1 Introduction

Multiferroic materials have the unique properties with existence of both ferroelectricity and ferromagnetism phenomenon in a single-phase distorted system. Among all the multiferroic materials available so far, BiFeO3 (BFO) is the only material known today in which coupling between magnetic and ferroelectric ordering exist at room temperature [1]. This type of effect is known as magneto-electric (ME) coupling effect [2]. As a result, multiferroic materials have become an integral part of recent electronic or advance technology due to their applications in information storage, high-density ferroelectric random access memory (FeRAM), multiple-state memories, magnetic data-storage media, actuators, transducers, sensors, quantum electromagnets, microelectronic devices and spintronics devices [3–5]. The structure of BFO has rhombohedral (i.e., distorted perovskite of ABO3) with space group R3c [6], where the A-sites could be filled by rare-earth, alkaline earth, alkali or other large ions and the B-sites always filled by transition metal cations. BFO shows simultaneous ferroelectricity [with high Curie temperature (Tc ~ 1103 K)], and ferromagnetism [with high Neel temperature (TN ~ 630 K)] in distorted perovskite [7, 8]. Because of the multifunctional applications of ME materials, they simultaneously exhibit spontaneous polarization and spontaneous magnetization. In BFO ferroelectric properties is due to the presence of 6 s lone pair electrons of Bi3+ ions, and thus ferromagnetic order is due to Fe3+ ions [9]. Unfortunately, the major problems of BFO and other materials of this family are their large leakage current density that limits the applications of the materials. Recently the researchers have developed few new single-phase and composite multiferroic materials with high ordering temperature, high coupling constant, low dielectric loss and low leakage current [10, 11]. The extensive literature survey on BFO structural compounds reveals that a lot of work has been carried out in rare-earth substituted BFO. The present work provides a new insight into the effect of substitution on rare-earth ions (R = Gd, Sm, Dy, Nd and Pr) at the Bi-site of BiFeO3 offers the opportunity to optimize several relevance’s of the material properties [12–15]. The polycrystalline samples of rare-earth modified bismuth ferrite (i.e., Bi0.75R0.25FeO3; R = Dy and Nd) (BRFO) were synthesized using a high-temperature solid-state reaction method. Dielectric, Complex impedance spectroscopy has been used to study electrical properties of these compounds. Magnetic analysis has been used to study the magnetic properties of these samples. The grain size and remnant magnetization have been shown to be influenced by rare-earth doping in BFO ceramics.

2 Experimental

The polycrystalline samples of BiFeO3, Bi0.75R0.25FeO3 (BRFO) (R = Dy and Nd) were prepared by using high-purity (>99.9 %) ingredients: Bi2O3, Fe2O3, Dy2O3 and Nd2O3 (M/s Loba Chemie Pvt. Ltd., India). The materials were first stoichiometrically weighed and mixed thoroughly using agate mortar and pestle in dry medium (i.e., in air atmosphere) for 1 h, and wet medium (methanol) media for about 2 h. The homogeneous mixtures of the above compounds were calcined at optimized temperatures in the range of 780–830 °C in alumina crucible for 4–5 h in air atmosphere. The process of grinding and calcination was repeated until the formation of the compounds was confirmed. The calcined powders were pressed into small disc of diameter = 10 mm and thickness = 1–2 mm at a pressure of 4 × 106 N/m2 with a binder [polyvinyl alcohol (PVA)]. The pellets were then sintered at different temperature (820–870 °C) for 4 h.

The preliminary structural analysis was performed by means of X-ray diffraction using (XRD; Rigaku Mini flex, Japan). The XRD pattern of the calcined powder was recorded at room temperature with CuKα (λ = 1.5405 Å) in a wide range of Bragg angles θ (20 ≤ 2θ ≤ 80o) at a scanning rate of 3 o/min. The microstructures of sintered pellets were recorded by scanning electron microscopy (SEM; JEOL-JSM, model 6510). For the measurements of electrical properties, a pellet was carefully polished and subsequently painted with high purity Ag-paste on its both sides to work as electrode. The frequency dependence of the capacitance, dissipation factor, phase angle and impedance were measured at different temperatures using a phase sensitive LCR/impedance meter (PSM 1735, N4L). The I-V characteristics of BRFO was measured as a function of voltage (1–100 V) with an interval of 25 °C starting from room temperature (25 °C) up to 400 °C using a programmable electrometer (Keithley, model 6517B). A superconducting quantum interference device (SQUID) magnetometer was used to study the magnetic properties of the materials at room temperature.

3 Results and discussion

3.1 Structural analysis

Figure 1 shows the XRD pattern of BiFeO3 and Bi0.75R0.25FeO3 (R = Dy and Nd). All the peaks of the XRD pattern were indexed using standard computer software program ‘PowdMult’ [13]. The lattice parameters of the samples were analyzed using computer software. From the best agreement between observed (dobs) and calculated (dcal) inter-planner distance d [Σ (dobs−dcal = minimum)], unit cell lattice parameters and volume of BRFO, structure of the compounds were found to be rhombohedral. There is no indication of change in basic structure of BFO on substituting Nd and Dy. Table 1 shows the comparison of lattice parameters a, c and volume (Å) with estimated standard deviation (in parenthesis) of BiFeO3, Bi0.75Nd0.25FeO3 and Bi0.75Dy0.25FeO3 samples.

Fig. 1
figure 1

XRD pattern of BFO, BNFO and BDFO at room temperature

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Table 1 Comparison of lattice parameters a, c and volume (Å) with estimated standard deviation (in parenthesis) of BiFeO3, Bi0.75Nd0.25FeO3 and Bi0.75Dy0.25FeO3 samples
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The scanning electron micrograph of BFO, BNFO and BDFO pellets are shown in Fig. 2a–c. It is observed that the grain size of BFO is significantly higher than that of rare-earth doped samples. Grains being smaller, aggregate to form bigger clusters is observed, which is consistent with our XRD study. It is also observed that rare-earth substitution densely less packed samples (i.e. creation of less porosity).

Fig. 2
figure 2

a–c SEM micrograph for BFO, BNFO and BDFO ceramics

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3.2 Electrical properties

Figure 3 shows the frequency dependence of relative permittivity (εr) and tangent loss (tan δ) for BFO, BNFO and BDFO at few selected temperatures. It can be observed that all the compounds exhibit dielectric dispersion where dielectric constant decreases rapidly with increasing frequency in the low-frequency region while it approaches almost frequency independent behaviour in the high-frequency region. The dielectric dispersion curve can be explained based on Koop’s theory [14, 15]. This model is based on the Maxwell–Wagner model for the inhomogeneous double structure in determining the dielectric properties of the ferrite materials. The dielectric loss gives the loss of energy from the applied field into the sample. This is caused by domain wall resonance. It is observed from Figure 3 that Nd-modified samples have low dielectric losses as compared to that of BFO and BDFO. The dielectric loss in the oxides of perovskite family is also caused by oxygen vacancies. At lower frequencies the value of tan δ is larger for BDFO as compared to BFO and BNFO. Generally, tan δ has very low value at high frequency. It indicates that dipoles with small effective masses (i.e., electrons and ferroelectric domains) mainly contribute to dielectric constant instead of charge defects with large effective masses (i.e., oxygen vacancies). Moreover, the higher values of tan δ in BDFO material generally represent large leakage current due to higher conductivity.

Fig. 3
figure 3

Variation of εr and tanδ with frequency for BFO, BNFO and BDFO at few selected temperatures

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Complex impedance spectroscopy (CIS) is an effective and unique method to establish correlation between the electrical properties of dielectric and ionic materials with their microstructures. In this technique, an ac signal is applied across the pellet sample and the output response is measured. The complex impedance data can ideally be analyzed by using an equivalent circuit consisting of two parallel combinations of R (resistance) and C (capacitance) components in a circuit. The frequency dependence of electrical parameters of a material is often represented in terms of complex impedance (Z*), electric modulus (M*), complex admittance (Y*) and complex dielectric constant (ε*) as given below: Complex impedance

$$Z^{*}\left( \omega \right) = Z^{\prime} - jZ^{ \prime \prime }$$

Where \(Z^{\prime} = R/1 + \, \left( {\omega CR} \right)^{2}\) and \(Z^{\prime\prime} = R^{2} \omega C/1 + \left( {\omega CR} \right)^{2}\)

$$\varepsilon^{*} = 1/M^{*} = \varepsilon^{\prime} - \, j\varepsilon^{\prime\prime}$$
$$tan\delta = \varepsilon^{\prime}/\varepsilon = M^{\prime\prime}/M^{\prime} = - Z^{\prime}/Z^{\prime\prime} = Y^{\prime\prime}/Y^{\prime}$$

Figure 4 shows the variation of Z″ with Z′ (usually referred as Nyquist plot [16]) for BFO, BNFO and BDFO at few selected temperatures. The Nyquist plots typically comprise a single semicircular arc, which is gradually resolved or distorted with rare-earth substituted BFO. The formation of a single arc is mainly due to the grains, and relaxation is due to Debye. For Nd and Dy-substituted BFO, there is a clear indication of formation of second semicircle in the low-frequency region. The formation of second semi-circle is due to grain boundary effect in the material [17]. For ideal Debye—like response, an equivalent circuit consists of parallel combination of (CQR) and (CR) where Q is said to a constant phase element (CPE).

Fig. 4
figure 4

Variation of Z′ with Z″ for BFO, BNFO and BDFO at few selected temperatures

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The admittance of Q is defined as Y (CPE) = A0 (jω)n = Aωn + jBωn where A = A0cos(nπ/2) and B = A0sin(nπ/2), where A0 and n are frequency independent but temperature dependent parameters. A0 determines the magnitude of the dispersion. The value of n is in between zero and one (0 ≤ n≤1). For an ideal capacitor n = 1 and for ideal resistor n = 0 [18].

The presence of non-semicircle, asymmetric or distorted semi-circle suggests the existence of non-Debye type of relaxation in the samples [19]. For an ideal Debye-type relaxation, a perfect semicircle with its center at Z′-axis is observed. The above observation clearly suggests that the studied materials do not obey Debye-type of relaxation. The observed depressed semicircular arcs having center lies below the real impedance (Z′) axis is due to the presence of distributed phase elements. The relaxation process associated with this observation is non-ideal in nature. This non-ideal behaviour may be originated from the several factors such as grain orientation, grain size distribution, grain boundaries, atomic defect distribution, and stress–strain phenomena [20]. The intercept of each semi-circle on real Z′-axis gives the value of bulk and grain boundary contributions in the resistance. The decrease in area of the semicircles is observed on increasing temperature in all the compounds. It reveals that the bulk relaxation of the sample varies with temperature in the studied frequency range. The value of bulk resistance (Rb) decreases with rise in temperature as well as Nd and Dy-modified BFO and indicate the presence of negative temperature coefficient of resistance (NTCR), as observed in semiconductors [21, 22]. The semi-circle of impedance spectra have a characteristic peak occurring at a particular frequency usually referred as relaxation frequency (fr). It can also be expressed as ωr RbCb = 1 and thus fr = 1/2πRbCb. The relaxation time due to bulk effect (τb) has been calculated using the equation ωrτb = 1 or, τb = 1/2πfr. The impedance value of BNFO and BDFO decreases, which shows more conductivity contribution for rare-earth modified BFO.

Figure 5 shows the frequency dependence of ac conductivity for BFO, BNFO and BDFO at few selected temperatures. The ac conductivity was calculated using an empirical dielectric relation:

$$\sigma_{\text{ac}} = \omega \, \varepsilon_{\text{r}} \varepsilon_{\text{o}} { \tan }\delta$$

(ω = angular frequency, εo = vacuum permittivity). Study of ac conductivity is important to understand the frequency dependence of electrical transport properties of the material. The frequency dependence of ac conductivity is explained using Jonscher’s power law [23],

$$\sigma_{\rm T} \left( \omega \right) = \sigma \left( {\text{o}} \right) + \sigma_{ 1} \left( \omega \right) = \sigma_{\text{o}} + {\text{a}}\omega^{\text{n}}$$

σT (ω) is the total ac conductivity, σ(o) is the dc conductivity term (i.e., independent of frequency), σ1(ω) is the dispersive component of ac conductivity, n is frequency independent but temperature dependent parameter. The value of n lies between zero and one. According to Jonscher’s power law, the relaxation is due to the mobile charge carriers. When a mobile charge carrier jumps from its original position to a new-site, which is in a state between two potential energy minima. If n < 1, the motion is translational, and n > 1 the motion is localized [24]. From Fig. 5, it is evident that the magnitude of ac conductivity rises with rise in temperature (i.e., thermally activated process). This is in good agreement with the observation made from the impedance variation with frequency. At high temperature (325 °C) and at low frequency domain, a plateau is observed indicating presence of dc conductivity. The plateau region extends to higher frequencies with increasing temperature. This feature is compatible with the development of a delocalized or low-localization conduction mechanism. In the dispersion region, there is a frequency at which a change in the slope of conductivity spectrum appears. This is conventionally known as the hopping frequency (ωp). The values of hopping frequencies as observed increased with the rise of temperature suggesting a possible enhancement in the carrier hopping rate of the mobile charge carriers with rise in temperature. The deviation from low frequency plateau at lower temperatures may be assigned to electron conduction through quantum mechanical tunneling between localized states. The power law regime of all the spectra is much less temperature dependent than the dc conductivity. At low temperatures σac is found to obey a power relation. These indicate that the transport mechanism is due to hopping of carriers via localized electron states [25].

Fig. 5
figure 5

Variation of ac conductivity (σac) with frequency for BFO, BNFO and BDFO at few selected temperatures

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Figure 6 shows the current density verses electric field for BFO, BNFO and BDFO at room temperature. The value of current density increases rapidly on increasing the electric field. It is found that Nd, Dy-modified BFO has high leakage current as compared to that of BFO. The increase of leakage current density in BNFO and BDFO is attributed to the effective reduction of the defect by the substituting Nd and Dy ions into the BFO. In the low-field region, curves of BNFO and BDFO obey the Ohmic-type of conduction behaviour (J α E γ : γ ~ 1) [26]. In the high-electric field region, the curve obeys Child’s law (J α E γ : γ > 1). Thus, in the high-electric field and high-temperature region, the conduction mechanism may be considered as combination of space charge limited, Schotty and Frenkel-Poole [27].

Fig. 6
figure 6

Variation of current density verses electric field for BFO, BNFO and BDFO at room temperature

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3.3 Magnetic properties

The field dependence of magnetization of BFO, BNFO and BDFO samples were measured between the fields of −2.5–2.5 T at room temperature using (SQUID). Figure 7 shows the M–H hysteresis loops of all the samples. The M-H loops indicate that the magnetic properties of doped samples are much more enhanced as compared to that of BFO. BFO has non-zero remnant magnetization (M r ) of the value of 4.72 × 10−4 emu/g at a coercive magnetic field (Hc) of 4.24 × 10−6 kOe. This is because BFO is known to be anti-ferromagnetic having a G-type magnetic structure but has a residual magnetic moment due to a canted spin structure (weak ferromagnetic) [28]. However, the switching behaviour as observed in BDFO, at low fields is the most interesting and exchange observations of the field dependence of magnetization which occurs at room temperature. It is noted that the enhancement of magnetic moment or magnetization is also accompanied with the decrease of grain size as evident from our SEM micrograph [29]. From the above Fig. 7, magnetization value is highest for BDFO sample because the effective Bohr magneton of Dy (9.8μB) is larger as compared to that of BFO and BNFO.

Fig. 7
figure 7

M–H hysteresis loops for BFO, BNFO and BDFO ceramics at room temperature

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4 Conclusion

The polycrystalline samples BFO, BNFO and BDFO were prepared by a solid-state reaction method. The XRD analysis of the samples suggests that there is no effect on crystal structure for Nd and Dy-modified BFO. The SEM micrographs showed that on substituting Nd, Dy- content in BFO, the grain size of the samples decreases and distributed uniformly. We also observed a remarkable change in the dielectric and conduction properties of the Nd, Dy- modified BFO. The impedance studies exhibit the presence of grain (bulk) and grain boundary effects, and existence of a negative temperature coefficient of resistance (NTCR) in the materials. It has been observed that Dy-substitution in BFO enhances the ferromagnetic properties and exhibits anti-ferromagnetic behavior of the samples at room temperature.