Abstract

We calculate the degree to which absorption can be enhanced by light trapping in a material whose thickness is on the order of a wavelength of light. The calculation makes use of an extension of the radiance theorem in the wave domain and represents the ultimate upper limit as determined by the laws of thermodynamics. It is a general upper limit, based solely on considerations of the modal structure of the device and independent of the technique used to couple light into the material. We assume only that the coupling mechanism is isotropic. An important parameter in our result is the density of guided modes within the planar structure, which we calculate. We find that even for structures supporting only a small number of guided modes, significant enhancements in absorption are possible.

© 1997 Optical Society of America

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References

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  1. J. H. Werner, R. Bergmann, R. Brendel, “The challenge of crystalline thin film solar cells,” in Festkörperprobleme/Advances in Solid State Physics, R. Helbig, ed. (Vieweg, Braunschweig, Germany, 1994), Vol. 34, pp. 115–146.
  2. M. S. Ünlü, S. Strite, “Resonant cavity enhanced photonic devices,” J. Appl. Phys. 78, 607–639 (1995).
    [Crossref]
  3. H. R. Stuart, D. G. Hall, “Absorption enhancement in silicon-on-insulator waveguides using metalisland films,” Appl. Phys. Lett. 69, 2327–2329 (1996).
    [Crossref]
  4. E. Yablonovitch, G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Devices ED-29, 300–305 (1982).
    [Crossref]
  5. E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am. 72, 899–907 (1982).
    [Crossref]
  6. For a discussion of the limitations to absorption enhancement by RCE see K. Kishino, M. S. Ünlü, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoç, “Resonant cavity-enhanced (RCE) photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
    [Crossref]
  7. I. M. Bassett, “Limits to concentration by passive means,” Phys. Rev. Lett. 54, 2014–2017 (1985).
    [Crossref] [PubMed]
  8. The intensity enhancement achievable by a nonisotropic coupling device (e.g., a grating) will always exceed that achieved by isotropic coupling but with an accompanying trade-off in the angular response of the device.
  9. R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), pp. 16, 42.
  10. R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), p. 32.
  11. This results from the nonisotropic distribution of the modal spectrum within the waveguide. This nonisotropic distribution suggests that a random surface texturing, even if it were practical, may not be the optimal means for coupling light into the thin layer for maximum absorption enhancement.
  12. R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), p. 31.
  13. The results for the case where h≫λ yielda similar interpretation. See T. Tiedje, E. Yablonovitch, G. D. Cody, B. G. Brooks, “Limiting efficiency of silicon solar cells,” IEEE Trans. Electron. Devices ED-31, 711–716 (1984).
    [Crossref]
  14. P. Rouard, A. Meesen, “Optical properties of thin metal films,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. VX, p. 77.
  15. For a classical treatment of the problem, see G. W. Ford, W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984). For a quantum treatment, see J. M. Wylie, J. E. Sipe, “Quantum electrodynamics near an interface,” Phys. Rev. A 30, 1185–1193 (1984).
    [Crossref]
  16. R. J. Warmack, S. L. Humphrey, “Observation of two surface-plasmon modes on gold particles,” Phys. Rev. B 34, 2246–2252 (1986).
    [Crossref]
  17. V. V. Truong, G. D. Scott, “Optical constants of aggregated gold films,” J. Opt. Soc. Am. 66, 124–131 (1976).V. V. Truong, G. D. Scott, “Optical properties of aggregated noble metal films,” J. Opt. Soc. Am. 67, 502–510 (1977).
    [Crossref]
  18. We used the data in Ref. 17 for gold and copper islands and several values of the effective thickness ranging from 10 to 20 nm.
  19. M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), p. 41, Eq. (2.133). This relation is derived under the assumption of a weakly guiding structure but is easily be shown to be true in general.

1996 (1)

H. R. Stuart, D. G. Hall, “Absorption enhancement in silicon-on-insulator waveguides using metalisland films,” Appl. Phys. Lett. 69, 2327–2329 (1996).
[Crossref]

1995 (1)

M. S. Ünlü, S. Strite, “Resonant cavity enhanced photonic devices,” J. Appl. Phys. 78, 607–639 (1995).
[Crossref]

1991 (1)

For a discussion of the limitations to absorption enhancement by RCE see K. Kishino, M. S. Ünlü, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoç, “Resonant cavity-enhanced (RCE) photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
[Crossref]

1986 (1)

R. J. Warmack, S. L. Humphrey, “Observation of two surface-plasmon modes on gold particles,” Phys. Rev. B 34, 2246–2252 (1986).
[Crossref]

1985 (1)

I. M. Bassett, “Limits to concentration by passive means,” Phys. Rev. Lett. 54, 2014–2017 (1985).
[Crossref] [PubMed]

1984 (2)

The results for the case where h≫λ yielda similar interpretation. See T. Tiedje, E. Yablonovitch, G. D. Cody, B. G. Brooks, “Limiting efficiency of silicon solar cells,” IEEE Trans. Electron. Devices ED-31, 711–716 (1984).
[Crossref]

For a classical treatment of the problem, see G. W. Ford, W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984). For a quantum treatment, see J. M. Wylie, J. E. Sipe, “Quantum electrodynamics near an interface,” Phys. Rev. A 30, 1185–1193 (1984).
[Crossref]

1982 (2)

E. Yablonovitch, G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Devices ED-29, 300–305 (1982).
[Crossref]

E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am. 72, 899–907 (1982).
[Crossref]

1976 (1)

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), p. 41, Eq. (2.133). This relation is derived under the assumption of a weakly guiding structure but is easily be shown to be true in general.

Arsenault, L.

For a discussion of the limitations to absorption enhancement by RCE see K. Kishino, M. S. Ünlü, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoç, “Resonant cavity-enhanced (RCE) photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
[Crossref]

Bassett, I. M.

I. M. Bassett, “Limits to concentration by passive means,” Phys. Rev. Lett. 54, 2014–2017 (1985).
[Crossref] [PubMed]

Bergmann, R.

J. H. Werner, R. Bergmann, R. Brendel, “The challenge of crystalline thin film solar cells,” in Festkörperprobleme/Advances in Solid State Physics, R. Helbig, ed. (Vieweg, Braunschweig, Germany, 1994), Vol. 34, pp. 115–146.

Boyd, R. W.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), pp. 16, 42.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), p. 32.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), p. 31.

Brendel, R.

J. H. Werner, R. Bergmann, R. Brendel, “The challenge of crystalline thin film solar cells,” in Festkörperprobleme/Advances in Solid State Physics, R. Helbig, ed. (Vieweg, Braunschweig, Germany, 1994), Vol. 34, pp. 115–146.

Brooks, B. G.

The results for the case where h≫λ yielda similar interpretation. See T. Tiedje, E. Yablonovitch, G. D. Cody, B. G. Brooks, “Limiting efficiency of silicon solar cells,” IEEE Trans. Electron. Devices ED-31, 711–716 (1984).
[Crossref]

Chyi, J. I.

For a discussion of the limitations to absorption enhancement by RCE see K. Kishino, M. S. Ünlü, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoç, “Resonant cavity-enhanced (RCE) photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
[Crossref]

Cody, G. D.

The results for the case where h≫λ yielda similar interpretation. See T. Tiedje, E. Yablonovitch, G. D. Cody, B. G. Brooks, “Limiting efficiency of silicon solar cells,” IEEE Trans. Electron. Devices ED-31, 711–716 (1984).
[Crossref]

E. Yablonovitch, G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Devices ED-29, 300–305 (1982).
[Crossref]

Ford, G. W.

For a classical treatment of the problem, see G. W. Ford, W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984). For a quantum treatment, see J. M. Wylie, J. E. Sipe, “Quantum electrodynamics near an interface,” Phys. Rev. A 30, 1185–1193 (1984).
[Crossref]

Hall, D. G.

H. R. Stuart, D. G. Hall, “Absorption enhancement in silicon-on-insulator waveguides using metalisland films,” Appl. Phys. Lett. 69, 2327–2329 (1996).
[Crossref]

Humphrey, S. L.

R. J. Warmack, S. L. Humphrey, “Observation of two surface-plasmon modes on gold particles,” Phys. Rev. B 34, 2246–2252 (1986).
[Crossref]

Kishino, K.

For a discussion of the limitations to absorption enhancement by RCE see K. Kishino, M. S. Ünlü, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoç, “Resonant cavity-enhanced (RCE) photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
[Crossref]

Meesen, A.

P. Rouard, A. Meesen, “Optical properties of thin metal films,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. VX, p. 77.

Morkoç, H.

For a discussion of the limitations to absorption enhancement by RCE see K. Kishino, M. S. Ünlü, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoç, “Resonant cavity-enhanced (RCE) photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
[Crossref]

Reed, J.

For a discussion of the limitations to absorption enhancement by RCE see K. Kishino, M. S. Ünlü, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoç, “Resonant cavity-enhanced (RCE) photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
[Crossref]

Rouard, P.

P. Rouard, A. Meesen, “Optical properties of thin metal films,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. VX, p. 77.

Scott, G. D.

Strite, S.

M. S. Ünlü, S. Strite, “Resonant cavity enhanced photonic devices,” J. Appl. Phys. 78, 607–639 (1995).
[Crossref]

Stuart, H. R.

H. R. Stuart, D. G. Hall, “Absorption enhancement in silicon-on-insulator waveguides using metalisland films,” Appl. Phys. Lett. 69, 2327–2329 (1996).
[Crossref]

Tiedje, T.

The results for the case where h≫λ yielda similar interpretation. See T. Tiedje, E. Yablonovitch, G. D. Cody, B. G. Brooks, “Limiting efficiency of silicon solar cells,” IEEE Trans. Electron. Devices ED-31, 711–716 (1984).
[Crossref]

Truong, V. V.

Ünlü, M. S.

M. S. Ünlü, S. Strite, “Resonant cavity enhanced photonic devices,” J. Appl. Phys. 78, 607–639 (1995).
[Crossref]

For a discussion of the limitations to absorption enhancement by RCE see K. Kishino, M. S. Ünlü, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoç, “Resonant cavity-enhanced (RCE) photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
[Crossref]

Warmack, R. J.

R. J. Warmack, S. L. Humphrey, “Observation of two surface-plasmon modes on gold particles,” Phys. Rev. B 34, 2246–2252 (1986).
[Crossref]

Weber, W. H.

For a classical treatment of the problem, see G. W. Ford, W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984). For a quantum treatment, see J. M. Wylie, J. E. Sipe, “Quantum electrodynamics near an interface,” Phys. Rev. A 30, 1185–1193 (1984).
[Crossref]

Werner, J. H.

J. H. Werner, R. Bergmann, R. Brendel, “The challenge of crystalline thin film solar cells,” in Festkörperprobleme/Advances in Solid State Physics, R. Helbig, ed. (Vieweg, Braunschweig, Germany, 1994), Vol. 34, pp. 115–146.

Yablonovitch, E.

The results for the case where h≫λ yielda similar interpretation. See T. Tiedje, E. Yablonovitch, G. D. Cody, B. G. Brooks, “Limiting efficiency of silicon solar cells,” IEEE Trans. Electron. Devices ED-31, 711–716 (1984).
[Crossref]

E. Yablonovitch, G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Devices ED-29, 300–305 (1982).
[Crossref]

E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am. 72, 899–907 (1982).
[Crossref]

Appl. Phys. Lett. (1)

H. R. Stuart, D. G. Hall, “Absorption enhancement in silicon-on-insulator waveguides using metalisland films,” Appl. Phys. Lett. 69, 2327–2329 (1996).
[Crossref]

IEEE J. Quantum Electron. (1)

For a discussion of the limitations to absorption enhancement by RCE see K. Kishino, M. S. Ünlü, J. I. Chyi, J. Reed, L. Arsenault, H. Morkoç, “Resonant cavity-enhanced (RCE) photodetectors,” IEEE J. Quantum Electron. 27, 2025–2034 (1991).
[Crossref]

IEEE Trans. Electron. Devices (2)

E. Yablonovitch, G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Devices ED-29, 300–305 (1982).
[Crossref]

The results for the case where h≫λ yielda similar interpretation. See T. Tiedje, E. Yablonovitch, G. D. Cody, B. G. Brooks, “Limiting efficiency of silicon solar cells,” IEEE Trans. Electron. Devices ED-31, 711–716 (1984).
[Crossref]

J. Appl. Phys. (1)

M. S. Ünlü, S. Strite, “Resonant cavity enhanced photonic devices,” J. Appl. Phys. 78, 607–639 (1995).
[Crossref]

J. Opt. Soc. Am. (2)

Phys. Rep. (1)

For a classical treatment of the problem, see G. W. Ford, W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984). For a quantum treatment, see J. M. Wylie, J. E. Sipe, “Quantum electrodynamics near an interface,” Phys. Rev. A 30, 1185–1193 (1984).
[Crossref]

Phys. Rev. B (1)

R. J. Warmack, S. L. Humphrey, “Observation of two surface-plasmon modes on gold particles,” Phys. Rev. B 34, 2246–2252 (1986).
[Crossref]

Phys. Rev. Lett. (1)

I. M. Bassett, “Limits to concentration by passive means,” Phys. Rev. Lett. 54, 2014–2017 (1985).
[Crossref] [PubMed]

Other (9)

The intensity enhancement achievable by a nonisotropic coupling device (e.g., a grating) will always exceed that achieved by isotropic coupling but with an accompanying trade-off in the angular response of the device.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), pp. 16, 42.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), p. 32.

This results from the nonisotropic distribution of the modal spectrum within the waveguide. This nonisotropic distribution suggests that a random surface texturing, even if it were practical, may not be the optimal means for coupling light into the thin layer for maximum absorption enhancement.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), p. 31.

J. H. Werner, R. Bergmann, R. Brendel, “The challenge of crystalline thin film solar cells,” in Festkörperprobleme/Advances in Solid State Physics, R. Helbig, ed. (Vieweg, Braunschweig, Germany, 1994), Vol. 34, pp. 115–146.

P. Rouard, A. Meesen, “Optical properties of thin metal films,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. VX, p. 77.

We used the data in Ref. 17 for gold and copper islands and several values of the effective thickness ranging from 10 to 20 nm.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), p. 41, Eq. (2.133). This relation is derived under the assumption of a weakly guiding structure but is easily be shown to be true in general.

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Figures (4)

Fig. 1
Fig. 1

Measured photocurrent enhancements due to the presence of a copper island film. Enhancement is defined as the ratio of the photocurrent of the device with the islands to that without the islands. The detector is fabricated in a 160-nm-thick silicon layer, which acts as the waveguide. The spacer layer is 30 nm of lithium fluoride. See Ref. 3 for details.

Fig. 2
Fig. 2

Light trapping works by coupling the incident beam into all of the internal modes, including the trapped modes. (a) In a thick slab of material, the trapped modes form a continuum. (b) When h becomes smaller, the trapped modes form a discrete set (the radiation modes still form a continuum). Planar-waveguide theory can be used to determine the characteristics of the trapped modes.

Fig. 3
Fig. 3

Thermodynamic limit to absorption when α h = 0.02 , plotted versus the normalized waveguide parameter V = ( 2 π / λ ) h n f 2 - n s 2 . The Yablonovitch limit (valid for large values of V) is shown for reference. The top curve is the general result, obtained from Eqs. (18) and (19). The lower curve is obtained by setting f rad = 0 in Eq. (19) and indicates the effect of ignoring the contribution of the radiation modes to the absorption process. The dashed line along the bottom indicates the absorption for normal incidence with no light-trapping or interference effects.

Fig. 4
Fig. 4

The two-dimensional k-space diagram for the allowed propagation constants of the guided modes.

Equations (35)

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ρ rad = Ω rad 4 π   ω 2 n f 3 π 2 c 3 .
Ω rad = 4 π 1 - 1 - n s 2 n f 2 .
ρ m = 1 2 π   β m   d β m d ω ,
U rad = hA ρ rad u ,
U m = A ρ m u ,
I = ( 1 / 4 ) ρ 0 uv g 0
r inc rad = IA   ρ rad ρ tot
r inc m = IA   ρ m h ρ tot ,
ρ tot = ρ rad + 1 h   m ρ m .
IA   ρ rad ρ tot = K rad Ah ρ rad u ,
IA   ρ m h ρ tot = K m A ρ m u .
K rad = K m = ρ 0 v g 0 4 ρ tot h K ,
r abs rad = L rad ( cos   θ d y d z ) α   d x cos   θ d Ω = α L rad Ah Ω rad
L rad = ( ρ rad u ν g rad ) / Ω rad .
r abs rad = α U rad ν g rad .
r abs m = H m ( cos   θ dy ) Γ m α   d z cos   θ d Ω 2 - d = 2 π Γ m α H m A .
H m = ( ρ m u   ν g m ) / 2 π
r abs m = α ( Γ m U m ) ν g m .
IA   ρ rad ρ tot = KAh ρ rad u rad + α Ah ρ rad v g rad u rad ,
IA   ρ m h ρ tot = KA ρ m u m + Γ m α A ρ m ν g m u m ,
u rad = I h ρ tot ( K + α v g rad ) ,
u m = I h ρ tot ( K + Γ m α ν g m ) .
f rad = r abs rad r inc rad = α α + 1 4 ρ tot v g rad ρ 0 v g 0 h ,
f m = r abs m r inc rad = Γ m α Γ m α + 1 4 ρ tot ν g m ρ 0 v g 0 h .
f tot = ρ rad ρ tot   f rad + m   ρ m h ρ tot   f m .
f tot = α α + 1 4 n f 2 h ,
k y , z = l y , z   π L l y , z = 1 ,   2 ,   3   .
k x 2 + β m 2 + n f 2 ω c 2 .
ρ m ( β m ) d β m = 1 L 2   1 4   2 π β m d β m ( π / L ) 2 ,
ρ m ( ω ) d ω = ρ m ( β m ) d β m ,
ρ m ( ω ) = 1 2 π   β m   d β m d ω .
c 2 v p v g = Γ m n f 2 + ( 1 - Γ m ) n s 2 ,
ρ m = ω 2 π c 2   [ Γ m n f 2 + ( 1 - Γ m ) n s 2 ] .
ρ 2 - D = ω π c 2   n 2
lim h λ 1 h   m ρ m + ρ rad = ω 2 n f 3 π 2 c 3 ,

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