Bleustein-Gulyaev waves in topological piezoelectric crystals

A topological insulator with piezoelectric properties is considered, and an action is proposed to find the equations of motion and the constitutive relations of the generalized coordinates of electrodynamic and elastic origin, paying special attention to the restructuring induced by the topological properties. The results are used to demonstrate that the electromechanical factor of the Bleustein-Gulyaev waves on the surface of a topological piezoelectric crystal of class C6v in contact with vacuum undergoes a second-order correction in the fine structure constant, associated with the topological properties.


Introduction
The low energy consumption and ultra-high response capacity of piezotronic devices based on topological insulators (TIs) have recently stimulated research work on these systems [1,2].TIs are materials that behave like an insulator in the bulk; however, they also present surface conduction states that are topologically protected [3,4].The first TIs were created in the form of quantum wells with materials exhibiting strong spin-orbit coupling, such as the quantum wells of mercury telluride and cadmium telluride (CdTe/HgTe/CdTe).By varying the dimensions of these wells, an inversion in the band structure is evidenced, which gives rise to surface states that are topologically protected against time-inversion symmetry [5].The presence of these states is manifested in the form of new polarizations and surface currents that do not undergo backscattering or dissipation.This implies that the low energy effective action used for the description of electromagnetic properties must include an axion term which couples the electric and the magnetic fields by means of a topological term, which is quantized and takes odd values of π for TIs [6,7].
Recent studies have shown that it is possible to change the band structure of different materials through the piezoelectric effect [8].Piezoelectricity is a phenomenon that occurs in crystals that lack central symmetry, such as zinc oxide (ZnO).When the material is subjected to mechanical stress, ion polarization is induced in the crystal, creating a piezoelectric potential.By controlling these polarization fields by applying potentials and stresses, it is possible to tune the electronic states and band structure of the material.This opens the possibility of a transition from conventional semiconductors to topological insulators [9], as is the case for the quantum wells of zinc oxide-cadmium oxide (ZnO/CdO), gallium nitride-indium nitride (GaN/InN/GaN), and gallium arsenide-germanium (GaAs/Ge/GaAs).This topological transition allows the design of piezotronic nanodevices such as memory and logic gates that increase their response capacity [10][11][12][13][14].The study of piezoelectric properties is not limited to this class of quantum materials.The piezoelectric response has recently been investigated in materials such as graphene [15] and in phononic topological plates [16].On the other hand, experimental evidence has been presented that demonstrates the coexistence of piezoelectric and topological properties in InX monolayers (X = Se and Te) [17].Similarly, piezoelectricity has been confirmed on the surface of Bi2Se3 nanosheets, a topological insulator.This opens perspectives for the development of nanoelectromechanical systems with piezoelectric topological insulators [18].
In piezoelectric materials, there is a coupling between elastic excitations and electrical excitations.For uniform deformations in the crystal, the induced polarizations vary proportionally with the applied stress through a parameter known as the piezoelectric coefficient.Conversely, it is possible to deform the material when an electrical potential is applied to the crystal [19].The ability of the material to transform mechanical energy into electrical energy is measured through the electromechanical coupling factor, in which the dielectric properties of the material are combined with the elastic properties and piezoelectric coefficient [20,21].Among the coupled electro-elastic modes, the Bleustein Gulyaev (B-G) modes, which are located on the surface of a piezoelectric and other materials, are of special importance.The B-G waves are shear horizontal surface acoustic waves (or SH-SAWs), in which the direction of the movement of the particles is perpendicular to the direction of propagation and is parallel to the surface of the piezoelectric [22,23].In the absence of a piezoelectric effect, the B-G waves transform into shear bulk waves.The B-G propagation modes are a key component in liquid detection applications because they do not radiate energy to the adjacent liquid and are sensitive to changes in mechanical, electrical, and liquid properties [24,25].
To our knowledge, B-G waves in piezoelectric materials with topological properties have not yet been studied, and the constitutive relations between the electromagnetic and elastic fields in such materials have not been discussed in detail.For this reason, this work focuses on the effects of the presence of the axion term on the electro-elastic response of a topological insulator with piezoelectric properties.In section 2, we propose an action to describe the electrodynamic and elastic properties of a non-magnetizable TI-piezoelectric.Then, the equations of motion and the constitutive relations of the material are obtained by varying the action with respect to the elastic and electrodynamic fields.In section 3, the results obtained in the previous section are used to examine the electromechanical coupling factor and the phase velocity of the B-G waves on the surface of a topological piezoelectric crystal of class C v 6 in contact with vacuum.

General relations
Let us consider a non-magnetizable TI characterized by a mass density ρ, dielectric permittivity tensor , the components A k of the vector potential, and the components u i of the elastic deformation are taken as generalized coordinates.For low elastic wave frequencies, electromagnetic fields behave like quasi-stationary fields [19].In the absence of free charges and conduction currents, the action S of the described system has the form: where ( ) is the strain tensor, P i is the component of the polarization vector, is the fine structure constant (with e the electron charge, c the speed of light and  the reduced Planck's constant), cdtdV dW= and e ijk is the Levi-Civita symbol.The first and the second integrals in equation (1) describe the action of the electromagnetic field in a polarizable medium in the quasi-stationary approximation [19].The third integral in equation (1) contains the magnetoelectric effects which arise as a result of the presence of surface states in the TI [3].The fourth integral involves the electro elastic properties of the material [19,20], which are defined by the elastic kinetic energy u i Varying the action (1) with respect to the scalar potential j we obtain Integrating by parts and using Gauss's theorem where S is the hypersurface that contains the four-volume .W The integral over S vanishes [19], and we find Similarly, the variation of the action S with respect to u i and A i leads to The variations in equations (2), (3), and (4) vanish when the corresponding subintegral functions are equal to zero.In particular, the condition S 0 The second expression in equation ( 5) is a constitutive relation which generalizes the conventional Hooke's law [26] in order to consider the stresses induced by the polarization associated with applied electric fields (inverse piezoelectric effect [20]).Note that the response of the material to the applied stresses is not altered by the polarizations associated with the surface states of the topological insulator.
Similarly, from the condition S 0 d = j in equation (2), we find Gauss's law modified by the joint presence of the piezoelectric and topological properties: where D i is the component of the electrical displacement with P i k 4 where H k denotes the magnetic intensity.The Cartesian components of the electric and magnetic fields are given by E i i j = - ¶ and B e A , i i j k j k = ¶ respectively [3,27].The constitutive relations for , ik s D i and H k involved in equations (5), (6), and (7), respectively, are the main results of the present work and generalize previous results reported in [1,3].Note that the polarizations associated with the surface states of the topological insulator are reflected in the restructuring of the electrical displacement and the magnetic intensity, as observed in equations ( 6) and (7), respectively.
As a result of the above considerations, a coupling appears between the electric and magnetic fields, which is mediated by the topological parameter and the fine structure constant .
a Furthermore, the polarization induced by the deformation of the crystal is evidenced in the presence of the term u i pq pq , b in the second relation in equation (6) (direct piezoelectric effect [20]).In the absence of a topological parameter 0, q = the constitutive relations involved in equations ( 6) and (7) are decoupled, and the constitutive relations are reduced to those of conventional piezoelectric materials [19,20].Note that the structure of equations ( 5)-( 7) is similar to the equations that govern the magneto-electro-elastic effects that appear in multilayer structures of conventional piezoelectric, piezomagnetic and magnetoelectric materials [28,29]

Bleustein-gulyaev waves in topological piezoelectrics
The above results can be used to study the coupling of elastic waves with electromagnetic fields (Bleustein-Gulyaev waves) in systems with topological properties.For this purpose, let us consider a piezoelectric crystal of mass density ρ which belongs to class C v 6 and has a topological parameter θ.The crystal occupies the region y 0 < and is limited to the xz plane, where z is the axis of symmetry of the crystal.The region y 0 > is occupied by vacuum.The non-zero components of the dielectric permittivity tensor , If the surface waves propagate along the x direction, then the relations involved in equation (8) imply that the non-zero components of the deformation vector u and the vector potential A are u z and A , z respectively.The homogeneity in z implies that A u 0 ).In this case, the electric field lies in the xyplane.The geometry of the system is shown in figure 1.
The constitutive equations which relate the stresses , ik s electric displacements D i and magnetic intensity H k to u , z j and A z are and the equations ( 5), ( 6) and (7) reduce to where , Because of the homogeneity of the system in the x direction, the differential equations involved in equations ( 12), (13), and (14) have solutions in the form of free waves with frequency w and wave vector k.Therefore, the y dependence of the fields can be written as

A y y D y E
D c and ( ) E v are the amplitudes of the fields, and the indices c and v indicate the regions occupied by the crystal and by the vacuum, respectively.
At the interface y 0 = the following boundary conditions for the fields , j A, D and H take place [26,27]: Additionally, the crystal is free of stresses on the surface, so ( ) Replacing the constitutive relations (9)- (11) and solutions (15)- (17) in the boundary conditions, the following system is obtained These relations lead to the following system of 3 3 ´homogeneous algebraic equations for the amplitudes ( ) A , c ( ) where the constant L d between k and k defines the electromechanical coupling factor.Equation (25) shows that the topological properties of the considered material affect the ability of the system to transform mechanical energy into electrical energy.Note that in the absence of the topological term, equation (25) reduces to the electromechanical coupling factor of the conventional Bleustein-Gulyaev waves [22,23].
The phase velocity v s of the B-G waves is given by the ratio .k w By inserting equation (25) in ̅ ( ) These results show that the presence of the topological term introduces second-order corrections in the fine structure constant a on the electromechanical coupling factor, which alters the original result reported in [22,23] for conventional piezoelectric materials.In particular, such corrections affect the phase velocity of B-G waves.
Figure 2 shows the dispersion relations for Bleustein-Gulyaev waves in topological piezoelectric crystals of GaN/InN (red line) and ZnO/CdO (blue line).The properties for these materials are reported in table 1.
In figure 2 it can be seen that the frequency grows more rapidly in GaN than in ZnO.This is due to the fact that, based on the parameters in table 1 and relation (26), the phase velocity in GaN is an order of magnitude greater than in ZnO.
On the other hand, figure 3 shows that the corrections induced by the topological parameter in the dispersion relations grow with the wave vector.Furthermore, it is observed that the topological correction grows more rapidly in ZnO.This behaviour is related to the fact, that the considered correction strongly depends on the inverse of the dielectric permittivity of the involved material.

Conclusions
An action was proposed for an insulator with a nontrivial topology, considering its elastic and electrodynamic properties.The polarizations associated with the surface states of the TI and the deformations of the crystal restructure the constitutive relations and produce a coupling between the elastic and electrodynamic fields that involve magnetoelectric effects.It was found that the response of the material to the applied stresses was not altered by the polarizations associated with the surface states of the topological insulator.On the other hand, the magnetoelectric coupling due to the axion term restructures the electrical displacement of piezoelectric materials.The obtained results were used to examine the propagation of Bleustein-Gulyaev waves at the interface between a topological piezoelectric crystal of class C v 6 and vacuum.It was shown that the electromechanical coupling factor and the phase velocity showed second-order corrections in the fine structure constant.

Future perspectives
Similar to piezoelectricity, which couples strain and electric polarization, there is another group of materials in which a coupling arises between the magnetic polarization of the crystal and the mechanical stresses applied to the material.These are known as piezomagnetic materials.Recent results [33], show that piezomagnetism arises in some materials with topological properties, which can be useful for spintronics applications.The action discussed in the present work can be extended to materials that jointly exhibit topological and piezomagnetic properties.The corresponding equations of motion and the extended constitutive relations obtained from such formalism can be used to investigate the effects produced by the presence of a non-trivial topology on the behavior of magneto-elastic waves that propagate in the surface of the crystal.
Additionally, the equations of motion and constitutive relations obtained in this work can be used to study the electro-elastic response in more complex configurations, such as multilayer systems composed of piezoelectric topological insulators and other materials [28,29].The topological parameter in these materials is assumed to be q p = [6, 7].

b
and topological parameter θ.The scalar potential , j

1 2 2 renergy u u iklm m l k l 1 2
 (where u i  are the deformation velocities), by the elastic potential l ¶ ¶ and by the term that couples the electric field and the crystal deformations through the tensor .lik , b tensor)[3,27].Finally, the condition S 0 A d = in equation (4) leads to Ampere's law for piezoelectric materials with non-trivial topological properties which has the form

ike
modulus of elasticity tensor iklm l and piezoelectric coefficient tensor l ik , piezoelectrically stiffened elastic constant[21,22] (c) refers to the region occupied by the crystal).

Figure 1 .
Figure 1.Schematic representation of the semi-infinite TI-piezoelectric crystal with topological parameter q in contact with vaccum.The crystal belongs to the C v 6 class with symmetry axis along z.The waves propagate in the x direction and u z represents the non-zero component of the deformation vector u associated to the surface wave.

p
For the existence of nontrivial solutions of equation(24), the propagation constants k and k must satisfy the condition

Figure 3 .
Figure 3. Corrections induced by the topological parameter on the dispersion relations.Red (dashed line): GaN.Blue (solid line): ZnO.