Lecture 6. Josephson junction circuits. Simple current-biased junction Assume for the moment that the only source of current is the bulk leads, and

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1 Lecture 6. Josephson junction circuits Simple current-biased junction Assume for the moment that the only source of current is the bulk leads, and I(t) its only destination is as supercurrent through the junction. Then according to the above, I(t) I(t) = I C sin ϕ(t), d dt ϕ(t) = 2 ( µl (t) µ R (t) ) = 2eV (t)/. Thus: ϕ(t) 1. if V (t) = 0, I(t) = const. I 0, get DC Josephson effect: ϕ(t) = const. = sin 1 I/I C. 2. if V (t) = const. 0, get AC Josephson effect: I(t) = I C sin(2ev t/ ). If V (t) = V 0 + V 1 cos ωt, (V 1 V 0 ), expect resonance (dc current) when ω = 2eV 0 /, Shapiro steps. More realistic model of junction: RSJ(C) model In real life, junction is likely to be shunted by resistance (or rather conductance) and capacitance. Application of Kirchoff s laws gives (if R assumed linear) C dv dt + V (t) R + I C sin ϕ(t) = I(t), or substituting the second Josephson equation and multiplying by Φ 0 /2π, ( ) 2 Φ0 d 2 ( ) 2 ϕ(t) Φ0 1 d ϕ(t) C + 2π dt 2 2π R dt I(t) R ϕ(t) C + I CΦ 0 2π sin ϕ(t) = Φ 0 2π I(t).

2 Note: apart from the term in R 1 this can be obtained from the Lagrangian L = C ( ) 2 Φ0 ( ϕ(t) ) 2 + EJ cos ϕ(t) + Φ 0 2 2π }{{ 2π I(t) ϕ(t). } washboard potential (but : are the minima really physically different?). Washboard potential V ( ϕ) = E J cos ϕ Φ 0 2π I ϕ. Note analog of pendulum in gravitational field with applied torque. Pendulum analogy: ϕ θ E J mgl V ( ϕ) ϕ(t) C(Φ 0 /2π) 2 ml 2 moment of inertia I (Φ 0 /2π) 1 N torque Becomes unstable when I 2πE J /Φ 0 I C ( lability point ). For I < I C (metastable) equilibrium value of ϕ is ϕ = sin 1 I/I C. ( ) Neglecting dissipation, small oscillations around ( ) are harmonic with frequency ( ω = ω J 1 I ) 1/4, ω J ω(i = 0) = 2π (E J /C) 1/2, I C Φ 0 ω J Josephson plasma frequency and damping β J ω J, where we define β J 1 ω J RC. l m θ

3 Typical values of I C, C and R: (TOB s) For given t, I C scales as junction area, so can be very large. Very small values of I C are not of (current) interest if E J k B T min, i.e. I C A. The capacitance C of a TOB also scales as junction area: for t 10 Å, ɛ 10 it is 0.1 F/m 2, thus for typical W 1 µm, C 100 ff. : Values of C much smaller than this should not be taken seriously, since they will be shorted out by bulk capacitance of leads. Note for given t, ω J is independent of junction area. Will always assume ω J / βj 1 ( / τ RC) 1/2 1. The shunting conductance G R 1 : at T T C this is mostly due to excited quasiparticles (Giaever tunneling) and is R N. However, at T T C this effect goes away as exp( const.t C /T ), so only shunting effect is presumably due to normal inclusions, etc. Nowadays can routinely obtain for this subgap resistance R S values R S 1 GΩ. Note: In context of quantum information, almost always interested in case of very large R S (β J 1). Classical dynamics in the washboard potential For I < I C, two steady states: (1) ϕ = sin 1 (I/I C ) f(t), V ϕ = 0 zero-voltage state. (2) ϕ = f(t), V = ϕ 0 running state. Note: in running state, σ R 1 likely to be strongly frequency-dependent with steep increase (Giaever) at ω 2 /. Since ω = 2eV/, this means that for I = I C +, V likely to saturate at /e. For I I C transport is mostly by normalquasiparticles (Giaever) tunneling through junction, so V IR N.

4 Outward I V characteristic (I(t) increasing with time). What about return chracteristic? (I(t) decreasing from above I C )? I = V/R N More complicated problem, see e.g. Chen et al., J. Appl. Phys. 64, I C 3119 (1988). Close to retrapping, characteristic frequency is ω J, so since we take /e V I ω J R should be R S in this regime. Retrapping current I r is determined by condition that energy dissipated/period = energy gain (= I C Φ 0 /2π). I r 2 = π 2Φ 0ω J RS 1 4 π β JI C I C. For I just above I r, get analytic result ( I I r = V ) 0 e V0/V + (higher order terms in e V0/V ), I r V V 0 ω J Φ 0. Thermal escape and retrapping Figure 1: Outward IV characteristic (top) and washboard potential (bottom). (from now on ϕ ϕ) For many real-life junctions, E J is not necessarily k B T, so thermal escape from washboard potential is nonnegligible: expected to lead to rounding of I V characteristics. Quite generally, in Arrhenius-Kramers theory, for escape from metastable well, I I C V

5 I C I I r /e V ( const. = f(damping) ). Γ th = const. ω J e V 0/k B T. In the case of the washboard potential ( ) Φ0 V (ϕ) = (Iϕ + I C cos ϕ), 2π we have as above V ( ϕ) ω J = (2π/Φ 0 )(E J /C) 1/2 (1 I/I C ) 1/4, and expanding V (ϕ) around the equilibrium value of ϕ, sin 1 I/I C, ( ɛ ϕ sin 1 (I/I C ) ), ω J ϕ(t) V (ϕ) V (ɛ) = Aɛ 2 Bɛ 3, (typical form near lability ) A I ( C 1 I ) 1/2, B 1 2 I C 6 I = 1 6 I C ( Φ 0 /2π), so that V 0 = 4 A 3 ( 2 27 B = E J 1 I ) 3/2. I C Thus, if I is held at a value close to I C, the escape rate is ( Γ th = const. 1 I ) 1/4 e E J(1 I/I C ) 3/2 /k B T. I C V 0

6 should probably not be taken too seriously for I I C, when V 0 k B T If we perform an experiment by sweeping the bias current I(t) up from 0 towards I C, we expect escape to be a stochastic function of I, given by (y I(t)/I C y(t)) P (y) = Γ(y) 1 y dy/dt e dy/dt 0 Γ(y)dy. If dy/dt = ω S is constant, say, as is usually the case, this expression has a maximum at the value of y for which dγ 1 /dy = ωs 1, with a distribution around that value. It is immediately clear that P (I) = P [ 1 k B T ( 1 I ) ] 3/2, I C P (I) T i.e. both the position of the peak relative to I C and its width scale as T 2/3. Well verified experimentally. Thermal retrapping of return current: (see Chen et al., op cit.). Theory : Γ (ret) th e const. E J 2k B T little experimental input as of now. (1 I/Ir) 2 ln(1 I/Ir) note : IC 0 I P (I)dI 1 I C

7 Flux qubits Take a Josephson junction with superconducting leads and join them up: result is flux qubit (pre-2000: rf SQUID ). If junction is described by RSJ(C) model, total circuit is as shown. The extra (new) energy is the selfinductance energy of the loop 1 2 LI2 : since the total flux Φ trapped through the loop is Φ ext + LI, we have E ind = (Φ Φ ext) 2. 2L ϕ Φ ext Typical loop inductance µ 0 R (dimension of loop) nh. L In (near-) equilibrium, the phase drop ϕ across the junction is related to the total flux (external + induced) through the loop by ( ) Φ ϕ = 2π, (+2nπ). Φ 0 Unlike in the case of a CBJ, values of Φ differing by nφ 0 are manifestly not to be identified. Hence simpler to work in terms of Φ. Then, neglecting dissipation, Lagrangian L(Φ, Φ) has form L = 1 2 C Φ 2 V (Φ), ) V (Φ) = E J cos (2π ΦΦ0 + (Φ Φ ext) 2. 2L For Φ ext close to Φ 0 /2 and β L 2πLI C /Φ 0 > 1, potential has double-well structure: R C

8 Most results for flux qubit can be read off from those for CBJ by replacement ϕ 2πΦ/Φ 0, I (Φ Φ ext )/L, e.g. classical damped equation of motion is C Φ+ Φ/R+E J sin(2πφ/φ 0 ) + (Φ Φ ext )/L }{{} - V/ Φ = 0. Φ

9 Miscellaneous notes on bulk and weak-link superconductivity 1. Effects on bulk superconductivity of proximity to vacuum and to normal metal The surprising (theoretical) conclusion: proximity to vacuum has (almost) no suppressive effect, proximity to normal metal is deleterious to superconductivity! Why? In case (a) single-particle wave functions have to accommodate to boundary, and do so over a range z kf 1. Once this is taken into account, there is no further energy necessary to bend the order parameter Ψ(z). condensation energy (E S E N ) insensitive to presence of boundary. In case (b), single-particle states extend across S N boundary, so order parameter must also do so. But according to GL theory, in N metal, a) b) a) b) Ψ Ψ S S z z =0 k 1 F V N z ξ N F (αn (T ) Ψ 2 + γ N Ψ 2) dτ, where (for no e e interaction in N) α N 1/ ln(ɛ c /T ) 1/ ln(ɛ c /T C ), γ N γ S, Ψ falls off as exp( z/ξ N ), ξ N (γ N /α N ) 1/2 and energy associated with N side is F ξ N Ψ 0 2 > 0, (Ψ 0 bulk S order parameter). (A sharp bending of Ψ similar to what occurs in S V case would cost

10 gradient energies ( γ S Ψ 2 dτ) which are over and above any already present in N phase). Thus, proximity to normal metal tends to suppress superconductivity.

11 2. Extended junction (Josephson, 1965) So far, except when treating the effect of an external t magnetic field, we treated the S junction as a point. However, for larger junctions we W x need to take into account selffield effects (i.e. currents flowing z through the junction may modify the field distribution). So: define ϕ(x, y) λ L ϕ 1 (x, y) ϕ 2 (x, y) so that by definition, ϕ ẑ: Pecked lines are at z 1, z 2 0 d t + 2λ J from z = 0, so d no current flows ϕ = (2e/ )(A 1 (x, y) A 2 (x, y)) or from B = A, ϕ x = 2e db y, 1 2 ϕ y = 2e db x, B i average of B i between 1 and 2. Now combine with Maxwell: ( B = µ 0 j + D ), both sides have only z component, t 2 ϕ = 2e ( µ 0d j z + D ) z. t However: j z = J C sin ϕ, D z = σ = CV, where J C = critical current / unit area, σ = surface charge / unit area. S λ J y

12 So with V = ϕ 2e t, 2 ϕ µ 0 d C 2 ϕ t 2 = defining a characteristic length λ J it is the Josephson penetration depth, and a characteristic velocity ( ) 2e µ 0dI C sin ϕ λ 2 J sin ϕ, ( ) 1/2, (usually λ L ), 2eµ 0 dj C c J (µ 0 d C) 1/2 = (µ 0 ɛɛ 0 d/t) 1/2 c(ɛd/t) 1/2 c, where c is the speed of light in vacuum. Thus for W λ J, current flows only through edges of junction. For W λ J, flows uniformly as assumed earlier.

13 3. How good is the RSJ(C) model? (for small junctions) (a) All the above analysis assumes that any charge surplus will accumulate directly across junction. But this will not be so, if the relevant capacitance (C J ) is geometrical capacitance (C G ) of rest of circuit (e.g. in flux qubit, of ring). In that case a better model is something like R C I R L C G C J (b) Especially for clean TOB junctions at low temperatures, where R R S GΩ, main mechanism of dissipation may not come from junction itself but from leads. Thus, in general it may be advantageous to consider the junction (itself a nonlinear circuit element) as shunted by an arbitrary linear admittance subject only to constraints imposed by causality, etc.:

14 so classical equation of motion is (ϕ q) V (ω) q = K(ω)q(ω), K(ω) iωy (ω). Y (ω)