Abstract

Raman spectra of three bulk 4H-SiC wafers with different free carrier concentration were measured at temperature from 80 K to 873 K. As temperature increases, Raman peaks of most optical phonon modes show monotonous down shift. An anomalous non-monotonous variation with temperature, was observed in the A1 longitudinal optical (LO) mode from doped samples. Two methods of theoretical fitting, one-mode (LO-plasma coupled (LOPC) mode) and two-mode (A1(LO) + LOPC) fitting, are employed to analyze this anomalous phenomenon. Theoretical simulations for temperature dependent Raman spectra by using two methods are critically examined. It turns out that one-mode method conforms well the experimental results, while two-mode method is untenable in physics. The non-monotonous variation of blue-red shifts with temperature for LOPC mode from doped 4H-SiC could be explained by the influence from ionization process of impurities on the process of Raman scattering. A quantitative description on temperature dependent Raman spectra for doped 4H-SiC is achieved, which matches well to experimental data.

© 2013 Optical Society of America

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References

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  1. S. Nakashima and H. Harima, “Raman investigation of SiC polytypes,” Phys. Status Solidi162(1), 39–64 (1997) (a).
    [CrossRef]
  2. W. S. Li, Z. X. Shen, Z. C. Feng, and S. J. Chua, “Temperature dependence of Raman scattering in hexagonal gallium nitride films,” J. Appl. Phys.87(7), 3332–3337 (2000).
    [CrossRef]
  3. M. S. Liu, L. A. Bursil, S. Prawer, and K. W. Nugent, “Temperature dependence of Raman scattering in single crystal GaN films,” Appl. Phys. Lett.74(21), 3125–3127 (1999).
    [CrossRef]
  4. S. C. Lien, “Raman studies of wide band gap semiconductors: GaN, InGaN and SiC,” Master thesis, National Taiwan University, 2007; S. C. Lien, Z. C. Feng, C. H. Kuan, Rusli, W. E. Collins, and W. Lu, “Special temperature dependence of carrier concentration in 4H-SiC,” Taiwan Association of Thin Films and Coating Technologies (TACT) 2007 annual meeting, Proceedings-CD, #10–2, 4-pages.
  5. R. Han, B. Han, M. Zhang, X. Y. Fan, and C. Li, “Temperature-dependent Raman scattering in round pit of 4H-SiC,” Diamond Related Materials20(9), 1282–1286 (2011).
    [CrossRef]
  6. R. Han, B. Han, D. H. Wang, and C. Li, “Temperature dependence of Raman scattering from 4H-SiC with hexagonal defects,” Appl. Phys. Lett.99(1), 011912 (2011).
    [CrossRef]
  7. V. V. Chaldyshev, F. H. Pollak, M. Pophristic, S. P. Gou, and I. Ferguson, “Micro-Raman investigation of the n-dopant distribution in lateral epitaxial overgrow GaN/sapphire,” J. Electron. Mater.31(6), 631–634 (2002).
    [CrossRef]
  8. R. M. Wang, G. D. Chen, J. Y. Lin, and H. X. Jiang, “Comparative analysis of temperature-dependent Raman spectra of GaN and GaN/Mg films,” Front. Phys. China1(1), 112–116 (2006).
    [CrossRef]
  9. J. C. Burton, L. Sun, M. Pophristic, S. J. Lukacs, F. H. Long, Z. C. Feng, and I. T. Ferguson, “Spatial characterization of doped SiC wafers by Raman spectroscopy,” J. Appl. Phys.84(11), 6268–6273 (1998).
    [CrossRef]
  10. H. Harima, T. Inoue, S. Nakashima, K. Furukawa, and M. Taneya, “Electronic properties in p-type GaN studied by Raman scattering,” Appl. Phys. Lett.73(14), 2000–2002 (1998).
    [CrossRef]
  11. R. R. Reeber and K. Wang, “Lattice parameters and thermal expansion of important semiconductors and their substrates,” Mat. Res. Soc. Symp. 622, T6.35.1- T6.35.6 (2000).
    [CrossRef]
  12. X. B. Li, Z. Z. Chen, and E. W. Shi, “Effect of doping on the Raman scattering of 6H-SiC,” Physica B405(10), 2423–2426 (2010).
    [CrossRef]
  13. A. J. Sievers and J. B. Page, “Generalized Lyddane-Sachs-Teller relation and disordered solids,” Phys. Rev. B Condens. Matter41(6), 3455–3459 (1990).
    [CrossRef] [PubMed]

2011

R. Han, B. Han, M. Zhang, X. Y. Fan, and C. Li, “Temperature-dependent Raman scattering in round pit of 4H-SiC,” Diamond Related Materials20(9), 1282–1286 (2011).
[CrossRef]

R. Han, B. Han, D. H. Wang, and C. Li, “Temperature dependence of Raman scattering from 4H-SiC with hexagonal defects,” Appl. Phys. Lett.99(1), 011912 (2011).
[CrossRef]

2010

X. B. Li, Z. Z. Chen, and E. W. Shi, “Effect of doping on the Raman scattering of 6H-SiC,” Physica B405(10), 2423–2426 (2010).
[CrossRef]

2006

R. M. Wang, G. D. Chen, J. Y. Lin, and H. X. Jiang, “Comparative analysis of temperature-dependent Raman spectra of GaN and GaN/Mg films,” Front. Phys. China1(1), 112–116 (2006).
[CrossRef]

2002

V. V. Chaldyshev, F. H. Pollak, M. Pophristic, S. P. Gou, and I. Ferguson, “Micro-Raman investigation of the n-dopant distribution in lateral epitaxial overgrow GaN/sapphire,” J. Electron. Mater.31(6), 631–634 (2002).
[CrossRef]

2000

W. S. Li, Z. X. Shen, Z. C. Feng, and S. J. Chua, “Temperature dependence of Raman scattering in hexagonal gallium nitride films,” J. Appl. Phys.87(7), 3332–3337 (2000).
[CrossRef]

1999

M. S. Liu, L. A. Bursil, S. Prawer, and K. W. Nugent, “Temperature dependence of Raman scattering in single crystal GaN films,” Appl. Phys. Lett.74(21), 3125–3127 (1999).
[CrossRef]

1998

J. C. Burton, L. Sun, M. Pophristic, S. J. Lukacs, F. H. Long, Z. C. Feng, and I. T. Ferguson, “Spatial characterization of doped SiC wafers by Raman spectroscopy,” J. Appl. Phys.84(11), 6268–6273 (1998).
[CrossRef]

H. Harima, T. Inoue, S. Nakashima, K. Furukawa, and M. Taneya, “Electronic properties in p-type GaN studied by Raman scattering,” Appl. Phys. Lett.73(14), 2000–2002 (1998).
[CrossRef]

1997

S. Nakashima and H. Harima, “Raman investigation of SiC polytypes,” Phys. Status Solidi162(1), 39–64 (1997) (a).
[CrossRef]

1990

A. J. Sievers and J. B. Page, “Generalized Lyddane-Sachs-Teller relation and disordered solids,” Phys. Rev. B Condens. Matter41(6), 3455–3459 (1990).
[CrossRef] [PubMed]

Bursil, L. A.

M. S. Liu, L. A. Bursil, S. Prawer, and K. W. Nugent, “Temperature dependence of Raman scattering in single crystal GaN films,” Appl. Phys. Lett.74(21), 3125–3127 (1999).
[CrossRef]

Burton, J. C.

J. C. Burton, L. Sun, M. Pophristic, S. J. Lukacs, F. H. Long, Z. C. Feng, and I. T. Ferguson, “Spatial characterization of doped SiC wafers by Raman spectroscopy,” J. Appl. Phys.84(11), 6268–6273 (1998).
[CrossRef]

Chaldyshev, V. V.

V. V. Chaldyshev, F. H. Pollak, M. Pophristic, S. P. Gou, and I. Ferguson, “Micro-Raman investigation of the n-dopant distribution in lateral epitaxial overgrow GaN/sapphire,” J. Electron. Mater.31(6), 631–634 (2002).
[CrossRef]

Chen, G. D.

R. M. Wang, G. D. Chen, J. Y. Lin, and H. X. Jiang, “Comparative analysis of temperature-dependent Raman spectra of GaN and GaN/Mg films,” Front. Phys. China1(1), 112–116 (2006).
[CrossRef]

Chen, Z. Z.

X. B. Li, Z. Z. Chen, and E. W. Shi, “Effect of doping on the Raman scattering of 6H-SiC,” Physica B405(10), 2423–2426 (2010).
[CrossRef]

Chua, S. J.

W. S. Li, Z. X. Shen, Z. C. Feng, and S. J. Chua, “Temperature dependence of Raman scattering in hexagonal gallium nitride films,” J. Appl. Phys.87(7), 3332–3337 (2000).
[CrossRef]

Fan, X. Y.

R. Han, B. Han, M. Zhang, X. Y. Fan, and C. Li, “Temperature-dependent Raman scattering in round pit of 4H-SiC,” Diamond Related Materials20(9), 1282–1286 (2011).
[CrossRef]

Feng, Z. C.

W. S. Li, Z. X. Shen, Z. C. Feng, and S. J. Chua, “Temperature dependence of Raman scattering in hexagonal gallium nitride films,” J. Appl. Phys.87(7), 3332–3337 (2000).
[CrossRef]

J. C. Burton, L. Sun, M. Pophristic, S. J. Lukacs, F. H. Long, Z. C. Feng, and I. T. Ferguson, “Spatial characterization of doped SiC wafers by Raman spectroscopy,” J. Appl. Phys.84(11), 6268–6273 (1998).
[CrossRef]

Ferguson, I.

V. V. Chaldyshev, F. H. Pollak, M. Pophristic, S. P. Gou, and I. Ferguson, “Micro-Raman investigation of the n-dopant distribution in lateral epitaxial overgrow GaN/sapphire,” J. Electron. Mater.31(6), 631–634 (2002).
[CrossRef]

Ferguson, I. T.

J. C. Burton, L. Sun, M. Pophristic, S. J. Lukacs, F. H. Long, Z. C. Feng, and I. T. Ferguson, “Spatial characterization of doped SiC wafers by Raman spectroscopy,” J. Appl. Phys.84(11), 6268–6273 (1998).
[CrossRef]

Furukawa, K.

H. Harima, T. Inoue, S. Nakashima, K. Furukawa, and M. Taneya, “Electronic properties in p-type GaN studied by Raman scattering,” Appl. Phys. Lett.73(14), 2000–2002 (1998).
[CrossRef]

Gou, S. P.

V. V. Chaldyshev, F. H. Pollak, M. Pophristic, S. P. Gou, and I. Ferguson, “Micro-Raman investigation of the n-dopant distribution in lateral epitaxial overgrow GaN/sapphire,” J. Electron. Mater.31(6), 631–634 (2002).
[CrossRef]

Han, B.

R. Han, B. Han, D. H. Wang, and C. Li, “Temperature dependence of Raman scattering from 4H-SiC with hexagonal defects,” Appl. Phys. Lett.99(1), 011912 (2011).
[CrossRef]

R. Han, B. Han, M. Zhang, X. Y. Fan, and C. Li, “Temperature-dependent Raman scattering in round pit of 4H-SiC,” Diamond Related Materials20(9), 1282–1286 (2011).
[CrossRef]

Han, R.

R. Han, B. Han, M. Zhang, X. Y. Fan, and C. Li, “Temperature-dependent Raman scattering in round pit of 4H-SiC,” Diamond Related Materials20(9), 1282–1286 (2011).
[CrossRef]

R. Han, B. Han, D. H. Wang, and C. Li, “Temperature dependence of Raman scattering from 4H-SiC with hexagonal defects,” Appl. Phys. Lett.99(1), 011912 (2011).
[CrossRef]

Harima, H.

H. Harima, T. Inoue, S. Nakashima, K. Furukawa, and M. Taneya, “Electronic properties in p-type GaN studied by Raman scattering,” Appl. Phys. Lett.73(14), 2000–2002 (1998).
[CrossRef]

S. Nakashima and H. Harima, “Raman investigation of SiC polytypes,” Phys. Status Solidi162(1), 39–64 (1997) (a).
[CrossRef]

Inoue, T.

H. Harima, T. Inoue, S. Nakashima, K. Furukawa, and M. Taneya, “Electronic properties in p-type GaN studied by Raman scattering,” Appl. Phys. Lett.73(14), 2000–2002 (1998).
[CrossRef]

Jiang, H. X.

R. M. Wang, G. D. Chen, J. Y. Lin, and H. X. Jiang, “Comparative analysis of temperature-dependent Raman spectra of GaN and GaN/Mg films,” Front. Phys. China1(1), 112–116 (2006).
[CrossRef]

Li, C.

R. Han, B. Han, M. Zhang, X. Y. Fan, and C. Li, “Temperature-dependent Raman scattering in round pit of 4H-SiC,” Diamond Related Materials20(9), 1282–1286 (2011).
[CrossRef]

R. Han, B. Han, D. H. Wang, and C. Li, “Temperature dependence of Raman scattering from 4H-SiC with hexagonal defects,” Appl. Phys. Lett.99(1), 011912 (2011).
[CrossRef]

Li, W. S.

W. S. Li, Z. X. Shen, Z. C. Feng, and S. J. Chua, “Temperature dependence of Raman scattering in hexagonal gallium nitride films,” J. Appl. Phys.87(7), 3332–3337 (2000).
[CrossRef]

Li, X. B.

X. B. Li, Z. Z. Chen, and E. W. Shi, “Effect of doping on the Raman scattering of 6H-SiC,” Physica B405(10), 2423–2426 (2010).
[CrossRef]

Lin, J. Y.

R. M. Wang, G. D. Chen, J. Y. Lin, and H. X. Jiang, “Comparative analysis of temperature-dependent Raman spectra of GaN and GaN/Mg films,” Front. Phys. China1(1), 112–116 (2006).
[CrossRef]

Liu, M. S.

M. S. Liu, L. A. Bursil, S. Prawer, and K. W. Nugent, “Temperature dependence of Raman scattering in single crystal GaN films,” Appl. Phys. Lett.74(21), 3125–3127 (1999).
[CrossRef]

Long, F. H.

J. C. Burton, L. Sun, M. Pophristic, S. J. Lukacs, F. H. Long, Z. C. Feng, and I. T. Ferguson, “Spatial characterization of doped SiC wafers by Raman spectroscopy,” J. Appl. Phys.84(11), 6268–6273 (1998).
[CrossRef]

Lukacs, S. J.

J. C. Burton, L. Sun, M. Pophristic, S. J. Lukacs, F. H. Long, Z. C. Feng, and I. T. Ferguson, “Spatial characterization of doped SiC wafers by Raman spectroscopy,” J. Appl. Phys.84(11), 6268–6273 (1998).
[CrossRef]

Nakashima, S.

H. Harima, T. Inoue, S. Nakashima, K. Furukawa, and M. Taneya, “Electronic properties in p-type GaN studied by Raman scattering,” Appl. Phys. Lett.73(14), 2000–2002 (1998).
[CrossRef]

S. Nakashima and H. Harima, “Raman investigation of SiC polytypes,” Phys. Status Solidi162(1), 39–64 (1997) (a).
[CrossRef]

Nugent, K. W.

M. S. Liu, L. A. Bursil, S. Prawer, and K. W. Nugent, “Temperature dependence of Raman scattering in single crystal GaN films,” Appl. Phys. Lett.74(21), 3125–3127 (1999).
[CrossRef]

Page, J. B.

A. J. Sievers and J. B. Page, “Generalized Lyddane-Sachs-Teller relation and disordered solids,” Phys. Rev. B Condens. Matter41(6), 3455–3459 (1990).
[CrossRef] [PubMed]

Pollak, F. H.

V. V. Chaldyshev, F. H. Pollak, M. Pophristic, S. P. Gou, and I. Ferguson, “Micro-Raman investigation of the n-dopant distribution in lateral epitaxial overgrow GaN/sapphire,” J. Electron. Mater.31(6), 631–634 (2002).
[CrossRef]

Pophristic, M.

V. V. Chaldyshev, F. H. Pollak, M. Pophristic, S. P. Gou, and I. Ferguson, “Micro-Raman investigation of the n-dopant distribution in lateral epitaxial overgrow GaN/sapphire,” J. Electron. Mater.31(6), 631–634 (2002).
[CrossRef]

J. C. Burton, L. Sun, M. Pophristic, S. J. Lukacs, F. H. Long, Z. C. Feng, and I. T. Ferguson, “Spatial characterization of doped SiC wafers by Raman spectroscopy,” J. Appl. Phys.84(11), 6268–6273 (1998).
[CrossRef]

Prawer, S.

M. S. Liu, L. A. Bursil, S. Prawer, and K. W. Nugent, “Temperature dependence of Raman scattering in single crystal GaN films,” Appl. Phys. Lett.74(21), 3125–3127 (1999).
[CrossRef]

Shen, Z. X.

W. S. Li, Z. X. Shen, Z. C. Feng, and S. J. Chua, “Temperature dependence of Raman scattering in hexagonal gallium nitride films,” J. Appl. Phys.87(7), 3332–3337 (2000).
[CrossRef]

Shi, E. W.

X. B. Li, Z. Z. Chen, and E. W. Shi, “Effect of doping on the Raman scattering of 6H-SiC,” Physica B405(10), 2423–2426 (2010).
[CrossRef]

Sievers, A. J.

A. J. Sievers and J. B. Page, “Generalized Lyddane-Sachs-Teller relation and disordered solids,” Phys. Rev. B Condens. Matter41(6), 3455–3459 (1990).
[CrossRef] [PubMed]

Sun, L.

J. C. Burton, L. Sun, M. Pophristic, S. J. Lukacs, F. H. Long, Z. C. Feng, and I. T. Ferguson, “Spatial characterization of doped SiC wafers by Raman spectroscopy,” J. Appl. Phys.84(11), 6268–6273 (1998).
[CrossRef]

Taneya, M.

H. Harima, T. Inoue, S. Nakashima, K. Furukawa, and M. Taneya, “Electronic properties in p-type GaN studied by Raman scattering,” Appl. Phys. Lett.73(14), 2000–2002 (1998).
[CrossRef]

Wang, D. H.

R. Han, B. Han, D. H. Wang, and C. Li, “Temperature dependence of Raman scattering from 4H-SiC with hexagonal defects,” Appl. Phys. Lett.99(1), 011912 (2011).
[CrossRef]

Wang, R. M.

R. M. Wang, G. D. Chen, J. Y. Lin, and H. X. Jiang, “Comparative analysis of temperature-dependent Raman spectra of GaN and GaN/Mg films,” Front. Phys. China1(1), 112–116 (2006).
[CrossRef]

Zhang, M.

R. Han, B. Han, M. Zhang, X. Y. Fan, and C. Li, “Temperature-dependent Raman scattering in round pit of 4H-SiC,” Diamond Related Materials20(9), 1282–1286 (2011).
[CrossRef]

Appl. Phys. Lett.

M. S. Liu, L. A. Bursil, S. Prawer, and K. W. Nugent, “Temperature dependence of Raman scattering in single crystal GaN films,” Appl. Phys. Lett.74(21), 3125–3127 (1999).
[CrossRef]

R. Han, B. Han, D. H. Wang, and C. Li, “Temperature dependence of Raman scattering from 4H-SiC with hexagonal defects,” Appl. Phys. Lett.99(1), 011912 (2011).
[CrossRef]

H. Harima, T. Inoue, S. Nakashima, K. Furukawa, and M. Taneya, “Electronic properties in p-type GaN studied by Raman scattering,” Appl. Phys. Lett.73(14), 2000–2002 (1998).
[CrossRef]

Diamond Related Materials

R. Han, B. Han, M. Zhang, X. Y. Fan, and C. Li, “Temperature-dependent Raman scattering in round pit of 4H-SiC,” Diamond Related Materials20(9), 1282–1286 (2011).
[CrossRef]

Front. Phys. China

R. M. Wang, G. D. Chen, J. Y. Lin, and H. X. Jiang, “Comparative analysis of temperature-dependent Raman spectra of GaN and GaN/Mg films,” Front. Phys. China1(1), 112–116 (2006).
[CrossRef]

J. Appl. Phys.

J. C. Burton, L. Sun, M. Pophristic, S. J. Lukacs, F. H. Long, Z. C. Feng, and I. T. Ferguson, “Spatial characterization of doped SiC wafers by Raman spectroscopy,” J. Appl. Phys.84(11), 6268–6273 (1998).
[CrossRef]

W. S. Li, Z. X. Shen, Z. C. Feng, and S. J. Chua, “Temperature dependence of Raman scattering in hexagonal gallium nitride films,” J. Appl. Phys.87(7), 3332–3337 (2000).
[CrossRef]

J. Electron. Mater.

V. V. Chaldyshev, F. H. Pollak, M. Pophristic, S. P. Gou, and I. Ferguson, “Micro-Raman investigation of the n-dopant distribution in lateral epitaxial overgrow GaN/sapphire,” J. Electron. Mater.31(6), 631–634 (2002).
[CrossRef]

Phys. Rev. B Condens. Matter

A. J. Sievers and J. B. Page, “Generalized Lyddane-Sachs-Teller relation and disordered solids,” Phys. Rev. B Condens. Matter41(6), 3455–3459 (1990).
[CrossRef] [PubMed]

Phys. Status Solidi

S. Nakashima and H. Harima, “Raman investigation of SiC polytypes,” Phys. Status Solidi162(1), 39–64 (1997) (a).
[CrossRef]

Physica B

X. B. Li, Z. Z. Chen, and E. W. Shi, “Effect of doping on the Raman scattering of 6H-SiC,” Physica B405(10), 2423–2426 (2010).
[CrossRef]

Other

R. R. Reeber and K. Wang, “Lattice parameters and thermal expansion of important semiconductors and their substrates,” Mat. Res. Soc. Symp. 622, T6.35.1- T6.35.6 (2000).
[CrossRef]

S. C. Lien, “Raman studies of wide band gap semiconductors: GaN, InGaN and SiC,” Master thesis, National Taiwan University, 2007; S. C. Lien, Z. C. Feng, C. H. Kuan, Rusli, W. E. Collins, and W. Lu, “Special temperature dependence of carrier concentration in 4H-SiC,” Taiwan Association of Thin Films and Coating Technologies (TACT) 2007 annual meeting, Proceedings-CD, #10–2, 4-pages.

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Figures (7)

Fig. 1
Fig. 1

Raman spectra of three 4H-SiC samples at temperature varying from 80 K to 873 K.

Fig. 2
Fig. 2

A1(LO) (S1) and A1(LO)-like (S2 and S3) mode peak positions of three 4H-SiC at temperature from 80 K to 873 K. Raman peak positions of S1, S2 and S3 at different temperatures denote by rectangle (black), circle (red) and inverted triangle (blue), respectively Solid lines are introduced to show the trend of temperature dependence of Raman peak position.

Fig. 3
Fig. 3

Fitting result of E1(TO) and E2(TO) mode peak positions of three 4H-SiC samples S1, S2 and S3.

Fig. 4
Fig. 4

Fitting results with two-mode method for (a) S1 and (b) S3 at 373 K

Fig. 5
Fig. 5

Temperature dependence of (a) plasma frequency ωp and (b) A1(LO) mode peak position ωs, obtained from the two-modes method. Solid lines are fitting results of three samples, which are introduced to reflect the variation of plasma frequency ωp and A1(LO) mode peak position ωs with respect to temperature.

Fig. 6
Fig. 6

Fitting results with one-mode method for (a) S1 and (b) S3 at 373 K

Fig. 7
Fig. 7

Temperature dependence of (a) plasma frequency ωp (b) experimental results (solid symbol) and calculation (hollow symbol) of LOPC mode peak positions, in one-mode method. Solid and short dot lines are fitting results, which are introduced to show the variation of plasma frequency ωp and Raman peak position with respect to temperature.

Equations (10)

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

ω(T)= Ω 0 + ω (1) (T)+ ω (2) (T),
ω (1) (T)= Ω 0 { exp[ κ 0 T ( α c (t)+2 α a (t) )dt ]1 },
ω (2) (T)= M 1 { 1+ i=1 2 1 e x i 1 }+ M 2 { 1+ j=1 3 [ 1 e y j 1 + 1 ( e y j 1) 2 ] },
I LOPC = d 2 S dωdΩ | A = 16π n 2 V 0 2 n 1 ω 2 4 C 4 ( dα dE )( n ω +1 )AIm( 1 ε ),
A=1+2C ω T 2 Δ [ ω p 2 γ( ω T 2 ω 2 ) ω 2 η( ω 2 + γ 2 ω p 2 ) ]+ C 2 ( ω T 4 Δ( ω L 2 ω T 2 ) ), ×{ ω p 2 [ γ( ω L 2 ω T 2 )+η( ω p 2 2 ω 2 ) ]+ ω 2 η( ω 2 + γ 2 ) }
Δ= ω p 2 γ[ ( ω T 2 ω 2 ) 2 + ( ωη ) 2 ]+ ω 2 η( ω L 2 ω T 2 )( ω 2 + γ 2 ),
ε= ε ( 1+ ω L 2 ω T 2 ω T 2 ω 2 iωη ω p 2 ω( ω+iγ ) ),
ω p 2 = 4πn e 2 ε m* ,
I A 1 (LO) = I 0 ( ω ω s Γ s ) 2 +1 ,
ω LOPC 2 = 1 2 { ( ω p 2 + ω LO 2 )+ [ ( ω p 2 + ω LO 2 ) 2 4 ω p 2 ω TO 2 ] 1 2 },

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