Elastic wave transmission at an abrupt junction in a thin plate with application to heat transport and vibrations in mesoscopic systems

Transcription

1 PHYSICAL REVIEW B, VOLUME 64, Elastic wave transission at an abrupt junction in a thin plate with application to heat transport and vibrations in esoscopic systes M. C. Cross Condensed Matter Physics 4-36, California Institute of Technology, Pasadena, California 95 Ron Lifshitz School of Physics and Astronoy, Rayond Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel Received 3 Noveber 000; published 8 August 00 The transission coefficient for vibrational waves crossing an abrupt junction between two thin elastic plates of different widths is calculated. These calculations are relevant to ballistic phonon theral transport at low teperatures in esoscopic systes and the Q for vibrations in esoscopic oscillators. Coplete results are derived in a siple scalar odel of the elastic waves, and results for long-wavelength odes are obtained using full elasticity theory. We suggest that thin-plate elasticity theory provides a useful and tractable approxiation to the three-diensional geoetry. DOI: 0.03/PhysRevB PACS nubers: 6.30.d, 63.., f I. INTRODUCTION The electronic properties of esoscopic systes have been studied experientally and theoretically for any years. More recently the behavior of other excitations, for exaple, lattice vibrations phonons and spin degrees of freedo, have coe under study in these systes. In this paper we present results relevant to the issues of heat transport by phonons and the dissipation of vibrational odes in esoscopic systes. The interest in heat transport by phonons in esoscopic systes arises because for easily fabricated devices the wavelength of a typical theral phonon becoes coparable to the diension of the theral pathway at accessible teperatures of order K. Thus quantized theral transport due to the discrete ode structure of the theral pathway should becoe evident. We and others,3 showed that this leads to a natural quantu unit of theral conductance k B T/h siilar to the role of e /h as a quantu of electrical conductance in one-diensional wires. 4 This quantu unit of theral conductance is predicted to be clearly observable at low enough teperatures where only the acoustic ( 0) vibrational odes are excited in the theral pathway the waveguidelike odes with nonzero frequency cutoffs at long wavelengths are populated with exponentially sall nubers: a universal theral conductance is predicted equal to N A k B T/3h with N A the nuber of acoustic odes four for a freely suspended bea of aterial, corresponding to a longitudinal ode, two bending odes, and a torsional ode. These predictions were recently verified in beautiful experients by Schwab et al. 5 Vibrational odes in esoscopic systes are found to have anoalously low-q values, copared to larger systes of the sae aterial. 6 At first sight, the dissipation ight be expected to becoe saller as the oscillator becoes saller, since defects such as dislocations are eliinated when the size gets less than a typical defect separation. Thus the observation of lower values of Q was a surprise. In addition, unexpected dependencies on teperature and agnetic field 7 have been observed and reain unexplained. Both the possibility of observing the universal theral conductance and explanations for the Q of sall resonators involve the properties of phonon excitations with a wavelength coparable to the syste size. Here we investigate a particular issue relevant to both these questions, naely, the coupling of vibrational odes across an abrupt junction between two blocks of the sae aterial but with different diensions. The geoetry typical of a nuber of experients is shown in Fig.. The bridge is ade of silicon, silicon nitride, or galliu arsenide, is freely floating, and is of rectangular cross section. The bridge is connected to two larger blocks of the sae seiconductor. In theral-conduction experients the block at one end, called the cavity, is also freely floating and is physically supported by four bridges and is of the sae thickness. The block at the other end provides both the echanical support and a theral reservoir. In recent experients the diensions were thickness t00 n, width b300 n, and length L5. In vibration experients the bridge ay be supported just at one end a cantilever 3 or at both ends a bea. 9, An iportant question in both theral transport and oscillator daping experients is the coupling of the vibrational odes of the bridge to odes in the supports how well the energy in a ode in the bridge is transitted to the supports, and vice versa. We have previously introduced a FIG.. Scheatic of possible experiental geoetry for the study of theral transport and oscillations in esoscopic systes /00/648/08534/$ The Aerican Physical Society

2 M. C. CROSS AND RON LIFSHITZ PHYSICAL REVIEW B siple scalar odel of the elastic waves to study this question in the context of theral transport. In this paper we give a ore realistic description of the vibrational odes. First, in Sec. II, we introduce an iproved scalar odel, using a better choice for the boundary conditions on the scalar field that provides a ore realistic approxiation to the waves in an elastic ediu. The scalar odel in its revised for provides a useful first guide to the expected behavior of the experiental syste, and a sipler environent in which to develop intuition and ethods of theoretical attack. With the scalar odel we perfor a coplete calculation of the scattering of the waves at the abrupt junction between the bridge and the supports for all the odes and at all wave vectors. We use the resulting transission coefficients to evaluate the effect of the abrupt junction on the theral conductance, particularly the universal low-teperature expression. In addition we introduce a siple ethod to calculate the transission coefficients for the long-wavelength acoustic odes, and copare the results with the general results. In the full elastic calculation we will not be able to calculate the transission coefficient for the full range of odes, but will be restricted to this type of long-wavelength calculation. Second we propose that the elasticity theory for a thinplate geoetry provides a useful seiquantitative description of the experiental geoetry. This is described in Sec. III. The full elasticity theory for the odes in the twodiensional, thin-plate geoetry is sufficiently tractable that a coplete ode spectru is readily calculated. On the other hand, a fully three-diensional elasticity theory can only be attacked purely nuerically. The two-diensional theory reproduces any iportant features of a fully threediensional elasticity theory, for exaple the ixing of bulk longitudinal and transverse waves by reflection at boundaries, the correct behavior of the dispersion relation at long wavelength and low frequency, including the bending odes with the unusual quadratic dispersion k at long wavelengths, and regions of negative dispersion in the ode spectra. Thus the results should be ore inforative than the naive scalar odel. The results should be accurate at sufficiently low teperatures where the odes with structure across the thickness are frozen out. We use the thin-plate odel to investigate the ode structure in the bea, and the coupling of these odes to the supports, also treated as thin plates of the sae thickness. Finally, in Sec. IV we apply the results to the issues of heat transport and oscillations in esoscopic systes. A. Heat transport A theral transport experient is shown in Fig.. Two theral asses are connected by four theral pathways of esoscopic diensions in which heat transport by phonons is the doinant echanis. One of the theral asses, which we call the cavity, is a freely suspended thin block of seiconductor, with resistive wires on the upper surface to act as heat source and theroeter. The four bridges act as the theral pathway to the outside world, as well as echanical supports. Conceptually, heat is added to the cavity by resistive heating, and the resulting teperature difference FIG.. Experiental geoetry of Tighe et al. Ref.. fro the reservoir is easured, yielding, for sall heating, the theral conductance of the bridges. In practice issues such as the theral contact between the electrons in the resistive heater and theroeter and the phonons, and other theral pathways to the reservoir such as through the electrical contacts to the resistive heaters, have to be considered. In this paper we will focus on the ideal situation where the phonon theral pathway of the bridge doinates the conductance. In esoscopic systes it is easy to cool to teperatures where the transverse diensions of the bea are coparable to or saller than the typical phonon wavelength hc/k B T where k B is Boltzann s constant, h is Planck s constant, c is a typical speed of sound in the aterial, and T is the teperature. When this condition is satisfied the discrete ode structure of the theral pathway becoes evident. Angelescu et al. showed that the theral conductance takes on a quasiuniversal for, largely independent of the aterial and ode structure of the bea, on the assuption that the contact between the odes in the bridge and the cavity and reservoir can be considered ideal. An ideal contact iplies that the right-going phonon odes in the bridge in Fig. are populated with a theral distribution at the teperature of the cavity, and the left-going odes at the teperature of the reservoir. In a theral-conductance easureent the cavity and reservoir are aintained at teperatures TT and T with teperature difference T sall copared to their ean teperature. If we first look at the transport by the right-oving phonons, the energy flux is H () dk vg k kn k, 0 where k is the wave vector along the bridge, (k) is the dispersion relation of the th discrete ode of the bridge, and v g d (k)/dk is the group velocity. Transforing the integral to an integral over frequencies yields an expression for the heat transport by right-oving phonons

3 ELASTIC WAVE TRANSMISSION AT AN ABRUPT... PHYSICAL REVIEW B H () d kn k, where is the cutoff frequency of the th ode, i.e., the lowest frequency at which this ode propagates. We have assued the th ode propagates to arbitrarily large frequencies. If a particular ode only propagates over a finite band of frequencies, the upper liit of the integral will be replaced by ax.) The key siplification in this result is that the group-velocity factor is cancelled by the density of states in transforing fro an integration over wave nubers to an integration over frequencies. For ideal coupling to the reservoirs the distribution function n( (k)) for the right-oving phonons in Eq. is evaluated as the Bose distribution at the cavity teperature TT. The theral conductance is given by subtracting the analogous expression H () for the left-oving phonons given by Eq., but now with the distribution n( (k)) given by the Bose distribution at the reservoir teperature T H () TTH () T K li. 3 T 0 T Finally, introducing the scaled frequency variable x /k B T gives the expression K k B T h I /k B T, where I is given by the integral Ix x y e y e y dy. Equation 4 deonstrates the iportant result that the properties of the bridge only enter through the ratio of the ode cutoff frequencies to the teperature /k B T. The quantity k B T/h plays the role of the quantu unit of theral conductance, analogous to the quantu of electrical conductance e /h for one-diensional wires. At very low teperatures the contribution to the theral conductance by the odes with nonzero cutoff frequency will be exponentially sall leaving a universal theral conductance 4 5 k B T KN a, 6 3h where N a is the nuber of acoustic odes, i.e., odes with frequency tending to zero at long wavelengths. Usually this will be four for the bea two transverse bending odes, one longitudinal copressional ode, and a torsional ode. Note that there is no dependence on the bridge properties in this expression. More generally we cannot assue perfect coupling between the odes in the bridge and the cavity and reservoir. This can be taken into account, following the Landauer approach to electrical conductance, 4 through a transission coefficient for energy to be transported across the interfaces. For exaple for iperfect contact at the cavity-bridge interface we would find a theral conductance returning to unscaled quantities in the integral for clarity K k B T T e e d, where T () is the energy transission coefficient fro the ode of the bridge at frequency into the cavity odes. Iperfect coupling at the bridge-reservoir junction, and elastic scattering due to iperfections in the bridge, can be siilarly included through a total transission atrix as in the electron case. A central issue in predicting the theral conductance is then to calculate the transission coefficient T (). This is particularly iportant in the question of the observability of the universal conductance at low teperatures, since the scattering of the long-wavelength phonons contributing to this quantity becoes strong indeed for the abrupt junction in Fig., T () 0 as 0 as we will see below. Although it is feasible in experient to sooth off the corners, as indeed was done in the experient of Schwab et al., 5 consideration of the worst-case abrupt junction provides insight into the iportance of geoetric scattering. B. Oscillator Q The Q of an oscillator is given by Q Ė E, where Ė is the rate of energy loss fro the ode at frequency containing energy E. If we consider the oscillations of a bea supported by two supports, or a cantilever with one support, and estiate the energy loss as the energy transitted into the supports, we find for the ode n Q n v g L T n, 9 where v g d/dk is the group velocity of a wave propagating in the bea, and L is the length of the bea. The exact evaluation of this quantity depends on the nature of the ode longitudinal, bending, etc.. For the longitudinal and torsional odes, which have a linear spectru ck, the frequency of the fundaental ode in a bea of length L is of order c/l, the group velocity is v g c, and so Q n T n n. 0 For the bending waves with a quadratic spectru k, the result is ore coplicated, but the Q values are siilar, and tend to this for for large n, so we will use this expression as a fairly accurate general estiate. We see fro Eq. 0 that good isolation of the ode T n 0 is a criterion for high Q. In practice this expression for the dissipation ay be an overestiate, since we are assuing that all the energy of the ode that enters the support either dissipates away, or propagates away to large distances

4 M. C. CROSS AND RON LIFSHITZ PHYSICAL REVIEW B so that the energy is not returned to the oscillations of the bea. If this is not the case, the transission of energy into the support is only one part of the proble we would also have to consider the behavior of the vibrational energy in the supports as well. II. SCALAR MODEL As a siple odel of the elasticity proble consider a single scalar field. This ight represent, for exaple, the scalar displaceent, and the vector stress would then be proportional to. We will suppose a twodiensional doain corresponding to the thin plate. This leads to a wave equation t c, with the two-diensional Laplacian and c giving the speed of propagation of the wave. Stress-free boundary conditions at the edges are then nˆ 0, with nˆ the noral to the edge. Note that this Neuann boundary condition allows the propagation of an acoustic ode (k 0) 0 in the bridge as we expect for elastic waves, whereas Dirichlet boundary conditions do not. An exaple of an elastic syste described by such a scalar odel is a stretched ebrane: would then be the displaceent noral to the ebrane and nˆ is the vertical force on a unit line in the ebrane noral to the direction nˆ. The scalar proble is sufficiently siple that we can calculate the transission across an abrupt junction such as the cavity-to-bridge junction in full detail. This allows us to gain insight into the ore coplicated elastic wave proble, and also allows us to illustrate and test approxiation schees that will be useful there. The odel, Eqs. and, was studied by Angelescu et al. using the ode atching ethod developed by Szafer and Stone 4 for the analogous electron wave calculation. However, Angelescu et al. iplicitly used a rather unnatural boundary condition (0) for the end of the cavity at the junction plane although Eq. was assued everywhere else, i.e., on the edges of the bea and cavity parallel to the propagation direction. We briefly review this work, explain how this boundary condition was introduced, and then treat the ore natural case Eq. everywhere using the sae ethods. This new treatent actually reoves a weak logarithic divergence found in the original treatent, and produces results at low frequencies that are ore consistent with the results of the full elasticity treatent. A. Model of Angelescu et al. Assue a siple two-diensional geoetry consisting of a rectangular bridge of transverse diension b connected to a rectangular cavity of transverse diension B, Fig. 3. In the FIG. 3. Geoetry for the calculation of the transission coefficient. Stress-free boundary conditions are assued on the edges as shown. Angelescu et al. used 0 boundary conditions on the end PQ and RS of the cavity. A better choice is to use stress-free conditions here as well. general three-diensional case, if the cavity and bridge have the sae thickness, there is no ixing of the z odes, and the proble separates into a set of two-diensional probles, one for each z ode. Here we will only consider the lowest ode with no structure across the z direction, which is the only ode excited at low enough teperatures. Let c (y) and (y) be orthonoral transverse odes satisfying the stress-free boundary conditions on the edges in the cavity and the bridge, respectively. For clarity we will denote cavity-ode indices by Greek letters, and bridge-ode indices by Roan letters. The solutions to the wave equation take the for x,y,t ye i(kxt) 3 for the bridge, where the frequency of the ode is given by c k with c/b the cutoff frequency of the th bridge ode. The for is siilar for the cavity odes with the cavity width B replacing the bridge width b. We will denote the frequency separation between bridge odes by : c/b. 4 Consider a phonon incident on the interface fro the cavity side (x0), in the ode of the cavity, and with longitudinal wave vector k (c). The solutions in the cavity and bridge, including the reflected waves in the cavity and the transitted waves in the bridge, are (c) (c) e ik (c) x r (c) e ik (c) x cavity, t e ik x bridge, 5 with r and t reflection and transission aplitudes to be deterined. In the above equations, k and k (c) are the

5 ELASTIC WAVE TRANSMISSION AT AN ABRUPT... PHYSICAL REVIEW B wave vectors of the transitted and reflected waves, respectively, given by the frequency atching condition c k (c) (c) c k c k (c) (c). 6 Note that the sus over and in Eq. 5 include evanescent waves iaginary k or k (c) ) although only the propagating odes will contribute to the energy transport. The field and the longitudinal derivative /x have to be atched in the ediu at x0, which leads to the equations (c) r (c) t, k (c) (c) r k (c) (c) t k. 7 Equations for the reflection and transission coefficients are extracted by integrating Eqs. 7 ultiplied by a transverse function, or (c), and using the orthogonality of the functions over the appropriate doain to extract relationships for the ode coefficients. We first ultiply one of the equations with a cavity ode (c), and integrate over the cavity width, aking use of the orthonorality relation dy (c) (c). In this section we follow Angelescu et al. and perfor this operation on the first equation i.e., the atching equation for the field. It is at this stage that the boundary condition on the cavity field at the face x0 for yb/ is introduced. The replaceent in the integration on the right-hand side B/ B/ dy (c) x0 b/ b/ dy (c) t 8 iplicitly forces the boundary condition (x0)0 for yb/. Multiplying the first equation in Eq. 7 by a cavity ode (c), integrating over the cavity width, and using the orthonorality of the cavity odes leads to r t a, 9 where a is the overlap of cavity and bridge transverse functions a b/ b/ dy (c). 0 Equation 9 ay now be plugged into the second equation in Eq. 7, and the result is k (c) (c) t a k (c) (c) t k, which, when integrated with over the bridge width, yields k (c) a n A n t n t k. This is a syste of equations that deterines t. In Eq. the kernel A n is given by A n a a n k (c). 3 These equations ay be solved for the t s, and then the flux transission probability fro the wave-vector k (c) state of cavity ode to bridge ode is given by T t k k. 4 (c) Now suing over all the cavity odes that are propagating at frequency leads to the transport transission coefficient fro the cavity to the th ode T () for )by T T t k (c) (c),, k, (c) 5 with k () and k (c) () given by Eq. 6. This also gives the energy transission coefficient fro the th bridge ode to the cavity, by the usual reciprocity arguents. Equations 3 and 5 involve sus over cavity odes. We ay either evaluate the sus directly for a chosen value of the width ratio B/b, or take the liit of a large cavity width B when the sus are replaced by integrals. We calculate the atrix A n for,nn ax with N ax soe upper cutoff for the nuber of bridge odes retained and invert the N ax N ax atrix syste nuerically to find t and hence T (). There is a siple approxiation 4 that provides an analytic for for the solution to Eq. that is in reasonably good agreeent with the exact solution. The approxiation derives fro three properties of a. First, a 0 unless and have the sae parity. In other words, even odes couple to even odes and odd odes to odd odes only. Second, as a function of, a is sharply peaked around B/b, the width of the peak being of order B/b. And third, a ust satisfy the copleteness relation a a n n. 6 The first two properties perit the key approxiation, naely, that A n n since the product of two functions peaked at different channels,n is very sall and A n is rigorously zero when and n are different parity odes. Then we only need the diagonal part of A A a k (c), 7 which is siply a weighted average of the coplex wave vector over the narrow range of reflected cavity odes for

6 M. C. CROSS AND RON LIFSHITZ PHYSICAL REVIEW B with and integers, with the special case 0 (y)/b and 0 c (y)/b. The a are easily calculated: FIG. 4. Transission coefficient T 0 coupling the lowest bridge ode to the cavity odes as a function of the reduced frequency of the ode / with the splitting between bridge odes at zero wave vector. Curves for cavity wave-nuber cutoff equal to 0, 40, 80, 00, and 500 ties b with b the width of the bridge show the weak dependence on this cutoff. which a is significant. Note a by copleteness. In this case Eq. separates into and then k (c) a A t k t 8 t k (c) a. 9 A k The flux transission probability fro cavity ode to bridge ode is given in this approxiation by T t k k 4k (c) k a (c) k K J, 30 where K Re A and J I A. The energy transission coefficient is T, (c) 4k (c) k a k K J 4K k k K J. 3 We now use the explicit for of the transverse odes to evaluate T (). With the boundary condition /y0 at the y boundaries we have y b cos y b even, 3 y c b sin y b odd, B cos y B even, B sin y B odd, 33 a 0, not both even or odd, 34 a Bsin b b B b B sin b B, even0, 35 b B a Bsin b b B b B sin b B, odd0, 36 b B a 0 Bsin b b B even0, 37 b B a 0 0 0, 38 a For large, both even or both odd we can approxiate a b B sin b B b B. 40 The a are indeed sharply peaked as a function of cavityode nuber. The large approxiation is essentially identical to the result in the electronic case. 4 However the sall, second ter in the braces in Eqs. 35 and 36 ignored in this approxiation appears with the opposite sign in our application. This turns out to render the su over the cavity odes appearing in Eq. 3 weakly logarithically divergent for large. Note that k (c) is iaginary here, so this divergent contribution is to IA n and to the copo

7 ELASTIC WAVE TRANSMISSION AT AN ABRUPT... PHYSICAL REVIEW B FIG. 5. Transission coefficient T 0 for the lowest bridge ode in the Angelescu et al. scalar odel calculated using the exact atrix diagonalization points and the diagonal approxiation crosses for a cavity-ode wave-vector cutoff 40b, with b the width of the bridge. The diagonal approxiation shows a soewhat stronger dependence on this cutoff than shown by the exact results, so that this coparison will vary as we change the cutoff assuption. nent J of the diagonal ters. We ust ipose soe upper cutoff to the su to achieve finite results. Physically we ight suppose such a cutoff ay coe fro the breakdown of the sharp corner approxiation at short enough scales, or ultiately, in a perfectly fabricated esoscopic syste, fro the atoic nature of the aterial leading to a finite nuber of odes. Results for the transission coefficient of the lowest bridge ode T 0 () are shown in Fig. 4. There is only a weak dependence on the cavity su cutoff. At sall frequency T 0 () is proportional to 3. The results shown were calculated for the case of an infinitely wide cavity sus over cavity odes replaced by integrals. Results for finite widths e.g., B/b0) are very siilar. The first bridge odes six even odes were retained in the atrix inversion for the results shown: increasing this nuber did not change the results significantly showing that A n indeed decreases rapidly for increasing n. Note that T 0 () rapidly approaches unity as the frequency grows. There is a sall (0%) decrease at the frequency where the second even bridge ode becoes propagating, and siilar features of reducing size occur at subsequent integral ultiples of. There is no coupling between even and odd odes for the syetric geoetry used. The coparison between the results fro the full atrix inversion and the diagonal approxiation is shown in Fig. 5. The diagonal approxiation gives results good to about 0%. However this coparison depends on the cavity wave-nuber cutoff, since the diagonal approxiation depends rather ore strongly on this paraeter than for the full results shown in Fig. 4. Note that the feature at due to the interaction between different bridge odes is absent in the diagonal approxiation. FIG. 6. Transission coefficient T 0 coupling the lowest bridge ode to the cavity odes as a function of the reduced frequency / in the scalar odel with stress-free boundaries. The ain graph is for an infinite cavity width. The inset shows the coparison with results for a finite cavity width (B0.7b). B. Stress-free ends. Mode atching calculation We now redo the scalar analysis, enforcing the boundary condition nˆ 0 on the end of the cavity x0, y b/. This corresponds to a stress-free boundary everywhere. The analysis proceeds as before up to Eq. 7. But now we first ultiply the second equation for the continuity of /x) by (c), and integrate over the cavity width. This enforces the boundary condition on the cavity face nˆ 0 for x0 and yb/. The orthogonality of the (c) gives 4 r k (c) k (c) t k a. 4 Use this equation to eliinate r fro the first of Eq. 7 (c) n t n k n /k (c) a n (c) t n n 43 and integrate with over the bridge width to yield where a n Ā n k n t n t, 44 Ā n a a n /k (c). 45 Again we can solve the equation for t, Eq. 44, nuerically or by using the diagonal approxiation for Ā n.itis easily seen that the extra inverse powers of k (c) in Ā n render the su over convergent, unlike the case for A n. The

8 M. C. CROSS AND RON LIFSHITZ PHYSICAL REVIEW B FIG. 7. Values of the transission coefficients fro the nth bridge ode to the cavity as a function of ( n )/ with n the cutoff frequency for the nth ode and the spacing between ode cutoff frequencies n n. The long-dashed line shows the square-root dependence near cutoff for the n0 odes. The shortdashed line is the linear dependence T 0 ()/ expected for the lowest ode. energy transission coefficient fro the th bridge ode reains given by Eq. 5. The diagonal approxiation Ā n Ā n now leads to 4K k T k K J k, 46 with J Re Ā and K I Ā. The result for T 0 () for the lowest bridge ode is shown in Fig. 6 taking the liit of an infinite cavity width the sus in Eq. 45, etc., evaluated as integrals. Again the transission coefficient grows rapidly, approaching close to unity as the frequency grows, for exaple reaching 0.9 by about 0.5. Note the iportant difference fro the previous scalar odel that the lowfrequency asyptotic behavior is linear, T 0 (), rather than cubic as obtained there, and so T 0 () approaches unity ore rapidly than anticipated in that work. A sall reduction in T 0 () by about 3%) near and by saller aounts at higher integral ultiples of this frequency are apparent. An analysis of the curve in this region shows a square-root dependence on, corresponding to a coupling in the full atrix calculation to the second bridge ode, and to the square-root growth in T () that occurs here. The diagonal approxiation not shown gives results that are indistinguishable on the figure for 0, but the sall decrease above does not appear in this approxiation. The inset in Fig. 6 shows the coparison with results for a finite cavity width. The results are quite surprising, with resonancelike features occurring whenever the frequency passes through the cutoff frequency of a cavity ode. This can be traced to the inverse power of k (c) occurring in Eq. 5 and Eq. 45. Since k (c) goes to zero as n (c) these singularities are integrable, and the features disappear in the liit of infinite cavity width. Soothing over the features e.g., taking the average over bins between successive n (c) ) gives points that follow the sooth curve for the infinite FIG. 8. Theral conductivity divided by teperature reduced by the zero-teperature universal value k B /3h as a function of the reduced teperature k B T/: solid curve full calculation including the transission losses due to the abrupt junction for stressfree face; long-dashed curve contribution fro the lowest acoustic ode, showing the reduction at low teperatures due to the scattering at the abrupt junction; dash-dotted curve ideal result fro all odes with full transission; dotted line lowteperature asyptotic slope predicted fro the low-frequency behavior of the transission coefficient. The short-dashed curves shows the result including scattering of the calculation of Angelescu et al. cavity width closely. Integrating the effect of the transission coefficient in the theral conductance over the theral factors in Eq. 7 will effectively perfor this soothing, so that the features will not be apparent in the theral easureents. The sharp features are presuably also soothed out if the junction is not perfectly abrupt. Results for T n () for other values of n are shown in Fig. 7. It is now straightforward to calculate the theral conductivity using Eq. 7. We focus on the low-teperature liit where the universal result for K/T applies in the ideal liit. With no reduction in the theral transport due to scattering at the junction the plot of K/T as a function of teperature develops a plateau at low teperatures at the universal value k B /3h see Fig. 8. In this regie the conductivity is doinated by the acoustic ode with 0 ask 0. Scattering at the junction reduces the transission of the sall,k waves, so that the value of K/T is reduced fro the no-scattering value. As can be seen fro Fig. 8 this reduction begins to occur as the teperature is lowered at about the sae teperature at which the plateau in the ideal case begins to develop. This suggests that the plateau at the universal value of K/T will not be well developed for the abrupt junction. The full calculation using Eq. 7 and the reduced transission coefficients of all the odes solid curve in Fig. 8 shows that this is the case including the effects of scattering at the abrupt junction yields a K/T curve that tends soothly to zero as T 0. This result clearly deonstrates the iportance of using sooth junctions between the bridge and the reservoirs, such as was done in the experients of Schwab et al., 5 if the universal value of K/T is to be apparent.. Long-Wavelength Calculation Although we have perfored the full calculation of the transission coefficient for the scalar case, this will not be

9 ELASTIC WAVE TRANSMISSION AT AN ABRUPT... PHYSICAL REVIEW B possible for the elasticity-theory calculation. We therefore introduce a siple analytic technique for establishing the low-frequency liit of T 0 () that can be extended to the full elasticity description. The approach relies on the poor transission T 0 () 0 for 0 long-wavelength waves are strongly affected by the abrupt junction to treat the transission perturbatively. First the bridge ode is calculated assuing perfect reflection, i.e., isolated fro the supports. A siple analysis shows that the appropriate boundary condition on the end of the bridge is zero displaceent rather than zero stress. Ifwe now reconnect the bridge to the supports, the stress fields at the end face of the bridge act as radiation sources of waves into the cavity. The total power in these waves for unit incident aplitude in the bridge gives us the transission coefficient. The total power radiated ay be readily calculated by integrating the product of the stress source and resulting velocity across the end face of the bridge. To establish the appropriate boundary conditions in the T 0 () 0 liit we first consider a finite cavity width, i.e., an abrupt junction between a bridge of width b for x0 and a cavity of finite width B b for x0 in the liit,k 0. It is siplest here to consider the transission fro bridge to cavity, and we have reversed the sense of the x coordinate copared to Sec. II B. For x0 we consider an incident wave of unit aplitude and a reflected wave of aplitude r e ikxit re ikxit, x0. 47 For x0 there is only the transitted wave te ikxit, x0. 48 Here the wave nubers k are fixed by the dispersion relation ck, which is the sae on both sides of the junction for the acoustic ode. The reflection and transission aplitudes can be calculated by a siple atching at x0. This is equivalent to a wave ipedance calculation. Matching the displaceent field gives rt and atching the total force gives 49 brbt. 50 Note that the force the integrated stress is conserved because there are no additional stresses on the cavity face for yb/: this atching would not be appropriate for the boundary condition used by Angelescu et al. Thus we find r Bb Bb, b t bb. 5 These expressions are good for kb,kb. In this liit the atching conditions can be applied outside of the region close to the junction where the fields are perturbed fro their asyptotic fors, Eqs. 47 and 48, but before the exponential phase factors of the wave propagation have significantly changed. For Bb these expressions reduce to r, t0, i.e., strong isatch and alost perfect reflection with a sign change. Note that at x0 this iplies 0, i.e., zero displaceent boundary conditions, together with /xik for unit incident aplitude. We now use this result as the basis of the calculation of the transission for k sall but nonzero and the liit B. The transission at k0 is calculated as a radiation proble, naely, via the power radiated by the end of the bridge into the cavity. The zeroth-order approxiation for the solution in the bridge is the perfect reflection result calculated above, giving the stress radiation source on the wall of the cavity sy /xikeit for yb/, 5 0 for yb/, where the second line is just the stress-free boundary condition /x0 for x0,yb/. 53 The proble of the radiation due to a stress source on the boundary of an elastic half space is known as the Lab proble in elasticity theory, and has been considered by any authors, for exaple, see Ref. 7. The radiation field can be calculated by standard Fourier-transfor techniques. The solution in the cavity for x0 can be written as e it e iqx e iy d, 54 with q k for k, ik 55 for k corresponding to propagation or exponential decay away fro the interface. The transfor () is given by atching to the known /x at x0 yielding iq sye iy dy. 56 The power radiated is the product of the stress /x and the velocity /t across the radiation source b/ P b/ t x dy x0, 57 where the denotes the tie average. For the fields varying as e it this gives P Re i b/ b/ ys*ydy x0 Inserting the Fourier expression for (x0), P 4 I d dy s*ye iy

10 M. C. CROSS AND RON LIFSHITZ PHYSICAL REVIEW B I d iq sye dy iy. 60 Since the transverse wave vector is liited by k for the integrand to be iaginary corresponding to q real, i.e., propagating waves in the cavity, and s(y) is nonzero only for yb/, an expansion in kb is given by the expansion sye iy dys 0 is, where s i are successive oents of the source s 0 sydy, 6 6 quite coplicated, involving the ixing of longitudinal and transverse coponents by reflection at the edges, explicit expressions for the transverse structure of the odes in ters of a finite su of sinusoidal functions can be written down. Copare this with a full three-diensional analysis, where a finite-diensional representation of the odes is not possible. In addition to the scattering at an abrupt junction pursued here, the thin-plate liit will perit an analysis of phenoena such as the scattering of the waves off surface and bulk iperfections, an issue that is pursued elsewhere. 5 We first review the general elastic theory for waves in a thin plate, and confir the expected dispersion relations for the acoustic odes of a rectangular bea i.e., the bridge in the long-wavelength liit, and then use the equations to study the coupling of long-wavelength odes across the abrupt bridge-support junctions. s syydy. 63 We will need only the leading-order ter, i.e., s 0 if the source is parity syetric in y, s if the source is antisyetric. In particular for the present case s 0 ikb so that P k 4k d 0 k. 64 Noralizing by the power in the incident wave bk gives the transission coefficient T 0 4kb k 0 k d. 65 The integral expression for T 0 can be easily understood physically as integrating over the power radiated into the continuu of waves, traveling at the wave propagation speed c, and propagating at all angles into the half space. For the scalar odel there is a single type of propagating wave. With the full elasticity theory we will see a siilar result, but with a nuber of propagating waves contributing to the power radiated. Perforing the integral gives T 0 kbb/c for kb. 66 This result confirs the linear dependence on for sall, as shown in Fig. 7. III. THIN-PLATE THEORY A useful odel of the esoscopic geoetry, that is ore tractable than a fully three-diensional elasticity calculation, is to assue a thin-plate geoetry. Thus we take the elastic structure to be carved fro a thin plate of unifor thickness d, which is taken to be sall with respect to the other diensions and also with respect to the wavelength of the elastic waves. In this odel the ode frequency cutoffs at k0, iportant in the theral conductance, can be readily calculated for ost of the odes they are given by siple analytic expressions. In addition, although the ode structure is A. Review of elastic theory and odes The elasticity of an isotropic solid is suarized by the relationship between the stress tensor T and the strain tensor T ij K ij ij. Here is the dilation and is the shear strain, u x x u y y u z z, ij u i x j u j x i 3 ij, 67 68a 68b with u(x) the displaceent and K and elastic constants. For a thin plate of thickness d in the xy plane, linear elasticity theory can be separated into equations for the noral displaceent u z w(x,y) of the plate and for the inplane displaceents averaged over the depth u(x, y) (u,v) with uu x (x,y,z) z and vu y (x,y,z) z, all functions of just two spatial variables. This is done by assuing, for in-plane wave vectors k such that kd, that the stresses in the vertical direction zj, which ust be zero at the nearby stress-free top and botto surfaces, ay be put to zero everywhere. This allows the variation of the strains across the thickness of the plates to be eliinated in ters of the variables u,v,w. For exaple, setting T zz to zero gives K u z z 3 u v K 4 3 x y. 69 Thus the odes separate into odes with in-plane polarizations, and odes with polarizations noral to the plane flexural odes. The full developent can be found in any standard text on elasticity, for exaple Landau and Lifshitz 6 or Graff. 8 Here we collect the ain results

11 ELASTIC WAVE TRANSMISSION AT AN ABRUPT... PHYSICAL REVIEW B In-plane polarization Using relationships such as Eq. 69 leads to an effective two-diensional elasticity theory for displaceents in the plane, suarized by the stress-strain relationship T () ij K () ij () ij, 70 where the indices i, j now run only over x and y, () and () are the two-diensional dilation and shear strain tensor () u x x u y y, 7a () ij u i u j x j x i () ij 7b and the effective two-diensional elastic constants are K, 3K K a 7b The elastic waves with in-plane polarization are then given by the equation of otion u t T () K u u, with the horizontal gradient operator. In a horizontally infinite sheet there are longitudinal and transverse waves with speeds K c L 4 K 3 K 4, 3 75a c T. 75b Alternatively, introducing Young s odulus E and the Poisson ratio with ) so that we have E K 3, E, c E L, c E T a 77b Rather than using the Poisson ratio directly it is convenient to introduce the paraeter for the ratio of wave speeds r c L c T. 78 In ters of the coponent displaceents u, v and the paraeter r the elastic wave equation can be written c T c T u u t r x u y r v xy, v t v x r v y r u xy. 79a 79b A result that we will find useful later is for the coponent displaceents in the solutions for the propagating waves u u 0,k Tx /k Ty e i(k T xt) transverse wave, u 0,k Ly /k Lx e i(k L xt) longitudinal wave. 80 We note for copleteness that in a three-diensional elastic ediu, waves polarized in the xy plane having no z dependence are also described by an effective twodiensional elasticity theory as in Eq. 70 but now with effective elastic constants K K 3, so that c L /c T ()/(). The difference in the effective elastic constants in the two cases arises because in the threediensional ediu restricted to no z dependence, there can be no expansion in the z direction to relieve the stresses set up by the strain in the xy plane. In any elasticity textbooks, the two-diensional elasticity and thin-plate theory discussed correspond to this case. We can relate these results to the thin-plate geoetry we are considering by appropriately transforing the effective elastic constants. For a finite plate we ust apply stress-free boundary conditions at the side edges nˆ T () 0, 8 with nˆ the noral to the edge. For waves propagating in the x direction in a long finite plate of width b the no-stress boundary condition at the edges of the plate at yb/ are therefore 0 () u y T xy v x, T () yy 0 r v y r u x. 8a 8b The boundary conditions have the effect of reflecting incident longitudinal waves into both longitudinal and transverse reflected waves, so that these waves becoe coupled in the finite geoetry, leading to a coplicated dispersion relationship. The solutions propagating in the x direction decouple into either an even or odd signature with respect to y reflection, and take the for using Eqs

12 M. C. CROSS AND RON LIFSHITZ PHYSICAL REVIEW B FIG. 9. Dispersion relation of in-plane polarized odes for the thin plate. The wave nubers are scaled by the bridge width b, and the frequencies by b/c T with c T the transverse wave speed in a large, thin plate. The dashed lines are the speeds of the transverse and longitudinal waves in a large thin plate. The value of the Poisson ratio is u (e) a T (e) cos T ya L (e) cos L ye i(kxt), 83a FIG. 0. Dispersion curves for the long-wavelength in-plane odes of a thin-plate bea of width b for Poisson ratio 3 wave speed ratio 3). Solid lines: nuerical results, long-dashed line: sall-k linear dispersion for copression ode, long-short-short dashed: linear dispersion for longitudinal wave in infinite plate, short dashed: linear dispersion for edge ode, and long-short dashed: quadratic dispersion for sall-k bending ode. Note the anoalous dispersion of the fourth ode at sall k. v (e) ik/ T a (e) T sin T y i L /ka (e) L sin L ye i(kxt) and u (o) a (o) T sin T ya (o) L sin L ye i(kxt), v (o) ik/ T a (o) T cos T y i L /ka (o) L cos L ye i(kxt), where T,L are given through the dispersion relation 83b 84a 84b For a given value of the speed ratio r these equations can be solved nuerically for (k). The spectru for the value of r3 corresponding to GaAs ( 3 ) is shown in Figs. 9 and 0. At large values of k the slopes of the curves, except for the lowest two, asyptote to c L or c T corresponding to the freely propagating waves in the plate. This is not yet fully evident at the values of,k plotted in Fig. 9, but is confired by extending the nuerics to higher values. The lowest two odes on the other hand asyptote to a slope c S less than both c T and c L. In this case for large k we have tan( T,L b/) i in Eqs. 86 and 87 so that the slope c S r S c T is given by the solution of K /c T k T r k L 85 4r S r S /r r S. 88 and we have defined K/c T, the wave nuber of a transverse wave at the frequency in an infinite plate. The values of T,L ay be real or iaginary. Note that each wave cobines both shear ( T ) and copressional ( L ) distortions, which are ixed by the reflection of the plane waves off the edges. (e),(o) The aplitudes a T,L ust be adjusted to satisfy the boundary conditions at yb/, Eqs. 8. Substituting into these conditions leads to a syste of two hoogeneous equations for a (e) T,a (e) L and two for a (o) T,a (o) L ) and so a consistency condition that leads to a transcendental equation for for each k, known as the Rayleigh-Lab equations. 0 For the even-signature odes the transcendental equation is k T tan Tb 4k T L tan Lb For the odd-signature odes the transcendental equation is 4k T L tan Tb k T tan Lb This is an edge wave analogous to the Rayleigh wave on the surface of a three-diensional slab of aterial. For r3, Eq. 88 gives c S /3c T. The values of the finite-frequency intercepts of the waveguide odes for k 0 can also be calculated analytically. For the even ode intercepts, Eq. 86 is satisfied at k 0 by tan( T b/)0 or tan( L b/). Siilarly Eq. 87 is satisfied by tan( L b/)0 or tan( T b/). Thus the zero wave-nuber intercepts are given by the siple expressions for transverse and longitudinal wave propagation n (T) nc T /b, n (L) rnc T /bnc L /b. 89 These siple results hold because for k0 there is no interconversion of longitudinal and transverse waves on reflection at the edges. The shapes of the curves between k0 and the large-k asyptotes are quite coplicated, with various ode crossings and regions of anoalous dispersion d/dk0. We are particularly interested in the long-wavelength odes k 0. The dispersion relation in this liit can be

13 ELASTIC WAVE TRANSMISSION AT AN ABRUPT... PHYSICAL REVIEW B found by Taylor expansion of the tan functions in Eqs. 86 and 87. The even ode tends to r c T ke/ k. 90 This agrees with the usual expression for the stretching ode of a rod. This is not the sae as the dispersion for the bulk longitudinal odes in the thin plate, c L k, but the speeds are quite close for On the other hand the odd ode gives a quadratic dispersion, characteristic of bending odes of beas, r 3r bc Tk. 9 This is given by expanding the tan functions up to cubic order. Rod-bending theory gives the expression /k EI/A with I the areal oent of inertia about the idline, and A the cross-section area. For the rectangular bea we have I/Ab /, and using (r )c T E/ shows the correspondence.. Flexural odes The flexural odes are ost easily derived by using relationships such as Eq. 69 to derive an expression for the energy of transverse displaceents 6 where F D w w x w y dxdy, D Ed3 w xy 9 93 is the flexural rigidity of the plate of thickness d. The equation of otion and boundary conditions are given by setting the variation of the energy with respect to displaceents w(x,y) to zero. For a region with rectangular boundaries at xa/ and yb/ the variation is FD 4 w dxdy dy w 3 w x 3 3 w w xy x dx w 3 w y 3 w 3 w y w y w x yb/ w x xa/ w y xa/ yb/ x y. 94 The first ter gives the effective force per unit area on the plate, and hence the equation of otion d w t D 4 w0. 95 The last two ters on the right-hand side are boundary ters given by integration by parts. Physically they give the work done at the boundaries by the vertical force per length of boundary V i against the vertical displaceent w and by the oent per unit length M i against the angular displaceent of the plate i w. Thus at xa/ we have V x D x 96a and at yb/ w x w, y M x D w x w y, V y D y w y w, x M y D w y w x. 96b 97a 97b For free edges, these quantities ust be set to zero. In addition to the force per unit length there are also point forces localized at the corners, e.g., at xa/,yb/ Ref. 9. F c D w xy. 98 These ust be included when we are calculating the total force acting across the width of the bea, for exaple, b/ F Vx dyf c b/f c b/ 99 b/ x b/ M x dy 00 b/ and the latter equality shows the consistency with the acroscopic equation for the rotational equilibriu of the bea see Sec. III B below. We now calculate the odes propagating in the x direction w (e),w (o) e i(kxt) in the bridge of width b. Again the odes have either even or odd signature with respect to y reflections. Since the wave equation is fourth order in the spatial derivatives, for each frequency there are two even or odd coponents. The solutions to the wave equation are even and odd w (e) a (e) cosh ya (e) cosh ye i(kxt)