Cooper Pairing

11 Key excerpts on "Cooper Pairing"

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  Superconductivity

  An Introduction

  • Reinhold Kleiner, Werner Buckel, Rudolf Huebener(Authors)
  • 2015(Publication Date)
  • Wiley-VCH

    (Publisher)

  Chapter 3 Cooper Pairing

  In Chapter 1 , we saw that superconductivity is intimately connected with the appearance of a macroscopic coherent matter wave constituted by electron pairs. Now we must ask how this pairing is accomplished, and how in the end it results in a macroscopic quantum wave with a well-defined phase. First, we will discuss conventional superconductors (see Section 2.1 ). In the second half of this chapter, we will turn to unconventional superconductors and in particular to high-temperature superconductors.

  3.1 Conventional Superconductivity

  3.1.1 Cooper Pairing by Means of Electron–Phonon Interaction

  Following our discussion in Chapter 1 , it appears relatively easy to arrive at a theory of superconductivity based on the microscopic interaction between the electrons themselves and between electrons and the surrounding crystal lattice. However, historically such a theory was confronted with extreme difficulties. Because of the striking change of the electrical conductivity and the magnetic effects during the onset of superconductivity, one could presume that essentially one is dealing with an ordering process within the system of conduction electrons. As we saw in Section 1.1 , because of the Pauli principle the conduction electrons have fairly large energies up to a few electronvolts (1 eV corresponds to thermal energy kB T at a temperature of about 11 000 K). However, the transition into the superconducting state occurs at only a few kelvins. Hence, one had to find an interaction that could lead to ordering within the electron system in spite of the high electron energies.

  There exists a multitude of possible interactions between the conduction electrons in a metal. It had been imagined [1] that Coulomb repulsion between the electrons could lead to a spatial ordering of the electrons in the form of a lattice. Also [2] a magnetic interaction seemed possible. The electrons propagating through the metal lattice with impressive velocities (electrons having energies near the Fermi energy can reach velocities near 1% of the velocity of light) generate a magnetic field because of the associated currents and can then interact with each other due to this magnetic field. Other interactions can result from the structure of the electron states (allowed energy bands, see Section 1.1

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  Introductory Solid State Physics

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

    (Publisher)

  Bardeen, Cooper and Schrieffer developed their theory on the premise that the elec- tron-photon interaction provides a means for creating Cooper pairs, but in principle the development of the general theory is independent of the origin of the interaction, provided that it leads to a net attraction. However, all the evidence, experimental and theoretical, that has accumulated since the publication of the theory in 1957 has confirmed the initial assumption of electron-phonon interaction as the true source of the attractive force between electrons in ordinary metallic superconductors. The heavy Fermion and high temperature cuprate superconductors are exceptional and at present there is no consensus regarding the origins of superconductivity in these materials. 13.8 Interacting Pairs In the case of ferromagnetism the occurrence of an atomic moment is a prerequisite, but it is not sufficient in itself to produce ferromagnetism; we need a mechanism for the coupling of the atomic moments on different atoms. Similarly, the creation of Cooper pairs is a necessary but not a sufficient condition for the creation of superconductivity. The pairs must interact in a cooperative fashion to produce a new stable electronic structure completely different from that of the ordinary metal. This new structure is the ground state of the superconductor—the condensate. The wave function of a given Cooper pair is so extended in space that it overlaps those of some 10 6 similar stationary pairs. The interactions between these overlapping pair states produce the new superconducting ground state. The ‘interaction’ may be described in terms of a transition between two degenerate electron configurations that differ only in that a pair state ↑k + , ↓k − changes to ↑kƍ + , ↓kƍ − ; clearly for this to happen the states k must initially be occupied and the states kƍ empty.

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  D-wave Superconductivity

  • Tao Xiang, Congjun Wu(Authors)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  As long as we know the characteristic energy scale of pairing interaction and the quasiparticle spectra function, we can accurately predict all dynamic and thermodynamic properties of superconducting states. This is the reason why we can still discuss and successfully predict physical properties of a high-T c superconductor without knowing clearly its microscopic pairing mechanism. 1.14 Classification of Pairing Symmetry Superconductors can be classified according to the internal symmetry of Cooper pairs. The wavefunction of a Cooper pair depends on both the spatial coordinates and the spin configurations of two electrons. In the absence of spin-orbit coupling or other interactions that break the spin rotational symmetry, the total spin is conserved and the pairing wavefunction can be factorized as a product of the spatial and spin wavefunctions  (σ 1 , r 1 ; σ 2 , r 2 ) = χ (σ 1 ,σ 2 )(R, r) , (1.131) where (σ 1 , r 1 ) and (σ 2 , r 2 ) are the spin and spatial coordinates of the first and second electrons, respectively. R = (r 1 + r 2 )/2 is the coordinate of the center of mass and r = r 1 − r 2 is the relative coordinate of the two electrons. A Cooper pair can be either in a spin singlet or in a spin triplet state depending on whether the total spin is 0 or 1. The spin wavefunction is antisymmetric, χ (σ 1 ,σ 2 ) = −χ (σ 2 ,σ 1 ), for the spin singlet state, and symmetric, χ (σ 1 ,σ 2 ) = χ (σ 2 ,σ 1 ), for the 34 Introduction to Superconductivity spin triplet state. Since the full pairing wavefunction,  (σ 1 , r 1 ; σ 2 , r 2 ), is always antisymmetric under the exchange of two electrons, the spatial wavefunction cor- responding to the spin singlet and triplet pairing states should be symmetric and antisymmetric, respectively. Under the exchange of two electrons, the coordinate of the center of mass R is invariant, but the relative coordinate r changes sign.

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  Superconductivity

  Fundamentals and Applications

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

    (Publisher)

  4 We note that the electrostatic repulsion is highly screened by the positive charges of the atomic ions. Fig. 3.4 Electron-electron interaction via phonons. 3.1 Conventional Superconductivity 117 ) p p p ( m 2 1 2 z 2 y 2 x k + + = ε ) k k k ( m 2 2 z 2 y 2 x 2 k + + = ε is active ranges between 100 and 1000 nm. 5) This length is referred to as the BCS coherence length x 0 of the Cooper pair. This length must not be mixed up with the Ginzburg-Landau coherence length x GL , which indicates the length scale within which the total system of Cooper pairs can change. The BCS coherence length x 0 can also be interpreted as the average size of a Cooper pair. In a highly simplified way we can say that in a pure superconductor a Cooper pair has an average size between 10 2 and 10 3 nm. This size is large compared to the average distance between two conduction electrons, which amounts to a few times 10 –1 nm. The Cooper pairs strongly overlap. Within the space of a single pair there exist 10 6 to 10 7 other electrons, each being correlated in pairs. Intuitively we expect that a system of such strongly overlapping particles must have unusual properties. This will be discussed in the next section. 3.1.2 The Superconducting State, Quasiparticles, and BCS Theory At least qualitatively we have seen that two electrons attract each other momentarily because of the electron-phonon interaction, and in this way they form a Cooper pair { k X , – k Y }. Next we deal with the question of how these pairs collectively can occupy the same quantum state. Here Bardeen, Cooper, and Schrieffer found an ingenious answer, which we want to outline for the case of zero temperature. Let us recall first the situation of the unpaired electrons. At T = 0 they occupy the lowest possible energy states.

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  Advanced Solid State Physics

  • Philip Phillips(Author)
  • 2012(Publication Date)
  • Cambridge University Press

    (Publisher)

  As we will see, it is the Pauli principle that makes BCS theory work so well. What do we mean by this? In BCS theory, it is assumed that electrons form Cooper pairs, and the pairs are strongly overlapping. Such a strong overlap would imply a strong correlation between pairs. In fact, it is the correlations between pairs that accounts for most of the ob-served properties of superconductors, for example the energy gap and the Meissner effect. In BCS theory, however, there is no explicit dynamical interaction between Cooper pairs. The only interaction, if it can be thought of in these terms, is that arising from the Pauli exclusion principle which precludes two Cooper pairs from occupying the same momentum state. That BCS theory works so well speaks volumes for the real nature of pair–pair correlations in met-als. It would suggest that real pair–pair interactions in a metal arise primarily from the Pauli exclusion principle, rather than from some additional dynamical interaction. It is primarily for this reason that the simple pairing hypothesis of BCS has had such profound success. 12.1 Superconductivity: phenomenology At the outset, we lay plain the experimental facts that any theory of superconductivity must explain. 189 190 Superconductivity in metals B ( z ) z B 0 superconductor vacuum Fig. 12.1 Fall-off of the magnetic field in a Type I superconductor. H c ( T ) H c (0) 7.19 4.15 1.2 T (K) Pb Hg Al N S Fig. 12.2 The dependence of the critical field as a function of temperature. The temperatures indicated on the horizontal axis represent the superconducting transition temperatures for a series of metals. (a) Zero resistance The typical signature of superconductivity is the vanishing of the electrical resistance below some critical temperature T c . The superconducting state is a thermodynamically distinct state of matter. Below T c , a current flows without any loss. Until the high-T c materials were made, Nb held the highest transition temperature at 9.26 K.

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  Introduction to Many-Body Physics

  • Piers Coleman(Author)
  • 2015(Publication Date)
  • Cambridge University Press

    (Publisher)

  Superconductivity involves an 14.4 Physical picture of BCS theory: pairs as spins 501 analogous quantity to spin, which we will call isospin, which describes orientations in charge space. The pairing field  can be regarded as a transverse field in isospin space. To bring out this physics, it is convenient to introduce the charge analogue of the electron spinor, the Nambu spinor, defined as ψ k =  c k↑ c † −k.↓  electron hole (14.61) with the corresponding Hermitian conjugate ψ † k =  c † k↑ , c −k↓  . (14.62) Nambu spinors behave like conventional electron fields, with an algebra {ψ kα , ψ † k  β } = δ αβ δ k,k  , (14.63) but instead of up and down electrons, they describe electrons and holes. These spinors enable us to unify the kinetic and pairing energy terms into a single vector field, analogous to a magnetic field, that acts in isospin space. The kinetic energy can be written as  k  k (c † k↑ c k↑ − c −k↓ c † −k↓ + 1) =  c † k↑ , c −k↓    k 0 0 − k   c k↑ ¯ c † −k↓  +  k  k , (14.64) where the sign reversal in the lower component derives from anticommuting the down- spin electron operators. The energy − k is the energy to create a hole. We will drop the constant remainder term ∑ k  k . We can now combine the kinetic and pairing terms into a single matrix:  k  σ c † kσ c kσ +  ¯ c −k↓ c k↑ + c † k↑ c † −k↓   =  c † k↑ , c −k↓    k  ¯  − k   c k↑ c † −k↓  = ψ † k   k  1 − i 2  1 + i 2 − k  ψ k = ψ † k [ k τ 3 +  1 τ 1 +  2 τ 2 ]ψ k , (14.65) where we denote  =  1 − i 2 , ¯  =  1 + i 2 and we have introduced the isospin matrices  τ = (τ 1 , τ 2 , τ 3 ) =   0 1 1 0  ,  0 −i i 0  ,  1 0 0 −1   . (14.66) By convention the symbol  τ is used to distinguish a Pauli matrix in charge space from a spin σ acting in spin space.

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  Superconductivity

  • Charles P. Poole, Horacio A. Farach, Richard J. Creswick(Authors)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  150 6 BCSTHEORY a square well electron-electron interac-tion potential, which is also the case treated in the original formulation of the theory. II. COOPER PAIRS One year before publication of the BCS theory, Cooper (1956) demonstrated that the normal ground state of an elec-tron gas is unstable with respect to the formation of bound electron pairs. We have used quotation marks here because these electron pairs are not bound in the ordinary sense, and the presence of the filled Fermi sea is essential for this state to exist. Therefore this is properly a many-electron state. In the normal ground state all one-electron orbitals with momenta k

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  A Course on Many-Body Theory Applied to Solid-State Physics

  • Charles P Enz(Author)
  • 1992(Publication Date)
  • WSPC

    (Publisher)

  The lattice displacement and charge density waves together can move rigidly through the lattice thus Chapter 4- Superconductivity © 179 resulting in a supercurrent. The solution of the general nonlinear problem of superconductivity was given by BCS in their gap equation (Eq. (2.39) of Ref. 5): It determines the energy gap A of the condensate formed by Cooper pairs ' of loosely coupled electrons localized close to the Fermi surface, as opposed to the Schafroth pairs ' which are quasi-bound states localized in physical space. The discovery of the new cuprate superconductors has again called into question this traditional phonon-mediated pairing mechanism; the presence of electron pairs, however, is confirmed by the observation of the charge 2e 15) . On the one hand, the coherence length £ 0 which is a measure of the spatial extension of the electron pairs (see Ref. 3, p. 156), is about 100 times smaller than in conventional superconductors , £ c ~ 1.5 - 4 A parallel and £ oi) ~ 14 - 30 A perpendicular to the c-axis ' (see also Batlogg in Ref. 17, p. 44). This suggests that the cuprates might be closer to Schafroth pairing and strong coupling, particularly along the c-axis, and raises the question of the existence of a pairing temperature T above the condensation temperature T c . But so far there is no indication of this eventuality. On the other hand, the discussion of pairing mechanisms in the cuprate superconductors has focussed essentially around the alternative of a Fermi-liquid or a non-Fermi-liquid description. The former further subdivides into a conventional Fermi liquid picture dominated by a charge-transfer mechanism between the conduction layers of the Cu0 2 planes and the out-of-plane charge-reservoir layers (see Jorgenson in Ref. 17, p. 34) and a non-conventional Fermi liquid picture which is dominated by strong antiferromagnetic correlations giving rise to spin bags (see Schrieffer in Ref. 17, p. 55).

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  Superconductors

  Properties, Technology, and Applications

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

    (Publisher)

  As mentioned in the introduction, this pair is called a ‘Cooper pair’. Once part of a Cooper pair, the two electrons can be described by a single quantum mechanical wave function ߰ ൌ ȁ߰ȁ݁ ௜ఝ . The spatial extent of this wavefunction, or the ‘size’ of a cooper pair, is known as the coherence length ( ξ ) and φ is the phase of the wavefunction. If the superconducting state is made possible by electrons ‘holding hands’ to form Cooper pairs, what happens to the superconductivity when the diameter of the nanowire the Cooper pairs are supposed to go through is smaller than ξ ? Do the Cooper pairs break up and is superconductivity destroyed in these quasi-1D systems? This dilemma lies at the root of why superconductivity in quasi-1D systems is different. It is also responsible for the plethora of novel phenomena seen on studying superconductivity in quasi-1D wires as described below. 3.1 Phase slips – Thermally activated and quantum In the superconducting state, the phase φ is spatially coherent. This means that if the phase at any one point is known, the phase at any other point can be predicted. Fluctuations (thermal and quantum) lead to loss of phase coherence in superconducting samples from time to time. The region where the coherence is lost becomes temporarily normal ( φ becomes ill defined). The spatial extent of such a region is given by ξ . For 3D or 2D samples, these normal regions without phase coherence do not affect the transport measurement as the charge carriers can bypass them. In a quasi-1D sample however, since the diameter (d) < ξ , the normal region encompasses the entire cross section of the wire and cannot be bypassed. Therefore, phase slips result in a loss of superconductivity in quasi-1D systems. Mathematically, this can be understood using Josephson relation:

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  Superconductors

  Materials, Properties and Applications

  • Alexander Gabovich(Author)
  • 2012(Publication Date)
  • IntechOpen

    (Publisher)

  The Chapter addresses the problem of the PG which is believed to appear most likely due to the ability of a part of conduction electrons to form paired fermions (so-called local pairs) in a high-T c superconductor at T ≤ T ∗ [6, 13, 22–27] 2. Theoretical background There are two different approaches to the question of the mechanisms for SC pairing of charge carriers in cuprates and therefore the physical nature of the PG [2, 27]. In the first approach, pairing of charge carriers in HTS’s is of a predominantly electronic character, and the influence of phonons is inessential [3–5, 28, 29]. In the second approach, pairing in HTS’s can be explained within the framework of the Bardeen-Cooper Shieffer (BCS) theory, if its conclusions are extended to the case of strong coupling [12, 30–32]. However, it gradually became clear that aside from the well-known electron-phonon mechanism of superconductivity, due to the inter-electronic attraction by means of phonon exchange [33, 34], other mechanisms associated with the inter-electronic Coulomb interaction can also exist in HTS’s [3–8, 10, 12, 35, 36]. That is why it is not surprising that the systems of quasiparticle electronic excitations, where factors other than phonons and excitons resulting in inter-electronic attraction and pairing are considered, are studied in a considerable number of theoretical investigations of HTS’s. Some examples are charge-density waves [7, 37, 38], spin fluctuations [3, 11, 39–41], ”spin bag” formation [42, 43], and the specific nature of the band structure -”nesting” [44]. The main distinguishing feature of these investigations compared to the conventional superconductors is the more detailed study of the models based on the existence of strong interelectronic repulsion in the Hubbard model which can result in anisotropic d-pairing [35, 45].

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  Superconductors

  New Developments

  • Alexander Gabovich(Author)
  • 2015(Publication Date)
  • IntechOpen

    (Publisher)

  Section 1 Experiment Chapter 1 Superconductivity and Physical Properties in the K x MoO 2-δ L. M. S. Alves, B. S. de Lima, M. S. da Luz and C. A. M. dos Santos Additional information is available at the end of the chapter http://dx.doi.org/10.5772/59672 1. Introduction In a normal metallic conductor the electrical resistivity decreases until it reaches a lower limit (˃ 0) when heat energy is removed from the system. In these materials electrons are scattered by the lattice and obey Fermi-Dirac statistics [1]. The behaviour of a gas of identical particles at low temperature depends on the spin of the particle. Fermions are formed by half-integral spin, and obey Pauli’s exclusion principle, wherein two of them cannot have the same quantum numbers [1]. However many metals, alloys and compounds exhibit the electrical resistivity dropping suddenly to zero and exclude magnetic flux completely when cooled down to a sufficiently low temperature [1]. This phenomenon is known as superconductivity. It was observed first by Kamerlingh Onnes in 1911, a couple of years after the first liquefied helium [2]. At a critical temperature T C , the material undergoes a phase transition from a normal state to a superconducting state. At that point an electron-electron attraction arises mediated by phonons and the pairs formation of electron is favourable. Electron pairs are named Cooper pairs [3,4]. Cooper pairs are a weakly bound pair of electrons, each having equal but opposite spin and angular momentum. It is interesting to note that when a superconductor is cooled below its critical temperature its electronic properties change appreciably, but its crystalline structure remains the same, as revealed by X-ray or neutron diffraction studies. Furthermore, the formation mechanism of photons that depends on the thermal vibrations of the atoms remains the same in the super‐ conducting phase as in the normal state.

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