In general, dielectric analysis take a vital role in enhancing knowledge of a material medium's electrical properties as a function of temperature and frequency. Furthermore, by doing the dielectric measurements, the dielectric analysis might be obtained for a material medium. The material's dielectric properties, such as the dielectric constant (εr׳), dielectric loss (εr״), dissipation factor (tanδ), and alternating current (ac), can be assessed based on the experimental results from these experiments. For the (1x) PbMg1/3Nb2/3 O3 (x) AlBiO3 crystalline medium, the frequency and temperature dependences of these parameters were defined and investigated in the current work. The conventional relations published in the literature were used to determine the number of parameters.

When an AC electric field acts on any dielectric, there is the dissipation of a certain quantity of electric energy in the form of heat. we can represent the dielectric constant in the complex form as:

$${\epsilon }^{*}={\epsilon }^{{\prime }}- j{\epsilon }^{{\prime }{\prime }}$$

1

The charging current is physically represented by ε' and the loss current by ε'', respectively. Additionally, both rely on frequency and temperature. The strength of the alignment of the dipoles in the dielectric is determined by the real part of the permittivity(ε'(ω)), which measures the energy stored from the applied electric field in the material. The energy lost in the dielectric due to frictional damping, which prevents displacements of the bound charge from staying in phase with field changes, makes up the imaginary portion of the permittivity or loss factor.

Electrical data measured as complex impedance (Z*) is used to calculate relative permittivities ε′r & ε′′r at the given frequencies by the relation below[12–14].

$${\epsilon }_{r}^{{\prime }}= \frac{{Z}^{{\prime }{\prime }}}{j\omega {C}_{o} {\mid Z\mid }^{2}}$$

2

$${\epsilon }_{r}^{{\prime }}= \frac{{Z}^{{\prime }{\prime }}}{2\pi f\frac{{\epsilon }_{o}A}{d}({Z}^{{\prime }2}+ {Z}^{{\prime }{\prime }2})}$$

3

Where, \({C}_{o}= \frac{{\epsilon }_{o}A}{d}\), Z′′ denotes imaginary impedance, |Z| represents the magnitude of total impedance, Co signifies the free air capacitance; A and d are the areas and the thickness of the sample respectively.

$${\epsilon }_{r}^{{\prime }{\prime }}= \frac{{Z}^{{\prime }}}{j\omega {C}_{o} {\mid Z\mid }^{2}}$$

4

$${\epsilon }_{r}^{{\prime }{\prime }}= \frac{{Z}^{{\prime }}}{2\pi f\frac{{\epsilon }_{o}A}{d}\left({Z}^{{\prime }2}+ {Z}^{{\prime }{\prime }2}\right)}$$

5

Dependences of the real dielectric constant (εr׳) and imaginary part of the dielectric constant (ε״) were individually investigated for (1x) PbMg1/3Nb2/3 O3 (x) AlBiO3 single crystals.

The maximum amount of energy that may be stored in the material is determined by the real dielectric constant(*ε**r*׳) and can be computed as follows.

U =\(\frac{1}{2}{\epsilon }_{r}^{{\prime }}{E}^{2}Ad\) (6)

The relative loss factor *(ε**r*״), measures the absorption of electrical energy by dielectric materials. Dissipation factors also affect how well materials can absorb an ac field.

Figures 1 (a), (b), (c), and 2 (a), (b), (c), demonstrate the temperature dependence of the dielectric constant (εr) and the dielectric loss *(ε**r*״) at various frequencies for compositions with x = 0.1, 0.2, and 0.3. respectively, In the temperature range of RT to 500oC, the temperature dependence of the dielectric constant (εr) was determined at three fixed frequencies for 10kHz, 350kHz, and 0.2MHz. At 10kHz, it is discovered that the maximum dielectric temperature for a composition of 0.7PMN-0.3AB is 86oC. The highest observed value of εr is 115 at a frequency of 10 kHz.The compositions of 0.7PbMg1/3Nb2/3 O3 – 0.3AlBiO3 had the widest dielectric peak. This suggests the diffuse phase transition-driven relaxor ferroelectric characteristics, which have also been explained as the outcome of short-range order. The concept of a coexistence area between phases is used to describe this behavior[15, 16].

Owing to the relaxation studies, the dielectric loss can be divided into three parts: conduction losses, dipole losses, and vibrational losses[17].

Ion migration over considerable distances is probably involved in the loss that was attributed to conduction. The motion that results from direct current conductions is the same as this one. The ions leap above the network's highest potential. The lattice receives heat from the moving ions, and the quantity of heat lost per cycle is proportional to σac (ω)/ω. Conduction losses are at their lowest level at low temperatures. Conduction losses increase along with the temperature as σac (ω) increases. As a result, the value of ε*"* increases with temperature. Naturally, the strong faulty structure that is the root of ions like space charges is what is to blame for the increase in dielectric loss. Dipole polarization is the cause of the increased dielectric loss at low frequencies. The ion vibrations, however, might be the only cause of dielectric loss at higher frequency values[18, 19].

For the (1x) PbMg1/3Nb2/3 O3 (x) AlBiO3 single crystals, the temperature dependence of the imaginary component of the dielectric constant was examined in the temperature range RT-500oc at various working frequencies of the applied ac field running from 100Hz to 1MHz. Figures 2a, b, and c show how the imaginary component of the dielectric constant varies with temperature. However, because the temperature has a similar influence on the imaginary component of the dielectric constant for all working frequencies examined here, these Figures include the behavior for specific working frequencies of 10 kHz, 350 kHz, and 0.2 MHz. In this graph, the imaginary portion of the dielectric constant is seen to climb significantly from room temperature (RT) to the temperature where the dielectric constant is at its highest at the operating frequencies[20].

The imaginary portion of the dielectric constant, however, gradually drops after reaching the temperature of the maximum dielectric constant and eventually shows no change. In the low-temperature region, as shown in Figs. 2(a), (b), and (c), the imaginary part of the dielectric constant of the (1x) PbMg1/3Nb2/3 O3 (x) AlBiO3 crystalline medium increases slightly with the increase in temperature, showing a slight dependence of the imaginary part of the dielectric constant on temperature. Additionally, the imaginary part of the dielectric constant exhibits a significant dependence on temperature in the high-temperature region, with its variation with temperature in the high-temperature region showing a greater dependence on temperature than its variation in the low-temperature region.

The (1x) PbMg1/3Nb2/3 O3 (x) AlBiO3 single crystals were studied for the frequency dependence of the real relative dielectric constant at frequencies ranging from 100Hz to 1MHz at 500oC. Figure 3 depicts how the actual relative dielectric constant varies with frequency. According to this Fig. 3, it can be seen that the dielectric constant εr gradually decreases as the applied frequency rises. Based on the oscillation of free dipoles in the presence of the alternating field, this can be explained. Dipoles follow the electric field at extremely low frequencies, but as the frequency increases, they start to lag behind the field and εr somewhat decreases[14]. Applying an ac electric field, also known as polarization, causes slow displacements of the charge carriers, which lead to a drop in the actual dielectric constant with increasing frequency in a dielectric material.

Electronic, ionic, dipolar (or orientation), and space charge are only a few of the well-defined electrical polarization components that contribute to the dielectric constant decreasing with frequency; (1) Electronic polarization (Pe), which appears at frequencies up to 1016 Hz and is caused by the valence electrons being displaced with the positive nucleus; (2) Ionic polarization (Pi), is developed as a result of the displacement of positive and negative ions relative to one another. Because they are heavier than electrons in this category, ions cannot be rapidly polarized, and as a result, the maximum frequency of ionic polarization is roughly 1013 Hz; (3) Dipolar polarization (Pd), is induced as a result of molecules' permanent electric dipole moments and regulates orientation change in the direction of the applied electric field. Dipolar polarization happens up to a frequency of around 1010 Hz, and (4) Space charge polarization (*P**s*) is due to the existence of the mobile charge carriers impeded by interfaces. Frequencies between 1 and 103 Hz are typically where space charge polarization takes place. Additionally, it is stated that the greater values of εr at low frequencies are brought on by the polarization of space charges or the buildup of charges at electrode and sample interfaces[21, 22]. This suggests that flaws like vacancies, dislocations, and other defects may be responsible for the larger value of the dielectric constant at low frequencies in the crystals under study.

The fact that εr is a multi-component of the polarizability, deformational (electronic and ionic), and relaxation (orientation & interfacial) polarization can be used to explain why εr decreases as frequency increases. Most polarization types are unable to orient quickly enough to maintain alignment with the applied field as the frequency of the applied field increases and as a result the dielectric constant drops. Figure 3 demonstrates that for x = 0.3, the (1x) PbMg1/3Nb2/3 O3 (x) AlBiO3 for x = 0.3 presents a larger value of εr, but for x = 0.2, it is relatively high.

For the (1x) PbMg1/3Nb2/3 O3 (x) AlBiO3 single phase crystals, the frequency dependence of the imaginary part of the dielectrics(ε″) was measured in the range of the applied ac field frequency from 100Hz to 1MHz at various working temperatures ranging from RT to 500oC. Figure 4 shows how the imaginary component of dielectrics (ε″) varies with the frequency of the applied ac field. However, due to the frequency's consistent impact on the imaginary component of the dielectric constant at all working temperatures examined here, this Figure depicts behavior for the chosen working temperature (500oC).

At all the working temperatures examined, Fig. 4 demonstrates a pronounced decrease in the imaginary part of the dielectrics (ε″) with an increase in frequency. The behavior of the (1x) PbMg1/3Nb2/3 O3 (x) AlBiO3 crystalline medium to the imaginary part of dielectrics (ε″) is clear from Fig. 4, and it is comparable to its behavior for the real part of dielectric constant. The behavior of the imaginary portion of the dielectrics (ε″) for the (1x) PbMg1/3Nb2/3 O3 (x) AlBiO3 crystalline medium, however, exhibits significant dependence on temperature and frequency and can be explained using frequency intervals.

The behavior indicates a dramatic decline in the imaginary portion of the dielectrics (ε″) with a frequency increase in the first interval, which is in the low-frequency range. As a result, in the low-frequency area, the imaginary portion of the dielectrics (ε″) of the (1x) PbMg1/3Nb2/3 O3 (x) AlBiO3 crystalline medium shows a strong frequency dependence. The initial frequency for such a working temperature is discovered to be where the imaginary component of dielectrics (ε″) reaches its maximum value. Therefore, compared to the dependency shown at the low-frequency region, the imaginary part of the dielectrics (ε″) of the (1x) PbMg1/3Nb2/3 O3 (x) AlBiO3 crystalline medium shows a weak dependence on temperature.

The large values of the imaginary part of the dielectric constant exhibited by the (1x) PbMg1/3Nb2/3 O3 (x) AlBiO3 single crystals at low frequencies may be attributed to space–charge polarization due to the crystalline defects. Dipole polarization is the cause of the increased dielectric loss at low frequencies. Figure 4 clearly shows that ε″ decreases with rising frequency over the tested range. The fact that the main source of the dielectric loss (ε″) at low frequencies is the migration of ions that lead to space charge polarization in (1x) PbMg1/3Nb2/3 O3 (x) AlBiO3 at the grain boundary interfaces may be the cause of the drop in ε″ with frequency. Due to the ion jump and conduction loss of ion migration, as well as the ion polarization or dipole polarization, ε″ at low and intermediate frequencies is consequently characterized by high values of ε″. However, the ion vibrations may be the only sources of dielectric losses at higher frequency values, hence (ε″) diminishes at higher frequency values. Additionally, the imaginary part of the dielectrics(ε″) grows together with the composition AlBiO3 in the PbMg1/3Nb2/3O3.