Download 2. Physics of Josephson Junctions: The Zero Voltage State 2.1 Basic

Chapter 2
Physics of Josephson Junctions:
The Zero Voltage State
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.1 Basic Properties of Lumped Josephson Junctions
• small spatial dimensions:
gauge invariant phase diff. & current density are uniform
 variations of supercurrent density on length scale larger than l L
≈ 10 nm – 1 ¹m
Josephson junction: ns strongly reduced:
l L→ l J ≈ 10 - 100 ¹m
2.1.1 The Lumped Josephson Junction
• JJ with spatially homogeneous supercurrent density and phase difference: lumped element
region of integration: junction area S
current-phase relation:
gauge invariant phase difference:
voltage-phase relation:
• uniform phase difference  total derivative
AS-Chap. 2 - 2
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.1.1 The Lumped Josephson Junction
„0“-junction
„¼“-junction
2.1.2 The Josephson Coupling Energy
• finite energy stored in JJ: overlap of macroscopic wave functions  binding energy
• initial current & phase difference: zero
• increase junction current from zero to finite value
 phase difference has to change
 voltage-phase relation: finite junction voltage
 external source has to supply energy (to accelerate the superelectrons)
 stored in kinetic energy of moving superelectrons
 integral of the power = Is ·V
(voltage during increase of current):
AS-Chap. 2 - 3
2.1.2 The Josephson Coupling Energy
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
with (0) = 0 and (t0) = :
integration:
Josephson coupling energy
• order of magnitude:
- typical Ic: 1 mA  EJ0 ≈ 3 x 10-19 J
- corresponds to thermal energy kBT around kB x 20 000 K
- junction with very small critical current Ic ≈ 1 µA  thermal energy ≈ kB x 20 K
AS-Chap. 2 - 4
2.1.3 The Superconducting State
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
–Ic < I < Ic  constant phase difference:
 zero junction voltage:
zero-voltage state / ordinary (S) state
• analysis of stability of (junction + current source) – system:
potential energy Epot of the system under action of external force: E – F·x
E: intrinsic free energy of the subsystem junction
F: generalized force: F = I
x: generalized coordinate: F· x/t  power flowing into subsystem (I·V):
 potential energy:
tilted washboard potential
~n
stable minima n, unstable maxima 
states for different n: equivalent
AS-Chap. 2 - 5
2.1.3 The Superconducting State
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• properties of the washboard potential
 0 for I/Ic  1
• close to Ic: a ≡ 1 - I/Ic << 1, we get the approximations:
• washboard potential extremely useful in describing junction dynamics for I > Ic
AS-Chap. 2 - 6
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.1.4 The Josephson Inductance
no minima for I > Ic
• energy storage in JJ  nonlinear reactance
• for small variations around Is = Ic sin: JJ equivalent to inductance
with
Josephson
inductance
• Josephson inductance:
Ls is negative for
(for V > 0: oscillating Josephson current)
AS-Chap. 2 - 7
2.1.4 Junction Fabrication
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
Cross-sectional views of two kinds of
junctions: R and RC-type.
JJ: upper Nb pattern of the
Nb/AlOx/Nb junction, RC: contact
between the junction and the
resistor, JC: contact between the
junction and the M4 layer
S. Nagasawa et al.,
Physica C: Superconductivity
Volumes 426–431, Part 2,
pp. 1525–1532 (2005)
AS-Chap. 2 - 10
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.1.5 Mechanical Analogs
• plane mechanical pendulum in uniform gravitational field:
phase difference :
supercurrent Is :
voltage V:
angle of pendulum with respect to equilibrium
torque
angular velocity of pendulum
• particle moving in tilted washboard potential
coordinate
x/ 
velocity
 v / d/dt / V
AS-Chap. 2 - 11
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.2 Short Josephson Junctions
• so far: zero-dimensional JJ (lumped elements)
 spatially homogeneous supercurrent density and phase difference
• now: extended junctions  spatial variations Js(r) and (r)
 consider magnetic field generated by the Josephson current itself (self-field):
• short Josephson junctions:
 self-field small compared to external field
• long Josephson junctions:
 self-field no longer negligible
• relevant length scale for transition from short to long Josephson junction:
Josephson penetration depth
density in weak coupling region
• JJ at finite voltage
 temporal interference  oscillation of Josephson current
• JJ at finite phase gradient  spatial interference  magnetic field dependence of
Josephson current
AS-Chap. 2 - 12
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.2.1 Quantum Interference Effects - Short JJ in an Applied
Field
• external magnetic field
 spatial change of gauge invariant phase difference (r)
 spatial interference of macroscopic wave functions in JJ
•
•
•
•
•
•
insulating barrier thickness: d
junction area: A = L¢W
W, L >> d (edge effects small)
electrode thickness > l L
magnetic field: Be=(0,By,0)
magnetic thickness:
tB = d+l L1+l L2
• effect of Be on Js :
- phase shift (P)-(Q) between point P and Q separated by dz
- line integral along red contour yields total phase change along closed contour: 2¼n
AS-Chap. 2 - 14
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.2.1 Quantum Interference Effects - Short JJ in an Applied
1
2
3
Field
4
gauge invariant phase gradient in the bulk superconductor:
gauge invariant phase difference across the barrier:
1 & 3 are differences acrosss the junction:
2 & 4 differences in the bulk, supercurrent equation for 𝛻𝜃:
AS-Chap. 2 - 15
2.2.1 Quantum Interference Effects - Short JJ in an Applied
Field
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• substitution:
• integration of A around closed contour  enclosed flux F
• integration of Js excludes insulating barrier  incomplete contour C‘:
• difference of gauge invariant phase differences (Q)-(P):
• line integral of supercurrent density Js :
- segments in x-direction cancel (separation: dz  0)
- segments in z-direction: deep inside SC (Àl L)  Js exponentially small
 therefore:
• total flux enclosed by the loop:
AS-Chap. 2 - 16
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.2.1 Quantum Interference Effects - Short JJ in an Applied
Field
• similar argument for P and Q separated by dy in y-direction
then
magnetic field component parallel to junction
plane induces phase gradient
• integration gives:
 0: phase difference at z = 0
• current phase relation:
with
• Js varies periodically with period Dz = 2¼/k = F0/tBBy
- flux through the junction within one period: F0
nota bene:
if t1<¸1 and t2<¸2:
AS-Chap. 2 - 17
Brief Summary
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1. Josephson equation:
2. Josephson equation:
• JJ with spatially homogeneous Js and : lumped element
Josephson coupling energy
tilted washboard potential
Josephson inductance
• extended junctions: Js = Js (r) and  = (r)
short JJ: no self-field ↔ long JJ: self-field
 Josephson penetration depth
• short junction in external field
field induces gradient of phase difference 
AS-Chap. 2 - 18
2.2.2 The Fraunhofer Diffraction Pattern
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• how does
depend on the applied field Be = (0,By,0)?
• integration in in y-direction:
• integral: complex, multiplication by ei0 does not change magnitude
•  maximum Josephson current: magnitude
magnetic field dependence of Is m
 Fourier transform of ic(z)
 analogy to optics
• Jc(y,z) homogeneous  ic(z) constant for  diffraction pattern of a slit:
Fraunhofer diffraction pattern:
flux though the junction
• experimental observation of Ims(F)  proof of Josephson tunneling of pairs
AS-Chap. 2 - 19
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.2.2 The Fraunhofer Diffraction Pattern
• spatially homogeneous maximum current density Jc(y,z)
 Fraunhofer diffraction pattern:
• maximum current
density integrated along
y-direction
• experiment: study of the homogeneity of the supercurrent flow in JJ
AS-Chap. 2 - 20
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.2.2 The Fraunhofer Diffraction Pattern
• intepretation of the shape of Is m(F):
spatial distribution of is (z) = ∫ Js (y,z)dz for
different applied fields
• zero field, F = 0:
(z) = 0
is (z) = const.
Josephson current maximum for 0 = - ¼/2:
Js (y,z) = - Jc(y,z)
• F = F0 /2:
 sinusoidal variation of supercurrent with z
difference between edges:
- half of a full oscillation period
- Josephson current maximum for 0 = - ¼/2
- linear increase of phase from -¼ at z = -L/2 to 0 at z = L/2
AS-Chap. 2 - 21
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.2.2 The Fraunhofer Diffraction Pattern
local Josephson current density in negative and
positive x-direction, but no net total current
 Josephson current tends to decrease with increasing field
spatial interference effect of
macroscopic wave functions:
plane of constant phase in
superconductor 2 is tilted by:
here: destructive interference
AS-Chap. 2 - 22
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.2.2 The Fraunhofer Diffraction Pattern
Josephson vortex
• closed current loop
• no penetration of applied field
into electrodes
• Josephson vortex
• no normal core
• vortex core in barrier region
AS-Chap. 2 - 23
2.2.2 The Fraunhofer Diffraction Pattern
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• arbitrary direction of the applied field within barrier plane:
AS-Chap. 2 - 24
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.2.3 Determination of the Maximum Josephson Current Density
• real JJ: inhomogeneities, e.g. spatially varying barrier thickness
- experimental determination of Jc(y,z) by measuring Is m(B) ???
 no access via inverse Fourier transform
 lack of phase information
• approximate ic(z) under certain assumptions
e.g.: symmetry to junction midpoint
n: number of zeros of |Is m(k)| between 0 and k
 information on Jc(y,z) on small length scale  high fields
 spatial resolution ≈ 1/By:
measurement of Is m(F) up to F/F0 = 1
 spatial resolution: junction length L
• tailored junctions:
- only central maximum desirable (x-ray detectors, suppression of supercurrent)
 Gaussian current distribution:
AS-Chap. 2 - 25
2.2.3 Determination of the Maximum Josephson Current Density
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• Is m(F)-dependence without side lobes:
• junction shape: should approache Gauss curve for homogeneous Jc(y,z)
 integrated current density in y-direction:
 Gaussian profile
AS-Chap. 2 - 26
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.2.3 Determination of the maximum Josephson current density
Additional topic:
supercurrent auto-correlation function
Comparison:
optical diffraction experiment ↔
field dependence of maximum
Josephson current:
• transmission function P0(z) ↔ ic(z)
• square root of light intensity Pt in
focal plane ↔ Is m(By)
backtransformation gives Pi (spatial
resolution given by # of diffraction
orders)
↔ phase is lost, backtransformation
of intensity (Icm)2 (By)
 autocorrelation function of
supercurrent distribution
AS-Chap. 2 - 27
2.2.3 Determination of the maximum Josephson current density
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• definition of auto-correlation function:
 overlap of ic(z) and same function shifted by ±
• Wiener-Khinchine theorem:
autocorrelation function of ic(z):
• spatial information of AC-function
depends on magnetic field interval
spatial resolution:
• recording 100 lobes in Is m(B)  spatial resolution 0.01 £ junction width
m
• statistical information in envelope of |Is (By)|2:
e.g. inhomogeneities with probability p(a) / 1/a (p £ length scale a = const.)
m
 |Is (By)|2 / 1/By („spatial 1/f noise“)
AS-Chap. 2 - 28
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.2.3 Determination of the maximum Josephson current density
• random distribution of filaments with
width a:
 envelop const. up to k = 2¼/a, then
≈ 1/By2
“spatial shot noise”
• analysis of AC gives statistical
information on current density
inhomogeneities
• YBa2Cu3O7-± grain boundary JJ
slope of envelop is about -0.65
m
 |Is (By)|2  1/By1.3
 p(a) / 1/a1.5
 small scale inhomogeneities are
more probable
AS-Chap. 2 - 29
2.2.4 Additional Topic: Direct Imaging of Supercurrent Distribution
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
scanning of JJ by focused electron / laser beam
 measure change dIsm(y,z) as function of beam position (y,z)
 ±Is m(y,z)  Jc(y,z)  2-dim. image of Jc(y,z)
 spatial resolution ≈ thermal healing length (≈ 1 ¹m)
AS-Chap. 2 - 30
2.2.5 Additional Topic: Short JJ - Energy Considerations
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
junction energy: E = ES + EI (electrode and barrier energy)
magnetic field energy + kinetic energy
Josephson coupling energy
Vi = Ai·d: insulator volume
EB: energy due to external field
EJ: Josephson coupling energy
short junction: EB À EJ
AS-Chap. 2 - 31
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.2.5 Additional Topic: Short JJ - Energy Considerations
define: short junction: EB >> EJ
for t1, t2 À ¸L the first integral dominates:
integration volume:
EJ for spatially homogeneous Jc(y,z):
for one flux quantum in the junction EB À EJ requires:
with Jc = Ic/WL:
(≈ Josephson penetration depth)
 (sec. 2.3)
AS-Chap. 2 - 32
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.2.6 The Motion of Josephson Vortices
• Josephson vortices: visualize Josephson current density
vortices moving in z-direction at constant speed vz
short junction: self-field negligible
 flux density in junction given by Be = (0,By ,0)
 gauge invariant phase difference:
• passage of 1 vortex changes phase by 2¼:
• current density pattern: moves at vz,
vortex with period p = L ©0/©
number of vortices in junction:
AS-Chap. 2 - 33
2.2.6 The Motion of Josephson Vortices
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• change of gauge-invariant phase difference:
2¼ £ # of vortices
• rate of vortex passage:
• with the voltage-phase relation:
constant velocity of vortices
 constant junction voltage / vortex rate
 application: single flux quantum pump
@ pump frequency f = dNV /dt
 V = f · F0
AS-Chap. 2 - 34
2.2. Summary – Short Josephson Junctions
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• short: lateral junction dimensions small compared to Josephson penetration depth
• effect of magnetic field parallel to junction electrodes
phase gradient
Josephson current density
• spatial distribution of is(z) = ∫ Js(y,z)dz
• Josephson vortex
AS-Chap. 2 - 35
2.2. Summary – Short Josephson Junctions
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• magnetic field dependence of maximum Josephson current:
flux though junction
Fraunhofer diffraction pattern
 analogy to optics (single slit diffaction)
no inverse Fourier transform
(missing phase)
autocorrelation function
• motion of Josephson vortices:
motion of single vortex across junction results in phase change of 2𝝅
constant motion of vortices  constant junction voltage / vortex rate
AS-Chap. 2 - 36
2.3 Long Josephson Junctions
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.3.1 The Stationary Sine-Gordon Equation
 generally valid
• now: magnetic flux density given by external and self-generated field
with B = ¹0H and D = ²0E:
• zero-voltage state:
• spatial derivative:
• assume: Jc(y,z) = const. and use Jx(y,z) = -Js(y,z)  Jx(z) = -Jc sin
 stationary Sine-Gordon equation (SSGE) (nonlinear differential equation)
Josephson penetration depth:
AS-Chap. 2 - 37
2.3.1 The Stationary Sine-Gordon Equation
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
two-dimensional stationary Sine-Gordon equation:
• relation between London and Josephson penetration depth
with
insert into expression for Josephson penetration depth
 l J corresponds to London penetration depth of weak coupling region with reduced
superelectron density ns *
AS-Chap. 2 - 38
2.3.1 The Stationary Sine-Gordon Equation
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
additional topic: analytical solutions of the SSGE
• small   linearization: sin  ≈ 
magnetic field along the junction:
¸J is a decay length
with
current is flowing at the edges of the junction
this Meißner solution is possible for Jx < Jc or By(z=0) ≤ µ0 Jc¸J
• for a small junction L ¿ ¸ J :
 short junction result
AS-Chap. 2 - 39
2.3.2 The Josephson Vortex
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
particular solution of the SSGE:
general solution: particular + homogeneous solution
important case: junction of infinite length, d/dz vanishes at z = §1
 particular solution ´ complete solution
AS-Chap. 2 - 40
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.3.2 The Josephson Vortex
• decay length for Js and By: ¸J
• maximum of Js
• integration: total current = 0, total flux = ©0
 Josephson vortex in a long JJ
AS-Chap. 2 - 41
2.3.2 The Josephson Vortex
additional topic: Energy of the Josephson Vortex Solution
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
energy stored in long JJ:
superconductor
insulator
stored energy for vortex solution, thick electrodes:
using
and integration over y and x:
now we use the vortex solution
AS-Chap. 2 - 43
2.3.2 The Josephson Vortex
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
additional topic: Energy of the Josephson Vortex Solution
energy per unit length of vortex:
EVortex > 0  we need:
external field and /or
current to supply energy
magnetic flux density Bc1 for first vortex entrance:
Bc1 ≈ magnetic flux density of a single flux quantum distributed over an area tB ¢ ¸J
AS-Chap. 2 - 44
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.3.3 Junction Types and Boundary Conditions
• junction geometry determines current flow  boundary conditions of SSGE
 magnetic flux density at junction edges:
• problem: B = Bex + Bel
 Bel not negligible and junction geometries are complicated
 current distribution in electrodes depends on current distribution in JJ itself
 boundary conditions depend on solution
 numerical iteration method required !!
J. Mannhart, J. Bosch, R. Gross, R. P. Huebener
Phys. Lett. A 121, 241 (1987)
• three basic types of junction geometries:
 overlap junction
 in-line junction
 grain boundary junctions
AS-Chap. 2 - 45
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.3.3 Junction Types and Boundary Conditions
A: overlap junction
- overlap of width W
- junction of length L extends in z-direction
perpendicular to current flow
- Bel is parallel to z  ? to the short side
 Fel = BelWtB negligibly small
B: inline junction
- overlap of length L
- junction of length L extends in z-direction
parallel to current flow
- Bel is perpendicular to z  ? to the long side
 Fel = BelLtB is not negligible
AS-Chap. 2 - 46
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.3.3 Junction Types and Boundary Conditions
C: grain boundary junction
- mixture of overlap and inline geometry
junction area is perpendicular to electrode currents
- junction area extends in yz-plane
perpendicular to current flow
- both y- and z-component of Bel are in junction plane
- Bzel has negligible impact since W << L
Felz = BelWtB << Fely = BelLtB
- finite inline admixture s = W/L ¿ 1
HTS bicrystal grain
boundary junction
AS-Chap. 2 - 47
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
2.3.3 Junction Types and Boundary Conditions
overlap junction:
highest zero field Is m ( junction area Ai = LW)
inline junction:
Is m saturates at 4Jc W¸J (Meißner screening, current flow at edges)
asymmetric inline junction: fields generated in bottom and top electrode point in the
same direction
 adds to/subtracts from external field
 increase or decrease of Ism with field
AS-Chap. 2 - 57
Summary (short junctions)
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
Josephson penetration depth
(short – long junctions)
Josephson coupling energy
nonlinear inductance
washboard potential
in-plane magnetic field By
spatial oscillations of the Jospehson current density:
integral Josephson current  FT of ic(z):
AS-Chap. 2 - 59
Summary (long junctions)
R. Gross, A. Marx, and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
SSGE: spatial distribution of gauge invariant phase difference:
self-consistent solution: boundary conditions depend on flux density at edges
3 basic types: inline, overlap and grain boundary junctions
particular solution of SSGE: Josephson vortex
AS-Chap. 2 - 60