VI.2 Efecto Josephson VI Superconductividad VI.1 Nociones generales y modelos teóricos VI.2 Efecto Josephson y circuitos superconductores con uniones Josephson VI.3 Aplicaciones de la Superconductividad VI.4 Prácticas Juan José Mazo Torres Agustín Camón Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 1 VI.2 Efecto Josephson Bibliografía – Charles Kittel. Introduction to Solid State Physics. Wiley. – N. W. Aschcroft and N. D. Mermin. Solid State Physics. Saunders College (1976). – Michael Tinkham. Introduction to Superconductivity (2nd Edition). Mc Graw Hill (1996). – Terry P. Orlando and Kevin A. Delin. Foundations of Applied Superconductivity. Addison-Wesley (1990). –A. Barone and G. Paternó. Physics and Applications of the Josephson Effect. Wiley (1982). –K. K.Likharev, Dynamics of Josephson Junctions and Circuits. Gordon and Breach (1986). Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 2 VI.2 Efecto Josephson Outline Basics – Josephson effects – Superconducting tunnel junction – Modeling Josephson arrays Localised excitations in Josephson arrays – – – – Vortices in 2D arrays Charge solitons in arrays of ultra-small junctions Kinks in parallel arrays Discrete breathers in JJA’s Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 3 VI.2 Efecto Josephson Josephson effect (B.D. Josephson. Possible new effect in superconductive tunneling. Phys. Lett. 1, 251 (1962)) Josephson effect in weakly coupled macroscopic quantum systems: • 9 Weakly coupled superconductors • • • • 9 A. Barone. Weakly coupled macroscopic quantum systems: likeness with difference, in Quantum Mesoscopic Phenomena and Mesoscopic Devices in Microelectronics, edited I.O. Kulik (Kluwer Academic, 2000) A. Barone and G. Paternò. Physics and applications of the Josephson effect, Jonh Wiley, 1982. K.K. Likharev, Dynamics of Josephson junctions and circuits, Gordon and Breach Science, 1984. T.P. Orlandoand K.A. Delin, Foundations of applied superconductivity, Addison Wesley, 1991. M. Tinkham, Introduction to superconductivity, Mc Graw Hill, 1996. Weak coupling of superfluids • O. Avenel and E. Varoquax, Josephson effect and quantum phase slippage in superfluids, Phys. Rev. Lett. 60, 416-419 (1988). 9 Weakly coupled Bose-Einstein condensates • A. J. Legget, Bose-Einstein condensation in the alkali gases: some fundamental concepts. Rev. Mod. Phys. 73, 307 (2001) Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 4 VI.2 Efecto Josephson Superconducting (tunnel) Josephson junction Superconductor (Nb, Al,...) Física de Bajas Temperaturas Insulating ~10Å (Al2Ox,...) -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 5 VI.2 Efecto Josephson Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 6 VI.2 Efecto Josephson 5 mm Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 7 VI.2 Efecto Josephson Superconducting (tunnel) Josephson junction Superconductor (Nb, Al,...) Insulating ~10Å (Al2Ox,...) Tunneling of Copper pairs between two superconductors? Before 1962: no possible. Tunneling of single electrons has low probability, so simultaneous tunneling of both electrons in a pair is statistically insignificant. After 1962: In 1962 Josephson(1) showed that the Cooper pair is like a single particle, is the macroscopic wave function that tunnels, so this is an observable process. Experimentally confirmed by P. Anderson and J. Rowell(2) in 1963. (1) B. D. Josephson. Possible new effects in superconductive tunneling. Physics Letters, 1, 251-253 (1962). (2) P. W. Anderson and J. M. Rowell. Probable observation of the Josephson superconducting tunneling effect. Phys. Rev. Lett. 10, 230 (1963) Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 8 VI.2 Efecto Josephson Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 9 VI.2 Efecto Josephson Different approaches 1.- Microscopic theory (Josephson) 2.- Quantum Macroscopic Model (Orlando) 3.- Quantum Macroscopic Model (Feymann) 4.-Ginzburg-Landau theory (for microbridges, Tinkham) Different types of junctions 1.- SIS Tunnel junctions 2.- Weak links: SMS junction, microbridges, point contacts… Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 10 VI.2 Efecto Josephson Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 11 VI.2 Efecto Josephson Superconducting (tunnel) Josephson junction Superconductor (Nb, Al,...) Ψ1 = n1 e iθ1 Ψ2 = n2 e iθ 2 Insulating ~10Å (Al2Ox,...) • Equations for the Josephson effect (1) I S = I C sin ϕ h dϕ V = 2 e dt r 2e 2 r ϕ = θ 1 − θ 2 − ∫ A( r , t ) ⋅ d l h 1 Física de Bajas Temperaturas • DC Josephson effect (V=0, ϕ=cte, I=cte≠0) • AC Josephson effect (V≠0, ϕ=2eVt/h+ϕ0, ac current with frequency ω=2eV/h) Φ0 = h / 2e ; flux quantum frequency: 483.6GHz / mV -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 12 VI.2 Efecto Josephson • Equations for the Josephson effect (2) The microscopic theory gives more complex relations that in the case of constant voltage V give: I ( t , V , T ) = I J 1 ( V , T ) sin ϕ + I J 2 ( V , T ) cos ϕ + I qp ( V , T ) ϕ (t) = ϕ 0 + ω f t with ω f = 2e V h I = I J 1 sin ϕ Si V = 0 then I qp = I J 2 = 0 Si V ≠ 0 then I = I J 1 sin ϕ + V σ ( 1 + ε cos ϕ ) y It includes an additional phase-dependent dissipative term reflecting an interference term between the pair and the quasi-particle currents. This term averages out to zero in many situations and in general is rather subtle to detect. Thus, is usually neglected. Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 13 VI.2 Efecto Josephson • Equations for the Josephson effect (3) • Biased by a constant current I<Ic, then ϕ=sin-1(I/Ic) and V=0 I S = I C sin ϕ h dϕ V = 2e dt • Biased by an ac current Idc+Iacsin(ω0t), then V(t)=Vdc+ harmonics and we can expect synchronization effects between the Josephson characteristic frequency ωJ=2eVdc/h and the biasing frequency [Vdc=(h/2e)ωJ=(h/2e)(n/m)ω0] which allows for a very precise determination of the e/h ratio. • This synchronization effect is also well observed when V=Vdc+Vaccosω0t, then ϕ=ϕdc+ωJt+(2eVac/hω0)sin(ω0t) and I s = I c ∑ (−1) n J n (2eVac / hω0 ) sin(ϕ dc + ω J t − nω0t ) Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 14 VI.2 Efecto Josephson SIS junction Física de Bajas Temperaturas SNS junction -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 15 VI.2 Efecto Josephson Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 16 VI.2 Efecto Josephson Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 17 VI.2 Efecto Josephson Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 18 VI.2 Efecto Josephson Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 19 VI.2 Efecto Josephson • Josephson energy U J = − EJ cosϕ EJ = hI c 2e We need Josephson energy to exceed thermal energy: EJ»kBT (Ic»2ekBT/h) • Temperature dependence of IC Ambegaokar-Baratoff equation for the temperature dependence of Ic in a tunnel junction [PRL 10, 486 (1963); erratum 11, 104 (1963)] I c Rn = (π ∆ 2e ) tanh (∆ 2 k B T ) T=0 ⇒ IcRn=π∆(0)/2e (∆(0)=1.764kBTc) After E. P. Balsamo at al., PRB 10, 1881 (1974) T→Tc ⇒ (2.34πkB/e)(Tc-T) (in SNS junctions Ic goes to zero at Tc and rises exponentially with decreasing temperature) Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 20 VI.2 Efecto Josephson Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 21 VI.2 Efecto Josephson Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 22 VI.2 Efecto Josephson Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 23 VI.2 Efecto Josephson • Current-voltage characteristics: The RCSJ model. I = I J + IC + I R I I J = I c sin ϕ ; I C = C ; with V dV and I R = R dt Equivalent circuit for the JJ I I c sin ϕ + 1 I = C V& + V + I c sin ϕ R V C R Φ ⎞ ⎛ ⎜V = 0 ϕ& ⎟ 2π ⎠ ⎝ i = ℵ(ϕ ) = ϕ&& + Γϕ& + sinϕ 1 1 where Γ = = 2 = Q βc Φ0 2π I c CR 2 I ; i= and ω p = Ic 2π I c Φ 0C Γ is the junction effective damping, Q is called quality factor (also Q=ωpRC) βc is called Stewart –McCumber parameter W.C. Stewart, Appl. Phys. Lett. 12, 277 (1968); D.E. McCumber, J. Appl. Phys. 39, 3113 (1968). Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 24 VI.2 Efecto Josephson Equivalent circuit for the JJ + I I c sin ϕ Non-linear pendulum τ app V l C R 1 C V& + V + I c sin ϕ = I R ϕ mg ml 2ϕ&& = τ app − D ϕ& − mgl sin ϕ ϕ&& + Γϕ& + sinϕ = I Particle in a washboard potential U (ϕ ) = − E J cos ϕ − (h I 2 e )ϕ (mass and viscous damping) m = (h 2 e ) C 2 Física de Bajas Temperaturas η = (h 2 e )2 (1 R ) -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 25 VI.2 Efecto Josephson [Overdamped case: V=R(I2-Ic2)1/2] 3 2 Γ=5 IC i=I/Ic 1 IV curves: RCSJ model vs experiments 0 -IC -1 -2 -3 -3 -2 -1 0 1 2 4 3 2 1 I/Ic 0 -1 -2 -3 -4 3 V/IcR=Γ<dϕ/dt> [Underdamped case] i=I/Ic 1 Γ=0.2 IC Iret 0 V/Vg -Iret -1 -1,5 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 g -IC -1,0 -0,5 0,0 0,5 1,0 V/IcR=Γ<dϕ /dt> Física de Bajas Temperaturas (V 1,5 (I ret = 2∆ e ) ≈ 4Γ π ) -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 26 VI.2 Efecto Josephson Nonlinear subgap resistance 4 3 2 1 I/Ic 0 -1 -2 -3 -4 C V& + 1 V + I c sin ϕ = I R (V ) ⎧ Rn if V > 2 ∆ (T ) / e R (V ) = ⎨ ⎩ R sg (T ) if V < 2 ∆ (T ) / e Rsg (T ) ≈ Rn e ∆ / k BT -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 V/Vg This quasiparticle tunneling current for kBT<<∆ and V<Vg=2∆/e is well approximated by 2G n − ∆ kT ⎛ 2 ∆ ⎞ ⎛ eV ⎞ ⎛ eV ⎞ = e ⎟ ⎟K0 ⎜ ⎟ ( eV + ∆ ) sinh ⎜ ⎜ e ⎝ 2 kT ⎠ ⎝ 2 kT ⎠ ⎝ eV + 2 ∆ ⎠ 12 I qp For some investigations and applications, it is convenient to shunt the junction by a small resistance, then the Γ>>1 and the overdamped limit of the RCSJ model is appropriate. This is also the case of SNS junctions. In other cases, especially when dealing with small junctions, in order to describe the behaviour of the system, it is essential to consider also the impedance of the external circuit. Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 27 VI.2 Efecto Josephson Thermal fluctuations + 1 ~ C V& + V + I c sin ϕ + I ( t ) = I R ⎛ ~ ⎜ I ( t ) = 0 and ⎝ I I c sin ϕ V C R ~ I(t) 2k T ⎞ I~ ( t ) I~ ( t ′ ) = B δ ( t ′ − t ) ⎟ R ⎠ - ~ i = ℵ(ϕ ) = ϕ&& + Γϕ& + sinϕ + i ω(I) ∆U(I) ⎛ ~ ⎜⎜ i (τ ) = 0 and ⎝ ⎞ kT ~ ~ i (τ ) i (τ ′) = 2 Γ δ (τ ′ − τ ) ⎟⎟ EJ ⎠ Overdamped junctions V. Ambegaokar and B.I. Halperin,, Phys. Rev. Lett. 22, 1364 (1969) Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 28 VI.2 Efecto Josephson Underdamped junctions Reduction of Ic by premature switching and increasing of Iret by premature retrapping. ⎧ ⎪ ⎪ Γ(I ) = ⎨ ⎪ ⎪⎩ ⎛ 18η ⎜ ⎝ 5π ⎞⎛ ∆ U ⎟ ⎜⎜ ⎠⎝ k B T ⎞ ⎛ ∆U ⎞ ⎟⎟ exp ⎜⎜ − ⎟⎟ ⎠ ⎝ k BT ⎠ IC ctu Flu ⎛ ∆U ⎞ ⎛ ωa ⎞ ⎟⎟ ⎜ ⎟ exp ⎜⎜ − ⎝ 2π ⎠ ⎝ k BT ⎠ ⎛ ωa2 ⎜ ⎜ 2πη ⎝ low damping I moderate damping ⎞ ⎛ ∆U ⎞ ⎟ exp ⎜ − ⎜ k T ⎟⎟ ⎟ B ⎝ ⎠ ⎠ on ati s Iret V high damping In a typical experiment: IV curve where I is ramped from zero at a given rate: ω a = ω p [1 − ( I / I c ) 2 ]1 / 4 ∆ U ≈ 2 E J (1 − I / I c ) 3 / 2 { [ Fluctuation effects cause a major reduction in Ic as soonn an kBT is as large as 5 percent of EJ ] } I c (T ) = I c 1 − (k B T / 2 E J ) ln (ω p ∆ t / 2π ) Física de Bajas Temperaturas 2/3 -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 29 VI.2 Efecto Josephson AC fields 1 I = C V& + V + I c sin ϕ = I dc + I ac cos( ω t ) R ℵ(ϕ ) = ϕ&& + Γϕ& + sinϕ = idc + iac cos(ω 'τ ) •Biased by an ac current Idc+Iacsin(ωt), then V(t)=Vdc+ harmonics and we can expect synchronization effects between the Josephson characteristic frequency ωJ=2eVdc/h and the biasing frequency. They are the so-called Shapiro steps. Overdamped case: no hysteresis Underdamped case: hysteresis Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 30 VI.2 Efecto Josephson Long Josephson-junction A long JJ is a junction which has one dimension (say x) long with respect to the Josephson penetration depth λJ. Then the phase difference is also a function of the spatial coordinate ϕ(x,t). The electrodynamics of the junction is described by a nonlinear partial differential equation that, neglecting dissipative effects, can be written as ϕ xx − ϕ tt = sin ϕ This is the sine-Gordon equation, which has coherent localised particle-like solutions or soliton solutions. ϕ(x) can be interpreted as the phase difference or the normalized magnetic flux. Then a soliton in the junction corresponds to solution for which the junction goes from 0 to 2π or the flux goes from 0 to Φ0, that is a fluxon of magnetic field. When losses and bias are includes the dynamics of the fluxon is described by the perturbed sine-Gordon equation ϕ xx − ϕ tt − sin ϕ = αϕ t − βϕ xxt − γ Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 31 VI.2 Efecto Josephson Quantum JJ Superconductor (Nb, Al,...) Ψ1 = n1eiθ 1 Ψ2 = n2 eiθ Insulating ~10Å (Al2Ox,...) 2 • Junction Hamiltonian Josephson energy Charging energy Q2 UC = 2C e2 = EC 2C Física de Bajas Temperaturas UJ = − Q2 H = − E J cos ϕ + 2C hIc cosϕ 2e EJ (Q ≈ d ϕ / dt ) -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 32 VI.2 Efecto Josephson To study the quantum dynamics of the junction we will follow the canonical quantization rules and treat ϕ and Q as operators which satisfy the usual communtation rule, [ϕ , Q ] = i 2e Then, in phase representation, ∂ Q ∂ ϕ = n→ i ⇒ H = − E cos − 4 E J C ∂ϕ 2 2e ∂ϕ 2 2 2 By analogy with the problem of a particle in a periodic potential the solutions of this Hamiltonian will take the form of Bloch functions. The ratio EJ/4EC measures the relative importance of the charging energy of the pairs. When EJ is dominant the quantum fluctuations in the phase are small. When EJ ∼EC, the kinetic term induce a delocalization of the phase. In the other limit EJ <<EC tunneling is weak, and large charge transfers are energetically prohibitive. In order to observe quantum behavior we need to have the thermal energy kBT<<EJ,EC and large tunnel resistances R>h/4e2=6.45kΩ Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 33 VI.2 Efecto Josephson Evidence of “Secondary” Quantum effects 9 Macroscopic quantum tunneling of the phase and energy level quantization [J. Clarke et al., Science 239, 992 (1988)] 9 Single Cooper-Pair tunneling [L.J. Geerligs et al., PRL 65, 378 (1990)] 9 Demonstration of Heisenberg’s uncertainty principle in a superconductor [W.J. Elion et al., Nature 371, 594 (1994)] 9 Dissipative phase transition in a single Josephson junction [Penttilä et al., PRL 82, 1004 (1999)] Operation of quantum states with single JJ and simple JJ circuits 9 Quantum superposition of distinct macroscopic states in a rf SQUID [J.R. Friedman et al., Nature 406, 43 (2000)] and a superconducting loop with 3 junctions [C.H. Van der Waal et al., Science 290, 773 (2000)] 9 Manipulation od the quantum state of superconducting tunnel junction Circuit [D. Vion et al, Science 296, 886 (2002)] 9 Generation and observation of coherent temporal oscillations between the macroscopic quantum states of a Josephson tunnel junction [Y. Yu et al, Science 296, 889 (2002)] 9 Josephson vortex qubit [A. Kemp et al, PSS B, 233, 472 (2002)] Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 34 VI.2 Efecto Josephson Modeling Josephson arrays Φ 0 dϕ 2π dt I J = I c sin ϕ V= Ψ2 = Ψ2 e iθ 2 Ψ1 = Ψ1 e iθ1 2e 2 r r ϕ = θ 1 − θ 2 − ∫1 A dl h E = E J (1 − cos ϕ ), with E J = SQUIDs I Parallel Series x x Física de Bajas Temperaturas I I I I I I x I Ladder Φ0 Ic 2π I I I 2D -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 35 VI.2 Efecto Josephson • Kirchoff’s conservation law (for the current and for the voltage) • Fluxoid quantization: for any loop l in the array (with at least one junction) the sum of all the phase differences around the loop is ∑ ϕ = 2π ( n − f ) j∈l j l l The integer nl is the vorticity of the loop and results from the multivaluedness of the superconducting phase variable (θ) The term fl accounts for the total (external plus induced) magnetic flux through the loop measured in terms of Φ0 (fl =Φl/ Φ0). In general, to compute the total induced flux in a cell one must take into account a full inductance matrix. However, in many cases we can work with the simplest approximation of taking into account only self-inductances (L) of each cell. The parameter λ= Φ0 /2πIcL measures the importance of induced fluxes 1 I loop ext ind ext We will leave out the nl terms and write ∑ ϕ j = − 2π ( f l + f l ) = − 2π f l − λ Ic j∈l Josephson circuits can be divided into two general categories. Circuits of the first type have λ>>1 so the induced flux in the loop is not important (these circuits are typically made of aluminium). Circuits of the second type have λ<<1 and induced flux causded by circulating currents is important (these circuits are typically made of niobium). Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 36 VI.2 Efecto Josephson Series Arrays N V ( t ) = ∑ϕ&k = F ( I L ( t )) k =1 I LOAD ϕ&&k+ Γϕ&k + sinϕ k + I L ( t ) = I [P. Hadley at el., PRB 38, 8712 (1988)] Search for junction dynamic synchronization for coherent radiation and definition of the voltage standard. Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 37 VI.2 Efecto Josephson rf-SQUID (the superconducting loop interrupted by a JJ) ϕ = −2π L Φ λ= 1 βL Ic = Φ Φ0 ⎛ Φ⎞ ⎟⎟ Φ = Φ ext − LI c sin⎜⎜ 2π Φ 0 ⎠ ⎝ ⎛ Φ⎞ ⎟⎟ I s = I c sin⎜⎜ 2π ⎝ Φ0 ⎠ 2 ⎛ ⎛ Φ ⎞ ⎞⎟ (Φ − Φ ext ) ⎜ ⎜ ⎟ U = EJ ⎜1 − cos⎜ 2π ⎟⎟ + 2L ⎝ Φ0 ⎠ ⎠ ⎝ Φ0 2πI c L Important for basic reasearch (Schrödinger cat problems, Caldeira-Legget model for quantun dissipation,…) and detection of magnetic fields. Operated at a radio frequency (typically between 20 and 30 MHz). An external circuit biased by an rf field and inductively coupled to the SQUID. Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 38 VI.2 Efecto Josephson dc SQUID ( a) neglecting inductive effects) Fluxoid quantization If inductive effects can be neglected, fluxoid quantization imposes a constraint to the equations and then reduces the number of independent variables of the system, 1 2 Iext (ϕ1 − ϕ 2 ) = − 2π Φ0 Current conservation Φ ext = −2πf 0 I ext = I1 + I 2 = I c sin ϕ1 + I c sin ϕ2 = 2I c cos(πf 0 ) sin(ϕ1 + πf 0 ) Behaves like a single junction with a critical current modulated by the external field !! Then the IV curve for the system, in the overdamped regimen (resistively shunted junctions) is given by: V=(R/2){I2-[2Iccos(πΦext/Φ0)]2}1/2 Measuring the critical current or the voltage change at a given bias current we can measure the external field. Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 39 VI.2 Efecto Josephson dc SQUID ( b) with inductive effects) Fluxoid quantization (ϕ1 − ϕ 2 ) = −2πΦ tot / Φ 0 = 1 = −2π (Φ ext + Φ ind ) / Φ 0 L Current conservation I1 = I c sin ϕ1 = − I m + I ext / 2 I 2 = I c sin ϕ1 = I m + I ext / 2 2 Φind = LI m = L( I 2 − I1 ) / 2 = LI c (sin ϕ2 − sin ϕ1 ) / 2 Iext If Iext is very small, then ϕ1 and ϕ2 are almost zero and we can write Φext=Φ+LIcsin(πΦ/Φ0) Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 40 VI.2 Efecto Josephson dc SQUID (a different approach) Current conservation Iext i1 = ϕ&&1 + Γϕ&1 + sinϕ1 = i mesh + iext ϕ1 I m B app ϕ i2 = ϕ&&2 + Γϕ& 2 + sinϕ 2 = − i mesh + iext 2 Fluxoid quantization (ϕ 1 − ϕ 2 ) = − 2π Φ0 L i mesh (Bapp A + LI mesh ) = − λ (ϕ 1 − ϕ 2 + 2π f 0 ) f 0 = Bapp A Φ 0 λ = Φ 0 2πLIc ϕ&&1 + Γ ϕ& 1 + sin ϕ 1 = − λ (ϕ 1 − ϕ 2 + 2π f0 ) + iext ϕ&&2 + Γ ϕ& 2 + sin ϕ 2 = λ (ϕ 1 − ϕ 2 + 2π f0 ) + iext Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 41 VI.2 Efecto Josephson 2-junctions in parallel τ app Iext ϕ1 2-coupled pendula Im ϕ B app λ 2 ϕ1 ϕ 2 L ϕ&&1 + Γ ϕ& 1 + sin ϕ 1 = − λ (ϕ 1 − ϕ 2 + 2π f0 ) + iext ϕ&&2 + Γ ϕ& 2 + sin ϕ 2 = λ (ϕ 1 − ϕ 2 + 2π f0 ) + iext Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 42 VI.2 Efecto Josephson JJ parallel array Iext Im Im Im Bapp Bapp ϕj −1 Bapp Im ϕj Im Bapp ϕj +1 B app Im Bapp • Set of junctions connected in parallel by superconducting wires. L τapp ϕ j −1 ϕ • All the junctions have the same dc voltage λ j ϕ j+1 ϕ&& j + Γ ϕ& j + sin ϕ j = λ (ϕ j +1 − 2ϕ j + ϕ j −1 ) + iext (Forced and damped Frenkel-Kontorova model or discrete sine-Gordon equation) Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 43 VI.2 Efecto Josephson JJ ladder array • Vertical junctions are connected in parallel by horizontal junctions. Nonlinear pendula coupled by nonconvex springs. • Not all the vertical junctions have the same dc voltage ξ j = ϕ jv + ϕ jt − ϕ jv+1 − ϕ jb + 2π f 0 = − 2π f jind (we have considered no external field) Φ0 =λ 2π I cy L λ Φ0 λx = = 2π I cx L h λy = Física de Bajas Temperaturas ℵ(ϕ j ) = − t λ ξj h ℵ(ϕ jv ) = λ (ξ j −1 − ξ j ) + iext ℵ(ϕ j ) = b h= λ h ξj I cx C x R y = = I cy C y Rx -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 44 VI.2 Efecto Josephson 2D array λ ext i ℵ(ϕ ijx ) = (ξ ij − ξ ij −1 ) + x h h ℵ(ϕ ijy ) = λ (ξ i −1 j − ξ ij ) + i yext ξ ij = − 2π f ijind = ϕ ijy + ϕ ijx+1 − ϕ iy+1 j − ϕ ijx + 2π f0 ( i , j + 1) i xext ϕ ij y ( i, j ) ( i, j ) ϕ ijx Φ0 =λ 2π I cy L λ Φ0 λx = = 2π I cx L h λy = ( i + 1, j ) i yext This is a model for a 2D array when only self-inductances are taken into account. Sometimes, depending on the problem to be studied, inductances are not needed at all; in other cases however a full inductance matrix is necessary. Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 45 VI.2 Efecto Josephson Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 46 VI.2 Efecto Josephson Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 47 VI.2 Efecto Josephson Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 48 VI.2 Efecto Josephson • In general, in a square 2D array we have to solve equations for the dynamics of 2N2-2N (for FBC) gauge-invariant phase differences, ϕij. However, when induced fields can be neglected (λ>>1 limit), flux quantization condictions imposes (N-1)2 constrains on these variables. Then, it is more convenient to express the system equation in terms of the phase in each island, θi, which are N2-1 independent variables. ϕ ij = θ i − θ j − r 2π j r A( r ) ⋅ d l ∫ Φ0 i the dynamical equations result after applying current conservation. • In the EJ>> EC limit the total Josephson energy is all the relevant energy: H J = − ∑ E J cos(θ i − θ j − Aij ) < ij > where Aij depend now only on the external magnetic field • In the opposite limit of ultrasmall junctions (EC>> EJ), the charging energy is the more important contribution: HQ = 1 ∑ij Q i C ij−1Q j 2 Qj stands for charge in island j. Cij for the capacitance matrix of the circuit. • The most interesting case is that of intermediate values where both terms should be considered and H = 1 ∑ij ( Q i + qi )C ij−1 ( Q j + q j ) − <∑ij > E J cos( θ i − θ j − Aij ) 2 qj are possible offset charges or charges induce by external sources (charge frustration). Charge and phase variables are canonically conjugate variables. Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 49 VI.2 Efecto Josephson Localised excitations in Josephson arrays The concept of coherent structures or coherent excitations has important consequences when applied to condensed matter systems. Spatially or temporally coherent structures appear in many nonlinear extended systems. Such structures usually can be characterized by marked particle-like properties. In the past few years, these notions have become fundamental for understanding many problems and their implications extend over different fields of the physics of continuous and discrete systems. JJ arrays are a very well example of an experimental system where such type of structures appear examples that we are going to present are: Vortices, kinks and breathers. – Vortices in 2D arrays – Charges and charge solitons in arrays of ultra-small junctions – Kinks in parallel arrays – Discrete breathers in JJA’s Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 50 VI.2 Efecto Josephson Vortices in 2D arrays [The two-dimensinal physics of Josephson junction arrays. R.S. Newrock, C.J. Lobb, U. Geigenmüller and M. Octavio. Solid State Physics, Vol 54, pp 263-512 (2000)] In classical 2D arrays of superconducting islands coupled by “large” JJs the relevant energy is the sum of the Josephson energies of the junctions HJ = − ∑ E J cos( θ i − θ j ) (no magnetic fields) < ij > • This sytem is a realization of the 2D XY model, a system which shows the KTB phase transition where vortex-antivortex pairs unbind. • The model supports.... ¾ “spin waves”: small amplitude and energy wave-like variations of the phase over the array ¾ vortices (and antivortices): localised excitations defined by the sum of the phase differences along a path containing the vortex whis is equal to 2π (-2π for antivortices). [Σ(θi−θj)=2πn] Vortices and antivortices are topological excitations, they behave large charges. Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 51 VI.2 Efecto Josephson Single vortex properties at zero temperature • Energy of the ground state E=-2N2EJ • Energy of an isolated vortex in a large square array of size L=Na→∞ E=πEJln(L/a)-2N2EJ • An external current can move the vortex F=(Φ0/2π)(Itot/Na)=(Φ0/2π) (I/a) • Energy barrier for a single vortex: in a square lattice EPN=0.2EJ ; for triangular lattices, EPN=0.043EJ • 2D PN potential. The potential along x direction UPN(x)=-(EPN/2) cos(2πx/a) [Vortex depinning current Icv=(π/Φ0) EPN=0.1Ic] • Effective equations for the motion of a vortex: external force (external current), viscous force (power dissipated in the resitive channels), kinetic energy (stored in the capacitors), potential energy (Josephson energy) Φ 0 d ( 2π x a ) Φ 0C d 2 ( 2π x a ) I = I cv sin( 2π x a ) + + 4π R dt 4π dt 2 Física de Bajas Temperaturas [From Newrock et al Sol. Stat. Phys. 54, 263 (2000)] -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 52 VI.2 Efecto Josephson Array properties at non zero temperatures • There are not free vortices at low enough temperatures [the energy to add a vortex is EV=πEJln(L/a) (valid for large L)] • At T≠0 the thermal generation of bound pairs of vortices is energetically favored [energy of a v-av pair Ep=2πEJln(r/a)] • Free vortices appear above KTB temperature, transition temperature TKT ∼ πEJ/2kB These results are affected by the inclusion of finite size effects, external magnetic fields, self-induced fields, disorder,... but in any of the cases the concept of vortices is essential to understand the physics of the array. Física de Bajas Temperaturas [From Mooij and Schön in “Single Charge Tunneling” Ed. By H. Grabert and M.H. Devoet, NATO ASI Series B: Physics Vol. 294,Plenum Press NY (1992)]. -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 53 VI.2 Efecto Josephson Charges in 2D arrays of ultrasmall junctions [H. J. Mooij and G. Schön, Single charges in 2-dimensional junction arrays. In “Single Charge Tunneling”, p. 275, ed. By H. Grabert and M.H. Devoret, Plenum Press NY (1992)] In arrays of ultrasmall tunnel junctions the more relevant energy is the charging energy HQ 1 = ∑ Q i C ij−1Q j 2 ij • Qi is the charge in the island. For metalic “normal state” islands this charge is an integer multiple of the electronic charge e. In the superconducting state, at low temperatures and voltages below the gap voltage, Cooper pairs are dominant and the charge appears in multiple of 2e. • In any of the cases at zero temperature no free charges are present in the array. The system is insulating. In arrays where mutual capacitance are dominant (C>>C0) the charges interact logarithmically over sufficient long distances. Then free charges are created by thermal activation through a KTB phase transition where pair of charges e, -e unbind and the array is resistive (Tcn=EC/4πkB, Tcs=EC/πkB, EC=e2/2C). • This transition is strongly affected by dissipation [measured by the coefficient αT=h/(4e2RT)], offset charges, ... Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 54 VI.2 Efecto Josephson Charge-vortex duality • In small capacitance superconducting junctions we have to include both the charging and the Josephson coupling. If we ignore quasiparticles the hamiltonian of the system is 1 H = Q i C ij−1Q j − E J cos( θ i − θ j ) 2 ij < ij > ∑ ∑ [we have ignored also charge or phase frustration (offset or induced charges and magnetic fields)] If charging energy can be ignored EC=0, the vortices undergo a KTB transition where vortex dipoles unbind. This transition separates a superconducting low temperature phase from a resistive high temperature phase. If the Josephson coupling is weak (EJ=0) the charges show a KTB transition where the dipoles, formed by a Cooper pair and a missing pair, unbind. The transition separates an insulating from a conducting phase. At finite EJ and EC both charge and vortex excitations have to be considered simultaneously. The charging energy provides a kinetic energy for the vortices, and the Josephson coupling allows the tunneling of cooper pairs and provides dynamics for the charges (If EC<<EJ or EJ<<EC perturbative approaches can be used). However, to fully understand the physics of the system we have to consider also the influence of the quasiparticle tunneling. We have a quantum dissiptive system. • We have also seen that vortices can be described as particles moving in a substrate potential with a mass given by the charging energy. For small junctions we can expect strong quantum mechanical efects. At finite temperatures vortex motion can be thermally excited. At low temperatures it is possible to observe quantum mechanical tunneling of vortices. Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 55 VI.2 Efecto Josephson Charge Solitons in 1D arrays [P. Delsing, One-dimensional arrays of small tunnel junctions. In “Single Charge Tunneling” Ed. By H. Grabert and M.H. Devoret, p. 249. Plenum Press NY (1992) and D.B. Haviland and P. Delsing, “Cooper-pair charge solitons: The electrodynamics of localized charge in a superconductor”, Phys. Rev. B, 54, R6857 (1996)] • Weak Josephson coupling EJ<<EC and large normal resistance R>h/4e2 • If a single electron (or single pair) is added to or substracted from an intermediate island the resulting localized state -the single charge plus the polarization cloud- is called charge soliton (or anti-soliton). It correspond to a localized voltage profile. Some of their properties have been studied with the use of a sine-Gordon equation. Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 56 VI.2 Efecto Josephson Kinks in parallel arrays [L.M. Floría and J.J. Mazo, Dissipative dynamics of the Frenkel-Kontorova model, Adv. Phys. 45, 505 (1996). O.Braun and Y. Kivshar, Nonlinear dynamics of the Frenkel-Kontorova model, Phys. Rep. 306, 1 (1998). A.V. Ustinov et al, Fluxon dynamics in one-dimensional Josephson-junction arrays, 47, 8357 (1993). S. Watanabe et al, Dynamics of circular arrays of Josephson junctions and the discrete sine-Gordon equation, Physica D 97, 429 (1996). F. Falo et al, Fluxon ratchet potentials in superconducting circuits, Appl. Phys. A 75, 263 (2002).] Iext Im Im Im Bapp Bapp ϕj −1 Bapp ϕj Im Im Bapp ϕj +1 B app Im Bapp L Discrete sine-Gordon equation or Frenkel-Kontorova model ϕ&& j + Γ ϕ& j + sin ϕ j = λ (ϕ j +1 − 2ϕ j + ϕ j −1 ) + iext Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 57 VI.2 Efecto Josephson The system supports different types of excitations: ¾ “linear waves”: small amplitude and energy wave-like variations of the phase over the array When the phases are small, the solution to the linearized form of the equations (for zero damping and current) are: ϕj(t) α exp i(ωkt-kj) with ωk2 = 1 + 4λ sin2 ( k / 2) ( −π < k < π ) They are linear waves with a frequency spectrum characterized by a finite band with gap ωmin=ω(k=0)=1 and maximun frequency ωmax=ωπ=(1+4λ)1/2 ¾ when the phases are not small the linear approximation is not valid and the dynamics is pretty rich supporting new type of localized excitations: kinks and breathers Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 58 VI.2 Efecto Josephson A kink(anti-kink) -discrete solitons, elementary discommensurations, ...corresponds to a solution where the phases go from 0 to 2π(−2π) along the array. Since (ϕj+1-ϕj)=-2πΦj/Φ0 ; then (ϕN-ϕ1)=2π=-2πΦtot/Φ0 ; one kink corresponds to one fluxon in the array. 9 They are the only excitations that can exist in the static state. 9 The existence of kinks do not depend crucially on the discreteness of the system (in many aspects they are similar to the solitons found in the continuous version of the equation, the sine-Gordon equation). Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 59 VI.2 Efecto Josephson 9 However the existence of a discrete lattice breaks the invariance under continuous translations of the solitons of the continuous model. The kink is invariant under discrete translations. Due to discreteness the kink is pinned to the lattice. Then there exist a minimun energy barrier the kink needs to overcome to move through the lattice. This energy is the socalled Peierls-Nabarro barrier EPN. Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 60 VI.2 Efecto Josephson We can go further and define not only a PN barrier but a PN potential for the kink. To compute the kink potential profile we let the kink configuration to relax (overdamped dynamics) from the saddle-point configuration to the minimun energy configuration. During the relaxation we work out the energy and center of mass of the configuration obtaining the potential profile E(XCM) ϕ& j = − sin ϕ j + λ (ϕ j +1 − 2ϕ j + ϕ j −1 ) • • • λ E = ∑ (1 − cos ϕ j ) + (ϕ j +1 − ϕ j ) 2 j 2 EJ X CM = C ± ∑ ϕ j j Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 61 VI.2 Efecto Josephson Then, in some cases is possible to identify the kink motion with the motion of a single particle over a (almost sinusoidal) periodic potential defined by: E PN (1 − cos X ) VPN ( X ) = 2 • In the presence of external bias, an effective equation of motion for the kink is given by E PN & & & mX + mΓX + sin X = i 2 Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 62 VI.2 Efecto Josephson The picture of the kink or fluxon as a single particle is useful in arrays which are larger than the fluxon width and are not driven or driven by small currents (far from the whirling mode). When the kinks move it radiates energy in form of small amplitude waves. This radiation is very strong for the case of underdamped arrays. There phonons are easily excited by the kink in its wake and resonances between the kink velocity and these waves appear. A particularly interesting configuration is a ring of JJ connected in parallel. There, once the array is superconducting magnetic field gets trapped as an integer number of fluxons. [After S. Watanabe et al, Phys. Rev. Lett. 74, 174 (1995)]. Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 63 VI.2 Efecto Josephson Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 64 VI.2 Efecto Josephson We have considered arrays where all the junctions are identical and all the cells have the same size. However, it is possible to design different arrays which are adequated for studying new physical problems. One example is the use of Josephson parallel arrays to study the dynamics of kinks subjected to substrate ratchet potentials [F. Falo et al, Fluxon ratchet potentials in superconducting circuits, Appl. Phys. A 75, 263 (2002)]. A ratchet potential is a periodic potential without inversion symmetry: V(x) = V(-x), then it’s easier to move a perticle in one direction than in the other. Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 65 VI.2 Efecto Josephson Examples of the PN potential for a fluxon trapped in a ring E/EJ 4,5 4,5 (a) 4,0 4,0 3,5 3,5 3,0 3,0 13 14 15 16 17 (c) 3,5 13 3,0 2,5 2,5 2,0 2,0 14 15 16 14 15 16 17 13 17 (d) 3,5 3,0 13 (b) 14 15 16 17 XCM Física de Bajas Temperaturas -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 66 VI.2 Efecto Josephson Fluxon depinning currents for the non-ratchet and the ratchet parallel arrays. 1,00 Ratchet array 0,2 0,75 ∆Idep/Ic Idep/Ic 0,1 0,50 Ratchet array 0,0 0,0 0,5 λ 1,0 1,5 Regular array 0,25 0,00 0,0 Física de Bajas Temperaturas 0,5 1,0 1,5 λ 2,0 2,5 3,0 -- Máster en Física y Tecnologías Físicas -- Curso 2006/07 67