Abstract

Dember and photo-electromotive-force (PEMF) currents are investigated in silicon photoconductive detectors both theoretically and experimentally. Dember photocurrents were found to dominate the response of high-purity silicon samples with top-surface electrodes to a moving interference pattern. The use of surface electrodes leads to shadowed regions beneath the electrodes, and Dember photocurrents appear under short-circuit conditions. A single-charge-carrier model of the Dember effect is in good qualitative agreement with experimental results. We also show theoretically that the PEMF effect in silicon is weak compared with other semiconductors because of its relatively high intrinsic conductivity.

© 2004 Optical Society of America

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References

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  1. S. M. Ryvkin, Photoelectric Effects in Semiconductors (Consultants Bureau, New York, 1964).
  2. M. P. Petrov, I. A. Sokolov, S. I. Stepanov, and G. S. Trofimov, “Non-steady-state photo-electromotive-force induced by dynamic gratings in partially compensated photoconductors,” J. Appl. Phys. 68, 2216–2225 (1990).
    [CrossRef]
  3. S. Stepanov, “Photo-electromotive-force effect in semiconductors,” in Semiconductor Devices, Vol. 2 of Handbook of Advanced Electronic and Photonic Materials and Devices, H. S. Nalwa, ed. (Academic, London, 2001), pp. 205–272.
  4. P. Bhattacharya, Semiconductor Optoelectronic Devices (Prentice Hall, Upper Saddle River, N.J., 1997).
  5. C. C. Wang, F. Davidson, and S. Trivedi, “Simple laser velocimeter that uses photoconductive semiconductors to measure optical frequency differences,” Appl. Opt. 34, 6496–6499 (1995).
    [CrossRef] [PubMed]
  6. G. J. Dunning, D. M. Pepper, M. P. Chiao, P. V. Mitchell, and F. M. Davidson, “Optimizing the photo-EMF response for high-speed compensation and broadband laser-based ultrasonic remote sensing,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 45–46.
  7. R. J. Dewhurst and Q. Shan, “Optical remote measurement of ultrasound,” Meas. Sci. Technol. 10, R139–R168 (1999).
    [CrossRef]
  8. N. A. Korneev and S. I. Stepanov, “Non-steady-state photoelectromotive force in semiconductor crystals with high light absorption,” J. Appl. Phys. 74, 2736–2741 (1993).
    [CrossRef]
  9. F. M. Davidson, C. C. Wang, C. T. Field, and S. Trivedi, “Photocurrents in photoconductive semiconductors generated by a moving space-charge field,” Opt. Lett. 19, 478–480 (1994).
    [CrossRef] [PubMed]
  10. J. A. Coy, D. D. Nolte, G. J. Dunning, D. M. Pepper, B. Pouet, G. D. Bacher, and M. B. Klein, “Asymmetric interdigitated metal-semiconductor-metal contacts for improved adaptive photoinduced-electromotive-force detectors,” J. Opt. Soc. Am. B 17, 697–704 (2000).
    [CrossRef]
  11. C. C. Wang, F. Davidson, and S. Trivedi, “Moving space-charge field effects in photoconductive semiconductors with interband optical excitation of free charge carriers,” J. Opt. Soc. Am. B 14, 21–26 (1997).
    [CrossRef]
  12. I. A. Sokolov and S. I. Stepanov, “Non-steady-state photoelectromotive force in crystals with long photocarrier lifetimes,” J. Opt. Soc. Am. B 10, 1483–1488 (1993).
    [CrossRef]
  13. N. Korneev, S. Mansurova, and S. Stepanov, “Non-steady-state photoelectromotive force in bipolar photoconductors with arbitrary level structure,” J. Opt. Soc. Am. B 12, 615–620 (1995).
    [CrossRef]
  14. Q. Liu, C. Chen, and H. Ruda, “Surface photovoltage in undoped semi-insulating GaAs,” J. Appl. Phys. 74, 7492–7496 (1993).
    [CrossRef]
  15. L. Kronik and Y. Shapira, “Surface photovoltage phenomena: theory, experiment, and applications,” Surf. Sci. Rep. 37, 1–206 (1999).
    [CrossRef]
  16. T. K. Woodward and A. V. Krishnamoorthy, “1-Gb/s integrated optical detectors and receivers in commercial CMOS technologies,” IEEE J. Sel. Top. Quantum Electron. 5, 146–156 (1999).
    [CrossRef]

2000 (1)

1999 (3)

R. J. Dewhurst and Q. Shan, “Optical remote measurement of ultrasound,” Meas. Sci. Technol. 10, R139–R168 (1999).
[CrossRef]

L. Kronik and Y. Shapira, “Surface photovoltage phenomena: theory, experiment, and applications,” Surf. Sci. Rep. 37, 1–206 (1999).
[CrossRef]

T. K. Woodward and A. V. Krishnamoorthy, “1-Gb/s integrated optical detectors and receivers in commercial CMOS technologies,” IEEE J. Sel. Top. Quantum Electron. 5, 146–156 (1999).
[CrossRef]

1997 (1)

1995 (2)

1994 (1)

1993 (3)

N. A. Korneev and S. I. Stepanov, “Non-steady-state photoelectromotive force in semiconductor crystals with high light absorption,” J. Appl. Phys. 74, 2736–2741 (1993).
[CrossRef]

Q. Liu, C. Chen, and H. Ruda, “Surface photovoltage in undoped semi-insulating GaAs,” J. Appl. Phys. 74, 7492–7496 (1993).
[CrossRef]

I. A. Sokolov and S. I. Stepanov, “Non-steady-state photoelectromotive force in crystals with long photocarrier lifetimes,” J. Opt. Soc. Am. B 10, 1483–1488 (1993).
[CrossRef]

1990 (1)

M. P. Petrov, I. A. Sokolov, S. I. Stepanov, and G. S. Trofimov, “Non-steady-state photo-electromotive-force induced by dynamic gratings in partially compensated photoconductors,” J. Appl. Phys. 68, 2216–2225 (1990).
[CrossRef]

Bacher, G. D.

Chen, C.

Q. Liu, C. Chen, and H. Ruda, “Surface photovoltage in undoped semi-insulating GaAs,” J. Appl. Phys. 74, 7492–7496 (1993).
[CrossRef]

Coy, J. A.

Davidson, F.

Davidson, F. M.

Dewhurst, R. J.

R. J. Dewhurst and Q. Shan, “Optical remote measurement of ultrasound,” Meas. Sci. Technol. 10, R139–R168 (1999).
[CrossRef]

Dunning, G. J.

Field, C. T.

Klein, M. B.

Korneev, N.

Korneev, N. A.

N. A. Korneev and S. I. Stepanov, “Non-steady-state photoelectromotive force in semiconductor crystals with high light absorption,” J. Appl. Phys. 74, 2736–2741 (1993).
[CrossRef]

Krishnamoorthy, A. V.

T. K. Woodward and A. V. Krishnamoorthy, “1-Gb/s integrated optical detectors and receivers in commercial CMOS technologies,” IEEE J. Sel. Top. Quantum Electron. 5, 146–156 (1999).
[CrossRef]

Kronik, L.

L. Kronik and Y. Shapira, “Surface photovoltage phenomena: theory, experiment, and applications,” Surf. Sci. Rep. 37, 1–206 (1999).
[CrossRef]

Liu, Q.

Q. Liu, C. Chen, and H. Ruda, “Surface photovoltage in undoped semi-insulating GaAs,” J. Appl. Phys. 74, 7492–7496 (1993).
[CrossRef]

Mansurova, S.

Nolte, D. D.

Pepper, D. M.

Petrov, M. P.

M. P. Petrov, I. A. Sokolov, S. I. Stepanov, and G. S. Trofimov, “Non-steady-state photo-electromotive-force induced by dynamic gratings in partially compensated photoconductors,” J. Appl. Phys. 68, 2216–2225 (1990).
[CrossRef]

Pouet, B.

Ruda, H.

Q. Liu, C. Chen, and H. Ruda, “Surface photovoltage in undoped semi-insulating GaAs,” J. Appl. Phys. 74, 7492–7496 (1993).
[CrossRef]

Shan, Q.

R. J. Dewhurst and Q. Shan, “Optical remote measurement of ultrasound,” Meas. Sci. Technol. 10, R139–R168 (1999).
[CrossRef]

Shapira, Y.

L. Kronik and Y. Shapira, “Surface photovoltage phenomena: theory, experiment, and applications,” Surf. Sci. Rep. 37, 1–206 (1999).
[CrossRef]

Sokolov, I. A.

I. A. Sokolov and S. I. Stepanov, “Non-steady-state photoelectromotive force in crystals with long photocarrier lifetimes,” J. Opt. Soc. Am. B 10, 1483–1488 (1993).
[CrossRef]

M. P. Petrov, I. A. Sokolov, S. I. Stepanov, and G. S. Trofimov, “Non-steady-state photo-electromotive-force induced by dynamic gratings in partially compensated photoconductors,” J. Appl. Phys. 68, 2216–2225 (1990).
[CrossRef]

Stepanov, S.

Stepanov, S. I.

I. A. Sokolov and S. I. Stepanov, “Non-steady-state photoelectromotive force in crystals with long photocarrier lifetimes,” J. Opt. Soc. Am. B 10, 1483–1488 (1993).
[CrossRef]

N. A. Korneev and S. I. Stepanov, “Non-steady-state photoelectromotive force in semiconductor crystals with high light absorption,” J. Appl. Phys. 74, 2736–2741 (1993).
[CrossRef]

M. P. Petrov, I. A. Sokolov, S. I. Stepanov, and G. S. Trofimov, “Non-steady-state photo-electromotive-force induced by dynamic gratings in partially compensated photoconductors,” J. Appl. Phys. 68, 2216–2225 (1990).
[CrossRef]

Trivedi, S.

Trofimov, G. S.

M. P. Petrov, I. A. Sokolov, S. I. Stepanov, and G. S. Trofimov, “Non-steady-state photo-electromotive-force induced by dynamic gratings in partially compensated photoconductors,” J. Appl. Phys. 68, 2216–2225 (1990).
[CrossRef]

Wang, C. C.

Woodward, T. K.

T. K. Woodward and A. V. Krishnamoorthy, “1-Gb/s integrated optical detectors and receivers in commercial CMOS technologies,” IEEE J. Sel. Top. Quantum Electron. 5, 146–156 (1999).
[CrossRef]

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

T. K. Woodward and A. V. Krishnamoorthy, “1-Gb/s integrated optical detectors and receivers in commercial CMOS technologies,” IEEE J. Sel. Top. Quantum Electron. 5, 146–156 (1999).
[CrossRef]

J. Appl. Phys. (3)

Q. Liu, C. Chen, and H. Ruda, “Surface photovoltage in undoped semi-insulating GaAs,” J. Appl. Phys. 74, 7492–7496 (1993).
[CrossRef]

M. P. Petrov, I. A. Sokolov, S. I. Stepanov, and G. S. Trofimov, “Non-steady-state photo-electromotive-force induced by dynamic gratings in partially compensated photoconductors,” J. Appl. Phys. 68, 2216–2225 (1990).
[CrossRef]

N. A. Korneev and S. I. Stepanov, “Non-steady-state photoelectromotive force in semiconductor crystals with high light absorption,” J. Appl. Phys. 74, 2736–2741 (1993).
[CrossRef]

J. Opt. Soc. Am. B (4)

Meas. Sci. Technol. (1)

R. J. Dewhurst and Q. Shan, “Optical remote measurement of ultrasound,” Meas. Sci. Technol. 10, R139–R168 (1999).
[CrossRef]

Opt. Lett. (1)

Surf. Sci. Rep. (1)

L. Kronik and Y. Shapira, “Surface photovoltage phenomena: theory, experiment, and applications,” Surf. Sci. Rep. 37, 1–206 (1999).
[CrossRef]

Other (4)

S. M. Ryvkin, Photoelectric Effects in Semiconductors (Consultants Bureau, New York, 1964).

S. Stepanov, “Photo-electromotive-force effect in semiconductors,” in Semiconductor Devices, Vol. 2 of Handbook of Advanced Electronic and Photonic Materials and Devices, H. S. Nalwa, ed. (Academic, London, 2001), pp. 205–272.

P. Bhattacharya, Semiconductor Optoelectronic Devices (Prentice Hall, Upper Saddle River, N.J., 1997).

G. J. Dunning, D. M. Pepper, M. P. Chiao, P. V. Mitchell, and F. M. Davidson, “Optimizing the photo-EMF response for high-speed compensation and broadband laser-based ultrasonic remote sensing,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 45–46.

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Figures (7)

Fig. 1
Fig. 1

Two configurations of electrodes on PEMF detectors: (a) side electrodes, (b) top-surface electrodes.

Fig. 2
Fig. 2

Theoretical plot of the magnitude of the ac Dember photocurrent, maximized with respect to the grating phase ϕ, as a function of phase-modulation frequency ω and grating wave-number K. The frequency axis is normalized to the carrier lifetime τ, and the wave-number axis is normalized to the inverse device length L. The ratio of diffusion length LD to the device length is 0.1.

Fig. 3
Fig. 3

Experimental setup used to characterize the Dember photocurrent from the detector samples. BS, beam splitter; M, mirror.

Fig. 4
Fig. 4

Frequency dependence of the measured ac photocurrent amplitude in a simple photoconductor configuration. The curves are theoretical plots that use a single value of 12 µs for the carrier lifetime. The signal and reference power levels were 5.65 and 4.45 mW for the circles, 2.90 and 2.30 mW for the squares, and 1.46 and 1.14 mW for the triangles.

Fig. 5
Fig. 5

dc Dember photocurrent under uniform illumination with increasing area. The illuminated area was controlled by the translation of a razor blade that went from covering the whole detector at zero to revealing the area between the two contacts at 2 mm. Circles, experimental data points; curve, a theoretical plot with LD=205 µm.

Fig. 6
Fig. 6

Amplitude of the ac Dember photocurrent as a function of grating wave number at a phase-modulation frequency of 1 kHz. The grating phase was adjusted to maximize the current at each wave number. Circles, experimental data points; curve, a theoretical plot. The signal and reference beam power levels were 2.70 and 2.25 mW, respectively.

Fig. 7
Fig. 7

Amplitude of the ac Dember photocurrent as a function of phase-modulation frequency for five values of the grating wave number. The grating phase was adjusted to maximize the current at each frequency. The signal and reference beam power levels were 2.65 and 2.20 mW, respectively.

Equations (78)

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I(x, t)=I0[1+m cos(Kx+Δ cos ωt)],
J(t)=Jω2exp(jωt)+Jω*2exp(-jωt)=Re[Jω  exp(jωt)],
Jω=m2Δ2σ0ED-jωτM1-ω2ττM+jω[τ+τM(1+K2LD2)].
|Jω|=m2Δ2σ0ED
×ωτM{(1-ω2ττM)2+ω2[τ+τM(1+K2LD2)]2}1/2.
|Jω|=m2Δ2σ0EDωτM{1+ω2[τM(1+K2LD2)]2}1/2.
|Jω|max=m2Δ4σ0ED.
|Jω|=m2Δ2σ0EDωτM1+ω2τ2.
|Jω|max=m2Δ2σ0EDτMτ.
J(x)=eμn(x)E(x)+kBTμdn(x)dx,
E(x)=J-kBTμdn(x)dxeμn(x).
E(x)=J-kBTμdΔn(x)dxσd+eμΔn(x),
x1x2E(x)dx=Jx1x2dxσd+eμΔn(x)-kBTe lnσd+eμΔn(x2)σd+eμΔn(x1)=0.
J=kBTe lnσd+eμΔn(x2)σd+eμΔn(x1)1x1x2dxσd+eμΔn(x) .
J=kBTμΔn(x2)-Δn(x1)x2-x1.
Δn1(x)=12g0τ[1-exp(x/LD)]+1-exp-xR+xLD,
Δn2(x)=12g0τ[1-exp(-xR/LD)]exp(-x/LD).
x1x2dxσd+eμΔn(x)=-xR0dxσd+eμΔn1(x)+0L-xRdxσd+eμΔn2(x)=I1+I2
I1
=LDσ01A×lnA-σdσ0 A-exp(-xR/LD)+σd+σ0σ0A+σdσ0 A+exp(-xR/LD)-σd+σ0σ0,
A2=σd+σ0σ02-exp(-xR/LD).
I2=LDσdlnσd exp[(L-xR)/LD]+σ02[1-exp(-xR/LD)]σd+σ02[1-exp(-xR/LD)].
Δn1(-xR)=12g0τ[1-exp(-xR/LD)]
Δn2(L-xR)=12g0τ[exp(xR/LD)-1]exp(-L/LD).
J=kBTe1I1+I2×lnσd+σ02[exp(xR/LD)-1]exp(-L/LD)σd+σ02[1-exp(-xR/LD)],
J=kBTeσ02L[1-exp(-xR/LD)]×{exp[-(L-xR)/LD]-1}.
Δn(x, t)t=g(x, t)-Δn(x, t)τ+1eJ(x, t)x,
D2Δn(x, t)x2-Δn(x, t)t-Δn(x, t)τ=-g(x, t),
I(x, t)=I0[1+m cos(Kx+ϕ+Δ cos ωt)],
I(x, t)I0+mI0[cos(Kx+ϕ)-Δ sin(Kx+ϕ)cos ωt]=I0+mI0 cos(Kx+ϕ)-mΔ2I0 sin(Kx+ϕ)exp(jωt)+c.c.,
Δn(x, t)=Δn0(x)+Δnω(x)2 exp(jωt)+c.c..
d2Δn0(x)dx2-Δn0(x)LD2=-g0τLD2[1+m cos(Kx+ϕ)],
d2Δnω(x)dx2-1+jωτLD2Δnω(x)=g0τmΔLD2 sin(Kx+ϕ).
Δn0(x)=12g0τ[1-exp(-L/LD)]exp(-x/LD)+12mg0τ1+K2LD2×{KLD sin ϕ+cos ϕ-exp(-L/LD)×[cos(-KL+ϕ)+KLD×sin(-KL+ϕ)]}exp(-x/LD)
Δn0(x)=12g0τ[exp(L/LD)-1]exp(x/LD)+12mg0τ1+K2LD2×{KLD sin ϕ-cos ϕ+exp(L/LD)×[cos(-KL+ϕ)-KLD sin(-KL+ϕ)]}×exp(x/LD)
Δn0(x)=-12g0τ{exp(x/LD)+exp[-(x+L)/LD]}+12mg0τ1+K2LD2×{(-cos ϕ+KLD sin ϕ)exp(x/LD)-[cos(-KL+ϕ)+KLD×sin(-KL+ϕ)]exp[-(x+L)/LD]}+g0τ+mg0τ1+K2LD2cos(Kx+ϕ)
Δnω(x)=12mΔg0τ1+K2LD2+jωτ KLD1+jωτ cos ϕ-sin ϕ+sin(-KL+ϕ)-KLD1+jωτ×cos(-KL+ϕ)exp(-L1+jωτ/LD)×exp(-x1+jωτ/LD)
Δnω(x)=12mΔg0τ1+K2LD2+jωτ KLD1+jωτ cos ϕ+sin ϕ-sin(-KL+ϕ)+KLD1+jωτ×cos(-KL+ϕ)exp(L1+jωτ/LD)×exp(x1+jωτ/LD)
Δnω(x)=mΔg0τ1+K2LD2+jωτ 12 KLD1+jωτ×cos ϕ+sin ϕexp(x1+jωτ/LD)+12 sin(-KL+ϕ)-KLD1+jωτ×cos(-KL+ϕ)exp[-(x+L)×1+jωτ/LD]-sin(Kx+ϕ)
J(t)=J0+Jω2 exp(jωt)+c.c.,
J0=kBTμΔn0(x2)-Δn0(x1)x2-x1
Jω=kBTμΔnω(x2)-Δnω(x1)x2-x1.
Δn0(0)=12g0τ[1-exp(-L/LD)]+12mg0τ1+K2LD2{KLD sin ϕ+cos ϕ-exp(-L/LD)[KLD sin(-KL+ϕ)+cos(-KL+ϕ)]},
Δn0(-L)=12g0τ[1-exp(-L/LD)]+12mg0τ1+K2LD2{exp(-L/LD)×(KLD sin ϕ-cos ϕ)+[-KLD sin(-KL+ϕ)+cos(-KL+ϕ)]},
Δnω(0)=12mΔg0τ1+K2LD2+jωτ KLD1+jωτ×cos ϕ-sin ϕ+exp(-L1+jωτ/LD)sin(-KL+ϕ)-KLD1+jωτ cos(-KL+ϕ),
Δnω(-L)=12mΔg0τ1+K2LD2+jωτ ×exp(-L1+jωτ/LD)×KLD1+jωτ cos ϕ+sin ϕ-sin(-KL+ϕ)+KLD1+jωτ cos(-KL+ϕ).
J0=kBTeσ02Lm1+K2LD2 KLD1+cos KLsin KL-1×[sin KL sin ϕ-(1-cos KL)cos ϕ].
Jω=kBTeσ02LmΔ1+K2LD2+jωτ×KLD1+jωτ1+cos KLsin KL-1×[sin KL cos ϕ+(1-cos KL)sin ϕ].
tan ϕopt=1-cos KLsin KL.
Jω=kBTeσ02LmΔ1+K2LD2+jωτ KLD1+jωτ-sin KL1+cos KL2(1+cos KL).
|I(ω)|=Idc1+ω2τ2,
I1=-xR0exp(x/LD)dx-σ02 exp(2x/LD)+(σd+σ0)exp(x/LD)-σ02 exp(-xR/LD).
I1=exp(-xR/LD)-(σd+σ0)/σ01-(σd+σ0)/σ02LDσ0 dvA2-v2.
Δn0(x)=C1 exp(r1x)+C2 exp(r2x),
d2ΔN0(x)dx2-ΔN0(x)LD2=-g0τLD2m cos(Kx+ϕ)
ΔN0(x)=A sin Kx+B cos Kx.
A=-mg0τ1+K2LD2 sin ϕ,
B=mg0τ1+K2LD2 cos ϕ.
ΔN0(x)=g0τ+mg0τ1+K2LD2 cos(Kx+ϕ).
Δn0(x)=C2 exp(-x/LD),
Δn0(x)=C5 exp(x/LD),
Δn0(x)=C3 exp(x/LD)+C4 exp(-x/LD)+g0τ+mg0τ1+K2LD2 cos(Kx+ϕ).
C2=12g0τ[1-exp(-L/LD)]+12mg0τ1+K2LD2×{cos ϕ+KLD sin ϕ-exp(-L/LD)×[KLD sin(-KL+ϕ)+cos(-KL+ϕ)]},
C3=-12g0τ+12mg0τ1+K2LD2(-cos ϕ+KLD sin ϕ),
C4=-12g0τ exp(-L/LD)-12 exp(-L/LD)mg0τ1+K2LD2[cos(-KL+ϕ)+KLD sin(-KL+ϕ)],
C5=12g0τ[exp(L/LD)-1]+12mg0τ1+K2LD2×{-cos ϕ+KLD sin ϕ+exp(L/LD)×[-KLD sin(-KL+ϕ)+cos(-KL+ϕ)]}.
Δnω(x)=C1 exp(r1x)+C2 exp(r2x),
r2-1+jωτLD2=0.
r1,2=±1+jωτLD2=±1LD22[(1+ω2τ2+1)1/2+j(1+ω2τ2-1)1/2].
ΔNω(x)=A sin Kx+B cos Kx.
ΔNω(x)=-mΔg0τ1+K2LD2+jωτ sin(Kx+ϕ).
Δnω(x)=C2 exp(-x1+jωτ/LD),
Δnω(x)=C5 exp(x1+jωτ/LD),
Δnω(x)=C3 exp(x1+jωτ/LD)+C4 exp(-x1+jωτ/LD)-mΔg0τ1+K2LD2+jωτ sin(Kx+ϕ).
C2=12mΔg0τ1+K2LD2+jωτ KLD1+jωτ cos ϕ-sin ϕ+sin(-KL+ϕ)-KLD1+jωτ×cos(-KL+ϕ)exp(-L1+jωτ/LD),
C3=12mΔg0τ1+K2LD2+jωτ KLD1+jωτ cos ϕ+sin ϕ,
C4=12mΔg0τ1+K2LD2+jωτ -KLD1+jωτ cos(-KL+ϕ)+sin(-KL+ϕ)×exp(-L1+jωτ/LD),
C5=12mΔg0τ1+K2LD2+jωτ KLD1+jωτ cos ϕ+sin ϕ-sin(-KL+ϕ)+KLD1+jωτ×cos(-KL+ϕ)exp(L1+jωτ/LD).

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