Electronic and magnetic properties of low dimensional system Co2TeO3Cl2

TL;DR

This study uses first-principles calculations to explore the electronic and magnetic properties of Co2TeO3Cl2, revealing antiferromagnetic interactions and the significance of spin-orbit coupling in its low-dimensional structure.

Contribution

It provides a detailed analysis of the exchange interactions and spin lattice of Co2TeO3Cl2 using density functional theory and Wannier functions, highlighting the role of spin-orbit coupling.

Findings

Nearest and next-nearest neighbor interactions are antiferromagnetic.

Inter-chain and intra-chain interactions dominate in the ab plane.

Spin quantization axis favors the b crystallographic direction.

Abstract

The electronic and magnetic properties of transition metal oxyhalide compound Co2TeO3Cl2 is investigated using first principle calculations within the framework of density functional theory. In order to find underlying spin lattice of this compound, various hopping integrals and exchange interactions are calculated. The calculations reveal that the dominant inter-chain and intra-chain interactions are in ab plane. The exchange path is visualised by Wannier function plotting. The nearest neighbour and next nearest neighbour exchange interactions are antiferromagnetic, making the system frustrated in low dimension. The importance of spin orbit coupling in this compound is also investigated. The spin quantization axis is favoured along the crystallographic b direction.

Tables

Table 1. Table 1: Hopping integrals (in meV) obtained from NMTO downfolding method for Co 2 TeO 3 Cl 2 . The interaction paths (t 1 , t 2 and t 3 , t 4 ) are indicated in Fig. 1(b)

Hopping	Atoms	Distance	orbital involved
Ing.		(Å)
				$|1⟩$	$|2⟩$	$|3⟩$
$t_1$	Co1-Co2	3.01	$⟨1^{\prime }|$	-22.6	0	-63.3
			$⟨2^{\prime }|$	28	15.4	30
			$⟨3^{\prime }|$	-38.4	-80.2	-5.1
				$|1⟩$	$|2⟩$	$|3⟩$
$t_2$	Co1-Co1	3.31	$⟨1|$	47.8	23.1	23
			$⟨2|$	23.1	-42.8	-42.9
			$⟨3|$	-23	42.9	-83.8
				$|1⟩$	$|2⟩$	$|3⟩$
$t_3$	Co1-Co2	4.12	$⟨1^{\prime }|$	-19.3	-29.3	21.1
			$⟨2^{\prime }|$	-31.7	-58.5	54.2
			$⟨3^{\prime }|$	-11.3	43.3	25
				$|1^{\prime }⟩$	$|2^{\prime }⟩$	$|3^{\prime }⟩$
$t_4$	Co2-Co2	5.05	$⟨1^{\prime }|$	-19.3	-31.7	-11.3
			$⟨2^{\prime }|$	-29.3	-58.5	43.3
			$⟨3^{\prime }|$	21.1	54.2	25

Table 2. Table 2: The relative energies per unit cell, magnetic moments, and band gaps for different magnetic configurations are listed here.

Magnetic	Band gap	$M_{Co1}$	$M_{Co2}$	Energy
Config.	(eV)	(${\mu }_B$)	(${\mu }_B$)	Difference (meV)
FM	2.1	2.65	2.66	0
AF1	2.1	2.65	2.66	$-$78.5
AF2	2.2	2.67	2.66	$-$125
AF3	2.2	2.67	2.63	$-$33.5
AF4	2.2	2.67	2.6	$-$135

Table 3. Table 3: The exchange interactions (meV) are listed here

Exchange	Distance	Type of Co	Total	$J^{\mathrm{AF}}$
path	(Å)	atom	Exchange	from NMTO
$J_1$	3.01	Co1-Co2	$-$6.01	$-$7.3
		intra-chain
$J_2$	3.31	Co1-Co1	$-$7.12	$-$8.7
		intra-chain
$J_3$	4.12	Co1-Co2	3.83	$-$2.4
		inter-chain
$J_4$	5.05	Co2-Co2	$-$4.76	$-$3.7
		inter-chain

Table 4. Table 4: The spin (orbital) moment of Co 2+ ions, energy difference (in meV) within GGA+U+SOC calculations (with U e ​ f ​ f = 4 subscript 𝑈 𝑒 𝑓 𝑓 4 U_{eff}=4 eV) ion axis are listed here.

Spin Quantization Axis	Co1 moment (${\mu }_B$)	Co2 moment	$\Delta E$
$\left(001\right)$	2.7 (0.21)	2.7 (0.22)	0
$\left(010\right)$	2.66(0.22)	2.67(0.22)	-2.20
$\left(100\right)$	2.66(0.22)	2.65(0.22)	-1.20

Equations

$$J^{\mathrm{AF}}=\frac{{\sum }_{ij}{t}_{ij}^2}{U+{\Delta }_{nn}}$$

$$|1⟩$$

$$|2⟩$$

$$|3⟩$$

$$|4⟩$$

$$|5⟩$$

$$|1^{\prime }⟩$$

$$|2^{\prime }⟩$$

$$|3^{\prime }⟩$$

$$|4^{\prime }⟩$$

$$|5^{\prime }⟩$$

Full text

I Introduction

Low dimensional spin systems have attracted much attention over the past few decades due to their variety of unusual magnetic properties.Lemmens et al. (2003); Bray et al. (1975); Johannes et al. (2006); Janson et al. (2012); Chakraborty and Dasgupta (2012) Effective low dimensionality means anisotropic exchange interactions in different direction of the crystal that arises due to interplay of geometry and chemical bonding. Due to enhanced quantum fluctuations in low dimension, exotic properties can be seen in these class of materials. Presence of magnetic frustration in low dimensional spin systems is particularly important. Many complex compounds with different spin lattices like spin ladder, kagome lattice, triangular lattice, spin chain show fascinating magnetic properties like spin gap state, spin liquid state, various type of complex magnetic order etc.Fåk et al. (2012); Jeong et al. (2011); Koteswararao et al. (2013, 2012); Chakraborty et al. (2016) Transition metal oxyhalide compounds comprising a p-element having a stereochemically active lone pair (such as Te4+, Se4+, Sb3+ etc.), form an attractive field of study. There is high probability of finding new low-dimensional and frustrated spin system in these class of materials. The example of such systems are: FeTe2O5Br, Cu2Te2O5Cl2, Ni5(TeO3)4Cl2, etc.Becker et al. (2006a); Rie Takagi, Mats Johnsson, Vladimir Gnezdilov, Reinhard K. Kremer, Wolfram Brenig and Lemmens (2006); Johnsson et al. (2003) Due to reduced dimensionality and geometric frustration, these materials often show complex magnetic order and interesting magnetic properties. Some of these materials also show multiferroic property.Pregelj et al. (2009); Chakraborty et al. (2013) Number of such oxyhalide compounds are synthesised and studied over time. However, the family of Co2+ containing oxohalogenides including a lone-pair cation is not studied much. Co2TeO3Cl2 is one such compound which is synthesized by Becker et al.Becker et al. (2006b) It has a layered structure. The magnetic properties measurements by them suggests long range antiferromagnetic interactions below 30 K and the Curie Weiss temperature is $-$97 K.Becker et al. (2006b) Neutron diffraction study of a powder sample suggests that Co spins are parallel to the crystallographic $b$ directions.Kashi et al. (2007) Exact spin lattice of the system is not known. In complex crystal structures, the microscopic understanding of the spin lattice remains a challenging problem. The knowledge of intra-chain and inter-chain exchange interactions are crucial to understand the magnetic behavior of these materials. ab initio calculations play important role in this direction.

In this paper, the electronic and magnetic properties of Co2TeO3Cl2 are studied using first principle calculations within the framework of density functional theory (DFT). To determine the underlying spin lattice of this compound, various hopping integrals and exchange interactions between Co2+ ions are calculated. In order to assess the importance of spin orbit coupling (SOC) in this system, the electronic structure calculations are also carried out with SOC and magnetic anisotropy energy is calculated. The remainder of the paper is organized as follows: the crystal structure and computational details are discussed in section II. The results of electronic structure calculations is discussed in section III. Finally the results are summarised in section IV.

II Crystal Structure and Computational details

The compound Co2TeO3Cl2 (see Fig. 1) has monoclinic structure with space group P21/m (No. 11). The lattice parameters are $a$ = 5.0472 Å, $b$=6.6325 Å, $c$= 8.3452 Å, and $\beta$=105.43∘.Becker et al. (2006b) There are two crystallographically different Co ions in this system. Co1 is in square planar environment with two oxygen ions at a distance of 2.053 Å  and other two oxygen ions at a distance of 2.084 Å. Further Co1 is coordinated by two chlorine atoms at a distance 2.497 Å, forming Co1O4Cl2 octahedron. The Co2 atom is coordinated by three oxygen atoms and to three chlorine atoms and forming Co2O3Cl3 octahedron. The Co1O4Cl2 octahedra share edges with Co2O3Cl3 octahedra and form layered structure. These layers are connected by Te ions. In a layer, the nearest Co1-Co2 distance is 3.06 Å. Fig. 1(b) shows the arrangement of Co ions in a layer.

In the present study, electronic structure calculations are carried out using two different basis sets: (i) the muffin-tin orbital based linearized muffin-tin orbital (LMTO) method within atomic sphere approximation (ASA) and Nth order muffin-tin orbital (NMTO) downfolding method as implemented in the Stuttgart codeAndersen and Jepsen (1984) and (ii) the plane wave based projector augmented wave (PAW)Blöchl (1994) method as implemented in the vasp code Kresse and Furthmüller (1996). For the TB-LMTO-ASA calculation, the space filling in the ASA is obtained by inserting appropriate empty spheres in the interstitial regions. The basis set for the self consistent electronic structure calculation for Co2TeO3Cl2 in the LMTO method includes Co ($s$, $p$, $d$), Te ($s$, $p$), Cl ($s$, $p$), and O ($s$, $p$) and the rest are downfolded. The self consistency is achieved by Brillouin-zone integrations in a $(8\times 8\times 8)$ ${\bf k}$ point mesh. NMTO downfolding method, where potentials are taken from self consistent LMTO method, is used to derive the low energy Hamiltonian. In this method, the low energy model Hamiltonian is constructed by selective downfolding method via integration process. The high energy degrees of freedom are integrated out from the all orbital calculation.

For the calculations using plane wave based method, the wave functions are expanded in the plane-wave basis with a kinetic energy cutoff of 500 eV. The exchange-correlation functional is chosen to be the Perdew-Burke-Ernzerhof (PBE) implementation of the generalized gradient approximation (GGA).Perdew et al. (1996) To properly describe the electron correlation associated with the 3$d$ states of Co, GGA+U method is used in the calculations.Dudarev et al. (1998) For electronic structure calculations with spin-orbit coupling (SOC), it is included in scalar relativistic form as a perturbation to the original Hamiltonian. The structural relaxations are carried out until the Hellman-Feynman forces became less than 0.01 eV/Å. The calculation revealed that there is very small deviation of the atomic position in the relaxed structure from experimental structure. Therefore all the calculations reported here have been done with experimental lattice parameters and atomic positions.

III Results and Discussions

III.1 Spin-unpolarized calculation

To begin with, the electronic structure of Co2TeO3Cl2 is studied without magnetic order. The total and orbital projected density of states (DoS) are displayed in Fig. 2(a). Non-magnetic DoS reveals that the Fermi level (EF) is dominated by partially filled Co-$d$ orbitals. The O-$p$ and Cl-$p$ states are fully occupied. The O-$p$ states overlap with Co-$d$ states indicating the strong hybridization of Co-$d$ states with O-$p$ states. As expected, the occupied Te-$s$ states lie far below the Fermi level and empty Te-$p$ states lie above EF, spreading in the energy range of 2 to 5 eV. The Te-5$s$ and Te-5$p$ states are strongly hybridized with the O-$p$ states, which in turn hybridize with Co-$d$ states near the Fermi level. The system is metallic for the spin unpolarized calculation within local density approximation (LDA).

The spin unpolarized band structure calculated within LMTO method is displayed in Fig. 2(b). The bands are plotted at the various high symmetry points of the Brillouin zone corresponding to the monoclinic lattice. The Fermi level is dominated by Co-$d$ states hybridized with O-$p$ and Cl-$p$ states.

III.2 Crystal Field Splitting and Magnetism

The NMTO downfolding calculations are performed in order to find the crystal field splitting of Co-$d$ orbitals and hopping interactions between Co ions. To calculate crystal field splitting, only Co-5$d$ orbitals are retained in the basis and the rest are downfolded. The diagonalization of the onsite block of 5 $\times$ 5 Hamiltonian gives the crystal field splitting for Co1-$d$ and Co2-$d$ states including the covalency with oxygens and chlorines. These energies are calculated to be $(-2.76,-2.69,-2.58,-1.72$ and $-1.41$ eV$)$ and $(-2.63,-2.61,-2.28,-1.83$ and $-1.43$ eV$)$ for Co1 and Co2 respectively. The corresponding eigenstates turn out to be of mixed character and nondegenerate. The eigen states are written in Appendix A. The level diagram of crystal field splitting and the filling of electrons (including spin polarization) in $d$ level of Co1 and Co2 are shown in Fig. 4. There are three partially filled orbitals of $d$-level of Co ions, in the low energy model only three of the $d$ orbitals of Co1 and Co2 are retained in the basis. The downfolded band structure compared with all orbital band structure is displayed in Fig. 3 The Fourier transformation of the low energy model Hamiltonian, $H_{R}=\sum_{ij}t_{ij}\left(c_{i}^{\dagger}c_{j}+H.C.\right)$ gives the hopping integrals between various Co atoms. The hopping integrals are displayed in Table 1. The next nearest neighbour (nnn) hopping interaction ($t_{2}$) turns out to be strongest. The nearest neighbour (nn) interaction $t_{1}$, between Co1-Co2, is compared to the nnn hopping interaction. $t_{3}$ and $t_{4}$ are smaller compared to $t_{1}$ and $t_{2}$ but non-negligible. Other hoppings in $ab$ plane are negligible. The hopping along crystallographic $c$ direction is also negligible. The strength of the intra-chain and inter-chain hoppings indicate that this system is low dimensional. The antiferromagnetic part of the exchange interaction can be calculated from the hopping integrals using the expressionKugel and Khomskii (1982):

[TABLE]

where, $U$ is the Coulomb interaction and $\Delta_{nn}$ is the onsite energy difference. $U$ = 4 eV and $\Delta_{nn}$ = 0.3 eV are taken in the calculation. The antiferromagnetic part of the exchange interactions are tabulated in the last column of Table 3.

In order to visualise the dominant interactions paths, the Wannier functions corresponding to the nearest neighbour Co1 and Co2 ions are plotted in Fig. 5. Fig. 5(a) reveals that the overlap between Co1- $d_{xy}$ effective Wannier functions and that between Co2-$d_{x^{2}-y^{2}}$ Wannier functions, placed at two nearest-neighbor Co sites within the chain. These two Co atoms are connected by three oxygens. The super exchange path of $t_{1}$ hopping is Co1-O-Co2. Additionally they are connected also by Cl atom. Each Co $d$ oribitals form strong $pd\sigma$ antibonding linkages with the neighboring O$p_{x}$/$p_{y}$ orbitals. The tails of the two Co Wannier functions overlap at the Cl site connecting two Co atoms, forming the crucial exchange path. Wannier function of Co2-$d_{3z^{2}-r^{2}}$ is plotted in Fig. 5(b).

III.3 Spin polarized Calculations

Next, the electronic structure of Co2TeO3Cl2 has been investigated with magnetic order. Fig. 6 displayed the spin polarized total and orbital decomposed density of states within GGA+U calculation with $U_{\text{eff}}$= 4 eV. DoS reveals that the majority spin of Co-$d$ states are completely occupied and minority spin states are partially occupied and this picture is consistent with high spin state of Co2+ ion. The magnetic moment of Co2+ ion is found to be 2.71 $\mu_{B}$ consistent with high spin configuration of Co2+ ion within GGA+U calculation. The Te-$s$ states are completely occupied and lies far below the Fermi energy. The O-$p$ and Cl-$p$ states are spread over the energy range $-$6 to 6 eV.

In the present case, the nearest neighbour exchange interaction is between edge sharing Co1 octahedron with Co2 octahedron and the angle is $\angle$Co1-O5-Co2 =81.93∘. Whereas nnn exchange interaction is between two corner sharing Co1 octahedra and the angle is $\angle$Co1-O5-Co1 = 105.5∘. Since the angles involved in the exchange paths are much deviated from 90∘ (or 180∘), Goodenough-Kanamori-Anderson (GKA) ruleGoodenough (1963); Kanamori (1959) can not be applied here. The nature of interaction between $d^{7}$-$d^{7}$ orbital mainly governed by the exchange path and angle with the bridging oxygen atom. In order to find out total magnetic exchange interactions between Co2+ ions, total energy calculation method is adopted here.Xiang et al. (2007); Chakraborty and Dasgupta (2012); Chakraborty et al. (2016); Koteswararao et al. (2012) In this method, the total energies are calculated for several ordered spin configurations of a system and then related the energy differences between these configurations to the corresponding energy differences expected from the Heisenberg spin Hamiltonian:$H=-\sum_{i,j}{\large J_{ij}}{\vec{S}_{i}}\cdot{\vec{S}_{j}}$. Here we have used plane wave basis set. From the hopping strengths, one can conclude that there are four dominant exchange interactions for this system as indicated in Fig. 1. Five different magnetic configurations (see Fig.7) are made to find out four dominant exchange interactions. 2$\times$1$\times$1 supercell is used to make different antiferromagnetic arrangements. In AF1 configuration, Co1 spins are antiferromagnetically coupled with Co2 spins , whereas in AF2 configuration, Co1 spins are antiparallel along $a$ direction and all four Co2 spins are parallel with each other. In AF3 configuration, Co2 spins are antiparallel along $a$ and $c$ directions and Co1 spins are antiparallel only along $b$ direction. AF4 configuration is similar to AF3 configuration except, Co1 spins are aligned antiferromagnetically along $a$ and $b$ directions in AF4 configuration. The energy difference of antiferromagnetic configurations with ferromagnetic configuration, band gap, magnetic moments in each configurations are displayed in Table 2. The calculated exchange energies are listed in Table 3.

The result of the calculations reveal that the nearest neighbour (nn) interaction $J_{1}$ between Co1 and Co2 is antiferromagnetic. Next nearest neighbour (nnn) interaction between Co1-Co2 is also antiferromagnetic and stronger than $J_{1}$ interaction. $J_{3}$ is the inter chain exchange interaction between Co1-Co2 and it is ferromagnetic type. The inter chain exchange interaction $J_{4}$ between Co2-Co2 is also antiferromagnetic and comparable to intra chain exchange interactions. Such strong antiferromagnetic interactions make the system frustrated in low dimension.

III.4 Spin orbit coupling

In order to investigate the importance of spin orbit coupling and magnetocrystalline anisotropy, the electronic structure calculations are carried out within GGA+U+SOC as implemented in vasp. The spin-quantization axis was chosen to be parallel to the direction of the crystallographic $c$ axis as well as perpendicular to the $c$ axis. The spin and orbital moment of Co2+ ions listed in Table 4. The value of orbital moment suggests the spin orbit coupling is important in this system. The magnetocrystalline anisotropy (listed in Table 4) is quite strong. The spin quantization is found to be favoured along crystallographic $b$ direction.

IV Summary and Conclusion

Ab initio density functional theory calculations are carried out to study the electronic structure and magnetic properties of oxyhallide compound Co2TeO3Cl2. In order to determine the underlying spin lattice, various hopping integrals and exchange interactions between Co2+ ons are calculated. Hopping integrals calculated using Nth order muffin-tin orbital (NMTO) downfolding method reveal that both intra-chain and inter-chain hoppings in $ab$ plane are comparable and therefore the system is low dimensional. The dominant exchange interactions are found to be antiferromagnetic. The nearest neighbour interaction $J_{1}$ and next nearest neighbour interaction $J_{2}$ are antiferromagnetic, making the system frustrated. $J_{1}$ is stronger than $J_{1}$. Results of the present work suggests that the system is two dimensional with competing magnetic interactions. The importance of spin orbit coupling in this compound is also investigated. The results of calculations reveal that magnetocrystalline anisotropy is strong in Co2TeO3Cl2 and $(010)$ is the easy axis as observed experimentally.

V Acknowdegement

I like to thank Dr. N. Ganguli for fruitful scientific discussions. Use of high performance computing facility of IISER Bhopal is gratefully acknowledged.

VI APPENDIX A

The eigen states of Co1 and Co2 obtained from NMTO method are following:

[TABLE]