Comparative Studies on Electronic Structures and Surface Acoustic Wave Properties of ( Ca , Sr , Pb ) 3 Ga 2 Ge 3 O 14 Piezoelectric Crystals

The electronic structural and surface acoustic wave (SAW) properties of Ca3Ga2Ge3O14 (CGG), Sr3Ga2Ge3O14 (SGG) and Pb3Ga2Ge3O14 (PGG) crystals have been investigated. The first-principle calculations show that CGG and SGG have similar electronic structure properties, the remarkable difference between PGG and another two crystals is the absence of lower part of valence band. Both the calculated effective charges and bond populations show that there is considerable covalency between cations and O atom. The calculated SAW results show that PGG has the lowest velocities because of its highest density, SGG has the highest electromechanical coupling coefficients, CGG and SGG have more zero power flow angles and negligible diffraction angles than PGG crystal. Some promising cut-types such as Y-cut 148 of SGG, X-cut 154 of CGG and Y-cut 0 of PGG have been selected for their SAW applications.

Since CGG crystal was first synthesized in 1980s, several groups have been doing a great amount of research on it.Mill, Klimenkova, Maximov, Molchanov, and Pushcharovsky (2007) have pointed out that the piezoelectric modulus d 11 of CGG crystal increases when the size of cations forming the framework increases, especially the size of dodecahedral ions .Based on this theory, different elements can be used to substitute the cations in CGG crystals to improve its piezoelectric property (Zhou, Xu, Hua, & Fan, 2004;Takeda et al., 1999).That is the main reason why CGG, SGG, and PGG crystals present different piezoelectric properties.In addition, Kaminskii et al. (1984) and Sorokin et al. (2004) have measured the material data including elastic, piezoelectric and dielectric constants.However, as far as we know, the theoretical studies of electronic structures for CGG, SGG, and PGG crystals have not been reported yet.For the SAW applications, there is a harsh requirement (Naumenko & Solie, 2001) that the substrates of SAW devices should have low propagation velocity (v), high electromechanical coupling coefficients (k 2 ), zero power flow angle (PFA) and other excellent SAW properties.So it is also critical to predict the optimal cut-types which possess these merits simultaneously by theoretical calculation.Since Campbell and Jones (1968) established the well-known partial wave method to estimate optimal crystal cuts and propagation directions for excitation of piezoelectric surface waves, many research groups (Bungo et al., 1999;Naumenko & Solie, 2001;Shibayama, Yamanouchi, Sato, & Meguro, 1976;Yamanouchi, Odagawa, Kojima, & Matsumura, 1997;Yakovkin, Taziev, & Kozlov, 1995;Puccio, Malocha, Saldanha, & da Cunha, 2007) have predicated the SAW properties of various piezoelectric materials by using this method, the theoretical results agree well with the experimental measurements (Bungo et al., 1999;Yakovkin, Taziev, & Kozlov, 1995).This paper consists of two main parts, in the first part, by using first-principles method (Lu, Lan, Li, W. C. Wang, & C. L. Wang, 2006;Lee, Tan, & Lim, 2006;Huang, Cheng, Xue, & Li, 2014;Huang, Yang, Lu, & Gao, 2012), we investigated the electronic structures of these three crystals and make a comparison with each other.In the second part, we calculated their SAW properties in X-cut, Y-cut and Z-cut and then selected several promising cuts for their SAW applications.

Crystal Structures and First-Principle Method
(Ca,Sr,Pb) 3 Ga 2 Ge 3 O 14 crystals have the same structures with La 3 Ga 5 SiO 14 (LGS)-type (Mill & Pisarevsky, 2000;Mill, 2001) which belongs to the space group P321 of trigonal system (Xin, Zheng, Kong, & Shi, 2008).There are four cation site types in this structure: a dodecahedral site locates at 3e position, an octahedral site (1a), a small sized tetrahedral site (2d) and a large sized tetrahedral (3f).For CGG and SGG crystal (Kaminskii et al., 1984), Ca 2+ and Sr 2+ occupy the 3e position, 2d position is occupied by Ge 4+ , 1a position and 3f position are both occupied by Ga 3+ and Ge 4+ .While for the PGG crystal according to Bezmaternykh et al. (2004), 1a position and 3f position are both occupied by Ge 4+ , Ga 3+ only occupy 2d position and Pb 3+ occupy 3e position.According to Kaminskii et al. (1984) and Mill et al. (2007), the content of Ga 3+ in 1a and 3f position equal to 30% and 50%.In order to conveniently investigate the electronic structure of CGG and SGG crystals, we constructed the super cells in which the occupations of Ga 3+ and Ge 4+ in 1a and 3f positions are equal (1:1).Figure 1

Surface Acoustic Wave Calculation Method
According to the well-known partial wave method (Campbell & Jones, 1968), the phase velocity can be obtained by solving Christoffel equation: In Equation ( 1), ( , 1, 2,3) ( 1,2,3) amplitude component, v is phase velocity, ρ is the density of crystal, l i is the direct cosine of wave propagation direction angle  , c, e and ε is elastic, piezoelectric and dielectric constant respectively.Free surface velocities (v f ) and metalized surface velocities (v m ) can be obtained by solving this Equation with the free and metalized boundary conditions at the surface of semi-infinite crystal.
Electromechanical coupling coefficient k 2 is given by Power flow angle (PFA)  reflects the discordance between the propagation direction  of energy flow and phase velocity: Anisotropy factor δ can reflect the SAW diffraction (the most optimal value is -1):

Electronic Structure
Figure 2 shows the band structures of CGG, SGG and PGG crystals.For SGG and CGG, the top of valence band and the bottom of the conduction band are in the same points, which means that they are direct band gap materials While the band feature of PGG crystal is a little different from SGG and CGG, it presents indirect band gap feature.Their energy gap values are listed in Table 1, it can be seen SGG has the highest energy gap, while PGG has the lowest energy gap.−19.72 eV (peak-2''), −17.21 eV (peak-3'), −12.32 eV (peak-4'), −1.95 eV (peak-5') and 5.22 eV (peak-6').
Compared with the DOS of SGG, we could find that the peaks of the valence band have a tendency of shifting to the lower energy.Moreover, Ca, Ga, Ge and O atom have the similar contribution to the whole energy band as compared with Sr, Ga, Ge, and O in SGG.

SAW Properties
In the SAW calculation, so the choice of material constants greatly affect the calculated results (Naumenko & Solie, 2001).Table 4 lists the elastic, piezoelectric, dielectric constants and material density of CGG, SGG and PGG crystals (Kaminskii et al., 1984;Sorokin et al., 2004).By solving Christoffel equation, the phase velocity, electromechanical coupling coefficient, power flow angle and anisotropy factor of X-cut, Y-cut and Z-cut of (Sr, Ca, Pb) 3 Ga 2 Ge 3 O 14 are obtained, the results are shown in Figure 4, Figure 5, Figure 6 and Figure 7.
As shown in Figure 4, it is clear to see that most phase velocities of these crystals are less than 3000 m s -1 , which are lower than that of traditional piezoelectric crystals, such as quartz (3200 m/s ≦ v ≦ 3800 m/s) and LiNbO 3 (3500 m/s ≦ v ≦4000 m/s) (Shibayama, Yamanouchi, Sato, & Meguro, 1976;Slobodnik Jr, 1976;Kovacs, Anhorn, Engan, Visintini, & Ruppel, 1990).In these three crystals, PGG has the lowest velocities (varied from 2100 m s -1 to 2500 m s -1 ) because of its highest density, which means it has the superiority of fabricating miniaturized SAW devices.The phase velocities of CGG and SGG are varied from 2400 m s -1 to 3000 m s -1 and 2300 m s -1 to 3100 m s -1 respectively.In addition, the velocity curves of Y-cut and Z-cut have a central symmetry axis at γ = 90°, while the curve of X-cut does not have this property.Figure 5 shows the electromechanical coupling coefficients of these three crystals, it can be seen that SGG has obviously higher k 2 in each cuts, the highest values in X-cut, Y-cut and Z-cut can attain 1.15%, 0.66% and 0.48% respectively.This outstanding feature may come from its excellent piezoelectric performance (Zhou et al., 2004), especially from the contribution of large e 11 .The k 2 of CGG and PGG are relatively lower, their highest values in X-cut, Y-cut, Z-cut are 0.38%, 0.25%, 0.15% and 0.14%, 0.10%, 0.03% respectively.In the field of design of interdigital transduce (IDT), the insertion loss of IDT is smaller if the substrate possesses lager electromechanical coupling coefficient (Lu, Zhu, Liu, Fang, & Wen, 2006).So it is reasonable to predicate that SGG can exhibit excellent performance in reducing insertion loss of devices when making SAW devices.
Figure 6 shows their power flow angles, we could see all of these three crystals have zero PFA angles, the PFA of SGG are relatively higher than that of another two crystals.In Y-cut and Z-cut, they all have three and seven zero PFA angles.While in X-cut, SGG and PGG have four zero PFA angles, CGG has only one zero PFA angles.
Figure 7 shows their anisotropy factors, many propagation angles with negligible diffraction (δ = −1) can be found.In Z-cut, they all have six negligible diffraction angles, while in Y-cut and X-cut, SGG and CGG have four and two negligible diffraction angles, PGG does not have any negligible diffraction angles.
In addition, comprehensively taking account into these various properties, we list the properties of zero PFA cuts for these crystals in Table 5.As we can see, Y-cut 148 o of SGG has low phase velocity (2282.03m s -1 ) and high electromechanical coefficient (0.46%), which exhibits superior SAW performance.Besides, Y-cut 0 o and Z-cut 30 o of SGG, X-cut 154 o of CGG, Y-cut 0 o of PGG can be given priority in the field of SAW application.

Conclusion
In this paper, the first-principle calculation based on density functional theory have been used to study the electronic structures of CGG, SGG and PGG crystals, their calculated energy gaps are 2.18, 2.86 and 2.09 eV respectively.By comparing the calculated results with each other, we found that CGG and SGG have similar electronic structure characteristic in TDOS spectrum, the atoms in CGG has similar contribution to the TDOS as compared with the corresponding atoms in SGG.However, the PDOS of PGG is quite different from that of another two crystals in the lower part of valence band.For these three crystals, we found that the states of cations and states of O are overlapped obviously at the upper parts of valence band and conduction band, which suggested that covalent bonds exist between cations and O atom.This property was also confirmed by the analysis of effective charges and bond population.The strength of Ga-O and Ge-O bond are obviously larger than that of (Ca, Sr, Ca)-O.
By solving the Christoffel equation with the free and metalized boundary on the surface of semi-infinite crystal, their phase velocities, electromechanical coupling coefficients, power flow angles and anisotropy factors in X-cut, Y-cut and Z-cut are obtained.By comparing the calculated SAW results with each other, we found that PGG has the lowest velocities (varied from 2100 m s -1 to 2500 m s -1 ), SGG has the highest electromechanical coupling coefficients, CGG and SGG have more zero power flow angles and negligible diffraction angles than PGG crystal.Some promising cut-types such as Y-cut 148 o of SGG, X-cut 154 o of CGG and Y-cut 0 o of PGG were listed, which may provide theoretical guidance for their SAW application.
(a) shows the super-cell structures of CGG and SGG; Figure 1 (b) shows the super-cell structures of PGG crystal.

Figure 3 .
Figure 3.The total and partial density of States of SGG(a), CGG(b) and PGG(c) crystal

Figure 4 .
Figure 4.The phase velocities of SGG, CGG and PGG in X-cut(a), Y-cut(b) and Z-cut(c)

Table 1 .
The calculated energy gap of SGG, CGG and PGG

Table 5 .
The SAW characteristics of zero PFA cut-types