Abstract

We report how the photorefractive properties of cobalt-doped barium titanate (BaTiO3) are influenced by deep and shallow traps. The light-induced dark decay of the space-charge field has two distinct time constants. The faster decay is due to the thermalization of charges from shallow traps, whereas the longer decay is that from deep traps formed when BaTiO3 is doped with cobalt. This dopant substantially decreases the dark conductivity of BaTiO3 and thus increases the dark storage time. The power coefficient of the sublinear intensity dependence of the photorefractive response time progressively increases with the cobalt-dopant concentration. Apparently cobalt doping of the deep level eliminates the effect from the shallow level in BaTiO3. Although cobalt-doped BaTiO3 has an increased effective trap density and thus an increased electrooptic gain, the response time and absorption are also increased, which thus decreases the photorefractive sensitivity.

© 1995 Optical Society of America

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References

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  1. M. B. Klein and R. N. Schwartz, “Photorefractive effect in BaTiO3: microscopic origins,” J. Opt. Soc. Am. B 3, 293 (1986).
    [Crossref]
  2. P. G. Schunemann, D. A. Temple, R. S. Hathcock, H. L. Tuller, H. P. Jenssen, D. R. Gabbe, and C. Warde, “Role of iron centers in the photorefractive effect in barium titanate,” J. Opt. Soc. Am. B 5, 1685 (1988).
    [Crossref]
  3. P. G. Schunemann, T. M. Pollak, Y. Yang, Y.-Y. Teng, and C. Wong, “Effects of feed material and annealing atmosphere on the properties of photorefractive barium titanate crystals,” J. Opt. Soc. Am. B 5, 1702 (1988).
    [Crossref]
  4. D. Rytz, B. A. Wechsler, M. H. Garrett, C. C. Nelson, and R. N. Schwartz, “Photorefractive properties of BaTiO3:Co,” J. Opt. Soc. Am. B 7, 2245 (1990).
    [Crossref]
  5. M. H. Garrett, J. Y. Chang, H. P. Jenssen, and C. Warde, “High beam-coupling gain and deep- and shallow-trap effects in cobalt-doped barium titanate, BaTiO3:Co,” J. Opt. Soc. Am. B 9, 1408 (1992).
    [Crossref]
  6. F. M. Michel-Calendini and P. Moretti, “Electronic structures of Co(II) and Co(III) impurities in cubic perovskite hosts,” Phys. Rev. B 27, 763 (1983).
    [Crossref]
  7. P. Tayebati and D. Mahgerefteh, “Theory of the photorefractive effect for Bi12SiO20and BaTiO3with shallow traps,” J. Opt. Soc. Am. B 8, 1053 (1991).
    [Crossref]
  8. L. Holtmann, “A model for the nonlinear photoconductivity of BaTiO3,” Phys. Status Solidi A 113, k89 (1989).
    [Crossref]
  9. D. Mahgerefteh and J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
    [Crossref] [PubMed]
  10. G. A. Brost and R. A. Motes, “Origin of the sublinear photorefractive response time in BaTiO3,” Opt. Lett. 14, 1194 (1990).
    [Crossref]
  11. 11N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectric 22, 949 (1979).
  12. D. Mahgerefteh and J. Feinberg, “Erasure rate and coasting in photorefractive barium titanate at high optical power,” Opt. Lett. 13, 1111 (1988).
    [Crossref] [PubMed]
  13. D. Mahgerefteh, “The speed of the photorefractive effect, shallow traps, photogalvanic currents, and light-induced surface damage in barium titanate,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1990).
  14. F. P. Strohkendl, “Light-induced dark decays of photorefractive gratings and their observation in Bi12SiO20,” J. Appl. Phys. 65, 3773 (1989).
    [Crossref]
  15. V. M. Fridkin and B. N. Popov, “The anomalous photovoltaic effect and photoconductivity in ferroelectrics,” Ferroelectrics 21, 611 (1978).
    [Crossref]
  16. S. Ducharme and J. Feinberg, “Speed of the photorefractive effect in a BaTiO3single crystal,” J. Appl. Phys. 56, 839 (1984).
    [Crossref]
  17. J. Y. Chang, M. H. Garrett, H. P. Jenssen, and C. Warde, “Intensity dependent photorefractive properties in an n-type BaTiO3,” J. Appl. Phys. (to be published).
  18. R. S. Hathcock, “Optical and photorefractive properties of iron-doped barium titanate,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1989).
  19. P. Günter, “Holography, coherent light amplification and optical phase conjugation with photorefractive materials,” Phys. Rep.93, 199 (1982).
    [Crossref]
  20. G. C. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704 (1983).
    [Crossref]
  21. K. W. Blazey and K. A. Muller, “Paramagnetic resonance and optical absorption of Co+4in SrTiO3,” J. Phys. C 16, 5491 (1983).
    [Crossref]
  22. F. M. Michel-Calendini, P. Moretti, and G. Godefroy, “Excitation energies and optical absorption in Remeika BaTiO3:Co crystals,” Ferroelec. Lett. 44, 257 (1983).
    [Crossref]
  23. H.-J. Hagemann and D. Hennings, “Reversible weight change of acceptor-doped BaTiO3,” J. Am. Ceram. Soc. 64, 590 (1981).
    [Crossref]

1992 (1)

M. H. Garrett, J. Y. Chang, H. P. Jenssen, and C. Warde, “High beam-coupling gain and deep- and shallow-trap effects in cobalt-doped barium titanate, BaTiO3:Co,” J. Opt. Soc. Am. B 9, 1408 (1992).
[Crossref]

1991 (1)

1990 (3)

D. Rytz, B. A. Wechsler, M. H. Garrett, C. C. Nelson, and R. N. Schwartz, “Photorefractive properties of BaTiO3:Co,” J. Opt. Soc. Am. B 7, 2245 (1990).
[Crossref]

D. Mahgerefteh and J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
[Crossref] [PubMed]

G. A. Brost and R. A. Motes, “Origin of the sublinear photorefractive response time in BaTiO3,” Opt. Lett. 14, 1194 (1990).
[Crossref]

1989 (2)

F. P. Strohkendl, “Light-induced dark decays of photorefractive gratings and their observation in Bi12SiO20,” J. Appl. Phys. 65, 3773 (1989).
[Crossref]

L. Holtmann, “A model for the nonlinear photoconductivity of BaTiO3,” Phys. Status Solidi A 113, k89 (1989).
[Crossref]

1988 (3)

1986 (1)

1984 (1)

S. Ducharme and J. Feinberg, “Speed of the photorefractive effect in a BaTiO3single crystal,” J. Appl. Phys. 56, 839 (1984).
[Crossref]

1983 (4)

G. C. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704 (1983).
[Crossref]

K. W. Blazey and K. A. Muller, “Paramagnetic resonance and optical absorption of Co+4in SrTiO3,” J. Phys. C 16, 5491 (1983).
[Crossref]

F. M. Michel-Calendini, P. Moretti, and G. Godefroy, “Excitation energies and optical absorption in Remeika BaTiO3:Co crystals,” Ferroelec. Lett. 44, 257 (1983).
[Crossref]

F. M. Michel-Calendini and P. Moretti, “Electronic structures of Co(II) and Co(III) impurities in cubic perovskite hosts,” Phys. Rev. B 27, 763 (1983).
[Crossref]

1981 (1)

H.-J. Hagemann and D. Hennings, “Reversible weight change of acceptor-doped BaTiO3,” J. Am. Ceram. Soc. 64, 590 (1981).
[Crossref]

1979 (1)

11N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectric 22, 949 (1979).

1978 (1)

V. M. Fridkin and B. N. Popov, “The anomalous photovoltaic effect and photoconductivity in ferroelectrics,” Ferroelectrics 21, 611 (1978).
[Crossref]

Blazey, K. W.

K. W. Blazey and K. A. Muller, “Paramagnetic resonance and optical absorption of Co+4in SrTiO3,” J. Phys. C 16, 5491 (1983).
[Crossref]

Brost, G. A.

G. A. Brost and R. A. Motes, “Origin of the sublinear photorefractive response time in BaTiO3,” Opt. Lett. 14, 1194 (1990).
[Crossref]

Chang, J. Y.

M. H. Garrett, J. Y. Chang, H. P. Jenssen, and C. Warde, “High beam-coupling gain and deep- and shallow-trap effects in cobalt-doped barium titanate, BaTiO3:Co,” J. Opt. Soc. Am. B 9, 1408 (1992).
[Crossref]

J. Y. Chang, M. H. Garrett, H. P. Jenssen, and C. Warde, “Intensity dependent photorefractive properties in an n-type BaTiO3,” J. Appl. Phys. (to be published).

Ducharme, S.

S. Ducharme and J. Feinberg, “Speed of the photorefractive effect in a BaTiO3single crystal,” J. Appl. Phys. 56, 839 (1984).
[Crossref]

Feinberg, J.

D. Mahgerefteh and J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
[Crossref] [PubMed]

D. Mahgerefteh and J. Feinberg, “Erasure rate and coasting in photorefractive barium titanate at high optical power,” Opt. Lett. 13, 1111 (1988).
[Crossref] [PubMed]

S. Ducharme and J. Feinberg, “Speed of the photorefractive effect in a BaTiO3single crystal,” J. Appl. Phys. 56, 839 (1984).
[Crossref]

Fridkin, V. M.

V. M. Fridkin and B. N. Popov, “The anomalous photovoltaic effect and photoconductivity in ferroelectrics,” Ferroelectrics 21, 611 (1978).
[Crossref]

Gabbe, D. R.

Garrett, M. H.

M. H. Garrett, J. Y. Chang, H. P. Jenssen, and C. Warde, “High beam-coupling gain and deep- and shallow-trap effects in cobalt-doped barium titanate, BaTiO3:Co,” J. Opt. Soc. Am. B 9, 1408 (1992).
[Crossref]

D. Rytz, B. A. Wechsler, M. H. Garrett, C. C. Nelson, and R. N. Schwartz, “Photorefractive properties of BaTiO3:Co,” J. Opt. Soc. Am. B 7, 2245 (1990).
[Crossref]

J. Y. Chang, M. H. Garrett, H. P. Jenssen, and C. Warde, “Intensity dependent photorefractive properties in an n-type BaTiO3,” J. Appl. Phys. (to be published).

Godefroy, G.

F. M. Michel-Calendini, P. Moretti, and G. Godefroy, “Excitation energies and optical absorption in Remeika BaTiO3:Co crystals,” Ferroelec. Lett. 44, 257 (1983).
[Crossref]

Günter, P.

P. Günter, “Holography, coherent light amplification and optical phase conjugation with photorefractive materials,” Phys. Rep.93, 199 (1982).
[Crossref]

Hagemann, H.-J.

H.-J. Hagemann and D. Hennings, “Reversible weight change of acceptor-doped BaTiO3,” J. Am. Ceram. Soc. 64, 590 (1981).
[Crossref]

Hathcock, R. S.

P. G. Schunemann, D. A. Temple, R. S. Hathcock, H. L. Tuller, H. P. Jenssen, D. R. Gabbe, and C. Warde, “Role of iron centers in the photorefractive effect in barium titanate,” J. Opt. Soc. Am. B 5, 1685 (1988).
[Crossref]

R. S. Hathcock, “Optical and photorefractive properties of iron-doped barium titanate,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1989).

Hennings, D.

H.-J. Hagemann and D. Hennings, “Reversible weight change of acceptor-doped BaTiO3,” J. Am. Ceram. Soc. 64, 590 (1981).
[Crossref]

Holtmann, L.

L. Holtmann, “A model for the nonlinear photoconductivity of BaTiO3,” Phys. Status Solidi A 113, k89 (1989).
[Crossref]

Jenssen, H. P.

M. H. Garrett, J. Y. Chang, H. P. Jenssen, and C. Warde, “High beam-coupling gain and deep- and shallow-trap effects in cobalt-doped barium titanate, BaTiO3:Co,” J. Opt. Soc. Am. B 9, 1408 (1992).
[Crossref]

P. G. Schunemann, D. A. Temple, R. S. Hathcock, H. L. Tuller, H. P. Jenssen, D. R. Gabbe, and C. Warde, “Role of iron centers in the photorefractive effect in barium titanate,” J. Opt. Soc. Am. B 5, 1685 (1988).
[Crossref]

J. Y. Chang, M. H. Garrett, H. P. Jenssen, and C. Warde, “Intensity dependent photorefractive properties in an n-type BaTiO3,” J. Appl. Phys. (to be published).

Klein, M. B.

M. B. Klein and R. N. Schwartz, “Photorefractive effect in BaTiO3: microscopic origins,” J. Opt. Soc. Am. B 3, 293 (1986).
[Crossref]

G. C. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704 (1983).
[Crossref]

Kukhtarev, N. V.

11N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectric 22, 949 (1979).

Mahgerefteh, D.

P. Tayebati and D. Mahgerefteh, “Theory of the photorefractive effect for Bi12SiO20and BaTiO3with shallow traps,” J. Opt. Soc. Am. B 8, 1053 (1991).
[Crossref]

D. Mahgerefteh and J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
[Crossref] [PubMed]

D. Mahgerefteh and J. Feinberg, “Erasure rate and coasting in photorefractive barium titanate at high optical power,” Opt. Lett. 13, 1111 (1988).
[Crossref] [PubMed]

D. Mahgerefteh, “The speed of the photorefractive effect, shallow traps, photogalvanic currents, and light-induced surface damage in barium titanate,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1990).

Markov, V. B.

11N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectric 22, 949 (1979).

Michel-Calendini, F. M.

F. M. Michel-Calendini and P. Moretti, “Electronic structures of Co(II) and Co(III) impurities in cubic perovskite hosts,” Phys. Rev. B 27, 763 (1983).
[Crossref]

F. M. Michel-Calendini, P. Moretti, and G. Godefroy, “Excitation energies and optical absorption in Remeika BaTiO3:Co crystals,” Ferroelec. Lett. 44, 257 (1983).
[Crossref]

Moretti, P.

F. M. Michel-Calendini, P. Moretti, and G. Godefroy, “Excitation energies and optical absorption in Remeika BaTiO3:Co crystals,” Ferroelec. Lett. 44, 257 (1983).
[Crossref]

F. M. Michel-Calendini and P. Moretti, “Electronic structures of Co(II) and Co(III) impurities in cubic perovskite hosts,” Phys. Rev. B 27, 763 (1983).
[Crossref]

Motes, R. A.

G. A. Brost and R. A. Motes, “Origin of the sublinear photorefractive response time in BaTiO3,” Opt. Lett. 14, 1194 (1990).
[Crossref]

Muller, K. A.

K. W. Blazey and K. A. Muller, “Paramagnetic resonance and optical absorption of Co+4in SrTiO3,” J. Phys. C 16, 5491 (1983).
[Crossref]

Nelson, C. C.

Odulov, S. G.

11N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectric 22, 949 (1979).

Pollak, T. M.

Popov, B. N.

V. M. Fridkin and B. N. Popov, “The anomalous photovoltaic effect and photoconductivity in ferroelectrics,” Ferroelectrics 21, 611 (1978).
[Crossref]

Rytz, D.

Schunemann, P. G.

Schwartz, R. N.

Soskin, M. S.

11N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectric 22, 949 (1979).

Strohkendl, F. P.

F. P. Strohkendl, “Light-induced dark decays of photorefractive gratings and their observation in Bi12SiO20,” J. Appl. Phys. 65, 3773 (1989).
[Crossref]

Tayebati, P.

Temple, D. A.

Teng, Y.-Y.

Tuller, H. L.

Valley, G. C.

G. C. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704 (1983).
[Crossref]

Vinetskii, V. L.

11N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectric 22, 949 (1979).

Warde, C.

M. H. Garrett, J. Y. Chang, H. P. Jenssen, and C. Warde, “High beam-coupling gain and deep- and shallow-trap effects in cobalt-doped barium titanate, BaTiO3:Co,” J. Opt. Soc. Am. B 9, 1408 (1992).
[Crossref]

P. G. Schunemann, D. A. Temple, R. S. Hathcock, H. L. Tuller, H. P. Jenssen, D. R. Gabbe, and C. Warde, “Role of iron centers in the photorefractive effect in barium titanate,” J. Opt. Soc. Am. B 5, 1685 (1988).
[Crossref]

J. Y. Chang, M. H. Garrett, H. P. Jenssen, and C. Warde, “Intensity dependent photorefractive properties in an n-type BaTiO3,” J. Appl. Phys. (to be published).

Wechsler, B. A.

Wong, C.

Yang, Y.

Ferroelec. Lett. (1)

F. M. Michel-Calendini, P. Moretti, and G. Godefroy, “Excitation energies and optical absorption in Remeika BaTiO3:Co crystals,” Ferroelec. Lett. 44, 257 (1983).
[Crossref]

Ferroelectric (1)

11N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectric 22, 949 (1979).

Ferroelectrics (1)

V. M. Fridkin and B. N. Popov, “The anomalous photovoltaic effect and photoconductivity in ferroelectrics,” Ferroelectrics 21, 611 (1978).
[Crossref]

J. Am. Ceram. Soc. (1)

H.-J. Hagemann and D. Hennings, “Reversible weight change of acceptor-doped BaTiO3,” J. Am. Ceram. Soc. 64, 590 (1981).
[Crossref]

J. Appl. Phys. (2)

S. Ducharme and J. Feinberg, “Speed of the photorefractive effect in a BaTiO3single crystal,” J. Appl. Phys. 56, 839 (1984).
[Crossref]

F. P. Strohkendl, “Light-induced dark decays of photorefractive gratings and their observation in Bi12SiO20,” J. Appl. Phys. 65, 3773 (1989).
[Crossref]

J. Opt. Soc. Am. B (6)

J. Phys. C (1)

K. W. Blazey and K. A. Muller, “Paramagnetic resonance and optical absorption of Co+4in SrTiO3,” J. Phys. C 16, 5491 (1983).
[Crossref]

Opt. Eng. (1)

G. C. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704 (1983).
[Crossref]

Opt. Lett. (2)

D. Mahgerefteh and J. Feinberg, “Erasure rate and coasting in photorefractive barium titanate at high optical power,” Opt. Lett. 13, 1111 (1988).
[Crossref] [PubMed]

G. A. Brost and R. A. Motes, “Origin of the sublinear photorefractive response time in BaTiO3,” Opt. Lett. 14, 1194 (1990).
[Crossref]

Phys. Rev. B (1)

F. M. Michel-Calendini and P. Moretti, “Electronic structures of Co(II) and Co(III) impurities in cubic perovskite hosts,” Phys. Rev. B 27, 763 (1983).
[Crossref]

Phys. Rev. Lett. (1)

D. Mahgerefteh and J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
[Crossref] [PubMed]

Phys. Status Solidi A (1)

L. Holtmann, “A model for the nonlinear photoconductivity of BaTiO3,” Phys. Status Solidi A 113, k89 (1989).
[Crossref]

Other (4)

D. Mahgerefteh, “The speed of the photorefractive effect, shallow traps, photogalvanic currents, and light-induced surface damage in barium titanate,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1990).

J. Y. Chang, M. H. Garrett, H. P. Jenssen, and C. Warde, “Intensity dependent photorefractive properties in an n-type BaTiO3,” J. Appl. Phys. (to be published).

R. S. Hathcock, “Optical and photorefractive properties of iron-doped barium titanate,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1989).

P. Günter, “Holography, coherent light amplification and optical phase conjugation with photorefractive materials,” Phys. Rep.93, 199 (1982).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

Deep-and-shallow-trap energy-level model for hole-dominated photorefractive BaTiO3. ND is the total deep-trap density, NF is the uncompensated deep-trap density, and MT is the total shallow-trap density.

Fig. 2
Fig. 2

Experimental setup for light-induced dark-decay and response-time measurements. A beam from an argon laser is expanded and collimated. The intensity is controlled by the half-wave plate and a polarizer. The beam ratio of two writing beams is controlled by the two polarizers after the beam splitter. In light-induced dark-decay measurement, the space-charge field strength is probed by the Bragg-matched helium–neon laser beam, which is optically chopped. The diffracted beam is detected by a photomultiplier, which is connected to the lock-in amplifier. In the response-time measurement, the signal beam is detected by the P-I-N detector, M’s, mirrors; P’s, polarizers; BS, beam splitter; PMT, photomultiplier; Ip, pump beam; Is, signal beam; ID, diffracted beam.

Fig. 3
Fig. 3

Normalized diffraction efficiency dependence with time for the light-induced dark decay for undoped and 20-, 50-, and 75-ppm cobalt-doped BaTiO3.

Fig. 4
Fig. 4

Coasting dependence with intensity for undoped and 20, 50, and 75 ppm cobalt-doped BaTiO3.

Fig. 5
Fig. 5

Corresponding dark conductivity calculated by the dielectric relaxation-time equation dependence on cobalt-doping concentration in BaTiO3.

Fig. 6
Fig. 6

Grating-formation rate dependence on intensity for undoped and 20-, 50-, and 75-ppm cobalt-doped BaTiO3. Note that the straight lines are least-squares fits to a power law.

Fig. 7
Fig. 7

Proposed energy-level model of the nominally undoped and cobalt-doped BaTiO3. D and S represent the deep and shallow levels, respectively. F is the Fermi level. In the cobalt-doped BaTiO3, Co+3VO/Co+2VO is added, which is ∼2.4 eV above the valence band, and the Fermi level is pulled up close to the middle of the band gap.

Tables (1)

Tables Icon

Table 1 Photorefractive Sensitivity of Cobalt-Doped BaTiO3a

Equations (5)

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

C = 100 I D ( t = 0 ) I D ( t 1 / β ) I D ( t = 0 ) ,
n 0 = s D I 0 ( N DF N 0 ) τ D ( I 0 ) ,
τ D ( I 0 ) = 1 γ D ( N F + N 0 ) .
I s ( t ) = I s 0 exp { γ + d [ 1 exp ( t / τ ) ] } ,
S = γ eo λ 4 π α I τ I ,

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