⇦Chinese Physics Letters, 2020, Vol. 37, No. 2, Article code 027501 Stable Intrinsic Long Range Antiferromagnetic Coupling in Dilutely V Doped Chalcopyrite * Weiyi Gong (龚唯奕), Ching-Him Leung (梁正谦), Chuen-Keung Sin (冼传强), Jingzhao Zhang (张璟昭), Xiaodong Zhang (张小东), Bin Xi (席斌), Junyi Zhu (朱骏宜)** Affiliations Department of Physics, The Chinese University of Hong Kong, Hong Kong Received 3 October 2019, online 18 January 2020 *Supported by Chinese University of Hong Kong (CUHK) under Grant No. 4053084, University Grants Committee of Hong Kong under Grant No. 24300814, and Start-Up Funding of CUHK.
**Corresponding author. Email: jyzhu@phy.cuhk.edu.hk Citation Text: Gong W Y, Liang Z Q, Xian C Q, Zhang J Z and Zhang X D et al 2020 Chin. Phys. Lett. 37 027501    Abstract A stable and long-range antiferromagnetic (AFM) coupling without charge carrier mediators has been searched for a long time, but the existence of this kind of coupling is still lacking. Based on first principle calculations, we systematically study carrier free long-range AFM coupling in four transition metal chalcopyrite systems: ABTe$_2$ (A = Cu or Ag, B = Ga or In) in the dilute doping case. The AFM coupling is mainly due to the $p$–$d$ coupling and electron redistribution along the interacting chains. The relatively small energy difference between $p$ and $d$ orbitals, as well as between dopants and atoms in the middle of the chain can enhance the stability of long-range AFM configurations. A multi-band Hubbard model is proposed to provide fundamental understanding of long-range AFM coupling. DOI:10.1088/0256-307X/37/2/027501 PACS:75.50.Pp, 71.15.Mb, 75.30.Et © 2020 Chinese Physics Society Article Text Antiferromagnetic (AFM) order is essential for many spintronic devices, such as tunnel junctions, spin valves, Hall devices, AFM/FM bilayers.[1–3] Compared to ferromagnets, AFM materials have many advantages.[4] AFM order has been observed in various metallic alloys[5] or high magnetic concentration semiconductors,[6] in which magnetism tuning can be of challenge. Meanwhile, AFM order in diluted magnetic semiconductors (DMSs) has not been discovered although it is generally believed that tuning of magnetism is relatively easy in DMSs. In DMS, a relatively low concentration of transition metal atoms are doped, compared to typical magnetic semiconductors. Usually there is a large distance between magnetic dopants in DMS. Therefore, without the enhancement of charge carriers,[7–9] magnetic coupling in DMS is usually restricted to nearest neighbors. AFM coupling is often induced by superexchange,[10,11] where two cations are coupled by an anion in the middle. Practically, the magnetic dopants often form clusters due to the stable short-range AFM or FM coupling.[12] Therefore, it is unlikely to form long-range AFM structures. However, there are some long-range AFM couplings proposed in previous theoretical and experimental studies, which passes through an ABCBA chain structure, with A sites as magnetic dopants. For example, long-range couplings were proposed along Cu–O–La–O–Cu in La$_4$Ba$_2$Cu$_2$O$_{10}$,[13] Eu–O–Ti–O–Eu in EuTiO$_3$,[14] Fe–S/Se–Ba–S/Se–Fe in iron-based antiperovskite chalcohalides.[15] The general theory of super superexchange in magnetic solids was discussed in Ref. [16]. However, there are several potential problems in the aforementioned literature: (1) There appears to be lack of a stability comparison between the short range and long-range magnetic configurations, and the long-range one may not be stable in the dilute doping limit, where usually short range superexchange dominates. (2) All of these studies are based on oxides or sulfides, in which the long-range AFM couplings mediated by ABCBA are believed to be weak.[13] (3) Although suggested as a possible mechanism for AFM coupling in Ref. [14], the underlying physics is not systematically investigated or clearly demonstrated. Extended super exchange[13] or super superexchange[15,16] were proposed as possible mechanisms for long-range AFM coupling. However, these mechanisms are generally believed to be weak as high order perturbations, and the roles of B and C atoms on the magnetic coupling were not systematically investigated. Recently, the stepping stone mechanism has been proposed to explain the formation and stabilization of ferromagnetic coupling in Cr doped (Sb,Bi)$_2$Te$_3$.[17] The stepping stone mechanism is based on spin polarized $s$ lone pair states of cations on C sites of spin chains of ABCBA type. Whether stepping stone mechanism can be extended to spin polarized $d$ states that may stabilize the long-range AFM coupling is still unknown, except one report on a possible long-range AFM coupling in Mn doped LiZnAs,[18] which is relatively unstable. In addition, no quantitative model has been constructed to describe the stepping stone mechanism and fundamental physical understanding is poor. In this Letter, we investigate stable long-range magnetic coupling in four transition metal chalcopyrite systems. A multi orbital super-exchange model is constructed to explain the underlying mechanism that gives rise to the intrinsic AFM coupling in DMS. In contrast to the usual cases where shorter range coupling is favorable, our results suggest a new mechanism where the transitional metal atom on the C site can induce an electron redistribution, which induce a strong Coulomb repulsion between magnetic atoms and stabilize the long-range configuration. This mechanism is different from the similar mechanisms, such as the "super superexchange" proposed previously,[16] where the long-range magnetic interaction between two magnetic atoms was mediated by bonding $p$ states. Our mechanism is mediated by localized $d$ states or $s$ states,[17] which was previously largely ignored. This mechanism is due to the relatively strong correlation effect. In addition, in the context of diluted magnetic semiconductors, it is challenging to find systems that have the most stable magnetic configuration other than the nearest neighbor configuration. This microscopic mechanism may lead to realization of macroscopic AFM order, which can be checked by large scale simulations that are beyond the capability of DFT. Details of calculation setup are given in the Supplementary Materials. In order to explore the short-range and long-range magnetic couplings, we substitute two In or Ga atoms with two V atoms with increasing distance. All possible configurations in $2\times 2\times 1$ supercell are shown in the Supplementary Materials. An In/Ga atom at body center of the supercell is substituted by a V atom. Another V atom replaces an In/Ga atom at various neighboring sites to the center V atom, from the first nearest neighbor (NN) to the forth NN. Cu/Ag, In/Ga and V atoms have approximately local $T_d$ symmetry in the host cell, which is a property of chalcopyrite. The point group of this supercell is $\bar{4}2d$ ($D_{2d}$). Under this symmetry, the third NN has two nonequivalent configurations, while the other three have only one configuration for each. The stability of V doped ABTe$_2$ is characterized by its formation energy, which is defined as $$ E_{\rm f} = E({\rm doped}) - n_{_{\rm V}} \mu_{_{\rm V}} - E({\rm undoped}) + n_{_{\rm B}} \mu_{_{\rm B}},~~ \tag {1} $$ where $E$ are total energies of AB$_{1-x}$V$_x$Te$_2$ and ABTe$_2$, $\mu$ are chemical potentials of V and In/Ga, $n$ is the number of corresponding atoms. In our study, different configurations have the same number of In/Ga and V atoms, so the difference of formation energies between different configurations cancels the contribution of chemical potentials. We define this difference between each configuration and that of the most stable one as relative formation energy, which equals the difference of total energies between them. The results in CuInTe$_2$ in $3\times3\times1$ supercell and spin-orbit coupling (SOC) effect are given in Fig. 1. Although the distance between two V atoms are the same in (a) $2 \times 2 \times 1$ and (c) $3 \times 3 \times 1$ supercells, the concentration of V dopant is different. Since the calculation is based on a periodic boundary condition, there are more periodic images of V atoms due to the symmetry in $2 \times 2 \times 1$ supercell than in $3 \times 3 \times 1$ supercell. A smaller amount of images may help to decrease the relative formation energy. Nevertheless, our main conclusion still holds and stabilization of long-range AFM configuration is clear. Although the effect of spin orbit coupling (SOC) is ignored in the theoretical model in the following text, SOC may induce further splitting of energy levels of heavy atoms like Te. Due to the SOC effect, the $p$ level of Te will become lower, while this effect at Cu and In atoms is not so obvious. Hence, according to the perturbation theory, smaller energy difference will cause a larger magnetic coupling. The results of DFT calculation with SOC effect considered confirm this point. At the second nearest neighbor, AFM has energy 7.57 meV lower than that of FM without SOC effect, while this value becomes 10.74 meV with SOC considered. Although the magnetic coupling of the first NN changes from FM to AFM once SOC is considered, the general trends of formation energy as functions of neighboring sites are similar.
cpl-37-2-027501-fig1.png
Fig. 1. The relative formation energy for V doped CuInTe$_2$ as a function of various neighboring configurations in $2\times 2\times 1$ and $3\times 3\times 1$ supercells, without SOC and with static SOC. Red(blue) points are the relative formation energies of AFM(FM) configurations, and green ones are their difference.
To understand the mechanism for this kind of long-range carrier free AFM interaction, we calculate the projected density of states (pDOS) for the second NN configuration, as shown in Fig. 2. The V shows clearly polarized $p$–$d$ hybridized states right above the Fermi level. The polarized $p$ states of V atoms mainly come from the hybridization of neighboring Te atoms. These polarized states have similar energy and shape to the polarized $p$ states of Te atoms. However, the Cu atoms show no spin polarization. This happens because the Cu atom is at the middle of the chain and two halves of the chain polarize with the same magnitude but opposite directions.
cpl-37-2-027501-fig2.png
Fig. 2. The density of states of the first three atoms along the chain. Green, blue and red represent $s$, $p$ and $d$ states.
To see the spin polarization more clearly, we calculate the polarized spin density of AFM second NN, as shown in Fig. 3. The V atom at the body center and the other two at the face center are connected by three V–Te–Cu–Te–V chains and one V–Te–In–Te–V chain. We can see clearly polarized Cu $d$ orbitals in the middle, which serve as "stepping stones" to pass on the magnetism along the chain.
cpl-37-2-027501-fig3.png
Fig. 3. Front view and side view of the spin texture of the second NN V-doped CuInTe$_2$. Dashed lines show the supercell. For clarity, atoms that are not on the chain have been removed.
We note from the DFT results that the relaxed magnetic moment of V atoms is 2.33, which is slightly greater than 2 due to the electron redistribution from Cu atoms in the middle. This redistribution is driven by the large onsite Coulomb repulsion on Cu atoms, and will induce a repulsion between two V atoms. The Coulomb energy difference is estimated to be around 100 meV between the 1st NN and the 2nd NN (shown in the Supplementary Materials), which is of same order with the energy difference shown in Fig. 1. This explains the stabilization of long-range coupling. Also, we notice that there is a difference between the DFT results and our theoretical analysis (see the Supplementary Materials for details) due to the following reasons: (1) the figure shows the lowest interaction energy states, which are not the eigenstates of the Hamiltonian. Taking the hopping term into consideration, the eigenstates are essentially linear combination of several low interaction energy states, in which the number of $d$ electrons at A atoms is less than 3; and (2) the DFT calculations use orbitals that are not so localized, so the DFT results should be expected to be qualitatively consistent with theoretical prediction of our systems, which are strongly correlated. We also conduct energy mapping analysis[16] to fit the system into the Heisenberg model with the onsite anisotropic term. We get $J\approx 3.5$ meV, $A\approx 0.5$ meV. The details are given in the Supplementary Materials. A multi-bands Hubbard model is proposed to study the mechanism of long-range AFM coupling in an A–B–C–B–A like chain which possesses the same geometry as that of the V–Te–Cu–Te–V chain, which is the building block of this AFM structure. For simplicity of theoretical modeling, we will ignore the effect of SOC in the following analysis. Nevertheless, SOC effect can still be treated using perturbation theory to achieve more accurate results, which is beyond the scope of this study. We construct various spin configurations with parallel or anti-parallel spins on $e_{\rm g}$ states at the A(V) sites. Effective Hamiltonians in low energy subspace are derived for these two cases, and configurations with anti-parallel spins show relatively lower energy than that with parallel spins. This accounts for an effective AFM coupling between V atoms. Finally, an estimation of the effective magnetic coupling is given. We use the expression of onsite electron-electron interaction for $p$ and $d$ orbitals in cubic crystal field following the result from Ref. [19]. We neglect the quadrupole moment terms and derive (for details see the Supplementary Materials) $$\begin{split} \hat{V} &= \hat{V}_0 + \hat{V}_{\rm sf} + \hat{V}_{\rm ph},\\ \hat{V}_0 &= u \hat{n}^2 - v \hat{m}_z^2 - (u - v) \hat{n} + 8v \sum _{\alpha} \hat{n}_{\alpha \uparrow} \hat{n}_{\alpha \downarrow} ,\\ \hat{V}_{\rm sf} &= -2 v \sum _{\alpha \neq \beta, \sigma} \hat{c}_{\alpha \sigma}^† \hat{c}_{\alpha, -\sigma} \hat{c}_{\beta, -\sigma}^† \hat{c}_{\beta \sigma} ,\\ \hat{V}_{\rm ph} &= 2 v \sum _{\alpha \neq \beta} (\hat{n}_{\alpha \beta})^2 \\ &= -2 v \sum _{\alpha \neq \beta, \sigma} \hat{c}_{\alpha \sigma}^† \hat{c}_{\alpha, -\sigma}^† \hat{c}_{\beta \sigma} \hat{c}_{\beta, -\sigma}, \end{split}~~ \tag {2} $$ where $\alpha$ and $\beta$ are indices of local orbitals; $\sigma$ and $\sigma'$ are indices of spin; $\hat{n}$ is the electron number operator; $\hat{m_z}$ is the $z$ component of the magnetic moment operator; and $\hat{n}_{\alpha \beta}\,=\,\sum_{\sigma} \hat{c}_{\alpha \sigma}^† \hat{c}_{\beta \sigma}$ is the onsite hopping between two orbitals. Here the site label is omitted for simplicity. The parameters $u, v$ in Eq. (2) are defined as $$\begin{split} u &= \frac{1}{2}U - \frac{1}{4}J + \frac{5}{2}\Delta J ,\\ v &= \frac{1}{4}J - \frac{3}{2}\Delta J. \end{split}~~ \tag {3} $$ For $d$ orbitals, $U$ is the Coulomb interaction between $t_{\rm 2g}$ orbitals, $J$ is the average exchange splitting of $e_{\rm g}$ and $t_{\rm 2g}$ orbitals and $\Delta J$ is the difference of exchange splitting between $e_{\rm g}$ and $t_{\rm 2g}$ orbitals following the definitions in Ref. [19]. While for $p$ orbitals, $U$ and $J$ are the Coulomb interaction and exchange splitting of three $p$ orbitals, and $\Delta J\,=\,0$. Then, the Hamiltonian of the multi-orbital Hubbard model under tight binding approximation is $$\begin{split} \hat{H} = \,&\sum _{\langle i, j \rangle} \sum _{\alpha \beta \sigma} t_{\alpha \beta}^{ij} \hat{c}_{i \alpha \sigma}^† \hat{c}_{j \beta \sigma} + {\rm H.c.} \\ &+ \sum _{i \alpha \sigma} \epsilon _{i \alpha} \hat{c}_{i \alpha \sigma}^† \hat{c}_{i \alpha \sigma} + \sum _{i} \hat{V}_0^i, \end{split}~~ \tag {4} $$ where $i, j\,=\,A, B, C$ are atomic sites; $t_{\alpha \beta}^{ij}$ are hopping integrals between nearest neighbor atoms; $\sum_{i} \hat{V}^i \approx \sum_{i} \hat{V}_0^i$ is the interaction according to Eq. (2) on each site, and the spin flip and pair hopping have been ignored by approximation. For more detailed discussions, please see the latter part of this letter. A and C atoms have local $T_{d}$ symmetry, so local $d$ orbital splits to 2 fold degenerated $e_{\rm g}$ states and 3-fold degenerated $t_{\rm 2g}$ states. Te $p$ orbitals interact strongly with nearby $t_{\rm 2g}$ (A,C) states which have the same symmetry and form $\sigma$ bonds. Te $p$ orbitals form only weaker $\pi$ bonds with nearby $e_{\rm g}$ (A,C) states, which play less important role near the Fermi level,[20,21] and we ignore these terms in this model. The signs of the hopping integrals depend on the geometry of the chain. Using the Slater–Koster matrix elements,[22] the signs of hopping integrals can be determined, as shown in the Supplementary Materials. The magnitudes of hopping are approximated to be the same between A, B and B, C with $t\,=\, V_{pd\pi}/\sqrt{3}$. Starting from the neutral state, different configurations were obtained by nearest neighbor hopping among $p$ orbitals of B and $t_{\rm 2g}$ (A,C). The $e_{\rm g}$ (A,C) were fixed in this model. The relative position of each atomic level were taken to be the same as V(A), Te(B), Cu(C), and this relation can as well be applied to compounds with elements one row lower or higher. The total energies of this chain with parallel and anti-parallel initial V spins are calculated using Eq. (4) in the Supplementary Materials. The lowest energy states with parallel ($\psi_{0 \uparrow \uparrow}$) and anti-parallel ($\psi_{0 \uparrow \downarrow}$) $e_{\rm g}$ spins at site A are shown in the Supplementary Materials. In state $\psi_{0 \uparrow \downarrow}$, two V(A) atoms have half-filled $3d$ orbitals with anti-parallel spins, two Te(B) atoms have fully filled $5p$ orbitals and Cu(C) atom has empty $t_{\rm 2g}$ orbitals and fully filled $e_{\rm g}$ orbitals. The spin configuration of two A atoms in this state is $3d_{\uparrow}^5$, $3d_{\downarrow}^5$ ($S\,=\,\frac{5}{2}$) obeying Hund's rule. While in state $\psi_{0 \uparrow \uparrow}$, electron fillings at sites B and C are the same, but one V(A) atom has $3d_{\uparrow}^5$ filling and the other has $e_{g\uparrow}^2 t_{2g\downarrow}^3$ filling. The energy of $\psi_{0 \uparrow \uparrow}$ is $24 v_{_{\rm A}}$ larger than that of $\psi_{0 \uparrow \downarrow}$. This is because $\psi_{0 \uparrow \downarrow}$ state has larger local magnetic moment and results in a smaller energy. Although hopping directly between A and C is not allowed in this model, it can happen by virtual hopping. In this process, one spin on the $p$ state at site B hops to $t_{\rm 2g}$ state at site C first, then one spin on the $t_{\rm 2g}$ state at site A hops to B site, and vice versa. This is a typical process that happens in super-exchange mechanism.[10] We denote the intermediate states and states after one virtual hopping as $\psi_{1 \uparrow \downarrow}, \psi_{1 \uparrow \uparrow}$ and $\psi_{2 \uparrow \downarrow}, \psi_{2 \uparrow \uparrow}$. In the parallel case, two A atoms are not symmetric, so we denote the state after one virtual hopping of an up (down) spin from $t_{\rm 2g}$ orbitals at site A as $\psi_{2 \uparrow \uparrow}(\psi_{2 \uparrow \uparrow}')$. Relatively higher energy states are those with smaller local magnetic moment at site A, the $t_{\rm 2g}$ spins at site A can be different. There are huge energy barriers between these states and the two lowest energy states, and we focus only on the lowest energy states and their intermediate states, and consequently these states are not considered in the following analysis. Starting from lowest energy configurations, higher order configurations can be obtained by nearest neighbor hopping. We can apply perturbation theory to calculate the effective ground state energy. Applying the fourth order perturbation theory, we obtain $$ E_{\uparrow \uparrow} - E_{\uparrow \downarrow}\,=\,24 v_{_{\rm A}} - \frac{72 t^4 v_{_{\rm A}}}{(\Delta E_{10})^2(\Delta E_{20})^2} ,~~ \tag {5} $$ where $\Delta E_{10}\,=\,E_{1\uparrow \downarrow} - E_{0\uparrow \downarrow}\,=\, E_{1\uparrow \uparrow} - E_{0\uparrow \uparrow}$, $\Delta E_{20}\,=\,E_{2\uparrow \downarrow} - E_{0\uparrow \downarrow}\,=\,E_{2\uparrow \uparrow} - E_{0\uparrow \uparrow}$, see the Supplementary Materials for detailed explanations. We find that anti-parallel initial spins would have lower energy than that of the parallel one. So the system prefers long-range AFM interaction. This long-range AFM interaction is mediated by low-lying $d$ orbitals of Cu atoms. If we replace the Cu atoms on the chain by Ag atoms, the low energy configurations will not change much, while $\Delta E_{10}$ and $\Delta E_{20}$ will become larger due to higher $d$ level of Ag atoms. The energy difference becomes larger according to Eq. (5). Remember that two V atoms are connected by six chains, three to the left, three to the right. To further illustrate the stepping stone mechanism mediated by $d$ states, we replace the six Cu atoms by Na atoms at the stepping stone sites and find that the energy difference between FM and AFM configurations is dramatically reduced to 1.6 meV. This small residue energy is probably due to the magnetism mediated by In atoms. We also replace the six Cu atoms by Ag atoms at stepping stone sites and find that the energy difference between FM and AFM configurations changes from 7.57 meV for the Cu case to 8.89 meV for the Ag case, which confirms the validity of our model. The lattice constants are the same for the Cu and Ag cases. However, in AgInTe$_2$ doped with V, the lattice constant is significantly different from the two cases in CuInTe$_2$. Consequently, the overlaps of electronic orbitals from neighboring sites are different in the systems of the two materials, leading a significant difference in electro hopping terms. Therefore, it is not suitable to directly compare the effect of Ag that replaces the Cu on the stepping stone site in the CuInTe$_2$ with the magnetic coupling strength in the AgInTe$_2$. The actual energy difference is small due to the following reasons: (1) The two V atoms in this system are actually connected by three V–Te–Cu–Te–V chains. Three Cu atoms in the middle provide more hopping channels, which will enlarge the coefficients of the second term in Eq. (5). (2) Configurations with different local magnetic moments also have low energies. For example, energy of configurations with $3d_{\uparrow}^4 3d_{\downarrow}^1$, $3d_{\downarrow}^4 3d_{\uparrow}^1$ (spin distribution of V atoms is $S\,=\,\frac{3}{2}$) is 32$v_{_{\rm A}}$ larger than that of lowest energy state. Although these configurations are ignored due to huge energy barriers (see the Supplementary Materials for details), they also help to reduce the magnetic coupling. The ground state will be the superposition of these states, so that the actual magnetic moments of V atoms are smaller than that of the lowest energy configuration, which has the largest magnetic moments. In addition, the effect of spin flipping and pair hopping become more important when $S < 5/2$, because in this case there are both up spins and down spins at site A. Spin flipping happens when $d$ orbitals of A are half-filled and pair hopping happens when there are both doubly filled and empty orbitals. Spin flipping effect of the stepping stones induce a ferromagnetic superexchange when there is a 90$^\circ$ twisted chain structure.[11] Our model gives clues to finding stable long-range AFM order in DMS, and several guidelines can be formed. (1) To search for a long-range AFM order independent of RKKY interaction, the host cell should be carrier free. (2) There should be low lying states along the path between magnetic dopants which serve as the stepping stones to pass on the spin exchange process. Such states are relatively easy to be magnetized due to a large local Coulomb repulsion that may induce charge redistribution along the chain. As a result, long-range magnetic order emerges with the assistance of the charge redistribution. (3) In addition, the host cell should consist of large atoms because they may suppress neighboring superexchange and favor the long-range interactions. In summary, we have discovered an intrinsic long-range AFM structure in V doped CuInTe$_2$. This AFM coupling cannot be explained by RKKY interaction since there are no carriers in this system. It can rather be explained by a coupling in the A–B–C–B–A chain mediated by the localized electronic states on C site, which we named the stepping stone mechanism. Energy difference between each two of the three atoms, local Coulomb interaction, local exchange splitting, local orbital degeneracy and asymmetry between two halves of the chain help to induce an effective AFM coupling between two magnetic dopants. A multi-orbital Hubbard model is proposed and confirmed by further DFT calculations. Our model can also be applied to other long-range chain structures with different types of localized electronic states locating at the stepping stone sites. 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