First principles investigation of the structural, electronic, thermal and transport properties of new ternary auride $\hbox {X}_{3}$AuO (X = K and Rb) semiconductors

Abstract

We report the structural, electronic, thermal and transport properties of two new ternary aurides containing gold atom $\hbox {X}_{3}$AuO (X = K and Rb) using the full-potential linearized augmented plane wave method, based on the density functional theory. To describe the exchange-correlation potential, we have employed the generalized gradient approximation–Perdew–Burke–Ernzerhof (GGA–PBE) scheme of the GGA. The computed ground state properties are in good accordance with the experiments. Moreover, Tran–Blaha-modified Becke–Johnson (TB–mBJ) potential improves the electronic properties and gives accurate band gaps. Both anti-perovskites $\hbox {X}_{3}$AuO are semiconductors with an indirect band gap. Furthermore, the inclusion of spin–orbit coupling effects on the band structure along with TB–mBJ approximation splits the valence band of our compounds and reduces their band gap energy. The thermodynamic properties including heat capacity ($C_{\mathrm{V}})$, thermal expansion ($\alpha$) and Debye temperature ($\Theta _{\mathrm{D}})$ are also estimated. The transport properties as function of temperature are calculated using the BoltzTrap code; therefore, these two materials are very appropriate for thermoelectric devices at high temperatures.

1 Introduction

Cubic inverse perovskites have immense technological interest considering their application in giant magneto resistance [1,2,3,4,5] and nearly zero temperature coefficient of resistance [6]. These compounds pose a wide range of properties, such as superconducting, magnetic, semiconducting and metallic properties [7,8,9,10,11]. Thus, it is very indispensable to search new inverse perovskites with significant physical properties which are appropriate in modern semiconductor technology.

Recently, the existence of anionic gold $(\hbox {Au}^{-1})$ in some alloys has attracted attention due to its electron affinity, which is very similar to iodine [12]. The study of anionic gold $(\hbox {Au}^{-1})$ chemistry is of great importance because of its excellent properties, which are ranging from metallic for KAu to semiconducting for RbAu and CsAu with CsAu having a band gap of ${\approx }2.6$ eV [13,14,15]. Moreover, materials contain gold involved significant attention from the adjustment of the Schottky contact height at Au contacts with semiconductors [16] to the development of field-emission properties [17]. The presence of Au-anions is also potentially important in catalytic processes [18, 19]. Several compounds containing anionic gold $(\hbox {Au}^{-1})$ have also been reported, such as $\hbox {A}_{3}$AuO (A = K, Rb and Cs) [20, 21] and $\hbox {Ba}_{8}\hbox {As}_{5}$Au [22]. To attain a deeper and clear knowledge about the chemistry of auride and its bonding characters, Feldmann and Jansen [21] use an X-ray structure to synthesize novel $\hbox {Rb}_{3}$AuO and $\hbox {K}_{3}$AuO. The novel ternary oxides $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO contain auride anions according to the ionic description $(\hbox {M}^{+})_{3}\hbox {Au}^{-1}0^{2-}$ (M = K and Rb). An appropriate method for this purpose is X-ray absorption spectroscopy. In particular, the X-ray absorption near edge structure efficiently indicates the presence of auride anions in $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO [23, 24]. Using this method, they found that $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$Au0 crystallize in the ideal cubic anti-perovskite structure type $(\hbox {anti-CaTiO}_{3})$. Gold adopts the Ca-position, oxygen adopts the Ti-position with a = 5.240 Å for $\hbox {K}_{3}$AuO and a = 5.501 Å for $\hbox {Rb}_{3}$AuO [25]. The purpose of this work is to use ab-initio calculations to study the structural, electronic, thermodynamic and thermoelectric properties of two novel ternary oxides containing anionic gold atom $\hbox {X}_{3}$AuO (X = K and Rb) using the full-potential linearized augmented plane wave (FP-LAPW) method within generalized gradient approximation–Perdew–Burke–Ernzerhof (GGA–PBE) and Tran–Blaha-modified Becke–Johnson (TB–mBJ) (with and without spin–orbit coupling (SOC)) approaches.

Table 1 Calculated structural equilibrium lattice constant ${a}_{0}$, bulk modulus ${B}_{0}$, its pressure derivatives ${B}_{\mathrm{P}}$ and the volume of the cubic $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO using the GGA–PBE calculations in comparison with available experimental works.

2 Method of calculations

In this paper, we investigated the structural, electronic, thermodynamic and thermoelectric properties using the FP-LAPW method within the frame work of the density functional theory [26,27,28], implemented in the WIEN2K code [29]. The experimental lattice constants are 5.240 and 5.501 Å for $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO, respectively [25, 30]. Furthermore, for the exchange-correlation functional, we have used the GGA functional developed by PBE [31]. The TB–mBJ potential [32,33,34,35,36] was employed to achieve the most accurate band gaps. We have also performed a calculation including the spin–orbit effect on the band structure along with the TB–mBJ approximation. The considered ternary oxides $\hbox {X}_{3}$AuO (X = K and Rb) were studied in an ideal cubic anti-perovskite structure (space group Pm-3m). We have employed a mesh of 1000 k-points in the whole Brillouin zone to perform self-consistent field calculations. The energy separation between the core and valence states is maintained at $-6.0$ Ry. To achieve energy and charge convergence, the $R_{\mathrm{MT}}*K_{\mathrm{Max} }$ is equal to 7.0, where $R_{\mathrm{MT}}$ is the smallest muffin-tin (MT) radius and $K_{\mathrm{Max}}$ is the maximum value of the wave vector. The MT radii are taken as 2.50 Ry for Au, 1.8 Ry for O, 1.95 Ry for K and 2.29 Ry for Rb.

3 Results and discussion

3.1 Structural properties

To determine the structural parameters of the studied compounds, we have employed the GGA–PBE approach. The new ternary oxides $\hbox {X}_{3}$AuO (X = K and Rb) crystallize as anti-perovskites with cubic symmetry (space group Pm-3m). The calculated ground states including lattice parameters ($a_{0})$, bulk modulus ($B_{0})$ and its pressure derivative ($B_{\mathrm{p}})$ at zero pressure are given in table 1 together with the experimental results for comparison [25, 30, 37]. The total energies for different volumes are calculated and fitted to the Birch–Murnaghan’s equation of state around the experimental unit-cell volume [38, 39]. The obtained lattice parameters are $a_{0}$ = 5.242 Å for $\hbox {K}_{3}$AuO and $a_{0}$ = 5.512 Å for $\hbox {Rb}_{3}$AuO, which are in excellent agreement with the experimental results [25, 30]; $a_{0}$ = 5.240 and 5.501 Å for $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO, respectively. From table 1, one can observe that the lattice constant $a_{0}$ increases in the sequence as $a_{0}$ $(\hbox {Rb}_{3}$AuO) $> a_{0}$ $(\hbox {K}_{3}$AuO). Considering that the lattice constant increases with an increase in the atomic radius of alkali-metal ($R_{\mathrm{Rb}}$ = 1.72 Å > $R_{\mathrm{K}}$ = 1.64 Å) [40]. When comparing the obtained results of the volume within GGA–PBE, the computed results deviate from the measured one within 0.09% for $\hbox {K}_{3}$AuO and 0.50% for $\hbox {Rb}_{3}$AuO [25, 37]. No experimental or theoretical results for the bulk modulus ($B_{0})$ and its pressure derivative ($B_{\mathrm{P}})$ are available in the literature for the $\hbox {X}_{3}$AuO compounds to be compared with the present calculations.

3.2 Electronic properties

As the TB–mBJ seems to determine the most accurate band gap compared to the GGA–PBE, we used it to perform the calculations of the electronic structure of $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO at the obtained equilibrium lattice parameters. Figure 1 displays the calculated band structures for both compounds along high symmetry directions in the Brillouin zone using GGA–PBE and TB–mBJ approaches (without SOC). From figure 1a, b, d and e, we observed that the valence band maximum (VBM) occurs at the $\Gamma$-point, while the conduction band minimum (CBM) is situated at the X-point, therefore, $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO indicate an indirect band gap ($\Gamma$–X) values of 0.47 and 0.14 eV from GGA–PBE and 1.72 and 1.34 eV when using TB–mBJ, respectively. As is known, the presence of a heavy element in materials in the medium range of atomic numbers (up to about 54) [41] requires the use of the SOC effect. Then, we have calculated the band structure by including SOC effects along with the TB–mBJ approach and the results are presented in figure 1c and f. The inclusion of the SOC effect splits the valence band of the two materials. Therefore, the band gap energy between the CBM and VBM is reduced. To gain a deeper insight into the electronic band structure, the total and partial densities of states are computed and depicted in figure 2 using GGA–PBE and TB–mBJ approximations. The general features of the densities of states have similarity for both compounds, except for the band gap values and positions of the peaks. Applying the GGA–PBE method, we observe that the valence band of both studied compounds can be divided into two groups as shown in figure 2a–c. The top of the valence band is dominated by O-p states with weak hybridization with K-p (Rb-p) states. The second region is essentially dominated by Au-s and Au-d states. Whereas, when using the TB–mBJ approach, the valence band consists of three parts as shown in figure 2b–d. The first peak is composed predominantly of Au-d states, while the next peak corresponds to Au-s states and the topmost of the UVB states is mainly due to O-p orbitals. Moreover, the top of the conduction band is formed by Au-p states and the bottom band is essentially due to the unoccupied K-d (Rb-d) orbitals with a small contribution of Au-p for two compounds using both methods.

Fig. 1

Calculated band structures for (a–c) $\hbox {K}_{3}$AuO and (d–f) $\hbox {Rb}_{3}$AuO at equilibrium lattice constants, using GGA–PBE and TB–mBJ approaches (with and without SOC effect). The Fermi level is located at ($E_{\mathrm{F}}$ = 0.0 eV).

Fig. 2

Total and partial densities of states of the cubic anti-perovskites (a and b) $\hbox {K}_{3}$AuO and (c and d) $\hbox {Rb}_{3}$AuO using GGA–PBE and TB–mBJ approximations. The Fermi level is set to 0 eV.

Fig. 3

Temperature dependence of (a) the volume V (b) the specific heat at constant volume $C_\mathrm{V}$, (c) the thermal expansion coefficient $\alpha$ and (d) the Debye temperature $\varTheta _{\mathrm{D}}$ of $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO anti-perovskites.

Fig. 4

(a) Seebeck coefficient S, (b) electrical conductivity $\sigma$, (c) thermal conductivity K and (d) figure of merit ZT as functions of temperature for the two compounds of $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO.

3.3 Thermodynamic properties

Thermodynamic properties are being very crucial in explaining some characteristics of semiconductors and semiconductor devices, especially at higher temperatures and pressures. The quasi-harmonic Debye model as implemented in the Gibbs program [42] is applied to calculate the volume V, heat capacity at constant volume $C_{\mathrm{V}}$, thermal expansion $\alpha$ and Debye temperature $\Theta _\mathrm{D}$. The calculated results are obtained at temperature ranging from 0 K to a temperature below the melting point ($T_{\mathrm{m}})$, 799 and 850 K for $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO, respectively [43] and pressure from 0 to 20 GPa. Our results are shown in figure 3. Temperature dependences of the volume for $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO are displayed in figure 3a. It can be seen that both compounds have nearly similar characters with a small difference in volumes. We observe a very moderate increase in the volume with an increase in temperature for $T<100$, however, for $T>100$, the volume remains constant for different pressures. At room temperature and zero pressure, the calculated equilibrium primitive cell volume is 159.57 and 187.66 Å$^{3}$ for $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO, respectively. Heat capacity $C_{\mathrm{V}}$ vs. temperature for $\hbox {X}_{3}$AuO (X = K and Rb) are shown in figure 3b. For both anti-perovskite materials, heat capacity reveals similar features. From figure 3b, we can see that at low temperature ($T<300$), $C_{\mathrm{V}}$ increases rapidly, which is proportional to $T^{3}$ [44]. Whereas, a tardily increase at high temperature reaching to the petit Dulong limit [45], which is common to all solids at high temperatures. Moreover, the tendency of matter to change its size with a changing temperature is described by another important thermodynamic parameter, i.e., the thermal expansion ($\alpha$). The parameter $\alpha$ as a function of temperature for both compounds is presented in figure 3c. $\alpha$ increases rapidly with T at low temperature and converges to a constant value at high temperatures. The Debye temperature is a fundamental parameter that provides us valuable information about diverse physical properties including elastic constants, specific heat and melting temperature. The Debye temperature is the highest temperature that can be achieved due to a single normal vibration. The Debye temperature is given by [46]:

$$\begin{aligned} \Theta = \frac{\hbar }{K}~{\left[ {6{\pi ^2}{V^{1/2}}n} \right] ^{1/3}}f\left( \sigma \right) \sqrt{\frac{{B_{\mathrm{s}}}}{M}}, \end{aligned}$$

(1)

where M is the molecular mass per unit cell and $B_{\mathrm{s}}$ is the adiabatic bulk modulus, which is given by the following equation [47]:

$$\begin{aligned} {B_{\mathrm{s}}} \cong B\left( V \right) = V\left( {\frac{{{d^2}E\left( V \right) }}{{d{V^2}}}} \right) . \end{aligned}$$

(2)

From figure 3d, we can see the variations of the Debye temperature vs. temperature for $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO. It can be seen that the Debye temperature of two compounds decreases slightly and linearly as the temperature increases. From equations (1 and 2), we conclude that the Debye temperature depends only on the volume of the unit cell. Moreover, it is clear that for constant temperature, the Debye temperature of $\hbox {X}_{3}$AuO increases almost linearly with the decrease in the volume, (figure 3a).

3.4 Thermoelectric properties

Over the past decade, researchers are involved in finding more efficient materials for electronic refrigeration and power generation [48]. Thermoelectric materials become more important as alternative energy sources. A thermoelectric material is a material that can be used to convert thermal energy into electric energy or provide refrigeration directly from electric energy. The thermoelectric efficiency of a material is quantified by a figure of merit $(\mathrm{ZT} = \sigma S^{2}T/\kappa$) [49,50,51], which contains three physical quantities; Seebeck coefficient (S), electrical conductivity ($\sigma$) and thermal conductivity (k). The Seebeck coefficient tells us how many volts per degree temperature difference are generated. The electrical conductivity determines how well a material conducts electricity, and the thermal conductivity is a measure of how well heat is conducted. The calculated Seebeck coefficient (S), electrical conductivity $(\sigma$), thermal conductivity (k) and figure of merit (ZT), which are computed by the equations reported in refs [52, 53] and represented in figure 4a–c for two compounds $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO. Moreover, the BoltzTrap code [54] as implemented in the Wien2k code is applied to study the transport properties of $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO materials. To ensure the convergence of our transport properties, a dense k-mesh is used. In the present case, thermoelectric properties are computed within a temperature range of 50–900 K. Figure 4a shows the plots of the Seebeck coefficient (thermo-power) for $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO. The positive value of the Seebeck coefficient confirms the p-type nature of these materials. It can be seen that the Seebeck coefficient increases rapidly with temperature and reaches its maximum value 217.51 $\upmu \hbox {V K}^{-1}$ (205.31 $\upmu \hbox {V K}^{-1})$ at 250 K (350 K) for $\hbox {K}_{3}$AuO $(\hbox {Rb}_{3}$AuO), and thereafter, decreases to attain a value of 183.54 $\upmu \hbox {V K}^{-1}$ at 800 K for both compounds. The value of the Seebeck coefficient at room temperature is 215.68 and 203.83 $\upmu \hbox {V K}^{-1}$ for $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO, respectively. Therefore, the studied semiconductors exhibit good thermoelectric features. Figure 4b represents the electrical conductivities of $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO. The electrical conductivity increases linearly with temperature, besides at given temperature, the electrical conductivity is approximately the same for two anti-perovskites. At 800 K, $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO record a maximum value of electrical conductivity as $28.61\times 10^{18}$ and $30.07\times 10^{18}$ ($\Omega \hbox {ms})^{-1}$, respectively, which means that $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO compounds have excellent electrical conductivities.

In semiconductor materials, heat is transferred mostly because of lattice vibrations, whereas in metals, free electrons are the good source of thermal conductivity [55, 56]. The calculated thermal conductivities are displayed in figure 4c. From this figure, we show that thermal conductivity of the examined compounds remains nearly the same until 400 K and then increases gradually with an increase in temperature to achieve the maximum value of $11.78\times 10^{14}$ W $\hbox {mK}^{-1} \hbox {s}^{-1}$ $(10.31\times 10^{14} \hbox {W mK}^{-1} \hbox {s}^{-1})$ at 800 K due to the electronic vibrations. To examine the performance of our studied materials, we have also calculated another important parameter; the dimensionless figure of merit (ZT) for $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO and represented in figure 4d. We observe that ZT of $\hbox {K}_{3}$AuO $(\hbox {Rb}_{3}$AuO) increases rapidly with an increase in temperature to reach its maximum value of 0.75 (0.73) at 500 K, and remains constant with an increase in temperature, which indicates that $\hbox {K}_{3}$AuO is the best thermoelectric material. Thus, the considered anti-perovskites $\hbox {X}_{3}$AuO (X = K and Rb) indicate very promising materials for thermoelectric applications in a high-temperature range.

4 Conclusion

In summary, we systematically investigated the structural, electronic, thermal and transport properties of new ternary aurides containing gold atom $\hbox {X}_{3}$AuO (X = K and Rb) for the first time by means of the FP-LAPW method. The study reveals that the anti-perovskites $\hbox {X}_{3}$AuO crystallize in cubic symmetry (space group Pm-3m) and their structural parameters are in good agreement with the experimental data. The GGA–PBE and TB–mBJ approaches show that both anti-perovskites are indirect band gap semiconductors. Furthermore, the inclusion of SOC in the TB–mBJ approach splits the valence band of $\hbox {X}_{3}$AuO compounds and reduces their band gaps. The thermal properties are also computed using the quasi-harmonic Debye model and found that the heat capacity ($C_{\mathrm{V}})$ is close to the Dulong–Petit limit. Furthermore, the effects of temperature on thermoelectric parameters are investigated using the BoltzTraP code, and the positive Seebeck coefficient confirms that $\hbox {K}_{3}$AuO and $\hbox {Rb}_{3}$AuO are p-type semiconductors. These two materials are expected to have significant applications in thermoelectric and thermoelectric devices at high temperatures.