Phys 769: Selected Topics in Condensed Matter Physics Summer 2010 Lecture 10: Superconducting devices Lecturer: Anthony J. Leggett TA: Bill Coish Introduction So far, we described only the superconducting ground state. Before proceeding to the main topic of this lecture, we need to say a little about the simplest excited states. For this purpose it is easier to use the original representation of BCS, in which the condition of particle number conservation is relaxed and the ground state has the form of a product of states referring to the occupation of the pair of states (k ↑, −k ↓): in obvious notation, 2 2 ΨBCS = Φk, Φk = uk |00i + vk |11i , |uk| + |vk| = 1 (1) Yk where |00i is the state in which neither k ↑ nor −k ↓ is occupied, |11i is that in which both are, and uk,vk are complex coefficients (though the usual convention is to take uk real). From the identification (lecture 9, Eq. (30)) of the Fourier-transformed wave function Fk with the “anomalous average” hN − 2| a−k↓ak↑ |Ni (which in the BCS representation becomes simply ha−k↓ak↑i we find ∗ Fk = ukvk. (2) In this representation we can associate a contribution to the kinetic and potential energy with each pair state (k ↑, −k ↓) separately (but must remember not to double-count). The contribution hV ik to the potential energy (excluding as usual the Hartree and Fock terms) is simply ∗ hV ik = −2∆kFk. (3) In the case of the kinetic energy, it is most convenient to calculate it relative to the value in the normal ground state (hnki = θ(kF − k)), which is 0 for ǫk > 0 and −2|ǫk| for ǫk < 0. This gives (cf. Eq. (31) of lecture 9) 2 δ hTki = |ǫk|−|ǫk| /Ek (4) independently of the sign of ǫk. Using Fk = ∆k/2Ek, we find that the total change in the contribution to the ground-state energy of the pair state (k ↑, −vk, ↓) due to pairing is 1 |ǫk|− Ek. So far, we essentially have just an alternative notation for describing the ground state. However, it is clear that the occupation space of the pair (k ↑, −k ↓) is spanned by the states |00i , |01i , |10i, and |11i, and is thus 4-dimensional. Thus, there must be three other energy eigenstates besides the state Φk of Eq. (1), and it is clear that within the general framework of the BCS ansatz1 these may be taken to be the “broken-pair” (BP) states |01i and |10i and the (unique) combination of |00i and |11i which is orthogonal to Φk, namely ∗ ∗ ′′ vk |00i− uk |11i (“excited pair states, (EP)) (5) we wish to evaluate the energy of these three states relative to the ground state (“ground pair”, GP) states. In the case of the EP state the evaluation is straightforward, since it is easily verified that the transition from Eq. (5) to Eq. (3) involves the replacements of Fk with −Fk and of the − sign in Eq. (31) of lecture 9 by a + sign. Thus, the difference in energy between the GP and EP states is (again using Fk = ∆k/2Ek) 2 ∗ EEP − EGP = 2|ǫk| /Ek + 4∆kFk = 2Ek (6) In calculating the energy necessary to excite the “broken pair” states from the GP one, we need to use the fact that this state does not contribute at all to the pairing energy, while its contribution to the kinetic energy (relative to the normal ground state) is just |ǫk|. In view of Eq. (4), this means that the change in kinetic energy on going from the 2 GP to the broken-pair state is +|ǫk| /Ek, so that the total excitation energy is according to Eq. (3) 2 ∗ EBP − EGP = |ǫk| /Ek + 2∆kFk = Ek (7) A standard representation in the literature is to regard the BP states |01i and |10i as each containing a single “Bogoliubov quasiparticle” while the EP state contains two such quasiparticles. If one takes this point of view, then it is easy to see that (for example) the operator which, acting on the ground-pair states (Eq. (1)) creates the (correctly normalized 1i.e. an ansatz in which the density matrix is factorized into terms associated with the different pairs (k ↑, −k ↓). 2 state |10i has to be of the form2 † † αk↑ = ukak↑ − vka−k↓ (8) while that for |01i is † † α−k↓ = uka−k↓ + vkak↑ (9) Apart from the form of the Bogoliubov operators, which we will need later in the context of the Josephson effect, the most important conclusion of the above argument is simply that the minimum energy needed to break any Cooper pair is given by the quantity 2 2 1/2 Ek ≡ ǫk + |∆k| ≥|∆k|∼ ∆ (10) where at the last step we have assumed that the energy gap ∆ is approximately constant over the Fermi surface (true for the classic superconductors). Let us estimate the number of broken pairs (or, what is equivalent in this limit, the number of excited particles) in the limit kBT ≪ ∆ (fairly easily obtained in most of the classic (BCS) superconductors such as Al or Nb). In this limit we need not worry about the possibility of excitation of the EP state, and the probability of a broken pair(either |01i or |10i) is approximately exp −Ek/kBT . Hence the total number hnexci of excitations is hnexci≃ 2 exp −Ek/kBT Xk ∞ ∞ dn 2 2 1/2 −1 2 2 1/2 = Ω dǫ exp − ǫ + ∆ /kBT ∼ Nǫ dǫ exp − ǫ + ∆ /kBT (11) dǫ Z F Z −∞ −∞ 2 2 1/2 1 2 or on expanding (ǫ + ∆ ) ≃ ∆+ 2 ǫ /∆, 1/2 ∆kBT hnexci∼ N 2 exp −∆/kBT (12) ǫF where N is the total number of electrons in the superconductor. It is interesting to ask what is the condition to have nexc ≪ 1. Neglecting the rather weak prefactor, we see that a sufficient condition for this is kBT ≪ ∆/ ln N. So if we consider 3 e.g. a 1cm block of Nb, where ∆ ∼ 1.75kBTc ∼ 16 K, we see that the condition is roughly (ln 1023 ∼ 60) T ≪ 250 mK – almost routinely attainable with current cryogenic technology! One might worry whether this argument could be vitiated by the existence of excitations, for example associated with the edge of the sample, which have energies ≪ ∆. Such excitations 2It is important to choose (and keep!) a definite convention for the state |11i relative to |00i. I choose it † † here so that |11i ≡ ak↑a−k↓ |00i. 3 could of course not propagate into the bullk, but it is possible that they might be a nuisance in the operation of a superconducting device, which as we shall see relies in some sense on having a “pure” condensate. Probably the best answer to this objection is a combination of experiment and the arguments given in lecture 2: any such excitations do not seem to affect the classical motion of the relevant macroscopic variable (the phase or flux, see below) very much, so it is not surprising that neither do they affect the quantum-coherent behavior. One other point is worth mentioning: In the arguments of lecture 9, and above, it was implicitly assumed that the total number of electrons is even. What if it is odd? Actually, there are experiments which show that a well-insulated superconducting grain does “know the difference” between (e.g.) 109 and 109 + 1, and this is even exploited in some “Cooper- pair box” devices. However, in most realistic experimental setups the system is in electrical contact with some “normal” external reservoir – e.g. the metal of the normal leads – so that it is not the total particle number N but the chemical potential µ which is held constant. Under these conditions the energy necessary to add an extra Cooper pair is just 2µ, which is exactly the energy necessary to take two electrons out of the normal reservoir, so it is realistic to work in the grand canonical ensemble and regard the density matrix of the system as an incoherent mixture3 of states involving different even numbers of electrons. On the other hand, to add or subtract a single electron takes an energy µ + Ek, so single electrons approaching from the normal metal are Andreev-reflected and do not enter the superconductor. Thus the assumption that N =even is justified. Thus, in the remainder of this lecture it will be assumed unless otherwise stated that the bulk superconducting samples are in their pure quantum ground state, i.e. no electronic excitations whatsoever are present. The Josephson effect The Josephson effect occurs, rather generically, whenever two bulk superconductors are connected by a “weak link”, that is a region which prevents a much increased impedance to the transport of single electrons and hence a fortiori of Cooper pairs. There are many types of weak link (tunnel oxide junction, point contact, microbridge, SNS junction, . ) but the one which is mostly employed in superconducting devices is the original tunnel oxide junction discussed in Josephson’s paper, which compared to the other types is (a) better controlled and (b) subject to less dissipation. I will therefore confine the discussion to that. 3Not, as many textbooks would have us believe, a coherent superposition of such states. The latter is a figment of the BCS approach. 4 In the following it is convenient to introduce the notion of the “order parameter” Ψ of a bulk superconductor. We will consider a spatially uniform superconductor with no current flow. Then up to an overall constant factor which is a matter of convention, we can define the order parameter by the equation Ψ= Fk (13) Xk It is then easy to see that it is (proportional to) the Cooper-pair wave function F (r) evalu- ated at r = 0, i.e.