⇦Chinese Physics Letters, 2018, Vol. 35, No. 1, Article code 017501 Effect of Zn Substitution on Magnetic Properties of CuFe$_{2}$O$_{4}$: a High-Temperature Series Expansions Study S. Salmi1. R. Masrour2**, A. El Grini1, K. Bouslykhane1, A. Hourmatallah3, N. Benzakour1, M. Hamedoun4 Affiliations 1Laboratoire de Physique du Solide, Université Sidi Mohammed Ben Abdellah, Faculté des sciences Dhar Mahraz, BP 1796, Fes, Morocco 2Laboratory of Materials, Processes, Environment and Quality, Cady Ayyed University, National School of Applied Sciences, Route Sidi Bouzid-BP 63 46000 Safi, Morocco 3Equipe de Physique du Solide, Laboratoire LIPI, Ecole Normale Supérieure, BP 5206, Bensouda, Fes, Morocco 4MAScIR Foundation, Institut of Nanomaterials and Nanotechnologies, Materials & Nanomaterials Center, B.P. 10100Rabat, Morocco Received 11 September 2017 **Corresponding author. Email: rachidmasrour@hotmail.com Citation Text: Masrour S S R, Grini A E, Bouslykhane K, Hourmatallah A and Benzakour N et al 2018 Chin. Phys. Lett. 35 017501 Abstract The magnetic properties of spinel ferrites Cu$_{1-x}$Zn$_{x}$Fe$_{2}$O$_{4}$ are studied using high-temperature series expansions combined with the Padé approximates. The exchange interactions, inter and intra-sublattices $J_{\rm AA}$, $J_{\rm BB}$ and $J_{\rm AB}$ are obtained using a probability distribution law. The critical exponent $\gamma$ associated with the magnetic susceptibility is obtained. DOI:10.1088/0256-307X/35/1/017501 PACS:75.25.-j, 75.30.Cr, 75.10.-b © 2018 Chinese Physics Society Article Text Spinel ferrites with the general formula $M$Fe$_{2}$O$_{4}$ ($M$=Cu, Co, Ni, Mg, or Zn) have been the subject of extensive study because of their wide range of applications from microwave to radio frequency and of their importance in understanding the theories of magnetism. Among the ferrites, copper ferrite CuFe$_{2}$O$_{4}$ is an interesting material and has been widely used for various applications, such as catalysts for environment,[1] gas sensors,[2] and hydrogen production.[3] Magnetic and electrical properties of spinel ferrites vary greatly with the change chemical component and cation distribution. For instance, most of the bulk CuFe$_{2}$O$_{4}$ has an inverse spinel structure, with 85% Cu$^{2+}$ occupying B sites.[4] Zn-substitution results in a change of cations in chemical composition and a different distribution of cations between sites A and B. Consequently, the magnetic and electrical properties of spinel ferrites will change with changing cation.[5] However, nanocrystalline ZnFe$_{2}$O$_{4}$ shows mixed spinel structure, in which Zn$^{2+}$ and Fe$^{3+}$ are distributed over A and B sites being represented as (Zn$_{1-x}$Fe$_{1-x}$)$_{\rm A}$[Zn$_{x}$Fe$_{1+x}$]$_{\rm B}$O$_{4}$, where $x$ represents the inversion parameter and corresponds to the degree of cation distribution, i.e., Fe$^{3+}$ ions occupying A sites.[6] The addition of Zn$^{2+}$ ion in copper ferrite causes the Fe$^{3+}$ ions to migrate from A site to B site, the Zn$^{2+}$ ion being larger ionic radius (0.81 Å) than that of the Fe$^{3+}$ ion (0.67 Å). Some physical, magnetic and transport properties of Zn-substituted ferrite has been extensively studied by different researchers.[7] As this parameter plays an important role in technology, we aim at studying the effect of Zn$^{2+}$ ion substitution on the magnetic properties such as the Curie temperature. ZnFe$_{2}$O$_{4}$ is a promising semiconductor which is capable of sensitizing the other existing photocatalyst and has the ability to activate under visible light, due to its small band gap.[8] In previous work, the Monte Carlo simulation is used to investigate the magnetic properties of ferromagnetic superlattices through the Ising model.[9] Experimentally, the origin of the spin-glass transition in LiCoMnO$_{4}$ may be attributed to a spatial segregation between non-magnetic Co$^{3+}$ regions and spin-glass ordered regions of Mn$^{4+}$ ions[10] and magnetization results show the spin-reorientation for MnBi at about 91 K due to the variations of the anisotropy constants.[11] On the other hand, the magnetic and the electronic properties of the geometrically frustrated triangular antiferromagnet CuCrO$_{2}$ are investigated by first-principles through density functional theory calculations within generalized gradient approximations (GGA)+$U$ scheme.[12] In the present work, the values of exchange interactions $J_{\rm AB}$, $J_{\rm BB}$ and $J_{\rm AA}$ are given using a probability distribution law for Cu$_{1-x}$Zn$_{x}$Fe$_{2}$O$_{4}$. The Padé approximate[13] analysis of the high-temperature series expansions (HTSEs) of magnetic susceptibility has been shown to be a useful method for studying the critical region.[14,15] We have also used HTSEs to determine the critical temperature $T_{\rm C}$ and the critical exponent associated with the magnetic susceptibility $\chi(T)$. The HTSEs of the magnetic susceptibility $\chi(T)$ have been derived to the seventh order in the reciprocal temperature for spinel lattices including both nearest-neighboring (nn) and next-nearest-neighboring (nnn) interactions in the Heisenberg model.[14] Finally, we have also used the HTSE method to estimate the values of critical exponent $\gamma$ associated with the magnetic susceptibility for the Cu$_{1-x}$Zn$_{x}$Fe$_{2}$O$_{4}$ system. To determine the intra ($J_{\rm AA}(x)$ and $J_{\rm BB}(x)$) and the inter ($J_{\rm AB}(x)$)-sublattice exchange interactions in the whole range of concentration, a probability distribution law is used based on the distribution of ions in A and B sublattices. The exchange interactions $J_{\rm AA}^{\rm Fe^{3+}-Fe^{3+}}$, $J_{\rm AB}^{\rm Fe^{3+}-Fe^{3+}}$, $J_{\rm AB}^{\rm Fe^{3+}-Cu^{2+}}$, $J_{\rm BB}^{\rm Fe^{3+}-Fe^{3+}}$, $J_{\rm BB}^{\rm Fe^{3+}-Cu^{2+}}$ and $J_{\rm BB}^{\rm Cu^{2+}-Cu^{2+}}$, in the inverted pure spinel, were given in Refs. [15]. The exchange interactions depend on the dilution[16] $x$ and the obtained expressions of $J_{\rm AB}(x)$, $J_{\rm BB}(x)$ and $J_{\rm AA}(x)$ are $$\begin{align} J_{\rm AA} (x)=\,&[(1-x)^2J_{\rm AA}^{\rm Fe^{3+}-Fe^{3+}}],\\ J_{\rm AB} (x)=\,&\frac{1}{2}[(1-x^2)J_{\rm AB}^{\rm Fe^{3+}-Fe^{3+}}\\ &+(1-x)^2J_{\rm AB}^{\rm Fe^{3+}-Cu^{2+}}],\\ J_{\rm BB} (x)=\,&\frac{1}{4}[(1+x)^2J_{\rm BB}^{\rm Fe^{3+}-Fe^{3+}}\\ &+2({1-x^2})J_{\rm BB}^{\rm Fe^{3+}-Cu^{2+}}\\ &+(1-x)^2J_{\rm BB}^{\rm Cu^{2+}-Cu^{2+}}],~~ \tag {1} \end{align} $$ where $J_{ij}^{\rm ion-ion}$ corresponds to the exchange interaction in the opposite pure system CuFe$_{2}$O$_{4}$. To calculate magnetic susceptibility and to deduce transition temperature and critical exponent of the ferrimagnetic spinel, with two magnetic sublattices, we consider a semi-classical Heisenberg spin model given by the following Hamiltonian: $$\begin{align} H=\,&-2J_{\rm AA} \mathop \sum \limits_{i,j} {\boldsymbol S}_i {\boldsymbol S}_j -2J_{\rm BB}\mathop \sum \limits_{i,i'}{\boldsymbol \sigma}_i {\boldsymbol \sigma}_j\\ &-2J_{\rm AB}\mathop \sum \limits_{i,j}{\boldsymbol S}_i {\boldsymbol \sigma}_j-\mu_{_{\rm B}}h_{\rm ex}\Big(g_{_{\rm A}}\mathop \sum \limits_{i}{\boldsymbol S}_i^Z\\ &-g_{_{\rm B}}\mathop \sum \limits_{j}{\boldsymbol \sigma}_j^Z\Big),~~ \tag {2} \end{align} $$ where ${\boldsymbol S}$ and ${\boldsymbol \sigma}$ are spin operators of Fe and Cu ions, respectively, $g^{\rm Fe}= 2.091$ and $g^{\rm Cu}=2.13$ are the corresponding Lande factors. The symbol $ < \ldots>$ denotes summation over nearest neighbors, and $J_{\rm AA}$, $J_{\rm BB}$, and $J_{\rm AB}$ are the intra- and the inter-sublattice exchange interactions in ferrimagnetic (FerriM) spinels. The magnetization of the system is given by $$\begin{alignat}{1} \!\!\!\!M=M_{\rm A} -M_{\rm B} =\mu _{\rm B} ({g_{_{\rm A}} \sum\limits_i {\langle {S_i^z} \rangle -g_{_{\rm B}} \sum\limits_j {\langle {S_j^z} \rangle}}}).~~ \tag {3} \end{alignat} $$ After computing the first derivative of the magnetization $\chi =({\partial M/\partial h_{\rm ex}})_{h_{\rm ex} \to 0}$, we obtain the general expression of susceptibility for the collinear inverse ferrimagnetic spinel as follows: $$\begin{align} \chi =\,&\Big({\frac{\mu _{\rm B}^2}{3k_{\rm B} T}}\Big)(N_{\rm A} g_{_{\rm A}}^2{\boldsymbol S}^2+N_{\rm B} g_{_{\rm B}}^2{\boldsymbol \sigma}^2\\ &-g_{_{\rm A}}^2\mathop \sum \limits_{i\neq i'} {\boldsymbol S}_i {\boldsymbol S}_{i'}-g_{_{\rm B}}^2\mathop \sum \limits_{j\neq j'} {\boldsymbol \sigma}_j {\boldsymbol \sigma}_{j'}\\ &-2g_{_{\rm A}}g_{_{\rm B}}\mathop \sum \limits_{i,j} {\boldsymbol S}_i {\boldsymbol \sigma}_{j}),~~ \tag {4} \end{align} $$ where $N_{\rm A}$ and $N_{\rm B}$ are the number of ions and the spin value of each type of spin, respectively. The final expression of magnetic susceptibility is obtained as $$\begin{align} \chi =\,&\Big(\frac{\mu _{\rm B}^2}{3k_{\rm B} T}\Big)(N_{\rm A} g_{_{\rm A}}^2{\boldsymbol S}^2+N_{\rm B} g_{_{\rm B}}^2{\boldsymbol \sigma}^2-N_{\rm A} g_{_{\rm A}}^2{\it \Gamma}_{\rm AA}\\ &-N_{\rm B} g_{_{\rm B}}^2{\it \Gamma}_{\rm BB}-N_{\rm B} g_{_{\rm A}}g_{_{\rm B}}{\it \Gamma}_{\rm BA}). \end{align} $$ Following the process in Refs. [17–22], we compute the expression of spin correlation functions ${\it \Gamma}_{\rm AA}$, ${\it \Gamma}_{\rm BB}$ and ${\it \Gamma}_{\rm BA}$ in terms of powers of $\beta$ and mixed powers of $J_1 =2J_{\rm BB} (x){\boldsymbol \sigma}^2$, $J_2 =2J_{\rm AB}(x){\boldsymbol S}{\boldsymbol \sigma}$ and $J_3 =2J_{\rm AA} (x){\boldsymbol S}^2$ $$\begin{align} {\it \Gamma}_{\rm AA}=\,&\mathop \sum \limits_{q=1}^7 \mathop \sum \limits_{m=0}^q \mathop \sum \limits_{n=0}^{q-m} \mathop \sum \limits_{p=0}^{q-m-n} a({m,n,p,q})J_1^m J_2^n J_3^{\rm p} \beta ^q,\\ {\it \Gamma}_{\rm BB}=\,&\mathop \sum \limits_{q=1}^7 \mathop \sum \limits_{m=0}^q \mathop \sum \limits_{n=0}^{q-m} \mathop \sum \limits_{p=0}^{q-m-n} b({m,n,p,q})J_1^m J_2^n J_3^{\rm p} \beta ^q,\\ {\it \Gamma}_{\rm BA}=\,&\mathop \sum \limits_{q=1}^7 \mathop \sum \limits_{m=0}^q \mathop \sum \limits_{n=0}^{q-m}\mathop \sum \limits_{p=0}^{q-m-n} c({m,n,p,q})J_1^m J_2^n J_3^{\rm p} \beta ^q,~~ \tag {5} \end{align} $$ where nonzero coefficients $a(m, n, p, q)$, $b(m, n, p, q)$, and $c(m, n, p, q)$ up to order 7 in $\beta =1/k_{\rm B} T$ are given in Ref. [17]. The powerful Padé approximation method has been used to estimate the critical temperature $T_{\rm C}$, and the critical exponent $\gamma$ associated with magnetic susceptibility. The locations of singularities in the PA method to the HTSEs of the magnetic susceptibility determine the Curie point. The magnetic susceptibility $\chi (T)$ is characterized by critical exponent $\gamma$ in the neighborhood of critical temperature: $$\begin{align} \chi(T)\propto ({T_{\rm C} -T})^{-\gamma}.~~ \tag {6} \end{align} $$ The usual approach is to compute the series for the logarithmic derivative of $\chi (T)$ $$\begin{align} \frac{d}{dT}\log [{\chi (T)}]\approx \frac{-\gamma}{T-T_{\rm C}}.~~ \tag {7} \end{align} $$ This function has a simple pole $T_{\rm C}$ and should be well represented by the Padé approximates $[P, Q]$. The exponent $\gamma$ is then re-estimated from the approximation to $$\begin{align} ({T-T_{\rm C}})\frac{d}{dT}\log [{\chi (T)}],~~ \tag {8} \end{align} $$ evaluated at $T=T_{\rm C}$. A PA $[M, N]$ to a magnetic susceptibility $\chi (T)$ is a rational fraction $P_{M}/Q_{N}$, with $P_{M}$ and $Q_{N}$, polynomials of orders $M$ and $N$ in $\beta =1/k_{\rm B} T$ ($k_{\rm B}$ is the Boltzmann constant) such that $\chi (\beta) \approx (P_{M}/Q_{N})+O(\beta ^{M+N+1})$, where $M$ and $N$ are the degrees of $P$ and $Q$ polynomials, respectively. Estimates of $T_{\rm C}$ and $\gamma$ for Cu$_{1-x}$Zn$_{x}$Fe$_{2}$O$_{4}$ have been obtained using the PA method.[13] The simple pole corresponds to $T_{\rm C}$ and the residues to the critical exponents $\gamma$. In this study, we have given the proposed cation distribution of the Cu$_{1-x}$Zn$_{x}$Fe$_{2}$O$_{4}$ system as listed in Table 1, the expression of exchanges interactions $J_{\rm AB}(x)$, $J_{\rm BB}(x)$ and $J_{\rm AA}(x)$ interactions are also given.
Table 1. Phase and cation distribution for Cu$_{1-x}$Zn$_{x}$Fe$_{2}$O$_{4}$ systems.
$x$ Distribution ion in site A Distribution ion in site B
0 $({\rm Cu_{0.0}^{2+} Fe_{1.0}^{3+}})_{\rm A} $ $({\rm Cu_{1.0}^{2+} Fe_1^{3+}})_{\rm B} $
0.2 $({\rm Cu_{0.0}^{2+} Zn_{0.2}^{2+} Fe_{0.8}^{3+}})_{\rm A} $ $({\rm Cu_{0.8}^{2+} Zn_{0.0}^{2+} Fe_{1.2}^{3+}})_{\rm B} $
0.4 $({\rm Cu_{0.0}^{2+} Zn_{0.4}^{2+} Fe_{0.6}^{3+}})_{\rm A} $ $({\rm Cu_{0.6}^{2+} Zn_{0.0}^{2+} Fe_{1.4}^{3+}})_{\rm B} $
0.6 $({\rm Cu_{0.0}^{2+} Zn_{0.6}^{2+} Fe_{0.4}^{3+}})_{\rm A} $ $({\rm Cu_{0.4}^{2+} Zn_{0.0}^{2+} Fe_{1.6}^{3+}})_{\rm B} $
0.8 $({\rm Cu_{0.0}^{2+} Zn_{0.8}^{2+} Fe_{0.2}^{3+}})_{\rm A} $ $({\rm Cu_{0.2}^{2+} Zn_{0.0}^{2+} Fe_{1.8}^{3+}})_{\rm B} $
1 $({\rm Cu_{0.0}^{2+} Zn_{0.5}^{2+} Fe_{0.5}^{3+}})_{\rm A} $ $({\rm Cu_{0.0}^{2+} Zn_{0.5}^{2+} Fe_{1.5}^{3+}})_{\rm B} $
We have used the values of exchange interactions $J_{\rm AA}^{\rm Fe^{3+}-Fe^{3+}}=-14$ K, $J_{\rm AB}^{\rm Fe^{3+}-Fe^{3+}}=-28$ K, $J_{\rm AB}^{\rm Fe^{3+}-Cu^{2+}}=-28$ K, $J_{\rm BB}^{\rm Fe^{3+}-Fe^{3+}}=-9$ K, $J_{\rm BB}^{\rm Fe^{3+}-Cu^{2+}}=-10$ K and $J_{\rm BB}^{\rm Cu^{2+}-Cu^{2+}}=29$ K of CuFe$_{2}$O$_{4}$ given in the previous work[15] to calculate $J_{\rm AB}(x)$, $J_{\rm BB}(x)$ and $J_{\rm AA}(x)$ with different concentrations $x$. We see that all exchange interactions are antiferromagnetic except $J_{\rm BB}^{\rm Cu^{2+}-Cu^{2+}}$, which is ferromagnetic. Furthermore, with increase in the Zn content all the interactions decrease except $J_{\rm BB}^{\rm Cu^{2+}-Cu^{2+}}$ which increases. The obtained values are given in Table 1 and Fig. 1. The absolute values of exchange interactions $J_{\rm AB}(x)$ and $J_{\rm AA}(x)$ decrease with the increase of dilution $x$. This is explained by the substitution of the magnetic iron ions in the A site by the non magnetic Zn$^{2+}$ ions. The values of $J_{\rm BB}(x)$ increase in absolute value with the increasing $x$ due to the increase of Fe$^{3+}$ ions and the decrease of Cu$^{2+}$ ion concentration in the B site. The magnetic properties of spinel ferrites depend on the magnetic interaction and cation distribution in the two sublattices, i.e., tetrahedral (A) and octahedral (B) lattice sites.
cpl-35-1-017501-fig1.png
Fig. 1. The values of exchange interactions $J_{ij}$ ($ij$=AB, BB and AA) of Cu$_{1-x}$Zn$_{x}$Fe$_{2}$O$_{4}$.
cpl-35-1-017501-fig2.png
Fig. 2. The magnetic phase diagram of Cu$_{1-x}$Zn$_{x}$Fe$_{2}$O$_{4}$ with ferrimagnetic (FerriM) and paramagnetic (PM) phases.
The high-temperature series expansions extrapolated in combination with the Padé approximation method is shown to be a convenient method to provide valid estimations of the critical temperatures for a real system. By applying this method to the magnetic susceptibility $\chi (T)$, we have estimated the critical temperature $T_{\rm C}$ for each dilution in the cation distribution of Cu$_{1-x}$Zn$_{x}$Fe$_{2}$O$_{4}$ systems. The obtained values of critical temperature are given in Table 2 and Fig. 2. The obtained values by HTSEs are near the experimental data available.[23]
Table 2. The exchange integrals ($J_{\rm AB}$, $J_{\rm BB}$, $J_{\rm AA}$), the values of critical temperature obtained by HTSE and critical exponent $\gamma$ for Cu$_{1-x}$Zn$_{x}$Fe$_{2}$O$_{4}$ systems.
$x$ $J_{\rm AB}(k)$ $J_{\rm BB}(k)$ $J_{\rm AA}(k$) $T_{\rm C}$ (K) Experiment results[23] $T_{\rm C}$ (K) (HTSE) $\gamma$
0.2 $-$22.40 $-$3.40 $-$8.96 620 680 1.29
0.4 $-$16.80 $-$6.00 $-$5.04 500 538 1.36
0.6 $-$11.20 $-$7.60 $-$2.44 459 492 1.27
0.8 $-$5.60 $-$8.80 $-$0.56 438 410 1.32
The sequence of $[M, N]$ PA to series of $\chi (T)$ has been evaluated. The values of critical exponent $\gamma$ associated with the magnetic susceptibility $\chi (T)$ have been estimated (see Table 2). By examining the behavior of these PAs, the convergence is found to be quite rapid. The obtained values of $\gamma$ are somewhat similar to that of the known $XY$ model.[24] These values are insensitive to the dilution $x$.[25] In summary, the values of exchange integrals $J_{\rm AB}$, $J_{\rm BB}$ and $J_{\rm AA}$ for Cu$_{1-x}$Zn$_{x}$Fe$_{2}$O$_{4}$ systems have been obtained for different dilutions $x$. The HTSEs extrapolated with the Padé approximates are shown to be convenient to provide a valid estimation of parameters associated with the critical region ($T_{\rm C}$, $\gamma$) of the Cu$_{1-x}$Zn$_{x}$Fe$_{2}$O$_{4}$ systems. The obtained values of $T_{\rm C}$ are comparable with the experimental magnetic measurements. The values of critical exponents associated with magnetic susceptibility for the Cu$_{1-x}$Zn$_{x}$Fe$_{2}$O$_{4}$ systems are found for different Padé approximates. References Thermally synthesized nanosized copper ferrites as catalysts for environment protectionPreparation and gas-sensing properties of CuFe2O4 at reduced temperatureHydrogen production from dimethyl ether steam reforming over composite catalysts of copper ferrite spinel and aluminaComputational study of copper ferrite (CuFe2O4)Cation distribution in nanocrystalline ZnFe 2 O 4 investigated using x-ray absorption fine structure spectroscopyElectrical conductivity of Ni–Zn–Gd ferritesMagnetization of sol–gel prepared zinc ferrite nanoparticles: Effects of inversion and particle sizeEffect of exchange interaction in ferromagnetic superlattices: A Monte Carlo studyObservation of spin glass transition in spinel LiCoMnO 4Fabrication, Structural and Magnetic Properties for Aligned MnBiExchange coupling and helical spin order in the triangular lattice antiferromagnet CuCrO 2 using first principlesThe critical temperature and exchange interactions of an S = 5/2 Heisenberg antiferromagnet on an f.c.c. latticeExchange constants in spinel ferritesMagnetic properties of (ZnxFe1−x)A(Mn1−xFe1+x)BO4 materialsA study of the critical behaviour of a normal ferrimagnetic spinel by high-temperature series expansionsHigh-temperature series expansion of the spin correlation functions in B-spinel latticeCritical properties and exchange interactions of antiferromagnetic A-spinel lattice: a study through high-temperature series expansionsPhase transition in Heisenberg magnetic filmsCritical properties and exchange interactions of the antiferromagnetic tetrahedral spinelMagnetic Properties and Critical Behaviour of the B-Spinel CdCr2xIn2?2xS4 (0.9 ? x ? 1)Monte Carlo Simulations of Mixed Highly Frustrated SystemsSkyrmion Strings and the Anomalous Hall Effect in C r O 2
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