Introduction

Lithium tantalate (LiTaO3, LT) and lithium niobate (LiNbO3, LN) are two of contemporary materials science's most researched oxide compounds. The various applications of these materials in functional electronics have generated curiosity about their utilization in thin films, micro- and nanopowders, single or polycrystalline crystals, and ceramics. Additionally, they may differ in their chemical composition, be congruent or stoichiometric, and include different dopants. Such a wide range of types and applications necessitates extra study to adjust and improve material characteristics [1, 2].

\(Li{Nb}_{1-x}{Ta}_{x}{O}_{3}\) (LNT) solid solutions have lately received attention, and they present opportunities for combining the benefits of the two materials. According to the literature, they form solid solutions over the whole composition range. In this way, the physical properties of materials, particularly ferroelectric and nonlinear optical properties, can be tuned as a function of composition. For instance, tuning the birefringence and refractive index as a function of the Ta/Nb ratio. A previous study showed zero birefringence in a composition range of 0.93 ≤ \(x\)  ≤ 0.94 at 293 K [1, 2]. Also, an optically isotropic crystal (a crystal with zero birefringence) maintains the unique benefits of ferroelectrics, including substantial electro-optic and nonlinear optical coefficients [3]. The birefringence of LNT mixed crystals was tested by Wood et al. for various compositions and temperatures, and they discovered that it varied consistently up to the ferroelectric Curie temperature (Tc). In particular, it is possible to anticipate that the LNT solid solutions will display strong piezoelectric coefficients (comparable to LN) while at the same time exhibiting LT features that are stable at high temperatures [2, 4, 5].

The solid solutions of LNT may serve as the foundation for ceramic materials with ferroelectric, superionic, semiconductor, and even mixed properties. As a result, this quasi-binary material system is extensively investigated using various techniques [6]. The LNT solid solutions are also appealing as a model system for how composition affects the structural properties of mixed oxides.

LN and LT endmembers are isostructural, have comparable lattice and atomic positional parameters, and have similar ionic radii and valence states for Ta and Nb. However, the separation between solidus and liquidus lines in the LN-LT phase diagram is large [7, 8]. This makes the growth of homogeneous single crystals of LNT solid solution difficult. The melt around the crystal becomes increasingly LN-rich as LN is repeatedly rejected from the solid–liquid interface during crystal growth. This causes compositional inhomogeneity in the crystal along the growth direction, e.g., using the Czochralski (CZ) method [2, 3, 9,10,11]. In contrast, crystals with uniform Ta and Nb composition have been grown using an edge-defined film-fed growth method [11]. Still, these crystals were not high-quality, showing bubbles, microcracks, and sub-grain boundaries.

Several studies have reported that LNT's Ta/Nb ratio affects structural characteristics such as lattice parameters and phase transition temperature [2, 7,8,9,10]. Wood et al. reported absolute values of the birefringence of LNT solutions [2], and Shimura et al. [3] published the refractive indices of Ta-rich LNT crystals. Yun-Cheng Ge et al. studied ferroelectric phase transition using Raman analysis [12]. Bartasyte et al. [1] studied the effect of Ta/Nb compositional variation on lattice parameters. Roshchupkin et al. [5] studied the crystal for sensor and actuator applications. In our other work, we observed the variation of Tc with Ta/Nb covering the whole composition range [13].

Despite its importance, the LN-LT phase diagram has not been studied in detail. The first study on the phase diagram was conducted by Peterson et al. [7, 8]. Crystals were produced by heating the powder mixtures of LNT in a platinum/iridium crucible and allowed to crystallize by cooling. Their study was carried out on LN/LT congruent crystals, where the liquidus and solidus meet at the ends of LN and LT in the phase diagram. However, in their subsequent study [8], they used stoichiometric LN and LT and observed a different behavior of the phase diagram where the liquidus and solidus do not meet at the end members of LN and LT. This is obvious because the stoichiometric and congruent melting compositions do not coincide; hence, the connection between the stoichiometric compositions rather represents an isopleth section through the Li2O–Nb2O5–Ta2O5 ternary system. Therefore, it is important to re-investigate the phase diagram of the LN-LT system over the whole composition ranges (\(x\hspace{0.17em}=\hspace{0.17em}0 to x\hspace{0.17em}=\hspace{0.17em}1\)) between the congruent compositions, to obtain a truly quasi-binary phase diagram. We are particularly interested in the mixed crystal's growth, miscibility, and solidus–liquidus separation. In this work, we will also construct the phase diagram based on a thermodynamic model in which we consider the excess enthalpies of mixing.

Furthermore, LNT single crystal specimens used in this study were taken from large volume crystals measuring 5 cm in length and 1.5 cm in diameter. The previous LNT single crystals grown by Bartasyte et al. [3] had a length of 1.5 cm, and Roshchupkin et al. [7] grew crystals measuring 2 cm in length and diameter.

Experiment

Single crystals were grown from LNT melts using the Czochralski process with induction heating. A cylindrical crucible made from platinum (for niobium-rich compositions with low melting temperature) or iridium (for moderate to high tantalum contents) measuring 60 mm in height and 60 mm in diameter was employed. The latter process was conducted in a protective atmosphere, namely argon with a small (< 1 vol%) addition of oxygen, otherwise in air. Starting materials were mixtures of the congruently melting LN, respectively LT, prepared from lithium carbonate (Li2CO3, Alfa Aesar, 5N), niobium pentoxide (Nb2O5, H.C. Stark, 4N5), and tantalum pentoxide (Ta2O5, Fox Chemicals, 4N), all dried before use. The congruent compositions of LN and LT are at 48.38% Li2O [14] and 48.46% Li2O [12], respectively. We took the average 48.4 mol% Li2O for all compositions. All crystals were grown along the c-axis using LN, LT or LNT seeds, respectively, at 0.5 mm/h pulling rate for all compositions.

Energy dispersive micro X-ray fluorescence (μ-XRF) measurements determined the elemental (Nb, Ta) distribution in the grown single crystals. The measurements were carried out with a Bruker M4 TORNADO spectrometer at a low-pressure environment (20 mbar). The X-ray source was a tube with Rh anode working at 50 kV and 200 μA. Bremsstrahlung was focused at the sample surface using polycapillary X-ray optics, yielding a high spatial resolution (beam width) of about 20 μm, and the measurement time per pixel was 20 ms. The fluorescence signal was detected with a circular silicon drift detector. Quantification was done by the built-in routines of the spectrometer using the fundamental parameters database. The accuracy of the XRF analysis was confirmed on samples that were additionally analyzed by ICP-OES (Inductively Coupled Plasma Optical Emission Spectroscopy). The deviation between XRF and ICP results was less than 1 at%.

Differential Thermal Analysis (DTA) was used to determine the solidus and liquidus temperatures. Samples weighing typically a few milligrams were extracted from Czochralski-grown LNT crystals for DTA analysis. These samples were placed in the DTA apparatus, where both the sample and the reference material were simultaneously heated at a controlled rate of 10 K/min. The DTA experiment recorded the temperature difference between the sample and the reference material as the temperature increased. The onset temperature of the endothermic peak during heating corresponds to the solidus temperature, marking the melting event. The temperature at which the peak ends is taken as the liquidus temperature. These temperatures were analyzed for various crystal compositions to map the solidus and liquidus lines on the phase diagram. This method facilitated the construction of a precise phase diagram, illustrating the stability regions of different phases and their transitions.

In this study, a NETZSCH STA 449C “Jupiter” thermal analyzer was utilized to perform simultaneous thermogravimetry (TG) and differential thermal analysis (DTA). The DTA/TG sample holder, equipped with platinum crucibles, Pt/Pt90Rh10 thermocouples, and a flowing mixture of 20 ml/min Ar and 20 ml/min O2, enabled readings up to 1923 K. Figures 2a and 2b depict the size of crystals used for the DTA measurements.

For the thermodynamic assessment of the phase diagram, FactSage 8/8.2 was used. The JANAF [15,16,17] table does not contain the data for LN and LT end members. Thus, the thermodynamic data of LN and LT was calculated from the Fact-PS database from the Li2O + Nb2O5 or (Ta2O5) data, and both compounds were modified using measured Cp data [18, 19].

Results and discussion

Figure 1 shows the DTA results for LN and LT end members and LNT solid solutions of different compositions. The onset of DTA peaks corresponds to the melting point, and this temperature will be taken as a solidus in optimizing the phase diagram. The temperature at which the DTA curve returns to the baseline after the melting represents the liquidus in the phase diagram. The DTA measurements were taken twice; the first heating corresponds to the heating of the as-grown crystal. After the first heating, the crystal melts and recrystallizes upon cooling. The second heating is carried out on the recrystallized sample. As can be seen from Fig. 1a, the onset is sharp for end members (\(x\hspace{0.17em}=\hspace{0.17em}1\)) in both first and second heating. The liquidus temperature is not visible for pure LT in the DTA curve. This is because of the temperature limit of the experimental equipment. Therefore, for pure LT, the melting temperature is taken from the literature [13]. Moreover, the end members do not differ in melting and solidification temperatures for the first and second heating. This is expected because there is no component segregation in the end members.

Figure 1
figure 1

a DTA plots of LN, LT, and LNT solid solutions with first heating (blue curve) and second heating (black curve). The legend on the right side of the y-axis shows the molar ratio. b DTA plots of the samples represented by b (specimen 1, \(x\) = 0.52–0.60) show no clear onset but prolonged melting in both heating cycles. DTA plots corresponding to b (specimen 2, \(x\) = 0.48) show clear melting temperature in the first heating but a prolonged melting in the second heating.

Full size image

Figure 1a shows that the melting temperature of the solid solution increases with increasing LT fraction. This is the expected behavior since the melting point of LT is 400 K higher than that of LN. However, the width of the melting peaks and the separation between solidus and liquidus temperatures vary with composition. For solid solutions, melting extends over a significantly larger temperature range than in the case of the end members, with a maximum for compositions around \(x \hspace{0.17em}=\hspace{0.17em} 0.5\). The separation between solidus and liquidus decreases as the solid solution approaches one of the end members, i.e., for Nb or Ta-rich solutions. In the second heating, melting is even more prolonged in solid solutions. This is attributed to segregation that occurs during solidification, leading to more inhomogeneous specimens containing LN and LT-rich regions. As a result, in the second heating, Nb-rich regions will start melting at lower temperatures and slowly dissolve the remaining solids. This tendency is observed for the second heating of all compositions of solid solutions. For \(x \hspace{0.17em}=\hspace{0.17em} 0.94\), the second heating shows melting at a decreased temperature (1876 K) compared to the first heating (1890 K). The liquidus temperature for this composition cannot be clearly identified as shown in the DTA curve. The heat absorbed in the endothermic process of melting is a function of composition and mass of the sample. Therefore, the peak area is different for different compositions.

Figure 1b shows DTA curves for two selected samples with comparable compositions. It is important to note that the first heating of sample 1 shows highly inhomogeneous regions and prolonged melting. The Ta and Nb elemental agglomeration are clearly visible, as shown exemplarily in µXRF maps in Fig. 2b. Although the composition of the sample in Fig. 2b varies from (\(x\) = 0.20–0.60), the major part of the crystal is dominated by Ta-rich composition. However, the small Nb-rich segregated regions cause prolonged melting. On the other hand, the first heating of sample 2 in Fig. 1b shows a sharp melting at nearly the same temperature as sample 1 (red dotted line in Fig. 1b). The reason for this sharp melting is that the sample is less inhomogeneous, comparable to the sample depicted in Fig. 2a. Here, the compositional variation is (\(x\) = 0.46–0.51) with no pronounced agglomeration as seen in Fig. 2b. In the second heating, the onset temperature could not be identified. Since the composition of the two samples is similar, they show similar inhomogeneity after the first melting and resolidification, as can be seen from prolonged melting behavior in both samples (black curves). However, during subsequent heating, the partial pre-melting caused by segregation leads to different DTA curves. Therefore, for the optimization of phase diagram, it is necessary to take the solidus and liquidus temperatures determined from the first heating. Table 1 lists the solidus and liquidus temperatures of homogeneous crystals as represented by Fig. 1a. The solid solutions show some variation of solidus and liquidus temperatures which are indicated by error ranges. The error was estimated with respect to the variation in composition in the samples. It is important to note here that the specimens used in Fig. 1b are not the same as shown in Fig. 2. The µXRF maps as shown in Fig. 2 are representative images of comparable compositions used in Fig. 1b. Here, the important point is to define inhomogeneities in the crystal. As can be seen the gradient (∆x) of x in Fig. 2a is ∆x ≤ 5 (\(x\) = 0.46–0.51). On the other hand, for Fig. 2b this gradient is quite large ∆x = 40 (\(x\hspace{0.17em}\)= 0.20–0.60). For all other crystals grown at IKZ the composition gradient is shown in Table 1. It can be seen that for some samples, ∆x is less than 1. Therefore, crystals grown at IKZ show very high level of homogeneity as compared to some previous studies as referred earlier.

Figure 2
figure 2

XRF elemental mapping of LNT mixed crystals (z-cut). a Less segregated crystal (homogeneous). b Crystal with higher segregation between Ta and Nb (inhomogeneous).

Full size image
Table 1 Crystal composition and corresponding solidus and liquidus temperatures from Fig. 1a
Full size table

Thermodynamic model

Before proceeding with a solution model for the phase diagram calculation, the thermodynamic parameters for LN and LT end members were required in the database. This database was used to determine the thermodynamic parameters (specific heat capacity (Cp), standard enthalpy (∆H), and entropy (∆S)) of LiNbO3 and LiTaO3 based on the Neumann–Kopp (NK) rule from the Factsage mixer module [3]. As per the Neumann–Kopp rule, the molar heat capacity (Cp(T)) of a compound is calculated by adding, at a specific temperature, the molar heat capacities of its individual components, each multiplied by their respective quantities within the compound [20]. A private database was thus created in the Factsage “Compound module”, containing all relevant compounds and mixtures:

$$2LiNbO_{3} = Nb_{2} O_{5} + Li_{2} O$$
$$2LiTaO_{3} = Ta_{2} O_{5} + Li_{2} O$$

Table 2 lists the thermodynamic parameters of LN and LT obtained from the above process. To make the phases stable in the Li2O-(Nb,Ta)2O5 system, the initial values of Ho298 (∆HLN = -1080 kJ/mole, ∆HLT = -1140 kJ/mole) were corrected to final values listed in Table 2. The correction is necessary because the sole application of the NK rule disregards the formation enthalpy of the complex oxides LN and LT from the simple oxides Li2O, Nb2O5 and Ta2O5, respectively.

Table 2 Thermodynamic parameters of end members calculated from the database using Neumann-Kop rule
Full size table

Since the end members are lithium niobate and lithium tantalate, the following reaction is considered based on Hess's law [17]:

$$LiNb_{1 - x} Ta_{x} O_{3} = xLiTaO_{3} + \left( {1 - x} \right)LiNbO_{3}$$

The Gibbs excess energy Gexcess for LN-LT mixtures is assumed to be relevant only in the solid phase. This is because Nb and Ta possess very similar chemical properties, and the ionic radii of Nb5+ and Ta5+ in octahedral coordination are almost identical (78 pm). Indeed, it turned out that already the assessment of the solid mixture phase could reproduce the experimental results satisfactorily if the liquid phase was treated as ideal.

Many simple models exist for solid solutions of isostructural compounds. Although these models are for mineralogical systems the assumptions on which the models are based are similar to the ones developed in this study. For example, Wood et al. use the so-called Bragg-Williams model, which considers the substitution of Si by Al in Mg3Al2Si3O12 system [21, 22]. In this model complete mixing of end members is considered, and the charge balance is maintained during the cationic substitution. Therefore, this model is also applicable to our system. Another model is used by Ringwood et al. which considers the substitution of Fe+2 by Mg+2 in a Fe2SiO4—Mg2SiO4 system [23]. This model also considers complete cationic exchange while maintaining the charge balance. For several other binary solid solutions, Holland et al. [19] discuss seven models for various binary solid solutions. The model used in our study is based on the complete mixing of end members. Our assumptions include disregarding all octahedral distortion, keeping lithium oxide constant throughout, and ensuring complete correlation with Nb+5-Ta+5 occupancy sites to preserve charge balance. Given these considerations, we have chosen to use the Redlich–Kister-Muggianu [24] polynomial model, as it is the most suitable model for our binary solid solution [18].

Optimization of phase diagram

The experimental results of specific heat capacity are discussed in detail in our previous work [13]. A plot of specific heat capacities of LN and LT is shown in Fig. 3. The plots compare the specific heat capacity calculated from the Neumann–Kopp rule with experimental results and show they are in reasonable agreement. The agreement between calculated and experimental data is the first proof that the end members created from the mixer module are reliable and can be used to create a solid solution compound module. However, it is important to note that only the experimental results will be used during the optimization of phase diagram [24]. A polynomial expression (Eq. 1) was used to fit the data calculated from the NK rule:

Figure 3
figure 3

Specific heat capacity of a LiNbO3 and b LiTaO3 calculated from the thermodynamic database of elemental oxides (Neumann–Kopp rule) and experimental results of specific heat capacity. The ferroelectric transition of LiNbO3 lies outside the temperature measurement range.

Full size image
$${C}_{p}=a+b.T+C.{T}^{-2}+d.{T}^{2}+e.{T}^{-1}+f.\sqrt{T}$$
(1)

The CALPHAD method illustrates the Gibbs free energy as composition dependent on a polynomial expansion. Therefore, in Factsage, the Redlich–Kister-Muggianu model is used [25] for our binary system.

$${G}^{excess}=\left(1-x\right)x\sum_{J=0}^{n}Lj{\left(\left(1-x\right)x\right)}^{j}$$
(2)

Under constant temperature and pressure, the Redlich–Kister-Muggianu polynomial is used to model the excess molar Gibbs free energy in the context of solution thermodynamics, particularly in the study of non-ideal solutions. The polynomial is often employed to describe the composition dependence of Gibbs free energy in binary systems, Lj are the interaction coefficients determined through regression analysis based on experimental data, and j is the order of interaction. The Redlich–Kister-Muggianu polynomial provides a flexible means of representing the non-ideality in solution. The specific form of the Redlich–Kister-Muggianu polynomial may vary depending on the order chosen and the particular details of the system under consideration. Higher-order terms in the polynomial can capture more complex deviations from ideal behavior [26]. The excess enthalpy and entropy contributions, specific heat capacity of end members, heats of fusion of end members and DTA are all input parameters given to the model to proceed with calculations in Factsage. The heats of fusion of end members are given by DTA measurements of LN (103 kJ/mol at 1531 K) and LT (289 kJ/mol at 1913 K).

Figure 4. shows the phase diagram optimized in the FactSage’s “Phase Diagram” module. The shape of this pseudo-binary phase diagram is similar to the phase diagram determined by Peterson et al. [7]. Peterson’s phase diagram is based on crystal growth experiments; it shows a large separation between solidus and liquidus, indicating large segregation, and his work does not provide any thermodynamic modeling of the phase diagram.

Figure 4
figure 4

Optimized phase diagram based on experimental results of DTA (composition and corresponding solidus and liquidus temperature) as well as specific heat capacity and heats of fusion of end members (Blue curve). Crystal growth experimental data (black dots) and DTA results (red dots) also included the phase diagram.

Full size image

The solidus and liquidus temperatures were taken from the first heating of DTA experimental results. Data points with errors are shown with error bars in the phase diagram accounting for the fact that the solidus and liquidus temperatures are not sharp onsets in all cases. On the other hand, the errors in composition are hardly beyond 2.3% (Table 1), as the DTA samples were prepared from homogeneous areas of crystals with a well-known composition. In contrast, melt and solid compositions from crystal growth are considered to be located at the liquidus in the phase diagram and at its corresponding solid composition along an isotherm. The graph shows that the crystal growth solid compositions lie within the equilibrium region. This coincides with the well-known fact that the effective segregation during crystal growth is always closer to unity than the segregation in equilibrium. Note that crystal growth experiments comprised three compositions without DTA results. Here, the solid and liquid composition from XRF results and melt, respectively, were fitted to the solidus and liquidus lines on the phase diagram. The optimized phase diagram was then constructed by the FactSage “CALPHAD” module.

To construct the plot of Gibbs free energy for a LNT solid solution, we derive the standard-state enthalpy and entropy at any temperature T from their values at 298.15 K. The relationships are as follows:

$$H_{T}^{{0\left( {LT} \right)}} = H_{298.15}^{{0\left( {LT} \right)}} + \mathop \smallint \limits_{298.15}^{T} C_{{p\left( {LT} \right)}} dT\,{\text{and}}\,S_{T}^{{0\left( {LT} \right)}} = S_{298.15}^{{0\left( {LT} \right)}} + \mathop \smallint \limits_{298.15}^{T} C_{{p\left( {LT} \right)}} /TdT$$
$$H_{T}^{{0\left( {LN} \right)}} = H_{298.15}^{{0\left( {LN} \right)}} + \mathop \smallint \limits_{298.15}^{T} C_{{p\left( {LN} \right)}} dT\,{\text{and}}\,S_{T}^{{0\left( {LN} \right)}} = S_{298.15}^{{0\left( {LN} \right)}} + \mathop \smallint \limits_{298.15}^{T} C_{{p\left( {LN} \right)}} /TdT$$

where \({H}_{T}^{0(LT)}\) and \({H}_{T}^{0(LN)}\) are the standard-state enthalpies at temperature T for LT and LN, respectively, and \({S}_{T}^{0(LT)}\) and \({S}_{T}^{0(LN)}\) are the corresponding standard-state entropies.

The standard-state Gibbs free energy for LT and LN at temperature T is given by:

$$G_{T}^{{0\left( {LT} \right)}} = H_{T}^{{0\left( {LT} \right)}} - TS_{T}^{{0\left( {LT} \right)}} \,{\text{and}}\,G_{T}^{{0\left( {LN} \right)}} = H_{T}^{{0\left( {LN} \right)}} - TS_{T}^{{0\left( {LN} \right)}}$$

These expressions represent the standard-state chemical potentials at temperature T for LT and LN, respectively. The Gibbs free energy for the LNT solid solution is then expressed as:

$$G_{T}^{LNT} = xG_{T}^{{0\left( {LT} \right)}} + \left( {1 - x} \right)G_{T}^{{0\left( {LN} \right)}} + G^{mix} = xG_{T}^{{0\left( {LT} \right)}} + \left( {1 - x} \right)G_{T}^{{0\left( {LN} \right)}} + G^{ideal} + G^{excess}$$

where \({G}^{mix}\) is the Gibbs free energy of mixing, which can be decomposed into ideal and excess parts:

$${G}^{mix}= {G}^{ideal}+ {G}^{excess}$$
$$where\,\,G^{ideal} = - T\Delta S^{mix} and\,G^{excess} = \Delta H^{mix}$$

From the phase diagram, the values of standard enthalpies corresponding to the Lj coefficients in Eq. 2 were extracted. Utilizing these coefficients along with the Ta/Nb compositions in Eq. 2, a plot of \({G}^{excess}\) of the LNT solid solution was obtained, represented by the black curve in Fig. 5. Additionally, the plots of the \({G}^{mix}\) for temperatures corresponding to the melting temperatures of LT (blue curve) and LN (red curve) are also depicted in Fig. 5. For an ideal solution, \({G}^{excess}\) is zero. A negative \({G}^{excess}\) indicates that the interactions between end members in the real solution are more favorable than those predicted by ideal solution behavior. This suggests that there is an attractive interaction between end members, atoms or molecules, making the real solution more stable than an ideal one. In this case it is Nb and Ta ions. \({G}^{mix}\) indicates the overall thermodynamic favorability of mixing components. Negative values across all compositions imply stable and thermodynamically favorable mixing. The fact that \({G}^{excess}\) is less negative than \({G}^{mix}\) implies that while the real solution is more stable than an ideal solution, the total free energy change upon mixing is dominated by the entropy term at high temperatures. The excess Gibbs energy being less negative means that the deviation from ideality is small but still significant. Figure 5 corresponds to the sum of L0 (-104) J/mole, L1 (-303) J/mole and L2 (− 103) J/mole. While the shape of the curves is symmetric for all the three coefficients L0, L1 and L2, but L0 shows the dominant effect.

Figure 5
figure 5

A plot for \({G}^{excess}\) and \({G}^{mix}\) for LNT solid solution.

Full size image

The phase diagram is used to calculate the segregation coefficient (K). K is defined as the ratio of Ta composition in solid (Cs) to Ta composition in the melt (Cl).

$$K = Cs/C_{l}$$
(3)

From the phase diagram (Cs) composition corresponds to any point on the solidus. The corresponding melt composition can be determined by drawing an isotherm (horizontal tie line) on the liquidus (Cl) as shown in Fig. 4 (solid magenta line connecting solidus and liquidus). The vertical projection from the end points of the horizontal line on the x-axis corresponds to crystal and melt compositions (vertical dashed lines in Fig. 4). The results of K are plotted in Fig. 6. As defined by Eq. 3, the segregation coefficient approaches unity as the LNT composition approaches LT. For any composition, the Ta segregation coefficient is always higher than unity. Therefore, the melt must become Ta deficient as the crystal grows. The segregation behavior of silicon (Si) in silicon–germanium (Si–Ge) solid solution exhibits similarities to that of Ta in LNT solid solutions. Specifically, the segregation coefficient of Si tends to be highest in Si-deficient (Ge-rich) solutions and approaches unity in Si-rich solutions [27].

Figure 6
figure 6

LT Segregation coefficient as a function of Ta mole fraction in the melt.

Full size image

From the segregation coefficient, melt and solid composition during the solidification process can be determined using the Scheil Equation [28], assuming complete mixing in the melt [29]:

$$Cs = KVo\left( {1 - Vs} \right)^{K - 1}$$
(4)

where Cs is the solute concentration in the solid, Vs is the volume of a solidified fraction of the melt, and Vo is the volume of the liquid when growth commences. The Scheil plot in Fig. 7 for solidification of an LNT melt with x = 0.5 shows that as a consequence of the segregation, the initial Ta composition in the crystal is as high as \(x =\) 80 (red arrow), but decreases in the melt (slope of the blue curve) as well as in the solid (green curve) as solidification progresses. The liquidus temperature (\(x\) melt = 0.50, T = 1800 K) decreases to finally approach the growth temperature of pure LN (1523 K). Close to the latter point, LT is completely depleted in the melt, and thus, the very bottom of the crystal (when all melt is solidified) will also contain no Ta. The inevitable segregation effects must be considered and accounted for to grow crystals useful for application purposes.

Figure 7
figure 7

Scheil cooling of LNT solid solution with starting composition of x = 0.5. Blue curve: Ta composition decreases fast in the melt. Green curve: In the crystal, initial Ta content is higher, but also drops along growth axis given by the slope of the curve.

Full size image

Conclusion

The LNT solid solutions were investigated using Differential Thermal Analysis (DTA), proposing a detailed phase diagram based on a thermodynamic solution model. The proposed phase diagram shows complete miscibility over all compositions (\(x=0-1\)) and reveals a narrower separation between the solidus and liquidus lines compared to previously studied phase diagrams.

A plot of Gibbs's excess energy demonstrates a significant deviation from ideality. Additionally, the segregation coefficient of Ta, calculated from the phase diagram, is greater than unity for all compositions. This indicates that differences in composition upon solidification (e.g., Scheil cooling) must be considered to obtain useful and optimized crystals for the intended application.

Overall, these findings underscore the effectiveness of our parameterization approach and the reliability of the Gibbs excess energy parameters, as the optimized phase diagram aligns closely with experimental results without necessitating further adjustments for the liquid phase.