A long wave electrodynamic model of an array of small Josephson junctions

Transcription

1 A long wave electrodynamic model of an array of small Josephson junctions Frontiers in nonlinear waves, Tucson, March Jean guy Caputo Laboratoire de Mathématiques INSA de Rouen, France Lionel Loukitch

2 Parallel array of small Josephson junctions (Microwave mixer) F. Boussaha and M. Salez, Observatoire de Paris l = 1µm,w = 7µm, l j = 1µm, w j = 1µm. Interaction between junction and microstrip Coupling between junctions Design of devices

3 Resistively Shunted Junction Model P P C φ φ y P tt = Inline current feed (;w) L P I (;) Cφtt φ + Jc sin φ + αφ t = L (l;w) I (l,) x CΦ tt ( 1 L Φ) + J c sin( Φ Φ ) + Φt R =. L = µ d branch inductance, C capacity / surface, J c critical current density, α = 1/R quasiparticles, Φ = 2e ν = (1) inline (overlap) Boundary conditions Φ x x {;l} = d H (1 ν) L 2w I Assume L constant Φ y y {:w} = ν L 2l I

4 Maximum current for static solution : experiment 3 M. Salez et al, J. of Appl. Physics (27) 25 Maximum static current H A device : F. Boussaha and M. Salez Device w a 1 a 2 a 3 a 4 a 5 A in µm, junctions 1 1

5 Normalized 2D model P P C φ φ y P tt = Inline current feed (;w) L P I (;) Cφtt φ + Jc sin φ + αφ t = L (l;w) I (l,) x ϕ tt ϕ + g(x, y)[κϕ tt + sin(ϕ) + αϕ t ] = Units of length, time and flux λ J = g = 1,,,κ = C C P 1, α = 1 R Boundary conditions ϕ x x=,l = H (1 ν) I 2w, Φ J cc P. Φ J cl, 1 ω p = ϕ y y=,w = ν I 2l C P Φ J c What to do? Global analysis, 2D numerics, Reduced model

6 Outline of talk 1 The 1D discrete/continuous model 2 Static solutions I max (H) Analysis 3 4

7 1D continuous model Expand ϕ on transverse Fourier modes ( y w 2 ϕ(x, y, t) = νi 2wl mode ) 2 + n= φ n(x, t)cos ( nπy w φ tt φ xx + w j w g(x, y = )(κφ tt + αφ t + sinφ) = ν γ l γ = I w w j /w rescaling of λ J (= 1) into λ eff = w w j > 1 JGC et al, J. of Modern Physics C, vol. 7, No. 2, , (1996). ),

8 1D continuous model and 1D continuous/discrete model g d (x) = φ tt φ xx + g d [sin(φ) + αφ t ] = γ/l. { w 2 j /(dw) x a i d/2 elsewhere Total current of junction constant : l g ddx = w 2 j /w. φ with different d, γ =.5, static regime. φ Josephson junction, l = 2,1,.5, 4 5 x l=1 l=.5 l= 1.5 With 3 Josephson junctions. 4 x 6 1 lim d R a+d/2 a d/2 φtt φxx + g d(sin(φ) + αφ t)dx = [φ x] a+ a + w2 j w (sin(φ a) + αφt a)

9 1D continuous/discrete model n junctions device n φ tt φ xx + d i δ(x a i )[κφ tt + sin(φ) + αφ t ] = ν γ l with, i=1 { φx (;t) = H (1 ν)γ/2, φ x (l;t) = H + (1 ν)γ/2. Advantages : and d i = w 2 i /w., 1 Evolution of φ is not neglected in the linear part. 2 Interaction between the junction and microstrip is considered in time and in space. 3 Approach remains coherent when we change, independently, the position and area of each junctions. 4 Non uniform devices, π junctions..

10 Maximum static current Maximal γ for H given. n φ xx + d i δ(x a i )sin(φ) = ν γ l i=1, { φ with, (;t) = H (1 ν)γ/2, φ (l;t) = H + (1 ν)γ/2. n i=1 d i sin(φ i ) = γ J. G. C. and L. Loukitch, SIAM J. of Appl. Math. (27)

11 Numerical methods shooting method : Polynomial by part depends on φ(a 1 ) and γ (H, ν fixed) P 1 (x) = νγ 2l (x2 a 2 1 ) + ( H (1 ν) γ 2) (x a1 ) + φ(a 1 ) P k+1 (x) = P 1 (x) + k m=1 d m sin(p m (a m ))(x a m ) Boundary value problem, nonlinear system : n = 5 φ = ψ ν γ l x 2 2 ψ f (x) H (1 ν) γ 2 ψ 2 ψ 1 a 2 a 1 + d 1 sin(ψ 1 f 1 ) =, ψ 3 ψ 2 a 3 a 2 + ψ 2 ψ 1 a 2 a 1 + d 2 sin(ψ 2 f 2 ) =, ψ 4 ψ 3 a 4 a 3 + ψ 3 ψ 2 a 3 a 2 + d 3 sin(ψ 3 f 3 ) =, ψ 5 ψ 4 a 5 a 4 + ψ 4 ψ 3 a 4 a 3 + d 4 sin(ψ 4 f 4 ) =, H (1 + ν) γ 2 + ψ 5 ψ 4 a 5 a 4 + d 5 sin(ψ 5 f 5 ) =.

12 The Periodicity γ max (H) = γ max (H + H p ) l j = a j+1 a j, and l min = min j (l j ) - Harmonic array : If l i is a multiple of l min for all i then H p = 2π l min - General case, if l j = p j /q j, p j and q j are integers, then H p = 2π LCM(q1;...;qn 1) HCF(p 1;...;p n 1) I max (H) inline overlap 3π 6π 9π 12π 15π 18π 21π a 1 = 1, a 2 = and a 3 = H p = 2πLCM(2; 3)/HCF(3; 5) = 12π.

13 Magnetic shift C1 {a 1,...,a n }, C2 {a 1 + c,...,a n + c}, same l, ν, d i γ max (H) C2 = γ max (H + H ν ) C1 avec H ν = νγ(a n + a 1 l)/(2l) I max (H) c= c= π -π π 2π l = 1, ν = 1, a 2 a 1 = 1.5, a 3 a 2 = 2.5, a 4 a 3 = 2

14 Effects of ν, l et d j : convergence inline overlap π 2π inline overlap π 2π inline overlap π 2π magnetic overlap inline 2π 4π magnetic overlap inline 2π 4π magnetic overlap inline 2π 4π l = 8, 16 et 64 d j = 1,.3 et.1. γ max (H) ( with ν γ max (H) with ν = (same circuit) ) when ν j d j /l Inline and overlap magnetic approximation for d j

15 Magnetic approximation, d j, φ(x) Hx + C, for d j 1 γ(h) = N j=1 d j sin(ha j + C), c max (H) = arctan ( PN i=1 d j cos(ha j ) P N j=1 d j sin(ha j ) ), γ max (H) = N j=1 d j sin(ha j + c max (H)) J. H. Miller et al, Appl. Phys. Lett. 59, 333, (1991).

16 Magnetic approximation, experiments M. Salez et al, J. of Appl. Physics (27) 3 Measured Simulated A 3 Measured Simulated B I max (H) (µma) I max (H) (µma) A and B devices : F. Boussaha and M. Salez Device w a 1 a 2 a 3 a 4 a 5 A B in µm, junctions 1 1

17 Symetric device : Symetric device : a j = a j et d j = d j, For junction j position a j and strength d j = w 2 j /w Properties 1 Moving junction unit does not change γ max (H) 2 γ max (H) = γ max ( H) 3 Current maximum for c max (H) = ±π/2, γ max (H) = d + 2 (n+1)/2 j=1 d j cos(ha j ) Three junctions : I max (H) = d + 2d 1 cos(ha j ) d 1 a 1 d a a 1 = 1, a =, a 1 = 1, d 1 = d 1 = 1 d = d = 1 d = 2 d = 3 (Likharev) π/2 γ max c max γ max c max d1 a1 γ max c max γ max c max π/2 π 2π π 2π π 2π π 2π

18 : position and sizes of junctions from γ max (H) J.G.C. and L. Loukitch, Inverse Problems, 28 Proposition Assume a γ g (H) even, periodic of period H p and strictly positive. The array is harmonic and the positions of the junctions are given by a i = i2π/h p where i is an integer. Their strengths d i are given by d i = d i = 1 Hp γ g (H)cos(Ha i )dh, i {,...,n/2}. (1) H p

19 Equidistant junctions I max (H) 1-4 A pulse uniform µm 14 2π H 2π µm 2

20 Non equidistant junctions triangle same d i µm π H 2π µm

21 : square γ max (H) magn. approx. inline 2π 3π/2 π π/2 π/2 π 3π/2 2π H i a i πd i 2h sin(2πh) sin(4πh) π 2π...

22 High voltage solution Rotating phase approximation φ vxx + αv n j=1 d jδ(x a j ) = ν γ l φ Vt + φ v (x) φ vx x= = H (1 ν) γ 2, φ vx x=l = H + (1 ν) γ 2 Solution exists if V = γ α P j d j Simplify problem ψ tt ψ xx + n j=1 d jδ(x a j ) ψ = φ φ v [ κψ tt + αψ t + sin(ψ + Ψ j ) Boundary conditions ψ x x= =, ψ x x=l = ] P γ n k=1 d = k

23 Adapted spectral problem ψ tt ψ xx + n j=1 d jδ(x a j )κψ tt = BCs ψ x,l = ψ(x, t) = e iωt ϕ(x) [ ϕ xx + ω ] n j=1 d jδ(x a j )κ ϕ =, κ = standard (harmonic) cosine Fourier modes κ new modes, non harmonic

24 κ =, single junction in cavity ) la + d 1 (αψ 1t + sin(ψ 1 ) γd1 A p + ( pπ l ) 2 Ap + 2d1 l c 1 p =, ) (αψ 1t + sin(ψ 1 ) γd1 =, ohmic mode : cavity driven by junction, αφ t a γ/d a =, [φ x ] a+ a = d a sin(φ a ) γ. junction mode : φ behaves the same in the whole circuit, sin(φ a ) + αφ t a γ/d a =, [φ x ] a+ a = γ. 16 numerical analytical 16 numerical analytical φ(a,t) t t φ(a,t) : junction mode ohmic mode

25 Dissipative kink solution κ, H 1 One junction only ] ψ tt ψ xx + d 1 δ(x a 1 ) [αψ t + sin(ψ) γ =, d1 Solution of wave equation Solution in junction ψ(x; t) = f (x ± t) f (ξ) αf + sin(f ) γ 2d 1 = ψ(x,t) = 2arctan (1 2α/(x ± t + C)) + 2kπ, for x ± t < C, π/2, for x ± t = C, 2arctan (1 2α/(x ± t + C)) + 2(k + 1)π, for x ± t > C, N. Velocity =1, width increases with α Kinks can exist for a few junctions if Ψ j 1 k

26 The 1D long wave continuous/discrete model Static solutions Imax (H) I V curves of the 5 junction device, κ = φ(x; t), x [; l], t =.8k, k N.16 5 jonc. π/2+4π φ π/2+2π γ.1.8 π/2.6 a1.4 a3 a4 a5 2 lim. junction : V = 3π/l, γ =.7 π/2+6π V in π/l I V curves of the device of the top photo, 5 junctions, di =.25, κ =, α =.3, H =, ν =. lim. soliton(v ) = Imax (H) =.125 lim. junction(v = h ) Pn i p j=1 dj cos(vaj ) αv (αv ) + supk {1,...,n} dk cos(vak ) Pn αv i=1 di 5 jonc. kink π/2+4π φ.2 a2 x lim. junction lim. soliton I-V π/2+2π π/2 a1 a2 a3 a4 a5 2 x lim. soliton : V = 2π/l, γ =.125

27 Same device but κ κ = 16 : κ = 64 : ω 2 +ω 4 ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 ω 7 ω 8 ω 9 ω 1 ω ω 13 ω 14 ω 15 ω 16 ω 17 ω 18 ω 19 2 ω 21 V en π/l I-V κ = 64, φ vs ε n φ(x;t), x [; l], t =.4k, k N 17π/2 13π/2 9π/2 5π/2 π/2 a 1 a 2 a 3 a 4 a 5 2 x φ ε 8 +3 V = ω , γ = π/2 13π/2 9π/2 ω 1 2 ω 3 4 ω 5 ω 6 7 ω ω9 8 ω 1 ω 11 ω ω13 12 ω 14 ω 15 ω 16 ω 17 ω 18 ω 19 2 ω 21 Associated linear problem ; φ tt φ xx + P n i=1 d jδ(x a j )κφ tt =. ω n eigenvalues : position of resonances, ε eigenvectors : adapted scalar product. 5π/2 π/2 φ ε 1 +3 a 1 a 2 a 3 a 4 a 5 2 x V = ω 1.328, γ =.781

28 Conclusion Statics well understood Localized sine nonlinearities (junctions) excite the whole spectrum of a cavity Small capacity miss-match (κ ) IV curve well understood (resonances, kinks) Large capacity miss-match (κ ) Linear problem dominates, new eigenfrequencies and eigenmodes