Introduction to Superconductivity

Transcription

1 Introduction to Superconductivity Vladimir Eltsov Aalto University, Finland Cryocourse 2016

2 CONTENTS Basic experimental properties of superconductors Zero resistivity Meissner effect Phase diagram Heat capacity Phenomenological description of superconductivity Thermodynamics Phase coherence and supercurrents London equation Flux quantization Energy gap and coherence length Clean and dirty Type I and type II superconductors Abrikosov vortices Introduction to the BCS theory Electron-electron attraction The Cooper problem Model Hamiltonian Bogolubov transformation Bogolubov quasiparticles The gap equation Non-uniform superconductors Bogolubov de Gennes equations Andreev reflection and Andreev bound states Scattering at the NIS interface Bound states in SNS, SIS junctions CdGM states in vortex cores Supercurrent with bound states Josephson effect DC and AC effect Shunted junction and washboard potential SQUID Quantum dynamics

3 QUANTUM MECHANICS AT A HUMAN SCALE Superconductivity was the first discovered macroscopic quantum phenomenon, where a sizable fraction of particles of a macroscopic object forms a coherent state, described by a quantum-mechanical wave function. Applications: High magnetic fields Filtering of radio signals Ultra-sensitive magnetic/temperature and other sensors Metrology Ultra-low-noise amplifiers Quantum engineering Quantum information processing

4 DISCOVERY OF SUPERCONDUCTIVITY Mercury sample, 1911 Heike Kamerlingh Onnes Liquefied helium in Resistivity of metals when T 0? Au, Pt: ρ(t ) = const Resistance, Ohm Temperature, K

5 BASIC EXPERIMENTAL PROPERTIES: Zero resistivity ρ n ρ ρ = 0 T c T Below the transition temperature T c, the resistivity drops to zero. Latest measurements put bound ρ < cm. Really zero! Dirty metals are good superconductors.

6 THE MEISSNER EFFECT T > T c T < T c Walther Meissner B B At T < T c the magnetic field is expelled from the superconductor even if the field was applied before reaching T c. Robert Ochsenfeld

7 PHASE DIAGRAM OF A SUPERCONDUCTOR Superconductivity is destroyed by sufficiently strong electric currents or by the magnetic field above the critical field H c. H H c (0) normal 1st order H c (T ) superconductor 2nd order T c T Empirically the dependence of H c on temperature is described reasonably well by H c (T ) = H c (0)[1 (T /T c ) 2 ]

8 THERMODYNAMICS OF THE SUPERCONDUCTING TRANSITION In the external field H = H c N S F n (T ) = F s (T ) + H 2 c (T ) 8π Entropy S = F/ T : S n S s = H c dh c 4π dt. Heat capacity C = T S/ T : C n C s = T 4π [ H c d 2 H c dt 2 + ( ) ] 2 dhc. dt The heat capacity jump at T c : C s (T c ) C n (T c ) = T c 4π ( ) 2 dhc. dt

9 HEAT CAPACITY C C s exp( /k B T ) C n = γ T 0 T c T Heat capacity jumps upward on transition to superconducting state. At T T c the electronic heat capacity approaches zero exponentially. Energy gap

10 PHASE COHERENCE Electrons form pairs, which are bosons and condense to a single state, described by the wave function ψ with the macroscopically coherent phase χ. ψ = ψ exp(iχ) j s Current with j s = e [ ψ ˆpψ + ψ ˆp ψ ] 2m ˆp = i h (e /c)a. We get for e = 2e, m = 2m and the density of pairs ψ 2 = n s /2: j s = (e ) 2 ( m c ψ 2 A hc ) ( e χ = e2 n s A hc ) mc 2e χ.

11 THE LONDON EQUATION Assuming density of superconducting electrons n s is constant: j s = e2 n s mc ( A hc ) 2e χ Using the Maxwell equation curl j s = e2 n s mc curl A = e2 n s mc h. j s = (c/4π) curl h Fritz London we get the London equation h + λ 2 L curl curl h = 0, where the London penetration depth Heinz London ( mc 2 ) 1/2 λ L =. 4πn s e 2

12 MEISSNER EFFECT h + λ 2 L curl curl h = 0 h = λ2 L 2 h [curl curl h = div h 2 h] h y S 2 h y x 2 λ 2 L h y = 0 h h y = h y (0) exp( x/λ L ) 0 λ L x Magnetic field penetrates into a superconductor only over distances ( mc 2 λ L = 4πn s e 2 ) 1/2 Typical metal with m m e and a 0 4 Å, n s a 3 0 λ L 30 nm

13 Integrating expression MAGNETIC FLUX QUANTIZATION j s = e2 n s mc ( A hc ) 2e χ along a closed contour within a superconductor mc n 1 e 2 s j s dl = curl A ds hc hc χ = 2e 2e 2πn l B S is the magnetic flux through the contour. = + 4π λ 2 L c j s dl = 0 n Quantum of magnetic flux 0 = π hc e Oe cm 2. In SI units, 0 = π h/ e = T m 2.

14 THE ENERGY GAP AND COHERENCE LENGTH Energy scale: Energy gap 0 Binding energy of Cooper pair 2 0, 0 k B T c. Energy gap (from the heat capacity) (T 0) = 0. Uncertainty principle: interaction time in the pair τ p h/ 0. Length scale: Coherence length ξ 0 : ξ 0 ξ 0 τ p v F hv F 0 v F

15 COHERENCE LENGTH AND THE CRITICAL FIELD Maximum phase gradient ( χ) max 1/ξ sets the maximum supercurrent j s = e2 n s mc ( A hc ) 2e χ (j s ) max hn s e m ( χ) max Maximum supercurrent is reached when screening the critical field H c Critical field H c 4πλ L c hn s e mξ = π hc e (j s ) max (c/4π)h c /λ L λ L 4πn s e 2 = 0 πξ mc 2 πξλ L E F Condensation energy [remember ξ 0 hv F / 0 ] 0 F n (0) F s (0) = H c 2(0) 8π n 2mvF [N(0) 0] 0 N(0) n/e F density of states, E F = mvf 2 /2 Fermi energy

16 LONDON AND PIPPARD REGIMES Two length scales: London penetration depth and coherence length. How do their values compare? Typical metallic superconductor [like Al] with T c = 1 K, the electron density n one per ion, lattice constant a 0 4 Å: n s n = a cm 3, v F = p F /m = ( h/m)(3π 2 n) 1/ cm/s ξ 0 = hv ( F mc 2 ) 1/ nm λ L = 30 nm 2πk B T c 4πn s e 2 ξ 0 ξ 0 H λ L H λ L T T c n s 0 λ L Pippard regime London regime

17 DIRTY AND CLEAN LIMITS Non-magnetic impurities (scattering centers) do not affect static properties of a superconductor (like T c ) [Anderson theorem]. But properties connected to the spatial variation of the superconducting state (in particular, supercurrents and coherence length) are strongly affected. Interaction time τ p h/ 0. ξ clean ξ dirty v F v F l Clean limit ballistic motion ξ clean = ξ 0 τ p v F hv F 0 Dirty limit diffusive motion mean free path l scattering time τ s = l/v F < τ p diffusion coefficient D vf 2 τ s = v F l ξ dirty = Dτ p = v F l h/ 0 = ξ clean l In dirty materials l a 0 λ L and ξ dirty < λ L

18 TYPE I AND TYPE II SUPERCONDUCTORS In bulk superconductor magnetic induction B = H + 4πM = 0. The magnetization and susceptibility are ideal diamagnetic M = H 4π ; χ = M H = 1 4π 4πM type I type II H c1 H c H H c2

19 INTERMEDIATE STATE OF TYPE I SUPERCONDUCTORS In the external field H < H c the sample is divided to normal and superconducting domains so that in the normal phase magnetic induction B = H c while in the superconducting domains magnetic field is absent. S N S B = 0 N S N S B = H c B = H < H c

20 ENERGY OF THE NS BOUNDARY h(x) λ L 0 magnetic loss cond. ξ ψ(x) type I: 2λL < ξ x h(x) type II: 2λL > ξ 0 magnetic ξ λ L gain cond. ψ(x) x Energy change compared to the uniform state at H = H c F = H 2 c 8π δ : ξ λ L δ ξ λ L > 0 δ 1.89ξ ξ λ L δ (λ L ξ) < 0 δ 1.104λ L Transition from the positive to the negative energy of the NS boundary is controlled by the Ginzburg-Landau parameter κ: κ = λ L 1, transition at κ =. ξ 2

21 THE GL PARAMETER FROM MATERIAL PARAMETERS Order-parameter variation scale coherence (healing) length ξ(t ) is not the Cooper-pair size ξ 0 (or ξ dirty = ξ 0 l): ξ(t ) : gradient energy [ξ 0 ] condensation energy [T -dependent] In the clean limit we have ξ(t ) = ξ 0 7ζ(3) 12 κ = 3c e h π 7ζ(3)N(0) [ 1 T T c ] 1/2, λ L = c 4 e v F k B T c v 2 F = 3π 2 14ζ(3) hc e 2 k B T c E F ] 3 1/2 [1 TTc πn(0) e 2 /a k BT c. E F E F For usual superconductors k B T c /E F < 10 3 κ 1 For HTSC k B T c /E F κ 1 For dirty superconductors with τ s /τ p 1: κ dirty κ clean (τ p /τ s ) 1

22 ABRIKOSOV VORTICES Magnetic field penetrates into type II superconductor in the form of Abrikosov vortices, which are topologically-protected linear defects of the order parameter: Order parameter is zero at the vortex axis and the phase of the order parameter winds by 2π on a loop around the vortex. Each vortex carries a single quantum of magnetic flux 0. B Alexei Abrikosov

23 ABRIKOSOV VORTICES Magnetic field penetrates into type II superconductor in the form of Abrikosov vortices, which are topologically-protected linear defects of the order parameter: Order parameter is zero at the vortex axis and the phase of the order parameter winds by 2π on a loop around the vortex. Each vortex carries a single quantum of magnetic flux 0. B Alexei Abrikosov 4πM 0 n v n v vortex density H c1 H c H H c2

24 2 0 ABRIKOSOV VORTEX STRUCTURE r/λ L κ = 5 r λl 1 r ξ ψ(r)/ψ eq h(r)/h c r/ξ r > λ L : f = ψ(r)/ψ eq = 1, h exp( r/λ L ) ξ < r < λ L : 1 f r 2, h ln(λ L /r) r < ξ (core): f r, h const Main vortex energy here: F v π 2 λ 2 ln κ L

25 PHASE DIAGRAM OF TYPE-II SUPERCONDUCTORS H H c1 : vortex energetically favorable Vortex H c Normal H c2 F v = (1/4π) 0 H c1 H c1 = 0 ln κ 4πλ 2 ln κ = H c L 2κ H c : thermodynamic H c1 Meissner T T c 0 H c = 2 2πλ L ξ H c2 : no stable SC regions H c2 = 0 2πξ = H 2 c 2κ For ξ 2 nm one gets H c2 80 T.

26 RESISTIVITY FROM VORTEX MOTION Lorentz force acting on vortex from electric current b f L f L = 0 c [jex b]. Per unit volume we have j ex F L = n v f L = c 1 [j ex (n v 0 b)] = c 1 [j ex B]. If vortex moves with friction η: v L = f L /η E = c 1 [B v L ] = ( 0 B/ηc 2 )j ex and resistivity ρ = E/j ex = 0 B/ηc 2. Magnetic field applications require pinning!

27 The BCS Theory John Bardeen Leon Cooper John Robert Schrieffer Nikolai Bogolubov

28 EXCITATIONS IN LANDAU FERMI LIQUID At T = 0 states with p < p F, E < E F are filled, others are empty. ε p n = p3 F 3π 2 h 3, p F hn 1/3 = h/a 0 p F holes particles p hole p particle p 0 p F 2 (p /2m) E F p ɛ p = { p 2 2m E F, E F p2 2m, p > p F p < p F ɛ p v F p p F f (ɛ p ) = 1 e ɛ p/k B T + 1 Constant density of states N(ɛ p = 0) = N(p = p F ) = N(0) = m p F 2π 2 h 3.

29 ELECTRON ATTRACTION: Polarization of the lattice E F ± hω D p 1 p 2 p 1 p 2 Atomic energy scale MωD 2 a2 0 e2 ( h/a 0 ) 2 a 0 m Debye frequency ω D E F h Length of electron tail m M p 1 p 2 p 1 p 2 v F ω 1 D M/m a 0 300a 0 p 1 ±p 2 p 1 p 2

30 p 2 ELECTRON ATTRACTION: Model p 1 p 1 p 2 q q II H e ph e I = 2 W q 2 h ω q < 0 ω 2 ωq 2 p 2 p 1 (1) (2) attraction if ω < ω q ω D p 1 p 2 Does not depend on the directions of p 1, p 2 orbital momentum L = 0 opposite spins Model: Two electrons with opposite momenta and spins attract each other with the constant amplitude W if ɛ p1 < E c and ɛ p2 < E c (where E c hω D ) and do not interact otherwise.

31 THE COOPER PROBLEM p F p F p p p p Schrödinger equation for the pair wave function (r 1, r 2 ) [ ] H ˆe (r 1 ) + Hˆ e (r 2 ) + W (r 1 r 2 ) (r 1, r 2 ) = E (r 1, r 2 ). Expansion in single-particle wave functions (r 1, r 2 ) = p c p ψ p (r 1 )ψ p (r 2 ) = p a p e ipr/ h = (r), r = r 1 r 2. ψ p (r 1 ) e ipr1/ h, ψ p (r 2 ) e ipr2/ h, a p = (r)e ipr/ h d 3 r

32 THE COOPER PROBLEM: Fourier transformed equation Schrödinger equation becomes 2ɛ p a p + p W p,p a p = Ea p. Interaction model: W, ɛ p and ɛ p < E c, W p,p = i.e. p F E c /v F < p(and p ) < p F + E c /v F 0, otherwise We thus have a p = W a p = W C E 2ɛ p E 2ɛ p p,ɛ p <E c 1 W = 1 (E) E 2ɛ p,ɛ p <E p c W 1 (E) E b 2ǫ 0 2ǫ n E

33 Bound state with E = E b < 0 THE COOPER PROBLEM: Bound state 1 W = 1. 2ɛ p,ɛ p <E p E b c Sum over momentum the integral over energy: 1 Ec ( ) W = 2N(0) dɛ p Eb + 2E c = N(0) ln. 2ɛ p + E b E b From this equation we obtain 0 E b = For weak coupling, N(0)W 1, we find For strong coupling, N(0)W 1, 2E c e 1/N(0)W 1 E b = 2E c e 1/N(0)W E b = 2N(0)W E c

34 THE BCS MODEL: Non-interacting Hamiltonian Non-interacting particles at k > k F and holes at k < k F H 0 = ɛ k c kσ c kσ + ɛ k h kσ h kσ. kσ,k>k F kσ,k<k F ξ k ǫ k F k k But h kσ = c k, σ and h kσ = c k, σ and we have H 0 = ɛ k c kσ c kσ + ɛ k c kσ c kσ kσ,k>k F kσ,k<k F = ɛ k c kσ c kσ ɛ k c kσ c kσ + ɛ k = ξ k c kσ c kσ + E 0. kσ,k>k F kσ,k<k F kσ,k<k F kσ Here we defined c kσ c kσ = 1 c kσ c kσ ξ k = sign(k k F )ɛ k = h 2 k 2 2m E F hv F (k k F ).

35 THE BCS MODEL: Pairing interaction Interaction (k, k ) (k, k) without affecting the quasiparticle spin. H = kσ ξ k c kσ c kσ + W kk c k c k c k }{{} c k }{{} kk A B Mean-field approximation: For two operators A and B AB = A B + A B A B + (A A )(B B ) The error is quadratic in fluctuations, which are relatively small in a macroscopic system.

36 H BCS = kσ ξ k c kσ c kσ + k THE BCS MODEL: Hamiltonian ( k c k c k + k c k c k k ) c k c k We defined k = W kk c k c k, thus k = k k W kk We want to diagonalize it with Bogolubov transformation c k c k H BCS = kσ E k γ kσ γ kσ + E cond Here γ kσ and γ kσ are new Bogolubov quasiparticles with spectrum E k. New operators mix particles and holes: γ k = u k c k + v k h k h k = c k γ k = u k c k vk h k h k = c k } } {γ kσ, γ k σ = δ kk δ σ σ, {γ kσ, γ k σ {γ } = kσ, γ k σ = 0 u k 2 + v k 2 = 1

37 THE BCS MODEL: Diagonal form The desired form is H BCS = kσ E k γ kσ γ kσ + E cond with E k = ξ 2 k + k 2 and u k 2 = 1 2 v k 2 = 1 2 ( 1 + ξ ) k E k ( 1 ξ ) k E k 1 0 v k 2 u k 2 k F Equal admixture of particle and hole k

38 E E F BOGOLUBOV QUASIPARTICLES: Energy spectrum E p = ɛ 2 p + 2 v 2 F (p p F ) holes particles p F p Landau criterion v c = min(e p /p) /p F

39 BOGOLUBOV QUASIPARTICLES: Group velocity For a given energy E > there are two possible values of ξ k : ξ ± k = ± E 2 2 = hv F (k ± k F ), k ± = k F ± 1 E2 hv 2. F ξ + k, k + particles, ξ k, k holes E k + k k k + particles holes holes particles k F 0 k F k v g = de dp = ˆk de h dk, E vparticles g = v 2 2 F ˆk, E E v holes g = v 2 2 F ˆk. E

40 BOGOLUBOV QUASIPARTICLES: Density of states Since E k = k N(0) ɛ 2 k + 2 we have dɛ k = N(0) ) d ( E 2k 2 = N(0) E k de k E 2k 2. Ns The density of states in the superconductor is 0, E, N s (E) = E N(0) E2, E >. 2 N(0) E

41 SELF-CONSISTENCY EQUATION We insert Bogolubov transformation [c kσ γ kσ (u k, v k ) ( k, E k )] into the gap definition and obtain equation for k [remember also E k ( k )] k = k W kk c k c k = [1 2f (E k )] 2E k k k W kk We used Fermi distribution γ kσ γ kσ = f (E k ), f (E) = 1 e E/k BT + 1. For our model interaction k dependence is simple { { W, ɛ k and ɛ k < E c, E k < Ec W kk = k = E c 0, otherwise 0, otherwise With these substitutions the gap equation becomes = W 1 2f (E k ) = W 1 2E k,ɛ k <E k 2 E c k,ɛ k <E k ( ) tanh E k( ) 2k B T c It has a trivial solution = 0 corresponding to the normal metal.

42 THE GAP EQUATION exp( 0 /k B T ) / = (π/γ )k B T c 1.76k B T c k B T c = (2γ /π)e c e 1/λ 1.13E c e 1/λ T /T c Ec 1 = λ (1 T /T c ) 1/2 de E2 2 tanh E 2k B T Interaction constant λ = N(0)W λ in practical superconductors

43 HEAT CAPACITY Only quasiparticles contribute to the entropy S = k B [(1 f (E k )) ln(1 f (E k )) + f (E k ) ln f (E k )] f (E k ) = The heat capacity kσ 1 e E k/k B T + 1, E k = ξk 2 + k(t ) 2 C = T ds dt = 2 k B k [ E 2 f (E k )(1 f (E k )) k T 1 2 2T S normal supercond. T c d k 2 ] dt T For T T c we have E k B T and f (E) e E/k BT. As a result C = 2 ( ) 3/2 ( 0 2πk B N(0) 0 exp ) 0. k B T k B T

44 NON-UNIFORM SUPERCONDUCTORS Real-space quasiparticle creation and annihilation operators (r, σ ) = k e ikr c kσ, (r, σ ) = k e ikr c kσ Non-interacting Hamiltonian corresponds to H 0 = ξ k c kσ c kσ H 0 = kσ σ d 3 r (r, σ ) ˆ H e (r, σ ) where the free particle Hamiltonian ξ k = h 2 k 2 2m E F Hˆ e = 1 ( i h e ) 2 2m c A + U(r) µ, which accounts for the magnetic field through the vector potential A and includes some non-magnetic potential U(r).

45 BCS THEORY IN COORDINATE SPACE Non-interacting add interaction ( ) mean-field theory diagonalize with Bogolubov transformation (r ) = k (r ) = k [ ] u k (r)γ k vk (r)γ k [u k (r)γ k + v k(r)γ k ] Here k can be any enumeration of states, not necessarily wave vector. [For uniform case u k (r) = u k e ikr and v k (r) = v k e ikr ] + completeness/orthogonality conditions on u k (r) and v k (r) from Fermi commutation relations for and γ.

46 Diagonalization condition BOGOLUBOV DE GENNES EQUATIONS { Self-consistency equation ˆ H e u k (r) + (r) v k (r) = E k u k (r) (r) u k (r) ˆ H e v k(r) = E k v k (r) Hˆ e = 1 ( i h e ) 2 2m c A + U(r) µ (r) = W k,ɛ k <E c u k (r)v k (r)[1 2f (E k)] + completeness/orthogonality conditions on u k (r) and v k (r) + Maxwell equations to connect current and field

47 Andreev reflection particle hole N S Alexander Andreev x

48 MODEL OF NS(NIS) INTERFACE Bogolubov de Gennes equations h 2 ( ie ) 2 2m hc A u(r) + [U ex(r) E F ] u(r) + (r)v(r) = ɛu(r), h 2 ( + ie ) 2 2m hc A v(r) [U ex(r) E F ] v(r) + (r)u(r) = ɛv(r). N S Use uniform bulk solutions on left and right U ex 0 x Join (u, v, x u, x v) at the interface 0 U ex (r) = Iδ(x) x

49 INCIDENT AND REFLECTED QUASIPARTICLES Incident particle excitation with energy ɛ > Amplitudes (for I = 0) a = A /A +, b = 0 c = 1/A +, d = 0 A ± = 1 2 (1 ± ɛ2 2 ɛ ) 1/2 k a k i k b θ y v ga v gd k d k c x k ±N = k F ± ɛ hv F ɛ k ±S = k F ± 2 2 hv F b ǫ a i N S d ǫ c k +N k N k +N k S k +S k F 0 k F k F 0 k F

50 SUBGAP STATES For an excitation coming from the normal side with ɛ < Andreev reflection probability a 2 = A /A + 2 = 1 A ± = 1 2 (1 ± i 2 ɛ 2 ɛ ) 1/2 v g incident particle - e + e reflected hole N p v g p -2 e penetrating particle p p annihilated hole S -2 e Cooper pair

51 ANDREEV BOUND STATES S N S = e iφ/2 p = e iφ/2 h ǫ < d/2 0 d/2 x ǫ Spectrum for short junction (d < ξ) p p ɛ = cos φ 2 0 π 2π φ

52 NEGATIVE ENERGIES Bogolubov de Gennes equations possess an important property: If ( ) u(r) is a solution for the energy ɛ v(r) then ( ) v(r) u(r) is a solution for the energy ɛ Thus formally we can introduce negative energies and Dirac sea of excitations. ǫ k x < 0 k x > 0 φ 0 π 2π

53 SUPERCONDUCTOR INSULATOR SUPERCONDUCTOR (SIS) CONTACT ǫ No barrier, T = 1 ǫ Barrier, T < 1 k x < 0 k x > 0 φ 2 1 T 0 π 2π 0 π 2π φ ɛ = ±ɛ φ = ± 1 T sin 2 (φ/2) Conduction channel transmission coefficient T 1 R N = T R 0, quantum of resistance R 0 = π h e k

54 BOUND FERMION STATES IN THE VORTEX CORE Caroli, de Gennes, Matricon 1964 ε n (p z = 0, µ) ε 0 (p z, µ) n max 1 n = 0 µ/p F hω 0 a y p z a x Andreev reflection y b x hole particle Radial quantum number n (n max a/ξ). Angular momentum µ = b p, quantized. µ/ h = Minigap ω 0 { m + 1/2, s-wave superconductors m, superfluid 3 He 1 h 2 h a p F E. F Anomalous (crossing zero) branch n = 0.

55 SUPERCURRENT VIA ANDREEV BOUND STATES S penetrating particle N bound state: particle p penetrating particle p S -2 e annihilated particle - e v g + e bound state: hole v g p p annihilated hole -2 e Cooper pair

56 SUPERCURRENT IN THE POINT CONTACT For the contact with resistance R N in the normal state supercurrent is I s = π sin(φ/2) er N tanh cos(φ/2) 2k B T T T c I T T c 2π T T c : I s I c sin(φ/2) sign cos(φ/2) I c = π er N 0 π φ T T c : I s I c sin(φ) I c = π 2 4k B T er N

57 Josephson effect and weak links e iχ 1 e iχ 2 (φ) φ = χ 2 χ 1 Brian Josephson DC Josephson effect I s = I c sin φ I c is the critical Josephson current AC Josephson effect h dφ dt = 2eV

58 V WEAKLY COUPLED SUPERCONDUCTORS (Feynman model) ϕ 2 = V /2 χ 2 = φ/2 I s χ 1 = φ/2 Wave functions of the Cooper-pair condensate are uniform ψ 1 = N 1/2 1 e iχ 1, ψ 2 = N 1/2 2 e iχ 2 N 1, N 2 number of Cooper pairs. When no coupling ψ α = const (t) ϕ 1 = V /2 i h ψ α t With coupling K between the superconductors = E α ψ α = 0 E α = 0, α = 1, 2 i h ψ 1 t i h ψ 2 t = (E 1 + e V /2)ψ 1 Kψ 2 = (E 2 e V /2)ψ 2 Kψ 1 Here e = 2e is the charge of the Cooper pair.

59 We obtain h dn 1 dt h dn 2 dt DC JOSEPHSON EFFECT = 2K N 1 N 2 sin(χ 2 χ 1 ) = 2K N 1 N 2 sin(χ 2 χ 1 ) This gives the charge conservation N 1 + N 2 = const together with the relation I s = I c sin φ where I s = e dn 2 = e dn 1 dt dt is the current flowing from the first into the second electrode, I c = 4eK N 1 N 2 / h is the critical Josephson current, while φ = χ 2 χ 1 is the phase difference.

60 For phases we find AC JOSEPHSON EFFECT Subtracting we get hn 2 dχ 2 dt hn 1 dχ 1 dt = ev N 2 + K N 1 N 2 cos(χ 2 χ 1 ) = ev N 1 + K N 1 N 2 cos(χ 2 χ 1 ) h(n 2 N 1 ) d(χ 2 χ 1 ) dt = 2eV (N 2 N 1 ) or h dφ dt = 2eV

61 I CAPACITIVELY AND RESISTIVELY SHUNTED JUNCTION R J C Mechanical analogue: I = C V t + V R + I c sin φ, V = h φ 2e t I = hc 2 φ 2e t + h φ 2 2eR t + I c sin φ M 2 φ t 2 Particle with coordinate φ and with the mass Medium with a viscosity η = h 2 4e 2 R = h 2 8E C RC, φ = η t U(φ) φ M = h 2 C 4e 2 = h 2 8E C E C = e2 2C Potential U(φ) = E J (1 cos φ) ( hi/2e)φ = E J (1 cos φ φi/i c ) E J = hi c 2e

62 WASHBOARD POTENTIAL U(φ) = E J (1 cos φ φi/i c ), U E J = hi c 2e φ Small oscillations around the minimum: 1 cos φ = 2 sin 2 (φ/2) φ 2 /2, U(φ) E J φ 2 2 Plasma frequency Quality factor ω p = EJ M = 8EJ E C h Q = ω p RC

63 EFFECTIVE INDUCTANCE I R J C I R L C φ 2π I I c φ V = h φ 2e t h I J 2eI c t V = 1 c t = L c 2 I L t The effective kinetic inductance of the Josephson junction L J = hc 2 2eI c Critical current I c depends e.g. on temperature detectors. Nonlinear inductance at large φ.

64 VOLTAGE BIAS I Let us consider the case when the constant voltage V is applied to the junction. In this case V R J φ t = 2e h V = ω J = const, φ = φ 0 + ω J t Total current I = V R + I c sin(φ 0 + ω J t) and the average current has simple ohmic behavior for any damping I = V /R. Thus for observation of non-trivial dynamics of Josephson junctions the current bias is essential.

65 DYNAMICS OF JOSEPHSON JUNCTIONS I > I c U I < I c 1 I < Ir φ 0 2 φ I > I r Large damping Q 1 ω 1 p ωp 2 2 φ t + 2 Q 1 ωp 1 φ t + sin φ = I I c ω 2 p ω J Q = RCω p ω p /ω J RC ω 1 J no inertia (capacitance) Small damping hysteretic behavior

66 V CURRENT VOLTAGE RELATIONS large Q small Q Small Q (no C) exact solution: V = R I 2 Ic 2 I r Ic I Large Q: rapid slide down with almost constant voltage V V. Expanding for small oscillations of phase φ = ω J t + δφ(t), ω J = 2e V / h, δφ 1 leads to V = IR, I r I c /Q

67 THERMAL FLUCTUATIONS: OVERDAMPED JUNCTION [ P U P + P ± = ω a exp U ] ± k B T [ U U + = ω a exp U ] 0 (π hi/2e) k B T I φ E J 2π = π hi U 0 2E J for I I c I c e 2π V I c I V = h h φ/ t = 2e 2e 2π(P + P ) = IR E ( J k B T exp 2E ) J k B T

68 SUPERCONDUCTING QUANTUM INTERFERENCE DEVICES dc SQUID: j s = e2 n s mc ( A hc ) 2e χ χ 3 χ 1 +χ 2 χ 4 2e ( 3 A dl + hc 1 (χ 2 χ 1 ) (χ }{{} 4 χ 3 ) = }{{} φ a φ b 2 4 = 0 ) A dl = 0 2e A dl = 2π hc Ia 1 2 Φ I 3 4 I b Total current ( ) ( π I = I a + I b = I c sin φ a + I c sin φ b = 2I c cos sin φ a π ) 0 0 The maximum current depends on the magnetic flux through the loop ( ) I c,squid = 2 π I c cos 0

69 INFLUENCE OF THE SQUID INDUCTANCE Extra flux from the circulating current I circ = I b I a = I [ ( c sin φ a 2π ) ] sin φ a in the SQUID loop = ext + LI circ c = ext β L 0 2π sin ( π 0 I a ) ( cos φ a π ) 0 I circ Φ I I circ I b Dimensionless parameter β L = 2eLI c = L 1 in practice hc 2 L J Ic,SQUID/2Ic β L = ext/ 0 Together with ( ) ( π I = 2I c cos sin φ a π ) 0 0 determines maximum current

70 SHAPIRO STEPS Microwave irradiation of the junction V = V 0 + V 1 cos(ωt) and t φ = 2e V (t ) dt = φ 0 + ω J t + a sin(ωt), h 0 The supercurrent When I = I c sin φ = I c n= ω J = 2e h V 0, a = 2e h ( 1) n J n (2eV 1 / hω) sin(φ 0 + ω J t nωt) ω J = nω, i.e. V 0 = n( hω/2e) the supercurrent has a dc component I n = I c J n (2eV 1 / hω) sin(φ 0 + πn). V V 1 ω 3hω/2e 2hω/2e hω/2e I

71 QUANTUM DYNAMICS OF JOSEPHSON JUNCTIONS Analogy Josephson junction particle in the washboard potential can be extended to quantum-mechanical description. If φ is the coordinate of the a particle, then the momentum operator is ˆp φ = i h φ The Shrödinger equation for the wave function is [ ] ˆp 2 φ Ĥ = 2M + U(φ) = E with M = h 2 C 4e = h 2, 2 8E C Thus the Hamiltonian is Ĥ = 4E C 2 U(φ) = E J (1 cos φ) ( hi/2e)φ φ 2 + E J (1 cos φ φi/i c )

72 REQUIREMENTS FOR JUNCTION PARAMETERS Kinetic energy The charging energy of the capacitor The operator of charge ˆQ on the capacitor: ˆp 2 φ 2M = ˆQ 2 2C, ˆQ = 2ie φ, [ ˆQ, φ] = 2e h [ ˆp φ, φ] = 2ie Quantum uncertainty in phase φ and in charge Q: φ Q 2e. Effects for Q e are important: E C = e2 2C k BT, E C R C C = ɛa 4πd h t = h RC 10 (100 nm)2 4π 1 nm F T 1 K R 2 h e 2 R 0 = π h e 2 L 1 Rext L S S 2 C ext C ext C, R 0 R ext R V

73 SUPERCONDUCTING QUANTUM ELECTRONICS Rapidly developing field. One of key players in quantum engineering and quantum information processing. U qubits and artificial atoms macroscopic quantum tunnelling φ

74 CONCLUSIONS Superconductivity is ubiquitous in conducting systems: metals, alloys, dirty and disordered systems, organic materials, 2D system... Superconductivity originates in the Cooper pairing of conduction carriers. In many systems the attractive interaction is mediated by phonons and pairing occurs in spin-singlet, orbital momentum zero state. But more and more unconventional systems are being discovered. Besides zero resistivity, superconductors possess broad range of interesting and practically important properties: magnetic, thermal, etc. Nanotechnology opened a new world in studies and applications of superconductors, in particular due to good matching to important physical length scales. In nanodevices Josephson effect and Andreev bounds states usually play key roles. Superconductors provide access to quantum-mechanical coherence at macroscopic length scales: a base for revolutionizing the world with quantum technologies.

75 LITERATURE M. Tinkham, Introduction to superconductivity. McGraw- Hill, New York. (1996); Dover Books (2004) P. G. de Gennes, Superconductivity of metals and alloys. W. A. Benjamin, New York (1966); Perseus Books (1999) A. A. Abrikosov, Fundamentals of the theory of metals. North Holland, Amsterdam (1998)