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Angular dependence of the superconducting nucleation field
PHYSICS
Volume 9, number 3
ANGULAR
DEPENDENCE
OF THE
LETTERS
SUPERCONDUCTING
15 April 1964
NUCLEATION
FIELD
M. TINKIIAM * Facultb ties Sciences,
Orsay
Seine et Oise , France
Received 11 March 1954 Saint-James and De Gennes l) have shown that, when the magnetic field parallel to the surface of a (potentially) superconducting material is decreased from a high value, the superconducting state first germinates in a surface layer of thickness - t(T). This germination occurs at a field Hc2 = 1.69 Hc2 = 2.4 K&b; where Hc2 is the usual ‘e/R1 upper critical field of a type II superconductor with the Landau-Ginzburg parameter K and thermodynamic critical field H,b. They also noted that if the magnetic field is normal to the surface, then germination first occurs at Hc2, the same field at which the whole volume can become superconducting in the usual vortex state. These predictions have received striking quantitative confirmation in the experiments reported by Hempstead and Kim 2) and by Tomasch and Joseph 2). The latter authors also report the measured values of the Fig. 1. Comparison of theoretical angulardependence critical nucleation field He as a function of the anfor nucleation field as given by eq. (2) (solid curve) with gle 0 between the field direction and the plane of elcperimentalpoints from ref. 3. Film no. SO-3is a 4.1 the film, a matter not treated theoretically by micron film of lead + 6.5 atomic percent thallium. Foil Saint-James and De Gennes. no. F 25 B is 127 microns thick and contains 10 atomic The purpose of this note is to point out that the percent thallium. He/Ht is the ratio of He to the perpendicular critical field, identifiedwith Hc3. measured values of He are in remarkably good agreement with those predicted by the simple anacrepancy between their film and foil data at large lytical interpolation formula angles is not understood, but the theoretical He cos 8 2 =I, curve well approximates the average behaviour. (I) > Hc,, Given this remarkable agreement, one is led proposed previously 4) for interpolating between to seek an explanation. Qualitatively, one can argue as follows: for fields in excess of Hc2, the parallel and perpendicular critical fields of a volume superconductivity is, by definition of Hc2, thin film. In the present case, however, Hc,, = H,g = 1.69 Hc2 and H,.. = f&2, so that h = H /Hc2 impossible. Since He > Hc2 for all angles (except is given by the universal eouation precisely 9Oo), in essentially all cases we are dealing with only a surface film of superconduct0.35 h2 cos20 + h sin e = 1 . (2) ing material. Generalising the argument of ref. 1, The result of solving (2) to obtain h(B) is shown in we expect the thickness of this layer to be - [(I’) fig. 1, superimposed on the experimental points and hence independent of 0 to a first approximafrom the figure of Tomasch and Joseph. In gention. Thus it is perhaps reasonable that the areral, the agreement is striking, considering that guments used in ref. 4 to obtain (1) for the case no adjustable parameters are involved. The disof a thin film of thickness d d 5 may apply here also. Finally, let us,recall the fundamental dimen* Permanent address: Departmentof Physics, University of California, Berkeley ’ sional argument which leads to (1). The argument
(
217
Volume 9, number 3
PHYSICS
LETTERS
is based on the fact that right at He, the superconducting order parameter is infinitesimal since the transition is assumed to be of second order. Hence the magnetic field is essentially uniform despite the presence of the film. In a thin film of thickness d, the parallel component of the field, H cos 6 = HII, will induce diamagnetic supercurrents J - cH,l d/X2, leading to a kinetic energy density term - A8J2/c8 - HII 2d2/A2. On the other hand, the perpendicular component will set up quantised vortices, as discussed in ref. 4. The requirement of unit fluxoid quantisation leads tc a vortex area ?7R2 - PO/H,,
where (p. is the fluxoia quantum hc/2e, and R is the characteristic linear dimension of the current patterns. Thus R - (cp,/H,)z. Since for fluxoid quantisation J - ccpo/RX2 - c(qq,H$/A2
,
we have a kinetic energy density term - X2J2/c8 The unusual first power dependence - &_&/x2. of the energy on H, is seen to result from the fact that the size of the current loops decreases as HI increases. Since the current patterns associated with HII and HL dre orthogonal, the energies should simply add. Normalising the coefficients of HII2 and HL by means of Hell and HcL, and postulating that the sum of the two energy density terms reach a constant critical value at the transition field He, we are lead to formula (1). As discussed in refs. 4 and 5, for thin films there is considerable experimental support for an interpolation formula of the type (1).
15 April 1964
In making the transcription to the present case, the weakest point in the argument is the need to assume that the effective thickness of the superconducting surface film is nearly constant, having a value - C;(T) at He, independent of 8. However, unless this thickness varies radically with angle, the finite slope (-1.43) of the initial drop of h with increasing 0 should be nearly correct since it is determined by the term in hl, which should be independent of the thickness. Also, h must approach unity when Q- $r. Thus, the region most sensitive to this assumption would appear to be the intermediate angles, near $IJ. This may have some bearing on the divergence of the experimental data on various samples which begins at angles in this range. It would be valuable to have further experimental tests of (2) for bulk superconductors of the second kind. It would also be desirable to extend the theory of Saint-James and De Gennes to the case of oblique fields so that its results could be compared with the simple interpolation scheme presented here. The author acknowledges valuable discussions with Dr. P. G. De Gennes and Mme C. Caroli concerning the content of this paper. References
1) D. Saint-James and P. G. De Gennes, Physics Letters 7 (1963) 306. 2) C. F.Hempateadand f. B.Kim, Phys. Rev. Letters 12 (1964) 145. 3) W. J.Tomaech and A. S. Joseph, Phys. Rev. Letters 12 (1964) 148. 4) M.Tinkham, Phys. Rev. 129 (1963) 2413. 5) M. Tinkham, Revs. Modern Phys., to appear.
*****
SOME
OBSERVATIONS ON STRUCTURAL TRANSFORMATION OF Ag8Se ALLOY FILMS S. K. SHARMA and G. L. MALHOTRA National
Physical
LaboratorJl
of India,
New Delhi
12
Received 23 March 1964 In an X-ray diffraction stud of crystalline silver -selenide (Ag8Se) , Rahlfs li characterised the a-phase of Ag8Se as orthorhombic or monoclinic which underwent a structural transformation to a body centred cubic (B-phase) at 128k 5oC. While studying the diffusion of thin layers of silver intc 218
selenium, Zorll 2) has examined the transmission electron diffraction patterns of the so formed Ag$e alloy and found that a-pha.se of Ag2Se has an orthorhombic structure which persisted up to 140°C, above which the structure was changed to (B-phase) body centred cubic. Chou-Ching Liang
Volume 9, number 3
ANGULAR
DEPENDENCE
OF THE
LETTERS
SUPERCONDUCTING
15 April 1964
NUCLEATION
FIELD
M. TINKIIAM * Facultb ties Sciences,
Orsay
Seine et Oise , France
Received 11 March 1954 Saint-James and De Gennes l) have shown that, when the magnetic field parallel to the surface of a (potentially) superconducting material is decreased from a high value, the superconducting state first germinates in a surface layer of thickness - t(T). This germination occurs at a field Hc2 = 1.69 Hc2 = 2.4 K&b; where Hc2 is the usual ‘e/R1 upper critical field of a type II superconductor with the Landau-Ginzburg parameter K and thermodynamic critical field H,b. They also noted that if the magnetic field is normal to the surface, then germination first occurs at Hc2, the same field at which the whole volume can become superconducting in the usual vortex state. These predictions have received striking quantitative confirmation in the experiments reported by Hempstead and Kim 2) and by Tomasch and Joseph 2). The latter authors also report the measured values of the Fig. 1. Comparison of theoretical angulardependence critical nucleation field He as a function of the anfor nucleation field as given by eq. (2) (solid curve) with gle 0 between the field direction and the plane of elcperimentalpoints from ref. 3. Film no. SO-3is a 4.1 the film, a matter not treated theoretically by micron film of lead + 6.5 atomic percent thallium. Foil Saint-James and De Gennes. no. F 25 B is 127 microns thick and contains 10 atomic The purpose of this note is to point out that the percent thallium. He/Ht is the ratio of He to the perpendicular critical field, identifiedwith Hc3. measured values of He are in remarkably good agreement with those predicted by the simple anacrepancy between their film and foil data at large lytical interpolation formula angles is not understood, but the theoretical He cos 8 2 =I, curve well approximates the average behaviour. (I) > Hc,, Given this remarkable agreement, one is led proposed previously 4) for interpolating between to seek an explanation. Qualitatively, one can argue as follows: for fields in excess of Hc2, the parallel and perpendicular critical fields of a volume superconductivity is, by definition of Hc2, thin film. In the present case, however, Hc,, = H,g = 1.69 Hc2 and H,.. = f&2, so that h = H /Hc2 impossible. Since He > Hc2 for all angles (except is given by the universal eouation precisely 9Oo), in essentially all cases we are dealing with only a surface film of superconduct0.35 h2 cos20 + h sin e = 1 . (2) ing material. Generalising the argument of ref. 1, The result of solving (2) to obtain h(B) is shown in we expect the thickness of this layer to be - [(I’) fig. 1, superimposed on the experimental points and hence independent of 0 to a first approximafrom the figure of Tomasch and Joseph. In gention. Thus it is perhaps reasonable that the areral, the agreement is striking, considering that guments used in ref. 4 to obtain (1) for the case no adjustable parameters are involved. The disof a thin film of thickness d d 5 may apply here also. Finally, let us,recall the fundamental dimen* Permanent address: Departmentof Physics, University of California, Berkeley ’ sional argument which leads to (1). The argument
(
217
Volume 9, number 3
PHYSICS
LETTERS
is based on the fact that right at He, the superconducting order parameter is infinitesimal since the transition is assumed to be of second order. Hence the magnetic field is essentially uniform despite the presence of the film. In a thin film of thickness d, the parallel component of the field, H cos 6 = HII, will induce diamagnetic supercurrents J - cH,l d/X2, leading to a kinetic energy density term - A8J2/c8 - HII 2d2/A2. On the other hand, the perpendicular component will set up quantised vortices, as discussed in ref. 4. The requirement of unit fluxoid quantisation leads tc a vortex area ?7R2 - PO/H,,
where (p. is the fluxoia quantum hc/2e, and R is the characteristic linear dimension of the current patterns. Thus R - (cp,/H,)z. Since for fluxoid quantisation J - ccpo/RX2 - c(qq,H$/A2
,
we have a kinetic energy density term - X2J2/c8 The unusual first power dependence - &_&/x2. of the energy on H, is seen to result from the fact that the size of the current loops decreases as HI increases. Since the current patterns associated with HII and HL dre orthogonal, the energies should simply add. Normalising the coefficients of HII2 and HL by means of Hell and HcL, and postulating that the sum of the two energy density terms reach a constant critical value at the transition field He, we are lead to formula (1). As discussed in refs. 4 and 5, for thin films there is considerable experimental support for an interpolation formula of the type (1).
15 April 1964
In making the transcription to the present case, the weakest point in the argument is the need to assume that the effective thickness of the superconducting surface film is nearly constant, having a value - C;(T) at He, independent of 8. However, unless this thickness varies radically with angle, the finite slope (-1.43) of the initial drop of h with increasing 0 should be nearly correct since it is determined by the term in hl, which should be independent of the thickness. Also, h must approach unity when Q- $r. Thus, the region most sensitive to this assumption would appear to be the intermediate angles, near $IJ. This may have some bearing on the divergence of the experimental data on various samples which begins at angles in this range. It would be valuable to have further experimental tests of (2) for bulk superconductors of the second kind. It would also be desirable to extend the theory of Saint-James and De Gennes to the case of oblique fields so that its results could be compared with the simple interpolation scheme presented here. The author acknowledges valuable discussions with Dr. P. G. De Gennes and Mme C. Caroli concerning the content of this paper. References
1) D. Saint-James and P. G. De Gennes, Physics Letters 7 (1963) 306. 2) C. F.Hempateadand f. B.Kim, Phys. Rev. Letters 12 (1964) 145. 3) W. J.Tomaech and A. S. Joseph, Phys. Rev. Letters 12 (1964) 148. 4) M.Tinkham, Phys. Rev. 129 (1963) 2413. 5) M. Tinkham, Revs. Modern Phys., to appear.
*****
SOME
OBSERVATIONS ON STRUCTURAL TRANSFORMATION OF Ag8Se ALLOY FILMS S. K. SHARMA and G. L. MALHOTRA National
Physical
LaboratorJl
of India,
New Delhi
12
Received 23 March 1964 In an X-ray diffraction stud of crystalline silver -selenide (Ag8Se) , Rahlfs li characterised the a-phase of Ag8Se as orthorhombic or monoclinic which underwent a structural transformation to a body centred cubic (B-phase) at 128k 5oC. While studying the diffusion of thin layers of silver intc 218
selenium, Zorll 2) has examined the transmission electron diffraction patterns of the so formed Ag$e alloy and found that a-pha.se of Ag2Se has an orthorhombic structure which persisted up to 140°C, above which the structure was changed to (B-phase) body centred cubic. Chou-Ching Liang