Rayleigh wave through half space semiconductor solid with temperature dependent properties

The article focuses on Rayleigh wave propagation in a homogeneous isotropic semi-conductor thermoelastic medium rotating with fixed angular frequency with temperature-dependent properties. The elastic constants depend upon the temperature function. The effects of temperature dependency parameter, time derivative, and fractional order are illustrated. By the theory of thermo-elasticity, waves result in the generation of thermal signals that propagate through the medium. A heat conduction model of three-phase lag (3PL) along with fractional order time derivative is used to analyze the thermal signals. The secular equations of Rayleigh waves are derived mathematically at the stress-free, carrier density and thermally insulated boundaries. Some specific properties like velocity, attenuation coefficient, specific heat loss and penetration depth for Rayleigh waves have been evaluated and presented graphically. The secular equations are computed numerically and depicted graphically using Matlab.


Introduction
Surface waves travel along the boundary of the medium and contain information about surface imperfections.These waves are commonly classified as either Love, Stoneley or Rayleigh waves [1].These are the waves, whose amplitude decreases exponentially with increasing distance from the surface of the medium.Rayleigh wave propagation in semiconductor solids is an important area of research in materials science, physics, and engineering [2][3][4].This phenomenon is an example of elastic wave propagation in materials that exhibit elasticity, which includes a wide range of materials including metals, ceramics, polymers and semiconductors.These waves are a type of surface wave that travels along the interface between two media with different elastic properties.In the context of semiconductor solids, Rayleigh wave propagation is of particular interest due to the unique electronic and mechanical properties of these materials, which have potential applications in a variety of fields including electronics, optoelectronics, and energy harvesting [5][6][7].
The theory of Rayleigh wave propagation in solids was first proposed by Lord Rayleigh in 1885 [8].According to this theory, surface waves are formed as a result of the interaction between two types of waves; compressional waves (P-waves) and transverse waves (S-waves).These waves have different velocities and amplitudes and their interaction leads to the formation of a surface wave with a characteristic displacement pattern.In semiconductor solids, the propagation of these waves is influenced by several factors, including the mechanical properties of the material (e.g.Young's modulus, Poisson's ratio), the electrical conductivity, and the doping concentration Xu et al found that a semiconductor solid crystal structure, which can be customized by procedures like epitaxy and ion implantation, determines its mechanical characteristics [9].In addition, the electronic properties of a semiconductor can be modified through doping with impurities, which can alter its electrical conductivity and other characteristics studied by Pustelny [10].
The propagation of Rayleigh waves in semiconductor solids can be studied using various experimental techniques, including laser-based acoustic microscopy, surface acoustic wave spectroscopy and Brillouin light scattering.These techniques allow researchers to measure the velocity, amplitude and dispersion of these waves in a given material, as well as the dependence of these properties on various external factors such as temperature, pressure and doping concentration studied by Salah et al [11].One of the key applications of this wave propagation in semiconductor solids is in the field of surface acoustic wave (SAW) devices.SAW devices are used in various electronic applications, including filters, oscillators and sensors.These devices operate based on the interaction between an electrical signal and a surface acoustic wave, which propagates along the surface of the semiconductor material.The properties of the SAW device are highly dependent on the characteristics of the Rayleigh wave, such as its velocity and attenuation, making Rayleigh wave propagation an important consideration in the design and fabrication of these devices by [12][13][14].
Ma et al have studied in addition to its applications in electronics, Rayleigh wave propagation in semiconductor solids has also been studied for its potential use in energy harvesting [15].By coupling a piezoelectric material to a semiconductor, it is possible to convert the mechanical energy of this wave into electrical energy, which can be used to power small electronic devices [16,17].This has potential applications in fields such as wireless sensor networks and internet of things (IoT) devices, which require low-power, selfcontained energy sources described by Sun et al [18].Rayleigh waves are a type of surface wave that propagates along the surface of solids.These waves are known for their ability to cause significant damage to buildings and other structures during earthquakes.In the field of materials science and engineering, it is important to understand the behavior of Rayleigh waves in various types of materials, including semiconductors.In this discussion, we will explore Rayleigh wave propagation in a semiconductor with temperature-dependent properties [19][20][21][22].
Rayleigh wave propagation in a semiconductor solid is the effect of defects and impurities on wave propagation.Defects and impurities in semiconductors can have a significant effect on the elastic properties of the material, which in turn can affect the propagation of Rayleigh waves.For example, the presence of defects can lead to scattering of the Rayleigh waves, which can result in a reduction in the amplitude of the waves as they propagate [23].The presence of external stresses and strains can also affect the propagation of Rayleigh waves in a semi-conductor solid.External stresses and strains can cause changes in the elastic properties of the material, which can in turn affect the wave speed and amplitude of Rayleigh waves.For example, the application of external stress can cause a change in the wave speed of Rayleigh waves, which can result in a shift in the frequency of the waves [24].A theory proposed by Abouelregal is a 3PL thermoelastic heat transfer model with a fractional order time derivative that can contain all the previous theories of thermoelasticity of heat conduction model simultaneously [25].The work has many applications in science and engineering, seismic engineering, seismology, nuclear reactors, aerospace, submarine systems, physical infrastructure, construction and nondestructive research.The theory of the fractional derivative is effectively applied to thermo-elasticity theories and numerous models of fractional order thermo-elasticity have been developed by various authors [26][27][28].In the current work, higher-order time-fractional derivatives of 3PL thermo-elastic heat conduction have been used on fractional calculus.A different approach to the construction model is created by extending the Raychaudhuri and Ezzat models with the time-fractional order expansion through the Taylor series [29].
In a semiconductor with temperature-dependent properties; the velocity and attenuation of Rayleigh waves can vary significantly as a function of depth into the material.This can lead to complex wave patterns that are difficult to analyze using simple models.By understanding the behavior of Rayleigh waves in these materials, it is possible to develop strategies for designing materials and structures that are more resistant to the damaging effects of earthquakes and other types of seismic activity [30].It is used for non-destructive testing, thin film characterization, surface acoustic wave devices and materials research and development.The use of advanced modeling and simulation techniques, along with experimental measurement, can provide valuable insights into the complex behavior of Rayleigh waves in semiconductors [31][32][33][34][35].
In this paper, we investigate Rayleigh wave propagation in a thermoelastic, isotropic, homogeneous, and semi-conducting solid with temperature-dependent properties.3PL heat transfer with fractional order derivative is applied to the considered medium.The Rayleigh wave secular equations will be mathematically derived at the thermally insulated, carrier density and stress-free boundaries.

Basic equation
Fourier's law of heat conduction is defined as the time rate of the heat transfer within a material that is directly proportional to the negative gradient of temperature and to the area perpendicular to the direction of heat flow.It is mathematically expressed as, Hence, q is the heat flow rate in the process of conduction and k is the thermal conductivity of the material and "T is a temperature gradient.In the theory of uncoupled thermoelasticity, the heat equation has a parabolic form, indicating the infinite speed of heat wave propagation.In which the modification made by Maxwell-Cattaneo that temperature gradient and heat flux occur at different times based on this representation [32], Tzou established the thermo-elasticity theory of DPL by encompassing two phase lag allied with heat flux and temperature gradient [36].The law of heat conduction of the DPL model can be expressed as; The extension of Tzou's idea was extended by Roychoudhuri [37] on Green-Naghdi model III with the addition of third phase lag.The thermoelasticity theory of 3PL is introduced as, Within this study, this theory is considered as developed by taking Taylor,s expansion of the time-fractional order i.e. α, established on the entire equation of 3PL heat conduction model and recollecting terms up till αorder terms in τ v and τ T and terms up till 2α-order terms for τ q .
The three-phase lag model having fractional order time derivative is given as follows, where, Therefore, τ T , τ q , τ v are expressed as the phase-lag of the temperature gradient, heat flux and thermal displacement gradient.Also, " 2 , α, T 0 represents the Laplace operator, the fraction order parameter and the reference temperature.
E ij ij represents specific heat, thermal conductivity and material constant.

Special cases
• By setting the fractional parameter to α = 1 and substituting it into the heat equation (1), we obtain the standard three-phase-lag model [24].
• The DPL equation can be obtained by setting the values of the parameters to α = 1, k * = 0, and τ v = 0 in equation (1) [22].
• When α → 1 and τ q = 0 = τ T = τ v are applied to the heat equation (1), the GN III model can be derived as a limiting case [38].
equation of motion and plasma transport equation for a semi-conductor thermoelastic solid rotating with fixed angular frequency W  is represented as, , where α ij is coefficient of linear thermal expansion.The terms Ω × (Ω × u) and 2Ω × u are the additional centripetal acceleration due to time-varying motion and Coriolis acceleration respectively.N is the carrier density,T is the temperature above the reference temperature T 0 , δ n is the difference of deformation potential, k is the coefficient of thermal conductivity, E g is the energy gap of semi-conductor, D E is the carrier diffusion coefficient, τ is the photo-generated carrier lifetime, N 0 is the carrier concentration at temperature T.
Stress-strain relationship in Cauchy theory in the semi-conductor thermoelastic medium is given by the following expression.

Formulation of the problem
Within this work, a continuum medium with homogeneous isotropic nature with temperature-dependent parameters is considered.The considered medium is having semiconductor properties and is half a space continuum.Governing equations are represented by using Cartesian coordinates systems Oxyz with domain The semi-conductor is considered rotating about y − axis i.e.
W .The field equations of the problem for homogeneous semi-conducting isotropic thermoelastic material are expressed in equation ( 1)-( 3)).
In general, the properties of semiconductor materials change with small temperature changes.The effect of temperature on solids becomes important when studying the problem of thermal stress, especially for testing the thermal conductivity of solids.To study the interaction of temperature-dependent parameters and thermoelastic waves, we consider the following parameters to be temperature-dependent, , , , , , , , ,  , ,  , , , ,  ,  , , 5 The condition (5), modifies the equation (4) as follows, where, f (T) is dimensionless temperature function.We now have various choices to assume the temperature function, keeping in view the simplicity and linearity, the temperature function is denoted as, where α * is an empirical material constant and T 0 is ambient temperature.The system of equations (1), ( 2) and (3) represented the problem are expressed as, The plane strain problem for the medium is considered as, Now by putting the, equation (10) in equations (7-9), yields.
This model assumes that the medium is in a state of rest initially, and that the reference temperature is maintained in the undisturbed state.The following are the initial and regular conditions: Following non-dimensionless parameters are introduced, Upon introducing the quantities defined equation (15) in equations (11)(12)(13)(14), and suppressing the primes, and yields.

Solution of the problem
Wave propagating through the medium is considered to propagate along x − axis with temporal time t.The amplitude of the wave varies exponentially along the z − axis with field variables converging to zero as z → ∞ .The solution of the Rayleigh wave can be represented as [41]: Therefore, m is the penetration depth, c = w x is the dimensionless phase velocity, ξ is the wavenumber and ω is the angular frequency.Now by putting the solution of Rayleigh wave in equation (16)(17)(18)(19), we get, The coefficients of the characteristic equation are expressed in appendix A. The characteristic equation (24) gives us four roots m 2 j , where j = 1, 2, 3, 4. As the concern is for the surface wave only so the motion is restricted to the free surface z = 0 of the half-space, satisfying the radiation condition  ( ) Re m 0 p .The displacement and temperature change can be written as [42], A e e Within this paper, A j ( j = 1, 2, 3, 4) are arbitrary constants and coupling constants are, The coefficients of the coupling constants are expressed in appendix B.

Boundary conditions
The stress-free surface of the semiconducting thermoelastic solid with temperature dependent properties locates at z = 0 and α * ≠ 0, which is where the boundary conditions are imposed.
(1) The surface is assumed to be both isothermal and torsion-free, (2) Carrier density boundary condition, here, s¢ represents the speed of recombination.

Derivation of secular equation
The secular equation of a Rayleigh wave propagating in a semiconductor solid with temperature-dependent properties can be derived by solving the equations of motion for a layered solid medium.The secular equation relates the wave number, frequency and material properties of the solid.For a semiconductor solid with temperature-dependent properties, the material properties such as density, elastic moduli and thermal conductivity may vary with temperature.Making use of equations (11)(12)(13)(14) and the equation of temperature and displacement change in equations (28)(29)(30)(31), we get a system of simultaneous homogeneous equations as, The system of equation (32) has non-trivial solution |k ij | 4×4 = 0, is the matrix of coefficients that relates horizontal and vertical components of the displacement and stress in each layer of the medium, the purpose of this equation is to derive the dispersion relation for Rayleigh waves, which establish a relationship between the wave number, frequency and phase velocity of the wave.
The obtained secular equation contains information about the wave number, attenuation factor, phase velocity, etc of the waves in the selected medium with temperature dependence.

Numerical results and discussion
In the numerical discussion we deal with different properties of Rayleigh wave obtain from equation (32).Theoretical results are calculated numerically for a specific medium, using selected elastic parameters and constants that correspond to silicon-type materials [43], By using MATLAB software we represent graphically the variation in phase velocity, specific heat loss, penetration depth and attenuation coefficient for different rotational frequency W  and fractional parameter α. Figure 1 illustrates the variation of phase velocity and attenuation factor concerning the empirical material parameter.Phase velocity and attenuation are inversely related to each other.From the graphical representation, it can be concluded that the empirical material parameter is directly proportional to the phase velocity while inversely proportional to the attenuation factor of it.By increasing the empirical material parameter of the medium along y − axis.Whereas α * is directly related to (ω) angular frequency of the wave.
Figure 2 depicts the variation of penetration depth and specific heat loss for Rayleigh wave concerning the empirical material parameter.According to the graphical representation, penetration depth increases by increasing the empirical material parameter while specific heat loss decreases.The convergence rate of factor also increases by increasing the α * .
Figure 3 describes the variations of phase velocity and attenuation factor concerning the rotational frequency and fractional order parameter.From the graphical representation, it can be concluded that rotational frequency and fractional order parameter along the specific ais results in an increase of phase velocity of Rayleigh wave propagation along x − axis, while a decrease in the attenuation factor of it.
Figure 4 describes the variation of penetration depth and specific heat loss concerning both rotational frequency and fractional order parameters.There is no direct relation between the specific heat loss of a medium and the penetration depth of the wave propagating through it.From a graphical point of view, both rotation and   fractional order parameter increases the penetration depth and it happens only if it results in a reduction of specific heat loss for a medium.

Conclusion
We can deduce the following findings from the analyses offered in this paper and the graphical results: A mathematical model has been developed to investigate the propagation of Rayleigh waves in a homogeneous isotropic semi-conducting thermoelastic medium subjected to the influence of empirical material parameter (α * ), rotation (Ω) and fractional order parameter (α).The model includes a derivation of the secular equation for surface waves.The properties of the wave, including its phase velocity and attenuation coefficient, have been determined and represented graphically.The graphs illustrate that the magnitude of several wave parameters is significantly affected by both the rotation and the fractional order parameter.
• Penetration depth increases by increasing the fractional order parameter while the specific heat loss reduces.
• Rotation frequency along the specific axis results in an increase of phase velocity of Rayleigh wave propagating along x − axis.while decreasing the attenuation factor of it.
• The penetration and specific heat loss are related to fractional order parameters in the same way as that of rotational frequency.i.e., penetration depth increases with the increase while specific heat loss reduces by increasing rotational frequency.
• Fractional order parameter is directly proportional to the phase velocity while inversely proportional to the attenuation factor.
• It is evident that the inclusion of the rotational component and fractional order parameter has a significant impact on the fields under investigation.
Since the destructive nature of earthquakes is significantly influenced by Rayleigh waves, the findings of this study may prove valuable in the investigation of earthquakes and associated phenomena.Moreover, elastic wave attenuation contains extensive information regarding the properties of the medium.Rayleigh waves in semiconductor solids can be used for non-destructive testing, providing valuable insights into the mechanical properties of the material and the detection of potential defects or surface irregularities.In addition, they are used in the characterization of thin films and determining their thickness, adhesion and uniformity, which is essential for the production of semiconductor components.Surface acoustic wave devices that utilize Rayleigh waves enable filtering, modulation, and sensing tasks, making them essential components in semiconductor applications such as RF filters and sensors.
The system of homogeneous equations from(20)(21)(22)(23) by equating the determinant of u * , w * , T * and N * to zero.We obtain the equation in m 2 as,

7Figure 1 .
Figure 1.Varying phase velocity and attenuation coefficient with empirical material parameter.

Figure 2 .
Figure 2. Varying phase velocity and attenuation coefficient with empirical material parameter.

Figure 3 .
Figure 3. Varying phase velocity and attenuation coefficient with rotational frequency and fractional order parameter. 26