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: http://www-qm.phy.cam.ac.uk/teaching.php 1
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 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 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
1908 Liquefaction of 4 He Timeline: 1911 Superconductivity in mercury 1925 Prediction of Bose-Einstein condensation (BEC) Kamerlingh Onnes 1927/38 1933 Superfluidity in 4 He Meissner effect 1950 1952/57 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
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
Superconducting elements: (from www.webelements.com - see also examples sheet) 7
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
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). 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
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
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
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 2 2 4 for < 0, the minimum is at = 0! order 13
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
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. 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