Superconductivity and Quantum Coherence

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1 Superconductivity and Quantum Coherence Lent Term 2008 Credits: Christoph Bergemann, David Khmelnitskii, John Waldram, 12 Lectures: Mon, Wed 10-11am Mott Seminar Room 3 Supervisions, each with one examples sheet This is a developing course feedback is welcome! Complete versions on course web site: 1

unified treatment of all three phenomena J. R.

textbook with deep discussions, including copper oxide

Tinkham: Introduction to Superconductivity traditional

2 Literature: J. F. Annett: Superconductivity, Superfluids and Condensates unified treatment of all three phenomena J. R. Waldram: Superconductivity of Metals and Cuprates modern textbook with deep discussions, including copper oxide superconductors M. Tinkham: Introduction to Superconductivity traditional textbook V. V. Schmidt: The Physics of Superconductors helpful insights C. J. Pethick/H. Smith: Bose-Einstein Condensation (BEC) in Dilute Gases BEC and superfluidity; recent developments 2

Outline: Phenomenology of the Superconducting State (4 lectures) Applications of Superconductivity (1) Bose-Einstein Condensates (1) Superfluidity in 4 He (1) Macroscopic Ginzburg-Landau

3 Outline: Phenomenology of the Superconducting State (4 lectures) Applications of Superconductivity (1) Bose-Einstein Condensates (1) Superfluidity in 4 He (1) Macroscopic Ginzburg-Landau Treatment Quantum Coherence and Bardeen- Cooper-Schrieffer (BCS) Theory (3) Microscopic Theory Superfluidity in 3 He and Unconventional Superconductivity in Exotic Materials (2) New Developments 3

Lecture 1: Historical overview Macroscopic manifestation of superconductivity:,, C/T Meissner effect and levitation Type-I and type-ii superconductivity

4 Lecture 1: Historical overview Macroscopic manifestation of superconductivity:,, C/T Meissner effect and levitation Type-I and type-ii superconductivity Superconductivity as an ordered state Landau theory as a precursor to Ginzburg-Landau theory Literature: Waldram ch. 4 (or equivalent chapters in Annett, Schmidt, or Tinkham) 4

superconductivity Abrikosov vortices 1957 BCS theory of superconductivity 1962/64 Josephson effect and SQUIDs 1971

5 1908 Liquefaction of 4 He Timeline: 1911 Superconductivity in mercury 1925 Prediction of Bose-Einstein condensation (BEC) Kamerlingh Onnes 1927/ Superfluidity in 4 He Meissner effect /57 Ginzburg-Landau theory of superconductivity Abrikosov vortices 1957 BCS theory of superconductivity 1962/64 Josephson effect and SQUIDs 1971 Superfluidity in 3 He 1970s-now Unconventional superconductors including high temperature superconductors 1990s-now BEC and BCS in atomic gases? 5

6 Examples of Superconductors Hg Nb NbTi Nb 3 Sn MgB 2 CeCu 2 Si 2, UBe 13 La 2-x Ba x CuO 4 YBa 2 Cu 3 O 7- HgBa 2 Ca 2 Cu 3 O 8+ Sr 2 RuO 4 UGe 2 first superconductor ever discovered highest T c amongst the elements used in superconducting magnets up to ~9 T used in superconducting magnets up to ~20 T highest T c amongst conventional superconductors first of the heavy-fermion superconductors first of the cuprate superconductors cuprate superconductor with T c above liquid nitrogen temperatures highest T c superconductor to date p-wave superconductor first ferromagnetic superconductor 4.1 K 9.3 K 10 K 24.5 K 39 K ~0.8 K ~35K 92 K 164 K 1.5 K 0.3 K 6

7 Superconducting elements: (from - see also examples sheet) 7

8 Basic experimental facts: The resistivity of a superconductor drops to zero below some transition temperature T c Immediate corollary: can t change the magnetic field inside a superconductor B t curl curl J 0, since 0 B = 0 B Switch on external B: zero field cooled 8

9 What if we cool a superconductor in a magnetic field and then switch the field off do we get something like a permanent magnet? B Experimentally, this does not work even when field cooled, the superconductor expels the field! field cooled B This is known as the Meissner effect and suggests that the superconducting transition is a true thermodynamic phase transition. field cooled 9

The Meissner effect leads to the stunning levitation effects that underlie many of the proposed technological applications of superconductivity (see examples sheet).

10 The Meissner effect leads to the stunning levitation effects that underlie many of the proposed technological applications of superconductivity (see examples sheet). The superconducting state is destroyed above a critical field H c Ideal magnetisation curve H c1 H c H c2 H and so-called type-ii superconductivity (which we ll discuss later) B M NB: These curves apply for a magnetic field along a long rod. 10

11 Picture credits: A. J. Schofield So, if we are really faced with a phase transition, we should have a look at the specific heat: anomaly at T c consistent with second order phase transition exponential low-t behaviour indicative of energy gap (explained by BCS) exponential in simple superconductors power-law behaviour at low-t in unconventional superconductors (to be discussed later) areas match to conserve entropy 11

12 From the form of C/T we find that the entropy vs temperature has the following form: S T c The superconducting state has lower entropy and is therefore the more ordered state. From what we know so far, the nature of the order parameter is unclear. However, a general theory based on just a few reasonable assumptions about the hypothetical order parameter is remarkably powerful. It describes not just BCS superconductors but also the high-t c s, superfluids, and BECs. This is known as Ginzburg-Landau theory. T 12

13 Landau Theory: For a second order phase transition, the order parameter vanishes continuously at T c. In the Landau theory one assumes that sufficiently close to T c the free energy can be expanded in a Taylor series in the order parameter, (assumed for now to be real): F( ) if F is an even function Where is the free energy minimum? for > 0, the minimum is at = 0! disorder for < 0, the minimum is at = 0! order 13

14 Picture credits: A. J. Schofield Free energy curves: > 0 < 0 F F 0 0 The phase transition takes place at (T c ) = 0. Thus, a power series expansion of (T) around T c may be expected to have the following leading form: a( T T ) (and is a constant) c This is enough to describe a second order phase transition, complete with specific heat jump (examples sheet). 14

15 This description is appropriate for, e.g., a magnetic phase transition where in the magnetization. In the Ginzburg-Landau (GL) theory, however, is assumed to be complex rather than real as is the case for a macroscopic wave function. We will see in a later lecture how a complex order parameter arises naturally from a microscopic theory. The assumptions in the GL theory are: can be complex-valued can vary in space but this carries an energy penalty proportional to 2 couples to the electromagnetic field in the same way as an ordinary wavefunction (Feynmann, Lectures III, ch. 21) Here, A is the magnetic vector potential and q is the relevant charge, which experimentally turns out to be q = 2e. 2 2 iqa / 4, 4 15

This provides the first clue that superconductivity has got something to do with electron pairs. The idea of electron pairing is central to the microscopic theory.

16 This provides the first clue that superconductivity has got something to do with electron pairs. The idea of electron pairing is central to the microscopic theory. A final part in the free energy that must not be forgotten is the relevant magnetic field energy density B M2 /2m 0, where B M =B-B E is due to currents in the superconductor and B E is due to external sources. (Note that when the material is introduced the total field energy density changes from B E2 /2m 0 to B 2 /2m 0, but B M B E /m 0 is taken up by the external sources (Waldram Ch.6)). So finally we arrive at the Ginzburg-Landau free energy density: We have written the free energy so that the gradient term involve an effective mass m = 2m e, which is consistent with q = 2e. 16

Superconductivity and Superfluidity

Superconductivity and Superfluidity Contemporary physics, Spring 2015 Partially from: Kazimierz Conder Laboratory for Developments and Methods, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland Resistivity