Critical Field

11 Key excerpts on "Critical Field"

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  RF Superconductivity

  Science, Technology, and Applications

  • Hasan Padamsee(Author)
  • 2009(Publication Date)
  • Wiley-VCH

    (Publisher)

  3 Surface Resistance and Critical Field 72 Fig. 3.25 (a) Increase of residual resistance due to baking, and restoration of the lower residual resistance due to HF rinsing [151] (courtesy of Saclay). (b) Growth of suboxides detected by XPS studies [175] (courtesy of UST). 3.5.1 H c1 , H c , and H c2 A brief explanation of Type I and Type II superconductors along with their var- ious Critical Fields is in order. Besides zero resistance to dc currents, the other hallmark property of super- conductors is the Meissner effect, which requires that there be zero magnetic field in the bulk of the superconductor. The Meissner condition prevails until the externally applied field reaches a Critical Field, when magnetic flux abruptly enters the superconductor. Thermodynamics determines the value of this criti- cal field. When electrons condense into Cooper pairs the free energy of the superconductor decreases by an amount referred to as the condensation energy. In the presence of an external field, the supercurrents which flow at the surface of the superconductor to shield the bulk from the external field increase the free energy. Work must be done to establish the currents to exclude the mag- netic field. At the field where the increase in magnetic free energy completely balances the decrease in condensation energy of the Cooper pairs, there is an abrupt transition to the normal state and all the external flux enters the super- conductor. This field is known as the thermodynamic Critical Field, H c . Super- conductors which show an abrupt transition at the thermodynamic Critical Field are called Type I superconductors. From a thermodynamic point of view it is possible for flux to enter a super- conductor below H c if the superconductor divides into alternating thin normal and superconducting regions, provided the thickness of the normal region is smaller than the penetration depth.

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  Superconductivity Of Metals And Alloys

  • P. G. De Gennes(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  2 MAGNETIC PROPERTIES OF FIRST KIND SUPERCONDUCTORS 2–1  Critical Field OF A LONG CYLINDER

  A long superconducting cylinder of radius r0 is placed in a solenoid of radius r1 > r0 (Fig. 2-1 ).

  A (weak) current I flows in the coil. The resulting field distribution h(r) is shown in Fig. 2-2 . Outside the sample, h(r) takes a constant value H. In the sample the field falls rapidly (in a depth λ ~ 500Å) to 0. We restrict our attention to a macroscopic specimen (r0 ≫ λ). Then, on the scale r0 , the field does not penetrate into the sample.

  This complete flux expulsion is observed for weak external fields H, but, when H reaches a critical value Hc , a radical modification occurs:

  (1)  The field becomes uniform across the entire section.

  (2)  The specimen is no longer superconducting (this we can see in principle by measuring the resistivity between the ends of the cylinder). With sufficiently perfect specimens, one can verify that this transition is reversible. If I is now decreased, the superconducting state reappears with total flux expulsion. Typical values of Hc are given in Table 2-1 . The Critical Field decreases with increasing temperature, roughly according to the law

  H c ( T ) = H c ( 0 ) [ 1 − T 2 T 0 2 ]

  (2-1)

  (For T > T0 the material is normal even in zero field; T0 is the transition temperature in zero field.) From Hc , we can calculate the difference in free energy between the normal and superconducting states:

  Figure 2-1

  Experimental geometry to measure the Critical Field. The sample is a long cylinder (of length L and radius r0 ). It is placed in a coil of radius r1 .

  (a)  When the cylinder is normal the field is uniform across the solenoid

  h = 4 π NI cL

  (2-2)

  (where N is the number of turns in the solenoid and L is its length, which is also the length of the specimen). The free energy of the system becomes

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  Magnetic Materials

  Fundamentals and Applications

  • Nicola A. Spaldin(Author)
  • 2010(Publication Date)
  • Cambridge University Press

    (Publisher)

  The science of superconductivity is extremely rich, and the details are beyond the scope of this book. However, in the remainder of this chapter we will give a brief overview of some of the fundamentals. 4.5.1 The Meissner effect If a metal such as lead, which is normally diamagnetic, is cooled in a magnetic field, then at some critical temperature, T c , it will spontaneously exclude all magnetic flux from its interior, as illustrated in Fig. 4.3. If B = μ 0 (H + M) = 0, then M = −H, and χ = M/H = −1 (in SI units). And the permeability μ = 1 + χ = 0, so the material is impermeable to the magnetic field. T c is also the temperature at which the material undergoes the transition to the superconducting state. The exclusion of flux is called the Meissner effect [14], and is the reason that superconductors are perfect diamagnets. The circulating currents which (by Lenz’s law) oppose the applied magnetic field are able to exactly cancel the applied field because the resistivity is zero in the superconducting state. This is the reason that the exclusion of flux coincides with the onset of superconductivity. 44 Diamagnetism 4.5.2 Critical Field Even below T c , the superconducting state can be destroyed if a high enough field is applied. The field which destroys the superconducting state at a particular tempera- ture is called the Critical Field, H c . At lower temperatures, the Critical Field is higher, and by definition it is zero at T c because the superconducting state is destroyed spontaneously. If the superconductor is carrying a current, then the field produced by the circulating charge also contributes to H c . Therefore there is a maximum allowable current before superconductivity is destroyed. The critical current depends on the radius of the conductor and is a crucial factor in determining the technological utility of a particular superconducting material. 4.5.3 Classification of superconductors Superconductors can be classified as type I or type II.

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  Superconductivity

  Fundamentals and Applications

  • Werner Buckel, Reinhold Kleiner(Authors)
  • 2008(Publication Date)
  • Wiley-VCH

    (Publisher)

  Since the critical current density represents the crucial quantity for the stability of the superconducting state, we expect that the transition cannot take place 4 For a wire in a perpendicular field, the demagnetization coefficient N M is 1 ⁄ 2 . Hence, at the lines with the highest field at the surface, we obtain B eff = 2 B a . Fig. 5.3 (a) Field distribution around a superconducting wire in the Meissner phase without a transport current. (b) Critical current of a wire with circular cross-section in external field B a oriented perpendicular to the wire axis. 5 Critical Currents in Type-I and Type-II Superconductors 274 in such a way that the current density is everywhere smaller than the critical value. This assumption is confirmed by experiment. If the critical current is exceeded, the superconductor enters the intermediate state, i. e., there appear normal conduct-ing domains. Several models have been proposed to describe this intermediate state. In this case, configurations of the normal conducting or superconducting domains are sought that yield the Critical Field B cth at all interfaces as much as possible. For the macroscopic structure of the intermediate state, this Critical Field value also yields the critical current density because of the fully developed shielding. A model of this kind is shown in Fig. 5.4 [2]. Since the magnetic field of the transport current consists of circular field lines, the phase boundaries must also run perpendicularly to the wire axis. Because of the requirement that for each radius the field must be equal to B cth , the current density must increase toward the wire axis. This is accomplished also by an increase of the thickness of the superconducting domains toward the wire axis. The geometrical details can only be found from a calculation. In this case certain additional assumptions are necessary. The various models of this current-induced intermediate state are different because of these additional assumptions.

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  Introduction to Superconductivity

  • A.C. Rose-Innes(Author)
  • 2012(Publication Date)
  • Pergamon

    (Publisher)

  So each superconductor has a different phase diagram (Fig. 4.3). From experiment it has been found that the Critical Fields fall off almost as the square of the temperature, so the Critical Field curves are closely ap-proximated by parabolas of the form 7x10 4 • 6 x10 4 -5x10 4 -4xt0 4 -3x10 4 -2x10 4 -1 ÷ 10 4 ; 800 700 600 500 „ (A 3 400 5 300 200 100 THE CRITICAL MAGNETIC FIELD 45 H C = H 0 [-(T/T c n (4.4) where H 0 is the Critical Field at absolute zero and T c is the transition temperature. Each superconductor can be characterized by its particular values of H 0 and T c , and, knowing these, we can use (4.4) to find the Critical Field at any temperature. Table 1.1 lists values of H 0 and T c for the superconducting elements. We may remark here that there is no fundamental significance in the relation between Critical Field and temperature being a parabola. It has merely been found experimentally that the variation can be conveniently described to within a few per cent by a parabolic curve. T h e experimen-tal curves are not in fact exactly parabolas, and to describe them ac-curately one would need a polynomial expression. For most calculations, however, it is sufficient to use the parabola given by (4.4). The Cryotron T h e existence of a critical magnetic field has been made use of in a controlled switch called a cryotron (Fig. 4.4). T h e current J which is to be controlled flows along a straight wire of tantalum called the gate. Around this, but insulated from it, is a niobium wire wound in a long single-layer coil called the control. When cooled to 4-2°K by immer-sion in liquid helium both the tantalum gate and the niobium control are superconducting, and the gate offers no resistance to the passage of the current / . If, however, a current i c is passed through the control coil, it generates a magnetic field along the gate, and, if the control current is large enough, the gate is driven normal by the magnetic field, and the appearance of resistance reduces the current , / . T h e control coil, however, remains resistanceless because the Critical Field of niobium is considerably higher than that of tantalum. Hence the current J through the gate can be controlled by a smaller current in the control, and the

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  Thermal Quantum Field Theory: Algebraic Aspects And Applications

  Algebraic Aspects and Applications

  • Faqir C Khanna, Adolfo P C Malbouisson, Jorge M C Malbouisson(Authors)
  • 2009(Publication Date)
  • World Scientific

    (Publisher)

  This is in contrast to other calculations that used modifications of the microscopic inter-actions in bulk superconductors, in order to explain the variations of the critical temperature with the thickness of the film. 19.4 Critical behavior of type-II superconducting films in a mag-netic field Until now, we have considered phase transitions in confined systems not taking into account the possible existence of an external magnetic field interacting with the order parameter of a type II superconducting film. In this respect, for the case of superconductors, we have neglected the minimal coupling with the vector potential when an external magnetic field is applied and, in its absence, the intrinsic gauge fluctuations. In this chapter, we investigate how the transition temperature for a film behaves as a function of its thickness and of the intensity of an applied magnetic field [306 , 307]. 19.4.1 Coupling-constant correction in the presence of an external magnetic field We shall take the uniform external magnetic field, H = H ˆ e 3 , normal to the film. To describe the critical curve of a type-II superconductor close to the upper Critical Field, we will neglect gauge fluctuations of the magnetic vector potential [293 , 294], that is, we take ∇ × A = H . Therefore, the Hamiltonian density is H = | ( ∇ -ie A ) φ | 2 + ¯ m 2 0 | φ | 2 + u ( | φ | 2 ) 2 , (19.55) 338 Thermal Quantum Field Theory: Algebraic Aspects and Applications where, as before, φ is an N -component vector and ¯ m 2 0 is the mass parameter which depends on L , H and T , in such way that lim L →∞ lim H → 0 ¯ m 2 0 ( L, H, T ) = m 2 0 ( T ) ≡ α ( T -T 0 ) . (19.56) Let us initially consider the Hamiltonian density (19.55) referring to a bulk superconductor.

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  AC Loss and Macroscopic Theory of Superconductors

  • W.J. Carr Jr.(Author)
  • 2001(Publication Date)
  • CRC Press

    (Publisher)

  A type-II superconductor also exhibits a Meissner effect in small fields, but when the Meissner effect breaks down and the flux begins to penetrate into the interior(ata lower Critical Field H C , where HQH C is typically in the range of 0.01 to 0.1 T) superconductivity is not destroyed. A n ideal type-II conductor continues to show little dc resistance up to a much higher upper Critical Field H C T, where superconductivity finally disappears. 4 IXQH C I in materials of primary interest is in the 10 T range. The voltage-bulk-current relationship of some type-II superconductors depends strongly on strain, as illustrated in Figure 2.2. Near the origin of these curves the resistance is difficult to detect. In theory a small supercurrent left unperturbed at low temperatures would noticeably decay only after an astronomical number of years. 5 However, at some point, as the current is increased through the application of a voltage, resistance begins to rise rapidly, leading to the concept of a critical current 1, or, more generally, a critical current density j c . The critical current density of a given material is a function of temperature, magnetic field, and strain. A plot which shows the dependence on the first two variables of three different superconductors is shown in Figure 2.3. NbjSn, which 'For other geometries, where the relevant demagnetizing factor is nonvanishing, an intermediate state is obtained in which normal and superconducting regions coexist. J Surface superconductivity persists up to a still higher Critical Field H ,j. 5 See Tinkham's book in the references A C Loss and Macroscopic Theory of Superconductors 19 Current (amperes) Figure 2.2 Plot of voltage versus current showing the onset of resistance in a multilila-ment conductor in a fixed magnetic field, 'file various curves correspond to different values of applied strain. (Taken from Clark and Ekin.'') was found to be superconducting by Matthias et al.

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  Introductory Solid State Physics

  • H.P. Myers(Author)
  • 1997(Publication Date)
  • CRC Press

    (Publisher)

  Very soon after its initial discovery, experiment also showed that the S state could be destroyed by the application of a sufficiently strong magnetic field. This gave rise to a rather arbitrary clas- sification into soft and hard superconductors, depending on whether the critical magnetic field strength B c0 was low or high. Here B c0 is the field required to quench superconductiv- ity at 0 K (Fig. 13.5 and Table 13.2). This classification is no longer used, and we speak of type I and type II superconductors, which we shall define later. The existence of a Critical Field dashed any hopes that one might have had at that time regarding the construction of superconducting solenoids. Figure 13.3 The variation of the heat capacity through the superconducting transition. (After Corak et al. 1956.) 458 Introductory Solid State Physics Figure 13.4 The variation of the thermal conductivity in the normal and the superconducting states for lead. (After Olsen 1952.) Perhaps the most significant experiment made before the war was that of Meissner and Ochsenfeld, who demonstrated in 1933 that an elemental superconductor always excludes all magnetic flux and is a perfect diamagnetic material. The experiment was very important because it showed that a superconductor, although having zero electrical resistance, is not the same as a perfect conductor. Figure 13.5 An external magnetic field shifts T c to lower values; the S state is quenched at 0 Κ in the field B c0 . The Critical Field satisfies . Superconductivity 459 Table 13.2 Some values of B c0 B c0 (G) Al 105 Ga 59 In 282 Sn 305 Pb 803 Ta 830 V 1410 Nb 2060 We may appreciate the difficulties in developing a theory of superconductivity in the fol- lowing qualitative fashion. Experimentally the pure metals have T c §5±5 K, and this means that the energy stabilizing the S relative to the N state is of order k B T c per atom and there- fore extremely small relative to the average energy of the conduction electrons.

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  Superconductivity Revisited

  • Ralph Dougherty, J. Daniel Kimel(Authors)
  • 2012(Publication Date)
  • CRC Press

    (Publisher)

  119 11 Type I Superconductivity Our discussion of type I superconductors will be include experimental observations and theoretical models that can account for the observations. Experimental observations are the foundation of the subject and the testing ground for all theoretical descriptions. EXPERIMENTAL TYPE I SUPERCONDUCTIVITY Kamerlingh Onnes’ 1911 discovery of the superconductivity of mercury at a temper-ature just below the boiling point of liquid helium [1,2] was the initiating event in this huge ongoing research effort. Data on the general subject of type I superconductivity is abundantly available, although the availability of specific details can be dependent on the trends that focus enquiry in science. C RITICAL M AGNETIC F IELD VERSUS T FOR E LEMENTAL S UPERCONDUCTORS Critical magnetic field, H c , as a function of temperature, T , was carefully examined in the early literature in the effort to obtain information on the mechanism of super-conductivity. A superconductor’s critical magnetic field is the maximum magnetic field that it can generate as a superconductor at a given temperature. It is also the maximum external magnetic field that the superconductor can be exposed to and still maintain superconductivity. Using data from the literature, critical magnetic field as a function of temperature is plotted for a number of elemental superconductors in Figures 11.1–11.4. The plots in Figures 11.1–11.4 follow the form of those presented by Shoenberg in 1960 [3]. Data for the plots came from the references for the figures, which are collected at before the four figures. As an aid to sorting out the periodic behavior of the critical magnetic field curves for this series of superconductors, we have reproduced the table on the T c of one-bar elemental superconductors as Table 11.1. When you scan Figures 11.1 through 11.4, you will notice the presence of families of curves in the data.

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  Applied Superconductivity

  Volume II

  • Vernon L. Newhouse(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  26. Relative critical current density versus temperature at constant magnetic field for NbaSn conductors. (Comparison not valid from one conductor to another.) 434 Y. IWASA AND D. B. MONTGOMERY 7 6 ω I-s oc o ï _ l LU v L 100 >v N. 150 ^ ^ 0 1 1 1 1 1 l > -1 L ^ 1 1 1 V 3 G 0 1 --J

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  Superconductors

  Properties, Technology, and Applications

  • Yury Grigorashvili(Author)
  • 2012(Publication Date)
  • IntechOpen

    (Publisher)

  The additional condition for new applications in these superconductors materials have been used to avoid any motion of flux lines. This can be achieved through the introduction of pinning centers in the superconductor, interacting, in most cases, with the normal conducting core of the flux lines. A flux line is composed of a normal conducting core regions and a surrounding circulating supercurrents area where the magnetic field and supercurrents fall off within penetration depth. The critical current density ( ) C J increases with the applied magnetic field ( ) H increases and penetration depth ( )  decreases. The force-distance (permanent magnets and superconductor) hysteresis loops during the descending and ascending process expanded with critical current density increases. Superconductors – Properties, Technology, and Applications 308 The mechanism of superconductivity in high temperature superconductor (HTS) is still controversially discussed. The Bardeen-Cooper-Schrieffer (BCS) concept of a macroscopic quantum state of Cooper pairs remains the common basis in almost all theories proposed for HTS. However, there is no consensus about the origin of the pairing so far. The lack of an isotope effect in HTS of highest C T would favor a non-phonon pairing mechanism as, for instance, d-wave pairing, whereas its presence in several HTS with lower C T suggests a phonon-mediated s-wave pairing. The most experimental features favor d-wave pairing. A direct test of the phonon mechanism for pairing in HTSs is to check the sign change in the energy gap ( ( )) g E k  as function of momentum ( ) k  . A d-state energy gap is composed of two pairs of positive and negative leaves forming a four-leaf clover. In contrast, the magnitude of an s-state may be anisotropic but always remains positive. 2. Levitation with superconductors Superconducting levitation by using the permanent magnet (PM) is a fascinating property of the magnetic behavior of these materials.

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