Ferroelectric Perovskites for High-Speed Memory

We start to understand classification of materials, from the viewpoint of electron migration. Dielectric is classified into insulator, where electron migration is inhibited. In parallel plate capacitor, electrodes are charged in the application of voltage. For a simplicity, instead of charged atom, point charge is introduced. Point charge is mathematical point with negative or positive charge, and its volume is neglected. In electromagnetics, the concept “electric field” is introduced. Electric field around charged plate is represented by Gauss’s law. Under three assumptions: (1) uniform allocation of point charges, (2) uniform electric field, (3) no electric field in perpendicular direction to electrode, electric field within parallel plate capacitor is formulated. By expressing voltage between charged plates using electric field, charge of plate is formulated. Then, vacuum capacitance is defined. Vacuum capacitance is useful index to represent how charge can be stored in a capacitor, though the definition lacks scientific accuracy. Polarisation implies the relative bias of positive and negative charges. In charge neutral system, polarisation vector is expressed using dipole model, where positive and negative point charges are paired. Polarisation vectors for NaCl solid, hydrogen molecule and water molecules are considered. However, as dipole model is not applicable for charged system, general formula of polarisation vector is formulated. Finally, we consider change of capacitance, when dielectric is inserted in parallel plate capacitor. Relative permittivity for dielectric is also defined.

Perovskites are applied in several fields of materials engineering: (1) capacitor, (2) secondary battery, (3) fuel cell, (4) photocatalyst, (5) photoluminescence, (6) solar cell dye. To enhance capacitance in parallel plate capacitor, dielectric perovskite is inserted between two electrodes. Depending on type of inserted dielectric, ceramic capacitor is classified into three types: (1) temperature compensating-type, (2) high permittivity-type, (3) semiconducting-type. Due to inflammability of liquid electrolyte in secondary battery, solid electrolyte using lithium ion-conducting perovskite has been explored. Lithium ion migrates through counter cation-vacancy. Recently, replacement of lithium has been also required from the viewpoint of resource constraint. Sodium ion-conducting perovskite has been expected. In solid oxide fuel cell, several perovskites are applicable as solid electrolytes with proton-conductivity or oxide ion-conductivity. In photocatalyst and photoluminescence, perovskite-type titanium oxide is used. Recently, lead iodide and bromide perovskites are applied as solar cell dye. General formula of simple perovskite is AMX₃, where A, M and X denote counter cation, metal and anion, respectively. Goldschmidt tolerance factor is useful index to evaluate crystal structure in AMX₃ perovskite. Shannon ionic radii are employed when calculating tolerance factor. However, as temperature is not taking into consideration, it is nothing more than estimate to represent tendency of cubic stability. In principle, perovskite exhibits dielectric property. It is because atomic displacements occur in the application of electric field. Finally, typical dielectric perovskites are briefly introduced.

Transistor is indispensable circuit element for electronic device. In present, the number of transistors integrated in integrated circuit (IC) has become double every 1.5–2 years, following Moore’s law. Microcontroller unit consists of central processing unit (CPU), main memory, input and output. In present main memory, data is temporarily stored in volatile memories such as Dynamic Random Access Memory (DRAM) and Static Random Access Memory (SRAM). In memory, binary number system is applied in order to express character and decimal number. Hence, two different states are recognised as 0/1 digital information. Volatility implies that digital data disappears, when stopping supply of electric energy. DRAM and SRAM are volatile. For data storage memory, floating gate memory such as Electrically Erasable Programmable Read-Only Memory (EEPROM) and Hard Disk Drive (HDD) are in practical use. Recently, Solid State Drive (SSD) is getting popular rather than HDD for personal computers. EEPROM, HDD and SDD are classified as non-volatile memory. In recent years, non-volatile memory for main memory has been actively developed for further high performance computing. The candidates include Ferroelectric Random Access Memory (FeRAM), Ferroelectric-gate Field Effect Transistor (FeFET), Magnetoresistive Random Access Memory (MRAM), Resistive Random Access Memory (ReRAM) and Phase Change Memory (PCM). Each non-volatile memory has unique operating mechanism. It is phenomenologically explained how to read and write 0/1 digital information.

In dielectric, charged atoms are displaced by applying electric field. Ferroelectric is the special dielectric. Even if electric field is off, displaced atoms are fixed at specific position. Polarisation implies relative bias of positively and negatively charged atoms. In ferroelectric, polarisation is retained as spontaneous polarisation. Due to non-volatility, Ferroelectric Random Access Memory (FeRAM) has been expected as next-generation high-speed memory. Curie temperature is the temperature, where ferroelectric transition occurs. It is empirically known that permittivity is inversely proportional to temperature. When sharp peak is observed around Curie temperature, it can be estimated that ferroelectricity appears below the temperature. In ferroelectric, electric field versus spontaneous polarisation plot exhibits hysteresis characteristic. Polarisation reversal can be explained by the use of ferroelectric hysteresis loop. After polarisation treatment, polarisation is saturated. Even if electric field is off, polarisation is still retained as spontaneous polarisation. Opposite spontaneous polarisation is given by applying reverse electric field. Ferroelectric perovskite is classified into displacement type ferroelectric. In principle, charged atom is displaced by applying electric field. In ferroelectric perovskite, some atomic displacements are inhibited, due to steric repulsion between atoms. From the meaning that covalent bond length varies, keeping covalent bonding formation, such atomic displacement is called Quantum Bonding (QB) motion. As counter cation-migration is combined, not only QB motion but also ion-conduction are taken into consideration. Polarisation reversal occurs in terms of polarisation domain. In ferroelectric BaTiO₃ perovskite, two types of polarisation domains (180° and 90° polarisation domains) are reported. Finally, downsizing limit for ferroelectric perovskite is explained.

In spite of discovery of hysteresis characteristic in Rochelle salt, the phenomenon was recognised as anomalous dielectric response. As magnitude of spontaneous polarisation varied by changing temperature, stable ferroelectric structure was ungiven. In potassium dihydrogen phosphate (KH₂PO₄), sharp peak appeared in temperature versus dielectric constant plot. Below Curie temperature, ferroelectric hysteresis loop was observed. Ferroelectric hysteresis characteristic was also confirmed in KH₂PO₄ family: KH₂AsO₄. At that time, theoretical investigation was also started. The situation of dielectric study dramatically changed in the 1940s. Ogawa discovered that at room temperature, anomalously high dielectric constant is shown in BaTiO₃ perovskite. The relation between spontaneous polarisation and crystal structure was investigated. It was concluded that tetragonal structure is responsible for spontaneous polarisation. Hippel et al. defined that ferroelectricity is the phenomenon that high dielectric constant maximum is connected with the concerted atomic displacements. After that, ferroelectricity has been used as scientific academic term in chemistry and physics. In BaTiO₃ perovskite, ferroelectric behaviour is unstable. Because phase transition temperature is overlapped with operation temperature. In the 1990s, alternative ferroelectric, lead zirconate titanate: PbZr_xTi_1-xO₃ perovskite (PZT) was employed as ferroelectric of Ferroelectric Random Access Memory (FeRAM). However, from the viewpoints of environment and health problems, Pb-free ferroelectrics have been required. Finally, other ferroelectric perovskites and hafnium oxide are shortly introduced.

Electron has an important role in ferroelectric behaviour. In electromagnetics, electron is normally approximated as point charge. However, in fact, electron acts as quantum particle having wave-particle duality. In classical manner, the ideal progressive wave is expressed using trigonometric function. In order to incorporate wave property into particle, the relational formula between matter wave and photon energy are introduced. Based on the formula, the equation for quantum particle is derived. It is called Schrödinger equation, where position and time are included as variable. In real atom-based system, no quantum particle is completely isolated from others. Hence, quantum interactions between quantum particles are inserted into Schrödinger equation as the form of potential. When quantum interactions are independent from time, Schrödinger equation can be expressed as time-independent form. In quantum chemistry, time-independent form is normally called Schrödinger equation. Operators of kinetic energy and potential are called Hamiltonian. When normalised quantum wave function is applied, the total energy is directly given by operating Hamiltonian operator to quantum wave function. Since Hamiltonian is categorised as Hermitian operator, the total energy is represented by real number. Quantum wave function itself does not represent figure in three-dimensional space. Instead, the square of quantum wave function represents density of quantum particle. Quantum particle, electron, spreads as wave. For better understanding the relationship between quantum particle and quantum wave function, quantum tiger law is introduced.

In Chap. 6, Schrödinger equation for quantum particle was derived by the introduction of wave-particle duality. Kinetic and potential energies were expressed as the general form. In this chapter, we consider the specific case: hydrogenic atom, where one-electron interacts with atomic nucleus. The Hamiltonian is constituted by operators of electron kinetic energy, nuclear kinetic energy and Coulomb potential energy between electron and atomic nucleus. Schrödinger equation is analytically solved in hydrogenic atom. Introducing polar coordinates, quantum wave function is separated into two parts. The operation is called separation of variables in differential equation. As the result, Schrödinger equation is transformed into two equations. One is for radial quantum wave function. The other is for angular quantum wave function. These functions are classified using three quantum numbers such as principal quantum number, angular momentum quantum number and magnetic quantum number. Shell type (K, L, M etc.) and subshell type (s, p, d etc.) are designated by principal quantum number and angular momentum quantum number, respectively. Though 2p_z atomic orbital is directly expressed by the use of spherical harmonics and radial quantum wave functions, 2p_x and 2p_y atomic orbitals are represented by linear combination of spherical harmonics. In the same manner, four of five 3d atomic orbitals are represented by linear combination of spherical harmonics. The total energy for hydrogenic atom is also analytically obtained. It is demonstrated how to estimate the excitation energy from the formula. Finally, pictorial representation of atomic orbital is explained.

In hydrogenic atom, Schrödinger equation can be analytically solved. Quantum wave function is concretely given. However, the situation dramatically changes in many-electron system. As analytical solution cannot be given, we need to solve Schrödinger equation numerically. In this chapter, necessary contents for numerical approach are explained. First, we consider representative two-electron system: helium atom. The Hamiltonian includes operators of kinetic energy of electron, kinetic energy of atomic nucleus, Coulomb interaction between electron and atomic nucleus, and Coulomb interaction between two electrons. In comparison with hydrogenic atom, Coulomb interaction between two electrons is added. In many-electron cluster or molecule, we need to incorporate the operator of Coulomb interaction between atomic nuclei. In Born–Oppenheimer approximation, the Coulomb interaction can be negligible. When considering more than two electrons, electron spin must be taken into account, by the introduction of spin function into quantum wave function. The quantum wave function in many-electron system is represented by the product of spatial orbital and spin function. Spin function satisfies orthonormality. As electron is categorised as fermion, inverse principle must be satisfied for total quantum wave function. It implies that total quantum wave function changes the sign, when the labels of any two identical fermions are exchanged. Hartree product does not satisfy the condition. In quantum chemistry, Slater determinant is introduced for the purpose. Finally, Slater determinant for two-electron system is shown.

In many-electron system having more than two electrons, mathematical method to obtain analytical solution in Schrödinger equation has not been discovered. Instead, we are able to obtain quantum wave function numerically. In quantum computational chemistry, Hartree–Fock equation is used for numerical calculation. Through minimization of total energy in Schrödinger equation, one-electron equation, Hartree–Fock equation is derived. In Schrödinger equation, Hamiltonian operates on total quantum wave function. On the other hand, in Hartree–Fock equation, Fock operator operates on quantum wave function (spin orbital). Fock operator consists of three operators: one-electron operator, Coulomb operator, exchange operator. Orbital energy is given as eigenvalue of Hartree–Fock equation. In closed shell system, two electrons with α and β spins are paired in the same spatial orbital. Total energy in Schrödinger equation is expressed by the use of one-electron operator, Coulomb integral, exchange integral and spatial orbitals, due to orthonormality of spin function. Orbital energy in Hartree–Fock equation is also expressed by the use of one-electron operator, Coulomb integral, exchange integral and spatial orbitals, due to orthonormality of spin function. The derivation processes are explained in details. From the viewpoint of restricted electron-allocation, the Hartree–Fock is called Restricted Hartree–Fock (RHF). In open shell system, different spatial orbitals are prepared for α and β electrons. In the same manner, we consider to express total energy in Schrödinger equation and orbital energy in Hartree–Fock equation. From the viewpoint of unrestricted electron-allocation, the Hartree–Fock is called Unrestricted Hartree–Fock (UHF).

One-electron equation called Hartree–Fock equation is derived by the minimization of total energy given in Schrödinger equation. When basis function is introduced in quantum wave function, the mathematical problem to obtain quantum wave function analytically is converted into calculating expansion coefficient numerically. In closed shell system, α and β electrons can occupy the same spatial orbital. One Hartree–Fock matrix equation is given. On the other hand, in open shell system, different spatial orbitals are prepared for α and β electrons. Two Hartree–Fock matrix equations are given. In this chapter, these Hartree–Fock matrix equations are derived. In atom, spatial orbital is expressed as superposition of basis functions. A set of basis functions for each atom is called basis set. In cluster, spatial orbital is expressed as combination of basis sets. In basis function, Gaussian-type exponential is applied, due to practical calculation with regard to two-electron integral. For correction of radial shape, contraction and split-valence are performed for basis function. In Hartree–Fock approximation, as the Coulomb interaction between electrons is estimated in average procedure, the deviation called electron correlation arises. To incorporate electron correlation, several calculation methods based on Hartree–Fock method have been explored: configuration interaction, coupled cluster etc. In another stream, density functional theory has been developed. By the use of density as variable, Schrödinger equation is converted into Kohn–Sham equation. By solving the equation, molecular orbital under consideration of electron correlation is obtained. When we start molecular orbital calculation, we need to consider three main factors: (1) basis set selection, (2) combination of basis set and calculation method, (3) modelling. The concept of initial atomic orbital (IAO) is useful for molecular orbital analysis. Chemical bonding rule is used to judge chemical bonding character: covalent bonding or ionic bonding. Atomic charge can be estimated based on Mulliken population analysis.

The purpose of this chapter is to understand ferroelectric behaviour in tetragonal BaTiO₃ perovskite from molecular orbital calculation results. In electromagnetics, ferroelectricity is explained based on point charge-based motion, where positively charged titanium atom and negatively charged oxygen atom are simply displaced in the same and reverse directions of electric field. However, the calculation results show that total energy monotonously increases in the motion. From the viewpoint of quantum chemistry, atomic displacement under electric field is reconsidered. Titanium-displacement to shorten Ti–O bond along long Ti–O–Ti and oxygen-displacement contribute energetic stabilisation. Based on the fact, we consider periodic and aperiodic atomic displacements patterns. As the result of quantum bonding (QB) motion, stabilisation energy is given. Note that QB motion is different from ion-conduction and chemical reaction. Though periodic atomic displacements in BaTi₈O₁₂ unit is necessary in the whole of tetragonal BaTiO₃ perovskite, the unit has a structural degree of freedom near surface. Hence, aperiodic QB motion occurs near surface. Aperiodic and periodic QB motions are smoothly connected in the junction. In principle, positively charged barium atom (barium cation) migrates in the application of electric field. Hence, we investigate the effect of barium cation migration on ferroelectric behaviour. Next, we discuss direction of QB motion. QB motion is energetically possible in different directions. However, ferroelectric QB motion is restricted along long Ti–O–Ti bond, giving large stabilisation energy. Finally, downsizing limit of ferroelectric BaTiO₃ nanoparticle is discussed.

PbTiO₃ perovskite is known as representative ferroelectric as well as BaTiO₃ perovskite. First, we investigate chemical bonding formation related to counter cation in several perovskites. In BaTiO₃ and La_2/3-xLi_3xTiO₃: LLTO perovskites, counter cation forms ionic bonding with constituent atoms. On the other hand, in PbTiO₃ perovskite, covalent bonding is formed between lead cation and constituent atoms. In BaTiO₃ perovskite, the ionic bonding is kept during barium cation migration around tetragonal centre, while in PbTiO₃ perovskite, chemical reaction, in the meaning of partial covalent bonding change, is combined during lead cation migration around tetragonal centre. Based on these scientific knowledge, ferroelectric quantum bonding (QB) motion in PbTiO₃ perovskite is investigated by performing molecular orbital calculation. Stabilisation energy and electrostatic moment (EM) value of PbTi₈O₁₂ unit are larger than BaTi₈O₁₂ unit. Near surface, aperiodic and periodic quantum bonding motions are smoothly connected through the junction. In the application of electric field, though lead cation cannot migrate between neighbouring PbTi₈O₁₂ units, lead cation can migrate near tetragonal centre. Hence, not only ion-conduction but also chemical reaction is combined in ferroelectric QB motion of PbTiO₃ perovskite. In ferroelectric engineering, zirconium doped PbTiO₃ perovskite: PbZr_1-xTi_xO₃ (PZT) perovskite has been employed as ferroelectric material. Finally, we discuss ferroelectricity in PZT perovskite.

Based on scientific knowledge gained from BaTiO₃ and PbTiO₃ perovskites, it is attempted to design new ferroelectric perovskite. They have tetragonal structure, which implies Ti–O–Ti bond length in one specific direction is different from other two directions. In quantum bonding (QB) motion, when stabilisation energy in one specific direction is larger than other directions, ferroelectric QB motion occurs. Hence, as our strategy, we consider to introduce different bond length in one specific direction. First, molecular orbital calculation is performed to investigate QB motion in ideal LaTiO₂N perovskite. Though Ti–N–Ti bond length is shorter than Ti–O–Ti bond length, ferroelectric QB motion along Ti–N–Ti bond occurs. Next, we consider SrTiO₂C perovskite, where carbon atoms are replaced by oxygen atoms in SrO layer. As same as LaTiO₂N perovskite, Ti–C–Ti bond length is different from Ti–O–Ti bond length. Covalent bonding is formed between titanium and carbon atoms. Hence, different type of QB motion along Ti–C–Ti bond is expected in the application of electric field. In the QB motion along Ti–C–Ti bond, Ti–C bond is elongated, instead of shrinking. In addition, though periodic QB motion is not stabilised, aperiodic QB motion occurs. On the other hand, in perpendicular QB motion, larger stabilisation energy is given. As the result, when controlling the magnitude of electric field, ferroelectric QB motion can be caused. Owing to smaller stabilisation energy, high-speed polarisation reversal, low electric power consumption and improvement of endurance can be expected in ferroelectric SrTiO₂C perovskite. QB motion in TiOC layer is also investigated. In this case, as larger stabilisation energy is given along Ti–C–Ti bond, the QB motion is advantageous.

It was demonstrated that ferroelectric quantum bonding (QB) motion occurs in tetragonal perovskite, from concrete molecular orbital calculation results. Further, new ferroelectric for high-speed memory was theoretically designed. In the final chapter, first, quantum bonding (QB) motion is summarised from the viewpoint of total energy variation. Limit of mathematical approach in electromagnetics is also considered. It is widely recognised that simple mathematical expression has a beauty in mathematics. However, we have to pay attention that mathematical trap exists in quantum phenomenon. At last, final remarks are mentioned. “Quantum Bonding Motion is different from chemical reaction. It is compulsory driven by our command”.