_{1}

^{*}

This comparative study acquaints the reader with some properties of the eighth and tenth new shear-horizontal surface acoustic waves (SH-SAWs) propagating along the free surface of the magnetoelectroelastic (6 mm) medium. These new nondispersive SH-SAWs cannot exist when the electromagnetic constant α is equal to zero. The piezoelectromagnetic SH bulk acoustic wave and the surface Bleustein-Gulyaev-Melkumyan (BGM) wave are also chosen for comparison. The main problem of this report is the demonstration of the fact that the new waves can propagate slower than the BGM wave. This problem can be very important due to the fact that among the other known SH-SAWs the BGM wave can propagate significantly slower than the corresponding SH bulk acoustic wave. Two new SH-SAWs are analytically and graphically studied in dependence on the electromagnetic constant α. For the graphical study, two (6 mm) composites are used: BaTiO
_{3}– CoFe
_{2}O
_{4} and PZT-5H–Terfenol-D. For the second composite it is solidly demonstrated that for small values of α, the eighth new SH-SAW cannot exist and its velocity starts with zero at some small threshold value of α rapidly reaching the BGM-wave velocity. This means that a weak magnetoelectric effect can dramatically slow down the speed of either new SH-SAW. As a result, the studied new SH-SAWs can be suitable for creation of new technical devices to sense the magnetoelectric effect. For the analytical study, extreme and inflexion points were evaluated in the velocities’ dependencies on the value of the electromagnetic constant α.

Two researchers, Bleustein [

In addition to the PE and PM solids there are piezoelectromagnetic (PEM) continua, also known as magnetoelectroelastics. The SAW peculiarities mentioned above for the PEs and PMs are also true for the PEMs. In the 6 mm PEM continua, the surface Bleustein-Gulyaev-Melkumyan (BGM) wave can propagate. The BGM wave was recently discovered by Melkumyan [_{3}) and magnetic (B_{3}) displacement components that are normal to the surface. The BGM wave can propagate under the following boundary conditions: mechanically free, electrically closed (φ = 0), and magnetically open (ψ = 0) surface of the piezoelectromagnetics. All the SH-waves mentioned above relate to pure waves [

Piezoelectromagnetics similar to piezoelectrics must be noncentrosymmetric monocrystals or two-phase materials to possess the piezoelectric effect. Besides, magnetoelectroelastics as a class of magnetoelectric materials can have the piezomagnetic and magnetoelectric effects. The smart magnetoelectric materials have mechanical, electrical, and magnetic subsystems and the last two subsystems can affect each other via the first one. This property makes these smart materials multi-promising for various technical applications, for instance, see in review papers [

The following section acquaints the reader with the analytical study of the nondispersive SH-waves. The main purpose is to investigate the eighth [

It is natural to first introduce the definition for the speed of the shear-horizontal bulk acoustic wave (SH-BAW), propagation of which is coupled with both the electrical and magnetic potentials. This is useful because the value of the SH-BAW velocity V_{tem} must be larger than the values of the corresponding SH-SAW velocities, for instance, the surface Bleustein-Gulyaev-Melkumyan (BGM) wave velocity V_{BGM}. The SH-BAW velocity V_{tem} can then be defined by the following well-known formula:

In Equation (1),

In definition (2) one can find the following independent nonzero material constants: the stiffness constant C, piezomagnetic coefficient h, piezoelectric constant e, dielectric permittivity coefficient ε, magnetic permeability coefficient μ, and electromagnetic constant α [

It is a natural choice to compare the velocity behaviors of the eighth and tenth new SH-SAWs recently discovered in papers [_{tem} and SH-SAW V_{BGM}. This is constructive because there is an assumption that the new SH-waves can propagate even slower than the surface BGM wave in dependence on the electromagnetic constant α. This statement must be demonstrated in the analysis developed below. It is essential to state right away that the SH-SAW V_{BGM} can propagate when the following mechanical, electrical, and magnetic boundary conditions are applied to the interface between the PEM solid and a vacuum: mechanically free surface, electrically closed surface (electrical potential φ = 0) and magnetically open surface (magnetic potential ψ = 0). For the same mechanical boundary condition, the propagation of the eighth and tenth new SH-SAWs [

So, let’s now introduce the formula for calculation of the BGM wave velocity V_{BGM} that can be derived in the following form:

where

The eighth new SH-SAW velocity V_{new}_{8} [

where

It is clearly seen in expressions (5) and (6) that the eighth new SH-SAW cannot exist for the case of zero value of the electromagnetic constant, α = 0, because _{tem}. Therefore, it is expected that the penetration depth of this new SH-SAW must be significantly larger than that for the other SH-SAWs.

In expression (6), µ_{0} is the magnetic constant for a vacuum and the following useful equalities were exploited:

In expressions (6) and (7), the coefficient of the electromechanical coupling (CEMC) denoted by

The other parameter denoted by

The second new wave for comparison called the tenth new SH-SAW [

where

It is also clearly seen in expressions (11) and (12) that the tenth new SH-SAW cannot exist for α = 0. This is like the eighth new SH-SAW introduced and in some measure discussed above. In expression (12), ε_{0} is the electric constant for a vacuum and the coefficient of the magnetomechanical coupling (CMMC,

Also, equality (8) and the following equality were used in expression (12):

It is clearly seen in expressions (5) and (6) that the velocity V_{new}_{8} can reach the SH-BAW velocity V_{tem} as soon as the following condition is completed:

Analyzing expressions (11) and (12), one can also find the following condition for the case of V_{new}_{10} = V_{tem}:

It is necessary to state that conditions (15) and (16) are also fulfilled when the _{BGM} = V_{tem} can also exist when

It is apparent that three conditions from (15) to (17) require corresponding large values of the electromagnetic constant α. It is also essential to provide the other conditions relating to the existence of the eighth and tenth new SH-SAWs. It is flagrant in definition (10) that

_{n}_{8} and b_{n}_{10} on the normalized value of α^{2}/εµ. The logarithmic scale is used for the later parameter in the figure to soundly demonstrate the peculiarity of the wave existence. The value of εµ is constant in the calculation. This means that only the value of the electromagnetic constant α is changed. The used material parameters of a vacuum are well-known: the magnetic permeability constant is

a vacuum. These parameters for the composite materials listed in

Composite material | C, 10^{10}, N/m^{2} | e, C/m^{2} | h, T | ε, 10^{?10}, F/m | μ, 10^{?6}, N/A^{2} |
---|---|---|---|---|---|

BaTiO_{3}?CoFe_{2}O_{4} | 4.40 | 5.80 | 275.0 | 56.4 | 81.00 |

PZT-5H?Terfenol-D | 1.45 | 8.50 | 83.8 | 75.0 | 2.61 |

Composite material | εμ, 10^{?16} | ρ, kg/m^{3} | |||

BaTiO_{3}?CoFe_{2}O_{4} | 4568.40 | 5730 | 2.4378 | 0.1356 | 0.0212 |

PZT-5H?Terfenol-D | 195.75 | 8500 | 0.9264 | 0.6644 | 0.1856 |

Composite material | V_{tem}, m/s | V_{BGM}, m/s | V_{new}_{8}, m/s | V_{new}_{10}, m/s | |

BaTiO_{3}?CoFe_{2}O_{4} | 0.1545 | 2977.450 | 2950.670 | 2976.023 | 2977.434 |

PZT-5H?Terfenol-D | 0.6817 | 1693.7503 | 1548.350 | 1687.238 | 1693.7500 |

BaTiO_{3}?CoFe_{2}O_{4}. It is necessary here to state that the magnetic permeability constant μ for the first composite is only twice as much in comparison with that for a vacuum. It is possible that this fact results in some peculiarities discussed below.

For comparison, the calculations are performed for the following two piezoelectromagnetic composite materials: PZT-5H?Terfenol-D and BaTiO_{3}?CoFe_{2}O_{4}. The material constants of the composites together with the other characteristics are listed in ^{2}/εµ. The eighth new SH-SAWs cannot exist when the value of α^{2}/εµ is smaller than ~6.4 ´ 10^{?3} for the PZT-5H?Terfenol-D composite because the parameter b_{n}_{8} becomes less than ?1, see also _{3}?CoFe_{2}O_{4}, the values of α^{2}/εµ must be smaller than ~5.0 ´ 10^{?7} to get b_{n}_{8} < ?1. This means that there is the threshold value of α^{2}/εµ to allow the eighth new SH-SAW to exist for either studied composite: α^{2}/εµ ~ 6.4 ´ 10^{?3} for the PZT-5H?Terfenol-D and α^{2}/εµ ~ 5.0 ´ 10^{?7} for BaTiO_{3}?CoFe_{2}O_{4}.

For the tenth new SH-SAW existence satisfying condition (19), the threshold values of α^{2}/εµ are significantly smaller, namely α^{2}/εµ ~ 4.8 ´ 10^{?8} for the PZT-5H?Terfenol-D and α^{2}/εµ ~ 5.0 ´ 10^{?9} for BaTiO_{3}?CoFe_{2}O_{4}. This means that the propagating waves can exist when the value of α is larger than the threshold value, α_{th}. In fact, a measured value of α is in general very small. This is not undesirable for a piezoelectromagnetic composite material concerning the existence of the eighth and tenth new SH-SAWs because a small proper value of α can cause a significant slowing down of at least one of the new SH-SAW velocities. This is the dramatic influence of the weak magnetoelectric effect on the existence and propagation of the new waves. It is also possible to shortly note that this theoretical report has an interest in a study of propagating (nondissipative) new SH-SAWs characterized by real speeds. Therefore, any dissipation corresponding to the case of α < α_{th} resulting in an imaginary speed is not treated here.

It is also possible to compare with some experimental data for the BaTiO_{3}?CoFe_{2}O_{4} composite: α^{2}/εµ ~ 4 ´10^{?5} [^{2}/εµ ~ 5 ´ 10^{?6} [_{3}Co_{2}Fe_{24}O_{41} Z-type hexaferrite [

However there is also one peculiarity for the eighth new SH-SAW existence for the PZT-5H?Terfenol-D composite that can be seen in ^{2}/εµ is larger than ~0.997, assuming that α^{2}/εµ < 1 must occur due to the following limitation: α^{2} < εµ [_{BGM} < V_{tem} should always occur. These facts are more clearly seen in figure 2 for the eighth new SH-SAW (thick solid black line) in the PZT-5H?Terfenol-D. Therefore, it is possible to analytically treat the possible cases of V_{new}_{8} = V_{BGM} and V_{new}_{10} = V_{BGM} similar to the cases of V_{new}_{8} = V_{tem} and V_{new}_{10} = V_{tem} considered above.

_{new}_{8}, V_{new}_{10}, V_{BGM}, and V_{tem} on the electromagnetic constant α (α^{2}/εµ with εµ = const) when the calculations are carried out for both the PZT-5H?Terfenol-D and BaTiO_{3}? CoFe_{2}O_{4} composites. It is clearly seen in the figure that for the first composite the velocities V_{new}_{8} (thick

solid black line) and V_{BGM} (dashed black line) can have a crossing point at a small value of the α. For a significantly smaller value of the α, the velocities V_{new}_{10} and V_{BGM} can also have a crossing point. Besides, _{new}_{8} and V_{BGM} can have two crossing points for the PZT-5H?Terfenol-D composite: at ^{2}/εµ ~ 1 when α^{2}/εµ < 1 [_{new}_{8} and V_{BGM} for the same composite because sole crossing point can exist. For BaTiO_{3}?CoFe_{2}O_{4}, only single crossing point at ^{2}/εµ < 1 is fulfilled in ^{2}/εµ > 1 that must be analytically illuminated.

So, let’s first treat the following case:

It is more convenient to treat the following equality instead of equality (20), see formulae (4) and (6):

Employing formulae (4) and (6), equality (21) leads to the following quadratic equation to find unknown values of the electromagnetic constant α:

So, two crossing points must exist in the common case because Equation (22) can have two equation roots. They can be calculated with the following expression:

To have real values of the electromagnetic constant α, it is apparent that the following inequality must be satisfied under the square root in expression (23):

It is obvious that the left-hand part in the inequality can always be larger than zero. The case of ε > 0 results in the fact that the right-hand part must be also larger than zero because the vacuum magnetic constant µ_{0} > 0. ^{2}/εµ. Therefore, inequality (24) can be surely written as follows:

Inequality (24) allows one to simplify formula (23). Indeed, formula (23) can be schematically introduced as

Let’s now analyze the existence of the crossing points between the velocities V_{new}_{10} and V_{BGM}. They occur when the following equality is satisfied:

Analogically, it is more convenient to use the following equality instead:

Utilizing formulae (4) and (12), equality (29) can be expanded to the following quadratic equation:

Accordingly, two equation roots can be inscribed as follows:

One has to deal here with the case of real roots of quadratic equation (30). This requires satisfaction of the following inequality:

It is even possible to write the following inequality because equality (28) occurs at a very small value of the electromagnetic constant α:

Using condition (33), equation roots (31) can be rewritten in the following simplified forms:

Comparing the cases of V_{new}_{8} = V_{BGM} and V_{new}_{10} = V_{BGM}, it is possible to conclude that the presence of the vacuum electric constant ε_{0} results in the smaller value of the electromagnetic constant α, at which there is the crossing point. Two crossing points can actually exist in either case but the second crossing point can be revealed at α^{2}/εµ > 1. Therefore,

The analysis carried out above is not complete because one can find several extreme points in the dependence of the eighth new SH-SAW velocity on the normalized value of α^{2}/εµ shown in _{new}_{8}(α) and V_{new}_{10}(α). It is well-known that there is a closely linear dependence around an inflexion point and this quasi-linear regime can be used in construction of different technical devices. Also, one can find that the V_{tem} and V_{BGM} velocities can have a smooth minimum in _{tem}(α) and V_{BGM}(α).

The dependence V_{tem}(α) is given by formula (1) at the beginning of this section. The first derivative of the SH-BAW velocity V_{tem} [

where the purely mechanical SH-BAW velocity V_{t} is defined right away after Equation (1).

This first derivative must be equal to zero at an extreme point. This can happen when the first derivative on the right-hand side is equal to zero. In expression (36), the first derivative of the coefficient of the magnetoelectromechanical coupling _{tem}(α) can be defined by solving the following equation:

where the coefficient

Utilizing expression (8), one can solidly find that there are two extreme points for the dependence V_{tem}(α). They are given by equalities (15) and (16). Indeed, the dependence

Concerning the dependence V_{BGM}(α) defined by expressions (3) and (4), its first derivative with respect to the α can be also borrowed from recently published paper [

Next, it is clearly seen in expression (38) that the dependencies V_{tem}(α) and V_{BGM}(α) actually have the same extreme points defined by equalities (15) and (16). This is so because this problem reduces to the treatment of Equation (37) for both the cases. However, one can also find an extra possibility given by the expression in the square brackets on the right-hand side of expression (38). Therefore, one must equal to zero the square brackets to check a possible existence of some extra extreme point. After several transformations, one can find that this problem is reduced to the following equality _{tem}(α) and V_{BGM}(α) can truly have only the same extreme points.

It is now possible to find the extreme points for both the dependences V_{new}_{8}(α) and V_{new}_{10}(α) defined by expressions (5) and (11), respectively. Expressions (6) and (12) depend on the coefficient

The existence of the extreme points requires that the first derivatives of the velocities V_{new}_{8}(α) and V_{new}_{10}(α) with respect to the electromagnetic constant α must be equal to zero. Therefore, the following expressions must be considered:

where

First of all, it is necessary to analyze expression (40). Using expressions (6) and (8), it is possible to mark that two terms on the right-hand side of Equation (40) have the same factor such as_{tem}(α) and V_{BGM}(α) analyzed above. This extreme point corresponds to the smooth minimum shown in

This equation can be simplified to the following form:

It is possible to exclude the factor of

As a result, the final equation representing a polynomial of the eight degree in the unknown parameter α must also have several extreme points. It reads:

where

For the analysis of expression (41), it is essential to use expressions (8) and (12). With expression (8), it is obvious that _{n}_{10} is proportional to the vacuum dielectric permittivity constant ε_{0} that is several orders smaller than the dielectric permittivity constant ε for either composite listed in the table. Therefore, it is possible to conclude that the first term in expression (41) is significantly larger than the second because the later term even has a factor of ε_{0}^{2}. This fact provides extreme points at the values of the electromagnetic constant α being close to those defined by equalities (15) and (16). Also, both terms on the right-hand side of Equation (41) have the same factor such as_{tem}(α) and V_{BGM}(α). However, it is not shown in ^{2}/εµ > 1 that is out of current interest.

All the extreme points can be revealed by solving the following homogeneous equation:

Proper transformations based on expressions (2), (8), (10), and (12) can lead to the following simplified form:

This form can be further simplified. For instance, the following equality must be employed for expression (12):

Thus, the reader has to cope with the following polynomial of the eighth degree in the unknown parameter α, where the function x(α) is defined by expression (48):

Unfortunately, polynomials (47) and (52) are not simple to analyze and therefore, they can be studied numerically. All the extreme points corresponding to the values of α^{2}/εµ from zero to unity are shown in ^{2}/εµ: (1) α^{2}/εµ ~ 0.143641 (maximum, V_{new}_{8} ~ 1687.238 m/s), (2) α^{2}/εµ ~ 0.278784 (minimum, V_{new}_{8} ~ 1684.997 m/s), and (3) α^{2}/εµ ~ 0.974169 (maximum, V_{new}_{8} ~ 2019.895 m/s) for PZT-5H? Terfenol-D; and (1) α^{2}/εµ ~ 0.000484 (maximum, V_{new}_{8} ~ 2976.023 m/s) and (2) α^{2}/εµ ~ 0.156816 (minimum, V_{new}_{8} ~ 2952.933 m/s) for BaTiO_{3}?CoFe_{2}O_{4}. It is worth noting that with a thorough analysis of the computation data for PZT-5H?Terfenol-D, the value of α^{2}/εµ ~ 0.278784 corresponding to the minimum of the velocity V_{new}_{8} is the same for the other velocities mention in this theoretical research such as V_{new}_{10}, V_{BGM}, and V_{tem}. It is interesting to compare this value of α^{2}/εµ ~ 0.278784 obtained in the numerical calculation with the value of α^{2}/εµ ~ 0.2793 calculated from equality ^{2}/εµ ~ 3.58 is calculated from equality _{3}?CoFe_{2}O_{4}, the value of α^{2}/εµ ~ 0.156816 also corresponds to the minimum for all these velocities: V_{new}_{8}, V_{new}_{10}, V_{BGM}, and V_{tem}. It is worth mentioned here anew that this minimum corresponds to the minimum in the dependence of the separated exchange coefficient

For the tenth new SH-SAW, the calculated extreme points can be also given here below. They correspond to the following values of the normalized parameter α^{2}/εµ: (1) α^{2}/εµ ~ 4.096 ´ 10^{−5} (maximum, V_{new}_{10} ~ 1773.2165 m/s) and (2) α^{2}/εµ ~ 0.278784 (minimum, V_{new}_{10} ~ 1684.997 m/s) for PZT-5H?Terfenol-D; and (1) α^{2}/εµ ~ 2.401 ´ 10^{-5} (maximum, V_{new}_{10} ~ 2979.395 m/s) and (2) α^{2}/εµ ~ 0.156816 (minimum, V_{new}_{10} ~ 2952.933 m/s) for BaTiO_{3}?CoFe_{2}O_{4}. It is also possible to write down the values of α^{2}/εµ calculated with conditions ^{2}/εµ ~ 0.156532 and α^{2}/εµ ~ 3.7395, respectively. One can see that there is a good correlation between the value computed at the minimum and that calculated with condition (15).

The second derivative can provide information on all possible inflexion points and therefore, reveal a linear regime around an inflexion point that can be useful for experimentalists and theoreticians. It is natural to start the analysis with the partial second derivative of the SH-BAW velocity V_{tem} with respect to the electromagnetic constant α. It can be also borrowed from work [

where

It is convenient to utilize the following definition to calculate the partial second derivative of the velocity V_{BGM}:

where the first and second derivatives of the

Regarding the partial second derivatives of the new SH-SAWs with respect to the material parameter α, they can be evaluated with the following complicated formulae:

where

The extreme points’ existence requires that the partial second derivatives of the functions V_{new}_{8}(α) and V_{new}_{10}(α) with respect to the electromagnetic constant α must be equal to zero. All the partial first and second derivatives on the right-hand sides of expressions from (56) to (59) are defined above. Note that expressions from (56) to (59) are as difficult as those obtained for the other new wave velocities studied in papers [

The other useful parameter must be also discussed in this paper. This parameter denoted by Δ was first introduced in book [_{tem}. As a result, the difference between the velocities can be as small as several meters per second or even less, or even often mm/s for a weak piezoelectromagnetics. So,

It is clearly seen in _{tem}. The parameter Δ_{M} never equals to zero because the BGM speed can never reach the SH-BAW speed V_{tem}, see the dashed lines in _{n}_{8} and Δ_{n}_{10} can be both larger and smaller than the parameter Δ_{M}. Indeed, the value of the parameter Δ_{n}_{8} (thick solid lines) can be significantly larger than the Δ_{M} value at small enough values of the electromagnetic constant α. For significantly smaller values of the α given in the context of Section 2 after existence conditions (18) and (19), this fact must be also true for the other parameter Δ_{n}_{10}. Also, it looks like that the Δ_{n}_{8} value can equal to zero for both the studied composites. This means that the speed of the eighth new SN-SAW can reach the SH-BAW speed V_{tem}. The dependencies Δ_{n}_{10}(α) look like they have one smooth minimum for each composite and the tenth new SH-SAW speed cannot touch the bulk wave speed.

So, it is possible to analytically consider the extreme and inflection points of the discussed parameters Δ. For the extreme points’ determination, the following equalities must be treated:

In expressions from (63) to (65), the partial first derivatives on the right-hand sides can be found in the previous sections. To find inflexion points in the dependencies Δ(α), one has to consider the following partial second derivatives with respect to the electromagnetic constant α:

The reader is already familiar with all the partial second derivatives present on the right-hand sides of equalities (66), (67), and (68). These derivatives are quite complicated and can be computed by using the theory developed in the previous section.

Exploiting the transversely isotropic (6 mm) magnetoelectroelastic composites such as BaTiO_{3}?CoFe_{2}O_{4} and PZT-5H?Terfenol-D, it was demonstrated that the magnetoelectric effect can dramatically affect the velocities of the studied nondispersive new SH-SAWs. This is true even in the case of a very small electromagnetic constant α because this small material parameter can result in dramatic slowing down the propagation speeds of the studied new SH-SAWs and even wave propagation loss. Also, analytical investigations of the studied new SH-SAW velocities were performed: the extreme and inflexion points were evaluated and discussed. The illuminated peculiarities can be useful for technical device construction based on the magnetoelectric effect and the effect of the slow speed can also find some practical applications, for instance, in filters such as delay lines, etc. The found peculiarities can be also involved in the study on better understanding of the magnetoelectric effect.

Aleksey AnatolievichZakharenko, (2015) Dramatic Influence of the Magnetoelectric Effect on the Existence of the New SH-SAWs Propagating in Magnetoelectroelastic Composites. Open Journal of Acoustics,05,73-87. doi: 10.4236/oja.2015.53007