Chapter  TypeII superconductors
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Chapter TypeII superconductors

1 Introduction For typeII superconductors sn 0 Superconductors will try to create as many SN boundaries SN domain walls down to the charactistic size and of the domain wall Indeed the and curves do not show perfect Meissner e64256ect Fig

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Chapter TypeII superconductors




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Chapter 4 Type-II superconductors 4.1 Introduction For type-II superconductors sn 0. Superconductors will try to create as many SN- boundaries (SN domain walls) down to the charactistic size and of the domain wall. Indeed, the )and ) curves do not show perfect Meissner effect. Figure 4.1: Magnetization curves of the type-II superconductor The penetration of the magnetic field in the type-II superconductor happens in a form of quantized vortices known as Abrikosov vortices. Each vortex has a normal core – long thin normal cylinder along the field direction.

The order parameter inside the core = 0. The radius of the normal core 59
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60 Chapter 4 Type-II superconductors Around the normal core, there is a circulating supercurrent. The direction of circu- lation is such that the direction of magnetic field created by this current coinsides with the direction of external magnetic field (along the normal core). The size of the region where the supercurrent circulates Each vortex carries one quantum of magnetic flux. The penetration of the vortices in the superconductor takes place at H>H .Vor- tices form a triangular

lattice because vortices of the same sign (direction of circula- tion) repell each other. Figure 4.2: Mixed state (Shub- nikov phase) of the type-II super- conductor. Vortices form a tri- angular lattice (Abrikosov lattice). Vortex cores are hatched. The so-called mixed state (Shubnikov-phase) is formed. Mixed state exist for H .As increases from to , the density of vortices increases (lattice period decreases). At the distance between vortices become | 0 and the second order phase transition to the normal state takes place. Figure 4.3: Mixed state in Nb ob- tained by Essmann and Tr auble

using decoration technique and SEM 4.2 Single Abrikosov vortex For the sake of simplicity we assume 1, i.e. . Thus for =1.Inthis region the 2nd GL Eq. (3.22b) can be written as rot rot (4.1)
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4.2 Single Abrikosov vortex 61 Taking int account that rot rot =rot , and taking another rot from both sides of (4.1) we get rot rot rot θ. (4.2) At any point of the vortex except its center rot = 0 because rot grad 0. In the center – singularity. To understand the behavior of rot at 0 we write rot θd θd =2 π, because each vortex carries one quantum of magnetic

flux. Thus rot is a function which is equal to zero everywhere eaxcept the center =0, and equal to at = 0, and its integral over the area including the center is equal to 2 rot is a -function: rot =2 where is a unit vector along the vortex. As a result, instead of (4.2) we get an equation rot rot = (4.3) The boundary condition at is 0. The solution of Eq. (4.3) is (4.4) where K is the so-called McDonald function with the following asymptotic behavior: ln(1 /z zz (4.5) From Eqs. (4.4) and (4.5) it follows that magnetic field diverges at 0. In reality this is not true, as the above

derivation is not valid in the region where psi 1. Qualitatively we should just cut o ) dependence at i.e. , assume that (0) )= ln κ. (4.6) Solving the problem, taking into account the variation of inthecoreregion,onearrives to the following result (0) = (ln 28) (4.7)
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62 Chapter 4 Type-II superconductors Figure 4.4: Single Abrikosov vor- tex in infinite type-II superconduc- tor. The behavior of the order- parameter and magnetic field 4.3 First critical field Let us find the first critical field i.e. , the field at which it is

energetically favorable to have a vortex in the superconductor. For this, we first find the free energy of a vortex . For the sake of simplicity we treat the case 1, i.e. . In this case we forget about and write only magnetic energy like in London theory magn. (rot kin. dV, (4.8) where the integral is taken over the space in between two planes =0and = 1 (the vortex is directed along -axis). Using the identity (rot rot rot div[rot we have rot rot dV div[rot dV. (4.9) Using the Gauss theorem we convert the second integral to the surface integral div[rot dV [rot =0 Since is to planes

const ,rot is in the plane const ,and[rot ]isalso in the plane, but is to the plane. Thus, the above integral is equal to zero and rot rot dV. (4.10) Since the vortex solution is a solution of Eq. (4.3), we can write dV (0) 4.6 ln κ. (4.11) In the above derivation we did not take into account the contribution to the energy asso- ciated with the normal core of the vortex. To estimate it, we note that in the normal core
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4.4 Interaction of vortices 63 the energy density is larger than in the surroundings by . The total contribution to the vortex energy can be estimated as

3.37 16 ln κ. More exact calculations give (ln +0 50) (4.12) The question is: at which field the formation of the vortices becomes favorable? BH dV (4.13) From (4.13) it is clear that may become negative if we apply magnetic field >H 4.12 (ln +0 50) (4.14) Comparing (0) (4.6) and we see that (0) .Since (0)( (4.15) is relatively weak field: one should create on the area 4.4 Interaction of vortices Figure 4.5: Interaction between two parallel vortices of the same sign. We should again calculate the energy magn. (rot kin. dV, (4.16)
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64 Chapter 4 Type-II

superconductors for which is a solution of equation rot rot = )+ )] (4.17) Following the same derivation as we did to calculate the energy of a single vortex, we get instead of Eq. (4.11) the following )+ )) (4.18) )= (0) + 12 ), )= (0) + 12 ). And =2 12 (4.19) The interaction energy )= 12 4.4 (4.20) and the force dU dx dB 12 dx (4.21) Since, according to the Maxwell equations dB 12 /dx 12 ), we have 12 )= ]= (4.22) Vortices of the same sign of circulation repel each other. Motion of magnetic flux (vortices) under the action of the transport current in per- pendicular direction. Flux

flow voltage. Flux flow resistivity. 4.5 Second critical field Mixed state — triangular Abrikosov vortex lattice. As we increase the field, the density of vortices increases, spacing decreases and when it reaches the 2nd order phase transition to the normal state takes place. The field at which it happens is the so-called second critical field Estimation: When the distance between vortices they are separated by a still superconducting interlayer (thin film) of the thickness . But the critical field of the thin film is known tf cm cm (4.23)

Exact calculations show that κH cm 3.37 (4.24) This result expresses the fact the distance between the vortices is of the order of Eq. (4.24) is used to determine in experiment. At , similar to ).
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4.6 Magnetization of type-II superconductor 65 4.6 Magnetization of type-II superconductor Problem: Find ) in the mixed state ( ). One can solve this problem using variational technique, but we will use an artificial trick. Consider a SC with a small gradient in i.e. . In this case all quantities, in particular , will become functions of . This means that there will be a

current dB dx flowing in -direction. But B/ ,so dM dx If it is so, than each vortex is exposed to a Lorenz force = dM dx On the other hand it is clear that the lattice cannot just move without any reason, so there should be another force which points in the opposite direction. This force is since is now -dependent. 4.11 dx ln d dx The second Newton’s law reads i.e. dM dx d dx Integration results in ln (4.25) where is the integration constant, which should be determined from the boundary con- ditions. Since grows with ,the decreases with and at some point )= .Atthispoint ) = 0, that is ),

see Eq. (4.25). On the other hand, since ,see(4.24),thenat we have πl = Dividing the last two expressions we get Finally, Eq. (4.25) becomes )= ln (4.26)
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66 Chapter 4 Type-II superconductors The magnetic induction is )= ln (4.27) which is valid (as all derivation above) for 1. For (4.28) What is the physical sense of cm or for type-II superconductor? In type-I it is the critical field of the bulk sample. In type-II, there are no any peculiarities on ) dependence at cm . For type-II superconductors is simply a measure of how profitable is the superconducting

state in comparison with the normal state. If can also say that the S N transition will happen when we spend the work cm to magnetize the sample. On the other hand this work is equal to MdH cm is the area under the ) curve! (4.29) Figure 4.6: ) curves for three superconductors with the same cm , but with different .Inall cases the area under the curve is cm 4.7 Surface superconductivity. Third critical field. If the external field is very high and decreases down to the superconducting islands nucleate close to the surface parallel to the fieled direction. At this

surface dψ/dx 0, so 1 looks like at the figure. Since the “superconducting film” is twice thinner, its critical field will be approximately twice higher. Exact calculations show that =1 69 — the third critical field (4.30) Note, that for the surface layer screens the magnetic field a little bit, therefore, there are currents flowing in the opposite directions along internal and external sides of the surface layer.
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4.7 Surface superconductivity. Third critical field. 67 Figure 4.7: Wave function of the island where the

superconductiv- ity nucleates. (a) In in bulk, the nucleation center has a size (b) Close to the surface the size of the nucleation center is Figure 4.8: The amplitude of the order parameter at the surface of the superconductor as a function of applied magnetic field for differ- ent values of As the field decreases down from the amplitude of the order parameter increases and the width ∆ of the surface layer grows, see Figs. Note, the surface superconductivity may exist even in the type-I superconductors: =1 69 =1 69 κH cm >H cm i.e. ,if κ> 69 42.
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68 Chapter 4 Type-II superconductors Figure 4.9: The width ∆ of the superconducting surface layer as a function of externally applied magnetic field for different values of
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4.7 Surface superconductivity. Third critical field. 69 Figure 4.10: Behavior of the crit- ical fields cm and as a function of temperature
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70 Chapter 4 Type-II superconductors 4.8 Surface barrier Up to now we believed that if it is energetically profitable for vortices to penetrate into SC at , then they really penetrate. More, accurate analysis

shows that to penetrate into superconductor, vortices should overcome a surface barrier Figure 4.11: The energy of a vor- tex as a function of its distance from the surface. At = 0 vortex is unstable and exists from SC. Why ) looks so? Image technique. Vortex is attracted by antivortex. Figure 4.12: Circulating currents of a vortex close to the surface of superconductor Now, in the presence of magnetic field. Figure 4.13: Meissner current cre- ated by the externally applied field repels a vortex from the surface The attraction force due to image image = dH dx
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4.8

Surface barrier 71 The repulsion force due to Meissner currents = x/ We would like to get ), i.e. )= fdx (2 )+ / C, where is the integration constant. At = ). Thus )= / (2 )+ (4.31) Figure 4.14: The energy of a vor- tex as a function of its distance from the surface for different val- ues of externally applied magnetic field The penetration takes place when dx =0 =0 i.e. ,when cm ! Bean-Levingston surface barrier. Experimentally results in a hys- teresis on the )curve. Figure 4.15: Hysteretic on the magnetization curve caused by the surface barrier If the surface of the

superconductor is rough, the penetration of the vortices may take placeatsmallerfields H cm The flux of a vortex close to the surface F Comparing with Eq. (4.31), and separating only dependent terms we get Φ= (1 /
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72 Chapter 4 Type-II superconductors