Abstract

The role of thin-film metal transducers in ultrafast laser-generated longitudinal acoustic phonons in Si (100) monocrystal substrates is investigated. For this purpose degenerate femtosecond pump-probe transient reflectivity measurements are performed probing the Brillouin scattering of laser photons from phonons. The influence of the metallic electron-phonon coupling factor, acoustical impedance and film thickness is examined. An optical transfer matrix method for thin films is applied to extract the net acoustic strain relative strength for the various transducer cases, taking into account the experimental probing efficiency. In addition, a theoretical thermo-mechanical approach based on the combination of a revised two-temperature model and elasticity theory is applied and supports the experimental findings. The results show highly efficient generation of acoustic phonons in Si when Ti transducers are used. This demonstrates the crucial role of the transducer’s high electron-phonon coupling constant and high compressive yield strength, as well as strong acoustical impedance matching with the semiconductor substrate.

© 2015 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  6. P. M. Norris, A. P. Caffrey, R. J. Stevens, J. M. Klopf, J. T. McLeskey, and A. N. Smith, “Femtosecond pump-probe nondestructive examination of materials,” Rev. Sci. Instrum. 74(1), 400–406 (2003).
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  7. J. Hohlfeld, S.-S. Wellershoff, J. Gudde, U. Conrad, V. Jahnke, and E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” Chem. Phys. 251(1–3), 237–258 (2000).
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  8. C. B. Scruby, R. J. Dewhurst, D. A. Hutchins, and S. B. Palmer, “Quantitative studies of thermally generated elastic waves in laser-irradiated metals,” J. Appl. Phys. 51(12), 6210–12368 (1980).
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  9. J. D. Aussel, A. Le Brun, and J. C. Baboux, “Generating acoustic waves by laser: theoretical and experimental study of the emission source,” Ultrasonics 26(5), 245–255 (1988).
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    [Crossref]
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    [Crossref] [PubMed]

2015 (1)

A. Devos, “Colored ultrafast acoustics: From fundamentals to applications,” Ultrasonics 56(2), 90–97 (2015).
[Crossref] [PubMed]

2014 (1)

G. D. Tsibidis, “Thermal response of double layered metal films after ultrashort pulsed laser irradiation: The role of nonthermal electron dynamics,” Appl. Phys. Lett. 104(5), 051603 (2014).
[Crossref]

2013 (1)

V. V. Temnov, C. Klieber, K. A. Nelson, T. Thomay, V. Knittel, A. Leitenstorfer, D. Makarov, M. Albrecht, and R. Bratschitsch, “Femtosecond nonlinear ultrasonics in gold probed with ultrashort surface plasmons,” Nat. Commun. 4, 1468 (2013).
[Crossref] [PubMed]

2012 (1)

G. D. Tsibidis, E. Stratakis, and K. E. Aifantis, “Thermoplastic deformation of silicon surfaces induced by ultrashort pulsed lasers in submelting conditions,” J. Appl. Phys. 111, 053502 (2012).
[Crossref]

2011 (1)

E. Pontecorvo, M. Ortolani, D. Polli, M. Ferretti, G. Ruocco, G. Cerullo, and T. Scopigno, “Visualizing coherent phonon propagation in the 100 GHz range: A broadband picosecond acoustics approach,” Appl. Phys. Lett. 98(1), 011901 (2011).
[Crossref]

2010 (1)

2009 (1)

P. E. Hopkins and P. M. Norris, “Contribution of ballistic electron transport to energy transfer during electron-phonon nonequilibrium in thin metal films,” J. Heat Transfer 131(4), 043208 (2009).
[Crossref]

2008 (1)

Z. Lin, L. V. Zhigilei, and V. Celli, “Electron-phonon coupling and electron heat capacity of metals under conditions of strong electron-phonon nonequilibrium,” Phys. Rev. B 77(7), 075133 (2008).
[Crossref]

2007 (1)

O. V. Misochko, T. Dekorsy, S. V. Andreev, V. O. Kompanets, Yu. A. Matveets, A. G. Stepanov, and S. V. Chekalin, “Effect of intense chirped pulses on the coherent phonon generation in Te,” Appl. Phys. Lett. 90(7), 071901 (2007).
[Crossref]

2006 (2)

T. Dehoux, M. Perton, N. Chigarev, C. Rossignol, J.-M. Rampnoux, and B. Audoin, “Effect of laser pulse duration in picosecond ultrasonics,” J. Appl. Phys. 100(6), 064318 (2006).
[Crossref]

E. Carpene, “Ultrafast laser irradiation of metals: Beyond the two-temperature model,” Phys. Rev. B 74(2), 024301 (2006).
[Crossref]

2005 (1)

J. K. Chen, W. P. Latham, and J. E. Beraun, “The role of electron–phonon coupling in ultrafast laser heating,” J. Laser Appl. 17(1), 63–68 (2005).
[Crossref]

2004 (2)

A. Devos and R. Cote, “Strong oscillations detected by picosecond ultrasonics in silicon: Evidence for an electronic-structure effect,” Phys. Rev. B 70(12), 125208 (2004).
[Crossref]

M. Lisowski, P. A. Loukakos, U. Bovensiepen, J. Staehler, C. Gahl, and M. Wolf, “Ultrafast dynamics of electron thermalization, cooling and transport effects in Ru (00l),” Appl. Phys., A Mater. Sci. Process. 78(2), 165–176 (2004).
[Crossref]

2003 (1)

P. M. Norris, A. P. Caffrey, R. J. Stevens, J. M. Klopf, J. T. McLeskey, and A. N. Smith, “Femtosecond pump-probe nondestructive examination of materials,” Rev. Sci. Instrum. 74(1), 400–406 (2003).
[Crossref]

2002 (1)

S. K. Sundaram and E. Mazur, “Inducing and probing non-thermal transitions in semiconductors using femtosecond laser pulses,” Nat. Mater. 1(4), 217–224 (2002).
[Crossref] [PubMed]

2000 (2)

J. Hohlfeld, S.-S. Wellershoff, J. Gudde, U. Conrad, V. Jahnke, and E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” Chem. Phys. 251(1–3), 237–258 (2000).
[Crossref]

D. H. Hurley, O. B. Wright, O. Matsuda, V. E. Gusev, and O. V. Kolosov, “Laser picosecond acoustics in isotropic and anisotropic materials,” Ultrasonics 38(1-8), 470–474 (2000).
[Crossref] [PubMed]

1998 (2)

A. P. Kanavin, I. V. Smetanin, V. A. Isakov, Yu. V. Afanasiev, B. N. Chichkov, B. Wellegehausen, S. Nolte, C. Momma, and A. Tunnermann, “Heat transport in metals irradiated by ultrashort laser pulses,” Phys. Rev. B 57(23), 14698–14703 (1998).
[Crossref]

A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998).
[Crossref] [PubMed]

1997 (1)

S. I. Anisimov and B. Rethfeld, “On the theory of ultrashort laser-pulse interaction with a metal,” Izvestiya Akademii Nauk Seriya Fizicheskaya 61, 1642–1655 (1997).

1995 (1)

O. B. Wright and V. E. Gusev, “Acoustic generation in crystalline silicon with femtosecond optical pulses,” Appl. Phys. Lett. 66(10), 1190–1192 (1995).
[Crossref]

1994 (2)

G. Tas and H. J. Maris, “Electron diffusion in metals studied by picosecond ultrasonics,” Phys. Rev. B Condens. Matter 49(21), 15046–15054 (1994).
[Crossref] [PubMed]

C. Sun, F. Vallée, L. H. Acioli, E. P. Ippen, and J. G. Fujimoto, “Femtosecond-tunable measurement of electron thermalization in gold,” Phys. Rev. B Condens. Matter 50(20), 15337–15348 (1994).
[Crossref] [PubMed]

1992 (1)

O. B. Wright and K. Kawashima, “Coherent phonon detection from ultrafast surface vibrations,” Phys. Rev. Lett. 69(11), 1668–1671 (1992).
[Crossref] [PubMed]

1991 (1)

H. N. Lin, R. J. Stoner, H. J. Maris, and J. Tauc, “Phonon attenuation and velocity measurements in transparent materials by picosecond acoustic interferometry,” J. Appl. Phys. 69(7), 3816–3822 (1991).
[Crossref]

1988 (1)

J. D. Aussel, A. Le Brun, and J. C. Baboux, “Generating acoustic waves by laser: theoretical and experimental study of the emission source,” Ultrasonics 26(5), 245–255 (1988).
[Crossref]

1986 (1)

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B Condens. Matter 34(6), 4129–4138 (1986).
[Crossref] [PubMed]

1980 (1)

C. B. Scruby, R. J. Dewhurst, D. A. Hutchins, and S. B. Palmer, “Quantitative studies of thermally generated elastic waves in laser-irradiated metals,” J. Appl. Phys. 51(12), 6210–12368 (1980).
[Crossref]

1974 (1)

S. I. Anisimov, B. L. Kapeliovich, and T. L. Perelman, “Electron emission from metal surfaces exposed to ultrashort laser pulses,” Sov. Phys. JETP 39(2), 375–377 (1974).

Acioli, L. H.

C. Sun, F. Vallée, L. H. Acioli, E. P. Ippen, and J. G. Fujimoto, “Femtosecond-tunable measurement of electron thermalization in gold,” Phys. Rev. B Condens. Matter 50(20), 15337–15348 (1994).
[Crossref] [PubMed]

Afanasiev, Yu. V.

A. P. Kanavin, I. V. Smetanin, V. A. Isakov, Yu. V. Afanasiev, B. N. Chichkov, B. Wellegehausen, S. Nolte, C. Momma, and A. Tunnermann, “Heat transport in metals irradiated by ultrashort laser pulses,” Phys. Rev. B 57(23), 14698–14703 (1998).
[Crossref]

Aifantis, K. E.

G. D. Tsibidis, E. Stratakis, and K. E. Aifantis, “Thermoplastic deformation of silicon surfaces induced by ultrashort pulsed lasers in submelting conditions,” J. Appl. Phys. 111, 053502 (2012).
[Crossref]

Albrecht, M.

V. V. Temnov, C. Klieber, K. A. Nelson, T. Thomay, V. Knittel, A. Leitenstorfer, D. Makarov, M. Albrecht, and R. Bratschitsch, “Femtosecond nonlinear ultrasonics in gold probed with ultrashort surface plasmons,” Nat. Commun. 4, 1468 (2013).
[Crossref] [PubMed]

Andreev, S. V.

O. V. Misochko, T. Dekorsy, S. V. Andreev, V. O. Kompanets, Yu. A. Matveets, A. G. Stepanov, and S. V. Chekalin, “Effect of intense chirped pulses on the coherent phonon generation in Te,” Appl. Phys. Lett. 90(7), 071901 (2007).
[Crossref]

Anisimov, S. I.

S. I. Anisimov and B. Rethfeld, “On the theory of ultrashort laser-pulse interaction with a metal,” Izvestiya Akademii Nauk Seriya Fizicheskaya 61, 1642–1655 (1997).

S. I. Anisimov, B. L. Kapeliovich, and T. L. Perelman, “Electron emission from metal surfaces exposed to ultrashort laser pulses,” Sov. Phys. JETP 39(2), 375–377 (1974).

Audoin, B.

T. Dehoux, M. Perton, N. Chigarev, C. Rossignol, J.-M. Rampnoux, and B. Audoin, “Effect of laser pulse duration in picosecond ultrasonics,” J. Appl. Phys. 100(6), 064318 (2006).
[Crossref]

Aussel, J. D.

J. D. Aussel, A. Le Brun, and J. C. Baboux, “Generating acoustic waves by laser: theoretical and experimental study of the emission source,” Ultrasonics 26(5), 245–255 (1988).
[Crossref]

Baboux, J. C.

J. D. Aussel, A. Le Brun, and J. C. Baboux, “Generating acoustic waves by laser: theoretical and experimental study of the emission source,” Ultrasonics 26(5), 245–255 (1988).
[Crossref]

Beraun, J. E.

J. K. Chen, W. P. Latham, and J. E. Beraun, “The role of electron–phonon coupling in ultrafast laser heating,” J. Laser Appl. 17(1), 63–68 (2005).
[Crossref]

Bovensiepen, U.

M. Lisowski, P. A. Loukakos, U. Bovensiepen, J. Staehler, C. Gahl, and M. Wolf, “Ultrafast dynamics of electron thermalization, cooling and transport effects in Ru (00l),” Appl. Phys., A Mater. Sci. Process. 78(2), 165–176 (2004).
[Crossref]

Bratschitsch, R.

V. V. Temnov, C. Klieber, K. A. Nelson, T. Thomay, V. Knittel, A. Leitenstorfer, D. Makarov, M. Albrecht, and R. Bratschitsch, “Femtosecond nonlinear ultrasonics in gold probed with ultrashort surface plasmons,” Nat. Commun. 4, 1468 (2013).
[Crossref] [PubMed]

Caffrey, A. P.

P. M. Norris, A. P. Caffrey, R. J. Stevens, J. M. Klopf, J. T. McLeskey, and A. N. Smith, “Femtosecond pump-probe nondestructive examination of materials,” Rev. Sci. Instrum. 74(1), 400–406 (2003).
[Crossref]

Cahill, D. G.

B. C. Daly, K. Kang, and D. G. Cahill, “Attenuation of picosecond ultrasonic pulses in a thin silicon wafer,” 1st Int. Symposium on Laser Ultrasonics: Science, Technology and Applications, July 2008, Montreal, Canada.

Carpene, E.

E. Carpene, “Ultrafast laser irradiation of metals: Beyond the two-temperature model,” Phys. Rev. B 74(2), 024301 (2006).
[Crossref]

Celli, V.

Z. Lin, L. V. Zhigilei, and V. Celli, “Electron-phonon coupling and electron heat capacity of metals under conditions of strong electron-phonon nonequilibrium,” Phys. Rev. B 77(7), 075133 (2008).
[Crossref]

Cerullo, G.

E. Pontecorvo, M. Ortolani, D. Polli, M. Ferretti, G. Ruocco, G. Cerullo, and T. Scopigno, “Visualizing coherent phonon propagation in the 100 GHz range: A broadband picosecond acoustics approach,” Appl. Phys. Lett. 98(1), 011901 (2011).
[Crossref]

Chekalin, S. V.

O. V. Misochko, T. Dekorsy, S. V. Andreev, V. O. Kompanets, Yu. A. Matveets, A. G. Stepanov, and S. V. Chekalin, “Effect of intense chirped pulses on the coherent phonon generation in Te,” Appl. Phys. Lett. 90(7), 071901 (2007).
[Crossref]

Chen, J. K.

J. K. Chen, W. P. Latham, and J. E. Beraun, “The role of electron–phonon coupling in ultrafast laser heating,” J. Laser Appl. 17(1), 63–68 (2005).
[Crossref]

Chen, W.

Chichkov, B. N.

A. P. Kanavin, I. V. Smetanin, V. A. Isakov, Yu. V. Afanasiev, B. N. Chichkov, B. Wellegehausen, S. Nolte, C. Momma, and A. Tunnermann, “Heat transport in metals irradiated by ultrashort laser pulses,” Phys. Rev. B 57(23), 14698–14703 (1998).
[Crossref]

Chigarev, N.

T. Dehoux, M. Perton, N. Chigarev, C. Rossignol, J.-M. Rampnoux, and B. Audoin, “Effect of laser pulse duration in picosecond ultrasonics,” J. Appl. Phys. 100(6), 064318 (2006).
[Crossref]

Conrad, U.

J. Hohlfeld, S.-S. Wellershoff, J. Gudde, U. Conrad, V. Jahnke, and E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” Chem. Phys. 251(1–3), 237–258 (2000).
[Crossref]

Cote, R.

A. Devos and R. Cote, “Strong oscillations detected by picosecond ultrasonics in silicon: Evidence for an electronic-structure effect,” Phys. Rev. B 70(12), 125208 (2004).
[Crossref]

Daly, B. C.

B. C. Daly, K. Kang, and D. G. Cahill, “Attenuation of picosecond ultrasonic pulses in a thin silicon wafer,” 1st Int. Symposium on Laser Ultrasonics: Science, Technology and Applications, July 2008, Montreal, Canada.

Dehoux, T.

T. Dehoux, M. Perton, N. Chigarev, C. Rossignol, J.-M. Rampnoux, and B. Audoin, “Effect of laser pulse duration in picosecond ultrasonics,” J. Appl. Phys. 100(6), 064318 (2006).
[Crossref]

Dekorsy, T.

O. V. Misochko, T. Dekorsy, S. V. Andreev, V. O. Kompanets, Yu. A. Matveets, A. G. Stepanov, and S. V. Chekalin, “Effect of intense chirped pulses on the coherent phonon generation in Te,” Appl. Phys. Lett. 90(7), 071901 (2007).
[Crossref]

Devos, A.

A. Devos, “Colored ultrafast acoustics: From fundamentals to applications,” Ultrasonics 56(2), 90–97 (2015).
[Crossref] [PubMed]

A. Devos and R. Cote, “Strong oscillations detected by picosecond ultrasonics in silicon: Evidence for an electronic-structure effect,” Phys. Rev. B 70(12), 125208 (2004).
[Crossref]

Dewhurst, R. J.

C. B. Scruby, R. J. Dewhurst, D. A. Hutchins, and S. B. Palmer, “Quantitative studies of thermally generated elastic waves in laser-irradiated metals,” J. Appl. Phys. 51(12), 6210–12368 (1980).
[Crossref]

Djurisic, A. B.

Elazar, J. M.

Ferretti, M.

E. Pontecorvo, M. Ortolani, D. Polli, M. Ferretti, G. Ruocco, G. Cerullo, and T. Scopigno, “Visualizing coherent phonon propagation in the 100 GHz range: A broadband picosecond acoustics approach,” Appl. Phys. Lett. 98(1), 011901 (2011).
[Crossref]

Fujimoto, J. G.

C. Sun, F. Vallée, L. H. Acioli, E. P. Ippen, and J. G. Fujimoto, “Femtosecond-tunable measurement of electron thermalization in gold,” Phys. Rev. B Condens. Matter 50(20), 15337–15348 (1994).
[Crossref] [PubMed]

Gahl, C.

M. Lisowski, P. A. Loukakos, U. Bovensiepen, J. Staehler, C. Gahl, and M. Wolf, “Ultrafast dynamics of electron thermalization, cooling and transport effects in Ru (00l),” Appl. Phys., A Mater. Sci. Process. 78(2), 165–176 (2004).
[Crossref]

Grahn, H. T.

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B Condens. Matter 34(6), 4129–4138 (1986).
[Crossref] [PubMed]

Gudde, J.

J. Hohlfeld, S.-S. Wellershoff, J. Gudde, U. Conrad, V. Jahnke, and E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” Chem. Phys. 251(1–3), 237–258 (2000).
[Crossref]

Gusev, V. E.

D. H. Hurley, O. B. Wright, O. Matsuda, V. E. Gusev, and O. V. Kolosov, “Laser picosecond acoustics in isotropic and anisotropic materials,” Ultrasonics 38(1-8), 470–474 (2000).
[Crossref] [PubMed]

O. B. Wright and V. E. Gusev, “Acoustic generation in crystalline silicon with femtosecond optical pulses,” Appl. Phys. Lett. 66(10), 1190–1192 (1995).
[Crossref]

Hohlfeld, J.

J. Hohlfeld, S.-S. Wellershoff, J. Gudde, U. Conrad, V. Jahnke, and E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” Chem. Phys. 251(1–3), 237–258 (2000).
[Crossref]

Hopkins, P. E.

P. E. Hopkins and P. M. Norris, “Contribution of ballistic electron transport to energy transfer during electron-phonon nonequilibrium in thin metal films,” J. Heat Transfer 131(4), 043208 (2009).
[Crossref]

Hurley, D. H.

D. H. Hurley, O. B. Wright, O. Matsuda, V. E. Gusev, and O. V. Kolosov, “Laser picosecond acoustics in isotropic and anisotropic materials,” Ultrasonics 38(1-8), 470–474 (2000).
[Crossref] [PubMed]

Hutchins, D. A.

C. B. Scruby, R. J. Dewhurst, D. A. Hutchins, and S. B. Palmer, “Quantitative studies of thermally generated elastic waves in laser-irradiated metals,” J. Appl. Phys. 51(12), 6210–12368 (1980).
[Crossref]

Ippen, E. P.

C. Sun, F. Vallée, L. H. Acioli, E. P. Ippen, and J. G. Fujimoto, “Femtosecond-tunable measurement of electron thermalization in gold,” Phys. Rev. B Condens. Matter 50(20), 15337–15348 (1994).
[Crossref] [PubMed]

Isakov, V. A.

A. P. Kanavin, I. V. Smetanin, V. A. Isakov, Yu. V. Afanasiev, B. N. Chichkov, B. Wellegehausen, S. Nolte, C. Momma, and A. Tunnermann, “Heat transport in metals irradiated by ultrashort laser pulses,” Phys. Rev. B 57(23), 14698–14703 (1998).
[Crossref]

Ishii, S.

Jahnke, V.

J. Hohlfeld, S.-S. Wellershoff, J. Gudde, U. Conrad, V. Jahnke, and E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” Chem. Phys. 251(1–3), 237–258 (2000).
[Crossref]

Kanavin, A. P.

A. P. Kanavin, I. V. Smetanin, V. A. Isakov, Yu. V. Afanasiev, B. N. Chichkov, B. Wellegehausen, S. Nolte, C. Momma, and A. Tunnermann, “Heat transport in metals irradiated by ultrashort laser pulses,” Phys. Rev. B 57(23), 14698–14703 (1998).
[Crossref]

Kang, K.

B. C. Daly, K. Kang, and D. G. Cahill, “Attenuation of picosecond ultrasonic pulses in a thin silicon wafer,” 1st Int. Symposium on Laser Ultrasonics: Science, Technology and Applications, July 2008, Montreal, Canada.

Kapeliovich, B. L.

S. I. Anisimov, B. L. Kapeliovich, and T. L. Perelman, “Electron emission from metal surfaces exposed to ultrashort laser pulses,” Sov. Phys. JETP 39(2), 375–377 (1974).

Kawashima, K.

O. B. Wright and K. Kawashima, “Coherent phonon detection from ultrafast surface vibrations,” Phys. Rev. Lett. 69(11), 1668–1671 (1992).
[Crossref] [PubMed]

Kildishev, A. V.

Klieber, C.

V. V. Temnov, C. Klieber, K. A. Nelson, T. Thomay, V. Knittel, A. Leitenstorfer, D. Makarov, M. Albrecht, and R. Bratschitsch, “Femtosecond nonlinear ultrasonics in gold probed with ultrashort surface plasmons,” Nat. Commun. 4, 1468 (2013).
[Crossref] [PubMed]

Klopf, J. M.

P. M. Norris, A. P. Caffrey, R. J. Stevens, J. M. Klopf, J. T. McLeskey, and A. N. Smith, “Femtosecond pump-probe nondestructive examination of materials,” Rev. Sci. Instrum. 74(1), 400–406 (2003).
[Crossref]

Knittel, V.

V. V. Temnov, C. Klieber, K. A. Nelson, T. Thomay, V. Knittel, A. Leitenstorfer, D. Makarov, M. Albrecht, and R. Bratschitsch, “Femtosecond nonlinear ultrasonics in gold probed with ultrashort surface plasmons,” Nat. Commun. 4, 1468 (2013).
[Crossref] [PubMed]

Kolosov, O. V.

D. H. Hurley, O. B. Wright, O. Matsuda, V. E. Gusev, and O. V. Kolosov, “Laser picosecond acoustics in isotropic and anisotropic materials,” Ultrasonics 38(1-8), 470–474 (2000).
[Crossref] [PubMed]

Kompanets, V. O.

O. V. Misochko, T. Dekorsy, S. V. Andreev, V. O. Kompanets, Yu. A. Matveets, A. G. Stepanov, and S. V. Chekalin, “Effect of intense chirped pulses on the coherent phonon generation in Te,” Appl. Phys. Lett. 90(7), 071901 (2007).
[Crossref]

Latham, W. P.

J. K. Chen, W. P. Latham, and J. E. Beraun, “The role of electron–phonon coupling in ultrafast laser heating,” J. Laser Appl. 17(1), 63–68 (2005).
[Crossref]

Le Brun, A.

J. D. Aussel, A. Le Brun, and J. C. Baboux, “Generating acoustic waves by laser: theoretical and experimental study of the emission source,” Ultrasonics 26(5), 245–255 (1988).
[Crossref]

Leitenstorfer, A.

V. V. Temnov, C. Klieber, K. A. Nelson, T. Thomay, V. Knittel, A. Leitenstorfer, D. Makarov, M. Albrecht, and R. Bratschitsch, “Femtosecond nonlinear ultrasonics in gold probed with ultrashort surface plasmons,” Nat. Commun. 4, 1468 (2013).
[Crossref] [PubMed]

Lin, H. N.

H. N. Lin, R. J. Stoner, H. J. Maris, and J. Tauc, “Phonon attenuation and velocity measurements in transparent materials by picosecond acoustic interferometry,” J. Appl. Phys. 69(7), 3816–3822 (1991).
[Crossref]

Lin, Z.

Z. Lin, L. V. Zhigilei, and V. Celli, “Electron-phonon coupling and electron heat capacity of metals under conditions of strong electron-phonon nonequilibrium,” Phys. Rev. B 77(7), 075133 (2008).
[Crossref]

Lisowski, M.

M. Lisowski, P. A. Loukakos, U. Bovensiepen, J. Staehler, C. Gahl, and M. Wolf, “Ultrafast dynamics of electron thermalization, cooling and transport effects in Ru (00l),” Appl. Phys., A Mater. Sci. Process. 78(2), 165–176 (2004).
[Crossref]

Loukakos, P. A.

M. Lisowski, P. A. Loukakos, U. Bovensiepen, J. Staehler, C. Gahl, and M. Wolf, “Ultrafast dynamics of electron thermalization, cooling and transport effects in Ru (00l),” Appl. Phys., A Mater. Sci. Process. 78(2), 165–176 (2004).
[Crossref]

Majewski, M. L.

Makarov, D.

V. V. Temnov, C. Klieber, K. A. Nelson, T. Thomay, V. Knittel, A. Leitenstorfer, D. Makarov, M. Albrecht, and R. Bratschitsch, “Femtosecond nonlinear ultrasonics in gold probed with ultrashort surface plasmons,” Nat. Commun. 4, 1468 (2013).
[Crossref] [PubMed]

Maris, H. J.

G. Tas and H. J. Maris, “Electron diffusion in metals studied by picosecond ultrasonics,” Phys. Rev. B Condens. Matter 49(21), 15046–15054 (1994).
[Crossref] [PubMed]

H. N. Lin, R. J. Stoner, H. J. Maris, and J. Tauc, “Phonon attenuation and velocity measurements in transparent materials by picosecond acoustic interferometry,” J. Appl. Phys. 69(7), 3816–3822 (1991).
[Crossref]

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B Condens. Matter 34(6), 4129–4138 (1986).
[Crossref] [PubMed]

Matsuda, O.

D. H. Hurley, O. B. Wright, O. Matsuda, V. E. Gusev, and O. V. Kolosov, “Laser picosecond acoustics in isotropic and anisotropic materials,” Ultrasonics 38(1-8), 470–474 (2000).
[Crossref] [PubMed]

Matthias, E.

J. Hohlfeld, S.-S. Wellershoff, J. Gudde, U. Conrad, V. Jahnke, and E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” Chem. Phys. 251(1–3), 237–258 (2000).
[Crossref]

Matveets, Yu. A.

O. V. Misochko, T. Dekorsy, S. V. Andreev, V. O. Kompanets, Yu. A. Matveets, A. G. Stepanov, and S. V. Chekalin, “Effect of intense chirped pulses on the coherent phonon generation in Te,” Appl. Phys. Lett. 90(7), 071901 (2007).
[Crossref]

Mazur, E.

S. K. Sundaram and E. Mazur, “Inducing and probing non-thermal transitions in semiconductors using femtosecond laser pulses,” Nat. Mater. 1(4), 217–224 (2002).
[Crossref] [PubMed]

McLeskey, J. T.

P. M. Norris, A. P. Caffrey, R. J. Stevens, J. M. Klopf, J. T. McLeskey, and A. N. Smith, “Femtosecond pump-probe nondestructive examination of materials,” Rev. Sci. Instrum. 74(1), 400–406 (2003).
[Crossref]

Misochko, O. V.

O. V. Misochko, T. Dekorsy, S. V. Andreev, V. O. Kompanets, Yu. A. Matveets, A. G. Stepanov, and S. V. Chekalin, “Effect of intense chirped pulses on the coherent phonon generation in Te,” Appl. Phys. Lett. 90(7), 071901 (2007).
[Crossref]

Momma, C.

A. P. Kanavin, I. V. Smetanin, V. A. Isakov, Yu. V. Afanasiev, B. N. Chichkov, B. Wellegehausen, S. Nolte, C. Momma, and A. Tunnermann, “Heat transport in metals irradiated by ultrashort laser pulses,” Phys. Rev. B 57(23), 14698–14703 (1998).
[Crossref]

Nelson, K. A.

V. V. Temnov, C. Klieber, K. A. Nelson, T. Thomay, V. Knittel, A. Leitenstorfer, D. Makarov, M. Albrecht, and R. Bratschitsch, “Femtosecond nonlinear ultrasonics in gold probed with ultrashort surface plasmons,” Nat. Commun. 4, 1468 (2013).
[Crossref] [PubMed]

Nolte, S.

A. P. Kanavin, I. V. Smetanin, V. A. Isakov, Yu. V. Afanasiev, B. N. Chichkov, B. Wellegehausen, S. Nolte, C. Momma, and A. Tunnermann, “Heat transport in metals irradiated by ultrashort laser pulses,” Phys. Rev. B 57(23), 14698–14703 (1998).
[Crossref]

Norris, P. M.

P. E. Hopkins and P. M. Norris, “Contribution of ballistic electron transport to energy transfer during electron-phonon nonequilibrium in thin metal films,” J. Heat Transfer 131(4), 043208 (2009).
[Crossref]

P. M. Norris, A. P. Caffrey, R. J. Stevens, J. M. Klopf, J. T. McLeskey, and A. N. Smith, “Femtosecond pump-probe nondestructive examination of materials,” Rev. Sci. Instrum. 74(1), 400–406 (2003).
[Crossref]

Ortolani, M.

E. Pontecorvo, M. Ortolani, D. Polli, M. Ferretti, G. Ruocco, G. Cerullo, and T. Scopigno, “Visualizing coherent phonon propagation in the 100 GHz range: A broadband picosecond acoustics approach,” Appl. Phys. Lett. 98(1), 011901 (2011).
[Crossref]

Palmer, S. B.

C. B. Scruby, R. J. Dewhurst, D. A. Hutchins, and S. B. Palmer, “Quantitative studies of thermally generated elastic waves in laser-irradiated metals,” J. Appl. Phys. 51(12), 6210–12368 (1980).
[Crossref]

Perelman, T. L.

S. I. Anisimov, B. L. Kapeliovich, and T. L. Perelman, “Electron emission from metal surfaces exposed to ultrashort laser pulses,” Sov. Phys. JETP 39(2), 375–377 (1974).

Perton, M.

T. Dehoux, M. Perton, N. Chigarev, C. Rossignol, J.-M. Rampnoux, and B. Audoin, “Effect of laser pulse duration in picosecond ultrasonics,” J. Appl. Phys. 100(6), 064318 (2006).
[Crossref]

Polli, D.

E. Pontecorvo, M. Ortolani, D. Polli, M. Ferretti, G. Ruocco, G. Cerullo, and T. Scopigno, “Visualizing coherent phonon propagation in the 100 GHz range: A broadband picosecond acoustics approach,” Appl. Phys. Lett. 98(1), 011901 (2011).
[Crossref]

Pontecorvo, E.

E. Pontecorvo, M. Ortolani, D. Polli, M. Ferretti, G. Ruocco, G. Cerullo, and T. Scopigno, “Visualizing coherent phonon propagation in the 100 GHz range: A broadband picosecond acoustics approach,” Appl. Phys. Lett. 98(1), 011901 (2011).
[Crossref]

Rakic, A. D.

Rampnoux, J.-M.

T. Dehoux, M. Perton, N. Chigarev, C. Rossignol, J.-M. Rampnoux, and B. Audoin, “Effect of laser pulse duration in picosecond ultrasonics,” J. Appl. Phys. 100(6), 064318 (2006).
[Crossref]

Rethfeld, B.

S. I. Anisimov and B. Rethfeld, “On the theory of ultrashort laser-pulse interaction with a metal,” Izvestiya Akademii Nauk Seriya Fizicheskaya 61, 1642–1655 (1997).

Rossignol, C.

T. Dehoux, M. Perton, N. Chigarev, C. Rossignol, J.-M. Rampnoux, and B. Audoin, “Effect of laser pulse duration in picosecond ultrasonics,” J. Appl. Phys. 100(6), 064318 (2006).
[Crossref]

Ruocco, G.

E. Pontecorvo, M. Ortolani, D. Polli, M. Ferretti, G. Ruocco, G. Cerullo, and T. Scopigno, “Visualizing coherent phonon propagation in the 100 GHz range: A broadband picosecond acoustics approach,” Appl. Phys. Lett. 98(1), 011901 (2011).
[Crossref]

Scopigno, T.

E. Pontecorvo, M. Ortolani, D. Polli, M. Ferretti, G. Ruocco, G. Cerullo, and T. Scopigno, “Visualizing coherent phonon propagation in the 100 GHz range: A broadband picosecond acoustics approach,” Appl. Phys. Lett. 98(1), 011901 (2011).
[Crossref]

Scruby, C. B.

C. B. Scruby, R. J. Dewhurst, D. A. Hutchins, and S. B. Palmer, “Quantitative studies of thermally generated elastic waves in laser-irradiated metals,” J. Appl. Phys. 51(12), 6210–12368 (1980).
[Crossref]

Shalaev, V. M.

Smetanin, I. V.

A. P. Kanavin, I. V. Smetanin, V. A. Isakov, Yu. V. Afanasiev, B. N. Chichkov, B. Wellegehausen, S. Nolte, C. Momma, and A. Tunnermann, “Heat transport in metals irradiated by ultrashort laser pulses,” Phys. Rev. B 57(23), 14698–14703 (1998).
[Crossref]

Smith, A. N.

P. M. Norris, A. P. Caffrey, R. J. Stevens, J. M. Klopf, J. T. McLeskey, and A. N. Smith, “Femtosecond pump-probe nondestructive examination of materials,” Rev. Sci. Instrum. 74(1), 400–406 (2003).
[Crossref]

Staehler, J.

M. Lisowski, P. A. Loukakos, U. Bovensiepen, J. Staehler, C. Gahl, and M. Wolf, “Ultrafast dynamics of electron thermalization, cooling and transport effects in Ru (00l),” Appl. Phys., A Mater. Sci. Process. 78(2), 165–176 (2004).
[Crossref]

Stepanov, A. G.

O. V. Misochko, T. Dekorsy, S. V. Andreev, V. O. Kompanets, Yu. A. Matveets, A. G. Stepanov, and S. V. Chekalin, “Effect of intense chirped pulses on the coherent phonon generation in Te,” Appl. Phys. Lett. 90(7), 071901 (2007).
[Crossref]

Stevens, R. J.

P. M. Norris, A. P. Caffrey, R. J. Stevens, J. M. Klopf, J. T. McLeskey, and A. N. Smith, “Femtosecond pump-probe nondestructive examination of materials,” Rev. Sci. Instrum. 74(1), 400–406 (2003).
[Crossref]

Stoner, R. J.

H. N. Lin, R. J. Stoner, H. J. Maris, and J. Tauc, “Phonon attenuation and velocity measurements in transparent materials by picosecond acoustic interferometry,” J. Appl. Phys. 69(7), 3816–3822 (1991).
[Crossref]

Stratakis, E.

G. D. Tsibidis, E. Stratakis, and K. E. Aifantis, “Thermoplastic deformation of silicon surfaces induced by ultrashort pulsed lasers in submelting conditions,” J. Appl. Phys. 111, 053502 (2012).
[Crossref]

Sun, C.

C. Sun, F. Vallée, L. H. Acioli, E. P. Ippen, and J. G. Fujimoto, “Femtosecond-tunable measurement of electron thermalization in gold,” Phys. Rev. B Condens. Matter 50(20), 15337–15348 (1994).
[Crossref] [PubMed]

Sundaram, S. K.

S. K. Sundaram and E. Mazur, “Inducing and probing non-thermal transitions in semiconductors using femtosecond laser pulses,” Nat. Mater. 1(4), 217–224 (2002).
[Crossref] [PubMed]

Tas, G.

G. Tas and H. J. Maris, “Electron diffusion in metals studied by picosecond ultrasonics,” Phys. Rev. B Condens. Matter 49(21), 15046–15054 (1994).
[Crossref] [PubMed]

Tauc, J.

H. N. Lin, R. J. Stoner, H. J. Maris, and J. Tauc, “Phonon attenuation and velocity measurements in transparent materials by picosecond acoustic interferometry,” J. Appl. Phys. 69(7), 3816–3822 (1991).
[Crossref]

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B Condens. Matter 34(6), 4129–4138 (1986).
[Crossref] [PubMed]

Temnov, V. V.

V. V. Temnov, C. Klieber, K. A. Nelson, T. Thomay, V. Knittel, A. Leitenstorfer, D. Makarov, M. Albrecht, and R. Bratschitsch, “Femtosecond nonlinear ultrasonics in gold probed with ultrashort surface plasmons,” Nat. Commun. 4, 1468 (2013).
[Crossref] [PubMed]

Thomay, T.

V. V. Temnov, C. Klieber, K. A. Nelson, T. Thomay, V. Knittel, A. Leitenstorfer, D. Makarov, M. Albrecht, and R. Bratschitsch, “Femtosecond nonlinear ultrasonics in gold probed with ultrashort surface plasmons,” Nat. Commun. 4, 1468 (2013).
[Crossref] [PubMed]

Thomsen, C.

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B Condens. Matter 34(6), 4129–4138 (1986).
[Crossref] [PubMed]

Thoreson, M. D.

Tsibidis, G. D.

G. D. Tsibidis, “Thermal response of double layered metal films after ultrashort pulsed laser irradiation: The role of nonthermal electron dynamics,” Appl. Phys. Lett. 104(5), 051603 (2014).
[Crossref]

G. D. Tsibidis, E. Stratakis, and K. E. Aifantis, “Thermoplastic deformation of silicon surfaces induced by ultrashort pulsed lasers in submelting conditions,” J. Appl. Phys. 111, 053502 (2012).
[Crossref]

Tunnermann, A.

A. P. Kanavin, I. V. Smetanin, V. A. Isakov, Yu. V. Afanasiev, B. N. Chichkov, B. Wellegehausen, S. Nolte, C. Momma, and A. Tunnermann, “Heat transport in metals irradiated by ultrashort laser pulses,” Phys. Rev. B 57(23), 14698–14703 (1998).
[Crossref]

Vallée, F.

C. Sun, F. Vallée, L. H. Acioli, E. P. Ippen, and J. G. Fujimoto, “Femtosecond-tunable measurement of electron thermalization in gold,” Phys. Rev. B Condens. Matter 50(20), 15337–15348 (1994).
[Crossref] [PubMed]

Wellegehausen, B.

A. P. Kanavin, I. V. Smetanin, V. A. Isakov, Yu. V. Afanasiev, B. N. Chichkov, B. Wellegehausen, S. Nolte, C. Momma, and A. Tunnermann, “Heat transport in metals irradiated by ultrashort laser pulses,” Phys. Rev. B 57(23), 14698–14703 (1998).
[Crossref]

Wellershoff, S.-S.

J. Hohlfeld, S.-S. Wellershoff, J. Gudde, U. Conrad, V. Jahnke, and E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” Chem. Phys. 251(1–3), 237–258 (2000).
[Crossref]

Wolf, M.

M. Lisowski, P. A. Loukakos, U. Bovensiepen, J. Staehler, C. Gahl, and M. Wolf, “Ultrafast dynamics of electron thermalization, cooling and transport effects in Ru (00l),” Appl. Phys., A Mater. Sci. Process. 78(2), 165–176 (2004).
[Crossref]

Wright, O. B.

D. H. Hurley, O. B. Wright, O. Matsuda, V. E. Gusev, and O. V. Kolosov, “Laser picosecond acoustics in isotropic and anisotropic materials,” Ultrasonics 38(1-8), 470–474 (2000).
[Crossref] [PubMed]

O. B. Wright and V. E. Gusev, “Acoustic generation in crystalline silicon with femtosecond optical pulses,” Appl. Phys. Lett. 66(10), 1190–1192 (1995).
[Crossref]

O. B. Wright and K. Kawashima, “Coherent phonon detection from ultrafast surface vibrations,” Phys. Rev. Lett. 69(11), 1668–1671 (1992).
[Crossref] [PubMed]

Zhigilei, L. V.

Z. Lin, L. V. Zhigilei, and V. Celli, “Electron-phonon coupling and electron heat capacity of metals under conditions of strong electron-phonon nonequilibrium,” Phys. Rev. B 77(7), 075133 (2008).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (4)

E. Pontecorvo, M. Ortolani, D. Polli, M. Ferretti, G. Ruocco, G. Cerullo, and T. Scopigno, “Visualizing coherent phonon propagation in the 100 GHz range: A broadband picosecond acoustics approach,” Appl. Phys. Lett. 98(1), 011901 (2011).
[Crossref]

G. D. Tsibidis, “Thermal response of double layered metal films after ultrashort pulsed laser irradiation: The role of nonthermal electron dynamics,” Appl. Phys. Lett. 104(5), 051603 (2014).
[Crossref]

O. B. Wright and V. E. Gusev, “Acoustic generation in crystalline silicon with femtosecond optical pulses,” Appl. Phys. Lett. 66(10), 1190–1192 (1995).
[Crossref]

O. V. Misochko, T. Dekorsy, S. V. Andreev, V. O. Kompanets, Yu. A. Matveets, A. G. Stepanov, and S. V. Chekalin, “Effect of intense chirped pulses on the coherent phonon generation in Te,” Appl. Phys. Lett. 90(7), 071901 (2007).
[Crossref]

Appl. Phys., A Mater. Sci. Process. (1)

M. Lisowski, P. A. Loukakos, U. Bovensiepen, J. Staehler, C. Gahl, and M. Wolf, “Ultrafast dynamics of electron thermalization, cooling and transport effects in Ru (00l),” Appl. Phys., A Mater. Sci. Process. 78(2), 165–176 (2004).
[Crossref]

Chem. Phys. (1)

J. Hohlfeld, S.-S. Wellershoff, J. Gudde, U. Conrad, V. Jahnke, and E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” Chem. Phys. 251(1–3), 237–258 (2000).
[Crossref]

Izvestiya Akademii Nauk Seriya Fizicheskaya (1)

S. I. Anisimov and B. Rethfeld, “On the theory of ultrashort laser-pulse interaction with a metal,” Izvestiya Akademii Nauk Seriya Fizicheskaya 61, 1642–1655 (1997).

J. Appl. Phys. (4)

H. N. Lin, R. J. Stoner, H. J. Maris, and J. Tauc, “Phonon attenuation and velocity measurements in transparent materials by picosecond acoustic interferometry,” J. Appl. Phys. 69(7), 3816–3822 (1991).
[Crossref]

G. D. Tsibidis, E. Stratakis, and K. E. Aifantis, “Thermoplastic deformation of silicon surfaces induced by ultrashort pulsed lasers in submelting conditions,” J. Appl. Phys. 111, 053502 (2012).
[Crossref]

C. B. Scruby, R. J. Dewhurst, D. A. Hutchins, and S. B. Palmer, “Quantitative studies of thermally generated elastic waves in laser-irradiated metals,” J. Appl. Phys. 51(12), 6210–12368 (1980).
[Crossref]

T. Dehoux, M. Perton, N. Chigarev, C. Rossignol, J.-M. Rampnoux, and B. Audoin, “Effect of laser pulse duration in picosecond ultrasonics,” J. Appl. Phys. 100(6), 064318 (2006).
[Crossref]

J. Heat Transfer (1)

P. E. Hopkins and P. M. Norris, “Contribution of ballistic electron transport to energy transfer during electron-phonon nonequilibrium in thin metal films,” J. Heat Transfer 131(4), 043208 (2009).
[Crossref]

J. Laser Appl. (1)

J. K. Chen, W. P. Latham, and J. E. Beraun, “The role of electron–phonon coupling in ultrafast laser heating,” J. Laser Appl. 17(1), 63–68 (2005).
[Crossref]

Nat. Commun. (1)

V. V. Temnov, C. Klieber, K. A. Nelson, T. Thomay, V. Knittel, A. Leitenstorfer, D. Makarov, M. Albrecht, and R. Bratschitsch, “Femtosecond nonlinear ultrasonics in gold probed with ultrashort surface plasmons,” Nat. Commun. 4, 1468 (2013).
[Crossref] [PubMed]

Nat. Mater. (1)

S. K. Sundaram and E. Mazur, “Inducing and probing non-thermal transitions in semiconductors using femtosecond laser pulses,” Nat. Mater. 1(4), 217–224 (2002).
[Crossref] [PubMed]

Opt. Express (1)

Phys. Rev. B (4)

Z. Lin, L. V. Zhigilei, and V. Celli, “Electron-phonon coupling and electron heat capacity of metals under conditions of strong electron-phonon nonequilibrium,” Phys. Rev. B 77(7), 075133 (2008).
[Crossref]

A. Devos and R. Cote, “Strong oscillations detected by picosecond ultrasonics in silicon: Evidence for an electronic-structure effect,” Phys. Rev. B 70(12), 125208 (2004).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 (a) Schematic diagram of excitation and detection scheme in the metal/Si systems, and (b) experimental setup for the femtosecond degenerate pump-probe differential reflectivity method.
Fig. 2
Fig. 2 (a) Transient reflectivity signals for three Ti thicknesses (Ti/Si samples). The inset shows the first and second acoustic echoes (53 nm Ti), (b) transient reflectivity signals for two Ag thicknesses (Ag/Si samples). The two signals are vertically offset for clarity. The inset shows the fast (less than 1 ps) and slower (a few ps) electronic signal decay for Ti and Ag metal films, respectively. In all cases, the period of the oscillations (~13 ps) corresponds to the Si Brillouin period.
Fig. 3
Fig. 3 Calculated strain pulse distribution in 25 nm metal/Si systems as a function of depth, at different time instants. Solid blue and dot-dashed red lines correspond to Ti/Si and Ag/Si cases, respectively. Vertical dashed line corresponds to the metal/Si interface position.
Fig. 4
Fig. 4 Calculated temporal evolution for the heat sources for (a) Ti, and (b) Ag.
Fig. 5
Fig. 5 Calculated temporal evolution of electron (solid red lines) and lattice (dashed blue lines) temperatures at the surface of (a) 25nm Ti thin film, and (b) 25nm Ag thin film, on Si substrates.
Fig. 6
Fig. 6 Calculated spatio-temporal evolution of lattice temperature of (a) 25nm Ti thin film, and (b) 25nm Ag thin film, on Si substrates.

Tables (1)

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Table 1 Calculated reflectance Rpr1, transmittance Tpr2 and strain probing efficiency peff

Equations (15)

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d opt =2 τ osc u Si = E T c n Si = λ n Si 1 τ osc = f Br = 2 n Si u Si λ
η i = H E
η s = ε o μ o N i cos θ i , η p = ε o μ o N i /cos θ i
[ B C ]=[ E a/m E m/Si H a/m E m/Si ]=[ cosδ isinδ η 2 i η 2 sinδ cosδ ]×[ 1 η 3 ]
R=( η 1 BC η 1 B+C ) ( η 1 BC η 1 B+C )
T= 4 η 1 Re[ η 3 ] ( η 1 B+C ) ( η 1 B+C )
A= 4 η 1 Re[ B C η 3 ] ( η 1 B+C ) ( η 1 B+C )
C e (2) T e (2) t = ·( k e (2) T e (2) )- G (2) ( T e (2) - T L (2) )+ U ee t
C L (2) T L (2) t = G (2) ( T e (2) - T L (2) )+ U eL t (3 λ (2) +2 μ (2) ) α (2) T L (2) j=1 3 ε ˙ (2) jj
C L (3) T L (3) t = ·( k L (3) T L (3) )-(3 λ (3) +2 μ (3) ) α (3) T L (3) j=1 3 ε ˙ (3) jj
ρ (i) 2 v (i) t 2 =( λ (i) +2 μ (i) ) 2 v (i) z 2 (3 λ (i) +2 μ (i) ) α (2) T L (i) t
ε (i) = v (i) z , σ z (i) =( λ (i) +2 μ (i) ) ε (i) (3 λ (i) +2 μ (i) ) α (2) ( T L (i) T 0 )
t { U ee U eL }= 2A ln2 J π ( hν ) 2 t p 0 t [ 1 1exp( dm α 1 +Λ ) 1 α 1 +Λ × exp( 4ln2 ( t t 0 t p ) 2 ) exp( 0 z 1 α 1 +Λ d z ) { H ee (t t ) H eL (t t ) } ]d t
k e = [ ( θ e ) 2 +0.16 ] 5/4 [ ( θ e ) 2 +0.44 ] θ e [ ( θ e ) 2 +0.092 ] 1/2 [ ( θ e ) 2 +η θ L ] 5/4 χ, θ e = T e T F , θ L = T L T F
k e = k e0 T e A e B L ( T e ) 2 + T L ,G= G 0 [ A e B L ( T e + T L )+1 ]

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