Abstract

A closed-form approximate expression for the optical conductivity of graphene is developed, which generates results with less than 0.8% maximum absolute error for λ>250nm. The expression takes wavelength, temperature, chemical potential, and hopping parameter into account and provides a fast, easy, and reliable alternative to well-known methods that include singular integrals. Numerical results confirm that the effective complex electrical permittivity derived from the optical conductivity successfully represents this one atom thick material in three-dimensional electromagnetic simulations and analyses.

© 2013 Optical Society of America

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  1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
    [CrossRef]
  2. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
    [CrossRef]
  3. F. Schwierz, Nat. Nanotechnol. 5, 487 (2010).
    [CrossRef]
  4. V. V. Cheianov, V. Falko, and B. L. Altshuler, Science 315, 1252 (2007).
    [CrossRef]
  5. L. Wu, H. S. Chu, W. S. Koh, and E. P. Li, Opt. Express 18, 14395 (2010).
    [CrossRef]
  6. A. Vakil and N. Engheta, Science 332, 1291 (2011).
    [CrossRef]
  7. F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, Nat. Photonics 4, 611 (2010).
    [CrossRef]
  8. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Phys. Rev. Lett. 96, 256802 (2006).
    [CrossRef]
  9. V. P. Gusynin and S. G. Sharapov, Phys. Rev. B 73, 245411 (2006).
    [CrossRef]
  10. N. M. R. Peres, F. Guinea, and A. H. C. Neto, Phys. Rev. B 73, 125411 (2006).
    [CrossRef]
  11. B. Wunsch, T. Stauber, F. Sols, and F. Guinea, New J. Phys. 8, 318 (2006).
    [CrossRef]
  12. E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 (2007).
    [CrossRef]
  13. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, J. Phys. Condens. Matter 19, 026222 (2007).
    [CrossRef]
  14. T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78, 085432 (2008).
    [CrossRef]
  15. Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, Nat. Phys. 4, 532 (2008).
    [CrossRef]
  16. A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, Phys. Rev. Lett. 100, 117401 (2008).
    [CrossRef]
  17. M. Bruna and S. Borini, Appl. Phys. Lett. 94, 031901 (2009).
    [CrossRef]
  18. E. Simsek, J. Liu, and Q. H. Liu, IEEE Trans. Antennas Propag. 54, 1742 (2006).
    [CrossRef]
  19. S. Chen, Q. Wu, C. Mishra, J. Kang, H. Zhang, K. Cho, W. Cai, A. A. Balandin, and R. S. Ruoff, Nat. Mater. 11, 203 (2012).
    [CrossRef]

2012 (1)

S. Chen, Q. Wu, C. Mishra, J. Kang, H. Zhang, K. Cho, W. Cai, A. A. Balandin, and R. S. Ruoff, Nat. Mater. 11, 203 (2012).
[CrossRef]

2011 (2)

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[CrossRef]

A. Vakil and N. Engheta, Science 332, 1291 (2011).
[CrossRef]

2010 (3)

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, Nat. Photonics 4, 611 (2010).
[CrossRef]

F. Schwierz, Nat. Nanotechnol. 5, 487 (2010).
[CrossRef]

L. Wu, H. S. Chu, W. S. Koh, and E. P. Li, Opt. Express 18, 14395 (2010).
[CrossRef]

2009 (1)

M. Bruna and S. Borini, Appl. Phys. Lett. 94, 031901 (2009).
[CrossRef]

2008 (3)

T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78, 085432 (2008).
[CrossRef]

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, Nat. Phys. 4, 532 (2008).
[CrossRef]

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef]

2007 (3)

E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 (2007).
[CrossRef]

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, J. Phys. Condens. Matter 19, 026222 (2007).
[CrossRef]

V. V. Cheianov, V. Falko, and B. L. Altshuler, Science 315, 1252 (2007).
[CrossRef]

2006 (5)

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Phys. Rev. Lett. 96, 256802 (2006).
[CrossRef]

V. P. Gusynin and S. G. Sharapov, Phys. Rev. B 73, 245411 (2006).
[CrossRef]

N. M. R. Peres, F. Guinea, and A. H. C. Neto, Phys. Rev. B 73, 125411 (2006).
[CrossRef]

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, New J. Phys. 8, 318 (2006).
[CrossRef]

E. Simsek, J. Liu, and Q. H. Liu, IEEE Trans. Antennas Propag. 54, 1742 (2006).
[CrossRef]

2004 (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Altshuler, B. L.

V. V. Cheianov, V. Falko, and B. L. Altshuler, Science 315, 1252 (2007).
[CrossRef]

Balandin, A. A.

S. Chen, Q. Wu, C. Mishra, J. Kang, H. Zhang, K. Cho, W. Cai, A. A. Balandin, and R. S. Ruoff, Nat. Mater. 11, 203 (2012).
[CrossRef]

Basov, D. N.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, Nat. Phys. 4, 532 (2008).
[CrossRef]

Bonaccorso, F.

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, Nat. Photonics 4, 611 (2010).
[CrossRef]

Borini, S.

M. Bruna and S. Borini, Appl. Phys. Lett. 94, 031901 (2009).
[CrossRef]

Bruna, M.

M. Bruna and S. Borini, Appl. Phys. Lett. 94, 031901 (2009).
[CrossRef]

Cai, W.

S. Chen, Q. Wu, C. Mishra, J. Kang, H. Zhang, K. Cho, W. Cai, A. A. Balandin, and R. S. Ruoff, Nat. Mater. 11, 203 (2012).
[CrossRef]

Carbone, F.

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef]

Carbotte, J. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, J. Phys. Condens. Matter 19, 026222 (2007).
[CrossRef]

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Phys. Rev. Lett. 96, 256802 (2006).
[CrossRef]

Cheianov, V. V.

V. V. Cheianov, V. Falko, and B. L. Altshuler, Science 315, 1252 (2007).
[CrossRef]

Chen, S.

S. Chen, Q. Wu, C. Mishra, J. Kang, H. Zhang, K. Cho, W. Cai, A. A. Balandin, and R. S. Ruoff, Nat. Mater. 11, 203 (2012).
[CrossRef]

Cho, K.

S. Chen, Q. Wu, C. Mishra, J. Kang, H. Zhang, K. Cho, W. Cai, A. A. Balandin, and R. S. Ruoff, Nat. Mater. 11, 203 (2012).
[CrossRef]

Chu, H. S.

Das Sarma, S.

E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 (2007).
[CrossRef]

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Engheta, N.

A. Vakil and N. Engheta, Science 332, 1291 (2011).
[CrossRef]

Falko, V.

V. V. Cheianov, V. Falko, and B. L. Altshuler, Science 315, 1252 (2007).
[CrossRef]

Ferrari, A.

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, Nat. Photonics 4, 611 (2010).
[CrossRef]

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Geim, A. K.

T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78, 085432 (2008).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Geng, B.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[CrossRef]

Grigorieva, I. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Guinea, F.

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, New J. Phys. 8, 318 (2006).
[CrossRef]

N. M. R. Peres, F. Guinea, and A. H. C. Neto, Phys. Rev. B 73, 125411 (2006).
[CrossRef]

Gusynin, V. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, J. Phys. Condens. Matter 19, 026222 (2007).
[CrossRef]

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Phys. Rev. Lett. 96, 256802 (2006).
[CrossRef]

V. P. Gusynin and S. G. Sharapov, Phys. Rev. B 73, 245411 (2006).
[CrossRef]

Hao, Z.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, Nat. Phys. 4, 532 (2008).
[CrossRef]

Hasan, T.

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, Nat. Photonics 4, 611 (2010).
[CrossRef]

Henriksen, E. A.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, Nat. Phys. 4, 532 (2008).
[CrossRef]

Hwang, E. H.

E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 (2007).
[CrossRef]

Jiang, D.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Jiang, Z.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, Nat. Phys. 4, 532 (2008).
[CrossRef]

Ju, L.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[CrossRef]

Kang, J.

S. Chen, Q. Wu, C. Mishra, J. Kang, H. Zhang, K. Cho, W. Cai, A. A. Balandin, and R. S. Ruoff, Nat. Mater. 11, 203 (2012).
[CrossRef]

Kim, P.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, Nat. Phys. 4, 532 (2008).
[CrossRef]

Koh, W. S.

Kuzmenko, A. B.

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef]

Li, E. P.

Li, Z. Q.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, Nat. Phys. 4, 532 (2008).
[CrossRef]

Liu, J.

E. Simsek, J. Liu, and Q. H. Liu, IEEE Trans. Antennas Propag. 54, 1742 (2006).
[CrossRef]

Liu, M.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[CrossRef]

Liu, Q. H.

E. Simsek, J. Liu, and Q. H. Liu, IEEE Trans. Antennas Propag. 54, 1742 (2006).
[CrossRef]

Martin, M. C.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, Nat. Phys. 4, 532 (2008).
[CrossRef]

Mishra, C.

S. Chen, Q. Wu, C. Mishra, J. Kang, H. Zhang, K. Cho, W. Cai, A. A. Balandin, and R. S. Ruoff, Nat. Mater. 11, 203 (2012).
[CrossRef]

Morozov, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Neto, A. H. C.

N. M. R. Peres, F. Guinea, and A. H. C. Neto, Phys. Rev. B 73, 125411 (2006).
[CrossRef]

Novoselov, K. S.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Peres, N. M. R.

T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78, 085432 (2008).
[CrossRef]

N. M. R. Peres, F. Guinea, and A. H. C. Neto, Phys. Rev. B 73, 125411 (2006).
[CrossRef]

Ruoff, R. S.

S. Chen, Q. Wu, C. Mishra, J. Kang, H. Zhang, K. Cho, W. Cai, A. A. Balandin, and R. S. Ruoff, Nat. Mater. 11, 203 (2012).
[CrossRef]

Schwierz, F.

F. Schwierz, Nat. Nanotechnol. 5, 487 (2010).
[CrossRef]

Sharapov, S. G.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, J. Phys. Condens. Matter 19, 026222 (2007).
[CrossRef]

V. P. Gusynin and S. G. Sharapov, Phys. Rev. B 73, 245411 (2006).
[CrossRef]

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Phys. Rev. Lett. 96, 256802 (2006).
[CrossRef]

Simsek, E.

E. Simsek, J. Liu, and Q. H. Liu, IEEE Trans. Antennas Propag. 54, 1742 (2006).
[CrossRef]

Sols, F.

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, New J. Phys. 8, 318 (2006).
[CrossRef]

Stauber, T.

T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78, 085432 (2008).
[CrossRef]

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, New J. Phys. 8, 318 (2006).
[CrossRef]

Stormer, H. L.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, Nat. Phys. 4, 532 (2008).
[CrossRef]

Sun, Z.

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, Nat. Photonics 4, 611 (2010).
[CrossRef]

Ulin-Avila, E.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[CrossRef]

Vakil, A.

A. Vakil and N. Engheta, Science 332, 1291 (2011).
[CrossRef]

van der Marel, D.

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef]

van Heumen, E.

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef]

Wang, F.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[CrossRef]

Wu, L.

Wu, Q.

S. Chen, Q. Wu, C. Mishra, J. Kang, H. Zhang, K. Cho, W. Cai, A. A. Balandin, and R. S. Ruoff, Nat. Mater. 11, 203 (2012).
[CrossRef]

Wunsch, B.

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, New J. Phys. 8, 318 (2006).
[CrossRef]

Yin, X.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[CrossRef]

Zentgraf, T.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[CrossRef]

Zhang, H.

S. Chen, Q. Wu, C. Mishra, J. Kang, H. Zhang, K. Cho, W. Cai, A. A. Balandin, and R. S. Ruoff, Nat. Mater. 11, 203 (2012).
[CrossRef]

Zhang, X.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[CrossRef]

Zhang, Y.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Appl. Phys. Lett. (1)

M. Bruna and S. Borini, Appl. Phys. Lett. 94, 031901 (2009).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

E. Simsek, J. Liu, and Q. H. Liu, IEEE Trans. Antennas Propag. 54, 1742 (2006).
[CrossRef]

J. Phys. Condens. Matter (1)

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, J. Phys. Condens. Matter 19, 026222 (2007).
[CrossRef]

Nat. Mater. (1)

S. Chen, Q. Wu, C. Mishra, J. Kang, H. Zhang, K. Cho, W. Cai, A. A. Balandin, and R. S. Ruoff, Nat. Mater. 11, 203 (2012).
[CrossRef]

Nat. Nanotechnol. (1)

F. Schwierz, Nat. Nanotechnol. 5, 487 (2010).
[CrossRef]

Nat. Photonics (1)

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, Nat. Photonics 4, 611 (2010).
[CrossRef]

Nat. Phys. (1)

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, Nat. Phys. 4, 532 (2008).
[CrossRef]

Nature (1)

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature 474, 64 (2011).
[CrossRef]

New J. Phys. (1)

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, New J. Phys. 8, 318 (2006).
[CrossRef]

Opt. Express (1)

Phys. Rev. B (4)

E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 (2007).
[CrossRef]

V. P. Gusynin and S. G. Sharapov, Phys. Rev. B 73, 245411 (2006).
[CrossRef]

N. M. R. Peres, F. Guinea, and A. H. C. Neto, Phys. Rev. B 73, 125411 (2006).
[CrossRef]

T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78, 085432 (2008).
[CrossRef]

Phys. Rev. Lett. (2)

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, Phys. Rev. Lett. 100, 117401 (2008).
[CrossRef]

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Phys. Rev. Lett. 96, 256802 (2006).
[CrossRef]

Science (3)

A. Vakil and N. Engheta, Science 332, 1291 (2011).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

V. V. Cheianov, V. Falko, and B. L. Altshuler, Science 315, 1252 (2007).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

(a) σr and (b) σi for μc=0eV (blue), μc=0.2eV (green), and μc=0.4eV (red) as a function of wavelength at T=45K. Solid lines are generated using Eqs. (1)–(7) and approximate results shown by markers are obtained using Eqs. (8)–(10). (c) and (d) show real and imaginary parts, respectively, in the visible region as a function of chemical potential.

Fig. 2.
Fig. 2.

σi calculated with Eq. (7) (blue dashed line), which has a singularity at ω=2μc, and with Eq. (9) (red solid line). ω is taken as 3 eV.

Fig. 3.
Fig. 3.

Transmissivity for normal incident as a function of λ for T=300K and μc=0.2eV where the top medium is vacuum and the bottom medium is either vacuum or glass. Solid lines and circles represent 2D and 3D approach results, respectively.

Equations (10)

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

σr=σ0[18(ω/t)2π12(3)]ψrκ,
ψr=tanh(ω+2μc4kBT)+tanh(ω2μc4kBT),
κ={1F(ω/2t)K(2ω/tF(ω/2t)),ω<2t12ω/tK(F(ω/2t)2ω/t),ω2t,
F(x)=(1+x)20.25(x21)2,
K(m)=01((1x2)(1mx2))1/2dx
σi=σ0π{4μcω[12(μc3t)2][1(ω6t)2]ϒ},
ϒ=log|ω+2μc||ω2μc|
κ={4.69362.897tanh(|ω2t|0.546),ω<2t4.6936exp(0.7714|ω2t|0.4727),ω2t,
ϒ=log(|ω+2μc|ψi|ω2μc|+ψi),
ψi=2kBT{tanh(|ω+2μc|4kBT)tanh(|ω2μc|4kBT)}.

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