Ferroelectricity in Simple Binary Crystals

Abstract

The origin of ferroelectricity in doped binary crystals, Pb_1−xGe_xTe, Cd_1−xZn_xTe, Zn_1−xLi_xO, and Hf_1−xZr_xO₂ is discussed, while no binary ferroelectrics have been reported except for two crystals, HCl and HBr. The ferroelectricity is induced only in doped crystals, which shows an importance of electronic modification in chemical bonds by dopants. The phenomenological and microscopic treatments are given for the appearance of ferroelectric activity. The discovery of ferroelectricity in binary crystals such as ZnO and HfO₂ is of high interest in fundamental science and also in application for complementary metal–oxide semiconductor (CMOS) technology.

1. Introduction

Ferroelectrics are expected as a key material for next-generation nonvolatile ferroelectric memories (FeRAM), piezoelectric actuators, high-k gate-materials for high-speed FET (field effect transistor), and optoelectronic devices [1,2,3,4,5,6]. Particularly, ferroelectric thin films such as perovskite PZT (PbZr_1−xTi_xO₃) and Bi-layered perovskite SBT (SrBi₂Ta₂O₉) have been investigated extensively for FeRAM, because of their excellent dielectric properties, i.e., high dielectric constant and large spontaneous polarization. However, it is not so easy to fabricate good quality ferroelectric thin films on silicon substrate and integrate into devices overcoming degradation of ferroelectric properties due to the so-called size effect. Many ferroelectrics have crystal structures consisting of more than three atoms. For example, BaTiO₃, known as a typical ferroelectric with a perovskite structure, consists of three atoms. New materials with a simple structure are not only preferable for understanding the microscopic origin of ferroelectricity, but are also easy for integrating into modern ferroelectric devices. No ferroelectrics with two atoms have been reported except for two molecular crystals, HCl and HBr (

Table 1. Phase diagram of HCl and HBr after Hoshino et al. [7].

	Phase III Ferroelectric	Phase II Paraelectric			Phase I Liquid
HCl	Orthorhombic-Bb21m	Cubic-Fm3m			Liquid
	Below 98 K	98~159 K			Above 159 K
HBr	Orthorhombic-Bb21m	Phase IIc Orthorhombic-Bbcm	Phase IIb Cubic	Phase IIa Cubic-Fm3m	Liquid
	Below 90 K	90~114 K	114~117 K	117~186 K	Above 186 K

) [7]. In 1969, Cochran developed the theory of lattice dynamics for alkali halide crystals such NaCl and discussed the possibility of ferroelectricity, which revealed a real alkali halide crystal is not a ferroelectric, because a short-range restoring contribution is about twice as great as a long-range Coulomb contribution as discussed later [8,9,10]. However, it is important to point out that these two contributions are the same order for alkali halide crystals. We will show in this article that this balance of these two contributions may be modified by introducing some dopants, strain or defects in crystals.

The electronic ferroelectricity was found in wide-gap semiconductor ZnO by introducing a small amount of Li dopants, although pure ZnO does not show any evidence of ferroelectricity [11,12,13,14,15]. ZnO has a simple binary AB structure with high-symmetry (wurtzite structure). Besides ZnO, Ge-doped PbTe, a IV-VI narrow-gap semiconductor [16], and Zn-doped CdTe, a II-VI wide-gap semiconductor [17], have been investigated as materials of binary crystals accompanying ferroelectricity (Figure 1). Moreover, recent works showed that thin films of HfO₂-ZrO₂ systems exhibit ferroelectricity [18,19]. Hafnia (HfO₂) and Zirconia (ZrO₂) have been well studied as high-k dielectric materials in CMOS (complementary metal–oxide semiconductor) technology. Pure HfO₂ crystal is monoclinic with space group P2₁/c at room temperature and atmospheric pressure. Only HfO₂ thin films doped with Si, Y, Al and Zr change the monoclinic crystal structure to a polar orthorhombic structure. The discovery of ferroelectricity in binary crystals such as ZnO and HfO₂ is of high interest in fundamental science and also in application fields. Our concern is to study the origin of this unexpected appearance of ferroelectricity in doped binary crystals. We will discuss why the ferroelectricity does not appear in pure systems but in doped crystals.

2. Ferroelectricity in Binary Semiconductors

Ferroelectrics, in general, have complicated crystal structures, which undergo a phase transition from a paraelectric high-temperature phase with decreasing temperature, breaking the symmetry of inversion. The dielectric constant (ε) and the spontaneous polarization (P_s) are characterized in the mean field approximation as

ε = C/(T − T_c)

(1)

P_s = P_o (T − T_c)^1/2

(2)

where T_c is a critical temperature. Ferroelectrics are generally classified into order-disorder, displacive and improper types, though multiferroic materials have been reported recently. According to these types, they show the following characteristic dielectric properties as summarized in

Table 2. Typical values of the Curie-Weiss constant (C), the dielectric constant around T_c (ε_max), and the spontaneous polarization (P_s) according to the type of ferroelectrics.

Type of Ferroelectrics	C [K]	εmax at Tc	Ps [µC/cm2]	Example
Order-disorder	1~3 × 103	103	3~5	TGS §
Displacive	105	104	10~30	BaTiO3
Improper	10	10	Small	ACS #
Electronic	-	21	0.9	ZnO

^§ TGS (NH₃CH₂COOH)₃H₂SO₄), ^# ACS ((NH₄)₂Cd₂(SO₄)₃).

[20].

Binary ferroelectric crystals show different but common dielectric behavior from those of the usual ferroelectrics described above; a small dielectric anomaly at T_c and relatively large P_s. We will review the ferroelectric properties of the ferroelectric binary semiconductors, PbTe, CdTe and ZnO, and a high-k dielectric HfO₂, briefly in this section. More detailed discussion should be referred in a monograph [15].

2.1. Narrow-Gap Ferroelectric Semiconductor PbTe

The PbTe-GeTe system has been investigated extensively about its ferroelectricity among IV-VI semiconductors [16], which has a rock-salt type structure (Fm3m, a = 6.46 Å) at room temperature. The energy gap (E_g) is 0.3 eV, which is comparable to the Lorentz field (4π/3)P. The ferroelectricity is observed in solid solution Pb_1−xGe_xTe. The stacking Pb²⁺ cation and Te²⁻ anion layers dimerize along the rhombohedral [111] direction, as shown in Figure 1. The crystal changes to a rhombohedral structure (R3m) which allows it to exhibit ferroelectric activity Pb_1−xGe_xTe with x = 0.003 shows a large dielectric anomaly at T = 100 K.

The large dielectric anomaly and the existence of the soft mode suggest the ferroelectric activity.

2.2. Wide-Gap Ferroelectric Semiconductor CdTe

Cd_1−xZn_xTe is a II-VI wide-gap semiconductor with E_g = 1.53 eV. Weil et al. discovered the ferroelectric activity in Cd_1−xZn_xTe, as shown in Figure 2 [17,21]. The cubic zinc-blende structure (space group $F\overline{4 }3m$, a = 6.486 Å) of pure CdTe crystal changes to a rhombohedral one (R3m, a = 6.401 Å, α = 89.94°) in Cd_1−xZn_xTe, as shown in Figure 1b. The dielectric anomaly at T_c (393 K) is smaller by two orders than that of typical ferroelectric BaTiO₃ (~14,000). The spontaneous polarization is about 5 µC/cm² along the rhombohedral [111] direction. Doped Zn ions locate at off-center positions [22] which cause rhombohedral distortion of about 0.01 Å in Cd_1−xZn_xTe [22].

No soft mode has been observed in Cd_1−xZn_xTe, and the dielectric anomaly is small. Although the behavior of off-center ions plays an important role in this ferroelectricity like Pb_1−xGe_xTe, the occurrence of phase transition seems to be driven in a different way from that of Pb_1−xGe_xTe.

2.3. II-VI Wide-Gap Semiconductor ZnO

Zinc Oxide (ZnO), a II-VI wide-gap semiconductor, is a well-studied electronic material with a large piezoelectric constant [23,24,25,26,27]. ZnO has been studied as materials for solar cells, transparent conductors and blue lasers [28,29]. This crystal has a wurtzite structure (P6₃mc) (Figure 3). This space group is non-centrosymmetric and is allowed to exhibit ferroelectricity, although no D-E loop has been observed until melting point. Introduction of a small amount of Li-dopants results in the ferroelectricity.

A dielectric anomaly in Zn_1−xLi_xO (x = 0.09) was found at 470 K (T_c) (Figure 4), though pure ZnO shows no anomaly from 20 K to 700 K. The small dielectric anomaly (ε_max = 21) is the same order with Cd_1−xZn_xTe (ε_max = 50).

The spontaneous polarization is 0.9 µC/cm² [30,31]. The phase diagram between T_c and x is shown in Figure 5, which reminds us of a phase diagram of quantum ferroelectrics such as KTa_1−xNb_xO₃ and Sr_1−xCa_xTiO₃. Raman scattering measurements showed no soft modes [32,33].

2.4. High-k Materials HfO₂ and ZrO₂

HfO₂ and ZrO₂ are well-known high-temperature dielectrics. The thin films of HfO₂ and ZrO₂ have been studied extensively as a high-k gate dielectric film in CMOS technology. The crystal structure is monoclinic with space group P2₁/c at room temperature and atmospheric pressure, which transforms to a tetragonal structure (P4₂/nmc) at ~1990 K, and then to a cubic Auorite structure (Fm3m) at ~2870 K. Other two orthorhombic phases have been reported under high pressure near 4 and 14 GPa [34]. Figure 6 shows a P-T phase diagram of HfO₂.

The monoclinic P2₁/c structure is centrosymmetric, which does not show any ferroelectric activity. Thin films of HfO₂ undergo a phase transition to a noncentrosymmetric orthorhombic structure, breaking the symmetry of inversion when the films are doped with Si, Y, Al, or Zr [18,19,35,36,37,38]. The X-ray diffraction patterns of HfO₂, Hf_0.5Zr_0.5O₂ and ZrO₂ thin films are shown in Figure 7 [38]. Four space groups (Pmn2₁, Pca2₁, Pbca, and Pbcm) are proposed for this orthorhombic ferroelectric phase. Among these space groups, it is considered that Pca2₁ is the most probable. The sequence of phase transitions and crystal structures are shown in Figure 8. The ferroelectricity was confirmed by D-E hysteresis measurements, which revealed P_s of 16 μC/cm² as shown in Figure 9. The Curie temperature (T_c) was estimated to be about 623 K [39,40].

3. Phenomenological Treatment for the Appearance of Ferroelectricity

Firstly following the Landau theory, we consider the free energy (F) for the paraelectric phase of binary crystals in terms of polarization P as

F = 1/2αP² + 1/4βP⁴ + …,

(3)

where α, and β are coefficients. In general, the only coefficient α depends on temperature as

α = α_o (T − T_o), α_o > 0

(4)

In the case for paraelectric dielectrics, the critical temperature T_o is considered to be lower than 0 K, because of the absence of phase transitions above 0 K. When some dopants are introduced to crystals, some structural changes due to a difference in atomic radii and bonding electrons are modified by dopants. These may induce some local changes in electronic distribution in crystals. These extra contributions could be added to the above free energy as

F = 1/2αP² + 1/4βP⁴ + gηP + 1/2α’η²…,

(5)

where the last term 1/2α’η² is the contribution by dopants, and gηP is an interaction term between the host crystal and extrinsic dopants. From the stability condition ∂F/∂η = 0,

η = −(g/α’)P,

(6)

The above free energy F can be rewritten as

F = 1/2(α − g²/α’)P² + 1/4βP⁴

(7)

Using Equation (4), we get

(α − g²/α’) = α₀ (T − T_c)

(8)

where

T_c = T_o + g²/α₀α’

(9)

As the critical temperature T_c increases when the coefficient α’ is positive, it should be possible to undergo a ferroelectric phase transition. T_c shows a rapid increase in the order of 10² K for large g and small α’ in the case of Pb_1−xGe_xTe, Cd_1−xZn_xTe, and Zn_1−xLi_xO. The appearance of ferroelectricity is realized in the doped binary crystals, while any phase transition does not occur in pure crystals.

4. Microscopic Consideration after Cochran’s Lattice Dynamical Theory

Although the above phenomenological theory can explain well the appearance of ferroelectricity, it is not so easy for us to understand what kind of phenomena occurs in real crystals. For binary crystals such as NaCl, Cochran proposed a lattice dynamical theory to elucidate the origin of ferroelectric phase transition based on the shell model [9,10]. We review simply Cochran’s soft mode theory for ferroelectrics at first.

Cochran calculated the frequencies of transverse and optic phonon modes in a diatomic cubic crystal using a shell model such as NaCl, as illustrated in Figure 10. The shell is originated from some local lattice deformation, electronic overlap forces, or covalency in chemical bonds. The core of the negative ion (charge Xe, mass m₂) will interact through an outer shell (charge Ye, mass ~0) with a force constant k. The positive ion (charge Ze, mass m₁) interacts with the shell by a short range force through a force constant R_o. The displacements are denoted as u₁, v₂ and u₂ for the positive ion, shell and negative core, respectively.

The frequencies of the transverse and longitudinal optic modes, ω_TO and ω_LO are calculated as

$$\mathsf{\mu }{\omega }_{TO}{}^2=R_o^{\text{'}}- \frac{4\pi \left({\epsilon }_{\infty }+2\right){(Z^{\prime }e)}^2}{9V}$$

(10)

$$\mathsf{\mu }{\omega }_{LO}{}^2=R_o^{\text{'}}+ \frac{8\pi \left({\epsilon }_{\infty }+2\right){(Z^{\prime }e)}^2}{9V{\epsilon }_{\infty }}$$

(11)

where

$$R_o^{\text{'}}= \frac{k{R}_o}{k+R_o}<R_o$$

(12)

$$Z^{\prime }=Z+\frac{Y{R}_o}{k+R_o}<Z$$

(13)

$$\mathsf{\mu }=\frac{m_1{m}_2}{m_1+m_2}$$

(14)

${\epsilon }_{\infty }$ and V are a high-frequency dielectric constant and the unit cell volume. Following the Lyddane-Sachs-Teller (LST) relation, the dielectric constant ε is given as

$$\frac{\epsilon }{{\epsilon }_{\infty }}=\frac{{\omega }_{LO}{}^2}{{\omega }_{TO}{}^2}$$

(15)

The ferroelectric phase is realized if ω_TO = 0 in Equation (10), because ferroelectric phase transitions generally accompany with a divergence of dielectric constant. The ferroelectricity is induced from the delicate balance between the short-range term R_o’ and the second dipolar Coulomb term in the right side of Equation (10).

Cochran showed that a real alkali halide crystal is not a ferroelectric, because R_o’ is about twice as great as the other while these two contributions are the same order for real alkali halide crystals. This is one reason why the ferroelectricity has been found only in HCl and HBr. Dopants will change local electronic distribution of chemical bond, i.e., the nature of covalency, particularly in mixed bonded crystals. Dopants also force it to displace to off-centered positions and induce local structural distortions. Particularly, the chemical bonds may be affected sensitively by dopants in ZnO, CdTe and HfO₂ where the degree of covalency (or iconicity) is nearly half, as shown in

Table 3. Dielectric constant and fractional degree of iconicity after J.C. Phillips, Bonds and Bands in Semiconductors [41].

Materials	ε at R.T.	Fractional Degree of Ionicity
NaCl	5.6	0.94
MgO	10.0	0.84
ZnO	8.8	0.62
CdTe	7.1	0.67
HfO2	25	0.8
ZrO2	30~46	0.8
TiO2	170	0.6
Si	3.6	0

. The balance between the restoring force R_o’ and the dipolar Coulomb force can be modified by dopants, defects or strain.

5. Discussion

5.1. Electronic Ferroelectricity in ZnO; Effect of Dopants

The replacement of host Zn ions by substitutional Li ions plays a primary role for the appearance of ferroelectricity in ZnO. To clarify the effect of dopants, structural size-mismatch and electronic models are studied: the introduction of small Be²⁺ ions (ionic radius 0.3 Å) should be effective than Li⁺ (ionic radius 0.6 Å) and Mg²⁺ (ionic radius 0.65 Å) ions if the ionic size-mismatch is important for ferroelectricity, while Mg²⁺ ions (1s²2s²2p⁶) should play a different role from the isoelectronic Li⁺ and Be²⁺ ions (1 s²) if the electronic configuration is important.

The series of dielectric measurements show that the introduction of Mg²⁺ ions suppresses T_c [42]. The appearance of ferroelectricity is primarily due to electronic origin.

5.2. Structural Modification by Dopants

The electronic distribution, especially the nature of d-p hybridization of paraelectric pure ZnO and ferroelectric Zn_1−xLi_xO at 19 K, was measured directly by X-ray diffraction. The main difference is observed in electronic distribution around Zn ion, as shown in Figure 11.

The negative distribution is observed around the Zn atom in Zn_1−xLi_xO, whose shape corresponds to Zn-3d_z²-orbital. This evidence shows that the Zn 3d-electrons disappear around Zn position in the doped ZnO.

Most crystals have a fraction of covalent and ionic bonding components. ZnO is bonded half by ionic and half by covalent forces, of which delicate balance is slightly changed by Li-dopants without d-electrons. Pure ZnO has large dipole moment [11], while this host dipole could not be reversible by an electric field. In Zn_1−xLi_xO, the dipole is reduced a little bit by the local electronic deformation along the polar (0 0 1) direction; this is, in other words, an introduction of negative dipoles in the host lattice. These negative dipoles are responsible for an electric field and behave as “hole dipoles” which are similar to “hole electrons” in n-type semiconductors. We should call this type of ferroelectrics as “n-type ferroelectrics”, while usual ferroelectrics are “p-type ferroelectrics”.

In the case of Hf_1−xZr_xO₂, the ionic radii (0.83 Å for Hf⁴⁺ and 0.84 Å for Zr⁴⁺) are almost the same. The main difference is an electronic structure, Hf (−4f¹⁴) and Zr (−4p⁶). As the high dielectric constants in these crystals (ε = 25 for HfO₂, ε = 30~46 for ZrO₂) mean the large splitting of ω_TO and ω_LO phonon modes, a ω_TO phonon mode is much lower compared with a ω_LO (LST relation, Equation (10)). In this sense, HfO₂ and ZrO₂ are incipient ferroelectrics as SrTiO₃. The electronic distribution in the Hf-O bond is perturbed by Zr dopants without f-electrons. This modification may stabilize an orthorhombic polar structure and results in the appearance of ferroelectricity in Hf_1−xZr_xO₂.

The above microscopic consideration gives us one perspective to understand the origin of ferroelectricity in doped binary crystals. If the condition that ω_TO approaches to zero at T_c holds exactly, soft mode should be detectable. However, no soft mode has been observed in Cd_1−xZn_xTe, Zn_1−xLi_xO and Hf_1−xZr_xO₂, except for Pb_1−xGe_xTe. This evidence suggests that the mechanism of the ferroelectric phase transition is not so simple in real crystals which are partially covalent and partially ionic. The small dielectric anomaly in Cd_1−xZn_xTe, Zn_1−xLi_xO and Hf_1−xZr_xO₂ reminds us of an improper type ferroelectrics, rather than the order-disorder-type and displacive-type of ferroelectrics. This problem has been left for our future studies.

6. Another Possible Ferroelectric TiO₂

Rutile TiO₂ is studied well by various techniques [43]. The structure is tetragonal with space group P4₂/mnm (a = 4.593659 (18) and c = 2.958682 (8) Å at 298 K, Z = 2) (Figure 12) [44]. Besides rutile, TiO₂ admits another two polymorphic forms in nature, i.e., anatase (I4₁/amd, a = 3.7845, c = 9.5143 Å, Z = 4) and brookite (Pbca, a = 5.4558, b = 9.1819, c = 5.1429 Å, Z = 8). Rutile TiO₂ is the most common of the three polymorphic forms. Under high pressure, TiO₂ undergoes a series of structural phase transitions.

Rutile TiO₂ has large refractive indices (n_c = 2.903, n_a = 2.616) and large static dielectric constants (ε_c = 170, ε_a = 86) at room temperature.

Parker measured the dielectric constant ε of TiO₂ which increases with decreasing temperature, but does not show any anomaly from 1.6 to 1060 K, as shown in Figure 13 [45,46]. Pure TiO₂ does not show any ferroelectric or antiferroelectric activity. The dielectric constant shows a plateau at low temperatures around 0 K, which reminds us of dielectric behavior in quantum paraelectrics, such as SrTiO₃.

Although there had been a long-running discussion concerning the covalency of the bonding in rutile, Gronschorek [47] and Sakata et al. [48] concluded that Ti-O bonding in rutile is largely covalent, as shown in Figure 14. If we can modify the nature of bonding by some dopants or stress, it may be possible to expect the appearance of ferroelectricity.

Recently, Montanari and Harrison proposed by density functional calculations that ferroelectric instability can be possible in rutile TiO₂ by applying a negative isotropic pressure [49]. The TO A2u mode, which is the c-axis ferroelectric mode, vanishes at −4 GPa, thereby leading to a crystal instability.

Similar calculations on binary oxides such BaO, CaO, MgO, EuO, SnO₂ have been reported recently [50,51], although these crystals are primarily ionic (the degree of ionicity is ~0.8). They showed that ferroelectricity can be induced even in simple alkaline-earth-metal binary oxides by using appropriate epitaxial strains in thin films or in nano-particles.

7. Conclusions

We discussed the origin of ferroelectricity in doped binary crystals, Pb_1−xGe_xTe, Cd_1−xZn_xTe, Zn_1−xLi_xO, and Hf_1−xZr_xO₂ on the basis of phenomenological and lattice dynamical treatments, while no ferroelectrics have been reported in pure binary crystals except for HCl and HBr. The delicate balance of the short-range restoring force and the long-range dipolar Coulomb force is tuned by dopants, particularly in binary crystals which are half covalent and half ionic. The modification of the electronic distribution in the chemical bond results in the local structural distortion, which may stabilize a non-centrosymmetric polar structure from a paraelectric structure with high-symmetry.

The discovery of ferroelectricity in doped binary crystals shows us a richness of structural science. Moreover, the ferroelectric HfO₂ is expected to be a promising candidate for FeRAM and high-speed FET. Ferroelectrics are a group of materials sensitive to small structural changes. We must take into account the electronic contribution in the case of simple binary crystals discussed here. Further precise structural and theoretical studies should be necessary to clarify the possibility of ferroelectricity in other doped binary crystals.