Lecture 6. Josephson junction circuits
Simple current-biased junction
∆ϕ(t)
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
I(t)
the above,
I(t) = IC sin ∆ϕ(t),
d
2
∆ϕ(t) = µL(t) − µR (t) ∼
= 2eV (t)/~.
dt
~
Thus:
1. if V (t) = 0, I(t) = const. ≡ I 6= 0, get DC Josephson effect:
∆ϕ(t) = const. = sin−1 I/IC .
2. if V (t) = const. 6= 0, get AC Josephson effect: I(t) = IC sin(2eV t/~).
If V (t) = V0 + V1 cos ωt, (V1 V0), expect resonance (dc current) when
ω = 2eV0/~ , “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)
R
dV
V (t)
I(t)
+
+ IC sin ∆ϕ(t) = I(t),
∆ϕ(t)
dt
R
or substituting the second Josephson
equation and multiplying by Φ0/2π,
2
2 2
C Φ0
Φ0 d ∆ϕ(t)
Φ0
1 d∆ϕ(t) IC Φ0
+
sin ∆ϕ(t) = I(t).
C
+
2π
dt2
2π R dt
2π
2π
C
Note: apart from the term in R−1 this can be obtained from the Lagrangian
2
2
C Φ0
Φ0
˙
L=
∆ϕ(t) + EJ cos ∆ϕ(t) + I(t)∆ϕ(t) .
2 2π
{z 2π
}
|
“washboard potential”
(but ⇑: are the minima really physically different?).
“Washboard potential”
Φ0
I∆ϕ.
2π
Note analog of pendulum in gravitational field with applied torque.
“Pendulum” analogy:
∆ϕ → θ
∆ϕ(t)
V (∆ϕ)
EJ → mgl
C(Φ0/2π)2 → ml2 ← moment of inertia
I → (Φ0/2π)−1N ← torque
Becomes unstable when I → 2πEJ /Φ0 ≡ IC (“lability point”).
V (∆ϕ) = −EJ cos ∆ϕ −
For I < IC (metastable) equilibrium value of ∆ϕ is
∆ϕ = sin−1 I/IC .
(∗)
Neglecting dissipation, small oscillations around (∗) are harmonic with frequency
1/4
I
2π
ω = ωJ 1 −
, ωJ ≡ ω(I = 0) = (EJ /C)1/2,
IC
Φ0
ωJ ← Josephson plasma frequency
and damping βJ ωJ , where we define
βJ ≡
1
.
ωJ RC
θ
l
m
“Typical” values of IC , C and R: (TOB’s)
For given t, IC scales as junction area, so can be very large. Very small values of IC are not of (current) interest if EJ . kBTmin, i.e. IC . 10−11 A.
The capacitance C of a TOB also scales as junction area: for t ∼ 10 Å,
∼ 10 it is ∼ 0.1 F/m2, 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, ωJ is independent of junction area. Will always assume
~ωJ /∆ ∼ βJ−1(~/∆τRC )1/2 1.
The shunting conductance G ≡ R−1: at T ∼ TC this is mostly due
to excited quasiparticles (Giaever tunneling) and is ∼ RN . However, at
T TC this effect goes away as exp(−const.TC /T ), so only shunting effect is presumably due to normal inclusions, etc. Nowadays can routinely
obtain for this “subgap” resistance RS values RS & 1 GΩ.
Note: In context of quantum information, almost always interested in
case of very large RS (βJ 1).
Classical dynamics in the washboard potential
For I < IC , two steady states:
(1) ∆ϕ = sin−1(I/IC ) 6= f (t), V ∝ ∆ϕ̇ = 0
“zero-voltage” state.
(2) ∆ϕ = f (t), V = ∆ϕ̇ 6= 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 = IC +, V likely to saturate at
∼ ∆/e. For I IC transport is mostly by normalquasiparticles (Giaever) tunneling through junction, so V ∼ IRN .
“Outward” I − V characteristic
(I(t) increasing with time).
What about “return” chracteristic?
(I(t) decreasing from above IC )?
I∼
= V /RN
More complicated problem, see
e.g. Chen et al., J. Appl. Phys. 64,
IC
3119 (1988).
Close to retrapping, characteristic fre- I
∆/e
V
quency is . ωJ , so since we take
ωJ ∆ R should be RS in this
regime. Retrapping current Ir is determined by condition that energy
dissipated/period = energy gain (=
Figure 1: “Outward” IV characteristic (top)
IC Φ0/2π). ⇒
and washboard potential (bottom).
4
2
Ir ∼
= 2 Φ0ωJ RS−1 ≡ βJ IC IC .
π
π
For I just above Ir , get analytic result
I − Ir
V0 −V0/V
=4 1+
e
+ (higher order terms in e−V0/V ),
Ir
V
V0 ≡ ωJ Φ0.
Thermal escape and retrapping
(from now on ∆ϕ → ϕ)
For many real-life junctions, EJ is not
IC
necessarily kBT , so thermal escape from washboard potential is nonnegligible: expected to lead to rounding of I − V characteristics. Quite I
generally, in Arrhenius-Kramers theory, for escape from metastable well,
V
IC
Ir
I
∆/e
V
Γth = const. × ωJ e−V0/kBT .
const. = f (damping) .
In the case of the washboard potential
Φ0
V (ϕ) = −
(Iϕ + IC cos ϕ),
2π
V (∆ϕ)
we have as above
ωJ = (2π/Φ0)(EJ /C)1/2(1 − I/IC )1/4, and
expanding V (ϕ) around the equilib
−1
−1
rium value of ϕ, sin I/IC , ≡ ϕ − sin (I/IC ) ,
ωJ
V0
∆ϕ(t)
V (ϕ) ≡ V () = A2 − B3, (typical form near “lability”)
1/2
1
1
I
IC
,B≡ I∼
A≡ √ 1−
= IC (×Φ0/2π),
IC
6
6
2
so that
√
3/2
3
4A
2
I
V0 =
= 4 EJ 1 −
.
27 B 2
3
IC
Thus, if I is held at a value close to IC , the escape rate is
1/4
√
I
− 4 3 2 EJ (1−I/IC )3/2 /kB T
Γth = const. × 1 −
e
.
IC
should probably not be taken too seriously for I → IC , when V0 . kBT
If we perform an experiment by sweeping the bias current I(t) up from
0 towards IC , we expect escape to be a stochastic function of I, given by
(y ≡ I(t)/IC ≡ y(t))
1 R y Γ(y)dy
Γ(y) − dy/dt
0
.
P (y) =
e
dy/dt
If dy/dt = ωS 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
kBT
1−
I
IC
3/2#
T↓ →
,
P (I)
i.e. both the position of the peak relative to IC and its width scale as T 2/3.
Well verified experimentally.
Thermal retrapping of return current: (see Chen et al., op cit.).
Theory—:
(ret)
Γth
note :
E
∼
(1−I/I )2
r
−const. 2k JT | ln(1−I/I
r )|
B
e
little experimental input as of now.
I→
IC
0
P (I)dI ≡ 1
IC
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 21 LI 2: since
the total flux Φ trapped through the loop is
Φext + LI, we have
Eind
(Φ − Φext)2
.
=
2L
∆ϕ
Φext
R
C
Typical loop inductance ∼ µ0R (dimension of
L
loop) ∼ 0.01 − 1 nH.
In (near-) equilibrium, the phase drop ∆ϕ across the junction is related
to the total flux (external + induced) through the loop by
Φ
, (+2nπ).
∆ϕ = 2π
Φ0
Unlike in the case of a CBJ, values of Φ differing by nΦ0 are manifestly
not to be identified. Hence simpler to work in terms of Φ.
Then, neglecting dissipation, Lagrangian L(Φ, Φ̇) has form
1
L = C Φ̇2 − V (Φ),
2
Φ
(Φ − Φext)2
V (Φ) = −EJ cos 2π
+
.
Φ0
2L
For Φext close to Φ0/2 and βL ≡ 2πLIC /Φ0 > 1, potential has double-well
structure:
Most results for flux qubit can be read off from
those for CBJ by replacement ϕ → 2πΦ/Φ0,
I → (Φ − Φext)/L, e.g. classical damped equation of motion is
C Φ̈+Φ̇/R+EJ sin(2πΦ/Φ0) + (Φ − Φext)/L = 0.
|
{z
}
-∂V /∂Φ
Φ
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 a)
S
is deleterious to superconductivity!
Why?
b)
In case (a) single-particle wave functions
S
have to accommodate to boundary, and do
so over a range ∆z ∼ kF−1. Once this is a)
taken into account, there is no further enΨ↑
ergy necessary to “bend” the order parameter Ψ(z).
⇒ condensation energy (ES − EN ) insen- b)
sitive to presence of boundary.
Ψ↑
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,
Z
2
2
F ∼
αN (T )|Ψ| + γN |∇Ψ| dτ,
z→
V
N
z=0
∼ kF−1
z→
∼ ξN
where (for no e − e interaction in N )
αN ∼ 1/ ln(c/T ) ∼ 1/ ln(c/TC ), γN ∼ γS ,
⇒ Ψ falls off as exp(−z/ξN ), ξN ∼ (γN /αN )1/2 and energy associated
with N side is
F ∼ ξN |Ψ0|2 > 0,
(Ψ0 ≡ bulk S order parameter).
(A “sharp” bending of Ψ similar to what occurs in S − V case would cost
R
gradient energies ( γS |∇Ψ|2dτ ) which are over and above any already
present in N phase).
Thus, proximity to normal metal tends to suppress superconductivity.
2. Extended junction (Josephson, 1965)
So far, except when treatt
ing the effect of an external
magnetic field, we treated the
S
S
junction as a point. HowW
↑x
y
ever, for larger junctions we
need to take into account selfz→
field effects (i.e. currents flowing through the junction may
modify the field distribution).
λL
So:
define ϕ(x, y) ≡
λJ
1
ϕ1(x, y) − ϕ2(x, y) so that by
definition, ∇ϕ ⊥ ẑ:
Pecked lines are at z1, z2 &
0
d ≡ t + 2λJ from z = 0, so
d
no current flows
⇒ ∇ϕ = (2e/~)(A1(x, y) − A2(x, y)) or from B = ∇ × A,
∂ϕ 2e
= dB y ,
∂x
~
2
2e
∂ϕ
= − dB x,
∂y
~
B i ≡ average of Bi between 1 and 2.
Now combine with Maxwell:
∂D
∇ × B = µ0 j +
, both sides have only z − component,
∂t
2e
∂D
z
⇒ ∇2ϕ = µ0d jz +
.
~
∂t
However: jz = JC sin ϕ, Dz = σ = C̃V ,
where JC = critical current / unit area, σ = surface charge / unit area.
So with V =
~ ∂ϕ
,
2e ∂t
2e
µ0dIC sin ϕ ≡ λ−2
J sin ϕ,
~
1/2
~
defining a characteristic length λJ ≡
, (usually λL),
2eµ0dJC
it is the Josephson penetration depth,
and a characteristic velocity
∂ 2ϕ
2
∇ ϕ − µ0dC̃ 2 =
∂t
cJ ≡ (µ0dC̃)−1/2 = (µ00d/t)−1/2 ≡ c(d/t)−1/2 . c,
where c is the speed of light in vacuum.
Thus for W & λJ , current flows only through edges of junction. For
W λJ , flows uniformly as assumed earlier.
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 (CJ ) is . geometrical capacitance (CG) of rest of circuit (e.g.
in flux qubit, of ring).
In that
case a better model is something like
R
R
←I
C
L
CJ
CG
(b) Especially for clean TOB junctions at low temperatures, where
R ∼ RS ∼ 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.:
so classical equation of motion is (ϕ → q)
∂V (ω)
= K(ω)q(ω),
∂q
Y (ω)
K(ω) ≡ −iωY (ω).