以第一原理計算研究鍶與碳族元素為基底的雙鈣鈦礦中的半金屬材料

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(1)國立臺灣師範大學物理學系研究所博士論文 Department of Physics National Taiwan Normal University. Doctoral Dissertation. 以第一原理計算研究鍶與碳族元素為基底的雙鈣鈦礦中的半金屬材料. First-Principle Studies of Half-Metallic Materials in Sr and IV-Group Element Based on Double Perovskites Structure Compounds. 劉雲平 Yun-Ping Liu. 指導教授:王銀國博士 Advisor: Yin-Kuo Wang Ph.D.. 中華民國 102 年 5 月 May, 2013. 1.

(2) 感謝曾經幫助、鼓勵與關心我的人願畢生的創作奉獻給世人. 2.

(3) Contents 1 Introduction ------------------------------------------------------------------------------------8 2 Density Functional Theory (DFT) and computational methods --------------------10 2.1 Born-Oppenheimer approximation 2.2 Density Functional Theory (DFT) 2.2.1 Hohanberg-Kohn theorems 2.2.2 Kohn-Sham equations 2.2.3 Exchange-correlation functionals 2.3 Projector Augmented Wave (PAW) method (VASP package) 2.4 L(S)DA(GGA)+U method 3 Half-Metallic (HM) Materials ---------------------------------------------------------------21 3.1 What is HM Materials? 3.2 The review of HM compounds research 3.3 Double perovskites structure and magnetic phase 3.3.1 Double perovskites structure 3.3.2 Structural optimization 3.3.3 Magnetic States 3.4 Calculation procedure 3.5 The formation of HM: Double-exchange couplings 4 HM Materials in Sr2BB′O6 (B/B′ = transition metal) ----------------------------------28 4.1 The searching groups 4.2 Group 1: 3B/4B transition metal pair with Cr, Co, V and Fe 4.3 Group 2: 6B/7B transition metal pair with Co, Cu and Ni 4.4 Group 3: 7B/8B/1B transition metal pair with Zn 4.5 Other Possible Space Group 5 HM Materials in A2BB′O6 (A = IVA group, B=Fe and Cr, B′ = Mo, Re and W) -59 5.1 IVA group elements on the A(Sr)-site position 5.2 A2FeMO6 5.3 A2CrMO6 6 Summary and Publication List --------------------------------------------------------------77 7 Bibliography ------------------------------------------------------------------------------------80 3.

(4) 論文摘要在這篇論文中，我們以鍶基底的雙鈣鈦(Sr-based Double Perovskites)結構，以第一原理計算找尋可能存在的半金屬。在Sr2BB′O6中(B,B’=過渡金屬)找到三個系列的半金屬組合，另外也A2Fe(Cr)MO6 (A=IVA族元素, M=Mo, Re and W)找到半金屬的一系列候選材料。我們使用的計算程式為VASP根據密度泛函(DFT)理論來計算材料的結構最佳化，從最初的四種磁相態出發：鐵磁(FM)、亞鐵磁(FiM)、反鐵磁(AFM) 與無磁性(NM)，其中使用廣義梯度近似(GGA)以及考慮庫倫排斥效應(GGA+U)。在第一章中，我們簡單介紹磁性半金屬過去的研究發展，以及我們找到那些可能的磁性磁半金屬候選者。在第二章中，我們簡單介紹相關的理論及計算方法，包括Born-Oppenheimer 近似、密度泛函理論(DFT)。其中包括Hohangberg-Kohn理論、Kohn-Sham方程式、交換關連效應、侷域密度近似(LDA)與廣義梯度近似(GGA)。使用的計算程式為 VASP ，其使用擴增平面波方式來計算。並且最後介紹庫倫電子關聯效應 (LDA/GGA+U)。在第三章中，我們簡單介紹磁性半金屬的特性，並且對過去十幾年來關於磁性半金屬材料的研究發展做一些簡介。並且詳細介紹雙鈣鈦結構以及初始的四種磁相態。最後，我們介紹整個研究的計算流程與計算的設定參數以及造成半金屬的重要物理機制-雙交換作用(double exchange)。在第四章中，將會詳細介紹在緦基底的雙鈣鈦結構(Sr2BB′O6, B,B’=過渡金屬) 中，我們找尋到三個系列的半金屬候選人，此分類的方式是根據BB′離子在週期表上分部的組合。第一系列為： Sr2Cr(Co)B′O6 (B′=Sc, Y, La, Ti, Zr 與 Hf) 以及 Sr2V(Fe)B′O6 (B′= Zr 與Hf)。在第一系列中最有可能成為半金屬的為Sr2CrScO6、 Sr2CrLaO6 、 Sr2CrTiO6 、 Sr2VZrO6 以及 Sr2VHfO6 這些材料。第二系列為 Sr2BB′O6 (B = Co, Cu 與Ni; B′ = Mo, W, Tc 與Re)，最有可能成為半金屬的是Sr2FeTcO6 、 Sr2CoWO6 與 Sr2NiTcO6。第三系列為Sr2ZnBO6 (B=Mn, Tc, Re, Fe, Ru, Os, Co, Ni, Pd 與Au)，其中Sr2ZnMnO6與Sr2ZnPdO6是半金屬材料的最佳選擇。整體的篩選是建構在比較不同磁相態的能量，並且同時在GGA與GGA+U兩種不同的情況皆為穩定才能脫穎而出。. 4.

(5) 在第五章中，我們基於Sr2FeMoO6可以將緦(Sr)置換為IVA族元素的想法，來發展出A2Fe(Cr)MO6(A=IVA族元素, M=Mo, Re 與W)的半金屬系列材料。這樣的想法是基於IIA(s2)族元素與IVA(p2)的外層價電子非常相似的緣故。結果顯示在A為錫(Sn) 與鉛(Pb)是較為穩定並且較有可能被合成的半金屬候選材料。. 最後，我們總結所有理論預測結果並且重述造成半金屬的物理機制。我們希望這篇論文可以在尋找半金屬材料方面的研究提供一些有用的訊息，希望對於未來合成半金屬材料的實驗能有所幫助。關鍵字：第一原理、半金屬材料、雙鈣鈦結構. 5.

(6) Abstract In this thesis, we thoroughly investigated three possible candidates series of half-metallic (HM) in the double perovskites structure Sr-based double perovskites Sr2BB′O6 (BB′=transition metal ions) and A-site substitution double perovskites A2Fe(Cr)MO6 (A=IVA group elements, M=Mo, Re and W). The calculation is based on the density functional theory (DFT) with full-structure optimization by generalized gradient approximation (GGA) and consideration of the strong correlation effect (GGA+U) and started with 4 types of initial magnetic states, i.e. ferromagnetic (FM), ferrimagnetic (FiM), antiferromagnetics (AF) and nonmagnetic (NM) using full-potential projector augmented wave (PAW) method within conjugate-gradient (CG) method implemented in VASP package (code). In the first chapter, we briefly introduced the previous researches of HM compounds and what series investigation that we had done. In chapter 2, we introduced the Born-Oppenheimer approximation, DFT (including Hohangberg-Kohn theorems, Kohn-Sham equations, exchange-correlation functional, local (spin) density approximation (L(S)DA) and generalized-gradient approximation (GGA)), as well as computational methods we used, including Projector Augmented Wave (PAW) method in VASP code and electron correlation effect (+U calculation). In chapter 3, we introduced its characteristics and properties of HM materials with reviewing the different structures that had been discovered. The detail of double perovskites structure and the initial magnetic states configurations are also introduced in this chapter. In the end, the calculation procedure with detailed setting parameter and double exchange mechanism of causing HM characteristics are schematic diagramed. In chapter 4, for Sr-based double perovskites Sr2BB′O6 (BB′=transition metal ions), we classified the possible HM compound into 3 groups according to the electronic configuration of the BB′ ion pairs. In Group 1: Sr2Cr(Co)B′O6 (B′=Sc, Y, La, Ti, Zr, and Hf) and Sr2V(Fe)B′O6 (B′= Zr and Hf), the most promising candidates are Sr2CrScO6, Sr2CrLaO6, Sr2CrTiO6, Sr2VZrO6 and Sr2VHfO6. In Group 2: Sr2BB′O6 (B = Co, Cu, and Ni; B′ = Mo, W, Tc, and Re), Sr2FeTcO6, Sr2CoWO6 and Sr2NiTcO6 are the most possible 6.

(7) HM candidates. And for Group 3: Sr2ZnBO6 (B=Mn, Tc, Re, Fe, Ru, Os, Co, Ni, Pd, and Au), the best choices for HM materials are Sr2ZnMnO6 and Sr2ZnPdO6. The selection is based on the energy differences between F(i)M and AF state in both GGA and GGA+U scheme. In chapter 5, for A-site substitution double perovskites A2Fe(Cr)MO6 (A=IVA group elements, M=Mo, Re and W), based on Sr2FeMoO6, we substituted the Sr ion with IVA group elements according to the similar valence electrons noting as IIA(s2) and IVA(p2). The results shows that choosing A= Sn and Pb can be able to synthesize stable HM double perovskites compounds. In the last chapter, we will make a summary of our work including the research method, the main results and the mechanism of causing HM compounds. We hope that this thesis on the searching HM compounds in double perovskites structures are useful for experimental research and bring up the research upsurge of HM materials.. Keyword: First-principle, Half Metallic Materials, Double Perovskites Structures. 7.

(8) Chapter 1 Introduction Half-metallic (HM) materials are important for spintronic devices. In one of the spin channels, an energy gap exists and indicates insulating behavior; in the other spin channel, metallic behavior occurs. Thus, the following three characteristic properties can be observed: (1) 100% spin polarization at the Fermi level (EF); (2) quantization of magnetic moment; and (3) zero spin susceptibility. Among these phenomena, HM materials offer potential technological applications, such as computer memory, magnetic recording, single-spin electron source, and high-efficiency magnetic sensors [1–3]. Therefore, the search for half-metallic compounds is a highly popular topic. HM materials include spinel Fe3O4 [2], rutile CrO2 [5], double perovskites Sr2FeMoO6 [3,5–7], La2VTcO6 [8] and La2VCuO6 [8],doped perovskite structure manganese oxide Ln0.5Ca0.5MnO3 [9] and Ln0.7Sr0.3MnO3 [1,10], spinel FeCr2S4 [11], and Mn-doping GaAs [12,13].. To find more suitable HM candidates, ordered double perovskites A2BB′O6 can be used as a basis structure because a variety of options are available for substituting the A-site or B-site elements. On the B-site elements, transition metal ions are used because of their diverse electronic configurations in d-orbitals. On the A-site elements, alkaline earth or rare-earth ions, such as Ca, Sr, Ba, and La, are often used as HMs. The differences in size and valence between the B and B′ ions are crucial in controlling the physical properties [14,15]. In Sr-based double perovskites Sr2BB′O6, research has shown that Sr2FeMoO6 [3,5–7], Sr2FeReO6 [5,16], Sr2FeWO6 [6], Sr2CrMoO6 [7], Sr2CrReO6 [16,17], and Sr2CrWO6 [5,18] are HM materials. These compounds exhibit cyclical behavior in the periodic table, which are controlled by the constitution of BB′ ion pairs. Therefore, we expand the research area following previous results and perform calculations of Sr-based double perovskites Sr2BB′O6. The calculation is based on the density functional theory (DFT) with full-structure optimization by generalized gradient approximation (GGA) and consideration of the strong correlation effect (GGA+U) and started with 4 types of initial magnetic states, i.e. ferromagnetic (FM), ferrimagnetic (FiM), antiferromagnetics (AF) and nonmagnetic (NM). As to the results, it is found that 8.

(9) these HM candidate compounds exhibit cyclical behavior in the periodic table, which are controlled by the constitution of BB′ ion pairs. As to the result, we classified the possible HM compound into 3 groups according to the electronic configuration of the BB′ ion pairs. The 3 groups are: Group 1: 3B/4B transition metal pair with Cr, Co, V and Fe Sr2Cr(Co)B′O6 (B′=Sc, Y, La, Ti, Zr, and Hf) and Sr2V(Fe)B′O6 (B′= Zr and Hf) Group 2: 6B/7B transition metal pair with Co, Cu and Ni Sr2BB′O6 (B = Co, Cu, and Ni; B′ = Mo, W, Tc, and Re) Group 3:7B/8B/1B transition metal pair with Zn Sr2ZnBO6 (B=Mn, Tc, Re, Fe, Ru, Os, Co, Ni, Pd, and Au). In order to find more suitable candidates of such magnetic electrodes in ordered double perovskites A2BB’O6, the A-site elements often been placed alkaline-earth or rare-earth ions such as Ca, Sr, Ba and La. In this work, by fixing the BB’ combinations as Fe(Cr)M (M=Mo, Re and W), we present the IVA group elements sitting on the A-site ion position, which is originated form the alkaline-earth elements (Ca, Sr and Ba) having the similar valence electrons with the IVA group elements noting as IIA(s2) and IVA(p2). The common exception in HM materials for A-site IVA elements is Carbon and Silicon, because covalent bond is too strong for and the valence is +4 rather than +2 that the double perovskites structure cannot be synthesized. We hope that these detailed calculations on the searching HM compounds in double perovskites structures are useful for experimental research and bring up the research upsurge of HM materials.. 9.

(10) Chapter 2 Density Functional Theory (DFT) and computational methods. Density Functional Theory (DFT) is a quantum mechanical modeling method, which is used in physics and chemistry to investigate the electronic structure of many-body systems, including atoms, molecules, and the condensed-matter. The name of “density functional theory” comes from the electron density calculation by using functional (functions of another function). DFT is the most popular and versatile method and it is applied in condensed-matter physics, computational physics and chemistry. In this chapter, we briefly introduce the Born-Oppenheimer approximation, Density Functional Theory (DFT), including Hohangberg-Kohn theorems, Kohn-Sham equations, exchange-correlation functional, local (spin) density approximation (L(S)DA) and generalized-gradient approximation (GGA). As well as computational methods we used, including Projector Augmented Wave (PAW) method implemented in the VASP code and electron correlation effect (+ U calculation),. 2.1 Born-Oppenheimer approximation There are nuclei and electrons in the system. Both nuclei and electrons parts constitute the total Hamiltonian of the system, which is. Hˆ = Hˆ e + Hˆ e!N + Hˆ N. (2.1). !2 1 1 Hˆ e = Tê + Uˆ e = !" # r2i + " 2 i$ j ri ! rj i 2m. (2.2). ZI Hˆ e!N = Uˆ e!N = " i,I ri ! RI. (2.3). !2 2 1 ZZ Hˆ N = TˆN + Uˆ N = !" # Ri + " I J 2 I$J RI ! RJ I 2M I. (2.4). 10.

(11) The ri is the coordinate of the electron i with mass m in the material, and RI is the coordinate of the nucleus I with mass MI and charge ZI. Hˆ e and Hˆ N represent the Hamiltonian of electrons and nuclei, respectively. Hˆ e!N represents the interactions between electrons and nuclei. Since MI is much heavier and slower than m, it can be thought as an adiabatic approximation motion [19]. For the electrons, Nuclei can be regard as the positive charge background in this approximation motion, i.e. RI, RJ are fixed positions. Under the Born-Oppenheimer approximation [20], the total Hamiltonian of the system is much simplifier. TˆN is zero and Uˆ N is constant. Thus, Hˆ can simply be rewrite as:. !2 1 1 Hˆ = Tˆ + Uˆ + Vêxt = !" # r2i + " + "Vext (ri ) 2 i$ j ri ! rj i 2m i. (2.5). The potential of the electron-nuclei interaction term ( Hˆ e!N ) has been replaced by an external potential Vext (ri ) in Eq.2.5, which includes more general fixed background potentials, such as the electric field.. 2.2 Density Functional Theory (DFT) Density functional theory (DFT) is the base idea of many-particle system, which can be thought as a functional of ground state particle density n0 (r) . This is originated form the Hohanberg-Kohn theorems [21]. W. Kohn and L. J. Sham [22] first suggested that the exchange and correlation interactions in the system could be approximated as functions of local density. This is so-called local (spin) density approximation (L(S)DA). The advanced improvements include the gradient of the density, i.e. generalized-gradient approximation (GGA)[23, 24].. 11.

(12) 2.2.1 Hohanberg-Kohn theorems First theorem: A given ground state density n0 (r) , and the relative external potential Vext (r) is determined uniquely, except for a constant. To prove this theory, first, we consider two different ground state wavefunctions !1 and ! 2 with different external potentials Vext1 (r) and Vext 2 (r) both have the same density n0 (r) . Two different external potential also lead two different Hamiltonian Hˆ 1 and Hˆ 2 . It is obviously that ! is not the ground state of Hˆ . It implies: 1. 2. E1 = !1 Hˆ 1 !1 < ! 2 Hˆ 1 ! 2 = ! 2 Hˆ 2 ! 2 + ! 2 Hˆ 1 " Hˆ 2 ! 2. (2.6). = E2 + # d 3rn0 (r) [Vext1 (r) "Vext 2 (r)] similarly,. E2 = ! 2 Hˆ 2 ! 2 < !1 Hˆ 2 !1 = !1 Hˆ 1 !1 + !1 Hˆ 2 " Hˆ 1 !1. (2.7). = E1 + # d 3rn0 (r) [Vext1 (r) "Vext 2 (r)]. Taking Eq.2.6 and Eq.2.7 summed together, it becomes E1 +E2 < E2 +E1. This result contradicts with initial assumption and proves that there cannot exist different external potentials, which share the same ground state density. That is, this theorem shows an one-to-one mapping between external potential and the ground state density, and it suggests that the density n(r) can be a good functional variable in quantum many-body system. It shows that the density have as much information as the wavefunction. The second theorem is that the functional E[n] can be defined for the density n in certain external potential Vext (r) , and the global minimum of E[n] is E[n0], where n0 is ground state density. Here, the proof of the theorem follows the way established by Levy and Lieb [25, 26]. Since there can be a class of different wavefunctions those have the same density n(r), the total energy for such density is. E = ! Tˆ + Uˆ ! + 12. 3. " d rn(r)V. ext. (r). (2.8).

(13) Levy and Lieb[41, 42]. Since there can be a class of different wavefunctions those have the same density n(r), the total energy for such density is. � ˆ ˆ E = �Ψ|T + U |Ψ� + d3 r n(r)Vext (r). (2.8) By varying the wavefunction and keeping the same density distribution, the energy By varying the wavefunction and keeping the same density distribution, the energy in Eq.2.8 can achieve the minimum, and there is the functional E[n]: in Eq.2.8 can achieve the minimum, and there is the functional E[n]: � ˆ ˆ E[n] = min �Ψ|T + U |Ψ� + d3 r n(r)Vext (r) Ψ→n(r) � = F [n] + d3 r n(r)Vext (r) = F [n] + Eext [n],. (2.9). (2.9) where the internal energy functional F [n] in Eq.2.9 is an ”universal functional”, which only depends upon the density function n(r). It is obviously that the functional E[n] has a global minimum while n = n0 . This statement not only proves the theorem butinternal also shows thatfunctional the universal be foundfunctional”, by minimizing the internal Where the energy F[n]:functional in Eq.2.9Fis[n]ancan ”universal energy with fixed density function. which only depends on the density function n(r). It is obvious that the functional E[n] has In the frame of DFT, the physical observables in quantum many-body systems can a global minimum while functionals. n = n0. ThisThat statement not only proves the theorem but also be density is shows that the universal functional F[n] can be found by minimizing the internal energy. ˆ with fixed density function. �Ψ|O|Ψ� = O[n(r].. (2.10). The idea of DFT also can be extended to spin-polarization system by including the In the frame of DFT,ofthe physicalFor observables in quantum many-body systems canspins are aligned spin degree freedom. the collinear spin system, all of the particle the quantization also be densityon functionals. That is: axis, which often determined by the external magnetic field. In that case the density function n(r) can be separated into n(r, σ =↑) and n(r, σ =↓). The ! Oˆ ! = O [ n(r)]. 5. (2.10). The idea of DFT can be extended to spin-polarization system by including the spin degree of freedom. For the collinear spin system, all of the particle spins are aligned on the quantization axis, which often determined by the external magnetic field. In that case the density function n(r), can be separated into n(r, σ =↑) and n(r, σ =↓). Thus the energy functional becomes E = E[n(r, ↑) , n(r, σ↓)], and the ground state of spin-polarization system can be estimated. For non-collinear case, there is no global quantization axis, but the local quantization axis still can be defined.. 13.

(14) 2.2.2 Kohn-Sham equations 1965, Kohn and Sham proposed an auxiliary single-particle-like (non-interactive) Schrödinger equation to solve the original interactive many-body system. In this way, the density of the many-body system can be thought as the total density of the occupied states of non-interactive auxiliary system. For frozen-nuclei system, N-electrons auxiliary Hamiltonian Hˆ is s. !2 2 ˆ ˆ ˆ ˆ ˆ H s = Ts + U s + Vext = ! " r + U s (r, ! ) + Vêxt 2m. (2.11). The subscript denotes that these operators only act on the non-interactive single particle system. For each spin channel σ, the eigenstates of the system !i (" ) (called Kohn-Sham orbitals) are filled by N(! ) electrons form the lowest level to the highest occupied state. Thus, N (! ). N = N(!) " N(#);n(r) = $ n(r, ! ) = $ $ "i (r, ! ) i,!. !. 2. i=1. (2.12). In Hartree-Fock approximation, the formalism of Eq.2.11 is orbital-independent, and it is easier to implement. The single-particle kinetic energy Ts is written as:. Ts = !. N (" ) !2 !2 2 !i (r, " ) " r2 !i (r, " ) = ! d 3r !i (r, " ) $ $ # 2m 2m " i=1. (2.13). The Uˆ s (r, ! ) term also implies the Hartree term EH and exchange-correlation term. E xc in such non-interactive system. Their definitions are. EH !. 1 ! 2 i, j,! i ,! j. 1 = 2. 3. 3. * i. !! d rd r!! (r, ! )! (r!, ! i. j. j. ). 1 !i (r, ! i )! j (r!, ! j ) r ! r". n(r)n(r!) "" d rd r! r ! r" 3. (2.14). 3. E xc ! F [ n ] " Ts [ n ] " EH [ n ]. 14. (2.15).

(15) Within the energy Eext from external potential, the total energy functional of the auxiliary system Es[n] is:. Es [ n(r)] = ! !i (r, " ) Hˆ s !i (r, " ) i,". = Ts [ n(r)] + EH [ n(r)] + E xc [ n(r)] + Eext [ n(r)] N (" ) !2 1 2 =" d 3r !i (r, " ) + ! ! # 2m " i=1 2. 3. 3. ## d rd r$. n(r)n(r$) + E xc [ n(r)] + # d 3rn(r)Vext (r) r " r$. (2.16) Since this single-particle system is expected to modal the original many-body system, to search the ground state density n0 in the many-body system is equivalent to find minimum Es in the density function. Applying the method to Eq.2.16, it gives Kohn-Sham equation:. Hˆ s!i (r, " ) = #i," !i (r, " ). " !Ts [ n ] !EH [ n ] !E xc [ n ] !Eext [ n ] % + + + $ '"i (r, ! ) = #i,! "i (r, ! ) # !n(r, ! ) !n(r, ! ) !n(r, ! ) !n(r, ! ) & " !2 2 % ) r +VH [ n ] +Vxc! [ n ] +Vext '"i (r, ! ) = #i,! "i (r, ! ) $( # 2m &. (2.17). The complicated internal energy in quantum many-body system has been transformed into single-particle kinetic energy and potentials in Eq.2.17. Similar to the Hartree-Fock approximation, we use self-consistent (SCF) calculations to find the ground state density of Eq.2.16 and Eq.2.12 (see Fig 2.1). The calculation starts with an initial density function, then it generates a Hamiltonian of Kohn-Sham equation. A serial of Kohn-Sham orbitals are solved by the equation. The orbitals form new density mixed with last density, and the new density becomes the new trial density of SCF calculation. The ground state density is achieved while the SCF calculation converges. Although the Kohn-Sham equation can produce the orbitals (eigenstates) and their energies (eigenvalues) for both occupied and unoccupied states of the system, only the 15.

(16) total density is meaningful in principle. Since Kohn-Sham orbitals are fictive non-interacting states (quasi-particles states). The band gap of the insulator or semiconductor systems cannot be predicted precisely, since energy functional derivative is discontinued between the case of electron-addition on the lowest unoccupied state and the electron-removal from the highest occupied state [27].. Fig. 2.1. Figure 2.1: Schematic flow chart diagram of self-consistent calculation.. Schematic flow-chart diagram of self-consistent calculation.. 8 16.

(17) 2.2.3 Exchange-correlation functionals Although the original quantum many-body system has been simplified to the Kohn-Sham equation (Eq.2.17), which is the single-particle-like Schrödinger equation, unfortunately, the exchange-correlation potential V !xc is not that easy to find out, and this is the major problem with DFT. The physical meaning of the exchange-correlation potential is the difference between the many-body internal energy F[n] and the single-particle contributions Ts [ n ] + EH [ n ] . Since the internal energy functional is a universal functional, it implies that the general form of the exchange-correlation potential can be fitted by other results, which are based on the exact many-body calculations, such as quantum Monte-Carlo simulations for uniform electron gas. Several approximations of exchange- correlation functionals have been developed. The most widely used approximation is the local-density approximation (LDA), where the functional depends only on the density at the coordinate where the functional is: E xcLDA [ n ] = ! ! xc (n)n(r)d 3r. (2.18). Where ! xc (n) is exchange-correlation energy density of uniform electron gas.. The local spin-density approximation (LSDA) is a straightforward generalization of the LDA to include electron spin: E xcLSDA [ n !, n "] = # ! xc (n !, n ")n(r)d 3r. (2.19). Generalized gradient approximations (GGA) are still local but also take account the gradient of the density at the same coordinate: E xcGGA [ n !, n "] = $ ! xc (n !, n ", # n!, # n" )n(r)d 3r. (2.20). The generalized-gradient approximation improves the L(S)DA functionals by including the density gradient term !n . The GGA functionals are expressed as the functions of n and !n , or saying F[n] = f (n, !n) . Perdew, Burke and Ernzerhof proposed a simplest GGA functional, GGA-PBE[28].. 17.

(18) 2.3 Projector Augmented Wave (PAW) method (VASP package) There are many methods to be developed for solving electronic structure problem efficiently, such as Pseudo-Potential (PP), Linear Combination of Atomic Orbital (LCAO), Linear Miffin Tin Orbital (LMTO), Linear Augmented Plane Wave (LAPW), Projector Augmented Wave (PAW) methods etc.… Since we used the Vienna Ab-initio Simulation Package (VASP) for structure optimization calculations in double Perovskites Structure Compounds material, we will briefly introduce the PAW method [29, 30]. PAW method is an approach that generalizes both PP and LAPW methods. The basic idea of PAW method is to divide the wavefunction into two parts: a partial wave expansion within an atom-centered sphere, and an envelope function outside. The two parts are matched smoothly at the sphere edge. The key point of the PAW method is to seek a linear transformation [29, 31] T, which maps computationally convenient pseudo ! ⟩ to true wave functions | ! ⟩. We are looking for smooth pseudo wave wave functions | ! functions, which are rapidly convergent planewave expansion. With such a PAW method is an approach that generalizes both PP and LAPW methods. The transformation, canofexpand the mehtod pseudo wave functions a convenient basicwe idea the PAW is to divide the into wavefunction intobasis two set, parts : a partial wave expansion withinallanphysical atom-centered sphere, and an envolope function outside. The such as plane waves, and evaluate properties. two parts are then matched smoothly at the sphere edge. The keypoint of the PAW method is to seek a linear transformation[45, 47] T which maps computationally conve˜ ! to true wave functions |Ψ�. We are looking for smooth nient pseudo wave functions |Ψ� (2.21) !n = T ! pseudo wave functions, which aren rapidly convergent plane-wave expansion. With such a transformation, we can expand the pseudo wave functions into a convenient basis set, such as plane waves, and evaluate all physical properties. The index n is the label for a quantum state, consisting of a band index and possibly a k-vector When we evaluate physical quantities, i.e. to evaluate ˜index. |Ψand � =spin T |Ψ � n. n. (2.21). expectation values of an operator A, which can be expressed in terms of either the true or The index n is the label for a quantum state, consisting of a band index and possibly the pseudo wave functions: a k-vector and spin index. When we evaluate physical quantities, i.e. to evaluate expectation values of an operator A, which can be expressed in terms of either the true or the pseudo wave functions: � � ˜ n |T † AT |Ψ ˜ n� �A� = fn �Ψn |A|Ψn � = fn �Ψ (2.22) (2.22) n. n. In the representation of pseudo wave functions we need to use transformed operators A˜ = T † AT . Remind that this equation only holds for the valence electrons. The core In the representation of pseudo waveasfunctions we need to use electrons are treated differently will be shown below. The transformed transformation takes us from of PP that for of the augmented wave methods, operators A!conceptionally =T†AT. Remind that the thisworld equation onlytoholds valence electrons. The which deal with the full wave functions. We will see that our pseudo wave functions, which are core electrons are treated differently parts as willofbethe shown transformation takes usinto the wave simply the plane-wave fullbelow. wave The functions, and translate of theofpseudopotential approach.wave methods, which deals with conception functions from the world PP to that of augmented Expectation values can be obtained either from the reconstructed true wave functions 18 functions[47]. or directly from the pseudo wave �A� =. �. fn �Ψn |A|Ψn �+. Nc �. �φcn |A|φcn �. =. �. ˜ n |T † AT |Ψ ˜ n �+ fn �Ψ. Nc �. �φcn |A|φcn � (2.23).

(19) |Ψn � = T |Ψn �. (2.21). two parts then matched smoothly at the edge. The keypoint of theconvePAW method is toare seek a linear transformation[45, 47]sphere T which maps computationally method seekfor a linear transformation[45, 47] which maps computationally conve˜ state, The index npseudo is is thetolabel a quantum consisting of aTband index andare possibly nient wave functions |Ψ� to true wave functions |Ψ�. We looking for smooth ˜ nient wave functions Ψ� true wave functions |Ψ�. are looking With for smooth a k-vector andpseudo spin index. When we |evaluate physical quantities, i.e. We to expansion. evaluate pseudo wave functions, which are to rapidly convergent plane-wave such pseudo wave functions, which are rapidly convergent plane-wave expansion. With such expectation values of an operator A, which can be expressed in terms of either the true a transformation, we can expand the pseudo wave functions into a convenient basis set, or the pseudo wave functions: a transformation, weand canevaluate expand the pseudo wave functions into a convenient basis set, such as plane waves, all physical properties. � � such as plane waves, and evaluate all physical properties. † pseudo the full We will see simply the ˜ n |Tour ˜ n � wave functions, which are (2.22) �A�wave = functions. fn �Ψn |A|Ψ fn �that Ψ AT |Ψ n� =. n ˜full plane-wave nparts |Ψn � of = the T |Ψ (2.21) n � wave functions, and translate into the wave functions of the ˜ |Ψ � = T | Ψ � (2.21) n n InPseudo-Potential the representation of pseudo wave functions we need to use transformed operators approach. † indexthat n isthis the equation label for only a quantum consisting of a band A˜ = T AT . The Remind holds forstate, the valence electrons. Theindex core and possibly The index n is the label for a quantum state, consisting of a band index possibly a k-vector and spin index. When we evaluate physical quantities, to evaluate electrons are treated differently as will be shown below. The transformation takesi.e. us and a k-vector andworld spin index. When we evaluate physical quantities, to the evaluate expectation values of of an operator A, can be expressed in which terms deal ofi.e. either true conceptionally from the PP to that of which augmented wave methods, expectation values ofWe an will operator A, our which can be expressed in which terms are of either the true thewave pseudo wave functions: with theor full functions. see that pseudo wave functions, values be full obtained either from reconstructed or plane-wave the pseudo wavecan functions: simply Expectation the of the wave andthe translate into the true wavewave �parts � functions, † ˜ ˜ �A� = from |A|Ψ = �fn �Ψn |T[31]. AT |Ψn � (2.22) �fn �Ψ functions pseudopotential approach. n � wavefunctions functionsoforthe directly thenpseudo ˜ nreconstructed ˜ n � true wave functions = ncan fbe � = n from fn �the Ψ |T † AT |Ψ (2.22) n �Ψ n |A|Ψneither Expectation�A� values obtained. n n or directly from the pseudo wave functions[47].. In the representation of pseudo wave functions we need to use transformed operators In=the representation of pseudo wave functions wefor need use transformed operators Nc that N c to A˜� T † AT . Remind this equation only holds the valence electrons. The core � � � † c c † c c ˜ ˜ ˜ A =fnT�ΨAT . Remind that this equation only holds for the valence electrons. The �A� =electrons |A|Ψ �+ �φ |A|φ � = f � Ψ |T AT | Ψ �+ �φ |A|φ � (2.23) nare treated n n be n shown below. n differently as will The transformation us n n n n (2.23) takescore n n=1 n n=1 electrons are treated differently be shown below. The transformation takes conceptionally from the world of as PPwill to that of augmented wave methods, which dealus conceptionally from ofstates PP that augmented wave methods, which deal theoccupations full wave functions. We will to see that pseudo wave are where fnwith are the ofthe the world valence and Nc of is our the number of corefunctions, states. c f Where are the occupations of the valence states and N is the number of core with the full wave functions. We will see that our pseudo wave functions, which are c simply the plane-wave parts of the full wave functions, and translate into the wave n over the valence states, and second over the core states |φn �. The first sum runs simply the plane-wave parts of approach. theforfull wavefunction functions, and Now functions we can decompose the matrix element a wave Ψ into its translate individual into the wave of the pseudopotential states. functions of the pseudopotential approach. contributions. Expectation values can be obtained either from the reconstructed true wave functions �the � either from the reconstructed true wave functions Expectation values can be obtained or directly wave l pseudo ˜ from ˜ l )|A| ˜ + functions[47]. ˜ l � )� �Ψ|A|Ψ� = � Ψ + (Ψ − Ψ Ψ (ΨlR� − Ψ (2.24) R pseudo R R or directly from the wave functions[47]. � R N N c c R � � � � c c † c c ˜ ˜ N N c c�φ |A|φ � (2.23) �A� = f �Ψ |A|Ψ �+ �φ |A|φ � = f � Ψ |T AT | Ψ �+ �they �n n � n n n n n aren n n Only the first part � is evaluated explicitly, c c † c c ˜ ˜ �A� = f f n n �Ψn |A|Ψn �+n=1 �φn |A|φn � = n n �Ψn |T AT |Ψn �+n=1 �φnc|A|φn � (2.23) The first sum runs over � the valence states, and second over the core states | ! n . l n=1 n=1 ˜ n Ψ� ˜ + ˜ l |A|Ψ ˜ l �)n �Ψ|A|Ψ� = �Ψ|A| (�ΨlR |A|Ψ − �valence Ψ (2.25) R R where ofR �the statesfunction and Nc Ψ is into the number of core states. n are the occupations Now we can fdecompose the matrix element for a wave its individual R c where are runs the occupations of thestates, valence states andover Nc is the number states. The firstfnsum over the valence and second the core statesof|φcore n �. c contributions. first sum over thethe valence states, andfor second the core states the rests The parts arewe neglected, because they vanish for sufficiently localover operators as into long Now canruns decompose matrix element a wave function Ψ its|φ individual n �. as the partial wave expansion is converged. Here we remind that for truly nonlocal Now we can decompose the matrix element for a wave function Ψ into its individual contributions. operatorscontributions. the rests parts would have � to be considered explicitly. � ˜ + �(Ψl − Ψ ˜ l )|A|Ψ ˜ + �(Ψl � − Ψ ˜ l � )� (2.24) (2.24) �Ψ|A|Ψ� = �Ψ R R Rl Rl l l ˜ ˜ ˜ ˜ �Ψ|A|Ψ� = �Ψ + R (ΨR − (2.24) 10ΨR )|A|Ψ + R� (ΨR� − ΨR� )� R�. R. Only part is evaluated Only the firstthe partfirst is evaluated explicitly,explicitly, they are they are Only the first part is evaluated � explicitly, they are ˜ Ψ� ˜ + �(�Ψl |A|Ψl � − �Ψ ˜ l |A|Ψ ˜ l �) �Ψ|A|Ψ� = �Ψ|A| R R R R l l l ˜ Ψ� ˜ + R (�Ψ |A|Ψ � − �Ψ ˜ |A|Ψ ˜ l �) �Ψ|A|Ψ� = �Ψ|A| R R R R R. (2.25). (2.25) (2.25). the rests parts are neglected, because they vanish for sufficiently local operators as long rests partswave are neglected, they vanish local as long asthethe partial expansionbecause is converged. Herefor wesufficiently remind that foroperators truly nonlocal as the partial wave expansion is converged. Here we remind that for truly nonlocal operators the rests parts would have to be considered explicitly. The rests parts are neglected, because they vanish for sufficiently operators the rests parts would have to be considered explicitly.local operators 10 Here we remind that for truly as long as the partial wave expansion is converged. 10 nonlocal operators the rests parts would have to be considered explicitly.. 19.

(20) φkG (r) =. � lm. α,k+G α Aα,k+G uαl (r, Elα )Yml (r) + Blm u´l (r, Elα )Yml (r), r ∈ Sα lm. 1 = √ exp i(k + G) · r, V. r∈I. (2.3. (2.3. With the LAPW thus defined, one constructs and solves the secular equation.. 2.4 L(S)DA(GGA) method by numerical integration of the radial Schrodinger equation on and u´αl +U are obtained radial mesh.. When we calculated electronic structures of some transition metal oxide. U method compounds, such2.5 as NiO,L(S)DA(GGA) by L(S)DA/GGA method, + it showed that is metal, however, it. is truly a insulator. is because L(S)DA/(GGA) calculation ignores the compounds, e When weThis calculated electronic structures of some transition metal oxide by L(S)DA(GGA) method, is metal, it is truely a insulat electron-electronNiO, correlation interaction. For the d,itf showed orbitals that electrons, duehowever to its strong This is because it ignore the electron-electron correlation interaction in L(S)DA(GG localization, we should consider these Coulomb repulsion energy of the electrons in the calculation. For d, f orbital electrons, due to its strong localization, we should consid orbital. The totalthese energy of the electrons’ system is: Coulomb repulsion energy. The system’s total energy is orbital ELDA+U = ELDA +. U − J �� 2 nmσ 2 m σ. (2.36). 12. Where U represents Coulomb repulsion energy, J represents Hund’s rule exchange energy, σ represents spin and n2mσ represents total electron number of specified orbital l (usually d or f orbital). Without +U, the energy gap and magnetic moment will be underestimated in those strongly correlated materials, transition-metal oxides, localized electron systems, insulators, etc. According to many experiences (calculations), after +U correction, these physical quantities are more matched with the experiments. Since the magnitude of U is difficult to calculate accurately, one usually varies U in the reasonable range to obtain better result. So L(S)DA/(GGA) +U method can be regarded as a semi-empirical calculation, but not exactly a ab-initial calculation. Nevertheless, GGA calculations can be corrected using a strong correlation correction denoted as GGA(LDA)+U method [32,33]. The GGA(LDA)+U scheme is a useful approach [34–36] that provides satisfactory results for many strongly correlated systems.. 20. (2.3.

(21) Chapter 3 Half-Metallic (HM) Materials. 3.1 What is HM Materials?. Fig. 3.1 A schematic diagram in density of state (DOS) of half-metallic (HM) material. A half-metal is any substance that acts as a conductor to electrons of one spin orientation, but as an insulator to those of the opposite orientation. (Fig. 3.1) In half-metals, the valence band related to one type of these electrons is fully filled and the other is partially filled. So only one type of electrons (the spin type with the partially filled band) can pass through it, because full bands have a net conductance of zero. In 1983, R. de Groot et al.[37] discovered HM ferromagnets (FM) by calculating the band structure of the magnetic semi-Heusler compounds NiReSb and PtReSb. HM materials have three characteristic properties: (1) quantization of the magnetic moment; (2) 100% spin polarization at the Fermi level (EF); and (3) spin susceptibility of zero. Due to their single-spin charge carriers,. 21.

(22) 3.2 The review of HM compounds research Due to their single-spin charge carriers, HM materials offer potential technological applications, such as computer memory, magnetic recording, single-spin electron source, and high-efficiency magnetic sensors [1–3]. The room-temperature colossal magnetoresistance (CMR) relies on the degree of spin polarized conducting. Therefore, the search for half-metallic compounds is a highly popular topic. The research of half-metallic compounds had been a great progress. Theoretically, the calculation of the band structure has been used to predict new compounds and at the same time, propose the mechanism of causing such feature. It provides lists of HM candidates for experimental research to synthesize and verify. From now, the HM materials, which are already been found, are in these kinds: (1.) Half-Heusler[38], full-Heusler alloys[39 40], such as NiMnSb, PtMnSb [37], FeVSb、. NiTiSn, CoMnsb[41], Co2MnSi, Co2MnGe [42] and Co2CrAl [43]. (2.) Rutile structure: such as CrO2 [44]. In the 70’s, CrO2 has been understood in the magnetic and optical behavior. But until 1986, Schwarz proposed the band structure of CrO2 and pointed out that it is ferromagnetic half-metallic compound. Also, the other work of band structure calculation gives the same result with the total magnetic moments is 2.0µB. Spinel structure: such as Fe3O4 [45]. (3.) Doped perovskite structure manganese oxide Ln1-xAxMnO3 (A=Ca, Sr and Ba), such as Ln0.5Ca0.5MnO3[9] and Ln0.7Sr0.3MnO3 [1,10]. Besides the half metallicity, these series compounds have abundant physical properties such as the semi-structure of high-temperature superconductivity. It has been a heat topic. (4.) Double perovskites structure A2BB′O6 (A=Ca, Sr and Ba; BB′=transition metal), such as Ca2FeMoO6、Sr2FeMoO6、Ba2FeMoO6、Ca2FeReO6 [3,46,47].. 22.

(23) 3.3 Double perovskites structure and magnetic phase 3.3.1 Double perovskites structure The main reason of choosing double perovskites structure AA′BB′O6 to search for HM materials is there are variety options for substituting the A-site or B-site elements. On the B-site elements, transition metal ions are used for its diversity of d-orbitals at the EF. And on the A-site elements, alkaline-earth or rare-earth ions are often been used such as Ca, Sr, Ba and La. Another important fact is that the double perovskites compounds are easy to synthesize. For example, in Sr-based double perovskite Sr2BB′O6, research has shown that Sr2FeMoO6 [3,5–7], Sr2FeReO6 [5,16], Sr2FeWO6 [6], Sr2CrMoO6 [7], Sr2CrReO6 [16,17], and Sr2CrWO6 [5,18] are HM materials. (Fig. 3.2). Fig. 3.2 An ideal ordered double perovskites structure Sr2BB′O6.. 23.

(24) 3.3.2 Structural Optimization In an ideal cubic structure ( Fm 3 m , No. 225) of the F(i)M state, the B and B′ ions are placed in the order of an NaCl configuration with a lattice constant 2a. Each B(B′) is coordinated by B′(B), and each has an O ion in between with the lengths of B-O and B′-O being equal. In an ideal cubic structure, the c/a ratio is. 2 . This ratio allows the structure to relax to a reduced symmetry during the structural optimization calculations with the consideration of a supercell (2 f.u.). After full structural optimization, several structures were reduced from a cubic structure ( Fm 3 m ) to a tetragonal structure ( I 4 / mmm , No. 139) with two non-equivalent types of O atoms. In the tetragonal structure ( I 4 / mmm ), two O1 atoms are located on the z-axis and four O2 atoms are located on the xy-plane (Fig. 3.2). The angle of B-O1-B′ remains at 180° during structural optimization; the angle of B-O2-B′ has been altered a little but still close to 180° during structural optimization . However, the lattice constant and the bond length changed. The symmetry reduction is deemed rather minor because the c/a ratio is very close to the ideal. 2 . For the AF state, the tetragonal structure ( P 4 / mmm , No. 123) remains the same during full structural optimization. The detail stable structure of each cubic value of. compound will relax into will be discussed in the following chapter.. 3.3.3 Magnetic States. Fig. 3.3 The schematic diagram of 4 magnetic states: FM, FiM, AF and NM. In the FM and FiM states, each B and B′ ion has similar spin states (that is, (B, B, B′, B′) = (m, m, m′,m′) = FM or (m, m, -m′, -m′) = FiM), which can lead to the assumption of the half-metallicity of the compound. Through the self-consistent process, most of the initial FM and FiM states all converge into one of the states. In the AF state, the spin state 24.

(25) of (B, B, B′, B′) can be described as (m, -m, m′, -m′). The induced equivalence in the charges is Q↑ [B (B′)] = Q↓ [B (B′)], which can be observed from the symmetry of the spin-up and spin-down in the total density of state (DOS). Thus, no half-metallicity can be observed. No spin polarization is observed in the NM state. Calculations for all four magnetic phases were performed to determine the most stable magnetic phase. (Fig.3.3) The calculations that took into account the spin polarization have always been more stable than those without. The self-consistent process with high convergence criteria was also performed to guarantee the accuracy of the result.. 3.4 Calculation procedure. Fig. 3.4 The calculation procedure of HM materials research, which includes the process of initial magnetic states, full structural optimization and electron correlation correction by the VASP code.. In the initial, 4 magnetic states (FM, FiM, AF and NM) are set in the ideal cubic structure ( Fm 3 m , No. 225). Through the full structural optimization in GGA scheme, the 25.

(26) FM and FiM state will converge to one of the state noted as F(i)M and others will remain in the state. But still, the exceptions exits, for example: the FM and FiM state doesn’t converge to one of the state but remain in the initial state; and another example is HM-AFM (Half-metallic antiferromagnetic), which has the characteristics of HM-FiM with the total magnetic moment is zero; that is (B, B, B′, B′). = (m, m, -m, -m) = FiM,. which can be noted as AFM). The full-potential projector augmented wave [24] method implemented in the VASP code [29,30] was used for the full structural optimization process (i.e., relaxation for both lattice constants and atomic positions). The conjugate-gradient method was used to find the stable ionic positions, and the energy convergence criteria for self-consistent calculations were set to 10-6 eV. The 8×8×6 k-point grids were set in the Brillouin zone, and the cut-off energy of the plane wave basis was set to 450 eV. Theoretical equilibrium structures were obtained when the forces and stresses acting on all the atoms were less than 0.03 eV/Å and 0.9 kBar, respectively. The Wigner–Seitz radius of the atom was set to 2.5 a.u. for Sr ion, to 2.1 a.u. for B(B′) ion, and to 1.4 a.u. for O. G1 After the structural optimization, the F(i)M usually will reduce to tetragonal structure ( I 4 / mmm , No. 139) and the AF state will be in tetragonal structure ( P 4 / mmm , No. 123). The energy difference between each state gives the massage of the stability of each compound, where the NM state will be unstable in most of the compounds. Also, the electronic structure of each compound can be obtained which can be presented in magnetic moments, the population of d electrons and density of states (DOS). Lately, the electron correlation correction (GGA+U) will be considered with the fixed structure. The effective parameter adopted in this study was Ueff = U - J, where U and J are the Coulomb and exchange parameters, respectively. In this work, the parameter Ueff was denoted as U for simplicity and functions on the d orbital. In transition metals, the near maximum values are selected from the reasonable range of U [48]. For example, the range of U for Fe is 3.0 eV to 6.0 eV. In this study, 5.0 eV was used in the calculation. The detailed U values will be listed in each section. The whole calculation procedure all schematic diagramed in Fig. 3.4.. 26.

(27) 3.5 The formation of HM: Double-exchange couplings. Fig. 3.5 The schematic diagram of Double-exchange couplings The half-metallic of double perovskites structure comes from both the the double exchange coupling of B(t2g)-O(2p)-B’(t2g) π-bonding, which has been called a generalized double exchange mechanism.[49, 50] It is similar to the double exchange mechanism of metallic ferromagnetism in colossal magnetoresistive magnets.[51] This non-local indirect exchange interaction enhances the HM exchange coupling of B and B′ ions. (Fig.3.5). 27.

(28) Chapter 4 HM Materials in Sr2BB′O6 4.1 The searching groups In the work of searching new HM materials form the Sr-based double perovskite Sr2BB′O6, it is found that these HM candidate compounds exhibit cyclical behavior in the periodic table, which are controlled by the constitution of BB′ ion pairs. As to the result, we classified the possible HM compound into 3 groups according to the electronic configuration of the BB′ ion pairs. The 3 groups are:. Group 1: 3B/4B transition metal pair with Cr, Co, V and Fe Sr2Cr(Co)B′O6 (B′=Sc, Y, La, Ti, Zr, and Hf) and Sr2V(Fe)B′O6 (B′= Zr and Hf). Group 2: 6B/7B transition metal pair with Co, Cu and Ni Sr2BB′O6 (B = Co, Cu, and Ni; B′ = Mo, W, Tc, and Re). Group 3: 7B/8B/1B transition metal pair with Zn Sr2ZnBO6 (B=Mn, Tc, Re, Fe, Ru, Os, Co, Ni, Pd, and Au). In the following section will present the calculation of the initial four magnetic states (FM, FiM, AF and NM) with full structural optimization and electron correlation correction in each group. This paper will carefully discuss the compounds from each group by its stable structure, local/total magnetic moments and density of states.. 28.

(29) 4.2 Group 1: 3B/4B transition metal pair with Cr, Co, V and Fe. Fig.4.1 The Group 1 HM compounds Sr2BB′O6 of BB′ ion pairs on the periodic table. Sr2Cr(Co)B′O6 (B′=Sc, Y, La, Ti, Zr, and Hf) and Sr2V(Fe)B′O6 (B′= Zr and Hf). Fig.4.1 shows the cyclical behavior of Group 1 HM compounds Sr2BB′O6 of BB′ ion pairs in the periodic table. Each pairs with the same color, for example, the red frame Cr(Co) pairs with the same color 3B4B element. And the blue frame V(Fe) pairs with Zr and Hf. As to the result, FM and FiM converge to FM state in tetragonal structure (I4/mmm, no. 139), except Sr2CrYO6 and Sr2CrLaO6, which remain in the ideal cubic structure ( Fm 3 m , no.225) during structural optimization. The detail parameters of the structure are listed in Table Table 4.1. Thus, we only present energy differences between the FM state and AF state (Table 4.2 and 4.3). Henceforth, we will use [BB′] to represent the entire chemical formula; for example, [CrY] stands for Sr2CrYO6.. 29.

(30) Table 4.1. Structural parameters in the fully optimized structure ( I 4 / mmm , no. 139 and. Fm 3 m , no. 255) where Sr(x,y,z) = (0, 0.5, 0.75), B(x,y,z) = (0, 0, 0), B′(x,y,z) = (0, 0, 0.5), O1(x,y,z) = (0, 0, O1z), and O2(x,y,z) = (O2x, O2y, 0.5). The asterisk (*) represents the ideal cubic structure ( Fm 3 m ), with O1 and O2 being equivalent. Sr2[BB′]O6. a. c/a. V0(Å3/f.u.). O1z. O2x. O2y. CrSc. 5.5735. 1.4142. 122.42. 0.2621. 0.2376. 0.2376. CrY*. 5.7379. 2. 133.62. 0.2300. -. -. CrLa*. 5.8759. 2. 143.46. 0.2240. -. -. CrTi. 5.4990. 1.4141. 117.57. 0.2537. 0.2470. 0.2470. CrZr. 5.6578. 1.4135. 128.00. 0.2394. 0.2607. 0.2607. CrHf. 5.6248. 1.4137. 125.80. 0.2414. 0.2587. 0.2587. VZr. 5.6783. 1.4141. 129.45. 0.2399. 0.2603. 0.2603. VHf. 5.6437. 1.4144. 127.12. 0.2420. 0.2582. 0.2582. CoSc. 5.5473. 1.4140. 120.69. 0.2622. 0.2377. 0.2377. CoY. 5.7043. 1.4148. 131.30. 0.2299. 0.2701. 0.2701. CoLa. 5.8614. 1.4143. 142.40. 0.2240. 0.2756. 0.2756. CoTi. 5.4555. 1.4136. 114.76. 0.2547. 0.2453. 0.2453. CoZr. 5.6052. 1.4136. 124.47. 0.2377. 0.2623. 0.2623. CoHf. 5.5726. 1.4138. 122.32. 0.2397. 0.2603. 0.2603. FeZr. 5.6146. 1.4140. 125.13. 0.2411. 0.6131. 0.6131. FeHf. 5.5810. 1.4141. 122.91. 0.2398. 0.2603. 0.2603. 4.2.1. Sr2Cr(V)B′O6 The FM states of all the compounds are more favored than the AF state from the energy difference by the order of 101 meV/f.u. to 102 meV/f.u. in the GGA scheme (Table 4.2). Unfortunately, in the GGA+U scheme, the AF states of [CrY], [CrZr], and [CrHf] are more stable than the FiM state in the order of 102 meV/f.u., i.e., 160.6, 138.6, and 116.8 meV/f.u., respectively. The NM state of each compounds are always be unstable. 30.

(31) Table 4.2. Calculated physical properties of Sr2[BB′]O6 (B=Cr and V; B′=Y, La, Zr and Hf) in double perovskite structure in the full structural optimization calculation of GGA(+U). Materials U(Cr,M) Sr2[BB′]O6 CrSc. CrY. CrLa. CrTi. CrZr. CrHf. VZr. VHf. Spin magnetic moment (µB/f.u.). d orbital electrons ↑/↓. N (EF). band gap. ΔE=FM-AF. MB. MB′. mtot. B. B′. states/eV/f.u.. eV. meV/f.u.. (0,0). 1.039. 0.026. 1.000. 2.641/1.620. 0.930/0.898. ↑4.720. ↓1.075. -180.9. (4,2). 1.333. 0.030. 1.000. 2.771/1.464. 0.989/0.865. ↑4.384. ↓1.800. -8.3. (0,0). 1.030. 0.017. 1.000. 2.630/1.617. 0.514/0.494. ↑4.332. ↓1.225. -71.9. (4,2). 1.157. 0.034. 1.000. 2.809/1.423. 0.503/0.482. ↑4.062. ↓3.046. 160.6. (0,0). 1.023. 0.019. 1.000. 2.629/1.625. 0.416/0.397. ↑4.445. ↓1.319. -79.2. (4,2). 1.361. 0.017. 1.000. 2.779/1.446. 0.394/0.376. ↑4.823. ↓2.381. -194.6. (0,0). 1.858. 0.084. 2.000. 3.049/1.219. 3.049/1.219. ↑4.162. ↓1.475. -99.0. (4,2). 2.105. 0.069. 2.000. 3.157/1.087. 1.383/1.287. ↑3.931. ↓2.050. -26.6. (0,0). 1.904. 0.034. 2.000. 3.062/1.188. 1.334/1.268. ↑4.660. ↓2.008. -70.8. (4,2). 2.212. 0.030. 2.000. 3.195/1.022. 0.769/0.735. ↑3.785. ↓2.897. 138.6. (0,0). 1.921. 0.021. 2.000. 3.066/1.175. 0.828/0.797. ↑4.531. ↓2.066. -79.9. (4,2). 2.220. 0.020. 2.000. 3.195/1.013. 0.801/0.774. ↑3.272. ↓3.023. 116.8. (0,0). 0.923. 0.068. 1.000. 2.214/1.307. 0.802/0.743. ↑3.442. ↓2.159. -42.9. (3,2). 1.043. 0.064. 1.000. 2.239/1.217. 0.785/0.722. ↑3.606. ↓2.707. -106.6. (0,0). 0.949. 0.048. 1.000. 2.220/1.287. 0.839/0.879. ↑3.968. ↓2.229. -56.3. (3,2). 1.066. 0.046. 1.000. 2.225/1.204. 0.812/0.766. ↑3.962. ↓3.034. -99.4. In the GGA scheme, the half metallic properties can be observed with the band gap at the spin-down channel and integer total magnetic moment (mtot) of 1.0 and 2.0µB. (Table 4.2). The results match those of a previous study on [CrZr] [52]. The configurations of the valence electron between 3B and 4B are identical, thus, we only present the DOS of B′ = Sc, Ti and Zr. The DOS are presented in Fig.4.2. The hybridization between the Cr(V) 3d and O 2p orbital mainly occurs in the energy region at approximately -6.0 eV to -4.0 eV and at -1.0 eV to 1.5 eV. At the Fermi level (EF), a strong spin splitting of the Cr(V) eg orbital results in the HM property. Furthermore, with the double exchange interaction, the O 2p and 3B/4B-d orbital hybrid interact in the same energy region. The weak magnetic moment under 0.070 µB of 3B/4B element induced by the spin spits the Cr(V) eg orbital at EF. Based on the calculated electron population, the electron configurations result in the following compounds: For [Cr-3B] pairs, the electron 31.

(32) configurations are Cr5+(3d1:t2g1eg0) at S = 1/2, 3B3+(3/4/5d0:t2g0eg0) at S=0; For [Cr-4B] pairs, the electron configurations are Cr4+(3d2:t2g2eg0) at S = 1, 4B4+(3/4/5d0:t2g0eg0) at S=0. And for [V-Zr/Hf] pairs, the electron configurations are V4+(3d1:t2g1eg0) at S = 1/2, Zr/Hf4+(4/5d0:t2g0eg0) at S=0.. Fig. 4.2 Calculated spin- and site-decomposed density of states of [CrSc], [CrTi] and [VZr] in the GGA scheme.. 4.2.2. Sr2Co(Fe)B′O6 The FM states of all the compounds are more favored than the AF state from the energy difference by the order of 101 meV/f.u. to 102 meV/f.u. in the GGA scheme (Table 4.3). Unfortunately, in the GGA+U scheme, [CoTi], [CoZr], [CoHf], [FeZr], and [FeHf], the AF state is more stable by 4.8 6.6, 10.8, 22.1, and 18.2 meV/f.u, respectively. However, the energy differences of [CoTi], [CoZr] and [CoHf] are extremely small; the AF state and F(i)M state in these compounds degenerate and can coexist. Table 4.3. Calculated physical properties of Sr2[BB′]O6 (B=Co and Fe; B′=Y, La, Zr and 32.

(33) Hf) in double perovskite structure in the full structural optimization calculation of GGA(+U). Materials U(Cr,M) Sr2[BB′]O6 CoSc. CoY. CoLa. CoTi. CoZr. CoHf. FeZr. FeHf. Spin magnetic moment (µB/f.u.). d orbital electrons ↑/↓. N (EF). band gap. ΔE=FM-AF. MB. MB′. mtot. B. B′. states/eV/f.u.. eV. meV/f.u.. (0,0). 1.157. -0.063. 2.000. 4.114/2.960. 0.913/0.953. ↓5.799. ↑1.100. -177.7. (6,2). 3.141. 0.130. 4.617. 5.036/1.904. 0.954/0.874 ↑2.463/↓2.408. -. 20.2. (0,0). 1.262. -0.034. 2.000. 4.156/2.898. 0.506/0.527. ↓5.092. ↑1.319. -178.0. (6,2). 1.166. -0.041. 2.000. 4.166/3003. 0.485/0.511. ↓6.933. ↑1.540. -199.8. (0,0). 1.392. -0.027. 2.000. 4.209/2.823. 0.406/0.420. ↓10.865. ↑1.378. -84.7. (6,2). 1.471. -0.028. 2.000. 4.267/2.803. 0.421/0.437. ↓4.614. ↑1.296. -207.3. (0,0). 0.707. -0.059. 1.000. 3.923/3.218. 1.332/1.375. ↓5.187. ↑0.700. -125.2. (6,2). 3.389. 0.177. 5.000. 5.153/1.777. 1.358/1.282. -. ↑↓2.975/0.075. 4.8. (0,0). 0.746. -0.028. 1.000. 3.936/3.194. 0.781/0.801. ↓5.116. ↑1.132. -70.8. (6,2). 3.469. 0.106. 5.000. 5.178 /1.722. 0.788/0.741. -. ↑4.353↓0.432. 6.6. (0,0). 0.732. -0.029. 1.000. 3.927/3.198. 0.819/0.836. ↓5.199. ↑1.155. -64.9. (6,2). 3.464. 0.130. 5.000. 5.173/1.721. 0.826/0.777. ↓0.318. ↑5.065. 10.8. (0,0). 1.695. -0.048. 2.000. 3.894/2.213. 0.772/0.806. ↓4.478. ↑1.447. -114.5. (5,2). 3.691. 0.062. 4.000. 4.807/1.149. 0.800/0.736. ↑1.386. ↓1.984. 22.1. (0,0). 1.681. -0.048. 2.000. 3.884/2.217. 0.810/0.838. ↓4.969. ↑1.482. -109.1. (5,2). 3.709. 0.052. 4.000. 4.811/1.135. 0.831/0.777. ↑1.343. ↓2.007. 18.2. The half metallic properties can be observed with the band gap at the spin-up channel and integer total magnetic moment of 1.0, and 2.0 µB (Table 4.3). The result matches that of a previous study on [FeZr] [53]. The DOS are presented in Fig. 4.3. With the Co(Fe)3d - O2p –3B4Bd hybridization at the spin-down channel, a weak magnetic moment under -0.05 µB is induced for 3B4B ions which can be regarded as zero. Their half metallic mechanisms are similar to that of [CrSc], so we can also consider them as FM–HM. Based on the calculated electron population, the electron configurations result in the following compounds: For [Co-3B] pairs, the electron configurations are Co5+(3d4:t2g4eg0) at S = 1, 3B3+(3/4/5d0:t2g0eg0) at S=0; for [Co-4B] pairs, the electron configurations are Co4+(3d5:t2g5eg0) at S = 1/2, 4B4+(3/4/5d0:t2g0eg0) at S=0. And for [Fe-Zr/Hf] pairs, the electron configurations are Fe4+(3d4:t2g4eg0) at S = 1, Zr4+/Hf4+(4/5d0:t2g0eg0) at S=0.. 33.

(34) Fig. 4.3 Calculated spin- and site-decomposed density of states of [CoSc], [CoTi] and [FeZr] in the GGA scheme.. 4.2.3. Exchange Correlation Correction The value of U is often based on the experimental results. Thus, choosing an appropriate U in finding new compounds is difficult. The effects of inducing the exchange correlation correction (GGA+U) are similar in all compounds (Fig. 4.4). The effect of the electron correlation correction will enhance the localization of the d orbitals and push unoccupied states into higher energy levels, thus increasing the energy gap and magnetic moment. For example, for BB′=Cr-3B, the energy gap reaches 1.80, 3.05 and 2.38 eV from 1.08, 1.22 and 1.32 eV, respectively. In the GGA+U scheme, the total magnetic moment remains the same, except for [CoSc] which lost its half-metalicity becoming conductor and for [CoTi], [CoZr], [CoHf], [FeZr], and [FeHf], which both gain from 1.0 µB to 5.0 µB and from 2.0 µB to 4.0 µB for B = Co and Fe, respectively. For B = Cr, Co, and V, the EF crosses the eg orbital in both GGA and GGA+U schemes. However, for [CoHf], [FeZr], and [FeHf], after inducing exchange correlation correction, the Fe(Co) t2g orbital dominates around EF. In addition, for [CoZr], the energy gap opens in both spin 34.

(35) states of 4.353 and 0.432 eV for spin-up and spin-down, respectively, and becomes a ferromagnetic insulator (FM-Is). Thus, the spin state of Co and Fe changes from low-spin state to high spin state. Based on the calculated electron population, the electron configurations result in the following compounds: For [Co-4B] pairs, the electron configurations are Co4+(3d5:t2g3eg2) at S = 5/2, Zr4+(4d0:t2g0eg0) at S=0. And for [Fe-Zr/Hf] pairs, the electron configurations are Fe4+(3d4:t2g3eg1) at S = 2, Zr/Hf4+(4/5d0:t2g0eg0) at S=0.. Fig. 4.4 Calculated spin- and site-decomposed density of states of [CrSc], [CrTi], [CoSc], [CoTi], [VZr], and [FeZr] in the GGA+U scheme.. 35.

(36) 4.2.3. Double Exchange Interaction. Fig. 4.5 The double exchange interaction configuration in the GGA and GGA+U schemes. Based on the electronic configuration in the previous discussion, we suggest that the double exchange interaction is important in these HM materials. In the GGA scheme, for B=Cr and V, the spin-up electrons transfer from occupied Cr d (S=1/2 or 1) to empty Y/La/Zr/Hf d (S=0) or occupied V d (S=1/2) to empty Zr/Hf d (S=0). This phenomenon results in the compounds appearing to be in FM state (Figs. 4.5 a,b). The magnetic moment on the B-site is spin-up and induces slight spin-up moments on the B’-site, which reflects the character of double exchange interaction. Furthermore, based on the DOS, the spin-up channel behaves as a conductor, and the energy gap lies in the spin-down channel. Conversely, the B ions (Co and Fe) are in a low-spin state, whereas the spin-down electron transfers between BB′ ions (Fig. 4.5 c,d), making the magnetic moment of B ions appear in a spin-up state and the magnetic moment of B′ ions appear in a spin-down state. Thus, the compound for B=Co (S=1/2 or 1) and Fe (S=1) appears to be in FiM state, but 36.

(37) the local magnetic moments of B′= Y/La/Zr/Hf (S=0) are extremely small, which are induced by the d-orbital of B ions. These conditions still satisfy the double exchange interaction. Thus, these compounds can still be regarded as in the FM state. Furthermore, in the DOS, the energy gap is located in the spin-up channel and the behavior of conductor is in the spin-down channel. In the GGA+U scheme, the exchange correlation correction is considered, and the repulsion between electrons rises. Thus, the B ions (Co and Fe) enter the high-spin state (Co: S=5/2; Fe: S=2), where the distance of each electron gains. Through double exchange interaction, the spin-up electron transfers between BB′ ions (Fig. 4.5 e,f), explaining why the energy gap switches spin sites in the GGA+U scheme for B=Co/Fe and B′=Zr/Hf. The double exchange interaction of B(t2g)-O(2p)-B’(t2g) configuration enhances the HM properties for each compound and provides reasonable explanations for the magnetic stable state, magnetic moment, and single-spin energy gap.. 37.

(38) 4.3 Group 2: 6B/7B transition metal pair with Co, Cu and Ni. Fig.4.6 The Group 2 HM compounds Sr2BB′O6 of BB′ ion pairs on the periodic table. Sr2BB′O6 (B = Co, Cu, and Ni; B′ = Mo, W, Tc, and Re, and BB′=FeTc) Fig.4.6 shows the cyclical behavior of Group 2 HM compounds Sr2BB′O6 of BB′ ion pairs in the periodic table. Each pairs with the same color, for example, the red frame Co(Ni) pairs with the same color 6B and 7B element. And the blue frame Cu pairs with Tc and Re. Combining with the previous research, for the pairs of B=Cr and Fe (yellow frame) can be also classified in Group 2 HM compounds where our calculations agrees with previous experimental and theoretical investigations[3, 5-7, 16-18]. Thus, in this paper, we focus on B = Co, Cu, and Ni and no longer discuss the series of B=Cr and Fe, except Sr2FeTcO6, which has been less discussed in previous work. During the self-consistent process, first, Sr2CuWO6 and Sr2NiTcO6 remain in their ideal cubic structure ( Fm 3 m ) and others relax into the tetragonal structure ( I 4 / mmm ). (Table 4.4) Second, the initial FM and FiM states converge into one of these states. For example, compounds of B = Cu converge into the FM state whereas compounds of B = Co converge into the FiM state. In the NM state, no spin polarization can be observed; thus, an absence of magnetic properties is obtained. Calculations that consider spin polarization are always more stable than those that do not. A self-consistent process with high convergence criteria is also performed to guarantee the accuracy of the calculation results. For brevity, we use [BB′] to indicate the chemical formula of the perovskites. For example, [FeTc] represents Sr2FeTcO6. Table 4.4 Structural parameters in the fully optimized structure ( I 4 / mmm , no.139 and 38.

(39) Fm 3 m , no.255) where Sr(x,y,z) = (0, 0.5, 0.75), B (x,y,z) = (0, 0, 0), B′(x,y,z) = (0, 0, 0.5), O1(x,y,z) = (0, 0, O1z) and O2(x,y,z) = (O2x, O2y, 0.5). The star-mark (*) represents the ideal cubic structure ( Fm 3 m ) with the O1 and O2 are equivalent. Sr2[BB′]O6. a. c/a. V0(Å3/f.u.). O1z. O2x. O2y. FeTc. 5.6054. 1.4131. 124.43. 0.2538. 0.2463. 0.2463. CoMo. 5.6077. 1.4145. 124.72. 0.2547. 0.2452. 0.2452. CoW. 5.6194. 1.4140. 125.46. 0.2573. 0.2426. 0.2426. CoTc. 5.5856. 1.4145. 123.24. 0.2549. 0.2452. 0.2452. CoRe. 5.6110. 1.4146. 124.95. 0.2575. 0.2426. 0.2426. CuMo. 5.6312. 1.4139. 126.24. 0.2578. 0.2423. 0.2423. CuW*. 5.6401. 126.86. 0.2582. -. -. NiMo. 5.6001. 1.4139. 124.15. 0.2526. 0.2473. 0.2473. NiW. 5.6097. 1.4141. 124.82. 0.2566. 0.2434. 0.2434. NiTc*. 5.5790. 122.79. 0.2549. -. -. NiRe. 5.6141. 125.09. 0.2536. 0.2464. 0.2464. 2. 2. 1.4139. The FiM states of compounds [FeTc], [CoMo], [CoW] and [NiTc] are more favorable than their AF states in the GGA scheme (Table 4.5). The results match those of a previous study on [CoMo] [54-56], [CoRe] [54], [CoW][57, 58] and [NiRe][59]. The AF states of [CoTc] and [CoRe] are more stable than their FiM states, with energies of 93.7 and 76.4 meV/f.u., respectively. The energy differences of [CuMo], [CuW], [NiMo], [NiW], and [NiTc] are extremely small at –2.2, –1.3, 13.0, 3.8, and 13.7 meV, respectively; that is, the AF and FiM states in these compounds degenerate and can coexist. In the GGA+U scheme, the AF states of [CoMo] and [CoRe] are more stable than their FiM states, with an energy difference in the order of 102 meV/f.u. More specifically, the energy differences between the AF and FiM states of [CoMo], and [CoRe] are 596.1, and 61.9 meV/f.u., respectively. The energy difference between the AF and FiM states of [CoTc] in the GGA+U scheme is about 2.2 meV, which implies that the FM state coexists with the AF state. For the rest of the compounds in the GGA+U scheme, the stable magnetic state resulting from the energy difference remains the same as in the GGA scheme. The value of U eventually depends on the experiment where it only can be evaluated by an reasonable range of U, where different value of U can product diverse results.. 39.

(40) Table 4.5 Calculated physical properties of possible HM of Sr2BB′O6 (B=Co, Cu and Ni; M=Mo, W, Tc and Re; and BB′=FeTc) in double perovskites structure in the full structural optimization calculation of GGA(+U). Materials U(Cr,M) Sr2[BB′]O6 FeTc. CoMo. CoW. CoTc. CoRe. CuMo. CuW. NiMo. NiW. NiTc. NiRe. Spin magnetic moment (µB/f.u.). d orbital electrons ↑/↓. N (EF). band gap. ΔE=FM-AF. MB. MB′. mtot. B. B′. states/eV/f.u.. eV. meV/f.u.. (0,0). 3.763. -1.038. 3.000. 4.849/1.122. 1.685/2.737. ↓0.572. ↑3.218. -455.9. (5,2.5). 4.192. -1.399. 3.000. 5.005/0.853. 1.503/2.909. ↓2.872. ↑2.159. -608.8. (0,0). 2.627. -0.123. 3.000. 4.851/2.240. 2.015/2.178. ↓6.186. ↑0.945. -74.7. (6,2). 2.787. -0.141. 3.000. 3.479/0.898. 1.764/2.442. ↓6.056. ↑1.856. 596.1. (0,0). 2.600. -0.036. 3.000. 4.851/2.266. 2.021/2.096. ↓7.422. ↑1.681. -78.7. (6,2.5). 2.745. -0.037. 3.000. 4.995/2.228. 1.971/2.026. ↓6.011. ↑2.731. -510.4. (0,0). 2.583. -0.740. 2.000. 4.826/2.264. 1.821/2.582. ↓6.064. ↑0.421. 93.7. (6,2.5). 2.78. -0.905. 2.000. 4.961/2.202. 1.715/2.624. ↓5.806. ↑1.494. 2.2. (0,0). 2.388. -0.459. 2.000. 4.777/2.407. 1.826/2.305. ↓4.671. ↑0.712. 76.4. (6,2). 2.440. -0.382. 2.000. 4.868/2.447. 1.818/2.208. ↓0.782. ↑1.424. 61.9. (0,0). 0.603. 0.019. 1.000. 4.861/4.257. 2.087/2.109. ↓1.748. ↑1.167. -2.2. (7,2). 0.596. 0.011. 1.000. 4.934/4.336. 2.065/2.094. ↓1.662. ↑1.879. -5.5. (0,0). 0.620. 0.030. 1.000. 4.871/4.250. 2.054/2.065. ↓1.808. ↑2.066. -1.3. (7,2.5). 0.607. 0.025. 1.000. 4.936/4.328. 2.044/2.021. ↓1.609. ↑2.906. -7.6. (0,0). 1.561. 0.061. 2.000. 4.868/3.314. 2.099/2.090. -. ↑1.027↓0.782. 13.0. (5,2). 1.734. 0.072. 2.000. 4.966/3.242. 2.084/2.045. -. ↑1.809↓2.171. 0.03. (0,0). 1.581. 0.07. 2.000. 4.882/3.306. 2.064/2.045. -. ↑1.903↓0.933. 3.8. (5,2.5). 1.750. 0.076. 2.000. 4.968/3.227. 3.017/1.973. -. ↑2.859↓3.105. 0.2. (0,0). 1.496. -0.627. 1.000. 4.395/3.353. 2.524/1.870. ↓8.119. ↑0.467. -28.4. (5,2.5). 1.723. -0.815. 1.000. 4.964/3.253. 1.754/2.579. ↓0.999. ↑1.564. -524.0. (0,0). 1.393. -0.467. 1.000. 4.817/3.431. 1.821/2.310. ↓7.975. ↑0.910. 13.7. (5,2). 1.746. -0.725. 1.000. 4.958/3.223. 1.679/2.413. ↓1.661. ↑2.462. -0.9. In the GGA scheme, the HM properties of compounds of B = Co, Cu, and Ni can be observed through the band gaps at the spin-up channel and the integer total magnetic moments (mtot) listed in Table 4.5. [NiMo] and [NiW] appear to be ferromagnetic insulators (FM-Is), having band gaps at both spin states, but retain magnetic properties. The valence electron configurations of Mo and W, as well as Tc and Re, are similar to 40.

(41) each other; thus, we only present the DOS of B′ = Mo and Tc in Fig. 4.7~4.8. Fig. 4.7 Calculated total, spin, and site-decomposed DOS of [FeTc], [CoMo], and [CoTc] in the GGA scheme. For B=Co and [FeTc], the hybridization between the Co(Fe) 3d and O 2p orbitals occurs mainly in the energy region from –7.0 eV to –1.0 eV in the spin-up channel and from –2.0 eV to 2.0 eV in the spin-down channel. Strong spin-splitting of the Co(Fe) eg orbital at the EF yields half-metallicity. Around the EF, the 4(5)d orbital of B′ (Mo, W, Tc, and Re) hybridizes with the O 2p orbital in the energy region from –2.0 eV to 2.0 eV in the spin-down channel, where the Co(Fe) orbital extends through double-exchange interactions. This interaction constructs a bridge of Co(Fe)3d–O2p-B′4(5)d for the conductance of the spin-down electron. Based on the calculated electron populations, the following electron configurations are obtained: [FeTc] (Fe3+(3d5:t2g3eg2) at S = 5/2, Tc5+(4d2:t2g2eg0) at S = -1), [Co-Mo/W] (Co3+(3d6:t2g4eg2) at S = 2, Mo/W5+(4/5d1:t2g1eg0) at S = -1/2), [Co-Tc/Re] (Co3+(3d6:t2g4eg2) at S = 2, Tc/Re5+(4/5d2:t2g2eg0) at S = -1). The mechanisms of FiM stabilization and half-metallicity of compounds of B = Co and [FeTc] can be described by p-d hybridization, as proposed by Terakura et al. [63]. 41.

(42) When a NM element is located between magnetic elements, such as Fe, the fully spin-polarized magnetic elements are denoted as d-states. By contrast, NM elements between spin-split d-states are denoted as the p-state as the EF goes through the p band. During p-d hybridization, the d-state pushes the p-state upward (downward) at the spin-up (spin-down) channel. The electron population switches spin states, inducing the NM elements to contribute negative moments and stabilize the FiM state to achieve a common EF value in both spin states. If p-d hybridization is adequately strong, the p-state can be pushed above the EF to the conduction band at the spin-up channel. During double-exchange, the band is extended to the spin-down channel with the EF, producing half-metallicity in the FiM states. In the current work, compounds of B = Co and Fe represent the spin-split d-state of magnetic elements, while those of B′ = Mo (W) and Tc (Re) represent the p-state of the NM elements.. Fig. 4.8 Calculated total, spin, and site-decomposed DOS of [CuMo], [NiMo], and [NiTc] in the GGA scheme. For compounds of B = Cu the DOS of the spin-up and spin-down channels are symmetrical, except for the orbitals around the EF (Fig. 4). The spin-splitting of the Cu t2g 42.