Optimizing SAW Device Performance Using Titanium-Doped Lithium Niobate Substrates

Abstract

This study introduces a new theoretical framework for the ferroelectric phase transition in lithium niobate (LiNbO₃), which explicitly incorporates electrostatic interactions between both first and second nearest-neighbor ions. This extended model is applied to estimate the inverse quality factor (Q⁻¹), the equivalent mechanical resistance (R_m), and the Curie temperature (Tc) of pure and titanium-doped lithium niobate (LiNbO₃:Ti). The proposed analytical expression for Tc is given by: $T_C^{*}={\displaystyle \frac{2\sqrt{\frac{-p^{*}}{3}}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{{\theta }^{*}}{3}\right)-\frac{B^{*}}{3}}{2\sqrt{\frac{-p}{3}}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\theta }{3}\right)-\frac{B}{3}}} T_C$. The analysis reveals that variations in Q⁻¹ and Tc are governed by factors such as ionic mass, charge, and defect structure. The theoretical predictions show good agreement with experimental data reported in the literature—particularly for Q⁻¹ in pure LiNbO₃ and for Tc in Ti-doped LiNbO₃—thus validating the reliability of the proposed model. Moreover, at constant temperature, both the inverse quality factor and the equivalent mechanical resistance decrease as the Ti concentration increases. This trend highlights that titanium doping enhances the acoustic performance of LiNbO₃ substrates, making them more suitable for high-performance surface acoustic wave (SAW) device applications.

1. Introduction

Lithium niobate (LiNbO₃, or LN) crystals occupy a central position among functional materials due to their remarkable piezoelectric, optical, and electrical properties [1,2,3,4]. Since their first growth by the Czochralski method in the 1960s [5], they have attracted considerable interest in a wide range of technological fields, including nonlinear optics [6,7], optical telecommunications, single-crystal nanodevices [8], acousto-optic components, sensors, and piezoelectric devices. However, to fully exploit the potential of these crystals, it is essential to understand the influence of their chemical composition as well as the role of structural defects on their intrinsic physical properties [9,10].

It is well established that point defects play a decisive role in the behavior of LN, particularly in its congruent form (cLN, Li/Nb ≈ 0.945) and stoichiometric form (sLN, Li/Nb = 1). In cLN, the significant presence of intrinsic defects such as niobium antisite ions (NbLi) and lithium vacancies (V_Li) strongly modifies the crystal structure and its electrical and optical properties [11,12]. Processing techniques such as vapor transport equilibration (VTE) are commonly employed to approach the ideal stoichiometry and reduce the concentration of these defects [13,14,15]. At room temperature, the solid solubility range of LiNbO₃ extends from 46% to about 50.4% molar Li₂O, and this range further evolves at high temperatures, reaching between 45% and 50% at approximately 1210 °C [16].

Lithium niobate undergoes a ferroelectric-to-paraelectric phase transition at a Curie temperature (Tc) of approximately 1200 °C in stoichiometric crystals [17]. Most theoretical models describing this transition consider only electrostatic interactions between first-nearest neighbors [18,19,20], which may not fully capture the experimental behavior, particularly in doped systems.

In this context, we propose a new theoretical approach that incorporates electrostatic interactions between the nearest neighbors and the most distant neighbors, which represents a significant extension of classical models. This methodology is applied to both pure LN and titanium-doped LN, using the vacancy model based on the substitution mechanism. The objective is to evaluate and analyze the influence of vacancy defects on the Curie temperature Tc, the quality factor Q, and the equivalent mechanical strength R_m. The results provide key information for optimizing titanium doping in LN, particularly for applications requiring high structural stability, strong electrical response, and improved optical performance, such as SAW resonators (Figure 1) and nonlinear optical modulators.

Figure 1. Basic structure of the SAW resonator [21].

2. Materials and Methods

2.1. Previous Theoretical Model (Th1)

In the pioneering work of Safaryan [19], the ferroelectric phase transition in lithium niobate (LiNbO₃) is attributed to the combined effect of ionic dynamics and thermal expansion of the crystal lattice, ultimately leading to a rearrangement of the ionic sublattices. His theoretical model captures the essential ferroelectric behavior of LiNbO₃ and reproduces the general features of its phase transition.

To make the analysis tractable, Safaryan simplified the complex three-dimensional ionic motion into a one-dimensional linear model of lattice vibrations, considering only electrostatic interactions between the nearest charged planes. Despite this simplification, the model provides meaningful insights into the origin of ferroelectricity in LiNbO₃.

$$T_C^{*}={\displaystyle \frac{{\omega }_2^{*2}}{{\omega }_2^2}}={\displaystyle \frac{{M_0}^{*}+{M_1}^{*}+{M_2}^{*}}{M_0+M_1+M_2}}{\displaystyle \frac{M_0{M}_1{M}_2}{{M_0}^{*}{M_1}^{*}{M_2}^{*}}}{\displaystyle \frac{{P_{2}P}_1^{*}}{P_1{P}_2^{*}}}{.T}_C$$

(1)

where the parameters $P_1$ and $P_2$ are defined as:

$$\begin{gathered} P_1=3q_0{R}_{12}^2 - q_1{R}_{20}^2- q_2{R}_{10}^2 \\ {P}_2={\displaystyle \frac{{{(R}_{10}{R}_{12})}^2}{q_1}} \left({\displaystyle \frac{1}{M_0}}+{\displaystyle \frac{1}{M_2}}\right)+{\displaystyle \frac{{{(R}_{20}{R}_{12})}^2}{q_2}} \left({\displaystyle \frac{1}{M_1}}+{\displaystyle \frac{1}{M_0}}\right)- {\displaystyle \frac{{{(R}_{10}{R}_{20})}^2}{{3q}_0}} ({\displaystyle \frac{1}{M_1}}+{\displaystyle \frac{1}{M_2}}) \end{gathered}$$

(2)

In these expressions, $M_i$ and $q_i$ represent the ionic masses and effective charges of Li⁺, Nb⁵⁺, and O²⁻ ions, while $R_{ij}$ corresponds to the equilibrium distances between interacting ions.

2.2. Our Theoretical Approach (Th2)

In the present work, we extend Safaryan’s model by including electrostatic interactions between both nearest and next-nearest neighboring ions. This refinement (denoted as Th2) aims to provide a more accurate and comprehensive description of the ferroelectric phase transition, especially in non-stoichiometric and doped LiNbO₃ systems, where long-range lattice distortions and defect-induced charge redistributions play a crucial role.

In the LiNbO₃ crystal structure, the lattice is formed by alternating planes of Li⁺, Nb⁵⁺, and O²⁻ ions that interact via long-range Coulomb forces. The equations of motion describing these ions include the restoring forces generated by electrostatic coupling between charged planes. When second-nearest-neighbor interactions are incorporated, additional terms appear in the equations to account for the extended range of interionic forces.

The lattice period $a$ is defined as the distance between two successive oxygen planes. Figure 2 schematically illustrates the spatial arrangement of these ions within the s-th unit cell and the corresponding displacements: U_s (Li⁺) for Li⁺, $V_s$ (B⁵⁺) for Nb⁵⁺ and $\xi$_s (O²⁻) for O²⁻.

Figure 2. Crystalline structure of LiNbO₃ showing atomic displacements along the c-axis. R_ij denotes the distance between the nearest-neighbor atoms, while R_ij′ represents the distance between the next-nearest neighbors.

The ferroelectric transition in LiNbO₃ is primarily driven by ionic displacements associated with the soft optical vibrational mode along the crystallographic c-axis—the polar direction where Li⁺, Nb⁵⁺, and O²⁻ ions are aligned. It is along this axis that spontaneous polarization develops due to the relative shifts in Li⁺ and Nb⁵⁺ with respect to the oxygen sublattice. Consequently, Coulomb interactions perpendicular to the c-axis are neglected, as they have a negligible contribution to the polarization mechanism responsible for the phase transition, as shown in Figure 3 [22].

Figure 3. Crystal structure of lithium niobate in ferroelectric and paraelectric phase. Adapted with permission from Ref. [22].

Considering the nearest-neighbor (interactions between ions in adjacent planes) and second-neighbor (ions separated by two planes) interactions, the equations of motion for the ions Li⁺, Nb⁵⁺, and O²⁻ can be written as follows:

$$\left\{\begin{array}{c}{M}_1\ddot{U_s}=C_{10}\left(\xi -U_s\right)+C_{21}\left(V_{s+1}-U_s\right)+{C\prime }_{10}\left({\xi }_{s+1}-U_s\right)+{C\prime }_{21}\left(V_{s+2}-U_s\right)\\ M_2\ddot{V_s}=C_{21}\left(U_s-V_s\right)+C_{20}\left({\xi }_s-V_s\right)+{C\prime }_{21}\left(U_{s+1}-V_s\right)+ {C\prime }_{20}\left({\xi }_{s+1}-V_s\right)\\ M_0\ddot{{\xi }_s}=C_{20}\left(V_s-{\xi }_s\right)+C_{10}\left(U_{s-1}-{\xi }_s\right)+{C\prime }_{20}\left(V_{s-1}-{\xi }_s\right)+{C\prime }_{10}\left(U_{s-2}-{\xi }_s\right)\end{array}\right.$$

(3)

Variables such as $U_s$, $V_s$ and ${\xi }_s$ represent the relative displacements of ions or groups of atoms within the S-th unit cell. Terms with negative or positive indices like $V_{s-1}$, $V_{s-2}$, ${\xi }_{s+1}$ or $V_{s+2}$ indicate displacements in neighboring cells located immediately before or after the cell S. Specifically $V_{s-1}$, denotes the displacement in the preceding cell to the left, while $V_{s-2}$ refers to the displacement in the next-nearest cell on that side. Conversely, $V_{s+1}$ and $V_{s+2}$ refer to displacements in the adjacent and subsequent cells to the right.

In this model, the terms $C_{{10}^{\prime }}$, $C_{{20}^{\prime }}$, and $C_{{21}^{\prime }}$ account for the contributions from second-nearest neighbors, i.e., ions located two planes apart along the trigonal c-axis. The masses $M_1$, $M_2$, and $M_0$ correspond to the respective masses of Li⁺, Nb⁵⁺, and O²⁻ ions. The interactions between ions in adjacent planes are characterized by parameters denoted $C_{ij}$.

It is important to emphasize that, although the notation $C_{ij}$ may resemble that of crystallographic elastic constants, in the present context, these quantities are not true elastic constants of the crystal. Instead, they represent effective coupling coefficients (or bonding force constants) between two ionic sublattices $i$ and $j$ located on adjacent planes along the c-axis. Physically, $C_{ij}$ quantifies the restorative electrostatic force opposing the relative displacement of the ions, and therefore reflects the strength of the interionic binding rather than the macroscopic elastic response of the crystal.

These effective force constants originate from Coulomb interactions between the ions’ effective charges and also incorporate the geometric arrangement of the lattice. In a simplified form, and up to a normalization factor, $C_{ij}$ can be expressed as [19]:

$$C_{ij}=-{\displaystyle \frac{e^2{q}_i{q}_j}{b\hspace{0.17em}{\left(R_{ij}^0\right)}^2}}\hspace{0.17em}n$$

(4)

where q_i and q_j are the electric charges of ions i and j, respectively; $R_{ij}^0$ is the equilibrium distance between these ions; b is the characteristic distance in the lattice, generally the distance between crystal planes; and n is the ionic density. Specifically, for the LiNbO₃ crystal, we have:

$$\begin{gathered} C_{Li-O}=C_{20}=3{\displaystyle \frac{e^2{q}_i{q}_j}{b{{{(R}_{20}}^0)}^2}}n;\hspace{1em}{C\prime }_{20}=3{\displaystyle \frac{e^2{q}_i{q}_j}{b{{{(R\prime }_{20}}^0)}^2}}n \\ {C}_{B-O}=C_{10}=3{\displaystyle \frac{e^2{q}_i{q}_j}{b{{{(R}_{10}}^0)}^2}}n;\hspace{1em}{C\prime }_{10}=3{\displaystyle \frac{e^2{q}_i{q}_j}{b{{{(R\prime }_{10}}^0)}^2}}n \\ {C}_{B-Li}=C_{12}=-{\displaystyle \frac{e^2{q}_i{q}_j}{b{{{(R}_{12}}^0)}^2}}n;\hspace{1em}{C\prime }_{12}=-{\displaystyle \frac{e^2{q}_i{q}_j}{b{{{(R\prime }_{12}}^0)}^2}}n \end{gathered}$$

(5)

Here, q₀, q₁, and q₂ are the charges of the ions O²⁻, Nb⁵⁺, and Lⁱ⁺, respectively, and b is the lattice constant. The parameters R_ij and R’_ij indicate the distances between the nearest and second-nearest neighbors.

The choice of solutions in the form of plane waves, $U_s=U{e}^{iwt}{e}^{iask}$ with $U_s$ = $V_s$,${\mu }_s$ or ${\xi }_s$.

After calculating the determinant (Δ = 0), which represents the mathematical condition indicating the existence of non-trivial solutions for the system of motion equations, we identify the natural frequencies of the crystal associated with its vibrational modes: acoustic and optical. Solving this equation allows us to determine the various dispersion branches as functions of the wave vector k, which we take as the wave number equal to zero in this context. This leads to the following equation:

$${\omega }^6+A{\omega }^4+{B\omega }^2+D=0$$

(6)

where the coefficients A, B and D depend on the interaction constants and the masses of the ions.

The coefficients A, B and D are given by:

$$\begin{gathered} A=-{\displaystyle \frac{1}{m_2}}\left(C_{12}+C_{20}+{C^{\prime }}_{12}+{C^{\prime }}_{20}\right)-{\displaystyle \frac{1}{m_1}}\left(c_{10}+c_{12}+{C^{\prime }}_{10}\right)-{\displaystyle \frac{1}{m_0}}\left(c_{10}+c_{20}\right) \\ B={\displaystyle \frac{1}{m_1{m}_2}}\left({C_{12}}^2+c_{10}{C}_{20}+C_{12}{C}_{20}+C_{20}{C^{\prime }}_{10}+C_{12}{C^{\prime }}_{12}+c_{10}{C^{\prime }}_{20}+C_{12}{C^{\prime }}_{20}+{C^{\prime }}_{10}{C^{\prime }}_{20}\right)+ \\ {\displaystyle \frac{1}{m_0{m}_2}}\left(c_{10}{C}_{12}+c_{10}{C}_{20}+C_{12}{C}_{20} +C_{10}{C^{\prime }}_{12}+C_{12}{C^{\prime }}_{20}+c_{10}{C^{\prime }}_{20}-C_{20}{C^{\prime }}_{20}-{{C^{\prime }}_{20}}^2\right)+ \\ {\displaystyle \frac{1}{m_1{m}_0}}(c_{10}{C}_{12}+c_{10}{C}_{20}+C_{12}{C}_{20}-C_{10}{C^{\prime }}_{10}+C_{20}{C^{\prime }}_{10}-{{C^{\prime }}_{10}}^2) \\ D=-{\displaystyle \frac{1}{m_0{m}_1{m}_2}}({C_{10}}^2\left(C_{20}-C_{12}-{C^{\prime }}_{12}+C_{20}\right)-{{C^{\prime }}_{10}}^2\left(C_{12}+{C^{\prime }}_{12}+{C_{20}}^2+{C^{\prime }}_{12}{C^{\prime }}_{20}-C_{12}{C^{\prime }}_{20}\right) \\ +c_{10}({C_{12}}^2+2C_{12}{C}_{20}-2C_{12}{C^{\prime }}_{10}+C_{20}{C^{\prime }}_{10}+C_{12}{C^{\prime }}_{12}+C_{20}{C^{\prime }}_{12}+2C_{12}{C^{\prime }}_{20} \\ -C_{20}{C^{\prime }}_{20}-{C^{\prime }}_{12}{C^{\prime }}_{20}-{{C^{\prime }}_{20}}^2)+C_{20}({C_{12}}^2+2C_{12}{C^{\prime }}_{12}-C_{12}{C^{\prime }}_{10})-C_{12}{{C^{\prime }}_{20}}^2) \end{gathered}$$

(7)

There are several methods for solving the characteristic Equation (4). A classic approach is the Cardan method [23,24].

We pose $\omega =z-{\displaystyle \frac{B}{3}}$ from which Equation (3) becomes

$$z^3+p\omega +q=0$$

(8)

where $p={\displaystyle \frac{3A-B^2}{3}}$ and $q={\displaystyle \frac{2B^3-9AB+27D}{27}}$.

We found that $\Delta =({{\displaystyle \frac{q}{2}})}^2+{\left({\displaystyle \frac{p}{3}}\right)}^3=-5.31{ \times 10}^{170}<0$, where there are three distinct real roots, which correspond to the fact that the system has three modes of vibration: an acoustic mode (the highest frequency) and two optical modes.

$$\left\{\begin{array}{c}{\omega }_1^2=2\sqrt{{\displaystyle \frac{-p}{3}}}\cos \left({\displaystyle \frac{\theta }{3}}\right)-{\displaystyle \frac{B}{3}}\\ {\omega }_2^2=2\sqrt{{\displaystyle \frac{-p}{3}}}\cos \left({\displaystyle \frac{\theta +2\pi }{3}}\right)-{\displaystyle \frac{B}{3}}\\ {\omega }_3^2=2\sqrt{{\displaystyle \frac{-p}{3}}}\cos \left({\displaystyle \frac{\theta +4\pi }{3}}\right)-{\displaystyle \frac{B}{3}}\end{array}\right.$$

(9)

where $r=\sqrt{{\displaystyle \frac{-p^3}{27}}}$ and $\theta ={\cos }^{-1}\left({\displaystyle \frac{-q}{2r}}\right)$.

By solving the system and substituting the physical parameters (expressed in the CGS unit system), we obtain the following values: ${\omega }_1^2=4.11\times {10}^{27} {\mathrm{s}}^{-1}$, ${\omega }_2^2=2.56\times {10}^{28} {\mathrm{s}}^{-1}$ et ${\omega }_3^2=1.20\times {10}^{29} {\mathrm{s}}^{-1}$.

These values are consistent with the general order of magnitude (${10}^{14}$ s⁻¹) typically reported for the oscillation frequencies in lithium niobate, as described by Safaryan [19]. Among these, the lowest frequency mode, ${\omega }_1^2$, corresponds to the soft mode associated with the ferroelectric phase transition.

For non-stoichiometric or doped lithium niobate, the corresponding soft-mode frequency can be written analogously as:

$${\omega \ast }_1^2=2\sqrt{{\displaystyle \frac{-p\ast }{3}}}\cos \left({\displaystyle \frac{\theta \ast }{3}}\right)-{\displaystyle \frac{B*}{3}}$$

(10)

Since the phase transition temperature is proportional to the square of the soft-mode frequency, the Curie temperature of non-stoichiometric or doped lithium niobate can be expressed as:

$$T_C^{\ast }={\displaystyle \frac{{\omega \ast }_1^2}{{\omega }_1^2}}{T}_C={\displaystyle \frac{2\sqrt{\frac{-p\ast }{3}}\cos \left(\frac{\theta \ast }{3}\right)-\frac{B\ast }{3}}{2\sqrt{\frac{-p}{3}}\cos \left(\frac{\theta }{3}\right)-\frac{B}{3}}}{T}_C$$

(11)

This relationship enables the evaluation of $T_C^{\ast }$ for different defect models and doping configurations considered in this study.

Here, $T_C\approx 1210$ °C represents the Curie temperature of stoichiometric lithium niobate [14], while the parameters with asterisks refer to non-stoichiometric or doped materials, for which the presence of dopants alters both charge distribution and lattice symmetry.

For titanium-doped lithium niobate (LiNbO₃:Ti), the ionic charge and mass parameters used in the calculations are: ${\mathrm{q}}_{\mathrm{Nb}}=+5, {\mathrm{q}}_{\mathrm{Li}}=+1, {\mathrm{q}}_{\mathrm{O}}=-2, {\mathrm{q}}_{\mathrm{Ti}}=+4,$ and ${\mathrm{M}}_{\mathrm{Nb}}=92.9 \mathrm{a}.\mathrm{u}., {\mathrm{M}}_{\mathrm{Li}}=6.94 \mathrm{a}.\mathrm{u}., {\mathrm{M}}_{\mathrm{O}}=48.0 \mathrm{a}.\mathrm{u}., {\mathrm{M}}_{\mathrm{Ti}}=47.86 \mathrm{a}.\mathrm{u}.$

The interplanar separations between the first and second nearest-neighbor planes at $\mathrm{T}=0 \mathrm{K}$ for Li, Nb, and O atoms are taken from reference [19].

The first-nearest-neighbor distances are: ${\mathrm{R}}_{\mathrm{Nb}–\mathrm{O}}=0.883 \mathrm{\AA }, {\mathrm{R}}_{\mathrm{Li}–\mathrm{O}}=0.680 \mathrm{\AA }, {\mathrm{R}}_{\mathrm{Li}–\mathrm{Nb}}=0.747 \mathrm{\AA }$, while the second-nearest-neighbor distances are: ${\mathrm{R}}_{\mathrm{Nb}–\mathrm{O}}^{\prime }\approx 1.766 \mathrm{\AA }, {\mathrm{R}}_{\mathrm{Li}–\mathrm{O}}^{\prime }\approx 1.360 \mathrm{\AA }, {\mathrm{R}}_{\mathrm{Li}–\mathrm{Nb}}^{\prime }\approx 1.494 \mathrm{\AA }$ [25].

2.3. Estimation of the Inverse Quality Factor, Electrical Conductivity, and Equivalent Mechanical Strength

2.3.1. Estimation of Electrical Conductivity

The electrical conductivity of LiNbO₃ is known to be temperature-dependent and follows the classical Arrhenius law [26]:

$$\sigma (T)={\sigma }_0{e}^{-{\displaystyle \frac{E_a}{K_{B}T}}}$$

(12)

where σ₀ is the pre-exponential factor, a characteristic constant specific to each LN sample, E_a is the activation energy, i.e., the energy required for an electron to move from one band to another or overcome an energy barrier, and K_B is Boltzmann’s constant (K_B = 8. 617 × 10⁻⁵ eV/K).

In doped or non-stoichiometric LiNbO₃, the introduction of point defects such as vacancies, antisite ions, or substitutional impurities modifies the local potential landscape and therefore alters the thermal activation of conductivity. To incorporate these effects, we introduce an effective temperature $T^{*}$, which accounts for the influence of lattice distortion and defect-induced energy perturbations. The modified expression for the conductivity then becomes:

$${\displaystyle \frac{{\sigma }^{*}\left(T^{*}\right)}{\sigma \left(T\right)}}=e^{-{\displaystyle \frac{E_a}{k_B{T}^{*}}}} e^{{\displaystyle \frac{E_a}{k_{B}T}}}, \mathrm{or} \mathrm{equivalently},{ \sigma }^{*}(T^{*})={\sigma }_0 e^{-{\displaystyle \frac{E_a}{k_B{T}^{*}}}}$$

(13)

Here, ${\sigma }^{*}\left(T^{*}\right)$ represents the effective electrical conductivity of doped or non-stoichiometric LiNbO₃, while $\sigma \left(T\right)$ corresponds to that of stoichiometric LiNbO₃. The distinction between $T$ and $T^{*}$ allows the model to reflect the structural disorder and modifications in the charge transport mechanisms introduced by impurities or deviations from stoichiometry.

For practical calculations, the modified expression can be rewritten as a function of the dopant concentration and the degree of non-stoichiometry:

$${\sigma }^{*}(T)={\sigma }_0 e^{-{\displaystyle \frac{E_a}{k_{B}f(x,y)T}}}$$

(14)

where $x$ and $y$ denote, respectively, the non-stoichiometric composition and the dopant concentration (in molar fraction). The function $f(x,y)$ quantifies the influence of these two parameters on the lattice dynamics and on the effective charge mobility.

Based on the vacancy model developed for titanium-doped lithium niobate, the general expression of $f(x,y)$ is given by:

$$f\left(x,y\right)={\displaystyle \frac{2\sqrt{\frac{-p*}{3}}\cos \left(\frac{\theta *}{3}\right)-\frac{B*}{3}}{2\sqrt{\frac{-p}{3}}\cos \left(\frac{\theta }{3}\right)-\frac{B}{3}}}$$

(15)

This relation establishes a direct connection between the defect structure (via $x$ and $y$) and the electrical conductivity, enabling a quantitative analysis of how dopant type and concentration affect charge transport in LiNbO₃.

2.3.2. Titanium-Doping Model

In this study, the incorporation of titanium (Ti) into the lithium niobate (LiNbO₃) lattice is described theoretically using the vacancy (defect) model, which captures both the substitutional behavior of dopant ions and the associated charge-compensation mechanisms. The chemical composition of Ti-doped lithium niobate can be expressed as:

$$[{\mathrm{Li}}_{({\mathrm{Li}}_{1-5x+y})}{\mathrm{Nb}}_{({\mathrm{Li}}_x)}{\square }_{({\mathrm{Li}}_{4x-y})}][{\mathrm{Nb}}_{({\mathrm{Nb}}_{1-y})}{\mathrm{Ti}}_{({\mathrm{Nb}}_y)}][{\mathrm{O}}_3]$$

where ${\square }_{\mathrm{Li}}$ denotes a lithium vacancy.

According to this model, Ti⁴⁺ ions are primarily incorporated on Nb⁵⁺ lattice sites (${\mathrm{Ti}}_{\mathrm{Nb}}$), due to their similar ionic radii and valence states. However, a small fraction of Ti⁴⁺ ions may also occupy lithium sites (${\mathrm{Ti}}_{\mathrm{Li}}$), particularly in non-stoichiometric LiNbO₃, where niobium antisite defects (${\mathrm{Nb}}_{\mathrm{Li}}$) are already present. To maintain charge neutrality, these substitutions are compensated by the creation of lithium vacancies (${\square }_{\mathrm{Li}}$).

It is important to note that, in the present work, Ti doping is not introduced experimentally. Instead, it is modeled theoretically through its effect on the defect structure, charge distribution, and resulting electrical conductivity of the LiNbO₃ crystal lattice.

Within this theoretical framework, the coefficients $A$, $B$, and $D$ in the vibrational model are replaced by modified expressions that reflect the changes in ionic mass and charge distributions arising from both non-stoichiometry (x) and titanium concentration (y). The corresponding quantities are defined as follows:

$$\begin{gathered} {M_0}^{*}=M_0 \mathrm{and} {q_0}^{*}=q_0 \\ {M_1}^{*}={\left(1-y\right)M}_1+y{M}_{Ti} \mathrm{and} {q_1}^{*}={\left(1-y\right)q}_1+y{q}_{Ti} \\ {M_2}^{*}=\left(1-5x+y\right)M_2+x{M}_1 \mathrm{and} {q_2}^{*}=\left(1-5x+y\right)q_2+x{q}_1 \end{gathered}$$

(16)

These modified parameters play a key role in determining the electrical conductivity and ferroelectric behavior of LiNbO₃ as functions of both defect chemistry ($x$) and dopant concentration ($y$).

The non-stoichiometry parameter $x$, which quantifies the concentration of niobium antisite defects (${\mathrm{Nb}}_{\mathrm{Li}}$), can be expressed as a function of the Ti doping level using the following relationship:

$$x={\displaystyle \frac{1-r}{r+5}}+{\displaystyle \frac{1+r}{r+5}}\hspace{0.17em}y$$

(17)

where $r$ denotes the Li/Nb molar ratio in the initial composition. Equation (15) indicates that the concentration of ${\mathrm{N}\mathrm{b}}_{\mathrm{L}\mathrm{i}}$ antisite defects in titanium-doped LiNbO₃ increases with the Ti content, revealing a direct correlation between dopant concentration and defect formation. The defect dynamics in LiNbO₃ are therefore not limited to a simple vacancy compensation mechanism but also involve the creation and redistribution of point defects depending on the dopant type and concentration.

2.3.3. Modeling of the Inverse Quality Factor

In this work, we investigate the inverse quality factor (Q⁻¹) of lithium niobate (LiNbO₃) in order to elucidate the influence of intrinsic structural defects and dopant-induced modifications on acoustic energy losses. Understanding these mechanisms is essential for optimizing the performance of LiNbO₃-based materials, especially when used as substrates for surface acoustic wave (SAW) resonators and interdigitated transducers (IDTs) (Figure 1).

The inverse quality factor quantifies the dissipation of acoustic energy and reflects the combined effects of electromechanical coupling, charge transport, and thermally activated relaxation. Its modeling provides a framework to link microscopic defect structures and doping effects to macroscopic performance indicators such as the Q factor.

To describe these loss mechanisms, the inverse quality factor can be expressed by a phenomenological relation that incorporates both electroacoustic relaxation and thermal activation processes. The general form is given by [27]:

$$Q^{-1}={\displaystyle \frac{K^2\omega {\tau }_C}{1+{\omega }^2{{\tau }_c}^2}}+{\Delta }_T{e}^{-\frac{E_C}{K_{B}T}}+C_0$$

(18)

In this equation, $K^2$ is the electromechanical coupling coefficient, which depends on the piezoelectric properties of the material. The angular frequency ω is defined as $\omega =2\pi f$, where f is the operating frequency. The relaxation time τc is given by ${\tau }_C={\displaystyle \frac{{\epsilon }_{ij}}{\sigma }}$, with ${\epsilon }_{ij}$ representing the dielectric permittivity and $\sigma$ the electrical conductivity. The second term of the equation accounts for thermally activated loss mechanisms, with $\Delta T$ being a fitting parameter and $E_C$ the activation energy. k_B is the Boltzmann constant, and T is the absolute temperature. The constant term $C_0$ represents residual losses, independent of both temperature and frequency.

For doped or non-stoichiometric materials, this model can be extended to include the effect of doping on the effective temperature. In this case, the temperature T is replaced by a modified temperature T*, expressed as a function of the dopant type and concentration: $T^{*}=f(x,y)\cdot T$. The modified inverse quality factor is then given by:

$$Q^{-1*}={\displaystyle \frac{K^2\omega {{\tau }^{*}}_C}{1+{\omega }^2{{{\tau }^{*}}_c}^2}}+{\Delta }_T{e}^{-{\displaystyle \frac{E_C}{K_{B}T*}}}+C_0$$

(19)

The modified relaxation time ${{\tau }^{*}}_C$ is calculated using ${\tau }_C={\displaystyle \frac{{\epsilon }_{ij}}{\sigma }}$, where the electrical conductivity.

Although Equation (18) was originally derived as an approximation valid for frequencies in the MHz range, it remains physically relevant and applicable to SAW devices that typically operate between hundreds of MHz and several GHz. This is because the parameters it contains, namely the electromechanical coupling coefficient ($K^2$), the dielectric constant (${\epsilon }_{11}$), and the elastic stiffness constant ($C_{44}$), are intrinsic material properties, independent of the specific acoustic propagation mode.

These quantities govern the velocity of acoustic wave propagation, the strength of piezoelectric coupling, and the extent of energy dissipation within the material. As such, the same physical principles that determine the Q factor in bulk acoustic wave systems apply analogously to surface modes in SAW devices. Consequently, Equation (18) can be used approximately to analyze the dependence of Q⁻¹ on temperature, frequency, and conductivity for LiNbO₃-based substrates, including those doped with Ti or other impurities.

This approach therefore provides a consistent and unified framework for understanding acoustic loss mechanisms and performance optimization in doped lithium niobate, across both bulk and surface acoustic applications.

2.3.4. Equivalent Mechanical Resistance

In surface acoustic wave (SAW) devices, the equivalent mechanical resistance (R_m) is a key parameter that quantifies the mechanical energy losses within the system. It is inversely proportional to the mechanical quality factor (Q), which is highly sensitive to the intrinsic properties of the piezoelectric material, temperature variations, and the presence of structural defects induced by doping.

The analytical expression of R_m as a function of Q is given by [28]:

$$R_m={\displaystyle \frac{1}{{\omega }_s{C}_0^{\prime }{K}_t^{2}Q}}={\displaystyle \frac{Q^{-1}}{{\omega }_s{C}_0^{\prime }{K}_t^2}}$$

(20)

where ${\omega }_s=2\pi f_s$ is the angular resonance frequency, $C_0^{\prime }$ represents the static capacitance of the interdigital transducer (IDT), $K_t^2$ is the electromechanical coupling coefficient of the material, and $Q^{-1}$ denotes the inverse of the mechanical quality factor.

In this study, we analyzed a SAW resonator based on titanium-doped lithium niobate (LiNbO₃:Ti⁴⁺) to investigate the evolution of R_m as a function of both dopant concentration and temperature. For the calculations, the following parameters were used ${\omega }_s=3.5 \mathrm{MHz},C_0^{\prime }=35.8 \mathrm{fF}$ [29], $K_t^2=0.57$.

3. Results and Discussion

3.1. Curie Temperature of Titanium-Doped Niobate

In this section, we analyze the evolution of the Curie temperature (T_C) as a function of titanium (Ti) concentration in lithium niobate (LiNbO₃:Ti), using two theoretical approaches: the Safaryan model (Th1) and our extended model (Th2) [30], which includes electrostatic interactions with second-nearest neighbors. The calculations are based on Equation (11) and the vacancy model adapted to describe charge-compensation mechanisms in Ti-doped LiNbO₃.

Figure 4a illustrates the variation in the Curie temperature with Ti molar concentration. A gradual decrease in T_C is observed as the titanium content increases. This behavior can be attributed to the substitution of Ti⁴⁺ ions for Nb⁵⁺ ions in the crystal lattice. Such substitution promotes the formation of Nb_Li antisite defects, which disrupt the balance of intrinsic lattice defects and enhance structural disorder (Figure 4b). This increased disorder weakens the stability of the ferroelectric phase, reduces long-range dipolar interactions, and consequently leads to a lower Curie temperature. Table 1 compares the experimental data with the theoretical predictions obtained from both models (Th1 and Th2), along with the corresponding deviations (ΔT_C). The results derived from the Th2 model exhibit better agreement with the experimental values, underscoring the importance of incorporating second-neighbor interactions in accurately describing the ferroelectric behavior of Ti-doped LiNbO₃. This improvement demonstrates that accounting for extended electrostatic coupling and charge redistribution within the lattice provides a more realistic representation of the soft-mode dynamics governing the ferroelectric transition.

Figure 4. (a): comparison between the theoretical evolution obtained by Th1 and Th2 and the experimental Curie temperature [31] of titanium-doped lithium niobate, and (b): concentrations of Nb_Li (x) in titanium-doped LiNbO₃ as a function of the concentration of Ti⁴⁺ (y) obtained from Equation (17).

Table 1. Comparisons between theoretical and experimental [31] results for the Curie temperature of titanium-doped niobate.

Ti%	${\mathit{T}}_{\mathit{c}}^{\mathbf{T}\mathbf{h}1} \mathbf{(}°\mathbf{C}\mathbf{)}$	${\mathit{T}}_{\mathit{c}}^{\mathbf{T}\mathbf{h}2} \mathbf{(}°\mathbf{C}\mathbf{)}$	${\mathit{T}}_{\mathit{c}}^{\mathit{E}\mathit{x}\mathit{p}} \mathbf{(}°\mathbf{C}\mathbf{)}$	${\mathsf{\Delta }\mathit{T}}_{\mathit{c}}^{\mathbf{T}\mathbf{h}1} \mathbf{(}°\mathbf{C}\mathbf{)}$	${\mathsf{\Delta }\mathit{T}}_{\mathit{c}}^{\mathbf{T}\mathbf{h}2} \mathbf{(}°\mathbf{C}\mathbf{)}$
0	1143.7	1141.53	1138.47	5.23	3.06
1	1110.07	1119.98	1122.54	−12.47	−2.56
2	1077.98	1099.93	1114.34	−36.36	−14.41
3	1047.32	1081.23	1100.68	−53.36	−19.45
4	1017.97	1063.76	1089.29	−71.32	−25.53

3.2. Inverse of the Quality Factor and Mechanical Strength of Lithium Niobate of Titanium-Doped Lithium Niobate

In this section, we analyze the variation in the inverse acoustic quality factor (Q⁻¹) as a function of inverse temperature for two types of lithium niobate (LiNbO₃): a stoichiometric sample with a molar ratio of r = [Li]/[Nb] = 0.99, and a congruent sample doped with titanium (LiNbO₃:Ti).

To validate our theoretical model, the calculated values of Q⁻¹ were compared with the experimental data reported by Éva Tichy-Rács et al. [27], using the material parameters listed in Table 2. The comparison for both samples is illustrated in Figure 5a.

Table 2. Adjustment parameters obtained from Q⁻¹ (T) adjustments of c.LN and s.LN samples [27].

	c.LN	s.LN
$\mathit{f}$	3.5 MHz	3.5 MHz
${\mathit{E}}_{\mathit{C}}$	(1.45 ± 0.07) eV	(1.53 ± 0.07) eV
$\mathsf{\Delta }\mathit{T}$	(8.90 ± 0.05)·10−3	(5.66 ± 0.05)·10−3
${\mathit{K}}^2$	0.57	0.57
${\mathit{E}}_{\mathit{T}}$	(0.10 ± 0.04) eV	(0.14 ± 0.04) eV
${\mathit{C}}_{\mathbf{0}}$	(1.2 ± 0.1)·10−3	(1.8 ± 0.1)·10−3
${\mathit{\sigma }}_{\mathbf{0}}$	(5.75 ± 0.07)·105 SKm−1	(1.16 ± 0.08)·106 SKm−1

Figure 5. (a) Inverse quality factor of pure lithium niobate as a function of inverse temperature; (b) Q⁻¹ of Ti⁴⁺-doped lithium niobate at a temperature of 200 °C and f = 3.5 MHz.

The variation in Q⁻¹ for stoichiometric LiNbO₃ (s-LN) and Ti⁴⁺-doped congruent LiNbO₃, measured at a frequency of 3.5 MHz, is presented in Figure 5a and Figure 5b, respectively. For the stoichiometric crystal, Q⁻¹ decreases with increasing inverse temperature, indicating that acoustic losses become more pronounced at higher temperatures. This trend is consistent with experimental observations and validates the thermal activation behavior predicted by the Arrhenius-type model. The theoretical predictions show good agreement with previously reported experimental data [27].

Furthermore, titanium doping induces a noticeable reduction in the inverse quality factor Q⁻¹ as shown in Table 3, corresponding to an increase in the overall quality factor (Q). This improvement suggests that Ti⁴⁺ incorporation effectively mitigates loss mechanisms, possibly by reducing defect-induced scattering and internal friction. Such behavior is particularly advantageous for surface acoustic wave (SAW) devices employing interdigitated transducers (IDTs), where LiNbO₃ serves as a substrate. In these devices, maximizing the quality factor is essential to minimize energy dissipation and enhance frequency stability and sensitivity. Therefore, Ti-doped LiNbO₃ can be considered a promising substrate material, particularly for low-temperature or high-frequency SAW applications. The results also provide valuable insights for the optimization of doped LiNbO₃, emphasizing the balance between conductivity, defect concentration, and acoustic losses required for advanced piezoelectric and electronic components.

Table 3. Theoretical values of the inverse quality factor Q⁻¹ for congruent lithium niobate (c-LN) and titanium-doped lithium niobate (LN:Ti).

c.LN (r = 0.9372)		LN:Ti
104/T(K−1)	Q−1	Concentration of Ti4+ (mol%)	Q−1 at 200 °C
9.43	0.005411	0	0.00162896
10	0.004368	1	0.00159652
12	0.003277	2	0.00156617
14	0.002813	3	0.00153781
16	0.002463	4	0.0015113
18	0.002190	5	0.00148654
20	0.001975	6	0.00146342

The data presented in Figure 6 reveal a clear and gradual decrease in the equivalent mechanical resistance $R_m$ with increasing titanium concentration in lithium niobate (LiNbO₃) at 200 °C. For the undoped sample (0% Ti⁴⁺), $R_m$ is measured at 10.04 Ω, whereas at a titanium concentration of 5% Ti⁴⁺, it decreases to 8.76 Ω. This downward trend indicates that the incorporation of titanium effectively reduces mechanical losses within the crystal. The reduction in $R_m$ reflects an enhancement of piezoelectric efficiency, likely due to the mitigation of internal friction and the suppression of dissipative mechanisms associated with lattice imperfections. In other words, Ti⁴⁺ doping contributes to a more coherent and energetically stable crystal structure, which favors the efficient propagation of acoustic waves. Therefore, titanium incorporation in LiNbO₃ can be regarded as beneficial for improving its acoustic performance, making Ti-doped lithium niobate a promising substrate for surface acoustic wave (SAW) devices where low energy dissipation and high signal stability are essential.

Figure 6. Mechanical R_m as a function of titanium dopant concentration at 200 °C.

4. Conclusions

In this work, we developed a theoretical framework based on ferroelectric phase transition theory to investigate the electrical and acoustic behavior of pure and titanium-doped lithium niobate (LiNbO₃:Ti). Our results reveal that the Curie temperature (Tc) decreases progressively with increasing titanium concentration. This trend is attributed to the structural disorder induced by Ti⁴⁺ substitution at Nb⁵⁺ sites, which promotes the formation of Nb_Li antisite defects and modifies the electrostatic balance within the lattice. The proposed model shows excellent agreement with experimental data reported in the literature, validating its predictive reliability for both stoichiometric and doped LiNbO₃. Moreover, titanium doping was found to reduce the inverse acoustic quality factor (Q⁻¹) and the equivalent mechanical resistance (R_m), indicating a decrease in energy dissipation and an enhancement in acoustic efficiency. Overall, these findings underline the promising potential of Ti-doped LiNbO₃ as a robust and tunable substrate for surface acoustic wave (SAW) devices. By combining improved thermal stability with enhanced electromechanical performance, Ti incorporation emerges as an effective route to optimize lithium niobate for next-generation SAW resonators and other high-frequency piezoelectric applications.