# Modelling Reflection and Transmission of Acoustic Waves at a Semiconductor: Fluid Interface

Authors: A. Sharma; J. N. Sharma; Y. D. Sharma
Journal: Advances in Acoustics and Vibration
 (2012)

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## Abstract

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.


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## Body

## 1. 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 [6–8]. Deresiewicz [9] studied the reflection of a plane waves from the stress- free 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.


## 2. Formulation of the Problem

We take the origin of rectangular Cartesian coordinate systemoxyz 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 tonly. 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].Figure 1
Geometry of the problem.n-type semiconductor(1)μ∇2u⃗+(λ+μ)∇∇⋅u⃗-λnN=ρu⃗̈,ρDn∇2N+ρ[1tn+-(1-tntn+)∂∂t-tn∂2∂t2]N-a2nT0λT∇⋅u⃗̇=0,p-type semiconductor(2)μ∇2u⃗+(λ+μ)∇∇⋅u⃗-λpP=ρu⃗̈,ρDp∇2P+ρ[1tp+-(1-tptp+)∂∂t-tp∂2∂t2]P-a2pT0λT∇⋅u⃗̇=0,
where ∇2=(∂2/∂x2)+(∂2/∂z2) is the Laplacian operator, N(x,z,t)=n-n0 is the electron concentration change, P(x,z,t)=p-p0 is the hole concentration change, and u⃗(x,z,t)=(u,0,w) is the displacement vector. Here λ,μ are Lame parameters;ρ is the density of the semiconductor; λn=(3λ+2μ)αN and λp=(3λ+2μ)αP are the elastodiffusive constants of electrons; αN,αP are the coefficients of linear electron and holes concentration expansions. Dn and Dp are the diffusion coefficients of electron and hole carriers; tn,tp and tn+,tp+are the relaxation and life times of the electron and hole charge carriers, respectively; n,p and n0,p0 are, respectively, the nonequilibrium and equilibrium values of electrons and holes concentrations of the semiconductors; T0 and λT=(3λ+2μ)αT are the uniform temperature and adiabatic thermomechanical coupling constant, respectively; a2n, a2p are the flux-like parameters. The superposed dot represents differentiation with respect to time.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 byn-type:(3)τzz=(λ+2μ)∂w∂z+λ∂u∂x-λnN,τxz=μ(∂u∂z+∂w∂x),p- type:(4)τzz=(λ+2μ)∂w∂z+λ∂u∂x-λpP,τxz=μ(∂u∂z+∂w∂x).
The basic governing equation for inviscid fluid medium is given by
(5)λL∇∇⋅u⃗L=ρLu⃗̈,
where λL is bulk modulus and ρL and u⃗L are the density of the fluid and velocity vector, respectively.We define the quantities(6)x′=ω*xc1,z′=ω*zc1,t′=ω*t,N′=Nn0,w′=ρω*c1λnn0w,u′=ρω*c1λnn0u,τij′=τijλnn0,tn′=tnω*,tn+′=tn+ω*,δL2=cL2c12,cL2=λLρL,δ2=c22c12,ω*=c12Dn,c12=λ+2μρ,c22=μρ,(7)εn=a2nT0λTλnρ(λ+2μ),uL′=ρω*c1λnn0uL,
where ω* is the elastodiffusive characteristic frequency and c1, c2 are, respectively, the longitudinal and shear wave velocities. Here εL is the thermomechanical coupling and cL 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)(8)δ2∇2u⃗+(1-δ2)∇∇⋅u⃗-∇N=u⃗̈,(9)∇2N-[tn∂2∂t2+(1-tntn+)∂∂t-1tn+]N-εn∇⋅u⃗̇=0,(10)τxz=δ2(∂u∂z+∂w∂x),τzz=(1-2δ2)∂u∂x+∂w∂z-N,(11)δL2∇∇⋅u⃗L=u⃗̈L.
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 functionsϕ and ψ through the relations(12)u=∂ϕ∂x+∂ψ∂z,w=∂ϕ∂z-∂ψ∂x.
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(13)∇2ϕ-ϕ̈-N=0,(14)∇2N-[tn∂2∂t2+(1-tntn+)∂∂t-1tn+]N-εn∇2ϕ̇=0,(15)∇2ψ=ψ̈δ2.
Similarly for p-type semiconductor, we have(16)∇2ϕ-ϕ̈-P=0,(17)∇2P-[tp∂2∂t2+(1-tptp+)∂∂t-1tp+]P-εp∇2ϕ̇=0,(18)∇2ψ=ψ̈δ2.
In case the semiconductors are of relaxation type, the life time and relaxation time become comparable to each other (tn≅tn+), and consequently, (14) get simplified. The stresses (10) in terms of potential functions ϕand ψ with the help of (12) to (18) become(19)τzz=ϕ̈-2δ2(ϕ,xx+ψ,xz),τxz=ψ̈+2δ2(ϕ,xz-ψ,xx)
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(20)uL=ϕL,X,wL=ϕL,Z.
Substituting (20) in (12) we get(21)δ2∇2ϕL-ϕ̈L=0.
This is the equation for waves in the inviscid fluid.


## 3. Boundary Conditions

Following sets of boundary conditions are assumed to hold at solid-fluid interfacez=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(22)ϕ̈δ2-2(ϕ,xx+ψ,xz)=ρLϕ̈Lρδ2.(2)
The tangential components of stress tensor of the solid should be zero, which implies that(23)ψ̈δ2-2(ψ,xx-ϕ,xz)=0.(3)
The normal component of displacement of the solid should be equal to that of fluid, which implies that(24)ϕ,z-ψ,x=ϕL,z.(4)
The electron and hole charge carrier fields satisfy the following conditions at the interfacez=0(25)∂N∂z+hn(1+tn∂∂t)N=0,∂P∂z+hp(1+tp∂∂t)P=0,
where hn=sn/c1, hp=sp/c1, sn and sp surface recombination velocities of electron and holes, respectively.


## 4. Reflection and Transmission of Plane Waves

We assume plane wave solution of the form(26)(ϕ,ψ,N,P,ϕL)=(A,B,C,D,E)×exp⁡{ιk(xsin⁡θ-zcos⁡θ-υt)},
where υ=ω/k, ωis 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 usn-type:(27)k12=a12ω2,k22=a22ω2,k32=ω2δ2,k42=a42ω2,p-type:(28)k1*2=a*12ω2,k2*2=a2*2ω2,k3*2=ω2δ2,k4*2=a4*2ω2,
where(29)a12+a22=1+αn*+ιω-1εn,a12a22=αn*,a32=1δ2,a42=1δL2,a1*2+a2*2=1+αp*+ιω-1εp,a12a22=αp*.


## 5. Reflection and Transmission in Case ofqP Wave Incidence

Let the suffixesi and r represent incident and reflected waves, respectively. Omitting the term exp(-ιωt), the solution (26) for the function ϕ, ψ, N, ϕL and P in case of incidence and reflected waves can be written as(30)ϕi=Aiexp⁡{ik1(xsin⁡θ1-zcos⁡θ1)},ϕr=∑j=12Ajexp⁡{ikj(xsin⁡θj+zcos⁡θj)},ψr=A3exp⁡{ik3(xsin⁡θ3+zcos⁡θ3)},Ni=S1Aiexp⁡⁡{ιk1(xsin⁡θ-zcos⁡θ)},Nr=∑j=12SjAjexp⁡{ιkj(xsin⁡θj+zcos⁡θj)},Pi=S1*Aiexp⁡{ιk1*(xsin⁡θ-zcos⁡θ)},Pr=∑j=12Sj*Ajexp⁡{ιkj*(xsin⁡θj+zcos⁡θj)},
where
(31)Sj=ω2-kj2=ω2(1-aj2),j=1,2,Sj*=ω2-kj*2=ω2(1-aj*2),j=1,2.
In the absence of electron field (N=0=εn) and hole carriers (P=0=εp),  we haven-type:a12=1,a22=αn*,a32=1δ2,a42=1δL2,p-type:a1*2=1,a2*2=αp*,(32)S1=0,S2=ω2(1-αn*),S1*=0,S2*=ω2(1-αp*).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 ofqP wave incidence at the interface are given by
(33)ϕ=ϕi+ϕr,N=Ni+Nr,P=Pi+Pr,ψ=ψr,ϕL=ϕLr.
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 andt, therefore from equations (A.1) we have
(34)k1sin⁡θ=k1sin⁡θ1=k2sin⁡θ2=k3sin⁡θ3=k4sin⁡θ4.
Upon using (27) in the above relation, we obtain
(35)θ=θ1,a1sin⁡θ1=a2sin⁡θ2=1δsin⁡θ3=a4sin⁡θ4.
This is modified form of the Snell’s law for the considered material. In the absence of electron field (N=0,εn=0), (35) becomes
(36)δsin⁡θ1=sin⁡θ3⟹sin⁡θ1c1=sin⁡θ3c2.
This is Snell’s law [6]. Solving the systems of equations (A.2) with the help of Gauss elimination method, the amplitude ratios RkqP(k=1,2,3) and T1qPare obtained as
(37)R1qP=1-2a2S2cos⁡θ2cos⁡22θ3a̅ΔL-ΔS,R2qP=-2S1a1a4δcos⁡θ4cos⁡θ1cos⁡22θ3ρ̅ΔL-a4δcos⁡θ4ΔS=-2S1a1cos⁡θ1cos⁡22θ3a̅ΔL-ΔS,R3qP=2a1a2δ2(a1S2sin⁡2θ1cos⁡θ2-a2S1cos⁡θ1sin⁡2θ2)cos⁡2θ3a̅ΔL-ΔS,(38)T1qP=2a1a2cos⁡θ1cos⁡θ2cos⁡22θ3(S2-S1)a4cos⁡θ4(a̅ΔL-ΔS)+a1a2δsin⁡2θ3(a1S2cos⁡θ2sin⁡2θ1-a2S1cos⁡θ1sin⁡2θ2)a4cos⁡θ4(a̅ΔL-ΔS),
where
(39)a̅=ρ̅a4δcos⁡θ4,(40)ΔL=cos⁡θ1cos⁡θ2cos⁡2θ3a1a2δ(S2-S1)+a1a2δ2(a1S2sin⁡2θ1cos⁡θ2-a2S1sin⁡2θ2cos⁡θ1)sin⁡θ3,(41)ΔS=[a1S1cos⁡θ1[cos⁡22θ3+a22δ2sin⁡2θ2sin⁡2θ3]-a2S2cos⁡θ2[cos⁡22θ3+a12δ2sin⁡2θ1sin⁡2θ3]].
In the absence of fluid medium (37) becomes
(42)R1qP=a1S1cos⁡θ1[cos⁡22θ3+a22δ2sin⁡2θ2sin⁡2θ3]ΔS+a2S2cos⁡θ2[cos⁡22θ3-a12δ2sin⁡2θ1sin⁡2θ3]ΔS,R2qP=-2a1S1cos⁡22θ3cos⁡θ1ΔS,R3qP=2a12a2δ2cos⁡2θ3[a2S1sin⁡2θ2cos⁡θ1-a1S2cos⁡θ2sin⁡2θ1]ΔS.
In the absence of fluid medium and electron field (N=0,εn=0), the expression (37) and (42) for amplitude ratios with the help of (32) becomes
(43)R1qP=δ2sin⁡2θ1sin⁡2θ3-cos⁡22θ3δ2sin⁡2θ1sin⁡2θ3+co⁡s22θ3,R2qP=0,R3qP=2δ2sin⁡2θ1cos⁡2θ3δ2sin⁡2θ1sin⁡2θ3+cos⁡22θ3.
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°=θ1) the amplitude ratios given by (37)-(39) with the use of Snell’s law provide us
(44)R1qP=-1,R2qP=0,R3qP=0.
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°=θ1), the corresponding values of reflection coefficients from (37)–(42) are again obtained as
(45)R1qP=-1,R2qP=0,R3qP=0=T1qP.
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 planeqSV wave for similar conditions on the boundary as in the previous section. For qSV wave, we have
(46)ψ=ψi+ψr=Aiexp⁡{ιk3(xsin⁡θ-zcos⁡θ)}+A3exp⁡{ιk3(xsin⁡θ3+zcos⁡θ3)},ϕ=ϕr=∑r=12Arexp⁡{ιkr(xsin⁡θr+zcos⁡θr)},N=Nr=∑r=12SrArexp⁡{ιkr(xsin⁡θr+zcos⁡θr)},ϕL=A4exp⁡{ιk4(xsin⁡θ4-zcos⁡θ4)}.
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 xand t, we have the relation
(47)k3sin⁡θ=k1sin⁡θ1=k2sin⁡θ2=k3sin⁡θ3.
This relation implies that
(48)θ=θ3,a1sin⁡θ1=a2sin⁡θ2=1δsin⁡θ3.
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
(49)R1qSV=-S2a2cos⁡θ2sin⁡4θ3a̅ΔL-ΔS,R2qSV=a1S1cos⁡θ1sin⁡4θ3a̅ΔL-ΔS,R3qSV=R(a̅ΔL-ΔS),T1qSV=2sin⁡2θ3ΔLρ̅ΔL-a4δcos⁡θ4ΔS.
where ℜ denotes {a̅ΔL-{a1a2δ2(a2S1cosθ1sin2θ2-a1S2cosθ2sin2θ1)×sin2θ3-(a1S1cosθ1-a2S2cosθ2)cos22θ3}}. Here ΔL and ΔS are defined in (40) and (41), respectively. In the absence of fluid medium (49) becomes
(50)R1qSV=S2a2cos⁡θ2sin⁡4θ3ΔS,R2qSV=-a1S1cos⁡θ1sin⁡4θ3ΔS,R3qSV=RΔS,
where ΔS is defined by (41). In the absence of fluid medium and electron field (εn=0=N), (49) and (50) reduce to
(51)R1qSV=-sin⁡4θ3cos⁡22θ3+δ2sin⁡2θ1sin⁡2θ3,R2qSV=0,R3qSV=δ2sin⁡2θ3sin⁡2θ1-cos⁡22θ3cos⁡22θ3+δ2sin⁡2θ3sin⁡2θ1,
Equations (51) agree with the corresponding equations in Achenbach [6] and Kino [7]. For grazing incidence (θ=90°=θ3) and in case of normal incidence (θ=0°=θ3),the expressions for reflection coefficients in (49)–(51) provide us
(52)R1qSV=0,R2qSV=0,T1qSV=0,R3qSV=-1.
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.


## 6. 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 incidentqSV wave incidence the particle velocities are(53)v̅=∂2ψi∂x∂t=ωk3sin⁡θAiexp⁡{ιk3(xsin⁡θ-zcos⁡θ-νt)},ϖ=∂2ψi∂z∂t=-ωk3cos⁡θ3Aiexp⁡{ik3(xsin⁡θ-zcos⁡θ-vt)}.
For reflected qSV waves,(54)v̅=∂2ψr∂x∂t=ωk3sin⁡θ3A3exp⁡{ik3(xsin⁡θ3+zcos⁡θ3-vt)},ϖ=∂2ψr∂z∂t=ωk3cos⁡θ3A3exp⁡{ik3(xsin⁡θ3+zcos⁡θ3-vt)}.
For reflected qP waves,(55)v̅=∂2ϕr∂x∂t=ωk1sin⁡θ1A1exp⁡{ik1(xsin⁡θ1+zcos⁡θ1-vt)}+ωk2sin⁡θ2A2exp⁡{ik2(xsin⁡θ2+zcos⁡θ2-vt)},ϖ=∂2ϕr∂z∂t=ωk1cos⁡θ1A1exp⁡{ik1(xsin⁡θ1+zcos⁡θ1-vt)}+ωk2cos⁡θ2A2exp⁡{ik2(xsin⁡θ2+zcos⁡θ2-vt)}.
For transmitted qP waves,(56)v̅=∂2ϕLr∂x∂t=ωk4sin⁡θ4A4exp⁡{ik4(xsin⁡θ4-zcos⁡θ4-vt)},ϖ=∂2ϕLr∂z∂t=-ωk4cos⁡θ4A4exp⁡{ik4(xsin⁡θ4-zcos⁡θ4-vt)}.
Taking the kinetic energy per unit volume as (1/2)ρ(v̅2+ϖ2), 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)c2cos⁡θ3k32=c1cos⁡θ2k12R12+c1cos⁡θ2k22R22+c2cos⁡θ2k32R32+ρLρcLcos⁡θ4k42R42.
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(58)1≅Z,
where
(59)Z=c2c1cos⁡θ2cos⁡θ3R12+R32-c2cLρLρcos⁡θ4cos⁡θ3R42.


## 7. (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, ϕL, and P by adopting the procedure of previous sections. The expressions of the reflection coefficients RKqP and RKqSV(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 ai2 (i=1,2) as the ai*2(i=1,2) in all the relevant relations and equations including Snell’s law.


## 8. 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 (n-type or p-type) under the assumption that semiconductor considered is of relaxation type so that tn,tn+ and tp,tp+ become comparable to each other in their values such that tn=tn+ and tp=tp+.Here the fluid chosen for the purpose of numerical calculations is water, the velocity of sound in which is given bycL=1.5×103 m/s and density is ρL=1000 kg/m3.The physical data for silicon material is given below in Table1.Table 1
Physical data of Silicon (Si) and Germanium (Ge) semiconductors.
CoefficientUnitValue (Ge)Value (Si)ReferenceλNm-20.48×10110.64×1011[17]μNm-20.53×10110.65×1011ρKgm-35.3×1032.3×103tn+s1 ps1 pstp+s1 ps1 psDnm2s-110-20.35×10-2Dpm2s-10.5×10-20.125×10-2αTK-15.8×10-62.6×10-6[21]n0=p0m-310201020[22]The values of reflection coefficientsRkqP(k=1,2,3), RkqSV(k=1,2,3), and transmission coefficients T1qP, T1qSV for incident qP and qSV waves have been computed from (37)–(39) and (49)-(50) for various values of the angle of incidence (θ) lying between 0°≤θ≤90° for silicon (Si) semiconductor.From Figure2, it is noticed that the magnitude of reflection coefficient (R1qP) 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 (R3qP) increases for 0°≤θ≤60° and attains a maximum value at θ=75°; that is, loss of energy by reflected wave (R1qP) is covered by reflected wave (R3qP). However, a meager amount of energy is associated with an electron wave (R2qP). It is also noticed that transmission coefficient T1qP 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 RkqP(k=1,2,3)and transmission coefficient T1qP 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].Figure 2
qP wave incidence at the interface of semiconductor and fluid.In Figure3, it is noticed that reflection coefficient R1qSV 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°≤θ≤45°. Thus shear wave incidence has a critical angle at θ=45°. Beyond this cutoff point, the amplitude of longitudinal wave component at the surface is finite, but no real power is associated with them; only decaying fields are associated with them. However, the reflection coefficient R2qSV increases sharply and attains a maximum energy at θ=30°, then decreases for 30°≤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 R3qSV 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. T1qSV 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.Figure 3
qSV wave incidence at the interface of semiconductor and fluid.Figure4, the variation of reflection coefficients RkqSV(k=1,3), and transmission coefficients T1qSV versus angle of incidence have been plotted. It is noticed that the reflection coefficient R1qSV 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 the incident shear wave power is converted into a reflected qSV wave so that |R3qSV|=1 approximately, although there is a π-phase shift of R3qSV as θ passes through the critical angle in case of qSV wave incident at the stress-free, isoconcentrated, or impermeable surface of silicon (Si) half-space [20]. Transmission coefficient T1qSV increases with increase in angle of incidence and attains a maximum value at θ=300; after that it starts decreasing and vanishes at θ=900. This implies that major portion of energy is carried by transmitted wave in comparison to R3qSV in qSV wave incidence at the interface of semiconductor and fluid under electron field equilibrium.Figure 4
qSV wave incidence at the interface of semiconductor and fluid under electron field equilibrium.From Table2 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.Table 2
Reflection/transmission coefficients in Germanium (Ge) semiconductor.
Angle of incidence(θ)Reflection/transmission coefficientsR1qSVR2qSVR3qSVT1qSV0°001015°0.49660.05380.02810.184830°0.25220.92080.12100.486345°0.20230.013130.87230.575960°0.06530.00060.93260.639175°0.29390.29800.58140.731390°00.075910.1094Table 3
Variation of energy coefficients (Z) in case of (qSV) wave incidence versus angle of  incidence (θ) in degree.
Angle of incidence(θ) in degree0153045607590Energy coefficients(Z)0.87201.35242.98890.994170.04371.19230.2224


## 9. Conclusions

It is noticed that the magnitude of reflection coefficient (R1qP) of qP wave decreases with increasing angle of incidence of qP wave in case of silicon (Si) semiconductor material half-space 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 RiqP(i=1,2,3) on ai(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.



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