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Superconductivity

- The Josephson effect describes the phenomenon of supercurrent flowing between two superconductors separated by a thin insulating barrier, known as a Josephson junction. - The key equations governing the Josephson effect relate the supercurrent (I) flowing through the junction to the phase difference (φ) of the superconducting wave functions on either side of the barrier. A voltage (V) develops proportionally to the rate of change of the phase difference. - Applications of the Josephson effect include SQUIDs for precision metrology, superconducting quantum computing using Josephson junctions as qubits, and superconducting tunnel junction detectors.

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UEEP2024 Solid State Physics Topic 6 Superconductivity
Superconductivity • The electrical resistivity of many metals and alloys drops suddenly to zero when the specimen is cooled to a sufficiently low temperature. This phenomenon is called superconductivity • Superconductor is not a normal conductor with zero electrical resistivity
Superconductivity • Two basic properties of superconductivity – 1. Zero resistivity – 2. Perfect diamagnetism • Missing any one of these two properties will make the superconducting phase thermodynamically unstable. Hence it needs both properties to prove a material is a superconductor.
Superconductivity • Zero resistivity – Below a certain temperature, the critical temperature Tc (property of the superconductor), resistivity of a superconductor will become exactly zero. The first superconductor was mercury, discovered by Onnes in 1911.
• Perfect diamagnetism – A superconductor expels magnetic field completely when it is in superconducting phase (T<Tc). This phenomenon was discovered by Meissner (and Ochsenfeld) in 1933, so it is called the Meisner effect. Superconductivity T<Tc T>Tc
Superconductivity • A sufficiently strong magnetic field will destroy superconductivity. A superconducting state will become normal when H>Hc(T)
Superconductivity Material Tc (K) Material Tc (K) Al 1.20 Nb3Sn 18.05 Hg 4.15 Nb3Al 17.5 Mo 0.92 V3Ga 16.5 Nb 9.26 La3In 10.4 Pb 7.19 LaBaCuO 35 Ta 4.48 YBCO 90 Ti 0.39 Tl2Ba2Ca2Cu3O10 125 Zn 0.88 Hg12Tl3Ba30Ca30 Cu45O125 138
Superconductivity Superconductors can be classified in accordance with several criteria that depend on our interest in their physical properties, on the understanding we have about them, on how expensive is cooling them or on the material they are made of. By their physical properties -type I superconductors -type II superconductors By the understanting we have about them -Conventional superconductors -Unconventional superconductors
Superconductivity By their critical temperature -Low temperature superconductors (those whose critical temperature is below 77K ) -High temperature superconductors (those whose critical temperature is above 77K ) By material -Pure element (Al, Pb, Mo, Hg, Ti, Zn……….) -Alloy (Nb3Al, NbN, Ti2Co ……) -Ceramics (YBa2Cu3O7, MgB2……….. )
Superconductivity • Type I superconductor Magnetization (M) versus applied magnetic field (H) for type I superconductor exhibits a complete Meissner effect (perfect diamagnetism). Above the critical field Hc the specimen is a normal conductor. The value of Hc are always too low for type I superconductors to have any useful technical application in coils for superconducting magnets
Superconductivity • Type II superconductor Type II superconductor has two critical fields, Hc1 and Hc2, being a perfect superconductor under the lower critical field (Hc1) and leaving completely the superconducting state above the upper critical field (Hc2). Between the lower critical field Hc1 and the upper critical field Hc2 the flux will penetrate the specimen (B ≠ 0) and the Meissner effect is said to be incomplete. In this region (between Hc1 and Hc2), the superconductor is said to be in the mixed state or Vortex state. The value of Hc2 is much higher than the value of the critical field Hc calculated from the thermodynamics of the transition.
Superconductivity Conventional superconductors: those that can be fully explained with the BCS theory or related theories. Unconventional superconductors: those that failed to be explained using such theories. This criterion is important, as the BCS theory is explaining the properties of conventional superconductors since 1957, but on the other hand there have been no satisfactory theory to explain fully unconventional superconductors. In most of cases type I superconductors are conventional, but there are several exceptions as niobium, which is both conventional and type II.
Superconductivity As described in the BSC theory developed by John Bardeen, John Schrieffer and Leon Cooper, the Cooper pair state is responsible for superconductivity. Cooper pair : A Cooper pair is the name given to electrons that are bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Cooper. Cooper showed that an arbitrarily small attraction between electrons in a metal can cause a paired state of electrons to have a lower energy than the Fermi energy, which implies that the pair is bound. Cooper pair is a composite boson as its total spin is integer (0 or 1).
Superconductivity How the Cooper pair is formed in a superconductor? An electron in a metal normally behaves as a free particle. The electron is repelled from other electrons due to their negative charge, but it also attracts the positive ions that make up the rigid lattice of the metal. This attraction can distort the positively charged ion lattice in such a way as to attract other electrons (electron–phonon interactions). At long distances this attraction between electrons due to the displaced ions can overcome the electrons' repulsion due to their negative charge, and cause them to pair-up.
Superconductivity • BCS theory predicts that at absolute zero the band gap is given by E=3.53kTc. When we considers the more realistic state consisting of many electrons forming pairs (Cooper pair fluid) we find that, due to quantum mechanics, the energy spectrum of the Cooper pair fluid possesses an energy gap ΔE, meaning that all excitations of the system must possess some minimum amount of energy. This gap to excitations ΔE leads to superconductivity. If ΔE is larger than the thermal energy of the lattice, given by kT, the fluid will not be scattered by the lattice since this small excitations (scattering of electrons) are forbidden • the Cooper pair fluid in a superconductor can flow without energy dissipations
Example Calculate the energy gap for superconducting tin and the minimum frequency that can be adsorbed given the critical temperature of the tin is 3.72 K.
Solution • E= 3.53kT= 3.53×1.38×10-23×3.72 = 1.81×10-22J. • Frequency of light absorbed • Note this frequency is 1000 times lower than that of visible light and corresponds to the microwave region of the electromagnetic spectrum. .1073.2 1063.6 1081.1 11 34 22 Hz h E         
Superconductivity • Normal conductor In a normal conductor, an electrical current may be visualized as a fluid of electrons moving across a heavy ionic lattice. The electrons are constantly colliding with the ions in the lattice, and during each collision some of the energy carried by the current is absorbed by the lattice and converted into heat, which is essentially the vibrational kinetic energy of the lattice ions. As a result, the energy carried by the current is constantly being dissipated. This is the phenomenon of electrical resistance.
Superconductivity Material Tc (K) ΔE(eV) ΔE/ (kTc ) Al 1.20 3.4 x 10-4 3.3 Hg 4.15 16.5 x 10-4 4.6 Mo 0.92 2.7 x 10-4 3.4 Nb 9.26 30.5 x 10-4 3.8 Pb 7.19 27.3 x 10-4 4.38 Ta 4.48 14 x 10-4 3.6 Ti 0.39 7.35 x 10-4 3.57 Zn 0.88 2.4 x 10-4 3.2 ΔE ~ 4kTc
Superconductivity (Isotope effect) • It has been observed that the critical temperature of superconductors varies with isotopic mass .constTM C  The experimental results within each series of isotopes may be a relation of the form : Original BCS model :  = 0.5 Material  Material  Zn Cd Pb ~0.45 ~0.32 ~0.49 Hg Mo Sn ~0.5 ~0.33 ~0.47
Superconductivity (Isotope effect) KTM C 0If Absence of lattice vibrations No superconductivity The importance of lattice vibrations suggests that an electron- phonon interaction is responsible for superconductivity. There is no other reason for the superconducting transition temperature to depend on the number of neutrons in the nucleus.
Superconductivity • The conditions for the Cooper pairs formation, in numbers large enough to allow superconductivity – Low T (limited random thermal phonons) – The interaction between an electron and a phonon are strong – The number of electrons in states lying just below the Fermi energy be large – The two electrons have “antiparallel” spins – The two electrons of a pair have linear momenta of equal magnitude but opposite direction Cooper pairs are weakly bound, they are constantly breaking up and then reforming. Also, they are large (~mm)
Josephson effect • The Josephson effect is the phenomenon of current flow across two weakly coupled superconductors, separated by a very thin insulating barrier. This arrangement—two superconductors linked by a non-conducting barrier—is known as a Josephson junction; the current that crosses the barrier is the Josephson current. The Josephson junction
Josephson effect The Josephson junction Al2O3 layer Al Glass slide Indium contact
Josephson effect the current-voltage relation for a Josephson junction
• The DC Josephson effect -- A DC current flows across the junction in the absence of any electric or magnetic field • The AC Josephson effect -- A DC voltage applied across the junction causes RF current oscillations across the junction. Further, an RF voltage applied with the DC voltage can then cause a DC current across the junction. • Macroscopic long-range quantum interference -- A DC magnetic field applied through a superconducting circuit containing two junctions causes the maximum supercurrent to show interference effects as a function of magnetic field intensity. Josephson effect
Josephson effect   te tV     2     )(sin tItI c  The basic equations governing the dynamics of the Josephson effect are (superconducting phase evolution equation) where V(t) and I(t) are the voltage and current across the Josephson junction, (t) is the "phase difference" across the junction, and Ic is a constant, the critical current of the junction. (Josephson or weak-link current-phase relation) magnetic flux quantum (1) (2)
Josephson effect • The Josephson effect has found wide usage, for example in the following areas: 1. SQUIDs, or superconducting quantum interference devices 2. In precision metrology, the Josephson effect provides an exactly reproducible conversion between frequency and voltage. Since the frequency is already defined precisely and practically by the caesium standard, the Josephson effect is used, for most practical purposes, to give the definition of a volt 3. superconducting single-electron transistor 4. Josephson junctions are integral in Superconducting quantum computing as qubits such as in a Flux qubit or others schemes where the phase and charge act as the Conjugate variables 5. Superconducting Tunnel Junction Detectors (STJs)
Josephson effect Y1 -- The probability amplitude of electron pairs in region 1 Y2 -- The probability amplitude of electron pairs in region 2 Region 1 Region 2 K
Josephson effect 1 2 2 1 , Y  Y Y  Y K t iK t i  With coupling by the weak link K and applying the time- dependent Schrodinger equation If a DC voltage V is applied across the junction, an electron pair experiences a potential difference 2eV, (1) can be rewritten as 12 2 21 1 , YY  Y YY  Y KeV t iKeV t i  (1) (2)
Josephson effect 2 2 21 2 1 , nn YYSince Thus, (2) can be rewritten as 1222 2111 12 2 2 2 2 21 1 1 1 1 2 1 2 1     iiii iiii enKeneV t eni t n e n i enKeneV t eni t n e n i                               (3) We can write )exp(,)exp( 222111  inin YY
Josephson effect From (3) we can get )( 212 2 2 2 )( 211 1 1 1 21 12 2 1 2 1                          i i ennKeVn t in t n i ennKeVn t in t n i   (4) Equate the real parts of (4) )cos( )cos( 21212 2 2 12211 1 1             nnKeVn t n nnKeVn t n   (5) (6)
Josephson effect Equate the imginary parts of (4) )sin( 2 )sin( 2 2121 2 1221 1         nnK t n nnK t n   (7) (8) From (7) and (8) and let (t)=2-1 ))(sin( 4 2121 t nneK dt dn e dt dn eJ    (9)
Josephson effect (10) It is more common to use current than current density. We can rewrite (9)    )(sin tItI c  Josephson super current Phase difference across the junction DC Josephson super current Constant phase difference across the junction
Josephson effect  eV t t 2)(    From (5) and (6) and let (t)=2-1                  2 1 1 2 12 ))(cos( 2 )( n n n n t KeV t t tt    (11) If n1=n2   te tV     2  or AC Josephson super current Oscillating phase difference across the junction (12)
Josephson effect       eVt II c 2 )0(sin  By integration of (12) that with a DC voltage across the junction  eVt t 2 )0()(  (13) The superconducting current is given by (10) with (13) for the phase The current oscillates with frequency (14)  eV2 
Josephson effect Josephson junction A Josephson junction B 1 2
Josephson effect BA   0   e AB 2 0  Let the phase difference between points 1 and 2 taken on a path through junction A(B) be A(B) the phase difference around a closed circuit which encompassed a total magnetic flux  Thus   ee oAoB  ,
Josephson effect                              e I ee I III oc ooc BAtotal cossin2 sinsin   The total current The current varies with  and has maxima when integerss e  , 
Superconducting magnets • Superconducting magnet is an application of superconductor. • Conceptually the simplest, and perhaps most obvious, application is to use the supercurrent to generate an intense magnetic field. • For long solenoid, the magnetic field produced in a solenoid is B = µonI where n is the number per unit length and I is the current in the wire. • What is the largest magnetic field that can be generated with this arrangement? • The answer is related to the current-carrying capacity of the wire, which in turn is determined by the amount of heat generated by the current. • For most copper wire , current density is about 400 A cm-2, and corresponds to maximum magnetic field of few millitesla. • By placing an iron core within the solenoid, this magnetic field can be enhanced by a factor about a thousand to produce a maximum field of about 2T. • Main disadvantage of this arrangement is that the iron core is extremely heavy and cumbersome.
Superconducting magnets • Similar structure using a superconducting wire. • It can achieve current density of 107 A cm-2, it can produced very large magnetic field. • An additional advantage is that no iron core is required. • This dramatically reduces the size and weight of electromagnet and open up a whole new range of applications. • Type I superconductors are usually unsuitable as they have critical fields of only a fraction of Tesla, but type II superconductors it is possible to achieve upper critical fields up to 50T.
Example • An electromagnet is formed by winding a solenoid with 150 turns per metre using copper wire of diameter 3 mm. If the maximum current density for copper is 400 A cm-2, determine the largest magnetic field that can be produced with this solenoid.
Solution • Maximum current • Maximum magnetic field     A mAmrJI 3.28 105.110400 23242 maxmax         T AmHmnIB o 3 116 max 1034.5 3.281501026.1     m
High-temperature Superconductors • The discovery of new superconductors with increasingly higher critical temperature has been slow and erratic. • By the mid 1980s the highest recorded critical temperature was 23K for the alloy Nb3Ge, and it was generally believed that the maximum possible critical temperature for any material was in the region of 40K. • All this changed in 1986 when Alex Muller and Georg Bednorn at IBM Zurich made a dramatic breakthrough. They discovered a material with a critical temperature of 35K. • The result was important not only because it represented a significant increase in the maximum critical temperature but also because the material lanthanum-barium-copper oxide (LBCO), belongs to a new class of superconducting materials.
High-temperature Superconductors • Up to the time of discovery, most known superconducting materials were metals. • Only a few groups in the world were investigating the superconducting properties of oxides and only a handful of superconducting oxides were known. • Over the next couple of years there was a dramatic increase in research into these materials, and the results were stunning. • In 1987 another oxide, yttrium-barium-copper-oxide(YBCO) was found to superconducting at temperature up to 90K, and the following year the highest critical temperature had been pushed up to 125K for a related material, thallium- barium-calcium-copper-oxide (TBCCO).
High-temperature Superconductors • So room temperature superconducting is not yet a reality, and is unlikely to be unless other new family of superconducting material is discovered. • However, the critical temperature of these new oxide superconductors comfortably exceeds another significant temperature- the boiling point of liquid nitrogen (at 77K). • This mean that the superconducting state can be achieved using liquid nitrogen, which is far cheaper and easier to handle than liquid helium.
High-temperature Superconductors • The high critical temperature is not the only unusual property of these oxide superconductors. • They also exhibit extreme type II behavior. • Estimates of the upper critical field vary widely, but even the most conservative calculation suggest that the material will remain superconducting in magnetic fields of 250T. • There are still some practical difficulties to be overcome before high temperature superconductors can be used commercially.