Van der Waals Ferroelectrics E-Book

Juras Banys

• Herausgeber: Wiley-VCH GmbH
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 755

Veröffentlichungsjahr: 2022

Beschreibung

The book provides in-depth insights, both from an experimental and a computational perspective, into the multiferroic properties of phosphorous chalcogenides and their applications.

Leseprobe

Van der Waals Ferroelectrics

Properties and Device Applications of Phosphorous Chalcogenides

Juras Banys

Andrius Dziaugys

Konstantin E. Glukhov

Anna N. Morozovska

Nicholas V. Morozovsky

Yulian M. Vysochanskii

Authors

Prof. Juras Banys

Vilnius University, Faculty of Physics

Institute of Applied Electrodynamics

and Telecommunications

Saulėtekio av. 9, III bld

10222 Vilnius

Lithuania

Dr. Andrius Dziaugys

Vilnius University, Faculty of Physics

Institute of Applied Electrodynamics

and Telecommunications

Saulėtekio av. 9, III bld

10222 Vilnius

Lithuania

Dr. Konstantin E. Glukhov

Uzhgorod National University

Institute Physics & Chemistry of Solid State

54 Voloshin Street

88000 Uzhgorod

Ukraine

Dr. Anna N. Morozovska

National Academy of Sciences of Ukraine

Institute of Physics

46, pr. Nauky

03028 Kyiv

Ukraine

Dr. Nicholas V. Morozovsky

National Academy of Sciences of Ukraine

Institute of Physics

46, pr. Nauky

03028 Kyiv

Ukraine

Prof. Yulian M. Vysochanskii

Uzhgorod National University

Institute Physics & Chemistry of Solid State

54 Voloshin Street

88000 Uzhgorod

Ukraine

Cover Image: © Science History Images/Alamy Stock Photo

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Introduction

After the discovery of graphene and its intriguing features, scientists have become increasingly interested in two‐dimensional (2D) crystals with strong in‐plane covalent bonds and weak van der Waals (vdW) interlayer interactions. The simplicity and rapid prototyping of micromechanical exfoliation technique, combined with a wide range of properties covered by these materials, have opened up a vast study field with the potential to impact a wide range of applications. The absence of a natural energy bandgap, on the other hand, limits the usefulness of graphene for high‐performance electrical switches and other optoelectronic devices. Here, alternative 2D materials, such as transition metal chalcogenophosphates (TMCPs), can play an essential role in this application. Natural bandgaps in these materials allow investigators to expand the scope of their research. TMCPs of the general formula MPX3 (M is transition metal or combination of metals: CuIn, AgIn, CuBi, CuCr, and AgCr, P is phosphorus and X = S or Se chalcogens) have stimulated the interest of the 2D material science community because they have multiferroic properties, making them easy to exfoliate with cleavage energies even lower than in graphite. The pursuit of long‐range ferroic ordering in vdW materials, such as ferroelectricity, ferromagnetism, and ferroelasticity, is on the rise with the fast development of vdW materials. Ferroelectricity is one of them, with applications in memory, capacitors, actuators, and sensors being researched extensively. The spontaneous ordering of electric dipoles below the Curie temperature (Tc) creates macroscopic polarization that may be changed by an external electric field. In condensed matter physics, vdW layered ferroelectric materials have emerged as a potential study area. Furthermore, their out‐of‐plane polarization is more suitable for nonvolatile memory and heterostructure‐based nanoelectronics/optoelectronics. CuInP2S6 (CIPS) is one of the most representative materials because of its room temperature ferroelectricity. CIPS is formed out of 2D flakes and is a strong piezoelectric material. The height of the flakes can be easily changed using the scotch‐tape method like for graphene. It is determined that these crystals have the largest nonlinear elastic properties and a large tense resistance. CuInP2S6 is only one known room temperature 2D semiconductor ferroelectric. The vdW gap is around 8 A, and it is compatible with modern nanotechnology. The presence of Cr, Bi, and Mn in the lattice allows the occurrence of magnetism (multiferroic). Cu+ is responsible for a high ionic conductivity in CIPS. The high ionic conductivity makes CIPS an ideal system to study the interplay with ferroelectricity. These materials are photosensitive semiconductors with a bandgap width of 0.3–3 eV. Besides, they are also characterized by a strong piezoeffect. CuInP2S6 layered crystals have the largest elastic nonlinearity of the known materials, which is important for acoustoelectronic and signal processing devices. Because of the electrostriction, the piezoeffect can be created by the constant external electric field, which is very important for the development of controllable electroacoustic transducers.

The density functional theory (DFT) is used for the ab initio simulation of electron and phonon subsystems of the considered family of layered compounds. A detailed description of spatial charge distribution elucidates the features of complex chemical bonding in both the tightly connected layers and between them. Special attention is paid to the interplay between the orbitals of [P2X6]4+ anion complex and different cations. Estimations of the energetics of dipole‐ or/and spin‐ordered states in materials under study allow explaining the peculiarities of the phase transitions in these structures, including the semiconductor–polar metal transition. Stability of different phases, patterns of atom displacements, phonon frequencies, and estimations of Raman intensities are obtained from the ab initio modeling of the lattice subsystem of a wide variety of representatives of the considered crystal family.

Using the Landau–Ginzburg–Devonshire phenomenology combined with electrostatic equations, elasticity theory, and finite element modeling, it is possible to analyze the influence of finite size, surface screening, flexoelectric effect, antiferroelectric–ferroelectric coupling, and polarization gradient on the formation of unusual domain morphology (such as labyrinthine and flux‐closure domains, versatile vortices, Bloch points, and more complex topological domain structures) and the physical origin of exotic domain wall properties (such as bright‐contrast domain walls in ferrielectric phases) in ferroelectric–antiferroelectric low‐dimensional layered chalcogenides, such as Sn2P2(S,Se)6 and CuInP2(S,Se)6 thin films and nanoparticles.

In this book, we focus mainly on the physical properties of CIPS family compounds. The unusual crystal structure and multiferroic order endow these crystals with intriguing characteristics and rich physics, which have attracted much interest in the past few years. We hope that this book will be interesting for the scientists' community who investigate TMCPs, ferroelectrics, ferroics, multiferroics, and phase transitions.

1Crystal Structure and Phase Transitions in Layered Crystals of Ternary Phosphorous Chalcogenides

1.1 Ferrielectric, Antiferroelectric, and Modulated Orderings in MM′P2X6 (M – Cu, Ag; M′ – In, Bi; X – S, Se)

Two‐dimensional crystals having strong in‐plane covalent bonds and weak van der Waals (vdW) interlayer interactions have attracted increasing attention of science community after the discovery of graphene and its interesting properties. Recent years have seen the emergence of relatively new ferroelectric materials belonging to the chalcogenophosphate family [1]. These compounds contain (P2X6)4− anions (X = S or Se), which are linked together by cations. Because the ethane‐like groups are able to withstand variations in PP and PX bond lengths, a large number of chalcogenophosphates have been prepared [2]. Moreover, the coordination preferences of different cations have led to two types of morphologies, i.e. either a three‐dimensional or lamellar structure. The nature of dipole ordering derives from bonding, Coulomb, and elastic effects peculiar to a given lattice and may thus be expected to be different in each class. There are six structures representing various cation sublattices possible in this broad class of materials. These include A22+[P2X6]4−, A4+[P2X6]4−, A4/33+ϒ2/3[P2X6]4− (where ϒ represents a vacant site), A21+B2+[P2X6]4−, and A1+B3+[P2X6]4− (here A and B are metals). The widest variety of symmetries is found within the sulfides. Almost every A22+[P2X6]4−‐type compound crystallizes into a C2/m monoclinic crystal structure. Fe2P2S6 is one of the representative examples for this class of materials. Mixed‐cation A1+B3+[P2X6]4−compounds have more variability in their structure. AgInP2S6 has a trigonal (13Pc) structure and is centrosymmetric at room temperature [3]. Most of mixed‐cation compounds are monoclinic. The 2D category is best represented by CuMP2S6 (M = Cr or In) in which copper is formally monovalent and M is trivalent [4–6]. These compounds consist of lamellae defined by a sulfur framework, which provides octahedral voids for metal cations and P–P pairs. Within a layer, Cu, M, and P–P form triangular patterns. Dipole ordering in these materials requires antiparallel displacements of the d10 cations, whereas the copper sublattice is antipolar in CuCrP2S6 at T < 150 K [6]. It is polar in CuInP2S6 below Tc = 315 K and coexists with the In3+ sublattice of unequal and opposite polarity. Cation off‐centering is attributable to the second‐order Jahn–Teller (SOJT) instability associated with the d10 electronic configuration; the lamellar matrix absorbs structural deformations via flexible (P2X)4− groups [4, 5]. Interestingly enough, Sn4+ is a d10 cation so that off‐centering in Sn2P2S6 might also be ascribed to a pseudo Jahn–Teller effect.

CuInP2S6 crystals are an uncommon example of the uncompensated anticollinear two‐sublattice ferroelectric system [5]. They exhibit the first‐order phase transition of an order–disorder type from the paraelectric to the ferrielectric phase (Tc = 315 K). The symmetry reduction at the phase transition (C2/c to Cc) occurs due to the ordering in the copper sublattice and the displacement of cations from the centrosymmetric positions in the indium sublattice. CuInP2S6 consists of lamellae defined by a sulfur framework in which the Cu and In cations and P–P pairs fill the octahedral voids and form triangular patterns within a layer (Figure 1.1) [7]. A spontaneous polarization arising at the phase transition to the ferrielectric phase is perpendicular to the layer planes. X‐ray investigations have showed that a Cu ion can occupy three types of positions: (i) Cu1 – quasitrigonal, off‐centered positions; (ii) Cu2 – octahedral, located in the octahedron centers; (iii) Cu3 – almost tetrahedral, penetrating into the interlayer space [4, 5]. The degree of occupation strongly depends on the temperature [5]. Moreover, two types of positions for Cu1 are distinguished: Cu1u is displaced upwards from the middle of the layer (centers of octahedrons) and Cu1d is displaced downwards. The ordering of Cu ions (hopping between Cu1u and Cu1d positions) in the double minimum potential is the reason for phase‐transition dynamics in CuInP2S6. At the temperature below Tc (315 K), the intersite copper mobility is limited, trapping the Cu1+ sublattice in its displaced state (Cu1, up) with a compensating shift in the opposite polarity In3+ sublattice, resulting in a noncentrosymmetric (Cc) ferrielectric phase.

Selenophosphate is other class of materials exhibiting ferroelectric properties. P2Se6 bonds have a higher degree of covalence than the P2S6 backbone, which is the main difference in this scenario. At low temperature, Cu+ ion displacement for CuInP2Se6 is only 1.17 Å [8] compared to 1.58 Å for CuInP2S6[5]. This is considered the main cause of the lower phase transitions of the CuInP2Se6 compound, which include the second‐order phase transition at Ti = 248 K and the first‐order phase transition at Tc = 236 K [9]. The occurrence of incommensurate, quasi‐polar phases, in which Cu+ cation displacement is modulated with a period different from the primary period of the crystal lattice, is attributed to the second‐order phase transition in this compound. Other family members show antiferroelectric (AFE) properties. CuBiP2Se6 and AgBiP2Se6 are two materials in the selenide family that demonstrate AFE ordering when Bi3+ is a trivalent cation [2]. Partially AFE ordering is exhibited in CuBiP2Se6 below 173 K, with 85% of Cu+ ions located in the well‐defined off‐center positions below 97 K. The higher displacement of Bi3+ compared to that of In3+ could indicate the presence of stereoactive lone pair electrons in Bi3+ but not In3+. At room temperature, AgBiP2Se6 exhibits a similar pattern [2].

Figure 1.1 (a) Three positions of copper in the cadge. (b) An in‐plane view of the layers in the ferrielectric state, Cu shifted up while In shifted down.

Source: Maisonneuve et al. [5] / American Physical Society.

Metal and chalcogen substitutions and alloying allow one to efficiently modify and introduce additional functions to the ferrielectric compound CuInP2S6. As a result, this vital subject has recently received a lot of attention. The typical strategy is to either change one of the two cation sublattices or to modify the P2S6 backbone by partial replacement of Se.

1.2 Relaxor and Dipole Glassy States on the Phase Diagram of CuInP2(SexS1−x)6 Mixed Crystals

Strongly disordered ferroelectrics such as ferroelectric solid solutions and ceramics can exhibit rather unusual “relaxor” properties among which the most known is a very slow relaxation of polarization. This feature had led to form the “ferroelectric relaxor” notion. Yet, the most important feature of this class of ferroelectric materials is that disorder destroys the ferroelectric transition in them. Thus, in the zero electric field in all temperature range down to T = 0, no spontaneous polarization or ferroelectric domains appear in relaxors as well as there are no changes in their (average) crystalline structure. So quite naturally the relaxors have just a broad maximum in the temperature dependence of dielectric susceptibility instead of a sharp peak, and the position of this maximum shifts to lower T at lower frequencies.

Despite the apparent absence of ferroelectric transitions in relaxors, ferroelectric polarization can be generated in them at low T by applying a suitably strong external field for a limited period. Otherwise, one can cool the sample in a strong enough field to a low T and then switch off the field to find that it acquires some polarization that is stable on laboratory time scales. Subsequent heating of the relaxor in the zero field demonstrates that such remanent polarization continues up to the threshold T. For the classical theory of phase transitions, the presence of stable remanent polarization in the material with no ferroelectric transition is a great contradiction and unexplained enigma. Initially, an attempt was made to overcome this dilemma using the concept of “diffuse phase transition,” in which distinct areas of a sample transform gradually into the ferroelectric phase across a temperature interval. This hypothesis appears to contradict X‐ray diffraction experiments that show no macroscopic polar areas in relaxors at any T in the zero field [10].

Dielectric materials are electrical insulators that can be polarized by applying an electric field, as opposed to conductors, which carry charges through the materials. When an external electric field is supplied to a dielectric material, the opposite direction field is induced inside the material. The total polarization of the material is the sum of all polarizations relevant to a specific system at the target frequency. It is commonly generated by electronic, ionic, dipolar, and interfacial processes. Orientational glasses are crystalline materials that undergo a transformation from a high‐temperature crystalline phase to a low‐temperature glassy state. Analogous to the spin glasses (for a review see Ref. [11]), randomly substituted impurity ions that carry a moment are located on a topologically ordered lattice. These moments have orientational degrees of freedom and they interact with one another. The dominant exchange interaction can be of an electrostatic dipolar, quadrupolar, or octupolar, or of an elastic quadrupolar nature. Here the interaction is mediated by lattice strains. The orientational disorder is cooperatively frozen‐in as a result of site disorder and anisotropic interactions. The term “glass‐state” implies some resemblance to canonical glasses. Indeed, the relaxation dynamics of orientational glasses are similar to those of canonical glasses.

CuInP2S6 crystals represent an unusual example of the collinear two‐sublattice ferrielectric system [5, 12]. Cooperative dipole effects play the main role in these lamellar chalcogenophosphates. The first‐order phase transition of an “order–disorder” type from the paraelectric to the ferrielectric phase is observed at Tc = 315 K. The phase transition reduces the symmetry C2/c → Cc, which occurs due to ordering in the copper sublattice and the displacement of cations from the centrosymmetric positions in the indium sublattice. These results were supported by the Raman investigation of CuInP2S6[13]. The spontaneous polarization is perpendicular to the layer planes. These thiophosphates consist of lamellae defined by a sulfur framework in which metal cations and P–P pairs fill the octahedral voids; within a layer, Cu, In, and P–P form triangular patterns [14]. The lamellar structure absorbs structural deformations via flexible P2S6 groups while forbidding the cations to antiparallel displacements that minimize the energy costs of dipole ordering. Cu ions can occupy several different positions in the lattice as is shown in Figure 1.2. Cu, In, and P–P form triangular patterns within the layer. Relaxational behavior is indicated by the temperature dependence of spectral characteristics, in agreement with X‐ray investigations. It was suggested that a coupling between P2S6 deformation modes and CuI vibrations enables copper ion hopping motions that lead to the loss of polarity and the onset of ionic conductivity in this material at higher temperatures [13].

1.2.1 XRD Investigations of CuInP2Se6

The selenium analog CuInP2Se6 is quite a new addition to this class of chalcogenophosphate materials. Single‐crystal XRD investigations were used to clarify the structure of CuInP2Se6 at various temperatures. CuInP2Se6 has a lamellar structure like Fe2P2Se6 and Mn2P2Se6 [15, 16]. Each layer is formed out of one [P2Se6]4− unit, and the structures contain well‐defined vdW gaps (Figure 1.2). Full data sets were collected in the temperature range from 100 to 300 K, and the structures were refined using the SHELX‐97 software [17]. The refined structural parameters are listed in Table 1.1. The indexing of the diffractograms showed that the phase of CuInP2Se6 belongs to the noncentrosymmetric space group P31C (No. 159) at 100 and 180 K (Table 1.1a). Therefore, it has a centrosymmetric space group P‐31C (No. 163) at 250 K (Table 1.1c). These results largely agree with those of the previous work [8].

Figure 1.2 Layered structures of CuInP2Se6 inferred from X‐ray diffraction. Structures at (a) 100 K and (b) 293 K. Blue atoms are Cu, pink atoms are In, yellow atoms are Se, and gray atoms are P. (c) View ([001] direction) of a single CuInP2Se6 layer showing the arrangement of Cu (blue), In (pink), P (gray), and Se (yellow). The atomic displacement of individual atoms of CuInP2Se6 at (d) 100 K, (e) 180 K, and (f) 250 K.

Cu+ and In3+ ions in CuInP2Se6 are octahedrally surrounded by Se atoms, where Cu+ can occupy a central (more probable) or a near‐edge position in the cage (Figure 1.2). The temperature change induces the ordering of Cu+ in the sublattice of this material. At room temperature, 43% of Cu+ ions occupy the central position and other 57% are found near the edges of the cage. This indicates copper hoping in the three‐well potential at T > Tc and an order–disorder type ferroelectric ordering in CuInP2Se6. At lower temperature, a probability to find copper located in the middle of the octahedral site decreases. At 100 K, about 93% of the Cu+ ions are in a well‐defined off‐center position, displaced by 1.38 Å along the c axis (Figure 1.2d). The remaining 7% of Cu+ ions are still disordered within the layer as before. In3+ shifts in the opposite direction when the temperature is lowered below Tc, by about 0.14 Å at 100 K (Figure 1.2d). Therefore, the material exhibits ferrielectric ordering, similar to that of CuInP2S6.

It has to be noted that selenides have a higher covalence degree of their bonds compared with that of the sulfide analog. Evidently, for this reason, the copper ion sites in the low‐temperature phase of CuInP2Se6 are displaced only by 1.38 Å from the middle of the structure layers in comparison with the corresponding displacement 1.58 Å for CuInP2S6 [5, 14]. Therefore, one can assume the potential relief for copper ions in CuInP2Se6 to be shallower than for its sulfide analog. Presumably, for this reason, the structural phase transition in the selenide compound is observed at lower temperature than for CuInP2S6.

Table 1.1 (a) Structural parameters of the single‐crystal CuInP2Se6 in the P31C (No. 159) phase at 100 K. A total of 145 603 reflections are collected. (b) Structural parameters of the single‐crystal CuInP2Se6 in the P31C (No. 159) phase at 180 K. A total of 14 831 reflections are collected. (c) Structural parameters of the single‐crystal CuInP2Se6 in the P‐31C (No. 163) phase at 250 K. A total of 11 996 reflections were collected.

(a)

T

 = 100 K

a =

 6.402(2) Å,

c =

 13.319(6) Å, and

V =

 472.8(5) Å

3

. The agreement factor

R

1

=

 2.66% was achieved by using 708 unique reflections with

I

 > 4

σ

and the resolution of

d

min

=

 0.65 Å. Anisotropic atomic displacement parameters were used for all elements

Site

x

y

z

Occupancy

U

eq

(Å

2

)

Cu

2

b

2/3

1/3

0.1474(2)

1

0.0173(4)

In

2

a

0

0

0.2591(1)

1

0.0087(2)

P

1

2

b

1/3

2/3

0.3256(3)

1

0.0057(4)

P

2

2

b

1/3

2/3

0.1579(2)

1

0.0057(4)

Se

1

6

c

0.3013(1)

0.3228(1)

0.1096(1)

1

0.0069(1)

Se

2

6

c

0.3552(1)

0.0054(1)

0.3706(1)

1

0.0077(2)

(b)

T

 = 180 K

a =

 6.410(8) Å,

c =

 13.337(20) Å, and

V =

 474.6

(1.3)

 Å

3

. The agreement factor

R

1

 = 4.58% was achieved by using 706 unique reflections with

I

 > 4

σ

and the resolution of

d

min

 = 0.65 Å. Anisotropic atomic displacement parameters were used for all elements

Site

x

y

z

Occupancy

U

eq

(Å

2

)

Cu

2

b

2/3

1/3

0.1526(4)

1

0.045(1)

In

2

a

0

0

0.2571(2)

1

0.0136(3)

P

1

2

b

1/3

2/3

0.3248(4)

1

0.0082(9)

P

2

2

b

1/3

2/3

0.1587(4)

1

0.0085(9)

Se

1

6

c

0.3059(2)

0.3248(2)

0.1097(1)

1

0.0119(3)

Se

2

6

c

0.3531(2)

0.0046(2)

0.3706(1)

1

0.0128(3)

(c)

T

 = 250 K

a

 = 6.397(1) Å,

c

 = 13.340(5) Å, and

V

 = 472.8(3) Å

3

. The agreement factor

R

1

=

 3.80% was achieved by using 538 unique reflections with

I

 > 4

σ

and the resolution of

d

min

=

 0.65 Å. Anisotropic atomic displacement parameters were used for all elements

Site

x

y

z

Occupancy

U

eq

(Å

2

)

Cu

1

2

d

2/3

1/3

1/4

0.354(6)

0.063(3)

Cu

2

4

f

2/3

1/3

0.333 4(6)

2 × 0.323(6)

0.063(3)

In

2

a

0

0

1/4

1

0.0217(3)

P

4

f

1/3

2/3

0.1662(1)

1

0.0133(3)

Se

12

i

0.33217(8)

0.33730(7)

0.12006(4)

1

0.0211(2)

The phase transitions in CuInP2(S,Se)6 crystals are caused by the cooperative freezing of intersite copper motions. This cooperative dipolar behavior is supposed [18] to arise from the presence of an off‐centering displacement caused by electronic instability in a form of the SOJT effect related to the d10 electronic configuration of cations Cu+. A SOJT coupling, involving the localized d10 states forming the top of the valence band (VB) and the s–p states of the bottom of the conduction band (CB), is predicted to yield such instability. Photoelectron spectroscopy measurements gave the evidence for a strong redistribution of the density of states at the top of the valence band [18]. By combining these data with band‐structure calculations, it was shown [18] that these changes are mainly ascribable to the redistribution of the Cu partial density of electron states, related to the off‐centered position of d10 cations in the ferrielectric phase.

1.2.2 Relaxor Phase in Mixed CuInP2(SxSe1−x)6 Crystals

Two very similar CuInP2(SxSe1−x)6 compounds of x = 0.2 and x = 0.25 are analyzed as they exhibit a peculiar dielectric behavior. Both compositions show just one peak of the real and imaginary part of dielectric permittivity in the temperature range 110–145 K at 10 kHz frequency [19]. A typical dielectric characteristic of relaxor ferroelectrics for both crystals is observed: diffused phase transition without the well‐defined Curie temperature. The dielectric permittivity of the CuInP2(Se0.75S0.25)6 crystal is shown in Figure 1.3. A broad peak of ε ′ (T) is observed. Tm (peak value of ε′) increases with decreasing the frequency of the applied field. A strong dielectric dispersion is detected in the radio frequency region around and below Tm at 1 kHz. The value of Tm (T of the maximum of ε″) is much lower than that of Tm at the same frequency. The position of the peak of ε ′ (T) is strongly frequency dependent and no certain static dielectric permittivity is obtained below and around the dielectric permittivity Tm at 1 kHz.

Such behavior can be described by the Vogel–Fulcher relationship

where k is the Boltzmann constant and Ef, ν0, and T0 are the parameters the values of which are presented in Table 1.2:

The dielectric dispersion of CuInP2(S0.25Se0.75)6 crystals shows a strong temperature dependence. At higher temperatures, the dielectric dispersion is only in the 107–1010 Hz region, and on cooling, it becomes broader and more asymmetric. A very broad and asymmetric dielectric dispersion is observed below Tm at 1 kHz. Therefore, the well‐known predefined dielectric dispersion formulas, such as Cole–Cole, Havriliak–Negami, or Cole–Davidson, cannot adequately describe the dielectric dispersion of the presented crystals. The Cole–Cole formula describes such dielectric dispersion only at higher temperatures due to the predefined symmetric shape of the distribution of relaxation times.

Figure 1.3 Temperature dependence of the complex dielectric permittivity of the CuInP2(S0.25Se0.75)6 crystals measured at several frequencies.

Source: Macutkevic et al. [20] / American Physical Society.

A more general approach must be used for the determination of a broad continuous distribution function of relaxation times f(τ) by solving the Fredholm integral equations,

(1.1a)

(1.1b)

Table 1.2 Parameters of the Vogel–Fulcher fit of the Tm dependence of frequency for the CuInP2(SxSe1−x)6 crystals with x = 0.2 and 0.25.

Composition

v

0

(GHz)

T

0

(K)

E

f

/k

, (K)

CuInP

2

(S

0.25

Se

0.75

)

6

38.34

 96.8

370

CuInP

2

(S

0.2

Se

0.8

)

6

10.96

134.5

150

Figure 1.4 Relaxation‐time distribution for the CuInP2(S0.25Se0.75)6 crystals at various temperatures.

Source: Macutkevic et al. [20] / American Physical Society.

the normalization condition:

(1.2)

The Tikhonov regularization [21] method is applied to solve this equation. The calculated distribution of the relaxation times of CuInP2(S0.25Se0.75)6 crystals is presented in Figure 1.4.

A high‐temperature region (T ≫ Tm) is characterized by a symmetric and narrow f(τ), while on cooling the f(τ) becomes broader and more asymmetric so that below Tm (at 1 kHz) the second maximum appears. Such behavior of the distribution of relaxation times has been already observed in the very well‐known relaxors: PMN, PMT, and SBN [21–23]. The most probable relaxation time τmp, the longest relaxation time τmax, and the shortest relaxation time τmin (0.1 level was chosen as sufficiently accurate) have been obtained. The shortest relaxation time τmin is about 0.1 ns for CuInP2(S0.25Se0.75)6 and about 0.01 ns for CuInP2(S0.2Se0.8)6; it increases slowly with the increase in temperature. The longest relaxation time τmax diverges according to the Vogel–Fulcher law. However, the most probable relaxation time τmp diverges with a good accuracy according to the Arrhenius law. The temperature dependence of the static dielectric permittivity ε(0) was fitted using the spherical random‐bond–random‐field (SRBRF) model

(1.3)

where J is the coupling constant and qEA is the Edwards–Anderson order parameter; if qEA = 0, then this equation appears to be the Curie–Weiss law. The Edwards–Anderson order parameter qEA for the relaxor can be determined by the equation [24]

(1.4)

where ΔJ is the variance of coupling and Δf