exhibiting (111 ) AgSbTe2 ∥ (0001 ) SbpTeq and 〈10¯〉 1AgSbTe 2 ∥ 〈¯ 2 110〉Sb pTeq orientation relationship, which was observed experimentally [100]. Both structures are displayed in **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 Sb2 Te3 / AgSbTe2 and Sb8 Te3 /AgSbTe2 interfaces are *γ* = 208 and 175 mJ m−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 (111) AgSbTe2 and (0001) SbpTeq crystallographic planes [100], which encourages formation of Sb2 Te3 or Sb8 Te3 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 *γ*3 [109].

**Figure 5.** Two slab models of (a) (Sb2 Te3 )2 /(AgSbTe2 )5 /(Sb2 Te3 )2 (40 atoms) and (b) (AgSbTe2 )3 /(Sb8 Te3 )6 /(AgSbTe2 )3 (90 atoms) including two Sb2 Te3 /AgSbTe2 and Sb8 Te3 /AgSbTe2 interfaces each, respectively. All interfaces, marked by arrows, are of ( 111 ) AgSbTe 2 ∥ ( 0001 ) Sb p Te q and 〈10 ¯ 1 〉 AgSbTe2 ∥ 〈 ¯2110 〉 Sb p Te q orientation relationship.

Interestingly, Sb2 Te3 phase exhibits lower value of formation energy and higher value of interfacial free energy compared to those of Sb8 Te3 phase. This implies that Sb8 Te3 is metastable phase that may form prior to the nucleation of Sb2 Te3 equilibrium phase [28, 29, 100–103]. Suggested nucleation sequence is, therefore, supersaturated-δ → supersaturated-δ + Sb8 Te3 → equilibrium-δ + Sb2 Te3 .

#### *<IP_ADDRESS>. Effects of Sb2 Te3 and Sb8 Te3 formation on electronic properties*

*<IP_ADDRESS>. The precipitation sequence: energetic aspects*

**Figure 4.** Thermoelectric power factor (PF) calculated for AgSbTe2

160 Thermoelectrics for Power Generation - A Look at Trends in the Technology

and Sb8

Te3

Te3

∥ (0001 )

phases from Sb-saturated δ-AgSbTe2

(R-3m) phases, calculated according to Eq. (5), are −62 and −56.6 kJ

Te3

∥ 〈¯ 2 110〉Sb pTeq

interfaces are *γ* = 208 and 175 mJ m−2, respectively. These values are

precipitates in the form of long lamellae along these

and Sb8

) 5 /(Sb2 Te3 ) 2

(90 atoms) forms are constructed, respectively, consisting of two interfaces

is more energetically favorable, assuming, that Sb2

/AgSbTe2

(pure AST; filled black squares) and LaAg7

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

(La-doped AST; empty black squares) alloys in temperature range 50–1000 K from first-principles applying Boltzmann

precipitates are adequately large, so that, interfaces do not play significant role. To address

/AgSbTe2

and 〈10¯〉 1AgSbTe 2

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 Sb2

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

and (0001)

planes [28, 29, 57, 58, 100–103]. These low values of interfacial free energy also initiate fast nucle-

SbpTeq

/(AgSbTe2

Te3

SbpTeq

Te3

which was observed experimentally [100]. Both structures are displayed in **Figure 5**.

AgSbTe2

ation, thanks to low activation energy for nucleation, which is proportional to *γ*3

Te3 ) 2 matrix has been observed,

interfaces are evalu-

orientation relationship,

(40 atoms) and (AgSbTe2

crystallographic planes [100], which

[109].

Te3

Sb8 Te16

> ) 3 /

Te3 /

Te3

and Sb8

mol−1, respectively. This implies that Sb2

the role of interfaces, free energies of Sb2

ated. To this end, two slab models of (Sb2

AgSbTe2

/AgSbTe2

Te3 or Sb8 Te3

) 3

Te3

/(AgSbTe2

and Sb8

encourages formation of Sb2

Te3

small atomic misfit between the (111)

each, exhibiting (111 )

Nucleation of Sb2

transport theory.

energies of Sb2

(Sb8 Te3 ) 6

AgSbTe2

In view of aforementioned prospect for the presence of either of Sb2 Te3 - or Sb8 Te3 -phases in AgSbTe2 -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 Sb2 Te3 phase are larger than those of Sb8 Te3 phase in wide temperature range, e.g., ca. 2100 S cm−1 and 85 µV K−1 for Sb2 Te3 compared to 1380 S cm−1 and 29 µV K−1 for Sb8 Te3 at 300 K, respectively. Moreover, comparison of these results with the data shown in **Figure 3** for AgSbTe2 -matrix implies that precipitation of Sb2 Te3 phase yields positive influence on TE performance: both electrical conductivity and Seebeck coefficient increase. It is strikingly indicated that the effects of Sb2 Te3 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-AgSbTe2 -matrix that is aged for a certain duration to form considerable amount of Sb2 Te3 phase. The effects of Sb8 Te3 phase on TE performance are, conversely, inferior to those of Sb2 Te3 phase. Sb8 Te3 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.

**Figure 6.** Electrical conductivity and Seebeck coefficient values calculated for Sb2 Te3 (filled black squares and blue circles, respectively) and Sb8 Te3 (empty black squares and blue circles, respectively) compounds in temperature range 50–1000 K from first-principles applying Boltzmann transport theory.

#### *4.1.3. Effects of off-stoichiometry on electronic properties of the AgSbTe2 phase*

An additional effect taking place during precipitation of any SbpTeq 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 Ag3 SbTe4 and AgSb3 Te4 , in addition to stoichiometric AgSbTe2 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 AgSbTe2 -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 Ag2−ySby Te1+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 SbpTeq 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.

**Figure 7.** Electrical conductivity and Seebeck coefficient values calculated for Ag3 SbTe4 (left-half-filled black squares and blue circles, respectively), (AgSbTe2 )2 (empty black squares and blue circles, respectively), and AgSb3 Te4 (right-halffilled black squares and blue circles, respectively) alloys in temperature range 50–1000 K from first-principles applying Boltzmann transport theory.

## **5. Experimental results**

*4.1.3. Effects of off-stoichiometry on electronic properties of the AgSbTe2*

**Figure 6.** Electrical conductivity and Seebeck coefficient values calculated for Sb2

162 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Te4

tivity due to additions of 2% AgTe to stoichiometric AgSbTe2

with *y*-values increasing from 1.26 up to 1.38 [101].

namely Ag3

respectively) and Sb8

SbTe4

Te3

ture range 50–1000 K.

and AgSb3

K from first-principles applying Boltzmann transport theory.

An additional effect taking place during precipitation of any SbpTeq

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,

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 tempera-

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 resis-

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,

alloys and found that electrical conductivity increases, while Seebeck coefficient decreases

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 treat-

Sb-supersaturated δ-matrix during heat treatments, Sb/Ag ratio in δ-matrix decreases. This

Zhang et al. reported on dependence of TE properties on composition for Ag2−ySby

ments (below Sb-solvus), or of TE generator during service. Since SbpTeq

 *phase*

, in addition to stoichiometric AgSbTe2 phase. These model

Te3

(empty black squares and blue circles, respectively) compounds in temperature range 50–1000

phase from

(filled black squares and blue circles,

Te1+y -based

phases nucleate from

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 AgSbTe2 -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 Sb2 Te3 precipitates [101–103]. These Sb2 Te3 precipitates, indeed, were not observed in Series A samples; however, La-rich precipitates having stoichiometry close to LaTe2 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 Sb2 Te3 phase, which is expected; however, LaTe2 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 Sb2 Te3 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**.

**Figure 8.** Differential scanning calorimetry (DSC) signals collected upon heating from La-free (continuous black curve) and La-doped (dashed red curve) samples.

Endothermic peak at around 630–650 K is observed for La-free material, which is associated with Ag2 Te + Sb2 Te3 → δ-AgSbTe2 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**

**Figure 8.** Differential scanning calorimetry (DSC) signals collected upon heating from La-free (continuous black curve)

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

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

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

precipitates are dissolved, so that

precipitates, in addition to La solute atoms, has positive

phase, which is expected; however, LaTe2

Te3

164 Thermoelectrics for Power Generation - A Look at Trends in the Technology

and La-doped (dashed red curve) samples.

amount of Sb2

Te3

first-principles, the presence of Sb2

for both La-free and La-doped samples, **Figure 8**.

effects on electronic transport.

**5.2. Thermal analysis**

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 <IP_ADDRESS>, 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 LaTe2 -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.

**Figure 9.** Electrical conductivity and Seebeck coefficient values measured for La-free (pure AST; filled black squares and blue circles, respectively) and La-doped (La-doped AST; empty black squares and blue circles, respectively) alloys of Series B in temperature range 300–673 K.

#### **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**.

Comparison between the results attained for alloys of Series A and Series B is very instruc-

"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

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

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

**Figure 9.** Electrical conductivity and Seebeck coefficient values measured for La-free (pure AST; filled black squares and blue circles, respectively) and La-doped (La-doped AST; empty black squares and blue circles, respectively) alloys of

tive. As noted, the samples of Series A exhibited formation of LaTe2

166 Thermoelectrics for Power Generation - A Look at Trends in the Technology

for electrical properties, **Figure 10**.

stability of La-free materials, as discussed above.

**5.4. Implications for thermoelectric power conversion**

Series B in temperature range 300–673 K.

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 AgSbTe2 alloy.

La-doping was tested by them for different compositions, where composition yielding the greatest PF values is AgSb0.99La0.01Te2 , 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.

**Figure 10.** Electrical conductivity and Seebeck coefficient values measured for La-free (pure AST; filled black squares and blue circles, respectively) and La-doped (La-doped AST; empty black squares and blue circles, respectively) alloys of Series A in temperature range 300–773 K.

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 AgSbTe2 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.

**Figure 11.** Thermoelectric power factor (PF) values evaluated for La-free (pure AST; filled black squares) and La-doped (La-doped AST; empty black squares) alloys of Series B in temperature range 300–673 K.

**Figure 12.** Thermoelectric figure of merit (ZT) values evaluated for La-free (pure AST; filled blue squares) and La-doped (La-doped AST; empty blue squares) alloys of (a) Series A and (b) Series B in temperature range 300–673 K.

#### **6. Summary and concluding remarks**

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

tion. 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 conduc-

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

**Figure 11.** Thermoelectric power factor (PF) values evaluated for La-free (pure AST; filled black squares) and La-doped

(La-doped AST; empty black squares) alloys of Series B in temperature range 300–673 K.

composi-

of La-doping are comparable to those of stoichiometric variations about AgSbTe2

168 Thermoelectrics for Power Generation - A Look at Trends in the Technology

tivity values [113].

materials hot-presses at 540°C.

This chapter introduced the following findings. Computationally, total energy calculations for different polymorphs of AgSbTe2 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 SbpTeq -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 Sb2 Te3 and Sb8 Te3 phases precipitate inside AgSbTe2 -matrix. It is found that both electrical conductivity and Seebeck coefficient values of Sb2 Te3 -phase are larger, than those of Sb8 Te3 -phase in wide temperature range, e.g., ca. 2100 S cm−1 and 85 µV K−1 for Sb2 Te3 compared to1380 S cm−1 and 29 µV K−1 for Sb8 Te3 at 300 K, respectively. Moreover, it is estimated that precipitation of Sb2 Te3 -phase in AgSbTe2 -matrix is expected to improve the total values of both electrical conductivity and Seebeck coefficient. Concerning nucleation sequence of Sb2 Te3 and Sb8 Te3 phases in AgSbTe2 , their molar formation energies and interfacial free energies were calculated, suggesting that Sb8 Te3 nucleates first as metastable phase, prior to the formation of equilibrium Sb2 Te3 phase. Second, to address the influence of deviations from AgSbTe2 stoichiometry on electron transport properties, off-stoichiometric model alloys Ag3 SbTe4 and AgSb3 Te4 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 Sb2 Te3 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+Sb2 Te3 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 (AgSbTe2 ) and quaternary (3 at.% La-AgSbTe2 ) alloys were synthesized by vacuum melting followed by quenching and hot-pressing. The appropriate conditions enabling formation of AgSbTe2 -matrix that dissolves La-atoms with no La-rich precipitates were established. DSC tests enable observation of Ag2 Te+Sb2 Te3 →δ-AgSbTe2 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 firstprinciples. 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.

## **Acknowledgements**

tivity 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

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

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,

Te3 and

and Sb8

Te3

SbTe4

alloy should increase with aging time, disre-

Te3

Te3

compared to1380 S cm−1

Te3

, their molar formation energies and interfacial free energies were cal-

were simulated. It is found that increase in Sb/Ag ratio results in decrease

phase. Second, to address the influence of deviations from AgSbTe2

precipitation accompanied by simultaneous decrease in Sb/Ag ratio

at 300 K, respectively. Moreover, it is estimated that precipitation

nucleates first as metastable phase, prior to the formation

of Sb-rich δ-phase to elevated temperatures results in precipitation of SbpTeq

Te3

conductivity and Seebeck coefficient. Concerning nucleation sequence of Sb2

stoichiometry on electron transport properties, off-stoichiometric model alloys Ag3

in electrical conductivity simultaneously with increase in Seebeck coefficient. Considering

in δ-matrix taking place with aging time at temperatures below δ-solvus, it is expected that

garding effects, such as electron boundary scattering. Interestingly, these two effects have opposite consequences regarding Seebeck coefficient, so that, it is difficult to assess resulting

Te3

temperature range, e.g., ca. 2100 S cm−1 and 85 µV K−1 for Sb2

Te3

each can be handled by DFT. To this end, we first consider the case in which Sb2

µV K−1 at room temperature.

phases precipitate inside AgSbTe2

170 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Te3

and Seebeck coefficient values of Sb2

Te3

Te3

electrical conductivity of two-phase δ+Sb2

and 29 µV K−1 for Sb8

phases in AgSbTe2

of equilibrium Sb2

both effects of Sb2

Seebeck coefficient.

and AgSb3

culated, suggesting that Sb8

Te4

Sb8 Te3

of Sb2 Te3 The author wishes to acknowledge generous support from the Israel Science Foundation (ISF), Grant no. 698/13, as well as, from the German-Israeli Foundation for Research and Development (GIF), Grant no. I-2333-1150.10/2012. Partial support from the Nancy and Stephen Grand Technion Energy Program (GTEP), the Russell Berrie Nanotechnology Institute (RBNI), Technion, and the Adelis Foundation for renewable energy research are greatly acknowledged, as well.

## **Author details**

Yaron Amouyal

Address all correspondence to<EMAIL_ADDRESS>
Department of Materials Science and Engineering, Technion—Israel Institute of Technology, Haifa, Israel

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#### **Nanostructured State-of-the-Art Thermoelectric Materials Prepared by Straight-Forward Arc-Melting Method Nanostructured State-of-the-Art Thermoelectric Materials Prepared by Straight-Forward Arc-Melting Method**

Federico Serrano-Sánchez, Mouna Gharsallah, Julián Bermúdez, Félix Carrascoso, Norbert M. Nemes, Oscar J. Dura, Marco A. López de la Torre, José L. Martínez, María T. Fernández-Díaz and José A. Alonso Julián Bermúdez, Félix Carrascoso, Norbert M. Nemes, Oscar J. Dura, Marco A. López de la Torre, José L. Martínez, María T. Fernández-Díaz and José A. Alonso

Federico Serrano-Sánchez, Mouna Gharsallah,

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65115

#### **Abstract**

[111] Chen Y, He B, Zhu TJ, Zhao XB.: Thermoelectric properties of non-stoichiometric

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[113] Su T, Jia X, Ma H, Yu F, Tian Y, Zuo G, Zheng Y, Jiang Y, Dong D, Deng L, Qin B, Zheng

2012;**45**:115302. doi:10.1088/0022-3727/45/11/115302

180 Thermoelectrics for Power Generation - A Look at Trends in the Technology

S.: Enhanced thermoelectric performance of AgSbTe2

based alloys with a small amount of GeTe addition. J. Phys. D: Appl. Phys.

thermo-

synthesized by high pressure and

AgSbTe2

Thermoelectric materials constitute an alternative to harvest sustainable energy from waste heat. Among the most commonly utilized thermoelectric materials, we can mention Bi2Te3 (hole and electron conductivity type), PbTe and recently reported SnSe intermetallic alloys. We review recent results showing that all of them can be readily prepared in nanostructured form by arc-melting synthesis, yielding mechanically robust pellets of highly oriented polycrystals. These materials have been characterized by neutron powder diffraction (NPD), scanning electron microscopy (SEM) and electronic and thermal transport measurements. Analysis of NPD patterns demonstrates near-perfect stoichiometry of above-mentioned alloys and fair amount of anharmonicity of chemical bonds. SEM analysis shows stacking of nanosized sheets, each of them presumably single-crystalline, with large surfaces parallel to layered slabs. This nanostructuration affects notably thermoelectric properties, involving many surface boundaries (interfaces), which are responsible for large phonon scattering factors, yielding low thermal conductivity. Additionally, we describe homemade apparatus developed for the simultaneous measurement of Seebeck coefficient and electric conductivity at elevated temperatures.

**Keywords:** thermoelectrics, nanostructuration, lattice thermal conductivity, thermopower, neutron powder diffraction

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **1. Introduction**

Thermoelectric materials possess the remarkable capability to transform temperature differen‐ ces between two ends of a material sample directly and reversibly into a electrical potential difference. Waste heat recovery, which implies around 70% of primary energy production, exploited as a new source of power generation, could mean significant progress worldwide [1, 2]. Thermoelectric generators are able to perform this task, but currently they are not yet cost‐ effective. Several advantages featured by thermoelectric power generation devices, such as the absence of moving parts, reliability, endurance, quiet operation and no pollutant emission, make these devices valuable from an energy and environmental point of view and useful in a wide range of applications.

The dimensionless figure of merit *ZT* = (*S*2 *σ*)*T*/*k*, where *S* stands for Seebeck coefficient, *σ* is the electrical conductivity and *κ* is the total thermal conductivity, evaluates thermoelectric performance of materials and serves as a reference value in thermoelectric materials research [3–8]. Maximization of ZT requires high Seebeck coefficient and low electrical resistivity and thermal conductivity values. This physical value is closely linked to the power generation efficiency of thermoelectric devices:

$$\varepsilon = \frac{T\_H - T\_c}{T\_H} \frac{\sqrt{1 + ZT\_M}^{-1} - 1}{\sqrt{1 + ZT\_M} + \frac{T\_c}{T\_H}},\tag{1}$$

where *TH*, *TC* and *TM* are the temperature of the hot and cold ends and the average temperature. From this, we can abstract, that larger mean ZT along with larger temperature differences return better conversion efficiencies. Current commercial devices based on BiTe alloys reach efficiencies of ~6%, while new materials based on recent advances to improve ZT are expected to reach ~12–17%. These approaches are mainly focused on lowering lattice thermal conduc‐ tivity by bulk nanostructuring and enhancing the power factor, S2 σ, by band engineering.

Experiments with nanostructured thermoelectric materials prove that highly efficient ther‐ moelectric energy conversion could be forthcoming [9, 10]. Bulk samples, containing nanoscale constituents or inhomogeneity, exhibit enhanced thermoelectric phenomena, which are connected with the latest advances in optimizing thermoelectric figure of merit. Materials featuring these characteristics have been found among compounds, where nano‐inclusions are inherently formed by using preparation methods to induce the nanostructured morphology. The main effect of nanostructuration is to affect the lattice thermal conductivity. Phonons are effectively scattered, when separation of defects or grain‐sizes is similar to phonons' mean‐ free path. Consequently, bearing in mind the difference in electronic scattering length, structural unit‐cells, comparable in size to heat carrying phonon wavelength, will improve the performance. On the other hand, quantum confinement effects could allow to treat *S*, *σ* and *k* quasi‐independently and achieve higher power factors, defined as *S*2 *σ* product [10–14].

Usually, thermoelectric nanocomposites are prepared initially and then assembled into bulk solids. Several methods for nanostructuring bulk materials have been developed; the most commonly used are spark plasma sintering (SPS), hot pressing, ball milling and wet chemical reactions [15]. All of them present different advantages and disadvantages, but they share drawbacks of long reaction and sample preparation times. For instance, it is expected, that the SPS method will be very beneficial for the reduction in lattice thermal conductivity due to retention of low-dimensional grains. On the other hand, it requires long annealing times and, as expected, results in more pronounced equiaxed morphology of powder particles with decrease in their size [16]. Chemical methods are convenient in terms of particle size, shape and crystallinity; nevertheless, removal of insulating organic capping ligands from nanocrystals is essential before consolidation into bulk pellets, and most of the chemically prepared materials present lower ZT values due to unsuitable charge carriers' concentrations and low intergranular connectivity achieved during compaction [17, 18].

Our present work deals with a straightforward and fast technique based on arc-melting synthesis. We have been able to prepare by this technique different families of thermoelectric materials including Bi2−xSbxTe3 and Bi2(Te1−xSex)3 alloys, SnSe and related alloys, PbTe and GeSe compounds [19–21]. This method yields highly nanostructured samples prepared in really short times, which require no further processing and are directly implementable into devices. Highly oriented polycrystalline pellets are obtained with extremely low thermal conductivity, probably linked to the nanostructured nature of polycrystalline domains. This chapter describes synthesis by arc-melting and structural and thermoelectric characterization of these materials. Structural characterization has been carried out by X-ray diffraction (XRD) and neutron powder diffraction (NPD), which complements the study of thermoelectric properties and is used as the basis for density functional theory (DFT) calculations. Therefore, we review transport results of various bismuth telluride and tin selenide-related alloys, as well as, some other alloys. Bi2Te3 forms the basis for the most widely used thermoelectrics near room temperature. Arc-melting is a good technique affording rapid production with various doping and alloying. But how good is the resulting material? Thermoelectric property measurements by various procedures can reveal it.

## **2. Experimental section**

**1. Introduction**

range of applications.

efficiency of thermoelectric devices:

e

182 Thermoelectrics for Power Generation - A Look at Trends in the Technology

tivity by bulk nanostructuring and enhancing the power factor, S2

quasi‐independently and achieve higher power factors, defined as *S*2

Thermoelectric materials possess the remarkable capability to transform temperature differen‐ ces between two ends of a material sample directly and reversibly into a electrical potential difference. Waste heat recovery, which implies around 70% of primary energy production, exploited as a new source of power generation, could mean significant progress worldwide [1, 2]. Thermoelectric generators are able to perform this task, but currently they are not yet cost‐ effective. Several advantages featured by thermoelectric power generation devices, such as the absence of moving parts, reliability, endurance, quiet operation and no pollutant emission, make these devices valuable from an energy and environmental point of view and useful in a wide

The dimensionless figure of merit *ZT* = (*S*2 *σ*)*T*/*k*, where *S* stands for Seebeck coefficient, *σ* is the electrical conductivity and *κ* is the total thermal conductivity, evaluates thermoelectric performance of materials and serves as a reference value in thermoelectric materials research [3–8]. Maximization of ZT requires high Seebeck coefficient and low electrical resistivity and thermal conductivity values. This physical value is closely linked to the power generation

1 1 ,

(1)

σ, by band engineering.

*σ* product [10–14].

+ +

where *TH*, *TC* and *TM* are the temperature of the hot and cold ends and the average temperature. From this, we can abstract, that larger mean ZT along with larger temperature differences return better conversion efficiencies. Current commercial devices based on BiTe alloys reach efficiencies of ~6%, while new materials based on recent advances to improve ZT are expected to reach ~12–17%. These approaches are mainly focused on lowering lattice thermal conduc‐

Experiments with nanostructured thermoelectric materials prove that highly efficient ther‐ moelectric energy conversion could be forthcoming [9, 10]. Bulk samples, containing nanoscale constituents or inhomogeneity, exhibit enhanced thermoelectric phenomena, which are connected with the latest advances in optimizing thermoelectric figure of merit. Materials featuring these characteristics have been found among compounds, where nano‐inclusions are inherently formed by using preparation methods to induce the nanostructured morphology. The main effect of nanostructuration is to affect the lattice thermal conductivity. Phonons are effectively scattered, when separation of defects or grain‐sizes is similar to phonons' mean‐ free path. Consequently, bearing in mind the difference in electronic scattering length, structural unit‐cells, comparable in size to heat carrying phonon wavelength, will improve the performance. On the other hand, quantum confinement effects could allow to treat *S*, *σ* and *k*

1

*T T H c ZTM TH*

*T T ZT H c M*

#### **2.1. Preparation by arc-melting**

Intermetallic alloys of different above-mentioned families were prepared in an Edmund Buhler Compact Arc Melter MAM-1 (**Figure 1a**). Stoichiometric amounts of grinded mixture of reacting elements were pelletized in a glove box. Pellets were molten under Ar atmosphere in water-cooled Cu crucible (**Figure 1b**), leading to intermetallic ingots (**Figure 1c**), which can be ground to powder for structural characterization or cut with a diamond saw in bar-shaped samples for transport measurements. Complete characterization of these novel materials has included structural study by XRD and NPD and detailed examination of thermoelectric parameters.

**Figure 1.** (a) Compact arc-melting furnace utilized for synthesis of nanostructured materials. (b) Water-cooled copper crucible, where sample can be quenched after melting process. (c) Typical aspect of as-grown ingots of intermetallic alloys.

#### **2.2. Structural characterization**

Initial characterization of products was carried out by laboratory XRD (Cu Kα, λ = 1.5406 Å). NPD diagrams were collected either at HRPT diffractometer of SINQ (The Swiss Spallation Neutron Source) spallation source at Paul Scherrer Institut or at D2B high-resolution diffractometer at Institut Laue-Langevin, Grenoble. Patterns were collected at room temperature with a wavelength of 1.494 Å (HRPT) or 1.594 Å (D2B). The high-flux mode was used (Δd/d ≈ 5 × 10−4); typical collection time was 2 h. For some selected samples (Sn0.8Ge0.2Se), temperaturedependent NPD experiment was also carried out at D2B diffractometer. About 2 g of the sample was contained in a vanadium can and placed in the isothermal zone of the furnace with a vanadium resistor operating under vacuum (PO2 ≈ 10−6 torr). Measurements were carried out upon heating at 25, 200, 420 and 580°C, and NPD data were collected in diffractometer D2B. Diffraction data were analyzed by Rietveld method with FULLPROF program [22]. Line shape of diffraction peaks was generated by pseudo-Voigt function. The following parameters were refined: background points, zero shift, half-width, pseudo-Voigt, scale factor and unit-cell parameters. Positional and occupancy factors and anisotropic displacement factors were also refined for NPD data. Coherent scattering lengths for Bi, Te, Sn, Ge and Se were 8.532, 5.800, 6.225, 8.185 and 7.970 fm, respectively. A preferred orientation correction was applied, considering platelets perpendicular to the [001] for Bi2Te3-related alloys and the [100] direction for SnSe-related alloys.

## **2.3. Microstructural characterization**

Surface texture of as-grown pellets is studied by field emission SEM (FE-SEM) in ZEISS 55 model. FE-SEM provides very focused energy electron beam, which improves greatly the spatial resolution and allows working at very low potentials, (from 0.02 to 5 kV); this helps to minimize effects of charge load on nonconductive samples and to avoid any damage to electron beam sensitive samples. It offers typical SEM image of surface topography of the sample with large depth of field. It is best suited for middle and low resolutions with high acceleration potential. It is mainly used to browse with low magnification looking for points of interest and to study samples with much topographical information. It also carries a secondary electron detector in lens: located inside the electron column, and it works with low-energy secondary electrons and offers higher resolution images. It is very sensitive to surface characteristics of the sample, so it is very suitable for surface characterization of any material.

#### **2.4. Transport measurements**

**Figure 1.** (a) Compact arc-melting furnace utilized for synthesis of nanostructured materials. (b) Water-cooled copper crucible, where sample can be quenched after melting process. (c) Typical aspect of as-grown ingots of intermetallic

Initial characterization of products was carried out by laboratory XRD (Cu Kα, λ = 1.5406 Å). NPD diagrams were collected either at HRPT diffractometer of SINQ (The Swiss Spallation Neutron Source) spallation source at Paul Scherrer Institut or at D2B high-resolution diffractometer at Institut Laue-Langevin, Grenoble. Patterns were collected at room temperature with a wavelength of 1.494 Å (HRPT) or 1.594 Å (D2B). The high-flux mode was used (Δd/d ≈ 5 × 10−4); typical collection time was 2 h. For some selected samples (Sn0.8Ge0.2Se), temperaturedependent NPD experiment was also carried out at D2B diffractometer. About 2 g of the sample was contained in a vanadium can and placed in the isothermal zone of the furnace with a vanadium resistor operating under vacuum (PO2 ≈ 10−6 torr). Measurements were carried out upon heating at 25, 200, 420 and 580°C, and NPD data were collected in diffractometer D2B. Diffraction data were analyzed by Rietveld method with FULLPROF program [22]. Line shape of diffraction peaks was generated by pseudo-Voigt function. The following parameters were refined: background points, zero shift, half-width, pseudo-Voigt, scale factor and unit-cell parameters. Positional and occupancy factors and anisotropic displacement factors were also refined for NPD data. Coherent scattering lengths for Bi, Te, Sn, Ge and Se were 8.532, 5.800, 6.225, 8.185 and 7.970 fm, respectively. A preferred orientation correction was applied, considering platelets perpendicular to the [001] for Bi2Te3-related alloys and the [100] direction

alloys.

**2.2. Structural characterization**

184 Thermoelectrics for Power Generation - A Look at Trends in the Technology

for SnSe-related alloys.

#### *2.4.1. Physical properties measurement system (PPMS)*

Three basic properties of thermoelectric materials (Seebeck coefficient, electrical resistivity and thermal conductivity) can be characterized simultaneously over a broad temperature range, between 2 and 400 K, by the thermal transport option (TTO) of the physical properties measurement system (PPMS) of Quantum Design Inc. This system allows for four electrical and thermal contacts with the sample. It uses two small thin-film temperature sensors (Cernox) mounted on small brass holders to measure both voltage and temperature drop across the sample. It uses another small brass piece with 2 kOhm resistive chip heater both to inject electrical current for resistivity measurement and to supply a known amount of heating power for Seebeck and thermal conductivity measurements. The fourth contact of the samples is the large brass baseplate of the sample holder, which thermally anchors it to the cryostat.

Measurements are carried out as follows: we use bar-shaped samples with 10 × 3 × 2 mm3 dimensions, prepared either by directly cutting from as-prepared ingots or by directly coldpressing it after arc-melting in a properly shaped die. Four copper leads are wrapped around the bar and then fixed with silver epoxy. We use the following thermal protocol: cool the sample from 300 K to low temperature (either 2 K or 10 K), then warm it to 395 K and then cool it again to base temperature. We use slow sweep rate of 0.3 K/min (over 2 days for each sample) and gather data continuously. Electrical resistivity is measured by applying sinusoidal current with typical frequency of 17 Hz and amplitude between 10 μA and 10 mA. Seebeck coefficient is then measured by establishing a temperature gradient of typically 3% of the sample sink temperature. Here, the only important source of experimental error may arise from the fact, that Cernox temperature sensors are not touching the sample directly, but are thermally connected to it via a few millimeter-length of copper wires. Thermal conductivity is measured by dynamically modeling the temperature gradient on the sample between two Cernox sensors as known heater power (between 10 μW and 50 mW, adjusted to achieve 3% gradient) is supplied, and then removed, to one end of the sample. Above ~150 K, errors related to radiative heat losses become important. These are ameliorated to a certain extent by careful application of the correction software of TTO. Another, less important, source of heat losses is conduction through copper wires of voltage/temperature gradient leads. Finally, convective heat losses are minimized by the high vacuum option of PPMS, establishing pressure below 10−5 torr.

For Hall coefficient measurements, we use thin pellets (~1 mm thick, 10 mm diameter) in van der Pauw geometry, with four contacts placed along the perimeter, and pass a DC current with alternating sign to eliminate thermoelectric voltages along diagonal. Measured voltage is still dominated by ohmic resistance, which is then removed by comparing values obtained in large positive and negative magnetic fields.

#### *2.4.2. High-temperature Seebeck coefficient measured in micro-miniature refrigerator (MMR) device*

Seebeck coefficient is the ratio of voltage difference produced from applied temperature gradient. In principle, this concept is easy to understand, and one might think that this thermoelectric property is easy to measure. However, it can be difficult to evaluate [23, 24]. Basically, a temperature gradient is established between hot and cold ends of the sample and then the voltage drop, appearing between them, is measured. If the temperature gradient is kept constant during measurement, the method is called steady-state method.

Here we describe a method used by the commercial system of MMR Technologies Inc. (MMR device). This system allows measuring Seebeck coefficient of semiconductors and metals between 70 and 730 K. This method employs two pairs of thermocouples. One pair is formed by junctions of copper and reference material (constantan) with a well-known value of Seebeck coefficient. The other pair is formed of junctions of copper and sample, in which Seebeck coefficient is to be determined (**Figure 2a**). This method requires that both materials, reference and sample, have similar thermal conductance in order to ensure similar thermal transport through both of them. Thus, constantan wires with different diameters are used as a reference to allow evaluating materials with very different thermal conductivities.

**Figure 2.** (a) Seebeck thermal stage supplied by MMR technologies. Black line indicates place of unknown sample and red line indicates that of reference sample. (b) Chamber with refrigerator, which provides temperature range between 70 and 730 K.

Another interesting characteristic of the method provided by MMR system is the double reference measurement technique, which gives more accurate and reproducible results. By using this feature, the equipment subtracts any instrumental offset voltage due to thermovoltage effects from wires or connections. Different stages, made up of alumina or polyamide, allow covering a broad temperature range of measurement from 70 to 730 K (**Figure 2a** and **b**).

#### *2.4.3. High-temperature transport device: high-temperature Seebeck measurements in homemade apparatus*

of the correction software of TTO. Another, less important, source of heat losses is conduction through copper wires of voltage/temperature gradient leads. Finally, convective heat losses are minimized by the high vacuum option of PPMS, establishing pressure below 10−5 torr. For Hall coefficient measurements, we use thin pellets (~1 mm thick, 10 mm diameter) in van der Pauw geometry, with four contacts placed along the perimeter, and pass a DC current with alternating sign to eliminate thermoelectric voltages along diagonal. Measured voltage is still dominated by ohmic resistance, which is then removed by comparing values obtained in large

*2.4.2. High-temperature Seebeck coefficient measured in micro-miniature refrigerator (MMR) device* Seebeck coefficient is the ratio of voltage difference produced from applied temperature gradient. In principle, this concept is easy to understand, and one might think that this thermoelectric property is easy to measure. However, it can be difficult to evaluate [23, 24]. Basically, a temperature gradient is established between hot and cold ends of the sample and then the voltage drop, appearing between them, is measured. If the temperature gradient is

Here we describe a method used by the commercial system of MMR Technologies Inc. (MMR device). This system allows measuring Seebeck coefficient of semiconductors and metals between 70 and 730 K. This method employs two pairs of thermocouples. One pair is formed by junctions of copper and reference material (constantan) with a well-known value of Seebeck coefficient. The other pair is formed of junctions of copper and sample, in which Seebeck coefficient is to be determined (**Figure 2a**). This method requires that both materials, reference and sample, have similar thermal conductance in order to ensure similar thermal transport through both of them. Thus, constantan wires with different diameters are used as a reference

**Figure 2.** (a) Seebeck thermal stage supplied by MMR technologies. Black line indicates place of unknown sample and red line indicates that of reference sample. (b) Chamber with refrigerator, which provides temperature range between

Another interesting characteristic of the method provided by MMR system is the double reference measurement technique, which gives more accurate and reproducible results. By using this feature, the equipment subtracts any instrumental offset voltage due to thermo-

kept constant during measurement, the method is called steady-state method.

to allow evaluating materials with very different thermal conductivities.

positive and negative magnetic fields.

186 Thermoelectrics for Power Generation - A Look at Trends in the Technology

70 and 730 K.

As commented above, measurement of Seebeck coefficient seems like one of the less challenging tasks: create temperature gradient, measure voltage … What could be simpler? Yet, at elevated temperatures, this poses a challenge, mainly due to large, hard to control, temperature gradients. The largest systematic errors arise from the fact, that it is virtually impossible to detect temperatures at exactly the same spot where voltage difference is measured. Additional difficulty is strong chemical and metallurgic reactivity of typical thermoelectric materials at temperatures above a few hundred degree celsius, limiting the choice of building blocks for any instrument. Even such precious workhorse material as Pt is out of the question.

There are two main approaches to measure Seebeck effect in thermoelectric materials: integral and differential methods [24]. In the integral method, one end is maintained at a fixed temperature *T*1, while another end is heated to induce, sometimes large, temperature gradient Δ*T* = *T*2 − *T*1. According to the definition of Seebeck coefficient, we can write:

$$V\_{TC} = V\_T(T\_1, T\_2) - V\_C(T\_1, T\_2) = \int\_{T\_1}^{T\_2} (S\_T(T) - S\_C(T)) dT,\tag{2}$$

where *VT* and *ST* are, respectively, Seebeck voltage and Seebeck coefficient of the thermoelectrics, while *VC* and *SC* are those of connecting lead. Normally, these conductors are metals with low Seebeck coefficient, such as Cu or Pt with known thermoelectric properties. Thus, it is possible to infer Seebeck voltage of interest, *VT*, from voltage measured, *VTC*, using the following expression:

$$V\_{TC} + V\_C(T\_1, T\_2) = V\_T(T\_1, T\_2) = \int\_{T\_1}^{T\_2} S\_T(T)dT.\tag{3}$$

In the differential technique, small thermal gradient is used to estimate Seebeck coefficient at mean sample temperature. We can distinguish two approaches to this technique. In the DC method, constant thermal gradient is maintained, while mean temperature is varied, whereas in the AC method mean temperature is stabilized followed by the change in temperature gradient, usually in a sinusoidal form [25].

In the following, we describe a high-temperature homemade instrument to measure Seebeck coefficient with both integral and differential techniques. We based our design on the system described in Ref. [25], with some modifications, which are indicated below. Typical sample dimensions are pellets of around 10 mm diameter and 2 mm thickness, also ideal for laser flash measurements of thermal diffusivity.

**Figure 3** shows the design scheme of measurement. This instrument is composed of two thermocouples, two blocks of niobium (Nb), two cartridge heaters with built-in thermocouples and a radiation furnace with yet another thermocouple. The whole assembly is placed in a vacuum chamber equipped with turbo-pump station and operates in a vacuum of around 7.5 × 10−7 torr and between 300 and 950 K. Baseplate can be water-cooled. We now describe important components of the homemade system.

**Figure 3.** The top panel shows the main pieces of homemade instrument: vacuum chamber with turbo-pump station, furnace assembly and electronics for temperature control and voltage measurements. Screenshot of data acquisition software shows typical data resulting in differential AC measurement.

#### *<IP_ADDRESS>. Thermocouples*

Thermocouples are used to measure both temperature of the sample at either side and Seebeck voltage. These consist of two different wires, one of chromel and another of niobium, of 0.1 mm diameter, inserted in a four-bore ceramic tube (1.2 mm diameter with 0.2 mm bores) in such a way that where these wires cross Nb ones make contacts with the sample [25]. Thus, the difference between temperature and voltage measurements is minimized to the diameter of Nb wire. These four-bore ceramic tubes are mounted on spring-loaded mechanisms to press them dynamically against the sample. Springs are located outside the hot-zone to maintain their elasticity. We found, that this system works well enough without the sophisticated strain gauge mechanism of Ref. [25]. Niobium is a good choice for its low Seebeck coefficient—we use two Nb leads to detect Seebeck voltage of the sample.

#### *<IP_ADDRESS>. Niobium blocks*

Niobium blocks form the basis of apparatus. These cylinders, whose diameter and length are 20 mm, have two functions: they house cartridge heaters (and thus heat the sample differen-

tially) and thermocouples, and press the sample during the measurement. After finding copper to be a poor choice due to its reactivity, we chose niobium for its chemical inertness against typical thermoelectric materials and for its good thermal conductivity. An advantage of using a typical metallic block instead of a ceramic one, as in Ref. [25], is that they can act as a pair of electrical contacts, to inject electrical current in 4-probe resistivity measurement, while ohmic voltage can be measured on contacts used for Seebeck voltage. Metallic blocks are also easier to machine. These blocks are electrically insulated from the rest of the equipment by ceramic (MACOR®) collars and are pressed against the sample by two sets of three springs on support bars.

#### *<IP_ADDRESS>. Cartridge heaters*

**Figure 3** shows the design scheme of measurement. This instrument is composed of two thermocouples, two blocks of niobium (Nb), two cartridge heaters with built-in thermocouples and a radiation furnace with yet another thermocouple. The whole assembly is placed in a vacuum chamber equipped with turbo-pump station and operates in a vacuum of around 7.5 × 10−7 torr and between 300 and 950 K. Baseplate can be water-cooled. We now describe

**Figure 3.** The top panel shows the main pieces of homemade instrument: vacuum chamber with turbo-pump station, furnace assembly and electronics for temperature control and voltage measurements. Screenshot of data acquisition

Thermocouples are used to measure both temperature of the sample at either side and Seebeck voltage. These consist of two different wires, one of chromel and another of niobium, of 0.1 mm diameter, inserted in a four-bore ceramic tube (1.2 mm diameter with 0.2 mm bores) in such a way that where these wires cross Nb ones make contacts with the sample [25]. Thus, the difference between temperature and voltage measurements is minimized to the diameter of Nb wire. These four-bore ceramic tubes are mounted on spring-loaded mechanisms to press them dynamically against the sample. Springs are located outside the hot-zone to maintain their elasticity. We found, that this system works well enough without the sophisticated strain gauge mechanism of Ref. [25]. Niobium is a good choice for its low Seebeck coefficient—we

Niobium blocks form the basis of apparatus. These cylinders, whose diameter and length are 20 mm, have two functions: they house cartridge heaters (and thus heat the sample differen-

important components of the homemade system.

188 Thermoelectrics for Power Generation - A Look at Trends in the Technology

software shows typical data resulting in differential AC measurement.

use two Nb leads to detect Seebeck voltage of the sample.

*<IP_ADDRESS>. Thermocouples*

*<IP_ADDRESS>. Niobium blocks*

Cartridge heaters create controlled temperature gradient across the sample. They are inserted into niobium blocks off-center and are in good thermal contact with them. Cartridge heaters (from Watlow Ltd, 6.3 mm diameter, 25 mm length, 150 W) have individually incorporated Jtype thermocouples. This enables us to smoothly control temperature gradient across the sample, even though cartridge heaters can deviate from the sample temperature by tens of degrees. We use programmable DC power supplies to control the temperature of cartridge heaters with high accuracy. Power supplies act as highly linear power amplifiers at the output of simple PID temperature controllers. Use of 0–5 V DC control logic helps us avoid introducing electronic noise into measurement, which would be rather troublesome in the typical solid state relay (SSR) scheme.

#### *<IP_ADDRESS>. Furnace*

The sample and Nb-blocks are surrounded by a small tubular furnace to establish the average temperature and crucially to reduce radiative heat losses. Homemade furnace is composed of an aluminum oxide tube of 34 mm internal diameter with rolled-up Kanthal wire. This heating element is surrounded by mineral fiber and stainless steel sheet with low emissivity. After placing the sample between the Nb-blocks, the whole furnace can slide up on three steel posts to position.

#### *<IP_ADDRESS>. Breakout connector*

The system has several thermocouples, and one of the crucial advantages of such an instrument, according to Ref. [25], is the possibility to easily swap and test different thermocouple wires for various samples. However, this poses technical challenge: these thermocouple wires must be run through the vacuum chamber. Although there exist commercial feedthroughs for thermocouples, these are costly and would eliminate flexibility to change the type of wire used. Therefore, we installed a break-out connector (DB-25) within the vacuum chamber and use copper wires from this post toward feedthrough (also DB-25, gold-plated pins) and outside toward electronics. As the break-out connector gets warm during operation (up to 80°C), we monitor its temperature with a resistive sensor (Pt100) and use this as "cold junction" in thermocouple measurements, in order to minimize spurious thermoelectric voltages.

#### *<IP_ADDRESS>. Electronics*

Seebeck voltage, thermocouples monitoring the sample temperature and any resistance measurements (such as Pt100 of breakout connector or the sample itself) are measured by Keithley-2700 scanning multimeter. Other thermocouples (cartridge heaters, furnace) are connected to three West-P6100 PID controllers. Their 0–5 V DC programming output acts on DeltaES150 DC power supplies. Smooth voltage output reduces electronic noise on sensitive Seebeck voltage and resistivity measurements from heating system.

#### *<IP_ADDRESS>. Data acquisition and temperature set-point control*

Data acquisition and temperature set-point control are handled by a LabVIEW program, which communicates via GPIB with multimeter and via RS485 serial protocol with PID controllers.

Salient features of this instrument are its operating range from slightly above room temperature up to 900 K and its flexibility to perform three different types of measurement schemes: quasi-integral, differential DC and differential AC. Each scheme has its own compromise between accuracy and overall required time for Seebeck coefficient measurements.

**•** *Quasi-integral* method is based on integral method and provides quick measurements, with low accuracy. We apply a step input to one heater, while the other is turned off. Although temperature of neither side of the sample is fixed, it is possible to extract the whole temperature-dependent Seebeck coefficient curve as polynomial function:

$$S(T) = S\_0 + S\_1 \cdot T + S\_2 \cdot T^2 + S\_3 \cdot T^3 + S\_4 \cdot T^4 + S\_5 \cdot T^5 + \dotsb \tag{4}$$

According to Eq. (3) Seebeck voltage is:

$$V\_T \left( T\_1, T\_2 \right) = S\_0 \cdot \left( T\_2 - T\_1 \right) + \frac{S\_1}{2} \cdot \left( T\_2^2 - T\_1^2 \right) + \frac{S\_2}{3} \cdot \left( T\_2^3 - T\_1^3 \right) + \frac{S\_3}{4} \cdot \left( T\_2^4 - T\_1^4 \right) + \dots \tag{5}$$

Therefore, from broad set of data, *Sn* polynomial coefficients can be numerically approximated. This technique does not generate particularly accurate results, but in as little as one hour, full *S*(*T*) curve may be obtained up to 900 K, which is very helpful for screening purposes, when faced with large number of samples. The peculiarity of this technique is that resulting *S*(*T*) curves are completely (and thus perplexingly) smooth, since they result from Eq. (4). Also, near the lowest and highest measured temperatures, *S*(*T*) curves behave anomalously due to the way the numerical fit works. In a sense, integral method is closest to real-life operation with large temperature differences present.

**•** *Differential DC* method is the closest to definition of Seebeck coefficient:

$$S(T) = \frac{dV}{dT} \,. \tag{6}$$

This method provides more accurate results than quasi-integral one, but is somewhat slower. We maintain more-or-less constant temperature gradient of a few degrees using cartridge heaters, while raising overall temperature smoothly with the furnace. This method also works without the surrounding furnace, but temperature differences between cartridge heaters and respective sample sides are quite large (several tens of degrees). It is also difficult to gauge any systematic errors arising from voltage and temperature offsets. Differential DC method is the closest operation mode to MMR system.

*<IP_ADDRESS>. Electronics*

Seebeck voltage, thermocouples monitoring the sample temperature and any resistance measurements (such as Pt100 of breakout connector or the sample itself) are measured by Keithley-2700 scanning multimeter. Other thermocouples (cartridge heaters, furnace) are connected to three West-P6100 PID controllers. Their 0–5 V DC programming output acts on DeltaES150 DC power supplies. Smooth voltage output reduces electronic noise on sensitive

Data acquisition and temperature set-point control are handled by a LabVIEW program, which communicates via GPIB with multimeter and via RS485 serial protocol with PID controllers.

Salient features of this instrument are its operating range from slightly above room temperature up to 900 K and its flexibility to perform three different types of measurement schemes: quasi-integral, differential DC and differential AC. Each scheme has its own compromise

**•** *Quasi-integral* method is based on integral method and provides quick measurements, with low accuracy. We apply a step input to one heater, while the other is turned off. Although temperature of neither side of the sample is fixed, it is possible to extract the whole tem-

> 2345 01 2 3 4 5 *ST S S T S T S T S T S T* ( ) = + ×+ × + × + × + × +L (4)

*S S S V TT S T T T T T T T T* (5)

Therefore, from broad set of data, *Sn* polynomial coefficients can be numerically approximated. This technique does not generate particularly accurate results, but in as little as one hour, full *S*(*T*) curve may be obtained up to 900 K, which is very helpful for screening purposes, when faced with large number of samples. The peculiarity of this technique is that resulting *S*(*T*) curves are completely (and thus perplexingly) smooth, since they result from Eq. (4). Also, near the lowest and highest measured temperatures, *S*(*T*) curves behave anomalously due to the way the numerical fit works. In a sense, integral method is closest to real-life operation

between accuracy and overall required time for Seebeck coefficient measurements.

( ) ( ) ( ) ( ) 1 2 22 33 44 3 12 0 2 1 2 1 2 1 2 1 , () 234 *T* = × - + × - + × - + × - +¼

perature-dependent Seebeck coefficient curve as polynomial function:

**•** *Differential DC* method is the closest to definition of Seebeck coefficient:

( ) = . *dV S T*

*dT* (6)

Seebeck voltage and resistivity measurements from heating system.

*<IP_ADDRESS>. Data acquisition and temperature set-point control*

190 Thermoelectrics for Power Generation - A Look at Trends in the Technology

According to Eq. (3) Seebeck voltage is:

with large temperature differences present.

**•** *Differential AC* is the slowest method, but also the most accurate, as described in Ref. [25]. It takes several hours (easily a full day) to obtain a few Seebeck coefficient data points, for example, every 50 K. This is because furnace and cartridge heater temperatures must be stabilized first and this takes a long time at high temperatures, and then cartridge heater temperatures are oscillated (with a period of tens of minutes) to obtain complete data sets (e.g., in **Figure 4**). The great advantage of this method is that any voltage offsets arising from stray thermoelectric emfs or temperature offsets, for example, due to miscalibration of thermocouples, are eliminated by linear regression to collected data. Differential AC method is reminiscent of the operation mode of PPMS.

**Figure 4.** Typical temperature-dependent Seebeck coefficient data gathered with both differential DC method (full symbols) and integral method (lines) for low Seebeck metals, aluminum and copper, commercial p-type Bi2Te3 ingot and test pellet of SnSe produced by arc-melting.

We end this section by presenting a few temperature-dependent Seebeck coefficient curves for test materials. We used pieces of commercial aluminum and copper to check effects of voltage and temperature offsets. These metals have minimal Seebeck coefficient values at room temperature, around 3.5 μV K−1 for Al and 6.5 μV K−1 for Cu. These would vary by only a few μV K−1 in the studied temperature range. We found, that differential DC method gives Seebeck coefficient values differing by less than 10 μV K−1, as absolute error. Relative error is less flattering, as for example, in the case of Al we measure negative values. Interestingly, integral method yields 5–6 μV K−1 for Cu, very close to tabulated value. Nevertheless, instrument is designed to study thermoelectric materials with large Seebeck coefficient values. We can take

±10 μV K−1 as a rough estimate for systematic errors of the instrument. Results for commercial ingot of p-type Bi2Te3 above 100 μV K−1, with characteristic maximum above 400 K, are similar to expected behavior. Finally, preliminary data from a pellet of SnSe produced by arc-melting and measured by quasi-integral method are also shown. These values are quite a bit lower than expected. The reason is still to be explored, but it is probably related to the fact, that data were recorded upon second heating run, and the sample may have suffered some chemical alteration. Whatever the case may be, it demonstrates the capabilities of the instrument.

#### *2.4.4. Thermal conductivity: laser flash thermal diffusivity method*

So-called laser flash thermal diffusivity technique is a useful method to determine thermal properties of bulk samples and also thin films. This method allows measuring thermal diffusivity (*α*) of a sample in very broad temperature range (80–2500 K) and diffusivity (0.001– 10 cm2 /s) range. For a given material, *α* is directly related to the speed at which the material can change its temperature. Thus, thermal conductivity of the sample is obtained from measurements of diffusivity (*α*), specific heat (*Cp*) and density (ρ) of the sample by means of this relation:

$$
\kappa = \alpha \cdot C\_P \cdot \rho. \tag{7}
$$

Commonly used systems, which measure *α*, usually allow also measuring *Cp*. In spite of this feature, it is recommended to measure specific heat by means of a separate technique, like differential scanning calorimetry (DSC), in order to obtain more accurate estimation of the final value. However, at high temperature, where the specific heat reaches a constant value, diffusivity provides the essential parameter to estimate thermal conductivity.

As shown in **Figure 5a**, use of this method to measure α implies illumination of one face of the sample by a laser pulse of length below 1 ms. An infrared (IR) detector placed behind rear face detects the signal, which is proportional to temperature rise. Thermal diffusivity value is obtained from IR signal rise against time. Example of IR signal profile vs. time is shown in **Figure 5b** corresponding to a graphite reference sample. Original Parker method [27] considers adiabatic conditions, and therefore, diffusivity value is obtained from thickness of the sample and time (t1/2), where IR signal profile reaches half of maximum rise:

$$\alpha = 0.1388 \left( \bigvee\_{l\_{1/2}}^{L^2} \right), \tag{8}$$

where *L* is the thickness of the sample. This method is valid only for adiabatic conditions; however, different methods have been designed in order to consider effects of finite laser pulse time and radiative losses in nonadiabatic conditions [27–30]. Correction fits and evaluation models based on this method are also provided by manufacturers of state-of-the-art thermal diffusivity measuring systems.

Nanostructured State-of-the-Art Thermoelectric Materials Prepared by Straight-Forward Arc-Melting Method http://dx.doi.org/10.5772/65115 193

**Figure 5.** (a) Schematic illustration of laser flash method (adapted from *Thermoelectric Handbook: Macro to Nano*) [26], (b) signal profile and model fitted of graphite reference sample obtained in Laser flash system Linseis LFA 1000.

In addition, laser flash method imposes certain requirements on samples to be measured. It is mandatory to prepare plate samples with flat parallel planes to ensure correct acquisition of temperature profile at the rear face of the sample. Another important point about the sample preparation is to ensure the highest emissivity/absorption in the rear/front surface of the sample. For this purpose, thin coating of graphite well adhered over the sample's surface is commonly used (see **Figure 6a**).

**Figure 6.** Different parts and sample holder (a, b), carrousel for six samples (c), and general view of laser flash equipment: Linseis LFA 1000 (d).

#### *2.4.5. Thermal conductivity: 3ω method*

±10 μV K−1 as a rough estimate for systematic errors of the instrument. Results for commercial ingot of p-type Bi2Te3 above 100 μV K−1, with characteristic maximum above 400 K, are similar to expected behavior. Finally, preliminary data from a pellet of SnSe produced by arc-melting and measured by quasi-integral method are also shown. These values are quite a bit lower than expected. The reason is still to be explored, but it is probably related to the fact, that data were recorded upon second heating run, and the sample may have suffered some chemical alteration. Whatever the case may be, it demonstrates the capabilities of the instrument.

So-called laser flash thermal diffusivity technique is a useful method to determine thermal properties of bulk samples and also thin films. This method allows measuring thermal diffusivity (*α*) of a sample in very broad temperature range (80–2500 K) and diffusivity (0.001–

> r

Commonly used systems, which measure *α*, usually allow also measuring *Cp*. In spite of this feature, it is recommended to measure specific heat by means of a separate technique, like differential scanning calorimetry (DSC), in order to obtain more accurate estimation of the final value. However, at high temperature, where the specific heat reaches a constant value,

As shown in **Figure 5a**, use of this method to measure α implies illumination of one face of the sample by a laser pulse of length below 1 ms. An infrared (IR) detector placed behind rear face detects the signal, which is proportional to temperature rise. Thermal diffusivity value is obtained from IR signal rise against time. Example of IR signal profile vs. time is shown in **Figure 5b** corresponding to a graphite reference sample. Original Parker method [27] considers adiabatic conditions, and therefore, diffusivity value is obtained from thickness of the sample

> 2 1/2

è ø *L*

where *L* is the thickness of the sample. This method is valid only for adiabatic conditions; however, different methods have been designed in order to consider effects of finite laser pulse time and radiative losses in nonadiabatic conditions [27–30]. Correction fits and evaluation models based on this method are also provided by manufacturers of state-of-the-art thermal

0.1388 , æ ö = ç ÷

= · ·. *CP* (7)

*t* (8)

ka

diffusivity provides the essential parameter to estimate thermal conductivity.

and time (t1/2), where IR signal profile reaches half of maximum rise:

a

diffusivity measuring systems.

/s) range. For a given material, *α* is directly related to the speed at which the material can change its temperature. Thus, thermal conductivity of the sample is obtained from measurements of diffusivity (*α*), specific heat (*Cp*) and density (ρ) of the sample by means of

*2.4.4. Thermal conductivity: laser flash thermal diffusivity method*

192 Thermoelectrics for Power Generation - A Look at Trends in the Technology

10 cm2

this relation:

Alternative procedure to determine thermal conductivity is the so-called 3ω (3 omega) method [31]. We sandwiched a thin gold wire (diameter *d* = 25 μm) between two identical, disk-shaped commercial Bi2Te3 samples. We used BN spray to electrically insulate the wire from the sample. This is a slight variation in the original method as heating/sensing wire is completely surrounded by the material. This method relies on applying AC current along the gold thread (heater) at frequency *ω*, which generates thermal oscillations finally resulting in a voltage at the third harmonic (3*ω*), which is closely related to thermal conductivity (κ):

$$V\_{3\alpha\phi} = -\frac{V\_{1\alpha\phi}^3 \cdot \mathcal{B}}{8\pi \cdot l \cdot \kappa \cdot R\_0} \cdot \left[ \ln(2\alpha) + \ln\left(\frac{d^2}{4\alpha}\right) - 2 \cdot \ln(2) \right] - i \cdot \frac{V\_{1\alpha\phi}^2 \cdot \mathcal{B}}{16 \cdot l \cdot \kappa \cdot R\_0} \,,\tag{9}$$

where *V*1*ω* is the amplitude of the ohmic voltage, *l* is the length of the heater, *β* is the temperature coefficient of gold, *R*0 is the nominal wire electrical resistance, *α* is the thermal diffusivity of the sample. This formula is valid as long as thermal penetration depth, 2 is greater than five times the wire radius. Performing linear fit to above expression, of *V*3*ω* vs ln(2*ω*), thermal conductivity can be determined.

Here we present the results of this frequency-dependent third harmonic measurement for commercial p-type Bi2Te3 at 300 K [31]. Performing linear fit to the expression indicated in this section, of *V*3*ω* vs ln(2*ω*), as shown in **Figure 7**, thermal conductivity obtained is *κ* = 1.32 W m−1 K−1 at 300 K, matching reasonably well the results with TTO method of PPMS.

**Figure 7.** 3ω method for extracting thermal conductivity of commercial p-type Bi2Te3 at room temperature, with frequency-dependent third harmonic voltage, V3ω (full symbols), and its phase shift (empty symbols) and with respect to logarithmic frequency (inset).

#### **2.5. DFT calculations**

We calculated the electronic density of states based on the experimentally determined unit cells in generalized gradient approximation (GGA) DFT scheme with Perdue-Burke-Emzerhof (PBE) pseudopotentials in CASTEP using Materials Studio package [32, 33]. We considered 1×2×2 minimal supercells replacing between 0 and 4 Sn atoms with Ge or Sb out of 16. For Sb alloying, we also considered structures with one Sn atom missing. We checked two different configurations for each composition.

## **3. Results and discussion**

## **3.1. Bi2Te3**

completely surrounded by the material. This method relies on applying AC current along the gold thread (heater) at frequency *ω*, which generates thermal oscillations finally resulting in a voltage at the third harmonic (3*ω*), which is closely related to thermal conductivity (κ):

> ( ) ( ) 3 2 2 1 1

é ù æ ö × × =- × + - × - × ê ú ç ÷

a

ç ÷ ×× × ê ú ×× × ë û è ø

where *V*1*ω* is the amplitude of the ohmic voltage, *l* is the length of the heater, *β* is the temperature coefficient of gold, *R*0 is the nominal wire electrical resistance, *α* is the thermal diffusivity of the sample. This formula is valid as long as thermal penetration depth, 2 is greater than five times the wire radius. Performing linear fit to above expression, of *V*3*ω* vs ln(2*ω*), thermal

Here we present the results of this frequency-dependent third harmonic measurement for commercial p-type Bi2Te3 at 300 K [31]. Performing linear fit to the expression indicated in this section, of *V*3*ω* vs ln(2*ω*), as shown in **Figure 7**, thermal conductivity obtained is *κ* = 1.32 W m−1 K−1 at 300 K, matching reasonably well the results with TTO method of