Superconduttività: il moto flussonico

Superconduttività:

il moto flussonico

Superconduttori di tipo II

T

fase normale

T ^c H

fase Meissner stato misto

H c1 H

B

H c2 H c1 (T)

H c2 (T)

Parziale penetrazione del flusso in forma di quanti di flusso o flussoni.

Due campi critici.

Superconduttori di tipo II

356 10 Superconductivity

relatively large critical parameters B _c2 (0) = 25 T and T _c (B = 0) = 18.7 K.

This is technically important because this compound can easily be processed and used as the material for wires in superconducting magnets. The novel high-T _c superconductors with their extremely high values of T _c and B _c2 will be considered separately in Sect. 10.5.4.

Both the critical field strengths B c1 and B c2 depend on temperature.

They have their largest value at T = 0, and vanish at T c . In Fig. 10.13a, the temperature variation of B _c2 of several type II superconductors is depicted. In Fig. 10.13b, the critical fields B c1 and B c2 , and the thermodynamic critical field B _c,th of an indium-bismuth alloy are shown. Obviously, all types of critical fields exhibit a similar temperature dependence.

0 5 10 15 20

Temperature T / K

0 10 20 30 40 50 60

M ag ne tic fie ld B / T

PbGd

_0,3

Mo

₆

S

₈

PbMo

_6.35

S

₈

Nb

₃

Sn V

₃

Ga NbTi

0 1 2 3 4

Temperature T / K

0 50 100

M ag ne tic fie ld B / m T

B _c1 B _c,th

B _c2

In:Bi

Fig. 10.13. (a) Upper critical field B

c2

versus temperature of several type II su- perconductors [465]. (b) Temperature dependence of the critical fields B

c1

, B

c,th

, and B

c2

of an indium-bismuth alloy [466]

In perfect crystals, the flux lines are regularly arranged in the so-called Abrikosov lattice, as is schematically shown in Fig. 10.14a. The first evidence for the existence of flux lines and their regular arrangement was obtained by decorating them on the surface of a sample with small particles of colloidal iron and making them visible with an electron microscope [467]. More re- cently, scanning tunneling microscopes have been used for the investigation of vortex lattices. A typical observation from this type of experiment is shown in Fig. 10.14b. Clearly, the flux lines in NbSe 2 are arranged in a hexagonal lattice. The contrast is due to the different work functions of the normal and superconducting phases. Based upon the different tunneling-current charac- teristics of the two phases and the high spatial resolution of this technique, it is possible to study the electronic density of states even inside the flux lines and in the superconducting neighborhood. In realistic materials, the regular

356 10 Superconductivity

relatively large critical parameters B c2 (0) = 25 T and T c (B = 0) = 18.7 K.

This is technically important because this compound can easily be processed and used as the material for wires in superconducting magnets. The novel high-T c superconductors with their extremely high values of T c and B c2 will be considered separately in Sect. 10.5.4.

Both the critical field strengths B c1 and B c2 depend on temperature.

They have their largest value at T = 0, and vanish at T c . In Fig. 10.13a, the temperature variation of B c2 of several type II superconductors is depicted. In Fig. 10.13b, the critical fields B _c1 and B _c2 , and the thermodynamic critical field B _c,th of an indium-bismuth alloy are shown. Obviously, all types of critical fields exhibit a similar temperature dependence.

0 5 10 15 20

Temperature T / K

0 10 20 30 40 50 60

M ag ne tic fie ld B / T

PbGd

_0,3

Mo

₆

S

₈

PbMo

_6.35

S

₈

Nb

₃

Sn V

₃

Ga NbTi

0 1 2 3 4

Temperature T / K

0 50 100

M ag ne tic fie ld B / m T

B _c1 B _c,th

B _c2

In:Bi

Fig. 10.13. (a) Upper critical field B

c2

versus temperature of several type II su- perconductors [465]. (b) Temperature dependence of the critical fields B

c1

, B

c,th

, and B

c2

of an indium-bismuth alloy [466]

In perfect crystals, the flux lines are regularly arranged in the so-called Abrikosov lattice, as is schematically shown in Fig. 10.14a. The first evidence for the existence of flux lines and their regular arrangement was obtained by decorating them on the surface of a sample with small particles of colloidal iron and making them visible with an electron microscope [467]. More re- cently, scanning tunneling microscopes have been used for the investigation of vortex lattices. A typical observation from this type of experiment is shown in Fig. 10.14b. Clearly, the flux lines in NbSe ₂ are arranged in a hexagonal lattice. The contrast is due to the different work functions of the normal and superconducting phases. Based upon the different tunneling-current charac- teristics of the two phases and the high spatial resolution of this technique, it is possible to study the electronic density of states even inside the flux lines and in the superconducting neighborhood. In realistic materials, the regular

cuprati: Hc2(0) ~ 100-300 T

Diagramma H-T H c2 (T)

Reticolo di Abrikosov

U. Essmann and H. Trauble Max-Planck Institute, Stuttgart Physics Letters 24A, 526 (1967) Bitter Decoration

Pb-4at%In rod, 1.1K, 195G

Scanning Tunnel Microscopy NbSe2, 1T, 1.8K

H. F. Hess et al.

Bell Labs

Phys. Rev. Lett. 62, 214 (1989)

Bitter Decoration MgB2 crystal, 200G L. Ya. Vinnikov et al.

Institute of Solid State Physics, Chernogolovka

Phys. Rev. B 67, 092512

(2003)

Corrente e reticolo di Abrikosov

  



   





























    



    





     





   

   



      

      

      

       



     













^























     

       

     





    

      















    



     



    

    

     





      





    



    



    





     





     









!























    









      





      









      







    

    















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

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TYPE II SUPERCONDUCTOR MAGNETIC VORTICES

CURRENT

CURRENT

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Copyright 1993 Scientific American, Inc.

B J

F

J

⇤ F = ⇤

J ⇥ ⇤ B

⇤ F = ⇤

J ⇥ ⇤

0

Sull’intero reticolo Forza (per unità di lunghezza)

sul singolo flussone

D. J. Bishop, P. L. Gammel, D. A. Huse, Scientific American 1993

⇥ 0 = ₀ ⇥ n F J

(analogia idrodinamica, energia corrente/corrente)

Una corrente di trasporto determina forze sui flussoni.

Il reticolo si muove perpendicolarmente a B e J.

Flussoni in moto: dissipazione

⇤ E = ⇤

B ⇥ ⇤ v _{f l}

F J

E Flussoni in moto: campo elettrico

nel core dei flussoni, dove quasiparticelle sono soggette a

scattering ––> dissipazione.

Flussoni in moto: dissipazione Flussoni immobili: resistenza nulla

⇤ F = ⇤

J ⇥ ⇤

0

Non è la corrente critica di depairing che limita

il trasporto, ma il “depinning” dei flussoni

Difetti e reticolo 10.2 Phenomenological Description 373 line is depicted in more detail in Fig. 10.21. The Cooper pair density, or order parameter, is zero in the center of flux lines. The rise to the bulk value n

_s

(∞) = |Ψ

∞

|

²

is determined by the coherence length ξ

_GL

. The mag- netic field has its maximum in the center where the shielding current j

_s

is zero. The spatial extent of both quantities is governed by the penetration depth.

B n

_s

j

_s

0 0

r j

_s

, B, n

_s

Fig. 10.21. Spatial variation of the Cooper pair density n

s

, the magnetic field B, and supercurrent density j

_s

in and close to a flux line

Single flux lines can be studied with the help of scanning tunneling mi- croscopy. The crucial quantity in measurements with this technique is the variation of the differential conductivity dI/dV with the voltage V applied between the tip of the microscope and the superconductor. As will be shown in Sect. 10.3.6, dI/dV ∝ D(E) and is thus reflecting the density of states of unpaired electrons. An example of this type, a measurement on NbSe

₂

, is shown in Fig. 10.22. The curve at the bottom represents the density of states without a magnetic field, i.e., in the Meissner phase. A gap in the density of states is found, followed by a maximum. This result is a character- istic of superconductors and will be discussed in detail within the framework of BCS theory (see Sect. 10.3). The curve at the top reflects the density of

-4 -2 0 2 4

Voltage V / mV

0 1 2 3 4 5

dI / dV (a rb itr ar y un its ) _{0 Å}

563 Å B = 0

Fig. 10.22. Differential conductiv- ity dI/dV at a flux line in NbSe

₂

ver- sus applied voltage. The curves are dis- placed vertically. The curve at the top was measured in the core of the vortex, the lower curves were obtained with in- creasing distance from the core up to 563 ˚ A. The lowest curve was taken in the Meissner phase [477]

Ogni flussone “paga” energia di

condensazione (per unità di lunghezza) : u _cond 1

2 µ ₀ H _c ² ⇥

Impedire il moto dei vortici:

“pinning”

Difetti: il parametro d’ordine è naturalmente depresso.

=> si risparmia energia di condensazione

=> centri preferenziali per i flussoni

Flussoni in moto: dissipazione

Flussoni immobili: resistenza nulla

Difetti e reticolo

difetto efficace: ~ξ difetto non efficace: << ξ

Collocandosi sul difetto il vortice risparmia energia di condensazione

(per unità di lunghezza) :

Compete con l’energia elastica del reticolo e del singolo flussone (deformazione dovuta alla

distribuzione non commensurata dei difetti):

=> reticolo: pinning puntiforme è conveniente. Il reticolo è completamente ancorato con (relativamente) pochi centri di pinning.

[per un insieme “liquido” ogni singolo flussone dovrebbe essere individualmente ancorato]

1

2 µ ₀ H _c ² ⇥ ²

Ancoraggio e irreversibilità. 1: LTS

Aggiungendo centri di pinning:

irreversibilità magnetica linea di irreversibilità

LTS: utilizzabili per applicazioni (quasi) fino a

H c2 .

T

fase normale

T ^c H

fase Meissner H ^c1 (T) H c2 (T)

ρ = 0,

Irreversibile.

H irr (T) Reversibile,

ρ ≠ 0.

stato misto

Un superconduttore ideale nello stato misto:

• è magneticamente reversibile

• presenta dissipazione

tifilamentary NbTi wires and hence has a similar composi- tion to the multifilamentary specimen. Results show that the width of the reversible region B_c2-B_irrfor this sample is very close to that of the wire at the same reduced temperature.

This result appears to support the notion that the irreversibil- ity line is a flux lattice melting line, since, in the creep model, complete flux penetration of the large bulk sample takes longer than for the fine filamentary samples. The irre- versibility line is thus expected to be higher for the larger of the two specimens, which is in contrast to the above obser- vation. However, the exponential creep rate means that the size effect will be difficult to detect, so that a creep model is not inconsistent with the data. As with the NbTa rod surface currents meant that the resistive transition did not coincide with the magnetic irreversibility line in a parallel field. No critical current was observed above B_c2 when the field was applied perpendicular to the axis of the sample.

The temperature dependencies of B_irr, determined from dc magnetization, ac susceptibility, and magnetoresistivity measurements, and B_c2, determined from dc magnetization measurements, are plotted in Fig. 6 for the NbTi wire. The figure illustrates an observable region of reversible flux mo- tion in the NbTi wires. The width of the reversible region, as measured by B_c2(T)-B_irr(T) or T_c(B)-T_irr(B), is quantita- tively similar to that reported by Suenaga et al.¹⁴ Interest- ingly, the two curves of B_irr(T) and B_c2(T) are ‘‘parallel’’ to each other over almost the entire experimental temperature range, except in the vicinity of T_c. Suenaga et al. and Schmidt, Israeloff, and Goldman¹⁵found that measured irre- versibility line data could be fitted well to the flux-line melt- ing model of Houghton, Pelcovits, and Sudbø⁴ for NbTi, Nb₃Sn multifilamentary wires, and Nb thin films. These au- thors suggested that the irreversibility line in low-T_c super- conductors is essentially the flux lattice melting line on the basis of their observations. The melting theory also suggests that the melting line predicted by the calculation is essen- tially parallel to the superconducting-normal phase boundary B_c2(T) over a wide range of field, except in the range close to T_c where B_irr follows a (1!T/T_c)² relation. This is in agreement with the data shown in Fig. 6. A fit of the data

given in Fig. 6 to the theory, however, is not very meaning- ful due to the relatively large experimental error in the data, particularly at temperatures close to T_c. Additionally, a lin- ear temperature dependence of B_c2 is assumed in the theory of Houghton, Pelcovits and Sudbø while the experimental data of B_c2 in Fig. 6 show nonlinear behavior which could be better described by the empirical relation B_c2(T)

"B_c2(0)!1!(T/T_c)²".

Also as shown in Fig. 6, the B_irrcurve tends to saturate at low temperatures, i.e., it has a slight negative curvature. Al- though this appears different from that seen in high-T_c ma- terials it is not entirely unexpected since on almost any model the irreversibility line cannot be far from the B_c2 line in low-T_c material.

It has been reported^14,15that the flux creep effect increases very rapidly near B_irr or T_irr confirming the easy motion of the flux lines in this field or temperature range. In particular Schmidt, Israeloff, and Goldman¹⁵analyzed their flux creep data within the framework of flux-line depinning theory¹and vortex glass transition theory²and found none of these theo- ries could account for the experimental data consistently. In their study, flux creep measurements were performed at se- lected temperatures. No detectable decay in magnetization was observed within the resolution of the experiment when the applied field was significantly less than B_irr. Flux creep, however, became apparent as the field was increased to B_irr which indicates that flux lines may be moved relatively eas- ily in this field region. Because of the large noise level com- pared with the variation of magnetization decay, it is difficult to say if the decay follows a ln t relation as seen in previous reports.^14,15

V. DISCUSSION

Consider first the NbTa. The magnetization curve would detect a critical current density of 5 A/cm² compared with 10⁴A/cm²at half B_c2. Within 10% of B_c2 we might expect J_cto be lower by a factor of about 100, so this is evidence of an irreversibility line. However, the most convincing evi- dence comes from the susceptibility measurements since the loss peak at the irreversibility line shows that there is a linear resistivity between this field and B_c2. The resistive measure- ments do not show an irreversibility line due to surface su- perconductivity and something that needs explaining is how a surface current which must be borne by pinned pancake vortices can carry a current so far above the irreversibility line of the bulk. It may be because the current is parallel to the field so there is little force on the pancakes. It is, how- ever, interesting to note that a similar effect has been seen in BSCCO crystals which have been shown to carry surface currents in fields well above the irreversibility field, although still much smaller than B_c2.³²

The very small size of the NbTi filaments make detecting an irreversibility line magnetically rather difficult. The loss peak was ill defined and J_c would need to be above 5000 A/cm² before the hysteresis would show up. However, the bulk sample showed similar behavior to NbTa and the irre- versibility field found in it was similar to the field at which the filaments became diamagnetic. But the most convincing evidence here comes from the resistive measurements. Here the resistive transition coincided with the magnetic irrevers- FIG. 6. Temperature dependence of Bc2 #determined from the

measured dc magnetization$ and Birr#determined from dc magneti- zation, ac susceptibility and resistivity$ for a NbTi multifilamentary wire sample.

15 434 D. N. ZHENG, N. J. C. INGLE, AND A. M. CAMPBELL PRB 61

Zheng et al., PRB 61 (2000)

H irr : prossima a H c2

Irreversibilità e levitazione

Sotto la linea di irreversibilità il flusso magnetico è bloccato dai centri di pinning:

Levitazione stabile

Rotazione stabile

Sospensione

Levitazione non Meissner

È l’ancoraggio dei flussoni che determina:

• alte correnti critiche (di “depinning”)

• levitazione stabile

I difetti possono favorire l’ancoraggio.

È possibile ingegnerizzare i difetti?

Scala di lunghezze:

devono bloccare un flussone, diametro ~ξ cuprati: ξ~2-5 nm

tradizionali: ξ~10-50 nm