Structural, electronic and magnetic properties of the CuFeO





2





 multiferroic compound

A. Maouhoubi; L. Ouzaroual; Y. Toual; S. Mouchou; F. Mezzat; A. Azouaoui; H. Ez-Zahraouy; A. Hourmatallah; K. Bouslykhane; N. Benzakour

doi:10.1016/j.arabjc.2023.105437

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01 2023

:17;

105437

doi:

10.1016/j.arabjc.2023.105437

Structural, electronic and magnetic properties of the CuFeO $_{2}$ multiferroic compound

A. Maouhoubi^a,⁎, L. Ouzaroual^b, Y. Toual^a, S. Mouchou^a, F. Mezzat^b, A. Azouaoui^a, H. Ez-Zahraouy^b, A. Hourmatallah^a,c, K. Bouslykhane^a, N. Benzakour^a

a Faculty of Sciences Dhar El Mehraz, Laboratory of Solid Physics, Sidi Mohamed Ben Abdellah University, BP 1796, Fez, Morocco

b Faculty of Sciences Rabat, Laboratory of Condensed Matter and Interdisciplinary Sciences, Mohammed V University, Av. Ibn Battouta, B.P. 1014, Rabat, Morocco

c Ecole Normale Supérieure, Sidi Mohamed Ben Abdellah University, BP 5206, Fes, Morocco

⁎Corresponding author. maouhoubi@gmail.com (A. Maouhoubi)

Received: 2023-7-21, Accepted: 2023-11-4,

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Abstract

The structural, electronic and magnetic properties of the delafossite material CuFeO $_{2}$ were studied using Density Functional Theory ( $D F T$ ) combined with Monte Carlo simulation ( $M C S$ ). The structural stability was verified in this work and showed that this material is antiferromagnetic (AFM). The density of states (DOS) carried out using the generalized Perdew–Burke–Ernzerhof gradient ( $P B E$ - $G G A$ ), Tran-Balaha modified Becke–Johnson exchange potentials ( $T B$ - $m B J$ ; $n m B J$ ) and the $G G A$ ＋ $U$ show that the compound has a metallic behavior in $P B E$ - $G G A$ and semiconductor behavior in $T B$ - $m B J$ , $n m B J$ and $G G A$ ＋ $U$ approximations. The obtained partial magnetic moment was used to study the magnetic behavior by Monte Carlo simulation, particularly, the evolution of magnetization as a function of temperature and phase transition. The results of $M C S$ showed the presence of two-phase transitions: ferromagnetic–antiferromagnetic ( $F M$ - $A F M$ ) first-order phase transition and antiferromagnetic–paramagnetic ( $A F M$ - $P M$ ) second-order phase transition. The Neel temperatures found are $T_{N_{2}} = 11.2$ K for the first transition and $T_{N_{1}} = 16.6$ K for the second transition.

Keywords

CuFeO2

GGA＋ U

TB-mBJ

Monte Carlo simulation

Antiferromagnetic

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1 Introduction

Multiferroics have received considerable attention as members of the multifunctional materials class (Vopson, 2015). For this reason, the multiferroics thanks to their belonging to this class of materials are the subject of ambitious research work (Vopson, 2015; Mao et al., 2022). In current research in condensed matter physics, these materials are gaining increasing attention due to their multiple properties (Seidel, 2016): ferromagnetic (Zurbuchen et al., 2005; Jungwirth et al., 2006), ferroelectric (Behera et al., 2021; Wang et al., 2023b) and ferroelastic (Jiao et al., 2022), where recent research focuses on simultaneous correlation properties (Szaller et al., 2014) such as magnetoelectric (Kimura et al., 2003). The magnetic and electrical order that appeared simultaneously (Wang et al., 2023a) has the advantage of combining properties that could make these materials useful for storage (Ahmed et al., 2021), processing (Rovillain et al., 2010) and information transmission (Wang et al., 2023a). The magnetic and elastic order (Maldovan and Thomas, 2006) namely magnetoelastic coupling links magnetostriction to piezomagnetism (Lines, 1979; Meng et al., 2023; Jaime et al., 2017 ) that can be useful in applications in the development of force sensors (Bieńkowski and Szewczyk, 2004). There is also the electrical and elastic order (Kuo et al., 2020) that can be used to create ferroelectric materials for applications in piezoelectric and optical devices (Alikin et al., 2022). The current challenge for scientists is to design the best devices that perform several functions simultaneously.

The multiferroic ${CuFeO}_{2}$ is an iron copper oxide mineral. It belongs to the delafossite group of the general formula $A B O_{2}$ where $A$ and $B$ are metals. The variation of the cations $A$ and $B$ allows for having new properties, particularly interesting, which are applied to both basic science and applications. ${CuFeO}_{2}$ has great potential applications in Lithium-ion battery anodes (Kim et al., 2020; Zhang et al., 2018; Dong et al., 2014 ), solar energy conversion (Liu et al., 2020; Li et al., 2024; Jin and Chumanov, 2016 ), water splitting catalyst and rhodamine B sensor (Mao et al., 2022). Various works have been done to synthesize the ${CuFeO}_{2}$ compound in several shapes and sizes such as nanoscale hexagonal plates and particles, powders, films and thin films (John et al., 2016; Jin and Chumanov, 2016; Ziani et al., 2021; Li et al., 2012; Wolff et al., 2020 ). At room temperature, it crystallizes in two different space groups (SG): $3 R$ - $p o l y t y p e$ (SG $R$ - $3 m$ ) and $2 H$ - $p o l y t y p e$ (SG $P 63 / m m c$ ) (John et al., 2016) (see Fig. 1). Iron atoms are trivalent, that is ${Fe}^{3 +}$ for $3 R$ - $p o l y t y p e$ ; an unambiguously approved experimental result by Mössbauer resonance (Ataoui et al., 2003), they occupy octahedral sites ( ${FeO}_{6}$ ) of shared edges, forming parallel layers alternately with layers of ${Cu}^{+}$ cations and separated by oxygen ions $O^{2 -}$ hence the formula ( ${Cu}^{+} {Fe}^{3 +} O_{2}^{2 -}$ ). ${CuFeO}_{2}$ is also present in another phase ( ${Cu}^{2 +} {Fe}^{2 +} O_{2}^{2 -}$ ) containing ${Fe}^{2 +}$ and ${Cu}^{2 +}$ crystallizing in SG $R$ - $3 m$ (John et al., 2016).

The present work is carried out in terms of computational methods, draws on recent work that has examined and illustrated the physical properties of various compounds, using density functional theory combined with Monte Carlo simulation. Among these works, the studies reported in Azouaoui et al. (2020b), Selmani et al. (2021), Idrissi et al. (2020a), Mouatassime et al. (2021), Idrissi et al. (2020b, 2019a), Azouaoui et al. (2021). In addition, recent works dealing with phase transition phenomena using the Monte Carlo simulation, such as the works reported in Mallick and Acharyya (2023), Idrissi et al. (2019b, 2021), Mtougui et al. (2019), Khalladi et al. (2019). We also add the high temperature series expansions method, which can be used to calculate the magnetic parameters, such as the critical temperature as mentioned in the Masrour et al. (2013), Azouaoui et al. (2020a). To our knowledge, no theoretical investigations have been performed using DFT combined with MCS were conducted for studying the three phases FM, AFM, and NM on the ${CuFeO}_{2}$ compound.

In this work, we have investigated the structural, electronic and magnetic properties of the delafossite ${CuFeO}_{2}$ material by the Density Functional Theory and Monte Carlo simulation. We have used several approximations to process the exchange and correlation energies, namely: the generalized Perdew–Burke–Ernzerhof gradient ( $P B E$ - $G G A$ ), Tran-Balaha modified Becke–Johnson exchange potentials ( $T B$ - $m B J$ ; $n m B J$ ) and the $G G A + U$ . We used the approximation $P B E$ - $G G A$ to optimize the lattice parameters of the two polytypes $2 H$ and $3 R$ of the multiferroic compound ${CuFeO}_{2}$ . We checked which of the following three phases is the most stable in the ground state: nonmagnetic ( $N M$ ), ferromagnetic ( $F M$ ) or antiferromagnetic ( $A F M$ ) configuration. Thus, the $P B E$ - $G G A$ , $T B$ - $m B J$ , $n m B J$ and $G G A + U$ approximations allowed us to calculate the density of states $D O S$ , energy gap ( $E_{g}$ ) and the partial magnetic moment in the $2 H$ - $p o l y t y p e$ structure. These results are used as input in Monte Carlo simulation to study the evolution of magnetization, magnetic susceptibility and the specific heat as a function of temperature to study the phase transitions and especially to determine the Neel temperatures $T_{N_{1}}$ and $T_{N_{2}}$ .

2 Computational method

Ab initio calculations based on density functional theory and implemented in $W i e n 2 k$ package (Blaha et al., 2019, 2020) were performed to study the structural, electronic and magnetic properties of ${CuFeO}_{2}$ compound, which crystallizes in two different space groups, $3 R$ - $p o l y t y p e$ (SG $R$ - $3 m$ ) and $2 H$ - $p o l y t y p e$ (SG $P 63 / m m c$ ) (see Fig. 1). The exchange and correlation energies were calculated with several different approximations: the generalized gradient approximation ( $G G A$ ) using the corrected Perdew–Burke–Ernzerhof gradient function ( $P B E$ ) (Perdew et al., 1996) and Tran-Balaha modified Becke–Johnson exchange potentials ( $n m B J$ - $G G A$ (Koller et al., 2012); $T B$ - $m B J$ - $G G A$ (Tran and Blaha, 2009)). We also performed $G G A + U$ calculations using various values of Hubbard $U$ parameter selections (Tran et al., 2006). The Brillouin zone integration was carried out by a Monkhorst–Pack mesh (Monkhorst and Pack, 1976) with $k$ - $p o i n t s$ 16 × 16 × 3 for $2 H$ - $p o l y t y p e$ and 23 × 23 × 1 for $3 R$ - $p o l y t y p e$ . The product of the maximum reciprocal vector module $K_{m a x}$ and the smallest radius of the atomic sphere $R M T$ is increased to a constant value ( $R M T \times K_{m a x} = 7$ Ry).

In this study, copper ( ${Cu}^{+}$ ) and oxygen ( $O^{2 -}$ ) ions are nonmagnetic. The magnetic interactions take into account magnetic iron ions ( ${Fe}^{3 +}$ ) and are modeled by the Ising Hamiltonian with spin- $\frac{5}{2}$ described by:

(1)

H = - J_{1} \sum_{〈 i, j 〉} S_{i} S_{j} - J_{2} \sum_{〈 i, k 〉} S_{i} S_{k} - J_{3} \sum_{〈 i, l 〉} S_{i} S_{l} - Δ \sum_{i} S_{i}^{2}

Where, the notations

〈 i, j 〉

〈 i, k 〉

and

〈 i, l 〉

indicate the summations run over the first, second and third neighbors, respectively.

Crystal structure of CuFeO 2 : ( a ) 2 H - p o l y t y p e in SG P 63 / m m c and ( b ) 3 R - p o l y t y p e in SG R - 3 m .

$S_{i}$ is a spin of the lattice, $S_{j}$ , $S_{k}$ and $S_{l}$ are the first, second and third nearest neighbors of spin $S_{i}$ , respectively;
$J_{1}$ , $J_{2}$ and $J_{3}$ are the first, second and third nearest neighbors interactions, respectively;
$Δ$ is the crystal field related to the presence of $d$ orbitals.

The study of the thermal and magnetic properties of ${CuFeO}_{2}$ compound in hexagonal structure is performed by Monte Carlo simulation ( $M C S$ ) with the Metropolis algorithm (Janke, 2008). The spin value of ${Fe}^{3 +}$ ions is $S = 5 / 2$ , with the electronic structure $[Ar]$ : $3 d^{5}$ . The spins are distributed in the sites of a hexagonal lattice (see Fig. 2), and can randomly have the values $\pm \frac{1}{2}$ , $\pm \frac{3}{2}$ and $\pm \frac{5}{2}$ . To take account of the $A F M$ configuration, the lattice is divided into two types of parallel plans, one is even ( $S^{'} = S$ ) and the other odd ( $S^{″} = S$ ). Our data was generated using $1 0^{6}$ Monte Carlo steps per spin Jabar et al. (2018), $1 0^{5}$ configurations are discarded to reach the equilibrium and account for fluctuations due to the thermalization process and numerical stability. $1 0^{7}$ Monte Carlo steps are used around the transition temperatures $T_{N_{1}}$ and $T_{N_{2}}$ to calculate them accurately. The size of the system is $N = N_{x} \times N_{y} \times N_{z}$ , with $N_{x} = N_{y} = 10$ and $N_{z} = 6$ are the numbers of occupied sites in the unit cell with directions $x$ , $y$ and $z$ , respectively. This corresponds to the total number of magnetic atoms $600$ . To simulate a bulk material, the choice is made on the periodic boundary conditions in the three directions. The average values of the calculated physical quantities are:

The J 1 , J 2 and J 3 interactions of a given site in a hexagonal lattice of a central S i spin interacting with the 6 S j , 6 S k and 2 S l .

The total energy of the system $E$ :

(2)

E = \frac{1}{N} 〈 H 〉;

The total magnetization

M

(3)

〈 M 〉 = \frac{1}{N} \sum_{i = 1}^{N} S_{i};

Average magnetization

m_{1}

of the even planes containing

S_{i}^{'}

spins:

(4)

〈 m_{1} 〉 = \frac{1}{N} \sum_{i = 1}^{\frac{N}{2}} S_{i}^{'};

Average magnetization

m_{2}

of the odd planes containing

S_{i}^{″}

spins:

(5)

〈 m_{2} 〉 = \frac{1}{N} \sum_{i = 1}^{\frac{N}{2}} S_{i}^{″};

Magnetic susceptibilities

χ

χ_{1}

and

χ_{2}

(6)

χ = \frac{1}{k_{B} T} [〈 M^{2} 〉 - {〈 M 〉}^{2}],

(7)

χ_{1} = \frac{1}{k_{B} T} [〈 m_{1}^{2} 〉 - {〈 m_{1} 〉}^{2}],

(8)

χ_{2} = \frac{1}{k_{B} T} [〈 m_{2}^{2} 〉 - {〈 m_{2} 〉}^{2}];

And specific heat

C_{v}

(9)

C_{v} = \frac{1}{k_{B} T} [〈 E^{2} 〉 - {〈 E 〉}^{2}] .

3 Results and discussion

3.1

3.1 Structural properties

The study of the structural properties of ${CuFeO}_{2}$ compound in hexagonal structures has been performed, it is based on the minimization of energy versus lattice parameters. Self-consistent ground-state calculations, without the polarized spin nonmagnetic configuration, and spin-polarized configurations of $F M$ and $A F M$ , were performed, separately by the $P B E$ - $G G A$ approximation, to determine the energies of each configuration by the variation of the volume and $c / a$ ratio. To ensure the accuracy of results that are interpolated through the Murnaghan equation (Murnaghan, 1944), we made the choice of k-points numbers 16 × 16 × 3 for $2 H$ - $p o l y t y p e$ and 23 × 23 × 1 for $3 R$ - $p o l y t y p e$ .

The parameters selected are those corresponding to the minimum energy of the three considered configurations. The obtained results by different calculations are shown graphically in Figs. 3 and 4. They show that both polytypes ( $2 H$ ; $3 R$ ) of ${CuFeO}_{2}$ compound are stable in AFM configuration. The optimized lattice parameters for the $N M$ , $F M$ and $A F M$ configurations are presented in Table 1. The obtained values corresponding to AFM configuration are $a = 3.0348$ , $c = 11.4687$ for $2 H$ - $p o l y t y p e$ and $a = 3.0395$ , $c = 17.1615$ for $3 R$ - $p o l y t y p e$ . The structural data of the ${CuFeO}_{2}$ delafossite have been revealed and confirmed by numerous experimental methods: X-ray diffraction and X-ray fluorescence measurements (Wolff et al., 2020); Mössbauer Spectroscopy of the Magnetic-Field-Induced (Nakamura et al., 2015; Siedliska et al., 2018); Raman Spectroscopy, Fourier Transform Infrared Spectroscopy (FT-IR), X-ray Photoelectron Spectroscopy (XPS), Field Emission Scanning Electron Microscopy (FESEM) and Energy Dispersive X-ray Spectroscopy (EDS) (Roble et al., 2019). The obtained results listed in Table 1 are in good agreement with the experimental and theoretical results reported in John et al. (2016), Shi et al. (2019), Zhao et al. (1995), Ataoui et al. (2003), Peng et al. (2021).

Table 1 The calculated values of

a

and

c

P B E

G G A

approximation together with experiment and theoretical results.

Alloys-Polytype	Config.	$a [1 0^{- 10} m]$	$c [1 0^{- 10} m]$	$E$ [Ry]
	$N M$	2.9960	11.7680	−12313.2628
${CuFeO}_{2}$ - $2 H$	$F M$	3.0327	11.4844	−12313.3201
	$A F M$	3.0348	11.4687	−12313.3242
	$E x p .$	3.0366 (John et al., 2016)	11.4550 (John et al., 2016)	–
	$N M$	3.0001	17.6151	−6156.6305
${CuFeO}_{2}$ - $3 R$	$F M$	3.0383; 2.93 (Shi et al., 2019) (Th.)	17.1749; 17.52 (Shi et al., 2019) (Th.)	−6156.6584
	$A F M$	3.0395	17.1615	−6156.6592
	$E x p$ .	3.0378 (John et al., 2016); 3.0351 (Zhao et al., 1995)	17.1806 (John et al., 2016); 17.166 (Zhao et al., 1995; Ataoui et al., 2003)	–
		3.0337 (Peng et al., 2021); 3.0345 (Ataoui et al., 2003)	17.1574 (Peng et al., 2021);	–

( a ) Energy versus volume at constant c / a ratio ( b ) Energy versus c / a ratio at constant volume, of 2 H - p o l y t y p e CuFeO 2 compound.

( a ) Energy versus volume at constant c / a ratio ( b ) Energy versus c / a ratio at constant volume, of 3 R - p o l y t y p e CuFeO 2 compound.

3.2

3.2 Electronic properties

The optimized lattice parameters corresponding to $A F M$ configuration calculated by $P B E$ - $G G A$ approximation are used to study the electronic properties of $2 H$ - $p o l y t y p e$ . The results of the total and partial $D O S$ ( $T D O S$ ; $P D O S$ ) obtained by $P B E$ - $G G A$ approximation are shown in Fig. 5. The $T D O S$ in Fig. 5(a) is perfectly symmetrical and has a continuous band at the Fermi level, showing that $2 H$ - $p o l y t y p e$ is an $A F M$ and has a metallic behavior which is confirmed by $P D O S$ of Iron atoms (see Fig. 5(b)). Copper’s TDOS and PDOS depicted in Figs. 5(c) and 5(a), respectively, are perfectly symmetrical that is due to the $3 d$ subshell contribution of ${Cu}^{+}$ ion which is completely filled ( ${Cu}^{+}$ : $[Ar] 3 d^{10}$ ). The electronic neutrality of the compound ${CuFeO}_{2}$ , as well as the perfect symmetry of the TDOS of the O atom (see Fig. 5(c)), are claimed because of the electronic configuration of the oxygen ions $O^{2 -} : [He] 2 s^{2} 2 p^{6}$ . So, copper and oxygen ions are nonmagnetic in this compound. Fig. 5(d) describes the lifting of the degeneracy of the $3 d$ iron ion subshell, which allows the calculation of the crystal field value $Δ = E_{e_{g}} - E_{t_{2 g}}$ to use as input in Monte Carlo simulation. The value of the crystal field energy is $Δ E_{c r y s t a l} = 0.3 eV$ .

It is known that $G G A$ approximation is overestimated to calculate the band gap ( $E_{g}$ ). For this reason, we have performed the calculation of $E_{g}$ of $2 H$ - $p o l y t y p e$ by $T B$ - $m B J$ , $n m B J$ and $G G A + U$ ( $2 \leq U \leq 9$ ) approximations. The obtained results show that the $T B$ - $m B J$ approximation is more accurate in calculating the energy gap according to distinct research works (Benaissa et al., 2023; Tedjini et al., 2021; Khan et al., 2022 ). The obtained DOS with these approximations are shown in Fig. 6. The values of the energy gaps ( $E_{g}$ ) and the partial magnetic moments ( $μ_{F e_{u p}}; μ_{F e_{d n}}$ ) of iron are presented in Table 2. The information from Fig. 6 affirms that $2 H$ - $p o l y t y p e$ is $A F M$ and exhibits semiconducting behavior. The energy gap is increased with U increases and depends on the approximations used. The value closest to the experimental is obtained by a potential $U = 9$ (see Table 2).

Table 2 The calculated energy gap and partial magnetic moment of iron by

T B

m B J

n m B J

and

G G A + U

approximations in

A F M

configuration together with experiment and theoretical results.

$A p p r o x i m a t i o n$		$E_{g}$ [eV]	$μ_{F e_{u p}}; - μ_{F e_{d n}} [μ_{B}]$
$G G A + U$	$(U = 2)$	0.558	1.10
	$(U = 3)$	0.664	1.11
	$(U = 3.38)$ (Hailouf et al., 2023)	0.731	1.13
	$(U = 4)$ (Zhang et al., 2011; Yoon et al., 2019)	0.805; 0.61 (FM) (Shi et al., 2019)	1.13; 3.34 (FM) (Shi et al., 2019)
	$(U = 5)$	1.004	1.15
	$(U = 6)$	1.084	1.15
	$(U = 7)$	1.097	1.16
	$(U = 8)$	1.172	1.15
	$(U = 9)$	1.252	1.15
$B T$ - $m B J$		0.879; 0.15 (FM) (Guo, 2015)	1.04; 4.11 (FM) (Guo, 2015)
$n m B J$		0.877	1.04
Experiment		1.33 (Jin and Chumanov, 2016)	1.25 (theoretical)

The total and partial density of states versus energy in approximation P B E - G G A .

The total and partial density of states versus energy in approximations m B J and G G A + U ( 2 ≤ U ≤ 5 ).

3.3

3.3 Magnetic properties and Monte Carlo simulation

The magnetic properties of the studied compound have been investigated in the ground state by $D F T$ . The results show that the total magnetic moment of iron atoms $μ_{F e} = μ_{F e_{u p}} + μ_{F e_{d n}}$ is zero and the partial magnetic moments $μ_{F e_{u p}} = - μ_{F e_{d n}}$ are presented in Table 2 for $T B$ - $m B J$ , $n m B J$ and $G G A + U$ ( $2 \leq U \leq 9$ ) approximations. The obtained values of $μ_{F e_{u p}}$ and $μ_{F e_{d n}}$ in $G G A + U$ for $2 \leq U \leq 5$ are increasing with $U$ and are vary between 1.10 to 1.15 $μ_{B}$ , from the value $U = 5$ , the magnetic moment $μ_{F e_{u p}}$ remains constant and equal to $1.15 μ_{B}$ . We can notice that the theoretical value of the average partial moment of iron atom ( ${Fe}^{3 +}$ : $[Ar] 3 d^{5}$ ) in the fundamental state is equal to $S / 2 = 1.25$ and for the Copper ( ${Cu}^{+} : [Ar] 3 d^{10}$ ), which indicates that ${Cu}^{+}$ ion is nonmagnetic, so, $μ_{C u} = 0 μ_{B}$ . The difference is due to $2 p$ - $5 d$ interaction between orbitals $2 p$ of Oxygen and $3 d$ of Iron that can give a very low moment for oxygen atoms. For the $T B$ - $m B J$ and $n m B J$ approximations, the obtained value is 1.04 $μ_{B}$ . Other theoretical results are reported in Shi et al. (2019), Guo (2015) have found a magnetic moment of iron atom in $F M$ phase (see Table 2).

The calculated coupling values $J_{1}$ , $J_{2}$ and $J_{3}$ are 5.26 meV, 1.01 meV and $- 0.79$ meV, respectively. $J_{1}$ and $J_{2}$ have positive values and $J_{3}$ is negative ( $J_{3} < 0$ ), it is obtained by calculating the energy difference $Δ E = E_{F M} - E_{A F M}$ of the AFM configuration $(S_{i}, S_{j}, S_{k}, S_{l}) = (5 / 2, 5 / 2, 5 / 2, - 5 / 2)$ (see Fig. 2) and FM configurations. The magnetic interactions $J_{1}$ , $J_{2}$ and $J_{3}$ take into account magnetic atoms of the same type, so these values are related by a relation of proportionality to interatomic distances: $| J_{3} | < | J_{2} | < | J_{1} |$ . This proportionality is taken by a classical approach to the dipole interaction between two magnetic moments separated by a distance $r$ . It is of orders $1 / r^{3}$ (Ashcroft and Mermin, 2022).

Monte Carlo simulation is used to study magnetic proprieties, namely: the evolution of total and partial magnetization ( $M$ , $m_{1}$ and $m_{2}$ ), total and partial magnetic susceptibilities ( $χ$ , $χ_{1}$ and $χ_{2}$ ) and specific heat at volume constant as a function of temperature. The magnetization curves (Fig. 7(a)) show the ferromagnetic behavior up to the temperature $T_{N_{2}}$ . This behavior is due to the weak effect of thermal agitation on the spins orientation. The spins are oriented in a direction favored by $d$ orbitals of iron ions ${Fe}^{3 +}$ , and the directions imposed by the super-exchange provided by oxygen ions $O^{2 -}$ . The crystal field $Δ$ contribute positively to the total magnetization, resulting an average value of magnetic moment of around $2.5 μ_{B}$ (S=5/2) (see black curve in Fig. 7(a)). The partial magnetization of the sub-lattices has average values of $m_{1}$ and $m_{2}$ around $1.25 μ_{B}$ . In the interval $[T_{N_{2}}, T_{N_{1}}]$ , the effect of temperature is comparable to the effects of coupling between spins. The spin sub-lattices system is much more stable in an anti-parallel orientation with zero total magnetic moment, which indicates the antiferromagnetic phase. The partial magnetization of the sub-lattices has average values $m_{1}$ and $m_{2}$ around $1.25 μ_{B}$ and $- 1.25 μ_{B}$ , respectively (see red and blue curves in Fig. 7(a)). Above temperature $T_{N 1}$ , the magnetic order can no longer occur without the application of an external magnetic field, the thermal agitation becomes predominant and spin disorientation leads to the paramagnetic phase with average values of total and partial magnetization are zero. The magnetic susceptibility and the specific heat curves (Fig. 7(b)and(c)) show a discontinuity (peak at $T = T_{N_{1}}$ ) which marks a second-order transition, the low peak at $T = T_{N_{2}}$ marks a first-order transition. The obtained values of $T_{N_{1}}$ and $T_{N_{2}}$ are in good agreement with the experimental results reported in (Song et al., 2016; Peng et al., 2021; John et al., 2016; Mekata et al., 1993; Xiong et al., 2015 )(see Table 3). The obtained curves of magnetization $M$ presented in Fig. 7(a) show that the material is $A F M$ . The $A F M$ - $P M$ phase transition is characterized by a temperature $T_{N_{1}} = 16.6$ K. The magnetic susceptibility and specific heat show a discontinuity for $T_{N_{1}}$ .

Total and partial magnetization, magnetic susceptibility and specific heat versus temperature of CuFeO 2 compound.

Table 3 The calculated Neel temperatures by

M C S

together with experiment and theoretical results.

	Our results	Experiment works	Theoretical works
$T_{N_{1}}$ (K)	16.6	14 (Song et al., 2016); 16 (Peng et al., 2021); 18.5 (John et al., 2016)	16 (Mekata et al., 1993)
$T_{N_{2}}$ (K)	11.2	11 (Song et al., 2016); 11.99 (Peng et al., 2021); 12 (Xiong et al., 2015);	11 (Mekata et al., 1993)

4 Conclusion

In conclusion, the study carried out with the help of the $W i e n 2 k$ package, allowed us to analyze and evaluate the structural, electronic and magnetic properties of the ${CuFeO}_{2}$ material delafossite by the density functional theory. We were able to verify the stability of the three ground state phases: $N M$ , $F M$ and $A F M$ by the $P B E$ - $G G A$ approximation. The obtained results of the ground state show that the compound is stable in the AFM phase. Thus, $T B$ - $m B J$ , $n m B J$ and the $G G A + U$ approximations show that this material has a semiconductor behavior, which proves our theoretical results are in good agreement with experimental results. The evolution of the magnetization and magnetic susceptibility as a function of temperature of multiferroic ${CuFeO}_{2}$ is studied by the Monte Carlo simulation. Two critical Neel temperatures are determined $T_{N_{2}}$ and $T_{N_{1}}$ with the values of 11.2 K and 16.6 K, respectively, which mark first and second order transitions. The results show that ${CuFeO}_{2}$ multifunctional is a promising material for magnetic recording applications, provided that a method of increasing the Neel temperatures using an adequate doping with trivalent atoms. The ${CuFeO}_{2}$ delafossite is a multiferroic material with phase transitions according to the evolution of magnetization or polarization or both at the same time. Theoretical or experimental studies are needed to verify and confirm these behaviors.

Acknowledgments

We thank P. Blaha and K. Schwarz for the $W i e n 2 k$ package and the $W i e n 2 k$ users for very useful discussions.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References

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Introduction

Computational method

Results and discussion

Structural properties
Electronic properties
Magnetic properties and Monte Carlo simulation

Conclusion

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[1] Ahmed S, Atif M, Rehman Atta Ur, Bashir S, Iqbal N, Khalid W, Ali Z, Nadeem M, . Enhancement in the magnetoelectric and energy storage properties of core-shell-like CoFe $_{2}$ O $_{4}$ BaTiO $_{3}$ multiferroic nanocomposite. J. Alloys Compd.. 2021;883:160875
[Google Scholar]

[2] Alikin Denis, Turygin Anton, Ushakov Andrei, Kosobokov Mikhail, Alikin Yurij, Hu Qingyuan, Liu Xin, Xu Zhuo, Wei Xiaoyong, Shur Vladimir, . Competition between ferroelectric and ferroelastic domain wall dynamics during local switching in rhombohedral PMN-PT single crystals. Nanomaterials. 2022;12(21)
[Google Scholar]

[3] Ashcroft Neil W., Mermin N. David, . Solid State Physics. Cengage Learning; 2022.

[4] Ataoui Khadija El, Doumerc Jean Pierre, Ammar Abdelaziz, Gravereau Pierre, Fournès Léopold, Wattiaux Alain, Pouchard Michel, . Preparation, structural characterization and Mössbauer study of the CuFe $_{1 - x}$ V $_{x}$ O $_{2}$ (0 $x$ 0.67) delafossite-type solid solution. Solid State Sci.. 2003;5:1239-1245.
[Google Scholar]

[5] Azouaoui A., Benzakour N., Hourmatallah A., Bouslykhane K., . Structural and magnetic properties of CrN: investigated by first-principles calculations, Monte Carlo simulation, and high-temperature series expansions. J. Supercond. Nov. Magn.. 2020;33:3113-3120.
[Google Scholar]

[6] Azouaoui A, Benzakour N, Hourmatallah A, Bouslykhane K, . Structural, magnetic properties and phase stability of CrAs in the zinc-blende structure: Investigated by first principle calculations and Monte Carlo simulation. Solid State Commun.. 2021;323:114101
[Google Scholar]

[7] Azouaoui A, Haoua M El, Salmi S, Grini A El, Benzakour N, Hourmatallah A, Bouslykhane K, . Structural, electronic, and magnetic properties of Mn $_{4}$ N perovskite: Density functional theory calculations and Monte Carlo study. J. Supercond. Nov. Magn.. 2020;33:1507-1512.
[Google Scholar]

[8] Behera Banarji, Sutar Bijuni Charan, Pradhan Nihar Ranjan, . Recent progress on 2D ferroelectric and multiferroic materials, challenges, and opportunity. Emerg. Mater.. 2021;4:847-863.
[Google Scholar]

[9] Benaissa N., Bentour H., Garmim T., Jouad Z. El, Louardi A., Hartiti B., Monkade M., Kenz A. El, Benyoussef A., . Experimental and DFT TB-mBJ calculations studies of structural, morphological, electronic, optical and electrical properties of copper oxide thin films. Opt. Mater.. 2023;136:113433
[Google Scholar]

[10] Bieńkowski Adam, Szewczyk Roman, . The possibility of utilizing the high permeability magnetic materials in construction of magnetoelastic stress and force sensors. Sensors Actuators A. 2004;113(3):270-276.
[Google Scholar]

[11] Blaha Peter, Schwarz Karlheinz, Madsen Georg K H, Kvasnicka Dieter, Luitz Joachim, Laskowsk Robert, Tran Fabien, Marks Laurence, Marks Laurence, . WIEN2k: An Augmented Plane Wave Plus Local Orbitals Program for Calculating Crystal Properties. Techn. Universitat; 2019. http://susi.theochem.tuwien.ac.at/reg_user/textbooks/usersguide.pdf

[12] Blaha Peter, Schwarz Karlheinz, Tran Fabien, Laskowski Robert, Madsen Georg K H, Marks Laurence D, . WIEN2k: An APW+ lo program for calculating the properties of solids. J. Chem. Phys.. 2020;152:74101.
[Google Scholar]

[13] Dong Yucheng, Cao Chenwei, Chui Ying-San, Zapien Juan Antonio, . Facile hydrothermal synthesis of CuFeO $_{2}$ hexagonal platelets/rings and graphene composites as anode materials for lithium ion batteries. Chem. Commun.. 2014;50(70):10151-10154.
[Google Scholar]

[14] Guo San-Dong, . X-ray emission spectra and gaps of CuFeO $_{2}$ with the modified Becke–Johnson potential. J. Magn. Magn. Mater.. 2015;377:226-228.
[Google Scholar]

[15] Hailouf Houssam Eddine, Gacem L., Gueddim A., Reshak Ali H., Obodo K.O., Bouhafs B., . Structural, electronic, magnetic, and optical properties of Fe-doped Na $_{2}$ ZnP $_{2}$ O $_{7}$ host: ab-initio calculation. Physica B. 2023;650:414554
[Google Scholar]

[16] Idrissi S., Bahmad L., Khalladi R., El Housni I., El Mekkaoui N., Mtougui S., Labrim H., Ziti S., . Phase diagrams, electronic and magnetic properties of the quaternary Heusler alloy NbRhCrAl. Chinese J. Phys.. 2019;60:549-563.
[Google Scholar]

[17] Idrissi S., Ziti S., Labrim H., Bahmad L., . Critical magnetic behavior of the rare earth-based alloy GdN: Monte Carlo simulations and density functional theory method. J. Mater. Eng. Perform.. 2020;29:7361-7368.
[Google Scholar]

[18] Idrissi S., Ziti S., Labrim H., Bahmad L., . The critical magnetic behavior of the new Heusler CoXO $_{2}$ alloys (X=Cu or Mn): Monte Carlo Study. Chinese J. Phys.. 2021;70:312-323.
[Google Scholar]

[19] Idrissi S., Ziti S., Labrim H., Bahmad L., El Housni I., Khalladi R., Mtougui S., El Mekkaoui N., . Half-metallic behavior and magnetic proprieties of the quaternary Heusler alloys YFeCrZ (z=Al, Sb and Sn) J. Alloys Compd.. 2020;820:153373
[Google Scholar]

[20] Idrissi S., Ziti S., Labrim H., Khalladi R., Mtougui S., El Mekkaoui N., El Housni I., Bahmad L., . Magnetic properties of the Heusler compound CoFeMnSi: Monte Carlo simulations. Physica A. 2019;527:121406
[Google Scholar]

[21] Jabar A., Masrour R., Tahiri N., . Ground state phase diagrams and magnetic properties of a bilayer hexagonal structure. Physica A. 2018;490:1019-1027.
[Google Scholar]

[22] Jaime M, Saul A, Salamon M, Zapf V S, Harrison N, Durakiewicz T, Lashley J C, Andersson D A, Stanek C R, Smith J L, Gofryk K, . Piezomagnetism and magnetoelastic memory in uranium dioxide. Nature Commun.. 2017;8:99.
[Google Scholar]

[23] Janke Wolfhard, . Monte Carlo methods in classical statistical physics. In: Fehske H., Schneider R., Weiß e A., eds. Computational Many-Particle Physics. Berlin, Heidelberg: Springer Berlin Heidelberg; 2008. p. :79-140.
[Google Scholar]

[24] Jiao Yinan, Qin Shengjian, Li Bing, Hao Weizhong, Wang Ye, Li Hao, Yang Zhanping, Dong Dejun, Li Xiangyu, Zhao Jinjin, . Ferroelasticity mediated energy conversion in strained Perovskite films. Adv. Electron. Mater.. 2022;8:2200415
[Google Scholar]

[25] Jin Yi, Chumanov George, . Solution synthesis of pure 2H CuFeO $_{2}$ at low temperatures. RSC Adv.. 2016;6:26392-26397.
[Google Scholar]

[26] John Melanie, Heuss-Aßbichler Soraya, Park So Hyun, Ullrich Aladin, Benka Georg, Petersen Nikolai, Rettenwander Daniel, Horn Siegfried R, . Low-temperature synthesis of CuFeO $_{2}$ (delafossite) at 70 $°$ C: A new process solely by precipitation and ageing. J. Solid State Chem.. 2016;233:390-396.
[Google Scholar]

[27] Jungwirth T., Sinova Jairo, Mašek J., Kučera J., MacDonald A.H., . Theory of ferromagnetic (III, Mn)V semiconductors. Rev. Modern Phys.. 2006;78(3):809-864.
[Google Scholar]

[28] Khalladi R., Labrim H., Idrissi S., Mtougui S., Housni I. El, Ziti S., Mekkaoui N. El, Bahmad L., . Magnetic properties study of the anti-perovskite Mn $_{3}$ CuN compound by Monte Carlo simulations. Solid State Commun.. 2019;290:42-48.
[Google Scholar]

[29] Khan Hukam, Sohail Mohammad, Rahman Nasir, khan Rajwali, ullah Asad, Hussain Mudassir, Khan Aurangzeb, Hegazy H.H., . Theoretical study of different aspects of Al-based fluoroperovskite AlMF $_{3}$ (M=Cu, Mn) compounds using TB-MBJ potential approximation method for generation of energy. Results Phys.. 2022;42:105982
[Google Scholar]

[30] Kim Jun-Hyuk, Kim Jae Young, Choi Yo Han, Youn Duck Hyun, . Facile CuFeO $_{2}$ microcrystal synthesis for lithium ion battery anodes via microwave heating. J. Mater. Sci., Mater. Electron.. 2020;31:9408-9414.
[Google Scholar]

[31] Kimura T, Goto T, Shintani H, Ishizaka K, Arima T, Tokura Y, . Magnetic control of ferroelectric polarization. Nature. 2003;426:55-58.
[Google Scholar]

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Structural, electronic and magnetic properties of the CuFeO 2 multiferroic compound