Abstract

The analysis of electron-distribution formation in a metal irradiated with a laser pulse is given on the basis of the explicit solution of the quantum kinetic equation. The angular distributions of electrons emitted under laser irradiation of a metal surface are associated with the non-Fermi electron distribution generated by the laser action. The anisotropy of photon absorption in an electron–phonon collision causes a two-leafed distribution structure, which qualitatively distinguishes this mechanism of laser-stimulated electron emission.

© 1998 Optical Society of America

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References

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  1. C.-K. Sun, F. Vallée, L. H. Acioli, E. P. Ippen, and J. G. Fujimoto, “Femtosecond tunable measurement of electron thermalization in gold,” Phys. Rev. B 50, 15337 (1994).
    [CrossRef]
  2. H. E. Elsayed-Ali and T. Juhasz, “Femtosecond time-resolved thermomodulation of gold films with different crystal structures,” Phys. Rev. B 47, 13599 (1993).
    [CrossRef]
  3. W. S. Fann, R. Storz, H. W. K. Tom, and J. Bokor, “Direct measurement of nonequilibrium electron-energy distributions in subpicosecond laser heated gold films,” Phys. Rev. Lett. 68, 2834 (1992).
    [CrossRef] [PubMed]
  4. D. Charalambidis, E. Hontropoulos, C. Fotakis, Gy. Farkas, and Cs. Toth, “High current, small divergence electron beams produced by laser-induced surface photoelectric effect,” J. Appl. Phys. 65, 2843 (1989).
    [CrossRef]
  5. A. M. Brodskii and Ju. Ia. Gurevich, The Theory of Electron Emission from Metals (Nauka, Moscow, 1973).
  6. J. M. Ziman, The Physics of Metals. I. Electrons (Cambridge University, Cambridge, 1969).
  7. G. Mahan, “Angular dependence of photoemission in metals,” Phys. Rev. Lett. 24, 1068 (1970).
    [CrossRef]
  8. A. V. Lugovskoy, T. Usmanov, and A. V. Zinoviev, “Laser-induced non-equilibrium phenomena on a metal surface,” J. Phys. D 27, 628 (1994).
    [CrossRef]
  9. A. T. Georges, “Theory of multiphoton photoelectric effect: a stepwise excitation process,” Phys. Rev. B 51, 13735 (1995).
    [CrossRef]
  10. E. M. Lifshitz and P. L. Pitaevskii, Physical Kinetics (Pergamon, Oxford, 1981).
  11. G. D. Mahan, “Theory of photoemission in simple metals,” Phys. Rev. B 2, 4334 (1970).
    [CrossRef]
  12. A. V. Zinoviev and V. B. Lugovskoy, “Nonequilibrium excitation of metal electrons by high-intensity monochromatic radiation,” Zh. Tekh. Fiz. 50, 1635 (1980); Sov. Phys. Tech. Phys. 25, 953 (1980).
  13. J. H. Bechtel, W. L. Smith, and N. Blombergen, “Two-photon photoemission from metals induced by picosecond laser pulses,” Phys. Rev. B 15, 4557 (1977).
    [CrossRef]
  14. A. V. Zinoviev, I. A. Kulagin, A. V. Lugovskoy, and T. Usmanov, “Nonlinear optical properties of metals in the strong electromagnetic field,” Zh. Eksp. Teor. Fiz. 107, 894 (1995); Sov. Phys. JETP 80, 508 (1995).
  15. Ju. P. Raiser, Gas Discharge Physics (Springer-Verlag, Berlin, 1991).
  16. V. F. Gantmaher and I. B. Levinson, Carrier Scattering in Metals and Semiconductors (North-Holland, New York, 1987).
  17. Ph. Dogusan, S. Guizard, K. Krastev, P. Martim, G. Petite, A. Dos. Santos, and A. Antonetti, “Direct observation of multiple photon absorption in a wide band insulator under strong laser irradiation,” Phys. Rev. Lett. 73, 2352 (1994).
    [CrossRef]
  18. A. Ducauze and P. C. Langer, “Repartition des particules emises pur focalisation d’un faisceau laser sur une cible solide,” R. Acad. Sc. Paris AB262, 1398 (1966).
  19. U. A. Arifov, V. V. Kazanskii, V. B. Lugovskoi, and V. A. Makarenko, “Observation of anisotropic electron emission at laser irradiation of metals,” Dokl. Akad. Nauk. SSSR 9, 18 (1969) (in Russian).
  20. G. Petite, P. Agostini, R. Trainham, E. Mevel, and P. Martin, “Origin of the high-energy electron emission from metals under laser irradiation,” Phys. Rev. B 45, 12210 (1992).
    [CrossRef]
  21. M. V. Fedorov Electrons in a Strong Light Field (Nauka, Moscow, 1991).
  22. H. Sambe, “Steady states and quasienergies of a quantum-mechanical system in an oscillatory field,” Phys. Rev. A 7, 2203 (1973).
    [CrossRef]
  23. R. H. Pantell and H. E. Puthoff, Fundamentals of Quantum Electronics (Wiley, New York, 1969).
  24. E. M. Epstain, “Electron scattering by phonons in a high-power radiation field,” Fiz. Tverd. Tela 11, 2732 (1969).

1995 (1)

A. T. Georges, “Theory of multiphoton photoelectric effect: a stepwise excitation process,” Phys. Rev. B 51, 13735 (1995).
[CrossRef]

1994 (3)

A. V. Lugovskoy, T. Usmanov, and A. V. Zinoviev, “Laser-induced non-equilibrium phenomena on a metal surface,” J. Phys. D 27, 628 (1994).
[CrossRef]

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

Ph. Dogusan, S. Guizard, K. Krastev, P. Martim, G. Petite, A. Dos. Santos, and A. Antonetti, “Direct observation of multiple photon absorption in a wide band insulator under strong laser irradiation,” Phys. Rev. Lett. 73, 2352 (1994).
[CrossRef]

1993 (1)

H. E. Elsayed-Ali and T. Juhasz, “Femtosecond time-resolved thermomodulation of gold films with different crystal structures,” Phys. Rev. B 47, 13599 (1993).
[CrossRef]

1992 (2)

W. S. Fann, R. Storz, H. W. K. Tom, and J. Bokor, “Direct measurement of nonequilibrium electron-energy distributions in subpicosecond laser heated gold films,” Phys. Rev. Lett. 68, 2834 (1992).
[CrossRef] [PubMed]

G. Petite, P. Agostini, R. Trainham, E. Mevel, and P. Martin, “Origin of the high-energy electron emission from metals under laser irradiation,” Phys. Rev. B 45, 12210 (1992).
[CrossRef]

1989 (1)

D. Charalambidis, E. Hontropoulos, C. Fotakis, Gy. Farkas, and Cs. Toth, “High current, small divergence electron beams produced by laser-induced surface photoelectric effect,” J. Appl. Phys. 65, 2843 (1989).
[CrossRef]

1977 (1)

J. H. Bechtel, W. L. Smith, and N. Blombergen, “Two-photon photoemission from metals induced by picosecond laser pulses,” Phys. Rev. B 15, 4557 (1977).
[CrossRef]

1973 (1)

H. Sambe, “Steady states and quasienergies of a quantum-mechanical system in an oscillatory field,” Phys. Rev. A 7, 2203 (1973).
[CrossRef]

1970 (2)

G. D. Mahan, “Theory of photoemission in simple metals,” Phys. Rev. B 2, 4334 (1970).
[CrossRef]

G. Mahan, “Angular dependence of photoemission in metals,” Phys. Rev. Lett. 24, 1068 (1970).
[CrossRef]

1969 (2)

E. M. Epstain, “Electron scattering by phonons in a high-power radiation field,” Fiz. Tverd. Tela 11, 2732 (1969).

U. A. Arifov, V. V. Kazanskii, V. B. Lugovskoi, and V. A. Makarenko, “Observation of anisotropic electron emission at laser irradiation of metals,” Dokl. Akad. Nauk. SSSR 9, 18 (1969) (in Russian).

Acioli, L. H.

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

Agostini, P.

G. Petite, P. Agostini, R. Trainham, E. Mevel, and P. Martin, “Origin of the high-energy electron emission from metals under laser irradiation,” Phys. Rev. B 45, 12210 (1992).
[CrossRef]

Antonetti, A.

Ph. Dogusan, S. Guizard, K. Krastev, P. Martim, G. Petite, A. Dos. Santos, and A. Antonetti, “Direct observation of multiple photon absorption in a wide band insulator under strong laser irradiation,” Phys. Rev. Lett. 73, 2352 (1994).
[CrossRef]

Arifov, U. A.

U. A. Arifov, V. V. Kazanskii, V. B. Lugovskoi, and V. A. Makarenko, “Observation of anisotropic electron emission at laser irradiation of metals,” Dokl. Akad. Nauk. SSSR 9, 18 (1969) (in Russian).

Bechtel, J. H.

J. H. Bechtel, W. L. Smith, and N. Blombergen, “Two-photon photoemission from metals induced by picosecond laser pulses,” Phys. Rev. B 15, 4557 (1977).
[CrossRef]

Blombergen, N.

J. H. Bechtel, W. L. Smith, and N. Blombergen, “Two-photon photoemission from metals induced by picosecond laser pulses,” Phys. Rev. B 15, 4557 (1977).
[CrossRef]

Bokor, J.

W. S. Fann, R. Storz, H. W. K. Tom, and J. Bokor, “Direct measurement of nonequilibrium electron-energy distributions in subpicosecond laser heated gold films,” Phys. Rev. Lett. 68, 2834 (1992).
[CrossRef] [PubMed]

Charalambidis, D.

D. Charalambidis, E. Hontropoulos, C. Fotakis, Gy. Farkas, and Cs. Toth, “High current, small divergence electron beams produced by laser-induced surface photoelectric effect,” J. Appl. Phys. 65, 2843 (1989).
[CrossRef]

Dogusan, Ph.

Ph. Dogusan, S. Guizard, K. Krastev, P. Martim, G. Petite, A. Dos. Santos, and A. Antonetti, “Direct observation of multiple photon absorption in a wide band insulator under strong laser irradiation,” Phys. Rev. Lett. 73, 2352 (1994).
[CrossRef]

Elsayed-Ali, H. E.

H. E. Elsayed-Ali and T. Juhasz, “Femtosecond time-resolved thermomodulation of gold films with different crystal structures,” Phys. Rev. B 47, 13599 (1993).
[CrossRef]

Epstain, E. M.

E. M. Epstain, “Electron scattering by phonons in a high-power radiation field,” Fiz. Tverd. Tela 11, 2732 (1969).

Fann, W. S.

W. S. Fann, R. Storz, H. W. K. Tom, and J. Bokor, “Direct measurement of nonequilibrium electron-energy distributions in subpicosecond laser heated gold films,” Phys. Rev. Lett. 68, 2834 (1992).
[CrossRef] [PubMed]

Farkas, Gy.

D. Charalambidis, E. Hontropoulos, C. Fotakis, Gy. Farkas, and Cs. Toth, “High current, small divergence electron beams produced by laser-induced surface photoelectric effect,” J. Appl. Phys. 65, 2843 (1989).
[CrossRef]

Fotakis, C.

D. Charalambidis, E. Hontropoulos, C. Fotakis, Gy. Farkas, and Cs. Toth, “High current, small divergence electron beams produced by laser-induced surface photoelectric effect,” J. Appl. Phys. 65, 2843 (1989).
[CrossRef]

Fujimoto, J. G.

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

Georges, A. T.

A. T. Georges, “Theory of multiphoton photoelectric effect: a stepwise excitation process,” Phys. Rev. B 51, 13735 (1995).
[CrossRef]

Guizard, S.

Ph. Dogusan, S. Guizard, K. Krastev, P. Martim, G. Petite, A. Dos. Santos, and A. Antonetti, “Direct observation of multiple photon absorption in a wide band insulator under strong laser irradiation,” Phys. Rev. Lett. 73, 2352 (1994).
[CrossRef]

Hontropoulos, E.

D. Charalambidis, E. Hontropoulos, C. Fotakis, Gy. Farkas, and Cs. Toth, “High current, small divergence electron beams produced by laser-induced surface photoelectric effect,” J. Appl. Phys. 65, 2843 (1989).
[CrossRef]

Ippen, E. P.

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

Juhasz, T.

H. E. Elsayed-Ali and T. Juhasz, “Femtosecond time-resolved thermomodulation of gold films with different crystal structures,” Phys. Rev. B 47, 13599 (1993).
[CrossRef]

Kazanskii, V. V.

U. A. Arifov, V. V. Kazanskii, V. B. Lugovskoi, and V. A. Makarenko, “Observation of anisotropic electron emission at laser irradiation of metals,” Dokl. Akad. Nauk. SSSR 9, 18 (1969) (in Russian).

Krastev, K.

Ph. Dogusan, S. Guizard, K. Krastev, P. Martim, G. Petite, A. Dos. Santos, and A. Antonetti, “Direct observation of multiple photon absorption in a wide band insulator under strong laser irradiation,” Phys. Rev. Lett. 73, 2352 (1994).
[CrossRef]

Lugovskoi, V. B.

U. A. Arifov, V. V. Kazanskii, V. B. Lugovskoi, and V. A. Makarenko, “Observation of anisotropic electron emission at laser irradiation of metals,” Dokl. Akad. Nauk. SSSR 9, 18 (1969) (in Russian).

Lugovskoy, A. V.

A. V. Lugovskoy, T. Usmanov, and A. V. Zinoviev, “Laser-induced non-equilibrium phenomena on a metal surface,” J. Phys. D 27, 628 (1994).
[CrossRef]

Mahan, G.

G. Mahan, “Angular dependence of photoemission in metals,” Phys. Rev. Lett. 24, 1068 (1970).
[CrossRef]

Mahan, G. D.

G. D. Mahan, “Theory of photoemission in simple metals,” Phys. Rev. B 2, 4334 (1970).
[CrossRef]

Makarenko, V. A.

U. A. Arifov, V. V. Kazanskii, V. B. Lugovskoi, and V. A. Makarenko, “Observation of anisotropic electron emission at laser irradiation of metals,” Dokl. Akad. Nauk. SSSR 9, 18 (1969) (in Russian).

Martim, P.

Ph. Dogusan, S. Guizard, K. Krastev, P. Martim, G. Petite, A. Dos. Santos, and A. Antonetti, “Direct observation of multiple photon absorption in a wide band insulator under strong laser irradiation,” Phys. Rev. Lett. 73, 2352 (1994).
[CrossRef]

Martin, P.

G. Petite, P. Agostini, R. Trainham, E. Mevel, and P. Martin, “Origin of the high-energy electron emission from metals under laser irradiation,” Phys. Rev. B 45, 12210 (1992).
[CrossRef]

Mevel, E.

G. Petite, P. Agostini, R. Trainham, E. Mevel, and P. Martin, “Origin of the high-energy electron emission from metals under laser irradiation,” Phys. Rev. B 45, 12210 (1992).
[CrossRef]

Petite, G.

Ph. Dogusan, S. Guizard, K. Krastev, P. Martim, G. Petite, A. Dos. Santos, and A. Antonetti, “Direct observation of multiple photon absorption in a wide band insulator under strong laser irradiation,” Phys. Rev. Lett. 73, 2352 (1994).
[CrossRef]

G. Petite, P. Agostini, R. Trainham, E. Mevel, and P. Martin, “Origin of the high-energy electron emission from metals under laser irradiation,” Phys. Rev. B 45, 12210 (1992).
[CrossRef]

Sambe, H.

H. Sambe, “Steady states and quasienergies of a quantum-mechanical system in an oscillatory field,” Phys. Rev. A 7, 2203 (1973).
[CrossRef]

Santos, A. Dos.

Ph. Dogusan, S. Guizard, K. Krastev, P. Martim, G. Petite, A. Dos. Santos, and A. Antonetti, “Direct observation of multiple photon absorption in a wide band insulator under strong laser irradiation,” Phys. Rev. Lett. 73, 2352 (1994).
[CrossRef]

Smith, W. L.

J. H. Bechtel, W. L. Smith, and N. Blombergen, “Two-photon photoemission from metals induced by picosecond laser pulses,” Phys. Rev. B 15, 4557 (1977).
[CrossRef]

Storz, R.

W. S. Fann, R. Storz, H. W. K. Tom, and J. Bokor, “Direct measurement of nonequilibrium electron-energy distributions in subpicosecond laser heated gold films,” Phys. Rev. Lett. 68, 2834 (1992).
[CrossRef] [PubMed]

Sun, C.-K.

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

Tom, H. W. K.

W. S. Fann, R. Storz, H. W. K. Tom, and J. Bokor, “Direct measurement of nonequilibrium electron-energy distributions in subpicosecond laser heated gold films,” Phys. Rev. Lett. 68, 2834 (1992).
[CrossRef] [PubMed]

Toth, Cs.

D. Charalambidis, E. Hontropoulos, C. Fotakis, Gy. Farkas, and Cs. Toth, “High current, small divergence electron beams produced by laser-induced surface photoelectric effect,” J. Appl. Phys. 65, 2843 (1989).
[CrossRef]

Trainham, R.

G. Petite, P. Agostini, R. Trainham, E. Mevel, and P. Martin, “Origin of the high-energy electron emission from metals under laser irradiation,” Phys. Rev. B 45, 12210 (1992).
[CrossRef]

Usmanov, T.

A. V. Lugovskoy, T. Usmanov, and A. V. Zinoviev, “Laser-induced non-equilibrium phenomena on a metal surface,” J. Phys. D 27, 628 (1994).
[CrossRef]

Vallée, F.

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

Zinoviev, A. V.

A. V. Lugovskoy, T. Usmanov, and A. V. Zinoviev, “Laser-induced non-equilibrium phenomena on a metal surface,” J. Phys. D 27, 628 (1994).
[CrossRef]

Dokl. Akad. Nauk. SSSR (1)

U. A. Arifov, V. V. Kazanskii, V. B. Lugovskoi, and V. A. Makarenko, “Observation of anisotropic electron emission at laser irradiation of metals,” Dokl. Akad. Nauk. SSSR 9, 18 (1969) (in Russian).

Fiz. Tverd. Tela (1)

E. M. Epstain, “Electron scattering by phonons in a high-power radiation field,” Fiz. Tverd. Tela 11, 2732 (1969).

J. Appl. Phys. (1)

D. Charalambidis, E. Hontropoulos, C. Fotakis, Gy. Farkas, and Cs. Toth, “High current, small divergence electron beams produced by laser-induced surface photoelectric effect,” J. Appl. Phys. 65, 2843 (1989).
[CrossRef]

J. Phys. D (1)

A. V. Lugovskoy, T. Usmanov, and A. V. Zinoviev, “Laser-induced non-equilibrium phenomena on a metal surface,” J. Phys. D 27, 628 (1994).
[CrossRef]

Phys. Rev. A (1)

H. Sambe, “Steady states and quasienergies of a quantum-mechanical system in an oscillatory field,” Phys. Rev. A 7, 2203 (1973).
[CrossRef]

Phys. Rev. B (6)

A. T. Georges, “Theory of multiphoton photoelectric effect: a stepwise excitation process,” Phys. Rev. B 51, 13735 (1995).
[CrossRef]

G. D. Mahan, “Theory of photoemission in simple metals,” Phys. Rev. B 2, 4334 (1970).
[CrossRef]

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

H. E. Elsayed-Ali and T. Juhasz, “Femtosecond time-resolved thermomodulation of gold films with different crystal structures,” Phys. Rev. B 47, 13599 (1993).
[CrossRef]

G. Petite, P. Agostini, R. Trainham, E. Mevel, and P. Martin, “Origin of the high-energy electron emission from metals under laser irradiation,” Phys. Rev. B 45, 12210 (1992).
[CrossRef]

J. H. Bechtel, W. L. Smith, and N. Blombergen, “Two-photon photoemission from metals induced by picosecond laser pulses,” Phys. Rev. B 15, 4557 (1977).
[CrossRef]

Phys. Rev. Lett. (3)

Ph. Dogusan, S. Guizard, K. Krastev, P. Martim, G. Petite, A. Dos. Santos, and A. Antonetti, “Direct observation of multiple photon absorption in a wide band insulator under strong laser irradiation,” Phys. Rev. Lett. 73, 2352 (1994).
[CrossRef]

W. S. Fann, R. Storz, H. W. K. Tom, and J. Bokor, “Direct measurement of nonequilibrium electron-energy distributions in subpicosecond laser heated gold films,” Phys. Rev. Lett. 68, 2834 (1992).
[CrossRef] [PubMed]

G. Mahan, “Angular dependence of photoemission in metals,” Phys. Rev. Lett. 24, 1068 (1970).
[CrossRef]

Other (10)

R. H. Pantell and H. E. Puthoff, Fundamentals of Quantum Electronics (Wiley, New York, 1969).

A. M. Brodskii and Ju. Ia. Gurevich, The Theory of Electron Emission from Metals (Nauka, Moscow, 1973).

J. M. Ziman, The Physics of Metals. I. Electrons (Cambridge University, Cambridge, 1969).

A. V. Zinoviev and V. B. Lugovskoy, “Nonequilibrium excitation of metal electrons by high-intensity monochromatic radiation,” Zh. Tekh. Fiz. 50, 1635 (1980); Sov. Phys. Tech. Phys. 25, 953 (1980).

E. M. Lifshitz and P. L. Pitaevskii, Physical Kinetics (Pergamon, Oxford, 1981).

A. Ducauze and P. C. Langer, “Repartition des particules emises pur focalisation d’un faisceau laser sur une cible solide,” R. Acad. Sc. Paris AB262, 1398 (1966).

M. V. Fedorov Electrons in a Strong Light Field (Nauka, Moscow, 1991).

A. V. Zinoviev, I. A. Kulagin, A. V. Lugovskoy, and T. Usmanov, “Nonlinear optical properties of metals in the strong electromagnetic field,” Zh. Eksp. Teor. Fiz. 107, 894 (1995); Sov. Phys. JETP 80, 508 (1995).

Ju. P. Raiser, Gas Discharge Physics (Springer-Verlag, Berlin, 1991).

V. F. Gantmaher and I. B. Levinson, Carrier Scattering in Metals and Semiconductors (North-Holland, New York, 1987).

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Figures (8)

Fig. 1
Fig. 1

Sample scheme for a photoemission experiment.

Fig. 2
Fig. 2

Electron momentum space. Electrons from the area between curves 1 [pz(p)=(p2 cotan2 θ+2mV0)1/2] and 1[pz(p)=(p2 cotan2(θ+dθ)+2mV0)1/2] refract at a metal surface and transit to states into the solid angle dΩ=sin θdθdφ (lower gray region). Here p2=px2+py2.

Fig. 3
Fig. 3

(a) Electron–electron and (b) electron–phonon collision rates versus an electron energy. Here τ4πωD-1(pFs/kT), ωD is Debye frequency, pF is the Fermi momentum, and s is the sound velocity.

Fig. 4
Fig. 4

Angular dependence of electron–phonon collision frequencies (a) νeph(1) and (b) νeph(2). The double arrows show the electric vector direction.  

Fig. 5
Fig. 5

Energy dependence of electron–phonon collision frequencies νeph(0) (solid curve), νeph(1) (dashed curve), and νeph(-1) (dotted curve).

Fig. 6
Fig. 6

Energy dependence of the electron distribution function calculated with Eqs. (16)–(23) at X=10-3. The dotted curves are the equilibrium Fermi function, calculated for the temperature T, which is obtained from the heat equation. The Fermi energy is 5 eV.

Fig. 7
Fig. 7

Direction diagrams of linear photocurrent for various angles between the field electric vector and the normal to the surface: (a) 90°, (b) 80°, (c) 45°, and (d) 0°.  

Fig. 8
Fig. 8

Direction diagrams from Ref. 19.

Equations (63)

Equations on this page are rendered with MathJax. Learn more.

jsnpzWpp(n)f0(εp-nω)d3p,
jV=1(2π)3 em Ppf(z=0, p)d3p,
px=px,py=py,pz2=pz2+2mV0,
pz2=(px2+py2)cot2 θ+2mV0,
djV=1(2π)3 em p1pf(z=0, p)d3p,
djVJVdΩ=1(2π)3 em φφ+dφθθ+dθ0×pf(z=0, p) pzpz p2dp sin θdθdφ=1(2π)3 em 0pf(z=0, p) pzpz p2dpdΩ,
jV(θ)=1(2π)3 em cos θ×0 p4(p2 cos2 θ+2mV02)1/2f(εp+V0)dp.
jV=ADTe2n=0K-1κ0nθn(t)Fϕ-nωTe+κ0KθK(t)×π26+(ϕ-Kω)22Te2-2KFϕ-(K-1)ωTe-FKω-ϕTe,
f(p)t=See+Seph+2π× n=-n0k|Ck|2[2N(k, t)+1]×Jn2(ak)[f(p+k)-f(p)]×δ(εp+k-εp-nω),
See=-νee(ε)[f(p, t)-f0(ε)],
νee(ε)1εF (kT)2+ε-εF2π2,
νee(ε)=8πL(EB/)nk-3,k=(2mε)1/2,
νeph(p)=2π k|Ck|2(2N(k, t)+1)×J02(ak)δ(εp+k-εp).
f(p)=fis(εp)+fan(p),
fis(εp)=14π 0π02πf(p)dφ sin θ dθ,
fan(p)=f(p)-fis(εp).
fis(ε)=f0(ε)+fis(1)(ε)+fis(2)(ε),
fis(1)(ε)=n=-n0 ν¯eph(n)(ε)νee(ε) [f0(ε+nω)-f0(ε)],
fis(2)(ε)=n=-n0 ν¯eph(n)(ε)νee(ε) m=-m0ν¯eph(m)(ε+nω)νee(ε+nω)× {f0[ε+(n+m)ω]-f0(ε+nω)}-ν¯eph(m)(ε)νee(ε) [f0(ε+mω)-f0(ε)]+12νee(ε) n=-n0k|Ck|2[2N(k, t)+1]×Jn2(ak)0π02πn=-m0×νeph(m)(p+q)-ν¯eph(m)(εp+q)νee(εp+q)+νeph(0)(εp+q)×[f0(εp+q+mω)-f0(εp+q)]-νeph(m)(p)-ν¯eph(m)(εp)νee(εp)+νeph(0)(εp) [f0(εp+mω)-f0(εp)]δ(εp+k-εp-nω)dΩ,
fan(p)=fan(1)(p)+fan(2)(p),
fan(1)(p)=n=-n0 νeph(n)(p)-ν¯eph(n)(εp)νee(εp)+νeph(0)(εp)×[f0(ε+nω)-f0(ε)],
fan(2)(p)=n=-n0 νeph(n)(p)-ν¯eph(n)(εp)νee(εp)+νeph(0)(εp)× m=-m0ν¯eph(m)(ε+nω)νee(ε+nω)× {f0[ε+(n+m)ω]-f0(ε+nω)}-ν¯eph(m)(ε)νee(ε) [f0(ε+mω)-f0(ε)]+2π n=-n0k|Ck|2[2N(k, t)+1]Jn2(ak)νee(εp)+νeph(0)(εp)×m=-m0νeph(m)(p+k)-ν¯eph(m)(εp+k)νee(εp+k)+νeph(0)(εp+k)×[f0(εp+k+mω)-f0(εp+k)]-νeph(m)(p)-ν¯eph(m)(εp)νee(εp)+νeph(0)(εp) [f0(εp+mω)-f0(εp)]δ(εp+k-εp-nω)
-14π 0π02πm=-m0νeph(m)(p+k)-ν¯eph(m)(εp+k)νee(εp+k)+νeph(0)(εp+k)×[f0(εp+k+mω)-f0(εp+k)]-νeph(m)(p)-ν¯eph(m)(εp)νee(εp)+νeph(0)(εp) [f0(εp+mω)-f0(εp)]}δ(εp+k-εp-nω)dΩ,
νeph(n)(p)=12 k|Ck|2[2N(k, t)+1]Jn2(ak)×δ(εp+k-εp-nω),
ν¯eph(n)(εp)=0π02πνeph(n)(p)dΩ,
νeph(n)(p)Wnν¯eph(0)(εp+nω),
e(ε)±ph±ωe(ε±ω).
e(ε)±ph±ωe(ε±ω)±ph±ω
e(ε±2ω),
e(ε)±ph±2ωe(ε±2ω).
νee(ε)νeph(0)(ε).
δf1δfn<(n!)2νeph(0)(ε-nω)νee(ε)n-11
νeph(0)(ε-nω)νee(ε)(n!)-2/(n-1).
f(p)f0(εp)+n=-n0 νeph(n)(p)νee(εp) [f0(ε+nω)-f0(ε)].
jsWK cos θE<jVWK νeph(0)(V0-Kω)νee(V0)νee(V0)cos θE<νeph(0)(V0-Kω),
Ψn,p(r, t)=Un,p(r, t)expi (pr-εn,pt),
Un,p(r+b, t)=Un,p(r, t),
Un,p(r, t+T)=Un,p(r, t),
φp=L-3/2 expipr-i2m 0tdτp-ec A0 sin ωτ2,
 
{apap+}=apap++ap+ap=δpp,
{apap}={ap+ap+}=0,
H=pεpap+ap+qωqcq++Heph+Hee,
Heph=p,qGq(t)ap+q+ap(cq++cq),
Hee=12 dpdpdqφ(q)ap+ap+ap+qap-q.
iap+ap/t=[ap+ap;H].
f(p)t=See+qGq(t)-tdτG-q(τ)(2N-q+1)×{[f(p+q)-f(p)]exp[i(εp-εp+q)×(τ-τ)/]+[f(p-q)-f(p)]×exp[-i(εp-εp-q)(τ-t)/]},
f(p)=nfn(p)exp(-inωt),
fnt-inωfn=See(n)+1 q|gq|2(2Nq+1)×[f0(p+q)-f0(p)]×mJn+m(qb)Jm(qb)Δ+i(εp-εp+q+mω)+Jn+m(qb)Jm(qb)Δ+i(εp+q-εp+mω),
f0t=See(0)+2π n=-q|gq|2(2Nq+1)×Jn2(qb)[f0(p+q)-f0(p)]×δ(εp+q-εp-nω),
f1t-iωf1=See(1)+q|gq|2(2Nq+1)×[f0(p+q)-f0(p)]×mJ1+m(qb)Jm(qb)Δ+i(εp-εp+q+mω)+J1+m(qb)Jm(qb)Δ+i(εp+q-εp+mω).
H0|m=εm|m,
[H0+VL(t)-it]|a=εa|a,
am=am|aaa, am+=aa|maa+,
aa=ma|mam, aa+=mm|aam+,
H=m,nm|H0+VL(t)+Vf|mam+am+m,nm,nm, n|W|m, nam+an+anam,
H=a,aa|H0+VL(t)+Vf|aaa+aa+a,βa,βa, β|W|a, βaa+aβ+aaaβ
m|mm|=1,α|αα|=1.
itaa=m(ita|mam+a|m[am, H]).
mit(α|m)am=βmitα|mm|βaβ=β,β[aα,β|-it|βaβ+aβ],
itaa=[aa, Hq],
Hq=a,aa|H0+VL(t)+Vf-it|aaa+aa+a,βa,βa, β|W|a, βaa+aβ+aaaβ,
=aεaaa+aa+a,aa|Vf|aaa+aa+a,βa,βa, β|W|a, βaa+aβ+aaaβ.

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