Critical-Field Ratio <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mfrac><mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>c</mi><mn>3</mn><mn></mn></mrow></msub></mrow></mrow><mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>c</mi><mn>2</mn><mn></mn></mrow></msub></mrow></mrow></mfrac></math></span> for Pure Superconductors outside the Ginzburg-Landau Region. II. <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi><mi>≃</mi><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math></span>

Chia-Ren Hu; Victor Korenman

doi:10.1103/PhysRev.185.672

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Critical-Field Ratio $\frac{H_{c 3}}{H_{c 2}}$ for Pure Superconductors outside the Ginzburg-Landau Region. II. $T ≃ T_{c}$

Chia-Ren Hu^*

Victor Korenman

Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742 and Department of Physics, University of Illinois, Urbana, Illinois 61803

Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742

^*Present address: Department of Physics, University of Southern California, Los Angeles, Calif. 90007.

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Phys. Rev. 185, 672 – Published 10 September, 1969

DOI: https://doi.org/10.1103/PhysRev.185.672

Abstract

In Paper I we found the appropriate generalization of Gor'kov's linearized gap equation for a pure, semiinfinite weak-coupling superconductor in a magnetic field, separated from vacuum or an insulator by a specularly reflecting surface. In that paper we used the gap equation to study the surface-nucleation critical field $H_{c 3}$ at $T ≃ 0 °$ K. Here we study the region $T ≃ T_{c}$ and find the first three nontrivial terms in an expansion of the ratio $\frac{H_{c 3} (T)}{H_{c 2} (T)}$ to be $1.695 [1 + 0.614 (1 - t) - 0.577 {(1 - t)}^{\frac{3}{2}}]$ , where $t \equiv \frac{T}{T_{c}}$ . The term linear in $1 - t$ has been found previously, but the last term is new. For $T$ close enough to $T_{c}$ we show that the system is accurately described by the linearized Ginzburg-Landau equation with the usual boundary condition and thus regain the results of Saint-James and de Gennes. At lower temperatures the pair wave function has a slowly varying component which satisfies a finite-order differential equation and a surface component which does not. An analysis of the surface component gives an effective boundary condition on the slowly varying part; from this condition the field $H_{c 3}$ is derived. In combination with the results of Paper I we propose an interpolation formula for the entire temperature range below $T_{c}$ . A comparison with the available experimental data is encouraging.

References (35)

C.-R. Hu and V. Korenman, Phys. Rev. 178, 684 (1969)
L. P. Gor'kov', Zh. Eksperim. i Teor. Fiz. 36, 1918 (1959) [English Transl.: Soviet Phys.—JETP 9, 1364 (1959)]
V. L. Ginzburg and L. D. Landau, Zh. Eksperim. i Teor. Fiz. 20, 1064 (1950)
D. Saint-James and P. G. de Gennes, Phys. Letters 7, 306 (1963)
L. P. Gor'kov', Zh. Eksperim. i Teor. Fiz. 37, 1407 (1959) [English trans].: Soviet Phys.—JETP 10, 998 (1960)]
L. Tewordt, Z. Physik. 184, 319 (1965) E. Helfand and N. R. Werthamer, Phys. Rev. 147, 288 (1966)
C.-R. Hu, thesis, University of Maryland, 1968 (unpublished)
A. A. Abrikosov, L. P. Gor'kov', and I. E. Dzyaloshinsky, R. A. SilvermanMethods of Quantum Field Theory in Statistical Physics, translated by (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1963)
[7]
Omitted endnote
[16] of 1
Omitted endnote
[4] Chap. VI of [7]
[6]
G. Lüders, Z. Physik 202, 8 (1967)
[15] G. Lüders, Theoretische Physikalisches Seminar, Gottingen University, 1967 (unpublished)
G. Lüders (private communication) Lüders in [15] [7] Ebneth and Tewordt [21]
P. G. de Gennes, Rev. Mod. Phys. 36, 225 (1964) P. G. de GennesSuperconductivity of Metals and Alloys, translated by P. A. Pincus (W. A. Benjamin, Inc., New York, 1966)
E. A. Shapoval {Zh. Eksperim. i. Teor. Fiz. 47, 1007 (1964) [English transl.: (Soviet Phys.—JETP 20, 675 (1965)]}
A. A. Abrikosov, Zh. Eksperim, i Teor. Fiz. 47, 720 (1964) [English transl.: Soviet Phys.—JETP 20, 480 (1965)]
G. Ebneth and L. Tewordt, Z. Physik 185, 421 (1965)
E. Schöler, thesis, Gottingen, 1966 (unpublished) G. Lüders: Z. Naturforsch. 21a, 680 (1966) ibid.21a, 1415 (1966) ibid.21a, 1425 (1966) ibid.21a, 1824 (1966) ibid.22a, 845 (1967) [15]
K.-D. Usadel and M. Schmidt (unpublished)
L. P. Gor'kov', Zh. Eksperim, i Teor. Fiz. 37, 833 (1959) [English transl.: Soviet Phys.—JETP 10, 593 (1960)]
E. Helfand and N. R. Werthamer, [6]
G. Eilenberger and V. Ambegaokar, Phys. Rev. 158, 332 (1967) E. D. Yorke and A. Bardasis, ibid. 159, 344 (1967)
G. W. Webb, Solid State Commun. 6, 33 (1968)
J. E. Ostenson and D. K. Finnemore, Phys. Rev. Letters 22, 188 (1969)
G. Fischer (private communication)
D. K. Finnemore, T. F. Stromberg, and C. A. Swenson, Phys. Rev. 149, 231 (1966)
N. R. Werthamer and W. L. McMillan, Phys. Rev. 158, 415 (1967)
P. C. Hohenberg and N. R. Werthamer, Phys. Rev. 153, 493 (1967)
D. E. Farrell, B. S. Chandrasekhar, and S. Huang, Phys. Rev. 176, 562 (1968)
[28]
G. Fischer, Phys. Rev. Letters 20, 268 (1968)

[c1] C.-R. Hu and V. Korenman, Phys. Rev. 178, 684 (1969)

[c2] L. P. Gor'kov', Zh. Eksperim. i Teor. Fiz. 36, 1918 (1959) [English Transl.: Soviet Phys.—JETP 9, 1364 (1959)]

[c3] V. L. Ginzburg and L. D. Landau, Zh. Eksperim. i Teor. Fiz. 20, 1064 (1950)

[c4] D. Saint-James and P. G. de Gennes, Phys. Letters 7, 306 (1963)

[c5] L. P. Gor'kov', Zh. Eksperim. i Teor. Fiz. 37, 1407 (1959) [English trans].: Soviet Phys.—JETP 10, 998 (1960)]

[c6] L. Tewordt, Z. Physik. 184, 319 (1965) E. Helfand and N. R. Werthamer, Phys. Rev. 147, 288 (1966)

[c7] C.-R. Hu, thesis, University of Maryland, 1968 (unpublished)

[c8] A. A. Abrikosov, L. P. Gor'kov', and I. E. Dzyaloshinsky, R. A. SilvermanMethods of Quantum Field Theory in Statistical Physics, translated by (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1963)

[c9] [7]

[c10] Omitted endnote

[c11] [16] of 1

[c12] Omitted endnote

[c13] [4] Chap. VI of [7]

[c14] [6]

[c15] G. Lüders, Z. Physik 202, 8 (1967)

[c16] [15] G. Lüders, Theoretische Physikalisches Seminar, Gottingen University, 1967 (unpublished)

[c17] G. Lüders (private communication) Lüders in [15] [7] Ebneth and Tewordt [21]

[c18] P. G. de Gennes, Rev. Mod. Phys. 36, 225 (1964) P. G. de GennesSuperconductivity of Metals and Alloys, translated by P. A. Pincus (W. A. Benjamin, Inc., New York, 1966)

[c19] E. A. Shapoval {Zh. Eksperim. i. Teor. Fiz. 47, 1007 (1964) [English transl.: (Soviet Phys.—JETP 20, 675 (1965)]}

[c20] A. A. Abrikosov, Zh. Eksperim, i Teor. Fiz. 47, 720 (1964) [English transl.: Soviet Phys.—JETP 20, 480 (1965)]

[c21] G. Ebneth and L. Tewordt, Z. Physik 185, 421 (1965)

[c22] E. Schöler, thesis, Gottingen, 1966 (unpublished) G. Lüders: Z. Naturforsch. 21a, 680 (1966) ibid.21a, 1415 (1966) ibid.21a, 1425 (1966) ibid.21a, 1824 (1966) ibid.22a, 845 (1967) [15]

[c23] K.-D. Usadel and M. Schmidt (unpublished)

[c24] L. P. Gor'kov', Zh. Eksperim, i Teor. Fiz. 37, 833 (1959) [English transl.: Soviet Phys.—JETP 10, 593 (1960)]

[c25] E. Helfand and N. R. Werthamer, [6]

[c26] G. Eilenberger and V. Ambegaokar, Phys. Rev. 158, 332 (1967) E. D. Yorke and A. Bardasis, ibid. 159, 344 (1967)

[c27] G. W. Webb, Solid State Commun. 6, 33 (1968)

[c28] J. E. Ostenson and D. K. Finnemore, Phys. Rev. Letters 22, 188 (1969)

[c29] G. Fischer (private communication)

[c30] D. K. Finnemore, T. F. Stromberg, and C. A. Swenson, Phys. Rev. 149, 231 (1966)

[c31] N. R. Werthamer and W. L. McMillan, Phys. Rev. 158, 415 (1967)

[c32] P. C. Hohenberg and N. R. Werthamer, Phys. Rev. 153, 493 (1967)

[c33] D. E. Farrell, B. S. Chandrasekhar, and S. Huang, Phys. Rev. 176, 562 (1968)

[c34] [28]

[c35] G. Fischer, Phys. Rev. Letters 20, 268 (1968)

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Critical-Field Ratio $\frac{H_{c 3}}{H_{c 2}}$ for Pure Superconductors outside the Ginzburg-Landau Region. II. $T ≃ T_{c}$

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Critical-Field Ratio Hc3Hc2 for Pure Superconductors outside the Ginzburg-Landau Region. II. T≃Tc

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References (35)

Critical-Field Ratio $\frac{H_{c 3}}{H_{c 2}}$ for Pure Superconductors outside the Ginzburg-Landau Region. II. $T ≃ T_{c}$