Modelling Reflection and Transmission of AcousticWaves at a Semiconductor : Fluid Interface

The paper concentrates on the study of reflection and transmission characteristics of acoustic waves at the interface of a semiconductor half-space underlying an inviscid liquid. The reflection and transmission coefficients varying with the incident angles are examined. Calculated results are verified by considering the quasilongitudinal (qP) and quasitransverse (qSV) waves. The special cases of normal and grazing incidence are also derived and discussed. Finally, the numerical computations of reflection and transmission coefficients are carried out with the help of Gauss elimination method by using MATLAB programming software for silicon (Si) and germanium (Ge) semiconductors. The computer simulated-results have been plotted graphically for Si and presented in tabular form in case of Ge semiconductors. The study may be useful in semiconductors, geology, and seismology in addition to surface acoustic wave (SAW) devices.


Introduction
Jeffrey's [1] and Gutenberg [2] considered the reflection of elastic plane waves at the surface of a solid halfspace.Sidhu and Singh [3] investigated the propagation of plane waves in a prestressed elastic solid possessing orthotropic symmetry and showed that the velocities of qL and qSV waves depend upon the angle of propagation.Rayleigh [4] considered the reflection and transmission of waves from an undulated boundary surface of an elastic solid.Knott [5] derived the equations for reflection and refraction of waves at plane boundaries.The reflection and refraction phenomenon of elastic waves in solids under different situations has been treated in detail as reported in books [6][7][8].Deresiewicz [9] studied the reflection of a plane waves from the stressfree boundaries of thermoelastic halfspace.Abo-dahab [10] studied the propagation of P waves from the stress-free surface of elastic half-space with voids under the influence of thermal relaxation and magnetic field.It is found that the angle of incidence (θ) significantly affects the reflection coefficients and the thermal relaxation time has negligible small effect on the amplitude of reflection coefficients.Madeo and Gavrilyuk [11] studied the propagation of acoustic waves in porous media including their reflection and transmission at pure fluid/porous medium permeable interface.A. N. Sinha and S. B. Sinha [12] studied the reflection of generalized thermoelastic waves from the free surface of a solid halfspace.Sharma et al. [13] studied the reflection of generalized thermoelastic waves from the boundary of a transversely isotropic halfspace.Abd-alla [14] considered the effect of relaxation time on reflection of generalized magnetothermoelastic waves.Lockett [15] studied the effect of thermal properties of a solid on the velocity of waves.Chadwick and Snedon [16] studied the reflection of plane waves in an elastic solid conducting heat.
Maruszewski [17] presented theoretical considerations of the simultaneous interactions of elastic, thermal, and diffusion of charge carrier fields in order to study surface waves in semiconductors.Sharma and Thakur [18] nondimensionalized the model [17] and studied the plane harmonic elastodiffusive surface wave in semiconductor material.Recently, J. N. Sharma and A. Sharma [19] studied the reflection of acoustodiffusive waves from the stress-free boundary of a semiconductor halfspace.As per knowledge of authors no study of reflection and transmission of waves based on the models of basic governing equations given in references [17,18] is available in the literature.Ultrasonic waves are reflected at boundaries due to the acoustic impedance mismatch of the materials on each side.Reflection and transmission coefficients are utilized for the conversion of longitudinal to shear waves and vice versa.This feature is extremely useful in the construction of shear wave transducers.Reflection coefficients also affect the response of a transducer to a sinusoidal signal and are useful in sonography as well as in signal processing.
Keeping in view the above and applications of semiconductors in acoustic devices, the present paper is an attempt to explore the reflection and transmission characteristics of elastic waves at the interface between elastic semiconductor (n or p-type) halfspace and inviscid liquid semispace.The mathematical model consisting of governing partial differential equations of motion and charge carriers' diffusion of n-type and p-type semiconductors has been solved both analytically and numerically in the study.The computer-simulated results so obtained with the help of MATLAB programming in respect of in case of silicon (Si) and germanium (Ge) semiconductors in contact with water have been illustrated graphically.

Formulation of the Problem
We take the origin of rectangular Cartesian coordinate system oxyz at a fixed point on the boundary of the semiconductor halfspace with positive z-axis directed normally into the solid medium and x-axis along the direction of propagation of waves; the y-axis is taken in the direction of the line of intersection of the plane wave front with the plane surface as shown in Figure 1.If we restrict our analysis to plain strain in the xz-plane, all the field variables may be taken as function of x, z, and t only.The basic governing equations of motion and diffusion of charge carrier fields for a homogeneous isotropic, elastic (n-type and p-type) semiconductors, in the absence of body forces and electromagnetic forces, are given as [17,18].n-type semiconductor p-type semiconductor where Further (1)-( 2) are subjected to following assumptions [17,18].
(i) All the considerations are made in the frame work of the phenomenological model.
(ii) The electric neutrality of the semiconductor is satisfied.
(iii) The magnetic field effect is ignored.
(iv) The mass of charge carriers is negligible.
(v) The electric field with in the boundary layer is very weak and can be neglected.
(vi) The recombination functions of electrons and holes are selected on the basis of facts that take care of the defects and hence the concentration values of the charge carrier fields.
The nonvanishing components of stress tensor in the semiconductor are given by n-type: Advances in Acoustics and Vibration 3 p-type: The basic governing equation for inviscid fluid medium is given by where λ L is bulk modulus and ρ L and u L are the density of the fluid and velocity vector, respectively.We define the quantities where ω * is the elastodiffusive characteristic frequency and c 1 , c 2 are, respectively, the longitudinal and shear wave velocities.Here ε L is the thermomechanical coupling and c L is the velocity of sound in the fluid.Such quantities in case of p-type semiconductor can be written from those in (6) by replacing the subscript/superscripts n with p and the quantity N with P. Upon using quantities ( 6) in ( 1)-( 4), and (5), we obtain (n-type semiconductor) The equations for p-type semiconductor can be written from ( 8) and ( 9) by replacing N with P and superscripts/subscripts n with p.
We introduce the elastic potential functions φ and ψ through the relations However, in general, such a decomposition of displacement vector is not possible in case of anisotropic materials [3].
Upon introducing (11) in ( 8) and ( 9), we get Similarly for p-type semiconductor, we have In case the semiconductors are of relaxation type, the life time and relaxation time become comparable to each other (t n ∼ = t + n ), and consequently, ( 14) get simplified.The stresses (10) in terms of potential functions φ and ψ with the help of ( 12) to (18) become for both n-type and p-type semiconductors.In the fluid medium, the nondimensional displacements are related to scalar and vector velocity potential through the relations given by Substituting (20) in (12) we get This is the equation for waves in the inviscid fluid.

Boundary Conditions
Following sets of boundary conditions are assumed to hold at solid-fluid interface z = 0 of the semiconductor halfspace [20].
(1) The magnitude of the normal component of stress tensor of the elastic half-space should be equal to the pressure of the liquid.This implies that φ (2) The tangential components of stress tensor of the solid should be zero, which implies that Advances in Acoustics and Vibration (3) The normal component of displacement of the solid should be equal to that of fluid, which implies that (4) The electron and hole charge carrier fields satisfy the following conditions at the interface z = 0 where h n = s n /c 1 , h p = s p /c 1 , s n and s p surface recombination velocities of electron and holes, respectively.

Reflection and Transmission of Plane Waves
We assume plane wave solution of the form where υ = ω/k, ω is circular frequency, and k is the wave number.Upon using (26) in ( 13)- (18) and in ( 21) we obtain a system of algebraic equations in unknowns A, B, C, D.
The condition for the existence of nontrivial solution of these systems of equations provides us n-type: p-type: where (29)

Reflection and Transmission in Case of qP Wave Incidence
Let the suffixes i and r represent incident and reflected waves, respectively.Omitting the term exp(−ιωt), the solution (26) for the function φ, ψ, N, φ L and P in case of incidence and reflected waves can be written as A j exp ik j x sin θ j + z cos θ j , S j A j exp ιk j x sin θ j + z cos θ j , where In the absence of electron field (N = 0 = ε n ) and hole carriers (P = 0 = ε p ), we have n-type: Case 1 (Quasilongitudinal (qP) wave incidence at an interface from the semiconductor).In this and the following sections, we shall confine our discussion to n-type semiconductor unless stated otherwise, and results in case of p-type semiconductor can be written from the expressions of various quantities obtained here by adopting the same procedure.
Because of coupling between various field functions the reflected fields in case of qP wave incidence at the interface are given by Upon using the above equations, we calculate the stresses from (19) and then employing the boundary conditions ( 22)-( 25) to obtain a system of four coupled algebraic equations given in the appendix.Since all the waves, incident, reflected, or transmitted, must be in phase at the surface z = 0 for all values of x and t, therefore from equations (A.1) we have Upon using (27) in the above relation, we obtain This is modified form of the Snell's law for the considered material.In the absence of electron field (N = 0, ε n = 0), (35) becomes This is Snell's law [6].Solving the systems of equations (A.2) with the help of Gauss elimination method, the amplitude ratios R qP k (k = 1, 2, 3) and T qP 1 are obtained as where +a 1 a 2 δ 2 (a 1 S 2 sin 2θ 1 cos θ 2 − a 2 S 1 sin 2θ 2 cos θ 1 ) sin θ 3 , (40) In the absence of fluid medium (37) becomes (42) In the absence of fluid medium and electron field (N = 0, ε n = 0), the expression (37) and (42) for amplitude ratios with the help of (32) becomes δ 2 sin 2θ 1 sin 2θ 3 − cos 2 2θ 3 δ 2 sin 2θ 1 sin 2θ 3 + co s 2 2θ 3 , 2 δ 2 sin 2θ 1 cos 2θ 3 δ 2 sin 2θ 1 sin 2θ 3 + cos 2 2θ 3 . (43) These relations are in complete agreement with the corresponding equations as given by Achenbach [6] in case of elastokinetics.In case of grazing incidence, (θ = 90 • = θ 1 ) the amplitude ratios given by ( 37)-(39) with the use of Snell's law provide us This shows that qSV wave and electron waves are not reflected and qP wave annihilates itself being 180 • out of phase with the incident wave.Similarly for normal incidence (θ = 0 • = θ 1 ), the corresponding values of reflection coefficients from (37)-( 42) are again obtained as Here qP wave gets reflected and transmitted in case of normal incidence.
Case 2 (Quasitransverse (qSV ) wave incidence at an interface from semiconductor).We now consider the reflection of a plane qSV wave for similar conditions on the boundary as in the previous section.For qSV wave, we have A r exp{ιk r (x sin θ r + z cos θ r )}, S r A r exp{ιk r (x sin θ r + z cos θ r )}, φ L = A 4 exp{ιk 4 (x sin θ 4 − z cos θ 4 )}. (46) Upon using solution (46) in the boundary conditions (22) and (25) at the surface z = 0 and assuming that all the incident or reflected waves are in phase at this surface for all values of x and t, we have the relation This relation implies that This is the modified form of Snell's law for the considered material, in this case.Upon solving (A.4) and (A.5), we get where

Advances in Acoustics and Vibration
Here Δ L and Δ S are defined in (40) and (41), respectively.In the absence of fluid medium (49) becomes where Δ S is defined by (41).In the absence of fluid medium and electron field (ε n = 0 = N), (49) and (50) reduce to Equations ( 51) agree with the corresponding equations in Achenbach [6] and Kino [7].For grazing incidence (θ = 90 • = θ 3 ) and in case of normal incidence (θ = 0 • = θ 3 ), the expressions for reflection coefficients in ( 49)-( 51) provide us Therefore, only shear wave is reflected as qSV wave in case of normal incidence, and the reflected qSV wave annihilates the incident qSV wave in case of grazing incidence case.The other two waves, namely, qP and N, are not reflected, and there is no transmitted wave in either case of normal or grazing incidence.

Energy Equations
From the principle of conservation of energy, the energy carried to the boundary by the incident wave must be equal to the energy carried away from the boundary by the reflected and refracted waves.For the incident qSV wave incidence the particle velocities are For reflected qSV waves, For transmitted qP waves, Taking the kinetic energy per unit volume as (1/2)ρ(v 2 + 2 ), we may calculate the energy flux for the waves mentioned above by multiplying the total energy per unit volume by the velocity of propagation and area of the wave front involved.Thus we may write the equality between the incident qSV wave energy and the sum of reflected qP, reflected qSV , and transmitted qSV -wave energies for the per unit area on the interface as (57) Here the cross-sectional areas of the incident, reflected, and transmitted waves are proportional to the cosines of the angles made by the ray directions of the waves with the normal to the interface.Hence the energy equation is given by where (59)

(p-Type) Semiconductor
The reflection coefficients and transmission coefficients in case of p-type semiconductor can be obtained by using boundary conditions ( 22)-(25) and solution (26) for the functions φ, ψ, φ L , and P by adopting the procedure of previous sections.The expressions of the reflection coefficients R qP K and R qSV K (k = 1, 2, 3) are again given by (37)-( 42) and ( 49)-(51) in case of qP-wave, qSV-wave incident at the surface of p-type semiconductor, respectively, with the replacement of a 2 i (i = 1, 2) as the a * 2 i (i = 1, 2) in all the relevant relations and equations including Snell's law.

Numerical Results and Discussion
In this section the reflection and transmission coefficients given by (37)-( 39) and ( 49)-(50) have been computed numerically for silicon (Si) and germanium (Ge) materials under the assumption of relaxation type semiconductor (ntype or p-type) under the assumption that semiconductor considered is of relaxation type so that t n , t + n and t p , t + p become comparable to each other in their values such that t n = t + n and t p = t + p .Here the fluid chosen for the purpose of numerical calculations is water, the velocity of sound in which is given by c L = 1.5 × 10 3 m/s and density is ρ L = 1000 kg/m 3 .
The physical data for silicon material is given below in Table 1.
From Figure 2, it is noticed that the magnitude of reflection coefficient (R qP 1 ) marginally increases in the range 0 • ≤ θ ≤ 10 • with increase in angle of incidence and decreases in the range 10 • ≤ θ ≤ 70 • ; that is, there is a sharp loss of energy which is noticed in the range 10 • ≤ θ ≤ 70 • before it sharply increases upto θ = 90 • in case of qP wave incidence at the interface of semiconductor and fluid.The magnitude of reflection coefficient (R qP 3 ) increases for 0 • ≤ θ ≤ 60 • and attains a maximum value at θ = 75 • ; that is, loss of energy by reflected wave (R qP 1 ) is covered by reflected wave (R qP 3 ).However, a meager amount of energy is associated with an electron wave (R qP 2 ).It is also noticed that transmission coefficient T qP 1 attains a maximum value at θ = 0 • and then varies linearly up to 0 • ≤ θ ≤ 15 • ; after that it starts decreasing up to θ = 90 • .This implies that in case of qP wave incidence at the interface of semiconductor and fluid, transmitted wave also travels with sufficient amount of energy.It is revealed that at grazing incidence θ = 90 • , the reflection coefficients R qP k (k = 1, 2, 3) and transmission coefficient T qP 1 of incident qP wave vanish, thereby meaning that reflected qP wave annihilates the incident qP wave.Whereas in contrast to this qP wave is reflected and transmitted as qP wave at normal incidence (θ = 0 • ), and other waves are not reflected or transmitted.Thus maximum energy is carried by longitudinal (qP) waves, reflected or transmitted, at normal and grazing incidence, though it is transported in a distributed manner among all coupled waves at other angles of incidence.The trend and nature of reflection/transmission coefficients in case of qP wave incidence in Figure 2 almost completely agree with those presented in Kino [7].
In Figure 3, it is noticed that reflection coefficient R qSV 1 increases with increase in angle of incidence in the range 0 • ≤ θ ≤ 30 • and attains a maximum value at (θ = 30 • ) due to stresses generated in the semiconductor material; after that it starts decreasing up to 30   for qSV wave decreases in the range 0 • ≤ θ ≤ 30 • , increases up to θ = 45 • and remains steady for 45 • ≤ θ ≤ 60 • and again decreases to attain its minimum value at θ = 72 • , and then sharply increases up to θ = 90 • ; that is, it recovers from the initial loss of energy.T qSV 1 increases in the range 0 • ≤ θ ≤ 30 • to attain its maximum value at θ = 45 • and decreases up to θ = 90 • .It is also noticed that about 40% of incident energy can be converted to a longitudinal wave in water at incident angles for which the reflected longitudinal wave is cutoff.Thus solid-liquid interface is a perfect reflector.increases sharply with increase in the angle of incidence and attains a significantly large value at θ = 30 • and after that it starts decreasing up to a critical angle θ = 45 • , which is θ = 36 • in case of stress-free boundary.This is attributed due to high stress generation in the material at this angle of incidence.Beyond this cutoff point whole of as θ passes through the critical angle in case of qSV wave incident at the stressfree, isoconcentrated, or impermeable surface of silicon (Si) half-space [20].Transmission coefficient T qSV 1 increases with increase in angle of incidence and attains a maximum value at θ = 30 0 ; after that it starts decreasing and vanishes at θ = 90 0 .This implies that major portion of energy is carried by transmitted wave in comparison to R qSV 3 in qSV wave incidence at the interface of semiconductor and fluid under electron field equilibrium.
From Table 2 it is noticed that the behaviors of reflection/transmission coefficients of various waves in germanium (Ge) semiconductor halfspace are almost similar to that in case of silicon (Si) semiconductor except some minor changes in their magnitudes.From Table 3, we concluded that the law of conservation of energy is valid.

Conclusions
It is noticed that the magnitude of reflection coefficient (R qP 1 ) of qP wave decreases with increasing angle of incidence of qP wave in case of silicon (Si) semiconductor material halfspace in the range 0 • ≤ θ ≤ 70 • , and the numerical results show that in case of qP wave incidence maximum energy is carried by transmitted longitudinal wave in the presence of electron wave which takes meager amount of energy.Thus energy transfer is by the phonon of the system, and partition of energy depends upon the angle of incidence.However in case of qSV wave incidence at the surface, transmitted wave becomes more prominent in the presence of electron field and energy is transported in distributed manner among the other waves.The dependence of R qP i (i = 1, 2, 3) on a i (1, 2) shows that the reflection coefficients do depend upon the frequency of waves and hence are dispersive in character.The study may find application in semiconductor, seismology, and signal processing devices in coated structures.

1 Figure 2 :
Figure 2: qP wave incidence at the interface of semiconductor and fluid.

Figure 3 :
Figure 3: qSV wave incidence at the interface of semiconductor and fluid.

Figure 4 ,
the variation of reflection coefficients R qSV k (k = 1, 3), and transmission coefficients T qSV 1 versus angle of incidence have been plotted.It is noticed that the reflection coefficient R qSV 1

1 Figure 4 :
Figure 4: qSV wave incidence at the interface of semiconductor and fluid under electron field equilibrium.
• ≤ θ ≤ 60 • ; this means that electron wave gets sufficient amount of energy before it dies out at θ = 90 • which is observed to be a new phenomenon here.The reflection coefficient R qSV 3

Table 3 :
Variation of energy coefficients (Z) in case of (qSV ) wave incidence versus angle of incidence (θ) in degree.