Understanding the role of 5d electrons in ferromagnetism and spin-based transport properties of K2W(Cl/Br)6 for spintronics and thermoelectric applications

In this article, we have systematically investigated the structural, electronic, magnetic, and spin-based thermoelectric properties of K2W(Cl/Br)6 by first-principles calculation. The obtained negative formation energy confirmed the thermodynamic stability of K2W(Cl/Br)6, while the tolerance factor calculation showed their cubic phase stability. In addition, we have estimated the elastic constants which confirmed the mechanical stability of K2W(Cl/Br)6. Further, the spin-polarized band structure and density of states calculations revealed the half-metallic nature with high Curie temperature (Tc) values of 613 K and 597 K for K2WCl6 and K2WBr6, respectively. Moreover, we have studied the temperature variation of thermoelectric properties such as kl, σ, ke, S, PF, and ZT. Such results showed that higher ZT values for spin-down channels are obtained from ultra-low ke, and high PF. Therefore, K2W(Cl/Br)6 are viable thermoelectric and spintronic materials.


Introduction
The advancement of spintronics and quantum computing technology has boosted the memory storage speed with multifunctional characteristics. This is a developing eld which reduces magnetic chip size and enhances the memory speed by utilizing the electron spin and its charge. 1,2 The advanced spintronic technology also has novel achievements in nonvolatile magnetic random-access memory. 3,4 These advancements can also be applied in electronic devices due to their low cost, faster data speed, and less power consumption. [5][6][7] In addition, the recent quantum technology takes advantage of electronic spin states in the digital display rather than the charge states of a typical electronic operation. The development of giant magnetic resistance (GMR) in 1998 boosted this technology, 8 where the electron spin can create an enormous difference in the resistance of alternative magnetic layers in the presence of external applied magnetic elds. Thus, the magnetic response of electronic charge and spin facilitate the low-powered, high-speed, non-volatile, and nano-size memory. 9,10 The present scientic achievements of spintronics technology gave rise to the improvement of magneto-resistive random-access memory (MRAM), magnetic sensors, magnetic valves, read heads of magnetic hard drives, and giant magneto-resistive effect (GMR). Materials exhibiting the high spin polarization (SP) are suitable candidates for spintronics technology. Rather, for the half-metallic ferromagnetism (HMF) materials, one channel is metallic and the other one is insulating, which produces 100% SP, demonstrating that they are spintronics materials. 11,12 The rst HMF was observed by de Groot et al. in Heusler alloy PtMnSb and NiMnSb in 1983. 13 Aer this, HMF was observed in many types of materials such as diluted magnetic semiconductors, perovskites, spinel chalcogenides, and double perovskites. 14,15 But, the major confront of spintronic materials are their phase instability at elevated temperature and clustering of magnetic ions greatly inuences the functionality of spin. In later research, the phase instability issue was resolved at higher temperature, but the problem of clustering remains to be addressed. Nonetheless, a lot of transition-metal doped alloys were formed at room temperature, still the issue of clustering limits their applications. 16 To solve the problem of spin segregation, numerous procedures were adopted, where the doping of nonmagnetic elements into alloys is prominent. The FM behaviour has been reported in Be 1−x C x S, 17 Sn 1−x Mg x O 2 , 18 etc. But these materials are expensive and have complicated production procurements.
Recently, Halide based double perovskites with chemical formula X 2 YZ 6 (X = Cs, Rb, K Y = Os, Nb, Ta, W; Z = Cl, Br, I) have attracted considerable attention for spintronics applications due to their low cost, high Curie temperature (T c ), and good stability. [19][20][21][22][23] In X 2 YZ 6 , the d-orbital electrons of Y atom contributes signicantly to the magnetic moment. Furthermore, it has been noted that the halide based double perovskites have ferromagnetism at high Curie temperature. However, there is a lack of detailed description on the physical properties of X 2 YZ 6 . In this article, we have taken K 2 W(Cl/Br) 6 compounds to investigate their magnetic and spin-based transport properties by rst-principles calculation. These K 2 W(Cl/Br) 6 4 and they have observed the cubic phase stability from the X-ray diffraction patterns. 24,25 However, existing literatures about these compounds are only limited to their structural studies. To the best of our knowledge, there is no detailed report available on magnetic and spin-based transport properties of K 2 W(Cl/Br) 6 . Our spin-polarized band structure and density of states calculations revealed the presence of half-metallic character in these materials with high Curie temperature. Thus, K 2 W(Cl/Br) 6 compounds are emerging spintronics materials.
Besides spintronics applications, the lead-free halide based double perovskites have attracted considerable attention for photovoltaic and thermoelectric technology. A good photovoltaic material should have high optical absorption coefficient and conductivity, with low reectivity. On the other hand, a material is efficient for thermoelectric technology, if it has high Seebeck coefficient (S), good electrical conductivity (s), and low thermal conductivity (k). 26 Many lead-free halide based double perovskites are reported to have suitable photovoltaic and thermoelectric properties. For instance, Haq et al. predicted that Rb 2 XGaBr 6 (X = Na, K) are promising photovoltaic and thermoelectric materials due to their optimum optical absorption coefficient and large ZT values, respectively. 52 Later, Iqbal et al. have shown the emerging photovoltaic and thermoelectric properties of Rb 2 AlInX 6 (X = Cl, Br, I) due to their narrow band gap. 53 Also, Nawaz et al. reported that Rb 2 YInX 6 (X = Cl, Br, I) are thermodynamically stable and they are suitable for photovoltaic and thermoelectric technology. 54 Instead of low thermal conductivity arising from the occupation of cations in the octahedral structure, it is very surprising that these halide based double perovskites are mainly studied for photovoltaic purposes. Only very few experimental studies were performed to investigate their thermoelectric properties and interestingly the research in the thermoelectric response is now growing. 27 In this article, we have carried out the spin-based thermoelectric properties of K 2 W(Cl/Br) 6 . We have computed the temperature variation of k l , s, k e , S, PF, and ZT. The higher ZT values for spindown channels have resulted from ultra-low k e , and high PF. Thus, K 2 W(Cl/Br) 6 are potential thermoelectric and spintronic materials.

Computational details
The electronic structure, magnetic properties and transport properties of K 2 W(Cl/Br) 6 were investigated by using Wien2k 28 and BoltzTraP code. 29 We have employed PBEsol approximation to optimize the crystal structure in FM and AFM states. 30 However, PBEsol approximation underestimates the electronic bandgap on which magnetic and transport properties are dependent. Therefore, we have employed TB-mBJ formalism, 31 which can accurately predict the bandgap. In addition, due to the presence of heavy elements, SOC coupling is signicant, that was added with TB-mBJ. The energy cut-off for geometry optimization was selected to be 520 eV. On the other hand, the average forces per ions were optimized to 0.002 eVÅ −1 . Furthermore, we have considered a k mesh of 12 × 12 × 12 for electronic calculation. The product of maximum wave vector and muffin tin radii is kept as R MT × K max = 8, along with angular momentum vector l = 10 and Gaussian factor = 10. The level convergence is achieved to be 10 −5 Ry self consistently by using the above-mentioned inputs. For transport properties calculation, we have considered a dense k-points of 150 000.

Structural and mechanical stabilities
The halide based double perovskites K 2 W(Cl/Br) 6 have a cubic phase with space group Fm 3m (225). 32 The perspective view of K 2 W(Cl/Br) 6 is shown in Fig. 1. The vacancies between the octahedra W(Cl/Br) 6 are occupied by K atoms whereas each octahedron is separated by the other octahedra through the 12fold coordination system of (Cl/Br). In this structure, each K atom is surrounded by 12 (Cl/Br) atoms, whereas every W atom is coordinated with 6 (Cl/Br) atoms. Moreover, each W(Cl/Br) 6 is located at the corner and face center of the cubic system. The K, W, and (Cl/Br) atoms in the unit cells of both systems are positioned at (0.25, 0.25, 0.25) (0, 0, 0) and (x, 0, 0), respectively. The atomic positions in the structure are corrected by minimizing strain throughout the optimization process.
The optimized energy versus volume plot is shown in Fig. 2(a and b). It is noticeable from Fig. 2(a and b), that K 2 W(Cl/Br) 6 compounds have positive energy difference between FM and AFM states indicates that the FM is more preferable because of more energy release in this process.
Curie temperature (T c ) of K 2 W(Cl/Br) 6 are predicted through the Heisenberg classical model. e., T c = 2DE/3xK B , where x is the contribution of W atom and DE is the energy difference between ferromagnetic and antiferromagnetic ground states, i.e. DE = E AFM − E FM . 33,34 The computed T c values are 613 K and 597 K for K 2 WCl 6 and K 2 WBr 6 , respectively. The high T c values make these compounds suitable for spintronic applications. The cubic phase stability of K 2 W(Cl/Br) 6 are investigated from Goldsmith tolerance fact or calculation as follows.
It is worthy to mention here that, the stable cubic phase should have s value in the range of 0.9-1.02. 35 Table 1 clearly shows that s values of K 2 W(Cl/Br) 6 are in this range demonstrating their cubic phase stability.
To investigate the synthetic possibility of K 2 W(Cl/Br) 6 , we have calculated the enthalpy of formation by using the following equation where, E total {K 2 W(Cl/Br) 6 }, E bulk K , E bulk W and E bulk (Cl/Br) represent the ground state energy of K 2 WX 6 , K, W, and (Cl/Br) in their bulk form. The predicted values of DH f are −31.12 eV and −27.12 eV for K 2 WCl 6 and K 2 WBr 6 , respectively. This negative DH f revealed the thermodynamic stability of K 2 W(Cl/Br) 6 .
The mechanical stability was studied from elastic constant values (C ij ), obtained by using the Cubic Elastic package. 56 It is observed from Table 2 that the values of the various components of C ij obeyed the Born criteria of mechanical stability, i.e. C 11 − C 12 > 0, C 44 > 0, C 11 + 2C 12 > 0, C 12 < B < C 11 . 37 Furthermore, we have estimated the Cauchy's pressure (CP = C 12 -C 44 ). The positive value of CP demonstrated the ductile properties of K 2 W(Cl/Br) 6 . The overall elastic study conrmed that K 2 W(Cl/ Br) 6 are mechanically stable. 36

Magnetic properties
The half-metallic ferromagnetic material having high spin polarization (P) is essential for spintronic applications. The spin polarization (P) can be obtained as where (N [ ) and (N Y ) represent the density of states of up and down the channels at Fermi level (E F ). The spin-polarized band structure is shown in Fig. 3. It is noticeable from Fig. 3 that, the quantum state of the spin-up channel overlapped with Fermi level (E F ), which demonstrates the metallic nature. However, the spin-down channel shows the insulating nature because of the existence of nite separation between valence band maxima (VBM) and conduction band minima (CBM). Therefore, the combined spin up and spin down showed the half-metallic ferromagnetism nature with 100% spin polarization (P = 1). The detailed investigation revealed that the valence band maxima (VBM) and conduction band minima (CBM) of spindown channels are located at the same k-point for both compounds, which indicates the direct band gap properties. We have obtained the band gap of 3.01 eV and 2.72 eV for K 2 WCl 6 and K 2 WBr 6 , respectively in the spin-down channel. Also, the estimated total magnetic moment is calculated to be 2m B for both compounds. The integer value of total magnetic moment implies that K 2 W(Cl/Br) 6 are half metallic ferromagnets. Thus, K 2 W(Cl/Br) 6 are emerging spintronics materials.
For the detailed investigation of band structure results, we have further calculated the total and partial density of states of K 2 W(Cl/Br) 6 as shown in Fig. 4(a-d). It is noticeable from Fig. 4(a and b) that the total DOS in the up spin showed the metallic behavior and the down spin presented the insulating behavior. Therefore, K 2 W(Cl/Br) 6 are half-metallic ferromagnetism (HMF) in nature. To investigate the origin of half-metallic ferromagnetism (HMF) in K 2 W(Cl/Br) 6 , we have further investigated the partial density of states (PDOS) which is shown in     states of W atoms are responsible for introducing HMF in K 2 W(Cl/Br) 6 . In the spin-up case, d-t 2g states of W atoms overlapped with Fermi level (E F ), which produces the metallic nature in K 2 W(Cl/Br) 6 . In the spin-down case, d-t 2g states move deep into the conduction band and thereby leaving a nite separation between the valence band and conduction band and therefore create insulating nature in K 2 W(Cl/Br) 6 . The magnetic behavior of K 2 W(Cl/Br) 6 has also been addressed by an exchange mechanism. The crystal eld due to the Coulomb interaction between W and (Cl/Br) atoms split the 5d states of W into t 2g and e g . The splitting of W-5d states is schematically shown in Fig. 5. It is clearly noticeable from Fig. 5 that the t 2g is triply degenerate (d xy , d yz , d zx ), which has lower energy than the double degenerate e g (d z 2 , d x 2 − y 2 ) states. The splitting of e g states results in the upward and downward shiing of d x 2 − y 2 and d z 2, respectively. At the same time, splitting of t 2g state results in the increase of d xy state energy and decrease of d yz and d zx states energies. The major magnetic response originated from the t 2g state of the W atom. This state is hybridized with the p state of (Cl/Br) via double exchange mechanism. Thus, we conclude that the double exchange mechanism between W atoms via (Cl/Br) atom is responsible for introducing HMF in K 2 W(Cl/Br) 6 .

Thermoelectric properties
Solid-state thermoelectric materials are gaining considerable attraction as novel materials for converting thermal to electrical energy. [38][39][40][41][42] Many mechanical and electronic devices release a huge amount of heat as waste. To take the advantage of wasted heat, efficient thermoelectric materials are needed which can directly convert the wasted heat into electricity. The thermoelectric performance of a material is given by a dimensionless parameter known as a gure of merit (ZT) given by; , where s, S, T, k e , and k l represent the electrical conductivity, Seebeck coefficient, temperature, electrical thermal conductivity, and lattice thermal conductivity, respectively. The material with a large ZT value is considered to be a potential candidate for thermoelectric technology. But, obtaining a large ZT value is challenging due to the strong correlation of these physical properties. According to a recent report, magnetic interaction is one of the suitable approaches to enhance ZT. [43][44][45][46][47] The interaction of charge carriers with local magnetic moments, can enhance the carrier effective mass (m*), consequently thermoelectric power (S 2 s) and corresponding gure of merit (ZT). It is worthy to mention here that such magnetic interaction is present in K 2 W(Cl/Br) 6 . Due to the Fig. 6 The temperature-dependent lattice thermal conductivity (K l ) of K 2 W(Cl/Br) 6 calculated through the Slack equation. presence of magnetic interaction in K 2 W(Cl/Br) 6 , it is expected that these systems will have large ZT values. However, there are no detailed studies on thermoelectric properties of K 2 W(Cl/Br) 6 . In this article, we have calculated the temperature variation of thermoelectric properties in K 2 W(Cl/Br) 6 . For this, we have employed BoltzTrap code. 24 During all calculations, the relaxation time (s) is xed by BoltzTraP code as s = 10 −14 s. We have separately calculated the thermoelectric parameters for spin-up and spindown congurations.
For thermoelectric calculation, the temperature was varied from 200 K to 600 K. The total thermal conductivity (k) is the sum of electronic (k e ) and lattice (k l ) thermal conductivities. A good thermoelectric material should have low k. The temperature variation of lattice thermal conductivity (k l ) was calculated by using the Slack equation. 55 The temperature variation of k l is shown in Fig. 6. It can be seen from Fig. 6 that k l has an inverse relationship with temperature, which is a typical feature of halfmetallic ferromagnetic materials. 48 It is worthy to mention here that K 2 W(Cl/Br) 6 compounds have low k l values.
The amount of charge ow per unit time inside of a compound can be understood from its electrical conductivity (s). The materials are categorized into insulator, semiconductor, and metal-based on their ability of charge ow. 49 A good thermoelectric material should have a large s value. 50 The temperature variation of s is calculated for K 2 W(Cl/Br) 6 as shown in Fig. (a) and Fig. 8(a), respectively. The detailed analyses revealed that the s value for spin-up congurations in K 2 W(Cl/Br) 6 decreases with temperature until achieving the lowest value of 1.97 × 10 5 U −1 m −1 (K 2 WCl 6 ) and 2.25 × 10 5 U −1 m −1 (K 2 WBr 6 ) at 600 K. On the other hand, the values of s for spin-up states in K 2 W(Cl/Br) 6 are almost constant in the entire temperature up to 400 K. However, there was a gradual increase of s is observed above 400 K. This trend of s for up and down channels is typical feature in HMF. 51 The detailed investigation reveals that spin-up channel s is dominant in both cases.
We have also examined the temperature variation of electronic thermal conductivity (k e ) as shown in Fig. 7(b) and 8(b) for K 2 WCl 6 and K 2 WBr 6 , respectively. It is noticeable that k e for spin-up conguration has a direct relation with temperature up to a certain limit where k e increases to 3.41 W K −1 m −1 and 4.1 2 W K −1 m −1 respectively for K 2 WCl 6 and K 2 WBr 6 , at 400 K. Above this temperature, k e decreases gradually for both systems. On the other hand, the values k e in spin-down states of K 2 W(Cl/ Br) 6 are 0.0 in the entire temperature up to 400 K. Above this temperature, k e increases abruptly in both cases. It is worthy to mention here that k e values for spin-down channels are very less compared to spin-up conguration.
The Seebeck coefficient (S) plays an important role to describe the thermoelectric performance. The S is dened by the ratio of a voltage difference to that of a temperature difference. It can also show the capability of a material to generate the thermo-electromotive force from a given temperature gradient. The S can be calculated by the following relation The absolute value of S for spin up conguration in K 2 WCl 6 (K 2 WBr 6 ) increases from 0.67 mV K −1 (1.01 mV K −1 ) at 200 K linearly up to 600 K. This increasing trend in up channel is due to its metallic nature. For metal, there is huge number of free electrons and hence applying a temperature gradient should lead to diffusion of more charge carriers towards cold end. As a result, the potential difference between two ends will increase  Fig. 9 The calculated ZT value of (a) K 2 WCl 6 and (b) K 2 WBr 6 as a function of temperature.
and therefore S will increase. In the spin-down channel, there is an abrupt decrease of S in K 2 WCl 6 (K 2 WBr 6 ) from a very high value of 1267 mV K −1 (1397 mV K −1 ) at 200 K beyond this temperature S decreases gradually. This, decrease trend in spindown channel can be described from eqn (4). The increasing of temperature can enhance the carrier concentration and thereby decreasing the value of S. The detailed investigation depicts that spin down channel Seebeck coefficient is dominant in both cases.
In addition, we have calculated the thermoelectric power factor (PF) as shown in Fig. 7(d) and Fig. 8(d). It is noticeable that the PF of spin-up conguration in K 2 WCl 6 (K 2 WBr 6 ) increases from 0.02 × 10 9 W mK −1 (0.01 × 10 9 W mK −1 ) at 200 K linearly up to 600 K. On the other hand, the PF of spin-down state is found to be 0.0 mW mK −2 for both systems in the entire temperature up to 400 K. Beyond this temperature, the PF increases gradually.
Motivated by large S, s, and PF with low k e , we have calculated the temperature variation of thermoelectric gure of merit (ZT) as Fig. 9(a and b). The detailed analysis revealed that the ZT for spin upstate in K 2 WCl 6 (K 2 WBr 6 ) increases from 0.001 (0.001) at 200 K to 0.05 (0.041) at 600 K. On the other hand, ZT values of a spin-down channel in K 2 WCl 6 (K 2 WBr 6 ) gradually decrease from 0.99 (0.884) at 200 K to 0.95 (0.845) at 600 K. It is noticeable that spin down channel ZT value is dominant in both cases. The higher ZT value of the spin-down channel is due to its very low thermal conductivity. Also, the large ZT values demonstrated that K 2 W(Cl/Br) 6 are potential thermoelectric materials.

Conclusion
To summarize this article, we have systematically investigated the structural, electronic, and thermoelectric properties of K 2 W(Cl/Br) 6 . The negative formation energy leads to the thermodynamic stability of K 2 W(Cl/Br) 6 . In addition, the spinpolarized band structure and density of states calculations revealed the presence of half-metallic character with higher T c values, which are 613 K and 597 K for K 2 WCl 6 and K 2 WBr 6 , respectively. Thus, K 2 W(Cl/Br) 6 are potential candidates for spintronics application. Furthermore, the origin of half-metallic ferromagnetism is discussed with the double-exchange mechanism. Finally, we have computed the temperature variation of k l ,s, k e , S, PF, and ZT. The higher ZT values for spin-down channels have resulted from ultra-low k e , and high PF. In short, K 2 W(Cl/Br) 6 are potential thermoelectric and spintronic materials.

Data availability
The datasets produced for current study would be available from Mr Mukaddar Sk on reasonable request.

Conflicts of interest
The authors have no conict of interest.