- Email: [email protected]
New angles of phonon refraction
Physica B 316–317 (2002) 366–368
New angles of phonon refraction M.E. Msalla,*, W. Dietscheb, K.-J. Friedlandc, Q.-Y. Tongd, B. Mohra a
Physics Department, Bowdoin College, 8800 College Station, Brunswick, ME 04011, USA b Max Planck Institut fur Stuttgart, Germany . Festkorperforschung, . c Paul Drude Institut fur Berlin, Germany . Festkorperelektronik, . d Research Triangle Institute, Research Triangle Park, NC 27709, USA
Abstract The true phonon analog of optical focusing is the redirection of flux at an interface, where continuity of interface strains requires the conservation of the wave vector component parallel to the interface on either side. Such refraction is observable only at extremely clean defect-free interfaces. We have conducted phonon imaging studies of a variety of wafer-bonded samples in which the bonded materials have a large lattice mismatch at the interface (e.g., [0 1 1] GaAs bonded to [0 0 1] GaAs). The materials on either side of the bond are of the same crystalline type but are acoustically distinct because of the deliberate misorientation of the sample faces. Phonon imaging reveals a combination of refraction and diffuse scattering at these interfaces. Comparison of the experimental images to computer simulations confirms the dominance of refraction without energy down-conversion at high quality interfaces and provides interesting evidence of inelastic phonon–defect interactions. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Wafer bonding; Phonon imaging; Solid–solid interfaces
Phonons diverge from a non-equilibrium source along trajectories that are not uniformly distributed in real space. When crossing an ideal material interface, phonon frequency should not change, nor should the wave vector component parallel to the plane of the interface. Within these constraints, phonons may still scatter into different wave vector and polarization states. This elastic scattering can cause significant changes in the distribution of phonon flux after the interface. Phonons refracted at the interface can result in converging phonon beams. The effects of an interface on phonon flux can be predicted with a computer simulation. [1] The *Corresponding author. Tel.: +1-207-725-3818; fax: +1207-725-3818. E-mail address: [email protected] (M.E. Msall).
simulation begins with an isotropic (in k-space) distribution of phonons of all polarization modes. The group velocities of these phonons are calculated and phonon trajectories simulated accordingly. At the interface, mode conversion allows trirefringent scattering with three reflected and three refracted states. These states (and their relative weights) are determined from the conservation of the component of the phonon wave vector that is parallel to the interface and of the total energy in the incoming wave packet, using the formalism introduced by Kinder and Weiss. [2] The computer simulation maps phonon trajectories for each of the new states and tallies any that intercept the far surface of the crystal. This process is illustrated in Fig. 1 for phonons that pass through the interface (reflection processes are not shown).
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 0 5 0 9 - 4
M.E. Msall et al. / Physica B 316–317 (2002) 366–368
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Fig. 2. Comparison of (a) an experimental, and (b) a simulated phonon image of 301 twist-bonded (0 0 1) silicon: The sample is composed of two hydrophobically bonded 3 mm-thick Si wafers. The images are 2 mm on a side. Image (b) is a grayscale representation of the 3D intensity plot on the top surface of Fig. 1. Fig. 1. Phonons originate in a non-equilibrium source region at the bottom surface of the cube. The intensity of the flux at the interface varies as shown in the 3D plot on the middle layer. After elastic scattering at the interface, the phonons propagate in a second piece of 301 twist-bonded (0 0 1) Si. The intensity of the flux at the top surface of the cube is shown in the upper 3D plot. The arrows illustrate representative paths of ST phonons that undergo intramode scattering at the interface and converge to form new focusing structures.
Converging phonons are best observed in phonon transmission experiments with defect-free interfaces. [3] High quality, wafer-bonded crystals can be produced with a large variety of refracting properties depending upon the crystallographic orientation of the two substrates. Fig. 2 shows a comparison of the simulated phonon flux pattern with an experimental phonon image of 301 twistbonded (0 0 1) silicon (i.e., two pieces of (0 0 1) Si bonded after one has been rotated by 301 about the z-axis). The sharp features of the phonon image (Fig. 2a) agree well with the simulation (Fig. 2b), indicating that a significant fraction of phonons undergo elastic scattering at the interface. The four-pointed star shape in this image is an example of a new focusing feature produced by converging phonons. The slow transverse (ST) phonons that make up the sides of the triangular star points were propagating along non-intersecting paths in the first piece of Si, but are brought together after refraction at the interface. The Si sample used to produce Fig. 2(a) was hydrophobically bonded; the surface was pro-
tected from oxidation by hydrogen termination before bonding. Hydrophobically bonded samples often have tiny voids at the interface. Some of the larger voids (B30 mm) are visible in the experimental phonon image as differences in the detected phonon intensity at symmetrically equivalent points. Smaller scale defects can cause an overall blurring of the image. The lack of sharpness in some of the expected focusing features raises the expectation that diffuse scattering should also play a role in determining the features of the experimental image. [4] In order to incorporate diffuse scattering in the simulation, the new wave vector at the interface is chosen randomly, without regard for the conservation conditions discussed earlier. All of our experimental samples include at least one piece of (0 0 1) crystal (either Si or GaAs) which has very strong focusing along the [0 0 1] direction. In this case, randomly chosen phonons have a high probability of propagating directly along [0 0 1], and the features in this model are largely determined by the distribution of phonon flux in the second piece. Thus, a diffuse scattering simulation of phonons in our 301 twist-bonded (0 0 1) Si results in a blurred version of the pattern from the interface layer. The lack of such a pattern in the experimental image indicates that there is little diffuse scattering in this sample. In contrast, the phonon image of a (0 1 1)/(0 0 1) GaAs sample exhibits a mixture of diffuse and
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M.E. Msall et al. / Physica B 316–317 (2002) 366–368
Fig. 3. Comparison of (a) a simulated (0 1 1) interface; (b) a simulated diffuse scattering; (c) a simulated elastic scattering; and (d) an experimental image of (0 1 1)/(0 0 1) GaAs. The sample is composed of two wafer-bonded 0.4 mm-thick wafers. The images are 2 mm on a side.
elastic scattering features. The simulated phonon image at the interface (after propagation through the (0 1 1) layer) is shown in Fig. 3a. The image produced by the diffuse scattering simulation (Fig. 3b) is a blurry reproduction of the interface image, as expected. An image from the elastic scattering simulation (Fig. 3c) retains the symmetry of the interface image but has new and unusual focusing structures. Most noteable is that the diffuse scattering model predicts a high concentration of phonons along the image center not present in the elastic scattering simulation. The experimental image shows concentrations of flux at the image center and remnants of fast transverse (FT) ridges that support the presence of diffuse scattering. However, the overall shape of the structures that develop from the initial [0 0 1] direction structures at the interface is broader and more
rounded and the slow transverse (ST) ramps are more narrow, in keeping with the elastic scattering model. This case demonstrates the necessity of choosing interface parameters carefully. The new focusing structures produced by elastic scattering at the (0 1 1)/(0 0 1) interface are relatively weak. Each of the displayed images is scaled so that the maximum intensity is represented by the same (dark) gray-scale value, but the absolute maxima in the diffuse simulation is roughly four times that in the elastic simulation. Thus, if diffuse and elastic scattering processes are equally probable at the interface, the diffuse scattering component will overwhelm the elastic scattering components in the experimental image. At other interfaces (including the 301 twist-bonded (0 0 1) Si shown in Fig. 1) this is not the case, and the diffuse scattering components are weaker than the elastic scattering components even when a comparable ratio of the two scattering processes is assumed. Thus, the clarity of observation of converging phonon beams is dependent upon the type of interface as well as the quality of the interface. Careful consideration of the maximum attainable focusing intensity for a given interface will be the key to successful applications of converging phonon beams. This work is partially supported by the AFOSR under Contract F49620-00-1-0328 through the MURI program.
References [1] J.P. Wolfe, Imaging Phonons: Acoustic Wave Propagation in Solids, Cambridge University Press, Cambridge, 1998. [2] H. Kinder, K. Weiss, J. Phys. Condens. Matter 5 (1993) 2063. [3] M. Msall, W. Dietsche, K.-J. Friedland, Q.-Y. Tong, Phys. Rev. Lett. 85 (2000) 598. [4] M.E. Msall, A. Klimashov, S. Kronm.uller, H. Kostial, W. Dietsche, K. Friedland, Appl. Phys. Lett. 74 (1999) 821.
New angles of phonon refraction M.E. Msalla,*, W. Dietscheb, K.-J. Friedlandc, Q.-Y. Tongd, B. Mohra a
Physics Department, Bowdoin College, 8800 College Station, Brunswick, ME 04011, USA b Max Planck Institut fur Stuttgart, Germany . Festkorperforschung, . c Paul Drude Institut fur Berlin, Germany . Festkorperelektronik, . d Research Triangle Institute, Research Triangle Park, NC 27709, USA
Abstract The true phonon analog of optical focusing is the redirection of flux at an interface, where continuity of interface strains requires the conservation of the wave vector component parallel to the interface on either side. Such refraction is observable only at extremely clean defect-free interfaces. We have conducted phonon imaging studies of a variety of wafer-bonded samples in which the bonded materials have a large lattice mismatch at the interface (e.g., [0 1 1] GaAs bonded to [0 0 1] GaAs). The materials on either side of the bond are of the same crystalline type but are acoustically distinct because of the deliberate misorientation of the sample faces. Phonon imaging reveals a combination of refraction and diffuse scattering at these interfaces. Comparison of the experimental images to computer simulations confirms the dominance of refraction without energy down-conversion at high quality interfaces and provides interesting evidence of inelastic phonon–defect interactions. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Wafer bonding; Phonon imaging; Solid–solid interfaces
Phonons diverge from a non-equilibrium source along trajectories that are not uniformly distributed in real space. When crossing an ideal material interface, phonon frequency should not change, nor should the wave vector component parallel to the plane of the interface. Within these constraints, phonons may still scatter into different wave vector and polarization states. This elastic scattering can cause significant changes in the distribution of phonon flux after the interface. Phonons refracted at the interface can result in converging phonon beams. The effects of an interface on phonon flux can be predicted with a computer simulation. [1] The *Corresponding author. Tel.: +1-207-725-3818; fax: +1207-725-3818. E-mail address: [email protected] (M.E. Msall).
simulation begins with an isotropic (in k-space) distribution of phonons of all polarization modes. The group velocities of these phonons are calculated and phonon trajectories simulated accordingly. At the interface, mode conversion allows trirefringent scattering with three reflected and three refracted states. These states (and their relative weights) are determined from the conservation of the component of the phonon wave vector that is parallel to the interface and of the total energy in the incoming wave packet, using the formalism introduced by Kinder and Weiss. [2] The computer simulation maps phonon trajectories for each of the new states and tallies any that intercept the far surface of the crystal. This process is illustrated in Fig. 1 for phonons that pass through the interface (reflection processes are not shown).
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 0 5 0 9 - 4
M.E. Msall et al. / Physica B 316–317 (2002) 366–368
367
Fig. 2. Comparison of (a) an experimental, and (b) a simulated phonon image of 301 twist-bonded (0 0 1) silicon: The sample is composed of two hydrophobically bonded 3 mm-thick Si wafers. The images are 2 mm on a side. Image (b) is a grayscale representation of the 3D intensity plot on the top surface of Fig. 1. Fig. 1. Phonons originate in a non-equilibrium source region at the bottom surface of the cube. The intensity of the flux at the interface varies as shown in the 3D plot on the middle layer. After elastic scattering at the interface, the phonons propagate in a second piece of 301 twist-bonded (0 0 1) Si. The intensity of the flux at the top surface of the cube is shown in the upper 3D plot. The arrows illustrate representative paths of ST phonons that undergo intramode scattering at the interface and converge to form new focusing structures.
Converging phonons are best observed in phonon transmission experiments with defect-free interfaces. [3] High quality, wafer-bonded crystals can be produced with a large variety of refracting properties depending upon the crystallographic orientation of the two substrates. Fig. 2 shows a comparison of the simulated phonon flux pattern with an experimental phonon image of 301 twistbonded (0 0 1) silicon (i.e., two pieces of (0 0 1) Si bonded after one has been rotated by 301 about the z-axis). The sharp features of the phonon image (Fig. 2a) agree well with the simulation (Fig. 2b), indicating that a significant fraction of phonons undergo elastic scattering at the interface. The four-pointed star shape in this image is an example of a new focusing feature produced by converging phonons. The slow transverse (ST) phonons that make up the sides of the triangular star points were propagating along non-intersecting paths in the first piece of Si, but are brought together after refraction at the interface. The Si sample used to produce Fig. 2(a) was hydrophobically bonded; the surface was pro-
tected from oxidation by hydrogen termination before bonding. Hydrophobically bonded samples often have tiny voids at the interface. Some of the larger voids (B30 mm) are visible in the experimental phonon image as differences in the detected phonon intensity at symmetrically equivalent points. Smaller scale defects can cause an overall blurring of the image. The lack of sharpness in some of the expected focusing features raises the expectation that diffuse scattering should also play a role in determining the features of the experimental image. [4] In order to incorporate diffuse scattering in the simulation, the new wave vector at the interface is chosen randomly, without regard for the conservation conditions discussed earlier. All of our experimental samples include at least one piece of (0 0 1) crystal (either Si or GaAs) which has very strong focusing along the [0 0 1] direction. In this case, randomly chosen phonons have a high probability of propagating directly along [0 0 1], and the features in this model are largely determined by the distribution of phonon flux in the second piece. Thus, a diffuse scattering simulation of phonons in our 301 twist-bonded (0 0 1) Si results in a blurred version of the pattern from the interface layer. The lack of such a pattern in the experimental image indicates that there is little diffuse scattering in this sample. In contrast, the phonon image of a (0 1 1)/(0 0 1) GaAs sample exhibits a mixture of diffuse and
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M.E. Msall et al. / Physica B 316–317 (2002) 366–368
Fig. 3. Comparison of (a) a simulated (0 1 1) interface; (b) a simulated diffuse scattering; (c) a simulated elastic scattering; and (d) an experimental image of (0 1 1)/(0 0 1) GaAs. The sample is composed of two wafer-bonded 0.4 mm-thick wafers. The images are 2 mm on a side.
elastic scattering features. The simulated phonon image at the interface (after propagation through the (0 1 1) layer) is shown in Fig. 3a. The image produced by the diffuse scattering simulation (Fig. 3b) is a blurry reproduction of the interface image, as expected. An image from the elastic scattering simulation (Fig. 3c) retains the symmetry of the interface image but has new and unusual focusing structures. Most noteable is that the diffuse scattering model predicts a high concentration of phonons along the image center not present in the elastic scattering simulation. The experimental image shows concentrations of flux at the image center and remnants of fast transverse (FT) ridges that support the presence of diffuse scattering. However, the overall shape of the structures that develop from the initial [0 0 1] direction structures at the interface is broader and more
rounded and the slow transverse (ST) ramps are more narrow, in keeping with the elastic scattering model. This case demonstrates the necessity of choosing interface parameters carefully. The new focusing structures produced by elastic scattering at the (0 1 1)/(0 0 1) interface are relatively weak. Each of the displayed images is scaled so that the maximum intensity is represented by the same (dark) gray-scale value, but the absolute maxima in the diffuse simulation is roughly four times that in the elastic simulation. Thus, if diffuse and elastic scattering processes are equally probable at the interface, the diffuse scattering component will overwhelm the elastic scattering components in the experimental image. At other interfaces (including the 301 twist-bonded (0 0 1) Si shown in Fig. 1) this is not the case, and the diffuse scattering components are weaker than the elastic scattering components even when a comparable ratio of the two scattering processes is assumed. Thus, the clarity of observation of converging phonon beams is dependent upon the type of interface as well as the quality of the interface. Careful consideration of the maximum attainable focusing intensity for a given interface will be the key to successful applications of converging phonon beams. This work is partially supported by the AFOSR under Contract F49620-00-1-0328 through the MURI program.
References [1] J.P. Wolfe, Imaging Phonons: Acoustic Wave Propagation in Solids, Cambridge University Press, Cambridge, 1998. [2] H. Kinder, K. Weiss, J. Phys. Condens. Matter 5 (1993) 2063. [3] M. Msall, W. Dietsche, K.-J. Friedland, Q.-Y. Tong, Phys. Rev. Lett. 85 (2000) 598. [4] M.E. Msall, A. Klimashov, S. Kronm.uller, H. Kostial, W. Dietsche, K. Friedland, Appl. Phys. Lett. 74 (1999) 821.