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ABSTRACT OF THESIS OF MASTER OF SCI ENCE

SURFACE ELASTIC WAVES ON (001) SURFACE OF NULL STRESS OFCUBIC CRYSTAL

The elastic surface in solids can be studied by continuum mechanics in which the wavelength ofacoustic wave is quite large than inter atomic spacing. There is only one frequency of an acousticmode in given direction of wave vector. The energy of surface wave is confined up to fewwavelengths below the surface. The amplitude of surface wave decays exponentially below thesurface. The existence of elastic wave confined to surface of elastic solid was first discovered forisotropic material. In isotropic material , there are two elastic constant: Youngs M odulus andPoissons ratio. Youngs modulus is a positive elastic constant for mate rials bu t Poissons ratio canhave value from -1 to 0.5. In isotropic material there exits three solution for wave velocity forsurface wave by using boundary condition at surface. Materials with Poissons ratio in range formost material 0.26 to 0.50, there is one real wave velocity (called Rayleigh wave) less thantransverse bulk wave and other two solution of wave velocity are complex. Since the phase velocityof a Rayleigh wave is lower than the slowest transversal bulk wave, it can t be phase match withany bulk wave. Complex solution of surface wave velocity gives rise to wave that decay in directionof propagation and are called leaky wave.In cubic crystal there are three elastic constants. In infinite bulk space of cubic material general

propagation direction there are three component of displacement vector and in general there arethree possible mode: longitudinal, shear horizontal, shear vertical. The bulk wave in symmetricdirection [100] , [110] and [111] can be degenerated to have same velocity. The bulk waves ingeneral propagation direction can be said to be quasi transverse and quasi longitudinal, in the sensethat, their net displacement vector make a small angle with longitudinal displacement direction or

perpendicular displacement direction. The bulk wave equation in general propagation direction will be coupled and three equation for component of displacement will have nontrivial solution only fora velocity for which dynamic equilibrium is satisfied by accelerat ion of particle and stress derivedfrom displacement field and will give in three velocity for general direction of propagation . On(001) surface, shear horizontal wave velocity remain decoupled and moves with velocity thatdepend on only on shear modulus and density. The in plane displacement remains coupled andwave velocity of quasi longitudinal and quasi transverse wave depend on angle of propagationdirection. The energy of bulk wave is not confined to surface since it radiates energy to the bulk.

There can be no elastic strain energy at surface if the displacement amplitude decays from surfaceto very low value with in few wavelength depth, This is possible with a surface wave whoseamplitude decay exponently with depth . The surface wave in general propagation direction will

have three partial wave. The combination of partial wave should satisfy the null stress at thesurface. The decay constants can be complex number depending upon anisotropy ratio. Positive real part of decay constant decays amplitude with depth while the imaginary decay constant gives propagation component in z direction. The in-plane quasi transverse bulk wave velocity on surfaceof (001) Silicon decreases as direction of propagation is increased from 0 to 45 degree. In wave

propagation direction [100] and [110] direction one can obtain explicit secular equation for surfacewave velocity . In other direction to obtain explicit equation is tedious task algebraically. In [100]direction on (001) surface wave is confined in a sagittal plane (plane containing wave vector andnormal vector to surface). Using symbolic toolbox of MATLAB we have calculated the threecomponent of amplitude for each partial wave till the surface wave velocity becomes equal to bulktransverse wave as direction of propagation of wave vector is rotated from [100] direction to [110]

direction . As approaches direction on (001) plane, Velocity of surface wave differs lessand less from slowest bulk wave on (001) in that particular direction. The penetration of surfacewave into solid increases and transverse component of particle motion at surface becomes more and

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more dominant. At , surface wave degenerates into transverse bulk wave , that alonesatisfy null stress boundary condition at surface. In direction there is also second and highervelocity which also satisfy null stress at surface in which particle motion is in sagittal plane , the

particle motion is elliptical with plane of ellipse in sagittal plane. This wave is end point of pseudo branch.

At around 30 degree from [100] direction, surface wave velocity becomes equal to quasi transversewave . The pseudo surface wave also appears near this angle with small imaginary part , this

wave also satisfy traction free boundary condition . Due to imaginary component of wavedecay also in direction of propagation . These wave have complex velocity and wave can becoupled with one or more bulk wave or in other sense these wave have complex wave propagationvector in direction of propagation and velocity higher than slowest bulk wave in that direction onsurface .

Rayleigh surface wave is dispersion free on null stress surface of infinite medium , it preserve its

shape during propagation , which means that an arbitrary surface pulse will propagated without anyshape change along surface because phase velocity does not depend on frequency .Velocities ofelastic waves are in range of 3000 m/sec, which is typical for transversal bulk waves in solids. As aconsequence, elastic wave propagate about 10 5 times slower than electromagnetic waves and hence

posses an accordingly smaller wave length at comparable frequency. Surface acoustic wave baseddelay lines, filters or resonators are about 10 5 times smaller in size than correspondingelectromagnetic device working at same frequency. Inter digital Transducer allow the efficientconversion of an electrical signal into elastic surface wave and then, in reverse process, thetransformation of acoustic energy back to an electric signal. The transducer consists of a system of

periodic metal electrode fingers deposited on a highly polished substrate surface of piezoelectricmaterials as quartz, LiNbO 3 or LiTaO 3. The periodic elastic deformation is produced by the

electrical field distribution arising in the substrate when a radio frequency voltage is supplied totransducer. With fields localized at free surface, the coupling can be made quite strong . Thetransducer possesses a maximum efficiency at the excitation frequency for which the surface wave

propagates exactly one transducer period in one radio frequency voltage period (synchronousfrequency). The crystal is oriented such that the charge on adjacent IDTs induce alternatingcompressive and tensile strains in piezoelectric material. The resulting periodic mechanicaldisplacements launch a propagating surface acoustic wave. Standard lithographic technique can beused to manufacture inter digitized patterns suitable for SAW generation in 10 MHz to 1 GHzfrequency range ( about ~1 micron periodicity ) . Wave trains of hundreds of wavelength long can

be generated and detected on crystal surface a few millimeter long, providing SAW with welldefined frequency . As wave propagate along the surface, they can be modulated , manipulated ,sampled between two IDT used as sender and receiver. Both the velocity and the attenuation ofSAW are sensitive to the acoustic properties of materials placed on top of propagation surface(acoustic wave sensors) and to the material properties of substrate to a depth of approximately aninverse acoustic wave number. Therefore, it is possible to deposit a piezoelectric film of sufficientthickness on a non piezoelectric substrate in such a device. As soon as wireless telecommunication application go beyond 1GHz, the interest in multilayer system with a

piezoelectric layer of ZnO or LiNbO 3 on a diamond layer and silicon substrate will grow. Sincediamond has the highest known acoustic velocity , feature sizes of approximately 1 micron aresufficient to reach 2-3 GHz , whereas the electrode spacing of IDT in Lithium niobate or Quartzwould be prohibitively small for current lithographic techniques ( 0.3 micron )

Kamal Kumar Gupta