Optical phonon mode assisted thermal conductivity in p-type ZrIrSb half-Heusler alloy: a combined experimental and computational study

Figure 3(a) represents the temperature-dependent S in the temperature regime of 300–950 K. At 300 K, the value of S is found to be ∼34.4 μV K⁻¹. The positive sign indicates the dominance of holes in the charge transport mechanism. With the increment in temperature, a linear increment in magnitude of S is noted till 500 K (denoted by a solid red line in figure 3(a)). Above 500 K, S increases non-linearly with temperature, reaching maximum value (S_max) ∼ 75.5 μV K⁻¹ at 950 K. The observed S_max is smaller compared to the XCoSb (X = Ti, Zr and Hf) series of alloys [17, 37, 38]. Under free electron theory approximation, S and m* of the charge carriers are directly related to each other and are given by the expression [39, 40]:

$$\begin{align}S = \frac{{8{\pi ^2}k_{\text{B}}^2}}{{3e{h^2}}}{m^*}T{\left( {\frac{\pi }{{3n}}} \right)^{2/3}},\end{align} \tag{ 1 }$$

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where k_B is the Boltzmann constant, e is the electronic charge and n is the charge carrier density. As T and n are positive quantities, this implies that the sign of S will be decided by the sign of m*. Generally, m* depends on the band structure of the materials. In addition, in the case of semiconductors, both types of charge carrier (electrons and holes) contribute to S. So, the sign of S will be decided by the charge carriers having larger m*. Thus, m* estimated from figure 2(a) will be useful to gain a qualitative understanding of the sign of the experimentally obtained value of S. Hence, the VBM and CBM near Г and X k-symmetry points are analyzed. The curvature of the energy curve at a k-point decides the value of m* and is given by [40],

$$\begin{align}{m^*} = \frac{\hbar }{{\left[ {\dfrac{{{{\text{d}}^2}E}}{{{\text{d}}{k^2}}}} \right]}}.\end{align} \tag{ 2 }$$

From the above equation, it can be inferred that at a given k point, the m* of a charge carrier for a broader energy curve is greater than the narrower energy curve. In order to see the m* contributions of holes and electrons along different high symmetry directions, m* has been estimated at Г point for holes (along K, W, X and L directions) and at X point for electrons (along Г, W, K and L). The calculated m* is given in table 2. In the table, K-Г-K represents the m* calculated at Г point along the Г-K direction. Similar notation is used for other directions as well. It is noted that band 1 is lighter with lesser m*, resulting in higher electrical conductivity of charge carriers in this band. However, bands 2 and 3 being degenerate along Г-X and Г-L directions have the same m* values. The estimated value of m* for holes is 0.6 m_e (of band 2 and 3) and 0.21 m_e (band 1) along X direction. Similarly, for electrons the maximum m* ∼ 0.6 m_e is found along Г direction, indicating higher mobility of electrons in the vicinity of CBM. This indicates that both holes and electrons have similar m* along X and Г directions, respectively. However, due to degeneracy of bands 1, 2 and 3 in the vicinity of Г point, there will be a larger contribution from the m* of holes in S. This is in agreement with our experimental results. In comparison to TiCoSb, ZrCoSb and HfCoSb, a lower S is observed in this alloy. This is due to the fact that there is contribution from only one Г-pocket in the latter case, whereas, for the other alloys four bands (at Г point) contribute to S. In addition, the presence of both anti-site disorder between Ir/Sb atoms and vacant sites may contribute to the lower value of S in ZrIrSb.

The temperature response of ρ of ZrIrSb in the temperature regime 300–950 K is shown in figure 3(b). At 300 K, the value of ρ is noted to be ∼16 mΩ cm. This value is quite small compared to the values noted for other XCoSb (X = Zr, Hf and Ti) HH alloys [17, 37, 38]. However, it is greater than those reported for other well-known TE materials (as given in table 3) [41–43]. It is observed that ρ decreases non-linearly as a function of temperature up to 370 K. Above 370 K, linear behaviour is noted. With further increment in temperature (above 750 K), an increasing trend in ρ value is observed (as shown in the inset of figure 3(a)). Changes in the slope of the resistivity curve indicate that different conduction mechanisms exist in the measured temperature regime. In addition, as discussed in section 3.3.1, an increment in S is noted as the temperature is increased. These observations support the fact that in ZrIrSb, the electrical transportpossibly occurs via delocalization of localized electrons. Therefore, to identify the mode of electronic transport, the obtained data are analyzed with different types of models available in the literature. One well-known mode of conduction is nearest neighbour hopping (NHH) [44]. There is constant activation energy (E_a) in this mechanism, which is responsible for the hopping of charge carriers to the nearest neighbour sites from their initial position. The ρ is expressed as [44],

$$\begin{equation}\rho \left( T \right) = {\rho _0}{ }{{\text{e}}^{\dfrac{{{E_{\text{a}}}}}{{{k_{{\text{B }}}}T}}}},\end{equation} \tag{ 3 }$$

where ρ₀ is the pre-exponential factor and T is the temperature. In the temperature regime of 370–640 K, the curve is fitted well with equation (3) (shown in the upper inset of figure 3(b)). The value of E_a is estimated to be ∼0.1 eV. However, below 370 K, as mentioned before, non-linear behaviour is observed. This suggests the presence of more than one E_a and points towards a different conduction mechanism. Generally, non-linear temperature variation of ρ is expressed as,

$$\begin{equation}\rho = {\rho _{0{ }}}{{\text{e}}^{{{\left( {\dfrac{{{T_0}}}{T}} \right)}^\gamma }}},\end{equation} \tag{ 4 }$$

where γ can take the value 1/4, 1/2 and 1/3 depending upon the different conduction models, and T₀ is the characteristic Mott's temperature. The curve in the regime 300–370 K is best fitted (using equation (4)) for γ = 1/4 (shown in the lower inset of figure 3(b)). This indicates that conduction in this regime is through Mott's variable range hopping (MVRH), where the hopping length decreases with increment in temperature. In addition, above 750 K, ρ displays metallic-type conduction (as shown in the inset of figure 3(a)). Hence, based on our analysis, it can be concluded that there is transformation from MVRH to NHH at around 370 K, followed by metallic-type conductivity above 750 K.

In addition, the lower magnitude of ρ (near 300 K) in comparison to ZrCoSb (given in table 3) can be attributed to the presence of anti-site disorder between Ir/Sb atoms and vacant sites in ZrIrSb. Due to this disorder, there is always the probability of finding Ir/Sb atoms at 4d position (vacant sites). This can alter the hybridization between Co and Ir atoms, which is responsible for the formation of band gap. It can either lead to a reduction in the band gap or the formation of in-gap electronic states near the VB and CB. A similar effect of anti-site disorder on ρ behaviour is also noted in ZrNiSn [19, 45]. The presence of this disorder is likely to reconcile the lower magnitude of ρ of ZrIrSb alloy in comparison with ZrCoSb. This, along with the non-stoichiometry of the alloy (as reported in section 3.1), can result in the formation of impurity bands near the CB, possibly leading to the observed hopping type of conduction.

The temperature response of κ_total is shown in figure 4(a). Near 300 K, the magnitude of κ_total is ∼9.3 W mK⁻¹. In comparison with XCoSb family (as listed in table 4), a significant reduction in room-temperature κ_total is noted. From figure 4(a), it can be inferred that κ_total decreases with temperature till 500 K. However, a slight increment in κ_total, followed by a hump is noted in 500–650 K temperature regimes. This hump may be due to the presence of bipolar effect, where both majority (holes) and minority (electrons) carriers contribute to κ_total [46, 47]. Interestingly, above 650 K, κ_total varies as a function of 1/T, reaching the minimum value of ∼2.79 W m⁻¹ K near 950 K.

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Heat transport mechanism in any material is primarily governed by the movement of charge carriers (κ_e) and through κ_L. However, in the present case, bipolar effect (κ_bp) also contributes. Hence, κ_total is defined as,

$$\begin{equation}{\kappa _{{\text{total}}}} = { }{\kappa _{\text{e}}} + { }{\kappa _{{\text{L }}}} + {\kappa _{{\text{bp}}}}.\end{equation} \tag{ 5 }$$

Each of these components is subtracted from κ_total to obtain the individual contributions to the thermal conductivity. The electronic contribution from κ_total is extracted using:

$$\begin{equation}{\kappa _{\text{e}}} = \frac{{LT}}{\rho },\end{equation} \tag{ 6 }$$

where L is the Lorentz number. Since ZrIrSb is a non-degenerate semiconductor, variable values of L are used (as shown in the left inset of figure 4(a)), and defined as [48],

$$\begin{equation}L = { }\left[ {1.5 + {\text{ex}}{{\text{p}}^{ - \left[ {\frac{{\left| S \right|}}{{116}}} \right]}}} \right].\end{equation} \tag{ 7 }$$

The temperature dependence of κ_e is presented in the right inset of figure 4(a). The value of κ_e at 300 K is estimated to be around 0.04 W m⁻¹ K, which also increases with temperature, reaching a maximum of ∼0.35 W m⁻¹ K near 950 K. As noted in section 3.3.2, above 750 K, ρ increases with increment in temperature, implying more contribution of charge carriers to κ_total at higher temperatures. In the present case, there is ∼11% contribution of charge carriers to κ_total at 950 K. To estimate the contribution of κ_bp to κ_total, the method proposed by Kitagawa et al is used [49]. At temperature above Debye temperature (θ_D), which is ∼297 K for ZrIrSb, the Umklapp phonon scattering mechanism is expected to be dominant, where κ_L obeys the following relation [48, 50]:

$$\begin{equation}{\kappa _{\text{L}}} = 3.5{ }\frac{{{k_{\text{B}}}}}{h}{ }\frac{{M{V^{1/3}}\theta _{\text{D}}^3}}{{{\gamma ^2}{ }T}},\end{equation} \tag{ 8 }$$

where M is the average mass per atom, V is the average atomic volume and γ is the Gruneisen parameter. From the above equation, it can be said that κ_L is proportional to T⁻¹. Therefore, κ_bp can be estimated by subtracting the linear term from κ_total− κ_e at higher temperatures. This method has been previously used in studying κ_total of CoSb_3−x Te_x , CoSb₃ systems [51, 52]. The κ_total − κ_e follows the linear behaviour of κ_L ∼ T⁻¹ quite well up to 500 K (as shown in the inset of figure 4(b)), which confirms the dominance of the Umklapp phonon scattering mechanism. This shows deviation from 500 K, and this temperature can be referred to as bipolar excitation temperature. At high temperature, κ_L is estimated by extrapolating the linear relationship of the κ_L ∼ T⁻¹. The resultant curve is shown in figure 4(b). The contributions of κ_e, κ_bp and κ_L to the κ_total are estimated as 11.36%, 2.38% and 86.26% near 950 K, respectively.

Temperature-dependent κ_L of ZrIrSb is also calculated using anharmonic first-principles phonon calculations. In the calculation, only three phonon processes are considered due to the presence of a negligible separation energy gap between the optical and acoustic phonon branches (discussed in detail in the next paragraph). The κ_L is estimated under SMRT approximation by solving using the LBTE method [33] and is defined as,

$$\begin{equation}{\kappa _{\text{L}}} = \frac{1}{{NV}}\sum\limits_\lambda {{C_\lambda }} {v_\lambda } \otimes {v_\lambda }\tau _\lambda ^{{\text{SMRT}}}.\end{equation} \tag{ 9 }$$

Here, N is the number of unit cells in the crystal, V₀ is the volume of the unit cell, c_λ is the mode-dependent specific heat, v_λ is the group velocity of the phonon mode and $\tau _\lambda^{{\text{SMRT}}}$ is the SMRT of phonon mode λ. Here, λ represents a phonon mode with a wavevector q, branch index j and $\tau _\lambda^{{\text{SMRT}}}$ is considered as the lifetime of the phonon mode τ_λ . The calculated κ_L is presented in figure 4(b). It can be observed that the calculated κ_L decreases with the increase in temperature, which is in agreement with our experimental observations. In the whole temperature range, the calculated value of κ_L deviates from the observed experimental data. The separation between the curves reduces towards lower temperature. This discrepancy can be due to the presence of defects or anti-site disorder in ZrIrSb.

As noted above, the transport of heat in ZrIrSb is mainly through the lattice via phonons. Thus, investigation of phonon properties is essential to understand the κ_L behaviour. Therefore, phonon dispersion along with its electronic structure is calculated using the computational method. This will help us to discern the contribution from the various phonon modes to the κ_L. The obtained phonon dispersion spectrum along high symmetry directions is presented in figure 4(c). From the figure, it is inferred that all the phonon frequencies are positive, which corroborates that the structural model of ZrIrSb is dynamically stable. For a primitive unit cell containing three atoms, there are three acoustic phonon modes (one longitudinal (LA) + two transverse (TA)) and six optical phonon modes (two longitudinal (LO) + four transverse (TO)) corresponding to those atoms. The phonon modes are classified into three categories: 0–3.5 THz (acoustic modes; black solid lines), 3.5–5.25 THz (low-frequency optical modes; red solid lines) and 5.5–7 THz (high-frequency optical modes; blue solid lines). From figure 4(c), it can be inferred that the separation gap between acoustic and low-frequency optical modes is very small ∼0.64 meV, whereas there is the presence of a large gap between low-frequency and high-frequency optical modes. Due to the presence of a lower energy gap between acoustic and low-frequency optical modes, it is expected that low-frequency optical modes will contribute to κ_L. The low-frequency optical modes lead to an increment in Umklapp processes, which tends to reduce κ_L.

In addition, to extract the contribution of different atoms to the acoustic and optical phonon modes, phonon TDOS (P-TDOS) along with the PDOS (P-PDOS) are calculated. P-TDOS per unit cell and P-PDOS per atom are shown in figure 4(d). In analogy to figure 4(c), in the P-TDOS plot, a separation gap between higher-frequency and lower-frequency optical phonon modes can be noted. From the plots of P-PDOS, it can be noted that higher-frequency optical phonon modes (∼5.5–7 THz) are mainly due to lighter Zr atoms. The considerable number of phonon states in the mid-frequency range (∼3.5–5.25 THz) are due to Sb and Ir atoms. However, the main contribution to lower-frequency acoustic phonon modes (∼0–3.5 THz) comes from heavier Ir atoms in the unit cell.

Moreover, based on our experimental results it is noted that the value of κ_L near room temperature is quite a bit lower [38]. This signifies that the presence of heavier elements has some significant impact on the phonon dispersion of ZrIrSb. The most remarkable difference is noted in the group velocities of acoustic phonon modes. The group velocities of phonon modes are estimated close to Г point. The estimated values are ∼2.6 km s⁻¹/2.6 km s⁻¹/5.2 km s⁻¹ (TA/TA/LA). In addition, flattening of the high-frequency optical phonon branches in the vicinity of Г point is shown in figure 4(c). This signifies that the group velocities (TA/LA) of high-frequency optical phonon modes are negligible. Therefore, the group velocities of acoustic phonon modes can be approximated as sound velocity of the alloy. The calculated values are comparable with the experimentally obtained longitudinal (v_L) and transverse (v_t) sound velocities (given in table 5). However, as noted from table 5, the values of v_L and v_t are smaller than those reported for TiCoSb, ZrCoSb and HfCoSb. The reduced sound velocity is due to the fact that, in ZrIrSb, in addition to acoustic phonon modes, the low-frequency optical phonon modes also play a significant role. In addition to this, a distinct variation can be seen in the frequency of optical phonon modes. As noted from figure 4(c), the lowest-frequency optical modes lie near 5 THz, whereas in TiCoSb, ZrCoSb and HfCoSb, they lie near 6.5, 5.9 and 5.2 THz, respectively [53]. Due to the low-lying optical phonon modes, it is expected that there will be strong coupling between optical and acoustic phonon modes, which is absent in the latter alloys. This leads to an increment in Umklapp processes of phonon scattering, which can be observed due to the presence of a large number of optical phonon excitations near the BZ boundary. Hence, it can be concluded that the lower κ_L noted in this alloy is due to the reduced velocity of phonons and enhancement in the Umklapp process of phonon scattering.

Ideally, in semiconductors or insulators, the acoustic modes play a significant role in the conduction of heat. However, as discussed above in the present case, the interaction of optical phonon modes with acoustic vibrations cannot be neglected. Hence, the contributions of different phonon modes to κ_L is analyzed through accumulated lattice thermal conductivity (κ_C). The accumulated thermal conductivity (κ_C) is defined as [54],

$$\begin{equation}{\kappa _{\text{C}}}\left( \omega \right) = \int\limits_0^\omega {{\kappa _{\text{L}}}} \delta \omega^ {\prime} - {\omega _\lambda }{\text{d}}\omega^{\prime} ,\end{equation} \tag{ 10 }$$

where κ_L is given by equation (9). The predicted normalized κ_C with respect to κ_L as a function of frequency is presented in figure 5. At 300 K, the contribution of acoustic phonon modes (0–3.5 THz) and lower optical phonon modes (3.5–5.25 THz) is found to be ∼50% and 45%, respectively. However, higher optical phonon modes contribute only 5% to κ_C. Hence, approximately 95% of the calculated κ_L is from delocalized low-frequency phonon modes, while higher optical modes contribute insignificantly to κ_L. This indicates that the phonon modes arising due to heavier elements dominate the heat transport mechanism. It is estimated to be ∼80% at 300 K. Interestingly, as the temperature increases from 300 to 500 K, κ_C/κ_L reduces due to the increase in phonon–phonon interactions and higher scattering rates, whereas the contribution from the Ir and Sb phonon modes increases. A similar variation is noted in the 500–700 K and 700–900 K temperature range.

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In addition, κ_L is also sensitive to the magnitude and frequency dependence of relaxation time (τ). Hence, phonon lifetime as a function of phonon frequency at 300, 500, 700 and 1000 K is represented in figure 6. The phonon lifetime is calculated from anharmonic three phonon–phonon scattering mechanisms or Umklapp processes. A sparse distribution of phonon modes can be observed in acoustic and lower optical mode regimes. At 300 K, the longest lifetime (τ_max) ∼ 32 ps is noted for phonons belonging to acoustic modes. τ is found to decrease with increment in frequency due to enhancement in phonon scattering events. A similar distribution of modes and trends in τ is noted in lower optical mode regimes. This indicates that phonons from both acoustic and optical modes contribute to κ_L, which is in agreement with previous observations. However, above 5.5 THz, the rate of decrement of τ is reduced. This behaviour is observed because the phonon number density increases (yellow/orange region) leading to higher scattering rates. Increment in scattering rate subsequently leads to phonon annihilation and ultimately decreases the lifetime of decay. Furthermore, as the temperature increases from 300 to 500 K, τ_max decreases, which indicates an enhanced phonon–phonon scattering rate at higher temperature. Hence, a reduced κ_L is noted. Similar behaviour is noted in the 500–700 K and 700–1000 K temperature regime. Therefore, based onour investigations, it can be said that coupling exists between the acoustic and lower-lying optical phonon modes originating due to Ir and Sb atoms, which plays a critical role in the thermal transport mechanism of ZrIrSb. This behaviour is quite different to raditional understanding of thermal transport. In our case, a lower κ_L is expected due to contribution from both modes. However, due to the large distribution of phonon modes in this regime (0–5.25 THz), a large κ_L is observed. In addition, the decrement in κ_L with increment in temperature can be attributed to decrement in τ of the acoustic and lower optical phonon modes, which is an outcome of enhanced phonon–phonon scattering rate.

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Finally, in order to evaluate the TE efficiency of the material, the ZT is estimated using the following equation [48]:

$$\begin{equation}{\text{ZT}} = \frac{{{S^2}T}}{{\rho {\kappa _{{\text{total}}}}}}.\end{equation} \tag{ 11 }$$

The ZT obtained experimentally for ZrIrSb is shown as a function of temperature in figure 7(a). The ZT value for ZrIrSb is ∼0.032 at 950 K, which is larger than those reported for similar alloys, such asTiCoSb, ZrCoSb and HfCoSb [37, 38]. The large value for ZrIrSb may be due to the lower value of ρ and κ_L. To enable this alloy to be used for TE conversion, an efficient method is to further decrease its κ_L. Various strategies, such as nano-structuring, construction of heterojunctions, introducing mass or strain effect, have been employed by researchers [55, 56]. Among these, we have further discussed the possibility of employing nano-structuring as an effective tool to decrease κ_L. In this strategy, the particle size is reduced, which in turn results in the decrement of the phonon mean free path (MFP). Hence, to explore this possibility, the size effect on κ_L is evaluated through the summation of contributions from all phonon modes in ZrIrSb. The ratio κ_C/κ_L plotted as a function of MFP is shown in figure 7(b). At 300 K, it is noted that 10 nm size crystals can reduce κ_L by ∼87% compared to the bulk, which in turn may increase the ZT substantially. Similar improvements are achievable at 500 K, 700 K and 900 K operating temperatures. Thus, it would be interesting to explore the TE properties of similar alloys through nano-structuring.

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