Structural and magnetodielectric properties of BiFeO3-GdMnO3 multiferroics

We report on structural, microstructural, spectroscopic, dielectric, electrical, ferroelectric, ferromagnetic, and magnetodielectric coupling studies of BiFeO3–GdMnO3[(BFO)1–x –(GMO) x ], where x is the concentration of GdMnO3 (x = 0.0, 0.025, 0.05, 0.075, 0.1, 0.15, and 0.2), nanocrystalline ceramic solid solutions by auto-combustion method. The analysis of structural property by Rietveld refinement shows the existence of morphotropic phase boundary (MPB) at x = 0.10, which is in agreement with the Raman spectroscopy and high resolution transmission electron microscopy (HRTEM) studies. The average crystallite size obtained from the transmission electron microscopy (TEM) and x-ray line profile analysis was found to be 20–30 nm. The scanning electron micrographs show the uniform distribution of grains throughout the surface of the sample. The dielectric dispersion behavior fits very well with the Maxwell-Wagner model. The frequency dependent phase angle (θ) study shows the resistive nature of solid solutions at low frequency, whereas it shows capacitive behavior at higher frequencies. The temperature variation of dielectric permittivity shows dielectric anomaly at the magnetic phase transition temperature and shifting of the phase transition towards the lower temperature with increasing GMO concentration. The Nyquist plot showed the conduction mechanism is mostly dominated by grains and grain boundary resistances. The ac conductivity of all the samples follows the modified Jonscher model. The impedance and modulus spectroscopy show a non-Debye type relaxation mechanism which can be modeled using a constant phase element (CPE) in the equivalent circuit. The solid-solutions of BFO-GMO show enhanced ferromagnetic-like behavior at room temperature. The ferroelectric polarization measurement shows lossy ferroelectric behavior. The frequency dependent magnetocapacitance and magnetoimpedance clearly show the existence of intrinsic magnetodielectric coupling. The (BFO)1–x –(GMO) x solid solutions with x = 0.025–0.075 show significantly higher magnetocapacitance and magnetoimpedance compared to the pure BFO.


Introduction
Multiferroic materials are attracting a great deal of interest because of their unusual interesting physical properties, such as coexistence of multiple switchable states (polarization, magnetization or strain), magnetoelectric (ME) coupling, structural phase transitions, and their potential applications in secure data storage, spintronics, and novel multifunctional devices [1][2][3][4][5][6][7][8]. The information stored by splitting data in multiferroic memory over two mediums (half stored electrically and half stored magnetically) could be encrypted in such a way that makes it for more secure than is currently possible, which will open up a new era in data storage.

Experimental details
Nanocrystalline ceramic solid solutions of (BFO) 1-x -(GMO) x (0.0 x 0.2) were synthesized using analytical grade chemicals of Fe(NO 3 ) 3 9H 2 O, Bi(NO 3 ) 3 5H 2 O, Gd 2 O 3 and (CH 3 COO) 2 Mn 4H 2 O with urea as a fuel. The homogeneous gel of these reagents were added stoichiometrically to this fuel and later were dissolved in nitric acid and water. The gel was heated to 400°C for 30 min and then the solvent was evaporated and auto-ignited to result in residues that were crushed later. The extracted material from this reaction was milled for homogeneity. The details of the auto-combustion synthesis process are described elsewhere [13,16,17]. The combustion residue of different concentration were calcinated at different temperatures for 3 h; for x = 0.0 at 550°C in air, 0.025 x 0.1 at 700°C for x = 0.15 and x = 0.2 at 750°C. After the calcination, the powder was press to 8 x 10 7 Kg/m 2 in cylindrical pellets utilizing polyvinyl alcohol as binder. The samples with concentration of 0.0 x 0.1 were sintered at 750°C, while samples with concentration of 0.15 x 0.2 were sintered at 780°C, for 6 h. The stoichiometry and phase purity of the samples were studied using x-ray diffraction (XRD) (Philips Analytical-PW3040) at 2°/min from 20° 2θ 80°using Cu-K α radiation (λ = 1.5405 Å). X-ray lineprofile analysis of the samples was used to calculated the crystallite size and lattice strain in the samples using BREADTH software [18]. Micro-Raman spectra of the samples were recorded on backscattering geometry using the 514.5 nm Ar-ion laser line using Renishaw micro-Raman spectrometer (model-INVIA). Microstructure, distribution of grain growth, and particle size were studied using field emission scanning electron microscope (SUPRA 35VT SEM) and transmission electron microscope (Techo C230 STwin TEM). For the electrical characterization, the pellets were polished and electroded with silver paint and dried at 150°C for 3 h.
Ferroelectric polarization measurements were carried out using Marine PE loop tracer. Dielectric and magnetodielectric measurements were carried out in a wide range of frequency (100 Hz-1 MHz) with the Hioki 3532-50 LCR meter and in-house ME measurement setup. Magnetic properties were carried out at room temperature using a vibrating sample magnetometer (Lakshore 142AVSM) up to a maximum field of ±2 T at RT.  3. Results and discussion 3.1. Structural and microstructural properties Figure 1 shows the XRD pattern of (BFO) 1-x -(GMO) x 0.0 x 0.2 solid-solutions. The XRD pattern of the samples was analyzed using the Rietveld refinement using FULLPROF 2.2.4 package [17]. The calculated and difference pattern with Bragg's position of the peaks were given for comparison. The peak shapes were refined using a pseudo-Voigt function and background was corrected using a six-coefficient polynomial function. In order to get a better refinement, zero correction, scale factor, background, unit cell parameters, atomic positions, thermal parameters, and half width parameters (U, V, and W) were varied. The occupancy of all the atomic sites was kept fixed during the refinement process. Rhombohedral crystal structure of undoped BiFeO 3 in R3c space group was used for Rietveld refinement. From the refinement analysis, it was found that BFO with up to 7.5% GMO concentration shows single phase with R3c space group, whereas above it shows a mixed phase (R3c + Pn2 1a ).
As can be seen from figure 1, a good agreement (goodness of fit χ 2 = 1.15-1.30) has been observed between the observed and the calculated patterns for all the samples. No traces of impurity phases (i.e., Bi 2 Fe 4 O 9 and Bi 25 FeO 39 ) were observed in the samples. The difference between the observed and calculated pattern and the Bragg's positions are given below the observed spectrum (figure 1) for comparison. The lattice constant, space group, and goodness of fit obtained from the Rietveld refinement of (BFO) 1-x -(GMO) x 0.0 x 0.2 samples are given in table 1.
The existence of both rhombohedral and orthorhombic structure becomes more prominent with increasing x. For x = 0.2, the XRD pattern contains the reflections, which are the characteristics of Pnma/Pbnm space group similar to RE manganites. The polar Pn2 1a space group is sub-space group of nonpolar Pnma space group. Therefore, Rietveld refinement was carried out with dual phase model (R3c + Pn2 1a ). For x = 0.1 composition, it is found that around 86% of rhombohedral R3c phase and remaining 14% that of orthorhombic Pn2 1a phase. Sample with x = 0.15 shows 42% of R3c and 58% of Pn2 1a phase, whereas x = 0.2, shows 24% of R3c and 76% of Pn2 1a phase. The transition from rhombohedral to a mix phase consists of rhombohedral plus orthorhombic phase is expected and can be explained by strong destabilization of R3c structure by Gd 3+ due to chemical pressure [17].
In order to establish the signature of compositional driven structural phase transition from R3c to R3c+Pn2 1a , the pseudo-cubic lattice parameter as a function of composition was plotted in figure 2. Pseudocubic lattice parameter is defined as the cube root of the perovskite unit cell volume. It offers a convenient approach because it brings down lattice parameter to the same scale of magnitude for better understanding and is a suitable probe to analyze structural phase transition. Figure 3 shows the average crystallite size of (BFO) 1-x -(GMO) x 0.0 x 0.2 samples. The average crystallite size of the samples were calculated using Fourier x-ray line profile analysis (XLPA) based on Double-Voigt method using the program package BREADTH [18]. The average crystallite size of (BFO) 1-x -(GMO) x 0.0 x 0.2 solid solutions varies in the range 20 nm to 30 nm. To further support the XLPA analysis, the microstructure of the sample x = 0.2 was investigated using high resolution transmission electron microscopy (HRTEM). Figure 4 shows the TEM micrograph (a) and selected area electron diffraction (SAED) pattern (b) of sample x = 0.2. The TEM micrograph (figure 4(a)) shows the average crystallite size of ∼30 nm with some agglomeration, which agrees well with the average crystallite size obtained from XLPA analysis. In order to get the structural insights and the presence of mixed phase, SAED pattern was taken for selected samples. For better the understanding of microstructure, the fractured surface of the samples were analyzed with scanning electron microscope (SEM). Figure 5 shows the scanning electron micrographs of selected (BFO) 1-x -(GMO) x samples (x = 0.0, 0.05, 0.1, and 0.2). As can be seen from the SEM micrographs, the samples show densely distributed grains with well defined grain boundary without defects and voids. Pure BFO (figure 5(a)) shows highly non-uniform grain size distribution ranging from 200 nm-2 μm, whereas samples with x = 0.05, 0.1, and 0.2 (figures 5(b)-(d)) show distribution of uniform grain size of average 200 nm. A decreasing in grain size was observed in solid solutions with increasing GMO concentration. This observation is not yet well understood, however, it can be attributed to the reduction of electrical resistivity due to the incorporation of rare earth oxide (GMO) phases into the semiconducting BFO phase which facilitate an enhanced diffusion process and decreasing gain size. The reduction in electrical resistivity by 10 orders of magnitude has been reported in BaTiO 3 with rare earth oxide incorporation [19].

Raman spectroscopy
In order to get the structural insight of the composition driven phase transition and changes in phonon response, micro-Raman spectra were recorded in backscattering geometry using Ar ion laser (λ = 514.5 nm). Figure 6 shows the comparison of Raman spectra of (BFO) 1-x -(GMO) x 0.0 x 0.2 samples at RT. According to the group theory selection rules, the rhombohedral structure of BFO with R3c (C 3v ) symmetry will give rise to 4A 1 + 9E Raman active modes [5,6]. The peaks at low temperature (81 K) in single domain single crystal observed at around 75, 265, 350, and 523cm −1 are assigned as A 1 modes, whereas peaks at around 79, 145, 175, 224, 277, 295, 371, 473, and 553 cm −1 are assigned as E modes [6]. Of these 13 single phonon modes of BFO, we observed only eight Raman active modes in (BFO) 1-x -(GMO) x 0.0 x 0.2 samples at RT. Some modes are not seen due to thermal broadening effect at RT. As can be seen from the figure 6, the A 1 modes are observed at around 75, 262, 344, and 530 cm −1 , whereas, four E modes are seen at around 139, 171, 370 and 470 cm −1 in agreement with BFO single crystal Raman spectra [5,6].
As GMO concentration increases, we observed some suppression of BFO phonon modes and appearance of three new phonon modes at around 293, 497, and 620 cm −1 associated with GMO. The peak at 620 cm −1 (B 2g (1) mode) appears from x = 0.025 is associated with the in-plane O 2 stretching and called as breathing mode. This mode is shifting to higher frequency with increasing GMO concentration. The mode at 293 cm −1 (A g (2) mode) appears from the sample x = 0.1 is related to the in-phase rotation (y-axis) of MnO 6 octahedra and O 2 motion along x-axis. The mode at around 497 cm −1 (A g (3) mode) appears in samples x 0.15 corresponds to bending of MnO 6 octahedra and O 2 anti-stretching [20]. As can be seen from figure 6, it is quite evident that (BFO) 1-x -(GMO) x up to x = 0.075 show phonon features dominated by BFO phase, while samples with higher concentration of GMO (x 0.1) show phonon anomalies indicating the existence of mixed structural phase. The observation of mixed phase at higher concentration of x agrees well with the XRD and TEM (SAED) studies.

Dielectric properties
In order to understand the dielectric behavior, the dielectric permittivity (ε), dielectric loss (tan δ), conductivity (σ ac ), and phase angle (θ) were measured as a function of frequency and temperature. Figure 7(a) shows the frequency variation of dielectric permittivity. The observation of decrease in permittivity with increasing frequency is the typical behavior of ferroelectrics. The high dielectric permittivity observed at low frequency can be explained by the presence of a potential barrier probably due to the existence of space charge polarization at the grain boundaries. This can lead to the accumulation of charges at the grain boundary, which leads to the high value of dielectric permittivity [21][22][23]. The decrease in dielectric permittivity with increasing frequency can be attributed to the reduction of space charge polarization [23]. The change in dielectric dispersion behavior (nonexponential) with increasing polar phase (Pn2 1a ) upon GMO substitution can be attributed to the presence of morphotropic phase boundary (MPB). No significant change is observed on the values and nature of dielectric dispersion at higher frequencies. However, at low frequency (<1 kHz) an increase in dielectric permittivity was observed with increasing GMO concentration. This can be explained by the change in conductivity or dipole relaxation phenomena [19].
To understand the nature of dielectric relaxation process, the dielectric dispersion behavior was analyzed with various dielectric relaxation models. As can be seen, the data fits very well with the modified Maxwell-Wagner (MW) relaxation model, as described in equation (1) [24][25][26]: ( ) a pt a pt ap where ò s is the dielectric permittivity at the lowest frequency, ò ∞ is the dielectric permittivity at the highest frequency, α is the measure of the polydispersity of the system, τ is the total relaxation time, and n is the dispersion parameter of frequency [27,28]. The main consideration of the MW effect is the charge accumulation at the interface and their different charge carrier relaxation times [24]. The MW effect is commonly observed in two-material interface, such as metal-insulator, metal-semiconductor, insulator-semiconductor, semiconductor-semiconductor, etc [24,27,28]. The ratio of dielectric constant to conductivity is the relaxation time associated with the spreading of the excess free carriers in the materials [24]. The solid line in figure 7(a) shows fits of the modified Maxwell-Wagner relaxation model with the experimentally observed data. The values of ò s , ò ∞ , α, n, and τ obtained from the MW-model for (BFO) 1-x -(GMO) x (0.0 x 0.2) solid solutions at RT are given in table 2. As can be seen, the relaxation time increases with increasing x. A significant increase in relaxation time was observed for samples with x 1, which can be associated with the increase of polar phase and MPB. Figure 7(b) shows the frequency variation of dielectric loss of BFO-GMO solid solutions. As can be seen, samples with x 0.05 show low dissipation compared to BFO in the measured frequency range. The sample with x = 0.075 shows low loss at higher frequencies compared to BFO. Samples with x 0.1 show no improvement in dielectric dissipation at lower frequencies, but shows slightly higher loss at higher frequencies.
The ac conductivity (σ ac ) of the (BFO) 1-x -(GMO) x samples obtained from the dielectric measurements using the formula [29, 30] σ = σ 0 ωò 0 ò tan δ, where σ 0 is the frequency independent conductivity, ω is angular frequency, ò 0 is vacuum dielectric permittivity, is shown in figure 7(c). Generally, σ ac of dielectrics increases with increasing both the frequency and the temperature. The frequency variation of ac conductivity was analyzed by universal power law proposed by Jonscher [31]: σ ac = Aω n . This power law is related to the dynamics of the ionic hopping transport between localized sites. The exponent n is the measure of the degree of interaction with the materials conduction environment. The exponent in the range of 0.6-1.0 indicates disordered or amorphous  Table 2. Fitting parameters obtained from Maxwell-Wagner model for (BFO) 1-x -(GMO) x 0.0 x 0.2 at RT for dielectric permittivity at lower frequencies (ò s ), dielectric permittivity at higher frequencies (ò ∞ ), polydispersity (α), dispersion parameter of frequency (n), and the relaxation time (τ). systems and the conduction mechanism is mostly associated with diffusion limited hopping. Exponent n ≈1 implies ideal long-rang (band-to-band) conduction process, which is normally observed at low temperature, whereas n > 1 indicates the presence of thermally activated hopping process between two sites separated by energy barrier [29]. It also could be due to the presence of two dispersion processes with different relaxation times [32][33][34][35]. As can be seen, the ac conductivity σ ac shows an excellent fit (solid line) to the modified Jonscher model, as defined in equation (2) [29,31]: where σ 0 is the frequency independent conductivity (dc conductivity), A 1 and A 2 are the coefficients, and n 1 and n 2 are critical exponents. The universal power law was modified with hopping relaxation model and conduction through grain boundaries. The first exponent is associated with the charge carriers motion due to dc conduction and dispersive ac response. The frequency at which slope changes is known as hopping frequency of the polarons (ω p ) and is temperature dependent. The plateau region corresponds to the frequency-independent dc conductivity. The rise in the conductivity value with frequency indicates a normal ferroelectric behavior [36].
The conduction parameters and critical exponents obtained from the fitting are given in table 3. As can be seen, n 1 varies from 0.48 to 0.90 and n 2 varies from 0.90 to 1.06 indicating presence disordered phase and hopping    type conduction process. The different n could be indicative of the charge carriers might have translational motion with a sudden hopping and localized hopping [37]. Figure 7(d) shows the frequency variation of phase angle (θ), the angle between the real (resistance, R) and imaginary part (reactance, X) of impedance Z = R + jX), of BFO-GMO samples. All the samples show a resistivity behavior at low frequency and gradually changes to highly capacitive nature at higher frequencies, which is consistent with the observation of very low dielectric dissipation at high frequency. Figure 8 shows the temperature variation of dielectric permittivity (ò r ) and loss (tan δ) of (BFO) 1-x -(GMO) x 0.0 x 0.2 at f = 1 MHz. The dielectric permittivity ( figure 8(a)) pure BFO shows an anomaly at around the Neel temperature (T N = 360°C). Samples with x = 0.025 and 0.05 show behavior similar to BFO showing a very weak anomaly near magnetic transition temperature. However, samples with higher concentration of GMO (x 0.1) show the shifting of anomaly towards lower temperature. Samples with x = 0.075, 0.10, 0.15, and 0.2 show anomaly at around 345, 275, 206, and 212°C, respectively. The observation of the shifting of the anomaly implies the shifting of the magnetic phase transition of the BFO towards the lower temperature. The dielectric loss ( figure 8(b)) as a function of temperature shows similar anomaly with increasing x. For pure BFO, the loss was almost constant up to T N and above T N it shows very slow increase. As the concentration of x increased in the solid solution the anomaly of dielectric loss shifted towards lower temperature with faster rate. The observed behavior of dielectric permittivity and loss in BFO is consistent with the reported results [38]. Figure 9 shows the ferroelectric polarization of some unpolled (BFO) 1-x -(GMO) x (x = 0.0, 0.025, and 0.05) samples at RT. The absence of saturation of polarization implies the lossy ferroelectric behavior. Decreasing in remanence and coercivity was found with in increasing GMO concentration. Samples with higher concentration of x (>0.05) show very similar PE loop as sample x = 0.05. The observed polarization is very low compared to the values obtained in single crystal and thin film, but it agrees well with the polycrystalline samples [38]. The polarization in BFO is associated with the Bi 6s lone pair electrons. However, the high polarization observed in thin films is attributed to the large strain-induced Fe-ion displacement relative to the oxygen octahedra in addition to the contribution of Bi 6s lone pair electrons [39]. The polarization in BFO is also highly dependent on structural phase [39]. The observation of suppression of polarization is attributed to the domain wall pinning [40]. BFO has very complex domain structures (twin 71°, twin 109°, and twin 180°domain walls) [40]. In polycrystalline samples, disorders play a significant role in suppressing polarization. Polycrystalline samples could not sustain the high electric field required for switching the magnetization to study the magnetoelectric effect. So, we focused our magnetoelectric (ME) coupling studies through magnetodielectric (MD) effect. Figure 10 shows the magnetic hysteresis loop of (BFO) 1-x -(GMO) x (0.0 x 0.2) solid solutions at RT (300 K) and 2 K. As can be seen from the RT hysteresis loop (figure 10(a)), pristine BFO and BFO with 2.5% GMO shows no hysteresis loop even up to field of 2 T. This observation is in agreement with the nature BFO magnetic ordering [1,2]. The magnetic nature of BFO is associated with the canted G-type antiferromagnetic ordering with spiral spin modulation of 60nm [41], due to the combined effect of exchange interaction and spin-orbit coupling. In the case of GMO, the ferromagnetically ordered Gd (T c = 293 K) spins significantly contribute to the net magnetic moment compared to the antiferromagnetically canted state of Mn (T N = 100 K) spins [15]. The solid solutions with higher concentration of x show ferromagnetic-like hysteresis loop at RT. This could be due to the change in magnetic ordering in BFO from the cycloidal spiral spin structure to a ferromagnetic ordering with increasing GMO concentration. Relatively high value of remanent magnetization and coercivity was observed in BFO with 10% GMO compared to other samples. The inset in figure 10(a) shows the variation of coercivity (H c ) and remanent (M r ) magnetization as the function of x at RT. Figure 10(b) shows the magnetic hysteresis (MH) loop at 2 K. As can be seen, no magnetic hysteresis is observed for pure BFO, which is consistent with its intrinsic magnetic ordering. We observed higher magnetization and saturation at 2 K in other samples and the trend is somewhat commensurate with the RT observation that the sample x = 0.1 shows higher coercivity compared to other samples. The improved magnetization and coercivity observed in samples with x 1.0 can be attributed to the synthesis of nanocrystalline solid solutions of BFO-GMO which might have introduced strain leading to the suppression of spin cycloid modulation of BFO and appearance of uncompensated spins [41][42][43]. The following possibilities might be responsible for the observed effect: (i) substitution of Gd 3+ at Bi-site might have facilitated to collapse of space modulated spin structure; (ii) substitution of magnetically active Mn 3+ at Fe-site of BFO favors ferromagnetic interaction. It is well known that the magnetic properties can be affected significantly by the nearest and next nearest neighbor exchange interactions. The samples with higher doping concentration GMO concentration (x 0.15) shows reduction in coercivity and remanence, which can be explained by the change in neighboring magnetic ions ordering from ferromagnetic to an antiferromagnetic (substitution of ferromagnetic spins of Fe by antiferromagnetic spin of Mn). This can reduce the exchange interactions resulting in lower magnetization.

Magnetic properties
In order to understand the nature of magnetic ordering and magnetization dynamics, temperature variation of field cooled (FC) and zero field cooled (ZFC) magnetization was carried out from RT down to 2 K as shown in figure 11. As can be seen, the FC and ZFC magnetization show very similar behavior indicating the absence of secondary magnetic phase and magnetic frustrations [44]. The sharp increase of magnetization at low temperature could be related to the presence of paramagnetic cluster or phase [45,46]. Since the 4f electrons of the rare earth (Gd 3+ ) are very localized, the direct exchange is not a dominating mechanism for the magnetic properties [45,46]. According to the Goodenough-Kanamori-Anderson (GKA) rules [47][48][49][50], the strong superexchange interaction between Mn 3+ and Mn 4+ through oxygen (O −2 ) anion (magnetic ion-ligandmagnetic ion are in 180°) results in antiferromagnetic ordering, whereas, a 90°superexchange interaction between Mn 3+ / Mn 3+ and Mn 3+ / Mn 4+ through vacancies could result in weaker ferromagnetic ordering. The antiferromagnetic coupling between Mn atoms is energetically favored [50]. The observation of ferromagneticlike behavior and non-saturation of magnetization at low temperature in BFO-GMO solid solution samples could be due to the presence of both the types of superexchanges.
3.6. Magnetodielectric properties 3.6.1. Impedance and modulus spectroscopy The dynamical features of dielectrics and ferroelectrics are accompanied by the motion of charges, e.g. orientation polarization of dipoles and/or the motion of ions. Impedance (Z) and modulus (M) spectroscopy are powerful methods of characterizing the electrical properties of materials and their interface with electrodes. The impedance spectroscopy (IS) allows the separation of resistances related to grains, grain boundaries, and electrode effects because of their different relaxation times that results in a separate semicircles in impedance spectra [26]. These techniques can also be used to study the dynamics of bound or mobile charge in the bulk or interfacial regions of solids and provide a better insight for understanding the relationship between electrical transport and microstructure in nanocrystalline samples.
In order to understand the dielectric relaxation, e.g. long-range conductivity (non-localized) and dipolar (localized) relaxation, we carried impedance spectroscopy of multiferroic solid solutions with and without magnetic field (H). Figure 12 shows the Nyquist plot of real ( ¢ Z ) and imaginary (Z″) part of the impedance of (BFO) 1-x -(GMO) x (0.025 x 0.2) solid solutions as a function of magnetic field at RT. As can be seen, the sample with x = 0.025 ( figure 12(a)) shows a non-linear behavior, but not semicircular, suggesting an insulating nature. The undoped BFO (not shown) shows very similar behavior as in the case BFO with 2.5% of GMO. The semicircular behavior gradually develops with increasing concentration of x. For x = 0.05, a semicircular arc starts forming and became prominent with increasing composition up to x = 0.075. Samples with composition x 0.1, exhibit the presence of two overlapping semicircular arcs with their centers below the real axis. The high frequency semicircle is attributed to the bulk (grain) property of the material, whereas the low frequency semicircular arc observed at intermediate frequency is attributed to the grain boundary effect. The intercept of the semicircular arcs on the real axis gives rise to grain and grain boundary resistance of the materials. The absence third semicircle rules out the presence of electrode-sample interface effect. As can be seen, all the samples show suppression in resistance with increasing magnetic field. For quantitative understanding of the impedance behavior and the effect of grain and grain boundary contributions, the data were simulated with the equivalent circuit using a brick-layer model for a polycrystalline material [26]. The high frequency semicircular arc was modeled to an equivalent circuit shown in inset figure 12(a). The equivalent circuit contains parallel combination of a bulk resistance (R 1 ), bulk capacitance (C 1 ) along with a constant phase element (CPE) [26], whereas the low frequency semicircular arc was modeled for parallel combination of grain boundary resistance (R 2 ) and grain boundary capacitance (C 2 ). The CPE admittance is ( . A 0 and n are frequency independent parameters usually depend on temperature, A 0 determines the magnitude of the dispersion and the n value is 0 n 1 [17]. The CPE describes an ideal capacitor for n = 1 and an ideal resistor for n = 0 [51][52][53]. Both the equivalent circuits are connected in series for fitting the impedance data. The solid line in figure 12 shows simulated fit of equivalent circuit to the data using an electrochemical impedance spectroscopy (EIS) data analysis software (ZSimpWin) [54]. As can be seen, a good agreement has been found with the equivalent circuit simulated data with the observed data. Tables 4 and 5 list the bulk capacitance and resistance, respectively, obtained from the Nyquist fittings using the equivalent circuits (Inset figure 12(a)) for (BFO) 1-x -(GMO) x (0.0 x 0.2) solid solutions. It is observed that for all the compositions the bulk capacitance decreases with increasing magnetic field, while bulk resistance increases, which indicates the presence of magnetoelectric coupling in BFO-GMO solid solutions.
For the better understanding of the relaxation process and dominating charge transport mechanisms, the dispersion of imaginary part of impedance (Z″) and modulus (M″) of (BFO) 1-x -(GMO) x (0.025 x 0.2) were measured and compared at a constant magnetic field (2 T) as shown in figure 13. As can be seen, no relaxation peak (maxima) was observed for x = 0.025 ( figure 13(a)), while samples with x 0.05 (figures 13(b)-(e)) shows dielectric relaxation in both impedance and modulus. It was found that the relaxation time (τ = 1/ pf 2 max , where f max is the frequency corresponds to peak maximum of (Z″) and (M″) decreases with increasing composition (x). The presence of dielectric relaxation peaks in impedance and modulus at different frequency implies non-Debye relation process which is dominated by the short-range hopping of charge carriers. In the case of long-range Debye type relaxation process the relaxation peak of impedance and modulus should appear at the same frequency [55][56][57]. Inserting a constant phase element (CPE) in the equivalent circuit modeled the observed impedance behavior very well.  Table 4. Bulk capacitance obtained from fitting of magnetic field dependent complex impedance plot of

Magnetocapacitance and magnetoimpedance measurement
In order to understand the magnetodielectric (MD) coupling, magnetocapacitance (MC) and magnetoimpedance (MI) of the BFO-GMO solid solutions were measured at RT. It has been predicted that an intrinsic magnetoresistance (MR) could enhance the MD effect from inhomogeneities. In ceramics the MR is dominated by the spin-polarized tunneling across the grain boundary. Maxwell-Wagner effect combined with MR could be the mechanism for magnetocapacitance without magnetoelectric coupling [58,59]. To rule out the MR related MD coupling effect, we carry out frequency dependent MC and MI at RT. The change in MC was measured with and without magnetic field as: MC(H) = [C(H)-C(0)]/C(0), where C(H) is the capacitance at the magnetic field H and C(0) is the capacitance at the zero magnetic field. Figure 14 shows the MC of (BFO) 1-x -(GMO) x (0.0 x 0.1) solid solutions at 100 Hz and 1 kHz in forward and reverse magnetic field (0-2 T) sweeps. Our study was focused on the compositions away from the MPB. As can be seen, the undoped BFO ( figure 14(a)) shows very weak MC change and magnetic hysteresis, which agrees with its intrinsic magnetic ordering of BFO. Samples with x = 0.025, 0.05, 0.075, and 0.10 (figures 14(b)-(e)) shows about 20%-25% MC and 25%-30% magnetic hysteresis at 2 T and 100 Hz. A strong magnetodielectric (MD) effect could be useful for device applications. All the samples show reduced MC and hysteretic behavior at 1 kHz. The observed negative MC in samples with x = 0.025, 0.075, and 0.10 in forward and reverse field sweeps indicates the presence of antiferromagnetic dominated magnetic ordering ar RT. The sample with x = 0.05 ( figure 14(c)) shows both the positive and negative MC change indicating the presence of both the ferromagnetic and antiferromagnetic orderings, which can be explained by the competing interaction between the ferromagnetically ordered Gd 3+ spins with the antiferromagnetically canted Mn 3+ spins. The observation of To establish the existence of intrinsic MD coupling, we carry out frequency dependent magnetoimpedance (MI) at RT. Figure 15 shows MI of (BFO)

Conclusion
In summary, nanocrystalline solid solutions of (BFO) 1-x -(GMO) x 0.0 x 0.2 have been synthesized successfully by auto-combustion method. The analysis of structural property by Rietveld refinement showed the existence of morphotropic phase boundary at x = 0.10, which is in agreement with the Raman spectroscopy and HRTEM studies. The dielectric dispersion behavior fits very well with the modified Maxwell-Wagner model. The frequency dependent phase angle study showed the resistive nature of solid solutions at low frequency, whereas it showed capacitive behavior at higher frequencies. The temperature variation of dielectric permittivity showed dielectric anomaly at magnetic phase transition temperature and it was shifted towards the lower temperature with increasing GMO concentration indicating the lowering of magnetic phase transition towards lower temperature. The Nyquist plot showed the conduction mechanism is mostly dominated by grains and grain boundary resistance. The ac conductivity of all the samples followed the modified Jonscher model. The impedance and modulus spectroscopy showed a non-Debye like relaxation mechanism which can be modeled using a CPE in the equivalent circuit.The unsaturated ferroelectric polarization loops indicated the lossy ferroelectric behavior. The BFO-GMO solid solutions showed enhanced ferromagnetic-like behavior at RT. The observation of frequency dependent magnetocapacitance and magnetoimpedance clearly showed the existence of an intrinsic magnetodielectric coupling. The (BFO) 1-x -(GMO) x solid solutions with x = 0.025-0.075 showed significantly high MC and MI compared to the pure BFO, which can be useful for practical device applications.