Using a microscopic model and the replica method as well as the Green’s function theory, we have investigated the relaxor and multiferroic behavior of CdCr2S4. The magnetization, the remanent polarization Pr and the real part of the dielectric function are studied theoretically as a function of temperature, with and without a magnetic field, respectively. The magnetization and the polarization exist together below the magnetic phase transition temperature. Pr decreases whereas increases and the peak shifts to smaller temperature values with increasing magnetic field h. Moreover, the temperature and electric field E dependence of the magnetization M is also discussed. A kink is observed around the ferroelectric transition temperature. The kink is deeper for stronger electric fields and anharmonic spin-phonon interactions.
The magnetic relaxor ferroelectric materials with the coexistence of relaxor ferroelectricity and magnetic order have been attracting considerable attention due to their peculiar properties and various applications    . Basic components of this family of materials are the chromium chalcogenide and the compounds that have the general formula ACr2X4, where A = Cd or Hg, X = S or Se. These crystals are generally found in the cubic spinel structure. It is well known that Cr3+ with spin S = 3/2 is responsible for the magnetic properties . As a relaxor ferroelectric, structural investigation indicates that the system is close to a structural instability and there is a possible off-center displacement of Cr3+ ions  . Such an off-center position, in conjunction with geometrical frustration, leads to a relaxor ferroelectric state . A spin-glass-like behavior in ferromagnetic phase of CCS is reported by Kitani et al. . The occurrence of a multiferroic state in various spinel systems, including CCS    , FeCr2S4 (FCS) , CdCr2Se4 (CCSe)  and HgCr2S4 (HCS)  is reported. In CdCr2S4 (CCS) the ferromagnetic and ferroelectric transitions were found at 84 and 57 K, respectively. An evident anomaly in the dielectric constant at TC = 84 K has been reported  , indicating the magnetoelectric coupling between magnetism and dielectric properties. Magnetic and electric field dependent dielectric and magnetic properties are studied in multiferroic CCS by Sun et al. . Concerning the microscopic origin of the polar moments in CCS and CCSe it can be assumed that the ferroelectric distortions result from an off-center position of the Cr3+-ions, which generates a locally polar but macroscopically isotropic cluster state . The results of Grimes et al.  revealed that in CCS and CCSe canonical ferromagnetism coexists at sizable ordering temperatures with a relaxor-ferroelectric state, characterized by a significant relaxational behavior. Both order parameters are strongly coupled. A scenario in which the relaxation mechanism interacts with magnetic order via exchange striction can be considered the most plausible.
There are not so many theoretical investigations of the properties of magnetic relaxor ferroelectrics. Fennie and Rabe  have studied the dielectric properties of the ferromagnetic spinel CCS from first principles. The observed results suggested that the observed anomalous dielectric behavior in CCS is not due to the softening of a polar mode. Oliveira et al.  have used the Landau theory to demonstrate that a linear coupling between the magnetic and polar order parameters (αPM) is sufficient to justify the appearance of magnetic cluster in the paramagnetic phase of CCS. Zhou et al. , based on the spherical random-bond-random-field model and the Heisenberg model, considering the coupling interaction between the relaxor ferroelectricity and magnetism, have calculated the third-order static nonlinear dielectric susceptibility as well as the scaled nonlinear susceptibility of magnetic relaxor ferroelectrics. The important role of the fluctuations of the spin-pair correlations in the change of the dielectric susceptibility around the magnetic phase transition temperature is shown by Xia and Jiang   based on the spherical random-bond-random-field (SRBRF) model, the Heisenberg model and a magnetoelectric coupling term. Using the same model Cao et al.  have investigated the influence of Fe substitution on the magnetic and static nonlinear dielectric response in the doped magnetic relaxor ferroelectrics CCS. Band structure calculations for ACr2X4 (A = Zn, Cd, Hg and X = O, S, Se) spinels are presented by Yaresko .
The aim of the present paper is to study the effect of an applied static magnetic field on the dielectric properties of magnetic relaxor ferroelectric CCS using a microscopic model and the replica method as well as the Green’s function theory.
2. The Model
Pioneering studies of magnetic semiconductors in the 1960s showed the coexistence of semiconduction and magnetism in some ferromagnetic semiconductors, such as europium chalcogenides and the spinel compounds CCS and CCSe . But in the last years they are considered as materials as magnetic relaxor ferroelectrics. The Hamiltonian which can describe their magnetic and relaxor ferroelectric properties is:
is the s-d Hamiltonian:
is the Heisenberg Hamiltonian for the ferromagnetically ordered d electrons,
where is the Heisenberg operator of Cr3+ at the lattice site i. Ferromagnetic order in CCSe and in CCS appears below TCm = 130 K and 84 K, respectively, as a result of competition between the direct Cr-Cr spin coupling and the near neighbour Cr-Se(or S)-Cr exchange interactions. The exchange interaction is a function of the lattice constant and stands for the nearest neighbors and next nearest neighbors . The frustration plays an important role in these materials. is an applied magnetic field.
Strong spin-phonon interaction is reported in CCS     :
is the normal coordinate of the lattice mode. F and R designate the amplitudes for coupling of phonons to the spin excitations in first and second order, respectively. h.c. represents the hermitian conjugate.
is the usual Hamiltonian of the conduction band electrons,
where and a are the Fermi creation and annihilation operators, is the chemical potential, and are the Bloch energies.
couples the two subsystems (3) and (4) by an intraatomic exchange interaction I
The Hamiltonian of the relaxor ferroelectric subsystem in the presence of an electric field can be written as 
means the summation over each distinct pair once, and is a dimensionless order parameter field. and are randomly frustrated bonds and random fields, respectively, which are satisfied with a Gaussian distribution. The last are characterized by the variance . is specified by its random average value and root-mean-square variance . is an applied electric field. A ferroelectric ordering in CCS occurs near TCf = 56 K.
There are four order parameters―the localized-spin magnetization M for arbitrary spin S, the conduction-electron magnetization , the polarization P and the spin glass order parameter q which are given by:
with the renormalized exchange interaction constants:
The magnetoelectric coupling term between the magnetic (3) and electric (7) subsystems in the magnetic relaxor ferroelectric CCS and CCSe is proposed to be linear in the electric order parameter and quadratic in the magnetic order parameter:
where g is the magnetoelectric coupling constant. It must be noted that Xia and Jiang   have chosen a biquadratic magnetoelectric coupling-quadratic in the electric and quadratic in the magnetic order parameters. But this coupling is suitable for multiferroics with very different phase transition temperatures, , for example in BiFeO3 or hexagonal RMnO3. But this is here not the case, because both temperatures are close to another, , similarly to RMn2O5, which corresponds to a linear magnetoelectric coupling.
The correlation functions are calculated using the Green’s functions and All quantities are renormalized due to the magnetoelectric and spin-phonon coupling and must be calculated selfconsistently.
3. Numerical Results and Discussion
The temperature dependence of the magnetization and polarization are numerically calculated taking parameters appropriated for CdCr2S4: J1 = 0.1 eV, J2 = −0.05 eV, I = 0.3 eV , S = 3/2, F = 4 cm−1, R = −0.35 cm−1, TCm = 84 K, TCf = 56 K, A = 0.1 eV, g = 0.3 eV.
The spin-phonon interaction taken up to second order and the magnetoelectric coupling g renormalize the exchange interaction constant to which is now temperature dependent:
where M is the magnetization and is the unrenormalized phonon energy.
By the calculation of the remanent polarization we have included an electric field of E = 100 kV/m because there is no ferroelectric transition except in applied electric fields what is typical for relaxor ferroelectrics . The results of M and Pr are presented in Figure 1 and Figure 2, respectively. They show a coexistence of ferromagnetism and relaxor ferroelectricity together below nearly T = 56 K, i.e. below the ferroelectric phase transition temperature TCf. There is a
Figure 1. Temperature dependence of the magnetization M in CCS for 500 Oe and different electric field values: E = 0 (1), 100 kV/m (2).
Figure 2. (Color online) Temperature dependence of the remanent polarization Pr in CCS for E = 100 kV/m and different magnetic field values: h = 0 (1), 500 Oe (2).
diffuse ferroelectric phase transition, which is characteristic for relaxor ferroelectrics. We have studied also the magnetic field dependence of the remanent polarization. Pr and TCf decrease with increasing magnetic field h (Figure 2, curve 2) what is evidence for a multiferroic behavior. The observed results are in good qualitative agreement with the experimental data of Hemberger et al. .
A strong decrease in the magnetization M is observed well below TCm = 85 K with increasing the external electric field E (Figure 1, curve 2). There is a kink around the ferroelectric phase transition TCf. The kink increases with increasing E in agreement with the experimental data of Sun et al. . This anomaly is due to the magnetoelectric coupling g. It is evidence for the strong correlation among magnetic spin, electric dipole and the lattice in the multiferroic CCS. It must be noted that with increasing spin-phonon interaction R the kink in the temperature dependence of M is deeper and clearer. This shows the importance of the anharmonic spin-phonon interaction and that it must be taken into account in order to obtain correct results. We have observed a similar kink in the temperature dependence of the magnetization in the multiferroic RMn2O5 , where is also valid .
We have calculated also the dielectric function from the expression  :
where Z is the electron charge and v is the volume, is the ferroelectric Green’s function.
The temperature and magnetic field dependence of the real part of is shown in Figure 3. The dielectric function of CCS becomes strongly enhanced in the region below the ferromagnetic transition temperature TCm = 84 K which is strongly correlated with the ferromagnetic ordering. Exchange striction  , which mechanically induces the strain, is associated with the ferromagnetic
Figure 3. (Color online) Temperature dependence of the real part of the dielectric function for E = 100 kV/m and different magnetic field values: h = 0 (1), 500 Oe (2).
ordering around TCm and local lattice distortion takes place inducing the enhancement of dielectric constant. It goes through a broad maximum, which indicates a diffuse phase transition. This broad peak of is typical for a relaxor ferroelectric. The observed behavior is in accordance with the experimental data for CCS . The low temperature peak is due to magnetic ordering, which shows that the magnetic and electric orders are coupled. All these characteristics are signatures of the relaxor state of magnetic relaxor ferroelectircs, in agreement with previous reports  .
The ferroelectric properties strongly depend on the external magnetic field h in multiferroics. Therefore, we have calculated also the magnetic field dependence of in CCS, see Figure 3, curve 2. The dielectric function increases with increasing h in agreement with the experimental data of Hemberger et al.  and Sun et al. . The maximum is shifted to smaller temperature values. A similar shift of is reported by Sun et al. . It must be noted that Hemberger et al.  and Krohns et al.  also observed an increase of the dielectric function with increasing the magnetic field and nonzero frequencies in CCS, but the peak shifts to higher temperature values. It must be noted that our calculations are for zero frequencies. In HCS Weber et al.  demonstrated also the strong increase of the dielectric constant in external magnetic fields.
The temperature and magnetic field dependence of relaxor characteristics such as remanent polarization Pr and real part of the dielectric function are investigated theoretically for CCS. The relaxor polarization and the magnetization appear together below T ~ 56 K, i.e. below the ferroelectric phase transition temperature TCf. A diffuse ferroelectric phase transition and a broad peak for are observed which are typical for relaxor ferroelectrics, such as CCS. The polarization decreases whereas the dielectric function increases and the maximum of shifts to higher temperature values with increasing the magnetic field h. It is shown that CCS is a multiferroic and that the relaxor behavior of magnetic relaxor ferroelectrics such as CCS can be controlled by the application of an external magnetic field.
The magnetization M decreases with increasing electric field E. Moreover, there is a kink around the ferroelectric transition temperature TCf = 56 K. The kink is stronger for higher electric fields E and anharmonic spin-phonon interaction constants R.
It would be of interest to investigate the influence of cation substitution on the magnetic and dielectric properties in the doped magnetic relaxor ferroelectric CCS. This will be made in a next paper.