Non-collinear magnetic structure and anisotropic magnetoelastic coupling in cobalt pyrovanadate Co2V2O7

TL;DR

This study reveals the complex non-collinear magnetic ground state and anisotropic magnetoelastic coupling in Co2V2O7, linking its magnetic structure to ferroelectricity and providing insights into its magnetic and structural properties.

Contribution

The paper uncovers the non-collinear magnetic structure of Co2V2O7 and demonstrates its role in magnetoelectric coupling, supported by neutron diffraction and ab initio calculations.

Findings

Co2V2O7 has a non-collinear magnetic ground state with a spin-chain-like structure.

The non-collinear magnetic structure is more stable than ferromagnetic states at low temperature.

Anisotropic magnetoelastic coupling is evidenced by opposite lattice parameter deviations below TN.

Abstract

The Co2V2O7 is recently reported to exhibit amazing magnetic field-induced magnetization plateaus and ferroelectricity, but its magnetic ground state remains ambiguous due to its structural complexity. Magnetometry measurements, and time-of-flight neutron powder diffraction (NPD) have been employed to study the structural and magnetic properties of Co2V2O7, which consists of two non-equivalent Co sites. Upon cooling below the Ne\'el temperature TN = 6.3 K, we observe magnetic Bragg peaks at 2K in NPD which indicated the formation of long range magnetic order of Co2+ moments. After symmetry analysis and magnetic structure refinement, we demonstrate that Co2V2O7 possesses a complicated non-collinear magnetic ground state with Co moments mainly located in b-c plane and forming a non-collinear spin-chain-like structure along the c-axis. The ab initio calculations demonstrate that the non-collinear magnetic structure is more stable than various ferromagnetic states at low temperature. The non-collinear magnetic structure with canted up-up-down-down spin configuration is considered as the origin of magnetoelectric coupling in Co2V2O7 because the inequivalent exchange striction induced by the spin-exchange interaction between the neighboring spins is the driving force of ferroelectricity. Besides, it is found that the deviation of lattice parameters a and b is opposite below TN, while the lattice parameter c and stay almost constant below TN, evidencing the anisotropic magnetoelastic coupling in Co2V2O7.

Tables

Table 1. Table 1:  Crystal structure of Co 2 V 2 O 7 ,Monoclinic Space group p ​ 2 1 / c 𝑝 subscript 2 1 𝑐 p2_{1}/c

Temperature		300 K					Temperature		2 K
a = 6.5932(2) Å		b = 8.3775(2) Å		c = 9.4782(2) Å			a = 6.5860(2) Å		b = 8.3681(2) Å		c = 9.4635(2) Å
$\alpha$ = 90∘		$\beta$ = 100.218(2)∘		$\gamma$ = 90∘			$\alpha$ = 90∘		$\beta$ = 100.191(2)∘		$\gamma$ = 90∘
Type	Site	x	y	z	Biso		Type	Site	x	y	z	Biso
Co1	4e	0.148(3)	0.123(3)	0.463(2)	0.45(2)		Co1	4e	0.149(3)	0.121(3)	0.469(2)	0.24(1)
Co2	4e	0.310(3)	0.382(3)	0.685(2)	0.45(2)		Co2	4e	0.308(3)	0.389(3)	0.680(2)	0.24(1)
V1	4e	0.34(2)	0.75(2)	0.53(2)	0.45(2)		V1	4e	0.35(2)	0.75(2)	0.53(2)	0.24(1)
V2	4e	0.19(2)	0.01(2)	0.81(2)	0.45(2)		V2	4e	0.18(2)	0.01(2)	0.82(2)	0.24(1)
O1	4e	0.604(1)	0.130(1)	0.123(1)	0.55(1)		O1	4e	0.604(1)	0.130(1)	0.124(1)	0.19(1)
O2	4e	0.428(1)	0.126(1)	0.396(1)	0.49(1)		O2	4e	0.427(1)	0.127(1)	0.395(1)	0.07(1)
O3	4e	0.170(1)	0.369(1)	0.461(1)	0.55(2)		O3	4e	0.176(1)	0.370(1)	0.461(1)	0.16(1)
O4	4e	0.255(1)	0.360(1)	0.183(1)	0.37(1)		O4	4e	0.257(1)	0.359(1)	0.183(1)	0.04(1)
O5	4e	0.678(1)	0.373(1)	0.347(1)	0.61(2)		O5	4e	0.679(1)	0.373(1)	0.348(1)	0.19(1)
O6	4e	0.028(1)	0.081(1)	0.246(1)	0.48(1)		O6	4e	0.028(1)	0.083(1)	0.246(1)	0.12(1)
O7	4e	0.852(1)	0.378(1)	0.006(1)	0.52(1)		O7	4e	0.852(1)	0.379(1)	0.007(1)	0.21(1)
300 K, 1.066Å		Rp = 4.03 , Rwp = 5.82 , $\chi$ = 0.50					2 K, 1.066Å		Rp = 4.47, Rwp = 6.32 , $\chi$ = 0.58
300 K, 3.731Å		Rp = 3.72 , Rwp = 5.35 , $\chi$ = 0.15					2 K, 3.731Å,		Rp = 5.39, Rwp = 8.46 , $\chi$ = 0.39

Table 2. Table 2:  The bond length and bond angle of CoO 6 octahedra.

Co1	300 K	Co2	300 K
O2	2.0521 Å	O1	2.1298 Å
O3	2.0675 Å	O2	2.0105 Å
O4	2.0878 Å	O3	2.1642 Å
O6	2.0956 Å	O4	2.0663 Å
O7	2.0649 Å	O5	2.0677 Å
O7	2.0746 Å	O6	2.0592 Å
O2-Co1-O3	85.16 ∘	O1-Co2-O4	96.40 ∘
O2-Co1-O7	94.63 ∘	O1-Co2-O5	87.08 ∘
O3-Co1-O7	94.65 ∘	O4-Co2-O6	88.93 ∘
O7-Co1-O7	86.44 ∘	O5-Co2-O6	87.67 ∘
O4-Co1-O6	173.43 ∘	O2-Co2-O3	174.35 ∘
Average Co-O bond length
300 K	Co1-O =2.0737	Co2-O =2.0829

Table 3. Table 3:  Nonzero basis vectors (BV) of the irreducible representations (IR) and positional coordinates for Co atoms that was used to describe the non-collinear magnetic structure of Co 2 V 2 O 7 with space group P ​ 2 1 / c 𝑃 subscript 2 1 𝑐 P2_{1}/c and propagation vector k = (0, 0, 0).

			BV Components			Positional coordinates
IR	BV	Atom	m∥a	m∥b	m∥c	x	y	z
$\Gamma$2	$\psi$4	1	1	0	0	$x$	$y$	$z$
		2	-1	0	0	$\overline{x}$+1	$y$+$\frac{1}{2}$	$\overline{z}$+$\frac{1}{2}$
		3	-1	0	0	$\overline{x}$+1	$\overline{y}$+1	$\overline{z}$+1
		4	1	0	0	$x$	$\overline{y}$+$\frac{1}{2}$	$z$+$\frac{1}{2}$
	$\psi$5	1	0	1	0	$x$	$y$	$z$
		2	0	1	0	$\overline{x}$+1	$y$+$\frac{1}{2}$	$\overline{z}$+$\frac{1}{2}$
		3	0	-1	0	$\overline{x}$+1	$\overline{y}$+1	$\overline{z}$+1
		4	0	-1	0	$x$	$\overline{y}$+$\frac{1}{2}$	$z$+$\frac{1}{2}$
	$\psi$6	1	0	0	1	$x$	$y$	$z$
		2	0	0	-1	$\overline{x}$+1	$y$+$\frac{1}{2}$	$\overline{z}$+$\frac{1}{2}$
		3	0	0	-1	$\overline{x}$+1	$\overline{y}$+1	$\overline{z}$+1
		4	0	0	1	$x$	$\overline{y}$+$\frac{1}{2}$	$z$+$\frac{1}{2}$

Table 4. Table 4:  Comparison of measured and calculated magnetic moments of two Co sites in Co 2 V 2 O 7 . The Co magnetic moments are in the unit of μ B subscript 𝜇 𝐵 \mu_{B} .

	Experiment			Constrained			Relaxed
	mx	my	mz	mx	my	mz	mx	my	mz
Co1	-0.260	1.773	0.968	-0.34	2.32	1.27	-0.30	2.33	1.25
	0.260	1.773	-0.968	0.34	2.32	-1.27	0.30	2.33	-1.25
	0.260	-1.773	-0.968	0.34	-2.32	-1.27	0.29	-2.34	-1.25
	-0.260	-1.773	0.968	-0.34	-2.32	1.27	-0.30	-2.34	1.24
Co2	0.348	2.376	1.271	0.34	2.33	1.24	0.37	2.37	1.15
	-0.348	-2.376	-1.271	-0.34	-2.33	-1.24	-0.37	-2.38	-1.14
	-0.348	2.376	-1.271	-0.34	2.33	-1.24	-0.36	2.37	-1.15
	0.348	-2.376	1.271	0.34	-2.33	1.24	0.37	-2.37	1.15

Table 5. Table 5:  Calculated total energy (meV/Co) of FM, AFM, and experimental non-collinear (denoted as NC) magnetic states with U = 0, 1.6, 3.2, and 4.8 eV using VASP.

U (eV)	0	1.6	3.2	4.8
FM	0.00	0.00	0.00	0.00
AFM	3.86	-8.59	20.78	18.72
NC	19.88	-7.17	-5.06	-3.18

Full text

I Introduction

Low-dimensional spin systems have attracted extensive interest as their low dimensionality and complex multiple-spin exchange interactions often lead to novel ground states as well as intriguing magnetic behavior, such as spinon excitations in one-dimensional (1D) spin chain system KCuF3 Tennant et al. (1993); Lake et al. (2005), and a spin liquid state in two-dimensional triangular lattice antiferromagnet NiGa2S4 Nakatsuji et al. (2005). The complex magnetic and structural characteristics of low-dimensional spin systems will also make its magnetic state highly sensitive to external perturbations including temperature and magnetic field. For instance, the field-induced quantum phase transition from the Néel ordered phase to the spin liquid one is observed in quasi-1D antiferromagnet BaCo2V2O8 Kimura et al. (2007), and a cascade of magntization plateaus in spin chain system Ca3Co2O6 Lampen et al. (2014); Kageyama et al. (1997); Kudasov (2006).

The family of cobalt vanadium oxides CoxV2O5+x with x = (1, 2 or 3) has attracted special attention due to their complex crystal structure and magnetic behavior, depending on the value of the subscript x He et al. (2009a); Drees et al. (2015); He et al. (2009b); Balakrishnan et al. (2004). The x = 1 compound CoV2O6 is a low-dimensional spin system which crystalizes in two structural polymorphs with one having a monoclinic unit cell and the other having a triclinic unit cell Mocala and Ziolkowski (1987), i.e. $\alpha$-CoV2O6 and $\gamma$-CoV2O6. Although two polymorphs possess different structural symmetries, both $\alpha$-CoV2O6 and $\gamma$-CoV2O6 are quasi-1D Ising spin-chain compounds. Application of external magnetic field leads to the observation of unusual 1/3 magnetization plateaus in both polymorphs He et al. (2009a); Lenertz et al. (2011); Nandi and Mandal (2016); Shu et al. (2016), which is attributed to the strong spin exchange and the strong coupling between orbital, magnetic and structural orders Wallington et al. (2015); Hollmann et al. (2014); Kimber et al. (2011). For x = 3, however, the Co3V2O8 consists of buckled Kagome layers of edge-sharing CoO6 octahedra separated by non-magnetic V5O4 tetrahedra, showing a geometrically frustrated lattice and a strong two-dimensional magnetic character Chen et al. (2006); Ramazanoglu et al. (2009). Interestingly, a dielectric anomaly is observed around the ferromagnetic (FM) transition temperature TC indicating the existence of strong coupling between spin and charges in Co3V2O8 Bellido et al. (2007).

When x = 2, Co2V2O7 adopts a dichromate K2Cr2O7-type structure, which crystallizes in a monoclinic system with space group $P2_{1}/c$ Sauerbrei et al. (1974). As shown in Fig. 1, Co2V2O7 possesses two different Co2+ sites with two differing local bonding geometries, which are linked in a geometry that can be referred to as a zigzag chain consisting of arrays of edge-shared CoO6 octahedra. The zigzag chains are interspersed with nonmagnetic V5+ in VO4 pyramid blocks, resulting in a quasi-1D structural arrangement He et al. (2009b). Touaiher et al. Touaiher et al. (2004) reported that Co2V2O7 shows a ferromagnetic ordering while Ni2V2O7 shows an antiferromagnetic (AFM) ordering. However, He et al. He et al. (2009c) argued that Co2V2O7 and Ni2V2O7 are likely possessing a similar magnetic ordering since they have the same crystal structure. Recently, two fascinating magnetization plateaus–a 1/2-like plateau and a 3/4 plateau, are observed in Co2V2O7, depending on crystallographic directions Yin et al. (2019); Chen et al. (2018a), which is quite different from the isostructural Ni2V2O7 with a 1/2 plateau Ouyang et al. (2018). The first-principles calculations show that the AFM ground state results in a \uparrow$$\downarrow$$\uparrow$$\downarrow-type of spin structure under the assumption that the spin configurations are collinear in the calculations Yin et al. (2019). In addition, magnetic field induced ferroelectricity was reported and found to be correlated with the magnetization plateaus in both Co2V2O7 and Ni2V2O7 Chen et al. (2018a). Moreover, the deviations of the dielectric constant with respect to lattice contributions is evidently observed in Co2V2O7 under zero magnetic field, suggesting the appearance of magnetostrictive effects induced by magnetic interactions Sanchez-Andujar et al. (2011). Thus, by investigating the magnetic ground state and the response of the lattice to the magnetic phase transitions, it is possible to develop a deeper understanding of the spin-lattice couplings in Co2V2O7 Chen et al. (2018b).

Although there are a few reports on Co2V2O7, its magnetic structure remains ambiguous because of the complexity of the structure, which is crucial to understand the field-induced magnetization plateaus, ferroelectricity as well as the magnetostrictive effect. In this paper, we report the structural and magnetic properties of Co2V2O7 by magnetization and neutron powder diffraction methods. Both nuclear and magnetic structures are determined through a refinement of high resolution neutron powder diffraction data at low temperature of 2 K. It is found that Co2V2O7 exhibits a non-collinear magnetic structure with Co spins forming a canted \uparrow$$\uparrow$$\downarrow$$\downarrow spin chain. The non-collinear magnetic ground state and electronic structure of Co2V2O7 can be well understood based on ab initio calculations. Moreover, according to the exchange striction model, the appearance of electric polarization can be predicted in Co2V2O7 along the chain direction.

II Experimental Methods

II.1 Sample preparation and characterisation

A Polycrystalline sample of Co2V2O7 was prepared using the sol-gel method by dissolving stoichiometric amounts of high purity Co(NO3)2 $\cdot$ 6H2O and NH4VO3 in deionized water. The X-ray powder diffraction measurement was carried out on a Huber diffractometer with Cu K$\alpha$ radiation $\lambda$ = 1.5406 Å and a graphite monochromator. A Quantum Design Physical Property Measurement System (PPMS) was used to characterize the magnetic properties of the Co2V2O7 powder sample in the temperature range from 2 to 350 K and the magnetic field range from 0 to 9 T. The temperature dependence of magnetization was measured during warming from 2 to 350 K under 1000 Oe after cooling down from 350 K without the applied field (zero field cooled-ZFC) or under the applied field (field cooled-FC), while the field dependence of magnetization was measured in magnetic field up to 9 T at selective temperatures.

To study the crystal and magnetic structure of Co2V2O7 systematically, time-of-flight (TOF) neutron powder diffraction (NPD) experiments were performed on the high resolution neutron powder diffractometer POWGEN Huq et al. (2011) at the Spallation Neutron Source (SNS) in Oak Ridge National Laboratory. About 5 gram of powder sample was loaded in an 8 mm diameter vanadium sample holder and then installed in the helium cryostat that can reach low temperatures down to 2 K. Neutron diffraction data were collected for the Co2V2O7 powder sample in the temperature range from 2 to 300 K. For each temperature the data were collected using two different center wavelengths, viz., 1.066 and 3.731 Å to cover a large d-space range of 0.5-12 angstrom. The representation analysis were carried out using the program SARAh Wills (2000), whereas the crystal and magnetic structure refinements were carried out from the NPD data using the Rietveld refinement program FullProf Rodríguez-Carvajal (1993).

II.2 Calculation details

We have calculated electronic structure, magnetic properties and total energies difference between FM, AFM, and the experimental non-collinear (NC) magnetic states using the Vienna ab initio simulation package (VASP) Kresse and Hafner (1993); Kresse et al. (1996). The nuclei and core electrons were described by the projector augmented wave (PAW) potential Kresse and Joubert (1999). The wave functions of valence electrons were expanded in a plane-wave basis set with cutoff energy of up to 520 eV. The VASP calculations were performed with 240 k points in IBZ. The correlation effects are also considered by using the generalized gradient approximation (GGA)+U method Dudarev et al. (1998). For transition metals, typical U values are between 1.5 to 6 eV Şaşıoğlu et al. (2011). Here, various U parameters between 0 to 4.8 eV are used to investigate the dependence of spin configuration on correlation effects. The experimental lattice parameters and atomic coordinates measured through neutron diffraction at 2K, as listed in Table I, were used in all calculations.

III Experimental results and discussion

III.1 Magnetization and magnetic phase transition

Figure 2 shows the temperature dependence of the magnetization of the Co2V2O7 polycrystalline sample under magnetic fields of 1000 Oe. With decreasing temperature, the magnetization increases smoothly in a temperature range from 350 K to 6.3 K, while it starts decreasing when the sample is cooled below 6.3 K. The phase transition at 6.3(1) K can be clearly observed as indicated by the maximum of the magnetization data. The temperature of 6.3(1) K can be denoted as the Néel temperature T**N where the paramagnetic state develops from the antiferromagnetic phase with increasing temperature, as will be demonstrated by neutron diffraction results in the following text.

Temperature dependence of the inverse susceptibility 1/$\chi$ of Co2V2O7 compound is also exhibited in Fig. 2. It is known that the magnetic susceptibility of localized noninteracting magnetic ions in high temperature region can be written as: $\chi(T)=\frac{C}{T-\Theta_{CW}}$, where C and $\theta_{CW}$ are the Curie constant and Curie-Weiss temperature, respectively. The Curie constant can be expressed as $C=NM^{2}_{eff}\mu^{2}_{B}/3k_{B}$, where N is the density of magnetic ions, $M_{eff}$ is the effective paramagnetic moment, $k_{B}$ is the Boltzmann constant, and $\mu_{B}$ is the Bohr magneton. As indicated in Fig. 2, the susceptibility strictly follows the Curie-Weiss behavior in high temperature range, while the inverse susceptibility 1/$\chi$ deviated from the Curie-Weiss estimation below 120 K, indicating the onset of considerable magnetic correlations. The effective paramagnetic moment is deduced to be 3.77(1) $\mu_{B}\,$ and it matches very well with the effective magnetic moment of Co2+, which can be obtained using $p_{eff}=g\sqrt{S(S+1)}$, where $g$ = 2 is the Landé g factor and S = 3/2. The fit of high temperature susceptibility by means of Curie-Weiss law gives a negative Curie-Weiss temperature of -16.1(5) K, indicating that the dominant magnetic interactions are antiferromagnetic in Co2V2O7. Moreover, by comparing the Curie-Weiss temperature $\theta_{CW}$ with the Néel temperature $T_{N}$, the frustration parameter Balents (2010) f = $\mid\Theta_{CW}\mid$/$T_{N}$ is obtained to be 2.5(1), suggesting a moderate spin frustration in Co2V2O7.

Figure 3 (a) shows magnetization measured as a function of magnetic field at 2 K. Although the magnetization increase monotonically upon the application of magnetic field, the saturated magnetization still could not be reached upon the application of magnetic field up to 9 T. It is also noticed that a kink takes place at 4.2 T in M-H curve, which suggestes that Co2V2O7 undergoes a spin-flop transition with increasing magnetic field. The field-induced spin-flop transition is also clearly emphasized in the derivative curve of the magnetization data and it can be considered as the result of competition between Zeeman energy E = -$\mu$0\vec{M}$$\cdot$$\vec{H} and magnetic anisotropy energy in Co2V2O7. The M-H curves at various temperatures are presented in Fig. 3(b). The spin-flop transition can still be observed at 5 K, while it is not discernible once temperature increased to above $T_{N}$. Nevertheless, considerable magnetization value, e.g. 2.85 $\mu_{B}/f.u.$ at 10 K and 9 T, can still be obtained above $T_{N}$, which indicates that the application of a magnetic field can not only change the ordering configuration but also enhances the magnetic ordering temperature of the Co2+ moment. With further increase in temperature, the magnetization of Co2V2O7 decreased and exhibited typical antiferromagnetic behavior.

III.2 Crystal structure and non-collinear magnetic ground state determined by NPD

In order to clarify the magnetic ground state of Co2V2O7, neutron diffraction patterns were collected in a wide temperature range from 2 to 300 K. Fig. 4 shows the NPD patterns of Co2V2O7 at 300 K and 2 K. Two different center wavelengths, i.e. 1.066 and 3.731 Å are applied during the measurements so that a great number of diffraction peaks can be obtained in a large d-space from 0.5 to 12 Å. Two neutron diffraction patterns collected with two different center wavelengths were simultaneously refined for each temperature. At 300 K, Co2V2O7 crystallizes in the monoclinic phase with space group $P2_{1}/c$. By taking advantage of large neutron scattering cross section of Oxygen, the accurate position of Oxygen atom can be easily determined. However, the atomic position of Vanadium can not be determined precisely as the nuclei of Vanadium have a small neutron scattering cross section. Therefore, X-ray diffraction pattern are collected for Co2V2O7 at 300 K to determine the atomic position of Vanadium since Vanadium is a strong scatterer of X-rays. The detailed structural information for Co2V2O7 at 300 K, as obtained from combining refinement of both NPD and XRD data, are given in Table I. The crystal structure of Co2V2O7 and the octahedra of CoO6 and are illustrated in Fig. 1. The bond length and bond angle of CoO6 octahedra at 300 K is shown in Table II. The average Co-O bond length of CoO6 for two nonequivalent Co sites, Co1 and Co2, are 2.074(5) and 2.083(5) Å respectively, indicating a slightly different degree of distortion for these two octahedra. The crystal structure consists two non-equivalent Co sites. With decreasing temperature, the monoclinic structure of Co2V2O7 persists down to 2 K accompanied by a shrinkage of lattice. The results of structural refinements at 2 K are also presented in Table I.

In contrast to the NPD patterns at 300 K, extra peaks with considerable intensities show up once the sample was cooled to 2 K, which is below the Néel temperature T**N = 6.3 K. Most of new emerging peaks possess relatively large d-value and can be assigned as magnetic Bragg peaks, indicating the formation of long range magnetic order of Co2+ moments. It is also found that all magnetic reflections can be indexed as commensurate reflections with the magnetic propagation wave vector k = (0, 0, 0), which means that the magnetic structure is commensurate in nature.

Symmetry analysis allows the determination of the symmetry-allowed magnetic structures based on the representation theory. Magnetic symmetry analysis was carried out for Co2V2O7 using the program SARAh Wills (2000) by which the allowed symmetry couplings in the form of irreducible representation and their basis vectors can be deduced. It is known that two Co2+ ions occupy two inequivalent crystallographic Wyckoff sites 2e in Co2V2O7 with the crystallographic space group P21/c. With the commensurate magnetic propagation vector k = (0, 0, 0), both Co ions are found to have four one-dimensional irreducible magnetic representations (IRs), i.e. $\Gamma$1, $\Gamma$2, $\Gamma$3, $\Gamma$4, and each of them is composed of three basis vectors (BV) with only real components. Assuming that Co atoms located at two nonequivalent crystallographic sites carry magentic moments with different amplitude as in Co3V2O8 Chen et al. (2006) counterpart, all possible IRs are adopted into the magnetic structural model and used to fit against the NPD pattern at 2 K.

As demonstrated in Fig. 4(d), a satisfactory fitting of all magnetic reflections can only be obtained by using IR $\Gamma$2 with Shubnikov space group of P21/c’, while other three representations can be ruled out due to poor fit of magnetic reflections. The decomposition of the magnetic representation $\Gamma$2 in terms of basis vectors are given in Table III. According to $\Gamma$2, the Co moments align antiferromagnetically in the magnetic unit cell and the refined vector components of magnetic moment are obtained to be different for two individual Co 2e sites, i.e. $m_{x}$ = -0.26(5) $\mu_{B}$, $m_{y}$ = 1.77(5) $\mu_{B}$ and $m_{z}$ = 0.97 (5) $\mu_{B}$ for Co1 site, $m_{x}$ = 0.35(5) $\mu_{B}$, $m_{y}$ = 2.38(5) $\mu_{B}$ and $m_{z}$ = 1.27 (5) $\mu_{B}$ for Co2 site, respectively. It is obvious that the canted Co moments ordered antiferromagnetically in b-c plane and form a spin-chain-like structure along c-axis, as illustrated in Fig. 5. The refined magnetic moments of Co in the spin chain are 2.06 (5) and 2.69 (5) $\mu_{B}$ for Co1 and Co2, respectively. The canted angle of Co moment toward c-axis is around 26(1)∘. Nevertheless, the tilt of Co spin does not follow the tilt of CoO6 octahedra, suggesting the negligible influence of electric multipolar field generated by oxygen neighbors on the orbitals of Co ions. The antiferromagnetic ground state of Co2V2O7 is apparently determined by the interplay between spin-spin exchange interaction and magnetocrystalline anisotropy, which are largely dependent on the electronic configuration of Co ions. It is known that the up–up–down–down spin arrangement along the alternating array of two magnetic sites can break the inversion symmetry, and the inequivalent exchange striction working between the up–up (or down–down) spin pair and the up–down (or down–up) can produce electric polarization along the chain direction Tokura et al. (2014). Obviously, the exchange-striction mechanism can be applied to Co2V2O7 with canted \uparrow$$\uparrow$$\downarrow$$\downarrow spin chains (see Fig. 5) and the electric polarization will emerge as a consequence.

III.3 Calculation results on magnetic ground state

The calculation started with the experimental spin structure to investigate the electronic structure and magnetic properties of Co2V2O7. Two sets of non-collinear calculations are performed: One was with a constraint that fixes Co magnetic moment directions during self-consistent calculations, and the other was without this constraint so that both the directions and magnitudes of each Co magnetic moment were relaxed. Table IV shows the magnetic moment components of each Co atom obtained from NC magnetic calculations with and without relaxing the spin directions. Experimental values are also included to compare. The calculations were carried out using DFT+U with U = 3.2 eV. The calculations show similar magnetic moments for two non-equivalent Co sites while the experiment found noticeably different magnetic moments between two Co sites. The further relaxation of spin direction only introduces small changes on the amplitudes and directions of Co moments, suggesting the obtained spin structure is rather stable.

Atom-projected partial densities of states (PDOS) of the relaxed spin configuration are shown in Fig. 6. The calculations were performed within GGA and GGA+U with U = 3.2 eV, respectively. In the case of U = 0 eV, there are sharp Co-DOS peaks near the Fermi energy. The bandgap sits between two sharp DOS peaks, which mostly consist of Co-d states that barely hybridize with other states and form flat bands. The bandgap is about 0.2 eV within GGA and depends on the applied U parameters with GGA+U. The PDOS with U = 3.2 eV are shown in Fig. 6 (b). As expected, a larger bandgap is obtained. It seems that U potential is able to rearrange Co states to promote the hybridization of Co-d states with other states so that NC magnetic structure becomes stable. It would be interesting to measure the bandgap, which may guide justifying the U parameter.

Considering that the relaxed spin configuration is not necessary the ground state spin configuration, here, we calculate and compare the total energies of FM, AFM, and NC magnetic structures using VASP. The dependence of these energy differences on the correlation parameter U is also investigated. The plain DFT calculations show that FM state has the lowest energy, which is 3.86 eV/Co and 19.88 eV/Co lower than the AFM state and the NC state, respectively. It is well known that LDA (GGA)+U method gives better agreement with experimental results than plain LDA (GGA) method for oxide gap materials because of strong correlation. We performed calculations with three different U parameters to understand their effects on the magnetic states. Table V shows the total energies for FM, AFM and NC magnetic structures with various U values. It shows that the magnetic ground state depends on the magnitude of U potential. A larger U parameter tends to stabilize the NC state.

III.4 Anisotropic magnetoelastic coupling and thermal expansion

The change of individual lattice parameters with temperature can be obtained through the refinement of NPD patterns collected at different temperatures. As shown in Fig. 7 (a)-(d), the onset of magnetoelastic distortions below T**N = 6.3 K is clearly visible for lattice parameters a and b, while the lattice parameter c and $\beta$ stay almost constant upon crossing the Néel temperature. It is worth noting that the sign of the deviation for a is opposite to that for b, indicating the existence of anisotropic magnetoelastic coupling in Co2V2O7. The corresponding unit-cell volume is shown in Fig. 7(e). Although a elongates and b shrinks upon cooling below TN, they compensate each other in the volumetric expansion. The unit-cell volume V shows no anomamalous change and increases monotonically with temperature.

Because the tilt of Co spin does not follow the tilt of CoO6 octahedra, we speculate that the influence of electric multipolar field generated by oxygen neighbors on the orbitals of Co ions is negligible. The antiferromagnetic ground state of Co2V2O7 is apparently determined by the interplay between spin-spin exchange interaction and single-ion anisotropy. In order to understand anisotropic magnetoelastic coupling and thermal expansion, we correlate the variation of distance between different Co ions in same spin-chain along the c-axis to antiferromagnetic phase transition. The change of distance between neighbouring Co sites at different temperatures is shown in Fig. 8(a). Both of the distance of Co1-Co1 and Co2-Co2 decrease with the increasing of temperature and remain almost constant above $T_{N}$. Figure 8(b) display the temperature dependence of the change of distance between different Co sites. According to the non-colliner magnetic structure illustrated in Fig. 5, two non-equivalent Co sites in same spin-chain form parallel or antiparallel spin configurations. The distance of Co1-Co2 with parallel spin ordering shows same behavior as the lattice parameter b, and the distance of Co1-Co2 with antiparallel spin ordering shows same behavior as the lattice parameter a. This is clearly evidence that the anisotropic magnetoelastic coupling and thermal expansion of lattice parameters are most likely associated with the different exchange interactions between two neighboring Co spins.

IV conclusions

In summary, we have studied the magnetic properties of Co2V2O7 by extensive approaches, such as magnetization measurements, neutron powder diffraction measurements and theoretical calculations. By modeling the temperature dependence of the magnetization with the Curie-Weiss law, we find that the Curie-Weiss temperature is negative, e.g., -16.1(5) K, indicating dominating antiferromagnetic coupling. A spin-flop transition was observed while magnetic field was applied below $T_{N}$= 6.3 K. By applying neutron powder diffraction methods, we confirm that Co2V2O7 remains monoclinic from 300 K down to 2 K. The magnetic ground state was determined unambiguously based on NPD results by using representation analysis combined with magnetic Rietveld refinements. Two non-equivalent Co sites with different magnetic moments form a spin-chain-like structure along c-axis and canted antiferromagnetic ordering in b-c plane. By comparing total energies between different magnetic states, we found that the theoretical magnetic ground state is sensitive to the U parameter and GGA+U calculations show that the non-collinear magnetic state possesses a lower energy than FM and AFM states. The ferroelectricity in Co2V2O7 can be understood based on exchange striction model, which can be applied to canted \uparrow$$\uparrow$$\downarrow$$\downarrow spin configurations. Moreover, the temperature dependence of lattice parameters in Co2V2O7 exhibits an anisotropic magnetoelastic coupling between the a, b and c below $T_{N}$.

V Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11874023) and Project Based Personnel Exchange Program (PPP) with China Scholarship Council (CSC Nos. 2016-6041) and German Academic Exchange Service (Project-ID 57219934). Part of the research conducted at SNS was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, US Department of Energy. WHJ would like to acknowledge financial support from the China Scholarship Council(CSC).