QUANTUM PARAELECTRICITY AND INDUCED FERROELECTRICITY BY GERMANIUM DOPING OF (Pb_ySn_1–y)₂P₂S(Se)₆ SINGLE CRYSTALS

Received 24 January 2020; revised 26 March 2020; accepted 6 April 2020

1. Introduction

For Sn₂P₂S₆ (SPS) ferroelectrics the second-order phase transition at T₀ ~ 337 K with a mixed displacive–order/disorder character occurred [1]. SPS is the most prominent example of ferroelectric ternary phosphorus chalcogenide crystals represented by the general formula M₂P₂X₆ (where M is transition or post-transition carbon group p metals, X is chalcogenides). Moreover, optical measurements have proven Sn₂P₂S₆ as a semiconductor with a relatively narrow bandgap and promising photorefractive, photovoltaic, electrooptic and piezoelectric features [2, 3]. Another feature, that makes them very attractive for the researchers, is an isovalent anion substitution into the crystal lattice producing series of (Pb_ySn_1–y)₂P₂(Se_xS_1–x)₆ mixed crystals. The impact of dopants on the ferroelectricity of crystals has a special attention as an effective engineering approach that introduces changes into the position or character of the phase transition and physical properties.

Sn₂P₂S₆ crystal is a uniaxial proper ferroelectric with a three-well local potential for order parameter fluctuations [4, 5]. The ferroelectric phase transition of Sn₂P₂S₆ is induced by stereoactivity of the Sn²⁺ cation 5s² electron lone pair: antibonding mixing of tin and sulfur orbitals (5s and 3p, respectively) interacts with the tin 5p orbitals, generating lower energy filled states Sn 5v (Sn 5s + S 3p). This formation of the Sn²⁺ lone pair electron cloud, together with the deformation of the nearest polyhedron formed by sulfur atoms, determines the origin of spontanous polarization [4, 6].

The experimental results of (Pb_ySn_1–y)₂P₂S₆ showed that the phase diagram of (Pb_ySn_1–y)₂P₂S₆ could be quite complicated with possible coexistence of ferroelectric-paraelectric states [7, 8]. This is in agreement with the thermodynamical description based on the Blume–Emery–Griffith (BEG) model with random bond and random field defects, where the introduction of Pb is responsible for the randomness [9]. One more feature of the phase diagram predicted by the BEG model is the presence of tricritical point (TCP). According to the experimental data, for the (Pb_ySn_1–y)₂P₂S₆ mixed crystals the paraelectric phase becomes stable above y > 0.7 at a normal pressure [10, 11]. Also, in previous papers it was shown that for y > 0.3 and below 220 K the hysteresis appears [8]. It was related to the coexistence of paraelectric and ferroelectric phases, and evidenced a discontinuous character of the phase transition [9]. The phase diagram of these mixed crystals with substitution of tin for lead or sulfur for selenium is really an interesting point that was investigated in-depth in several studies [1, 7–8, 12–13].

SrTiO₃ in a chemically pure form was reported as a typical quantum paraelectric featured by high dielectric permittivity ε and low dielectric loss tanδ, absent from ferroelectric transitions down to the lowest temperatures [14, 15]. This effect is attributed to large quantum fluctuations in the limit of absolute zero temperature [15]. SrTiO₃, CaTiO₃ and KTaO₃ are also classified as quantum paraelectrics or incipient ferroelectrics which possess a high dielectric constant [16]. Very similar effects are also observed in the chemically or isotopic substituted SrTiO₃ (SrTiO₃:Ca or SrTiO₃:O¹⁸) [17].

A key factor in the pure Pb₂P₂S₆ single crystal is ‘quantum paraelectric state’ [8], as it was classified according to the study of Müller et al. [14]. The phase diagram of (Pb_ySn_1–y)₂P₂S₆ mixed crystals was examined systematically. Quite often, the material can be tuned to a quantum critical point (QCP) by applying pressure or changing the chemical composition in mixed crystals [18]. QCP is a point at absolute zero temperature in a phase diagram, where the quantum fluctuations drive a phase transition. Thus, it may seem that the quantum phase transition is an abstract theoretical idea. However, the influence of the critical point extends over a wide range at T > 0: this is the regime of quantum criticality.

We present in this paper the results of dielectric spectroscopic studies carried out on undoped Pb₂P₂S₆ and Ge-doped lead-based crystals that revealed the presence of quantum paraelectricity and induced ferroelectricity.

2. Results and discussion

In Fig. 1, the results of the dielectric permittivity of Pb₂P₂S₆ single crystal are presented. It is shown that for the Pb₂P₂S₆ single crystal, the real part of dielectric permittivity increases monotonously with decreasing temperature in the measured temperature range. The saturation behaviour of the real part of dielectric permittivity is not observed. The dispersion of the real part of dielectric permittivity was not detected. In the quantum critical regime the usual Curie–Weiss law of inverse dielectric permittivity 1/ε(T) ~ T changes into 1/ε(T) ~ T² [18]. That is the most prominent criterion of quantum critical behaviour. Other quantities such as the thermal expansion coefficient and soft-mode frequencies are expected to vary in an unconventional manner. As shown in the inset of Fig. 1, the inverse dielectric permittivity 1/ε(T) exhibits the expected non-classical T² temperature dependence over the finite temperature range above approximately 50 K and below 250 K. Typically, there is no ferroelectric phase transition in quantum paraelectrics, but critical behaviour, manifested by the non-classical T² dependence, occurs [19].

In order to describe the temperature dependences of the dielectric permittivity of quantum paraelectrics, Barrett extended the Slater’s meanfield theory by including the quantum effect, and an equation was derived as

where C is the Curie–Weiss constant, θ_CW can be recognized as the classical paraeletric Curie temperature, ε₀ is the temperature-independent constant, T is the sample temperature, and T₁ is recognized as the temperature at which the instability begins. Approximately, it may be said that T₁ is the dividing point between the low temperature where quantum effects are important so ε(T) deviates from the Curie–Weiss law, and the high temperature region where a classical approximation and the Curie–Weiss law are valid [19].

In the low temperature region (T < T₁), the quantum effects are present. In many cases, θ_CW ≤ 0 K, which refers to the virtual transition temperature. Therefore, the material does not undergo a ferroelectric phase transition at any finite temperature because of the quantum effects. When θ_CW is finite and θ_CW < T₁, the quantum fluctuations that occur below T₁ break the long-range ferroelectric order and stabilize the quantum paraelectric state in the sample, and a probable ferroelectric transition occurs at θ_CW [20]. The dielectric data of Pb₂P₂S₆, (Pb_0.98Ge_0.02)₂P₂S₆, (Pb_0.7Sn_0.3)₂P₂S₆ + 5% Ge and (Pb_0.7Sn_0.3)₂P₂Se₆ + 5% Ge single crystals were analysed on the basis of Barrett’s formula (1). The results are presented in Figs. 1, 2, 4 and 5, respectively. The red (online) solid line is a data fitting with the Barrett’s formula. The best-fit parameter values of all measured samples are summarized in Table 1.

The Barrett’s formula has been reported as a fundamental model to describe the temperature behaviour of dielectric constant of the so-called incipient ferroelectrics (or quantum paraelectrics). However, discrepancies in the fitting with the Barrett’s equation were reported by several authors. Then, various alternative models, in addition to the Barrett’s model, have been proposed to explain the quantum paraelectric state. Unfortunately, disagreements still exist in the results of analyses and interpretations based on different models. The authors in the [16] study analysed the temperature dependences of dielectric permittivity of the most-studied quantum paraelectrics SrTiO₃ and KTaO₃ on the basis of three different models (Barrett’s model, Vendik model and quantum criticality) to determine the most appropriate standpoint. It was concluded that the dielectric permittivity at low temperatures cannot be described properly using the Barrett’s formula. The Vendik model was more appropriate in the low-temperature approximation. So, at this time, minor deviations of the temperature dependences of quantum paraelectrics exist for different models and there is no single model appropriate in all temperature ranges. This way, the Barrett’s model still remains as a starting point to determine quantum paraelectricity.

In the temperature dependences of the dielectric permittivity of Pb₂P₂S₆ single crystals the deviation from the Barrett’s equation starts around 50 K. The abovementioned notes of applicability of various models are suitable to explain such discrepancy. The obtained parameter values (T₁ = 190 K and θ_CW = –376 K) of pure single crystal Pb₂P₂S₆ demonstrate that the material does not undergo ferroelectric phase transition at any finite temperature.

Lead replaces tin and selenium takes the place of sulfur in the whole concentration range of (Pb_ySn_1–y)₂P₂(Se_xS_1–x)₆ mixed crystals. In addition, these crystals can be doped with germanium Ge, antimony Sb or tellurium Te, but only in small percentages. Germanium is the most important dopant because it takes the place of tin in the Sn₂P₂S₆-type crystals. It is known that tin ions play an important role in the ferroelectricity of the material. Introducing Ge atoms into the cation sublattice of SPS crystal enhances the stereoactivity of the cation sublattice, increasing the critical temperature and sharpening the phase transition character. An increase of the transition temperature has already been observed in the paper [21] measuring the temperature evolution of piezoelectric and pyroelectric coefficients. It is worth noting that when Pb substitutes for Sn, the hybridization becomes weaker, reducing the phase transition temperature. On the other hand, the Ge dopant in Pb₂P₂S₆-type crystals plays an opposite role: it enhances the total stereoactivity of metalic cations in the crystal.

It is known that a small amount of impurities in quantum paraelectrics could induce ferroelectricity [22, 23]. The transition to a polar state appears above some impurity critical concentration x. A widely accepted viewpoint is that the ferroelectricity in doped quantum paraelectrics is determined by an off-centre position of impurity ions which produces electric dipoles that induce polarization [22]. So, it is interesting to investigate how germanium impurities can affect the quantum paraelectric state of Pb₂P₂S₆. Figure 2 shows the temperature dependence of the real part of (Pb_0.98Ge_0.02)₂P₂S₆ dielectric permittivity. The inset of Fig. 2 confirms the non-clasical T² behaviour of inverse dielectric permittivity. It is needless to say that T² temperature dependence is not valid at low temperatures. For the doped system (Pb_0.98Ge_0.02)₂P₂S₆, both temperatures (T₁ and θ_CW) are finite (Table 1). Also, since θ_CW < T₁ for (Pb_0.98Ge_0.02)₂P₂S₆, it could be concluded that the long-range ferroelectric order in this Gedoped sample is broken due to quantum fluctuations below 207 K, and a probable ferroelectric transition occurs as manifested by a peak at 40 K for dielectric permittivity in Fig. 3. These differences of the Pb₂P₂S₆ and (Pb_0.98Ge_0.02)₂P₂S₆ crystals could be related with different ionic radii of Pb²⁺ (r_Pb = 1.33 Å) and Ge²⁺ (r_Ge = 0.87 Å). The smaller germanium ionic radius results in different hybridziation with the sulfur ions. Thus, doping with more stereoactive germanium cations induces some disorder effects and decreases dielectric permittivity value, below 75 K deviating from the Barrett’s fit (Fig. 2). It suggests that possible phase transition might occur in this region.

Temperature dependence of the dielectric permittivity of (Pb_0.98Ge_0.02)₂P₂S₆ at different frequences is presented in Fig. 3. The peak of dielectric permittivity is broad. The broadness of the phase transition is due to small compositional fluctuations. However, there are two peaks of the imaginary part of dielectric permittivity with a frequency dispersive behaviour, and the temperatures of loss peaks are around 50 and 100 K at 100 kHz.

The relaxation rate derived from the temperature dependence of the imaginary part of permittivity can be well fitted to the Arrhenius law

where ν₀ is the relaxation rate at the infinite temperature, E_A is the activation energy for the relaxation, k is the Boltzmann constant, and T is the temperature. The best-fitting value of activation energy is equal to 0.41 eV.

Pb₂P₂S₆ compound based mixed crystals have a stable paraelectric ground state till 0 K when lead concentration reaches y ~ 0.7 [8]. In previous papers it was shown that for (Pb_ySn_1–y)₂P₂S₆ mixed crystals with compositions y ~ 0.61 and y ~ 0.65 (which are close to the transition at zero temperature from the polar phase with y < 0.7 to the paraelectric one with y > 0.7) their dielectric susceptibility demonstrates the quantum critical behaviour in vicinity of the first-order transitions with T_C ~ 35 and 20 K, respectively [11]. To study the effect that the addition of Ge dopants has on the quantum paraelectric nature of Pb₂P₂S₆ compounds was precisely the aim of further investigations of (Pb_0.7Sn_0.3)₂P₂S₆ + 5% Ge (Fig. 4) and (Pb_0.7Sn_0.3)₂P₂Se₆ + 5% Ge (Fig. 5) samples. In these samples, Sn²⁺ sites codoping was realized by using two different ionic radius impurities – lead and germanium. It is worth reminding that the substitution for Sn has the strongest effect because the ferroelectric phase transition is induced by the stereoactivity of the Sn²⁺ cation 5s² electron lone pair. However, it is known that lead and germanium have very different influences on the phase transitions.

For (Pb_0.7Sn_0.3)₂P₂S₆ + 5% Ge, by fitting to the Barrett’s equation (1), T₁ = 69.8, θ_CW = –4.1 and C = 30677 K were obtained. From Fig. 4 it can be seen that the fitting data are in good agreement with the experimental data. The obtained values demonstrate that the sample does not undergo a ferroelectric phase transition at any finite temperature.

3. Conclusions

We have carried out a dielectric study of four single crystals Pb₂P₂S₆, (Pb_0.98Ge_0.02)₂P₂S₆, (Pb_0.7 Sn_0.3)₂P₂S₆ + 5% Ge and (Pb_0.7Sn_0.3)₂P₂Se₆ + 5% Ge with a special emphasis on low temperatures. The temperature dependences of dielectric permittivity were analysed in terms of the Barrett’s model. This detailed analysis provided indications that Pb₂P₂S₆-type crystals exhibit a quantum paraelectric state. By introducing small amounts (2%) of germanium dopants, the ferroelectric phase appears.

References

KVANTINIS PARAELEKTRIŠKUMAS IR INDUKUOTASIS FEROELEKTRIŠKUMAS ĮTERPIANT GERMANĮ Į (Pb_ySn_1–y)₂P₂S(Se)₆ KRISTALUS

Santrauka

Darbe tyrinėjamos keturių kristalų – Pb₂P₂S₆, (Pb_0,98Ge_0,02)₂P₂S₆, (Pb_0,7Sn_0,3)₂P₂S₆ + 5 % Ge ir (Pb_0,7Sn_0,3)₂P₂Se₆ + 5 % Ge – dielektrinės savybės intervale nuo kambario temperatūros iki 20 K. Atlikti tyrimai leido nustatyti kvantinę paraelektrinę būseną kristaluose, kuriuose buvo įterpta germanio priemaišų. Visose tirtose medžiagose stebėtas atvirkštinės dielektrinės skvarbos nukrypimas nuo Kiuri ir Veiso dėsnio, aprašytas T² temperatūrine priklausomybe. Tirtųjų kristalų dielektrinės skvarbos temperatūrinės priklausomybės analizuotos remiantis Barrett’o modeliu.