Download Chapter 1 Macroscopic Quantum Phenomena

Chapter 1
Macroscopic Quantum Phenomena
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
I. Foundations of the Josephson Effect
1.
Macroscopic Quantum Phenomena
1.1
The Macroscopic Quantum Model of Superconductivity
quantum mechanics:
- physical quantities (momentum, energy) are quantized
- usually thermal motion masks quantum properties
 no quantization effects on a macroscopic scale
- superconductivity:
 macroscopic quantum effects are observable
 e.g. quantization of flux through a loop
 why: electrons form highly correlated electron system
Chap. 1 - 2
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.1.1 Coherent Phenomena in Superconductivity
milestones:
1911
discovery of superconductivity
1933
Meißner-Ochsenfeld effect
1935
London-Laue-theory: phenomenological model describing observations
1948
London treats superelectron fluid as a quantum mechanical entity
 superconductivity is an inherent quantum phenomenon manifested on a
macroscopic scale
 macroscopic wave function Y(r,t) = Y 0 eiq(r,t)
 London equations
1952
Ginzburg-Landau theory
 description by complex order parameter Y(r)
 treatment of spatially inhomogeneous situations
1957
microscopic BCS theory (J. Bardeen, L.N. Cooper, J.R. Schrieffer)
 BCS ground state Y BCS (coherent many body state)
1962
prediction of the Josephson effect
Chap. 1 - 3
1.1.1 Coherent Phenomena in Superconductivity
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
Macroscopic quantum model of superconductivity:
macroscopic wave function
 describes the behavior of the whole ensemble of superconducting electrons
 justified by microscopic BCS theory:
- small portion of electrons close to Fermi level are bound to Cooper pairs
- center of mass motion of pairs is strongly correlated
- example:
wave function
every pair has momentum
or velocity
Chap. 1 - 4
1.1.1 Coherent Phenomena in Superconductivity
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
Basic quantum mechanics:
quantization of electromagnetic radiation (Planck, Einstein):
photons represent smallest amount of energy:
Luis de Broglie: describes classical particles as waves  wave particle duality
particle – wave interrelations:
Erwin Schrödinger: development of wave mechanics for particles
complex wave function describes quantum particle:
free particle without spin (Epot = 0  E = Ekin): start with wave equation
vph: phase velocity
Chap. 1 - 5
1.1.1 Coherent Phenomena in Superconductivity
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
Basic quantum mechanics:
quantization of electromagnetic radiation (Planck, Einstein):
photons represent smallest amount of energy:
Luis de Broglie: describes classical particles as waves  wave particle duality
particle – wave interrelations:
Erwin Schrödinger: development of wave mechanics
complex wave function (wave packet) describes quantum particle:
Chap. 1 - 6
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.1.1 Coherent Phenomena in Superconductivity
with
multiply by
use
since
Chap. 1 - 7
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.1.1 Coherent Phenomena in Superconductivity
with similar considerations Schrödinger postulated a general time dependent
Schrödinger equation (differential equation) for massive quantum objects :
Hamilton operator
• we restrict ourselves to systems with constant total energy (conservative systems)
 due to E = ħw also the frequency is constant
 prefactor of 2nd term on lhs of
is constant
 solutions can be split in a two parts depending only on space and time
Chap. 1 - 8
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.1.1 Coherent Phenomena in Superconductivity
 stationary Schrödinger equation:
Chap. 1 - 9
1.1.1 Coherent Phenomena in Superconductivity
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
Probability currents:
interpretation of complex wave function
(note: em fields represented as the real or imaginary part of a complex expression)
Schrödinger equation suggests that phase has physical significance
Max Born: interpretation of
absolute square as probability of
quantum object
conservation of probability density requires
(continuity equation)
describes evolution of probability in space and
time statement of conservation of probability
describes probabilistic flow of quantum object, not the motion of a charged particle in
an electromagnetic field (forces depending on the motion of the particle itself)
Chap. 1 - 10
1.1.1 Coherent Phenomena in Superconductivity
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• some math: derivation of probability current density
(Schrödiger equation)
(multiply from left by Y*)
subtracting the complex conjugate
yields
vector identity:
Chap. 1 - 11
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.1.1 Coherent Phenomena in Superconductivity
 time evolution of
Task: find
defines probability current:
for a charged particle in em field
start with classical equation of motion:
canonical momemtum (kinetic and field momentum)
Chap. 1 - 12
1.1.1 Coherent Phenomena in Superconductivity
Lorentz’s law:
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
Lorentz’s law with E and B expressed in terms of potential Φ and A
with:
 generalized potential:
Schrödinger equation:
insert expression for V(r,t)
Chap. 1 - 13
1.1.1 Coherent Phenomena in Superconductivity
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
Schrödinger equation of a charged particle in an electromagnetic field:
probability current of a charged particle in an electromagnetic field:
central expression in quantum description of superconductivity
 wave function of single charged particle will be replaced by
macroscopic wave function describing the whole entiry of all
superelectrons
Chap. 1 - 14
1.1.2 Macroscopic Quantum Currents in Superconductors
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
normal metals:
electrons as weakly/non-interacting particles
 ordinary Schrödinger equation:
where
is the complex wave function of a particle
in the stationary case:
 quantum behavior reduced to that of wave function phase
 Fermi statistics  different energies: different time evolution of phase
 phases are uniformly distributed, phase drops out for macroscopic quantities
Chap. 1 - 15
1.1.2 Macroscopic Quantum Currents in Superconductors
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
Superconductors
spin singlet Cooper pairs, S = 0 (in the simplest case)
 large size of Cooper pairs, wave functions are strongly overlapping
 pairs form phase-locked coherent state, description by single wave function with phase µ
 identical µ/t
 macroscopic variables (e.g. current) can depend on phase µ
 µ changes in quantum manner under the action of an electromagnetic field
 zero resistance, Meißner-Ochsenfeld effect, coherence effects
(flux quantization, Josephson effect)
similarity to electron orbiting around nucleus in an atom
Chap. 1 - 16
1.1.2 Macroscopic Quantum Currents in Superconductors
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
Central hypothesis of macroscopic quantum model:
there exists a macroscopic wave function
describing the behavior of all superelectrons in a superconductor
(motivation: superconductivity is a coherent phenomenon of all sc electrons)
normalization condition:
Ns and ns (r,t) are the total and local density of superconducting electrons
 charged superfluid (analogy to fluid mechanics)
 similarities in description of superconductivity and superfluids
 no explanation of microscopic origin of superconductivity
 relevant issue: describe superelectron fluid as quantum mechanical entity
Chap. 1 - 17
1.1.2 Macroscopic Quantum Currents in Superconductors
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
Some basic relations:
• general relations in electrodynamics:
electric field:
flux density:
A = vector potential
f = scalar potential
• electrical current is driven by gradient of electrochemical potential:
• canonical momentum:
• kinematic momentum:
Chap. 1 - 18
1.1.2 Macroscopic Quantum Currents in Superconductors
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• Schrödinger equation for charged particle:
electro-chemical potential
• insert macroscopic wave-function
Y  y,
q  qs ,
m  ms
• split up into real and imaginary part and assume
real part:
energy-phase relation
kinetic energy
imaginary part:
potential energy
current-phase relation
superelectron velocity vs  Js = ns qs vs
Chap. 1 - 19
1.1.2 Macroscopic Quantum Currents in Superconductors
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• we start from Schrödinger equation:
electro-chemical potential
• we use the definition
and obtain with
Chap. 1 - 20
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.1.2 Macroscopic Quantum Currents in Superconductors
Chap. 1 - 21
• equation for real part:
∆
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.1.2 Macroscopic Quantum Currents in Superconductors
energy-phase relation (term of order ²ns is usually neglected)
Chap. 1 - 22
1.1.2 Macroscopic Quantum Currents in Superconductors
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• interpretation of energy-phase relation
corresponds to action
 in the quasi-classical limes
the Hamilton-Jacobi equation
the energy-phase-relation becomes
Chap. 1 - 23
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.1.2 Macroscopic Quantum Currents in Superconductors
• equation for imaginary part:
continuity equation for probability
density
and
probability current density
conservation law for probability density
Chap. 1 - 24
1.1.2 Macroscopic Quantum Currents in Superconductors
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• energy-phase relation
1
• supercurrent density-phase relation
(London parameter)
2
 equations (1) and (2) have general validity for charged and uncharged superfluids
(i)
superconductor with Cooper pairs of charge qs = -2e
(ii)
neutral Bose superfluid, e.g. 4He
(iii)
neutral Fermi superfluid, e.g. 3He
 we use equations (1) and (2) to derive London equations
• note:
k drops out !!
Chap. 1 - 25
1.1.2 Macroscopic Quantum Currents in Superconductors
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
additional topic: gauge invariance
 expression for the supercurrent density must be gauge invariant
with the gauge invariant phase gradient:
the supercurrent is
London coefficient
London penetration depth
Chap. 1 - 26
1.1.2 Macroscopic Quantum Currents in Superconductors
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
Summary:
macroscopic wave function
describes whole ensemble of superelectrons with
the current-phase relation (supercurrent equation) is
gauge invariant phase gradient
the energy-phase relation is
Chap. 1 - 27
1.1.2 Macroscopic Quantum Currents in Superconductors
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
using current-phase and energy-phase relation we can derive:
• 1. and 2. London equation
• Fluxoid quantization
• Josephson equations
Chap. 1 - 28
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.1.3 The London Equations
Fritz London (1900 – 1954)
Chap. 1 - 29
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.1.3 The London Equations
• London equations are purely phenomenological
- describe behavior of superconductors
- starting point: supercurrent-phase relation
(London parameter)
2nd London equation and the Meißner-Ochsenfeld effect:
• take the curl
 second London equation:
Maxwell:
which leads to:
• describes Meißner-Ochsenfeld effect:
applied field decays exponentially inside superconductor,
 decay length:
(London penetration depth)
Chap. 1 - 30
1.1.3 The London Equations
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• example: plane surface extending in yz-plane, magnetic field Bz parallel to z-axis:
 exponential decay
 with

Chap. 1 - 31
1.1.3 The London Equations
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1st London equation and perfect conductivity:
• take the time derivative 
• use energy-phase relation:
• substitution and using
first London equation:
• neglecting 2nd term:
(linearized 1st London equation)
• for a time-independent supercurrent the electric field inside the superconductor
vanishes  dissipationless supercurrent
Chap. 1 - 32
1.1.3 The London Equations
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• Processes that could cause a decay of Js:
example: Fermi sphere in two dimensions in the kxky – plane
• T = 0: all states inside the Fermi circle are occupied
• current in x-direction  shift of Fermi circle along kx by ±dkx
• normal state:
relaxation into states with lower energy (obeying Pauli principle)
 centered Fermi sphere, current relaxes
• superconducting state: all Cooper pairs must have same center of mass moment
 only scattering around the sphere  no decay of supercurrent
normal state
superconducting state
Chap. 1 - 33
1.1.3 The London Equations
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
additional topic: Linearized 1. London Equation
• usually, 1. London equation is linearized:
 allowed if |E| >> |vs | |B|
condition is satisfied in most cases
kinetic energy of superelectrons
Note:
 the nonlinear first London equation results from the Lorentz's law and the second
London equation
 exact form of the expression describing the phenomenon of zero dc resistance in
superconductors
 the first London equation is derived using the second London equation
 Meißner-Ochsenfeld effect is the more fundamental property of
superconductors than the vanishing dc resistance
additional topic: The London Gauge (see lecture notes)
• rigid phase:
• no conversion of Js in Jn:
Chap. 1 - 34
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.2. Fluxoid Quantization
• direct consequence of macroscopic quantum model
Gedanken-experiment:
generate supercurrent in a ring
zero dc-resistance
 stationary state
• quantum treatment: stationary state determined by quantum conditions
• cf. Bohr‘s model for atoms: quantization condition for angular momentum
 no destructive interference of electron wave  stationary state
stationary
supercurrent:
macroscopic wave
function is not
allowed to interfere
destructively
 quantization
condition
superconducting cylinder
• derivation of quantization condition (based on macroscopic quantum model of superconductivity)
start with supercurrent density:
Chap. 1 - 35
1.2. Fluxoid Quantization
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• integration of expression for supercurrent density around a closed contour
• Stoke‘s theorem (path C in simply or multiply connected region):
• applied to supercurrent:
• integral of phase gradient:
• if r1  r2 (closed path), then integral → zero
but: phase is only specified within modulo 2p of its principal value [-p, p]: qn = q0 + 2p n
Chap. 1 - 36
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.2. Fluxoid Quantization
• then:
fluxoid is quantized
• flux quantum:
• quantization condition holds for all contour lines
including contour that has shrunk to single point
 r1 = r2 in limit r1 → r2  n = 0
• contour line can no longer shrink to single point
 inclusion of nonsuperconducting region in contour
 r1 ≠ r2 in limit r1 → r2  n ≠ 0 possible
Chap. 1 - 37
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.2.1 Fluxoid and Flux Quantization
• Fluxoid Quantization:
total flux = external applied flux + flux generated by induced supercurrent
must have discrete values
• Flux Quantization:
- superconducting cylinder, wall much thicker than ¸L
- application of small magnetic field at T < Tc
 screening currents, no flux inside
 application of Hext during cool down: screening current on outer and inner wall
 amount of flux trapped in cylinder: satisfies fluxoid quantization condition
 wall thickness >> l J: closed contour deep inside with Js = 0
 then:
flux quantization
remove field after cooling down: trapped flux: integer of the flux quantum
Chap. 1 - 38
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.2.1 Fluxoid and Flux Quantization
magnetic
flux
Hext
Js
Js
inner
surface current
r
r
outer
surface current
Chap. 1 - 39
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.2.1 Fluxoid and Flux Quantization
Bext > 0
Bext > 0
Bext = 0
T > Tc
T < Tc
T < Tc
Chap. 1 - 40
1.2.1 Fluxoid and Flux Quantization
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• Flux Trapping: why is flux not expelled after switching of external field
 Js /t = 0 according to 1st London equation: E = 0 deep inside
(supercurrent only on surface within l J )
• with
and
we get:
F: magnetic flux enclosed in loop
contour deep inside the superconductor: E = 0 and therefore
 flux enclosed in cylinder stays constant
Chap. 1 - 41
1.2.2 Experimental Proof of Flux Quantization
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• 1961 by Doll/Näbauer at Munich, Deaver/Fairbanks at Stanford
 quantization of magnetic flux in superconducting cylinder
 Cooper pairs with qs = - 2e
cylinder with wall thickness À l J
different amounts of flux are frozen in during cooling down in Bcool
measure amount of trapped flux
required: large relative changes of magnetic flux
 small fields and small diameter d
 for d = 10 µm we need 2x10-5 T for one flux quantum
 measurement of (very small) torque
D = ¹ x Bp due to probe field Bp
 resonance method
Bp
 amplitude of rotary oscillation ≈ exciting torque




quartz
thread
quartz
cylinder
Pb
d ≈ 10 µm
Chap. 1 - 42
1.2.2 Experimental Proof of Flux Quantization
3
4
R. Doll, M. Näbauer
Phys. Rev. Lett. 7, 51 (1961)
trapped magnetic flux (h/2e)
resonance amplitude (mm/Gauss)
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
4
2
1
 F0
0
-1
-0.1
0.0
0.1
0.2
0.3
0.4
B.S. Deaver, W.M. Fairbank
Phys. Rev. Lett. 7, 43 (1961)
3
2
1
0
0.0
Bcool (Gauss)
0.1
0.2
0.3
0.4
Bcool (Gauss)
prediction by Fritz London: h/e
 experimental proof for existence of Cooper pairs
qs = - 2e
Paarweise im Fluss
D. Einzel, R. Gross, Physik Journal 10, No. 6, 45-48 (2011)
Chap. 1 - 43
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.3 Josephson Effect
Brian David Josephson (born 04. 01. 1940)
Nobel Prize in Physics 1973
"for his theoretical predictions of the properties of a supercurrent through a tunnel barrier,
in particular those phenomena which are generally known as the Josephson effects"
(together with Leo Esaki and Ivar Giaever)
Chap. 1 - 45
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.3.1 Josephson Equations
• what happens if we weakly couple two superconductors?
- coupling by tunneling barriers, point contacts, normal conducting layers, etc.
- do they form a bound state such as a molecule?
- if yes, what is the binding energy?
• B.D. Josephson in 1962
(nobel prize with Esaki and Giaever in 1973)
 Cooper pairs can tunnel through thin insulating barrier
expectation:
- tunneling probability for pairs ≈ (|T|²)2
 extremely small ≈ (10-4)2
Josephson:
- tunneling probability for pairs ≈ |T|²
- coherent tunneling of pairs („tunneling of macroscopic wave function“)
 finite supercurrent at zero applied voltage
Josephson effects
 oscillation of supercurrent at constant applied voltage
 finite binding energy of coupled SCs = Josephson coupling energy
Chap. 1 - 46
• coupling is weak  supercurrent density is small  |ª|2 = ns is not changed
• supercurrent density depends on gauge invariant phase gradient:
• then:
- replace gauge invariant phase gradient °
by gauge invariant phase difference:
 g(x) dx,g(x) (arb. units)
• simplifying assumptions:
- current density is homogeneous
- ° varies negligibly in electrodes
- Js same in electrodes and junction area
 ° in superconducting electrodes
much smaller than in insulator I
1.0
ns(x)
0.5
ns /n
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.3.1 Josephson Equations
g(x)
0.0
S1
I
S2
-0.5
-1.0
-6
 g(x) dx
-4
-2
0
2
4
6
x (arb. units)
Chap. 1 - 47
1.3.1 Josephson Equations
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
first Josephson equation:
• we expect:
• for Js = 0: phase difference must be zero:
therefore:
Jc: critical/maximum
Josephson current density
general formulation of 1st Josephson equation
• in most cases: keep only 1st term (especially for weak coupling):
1. Josephson equation:
current – phase
relation
• generalization to spatially inhomogeneous supercurrent density:
derived by
Josephson for
SIS junctions
supercurrent
density varies
sinusoidally with
 = q2- q1 w/o
external
potentials
Chap. 1 - 48
1.3.1 Josephson Equations
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• other argument why there are only sin contributions to Josephson current
time reversal symmetry
• if we reverse time, the Josephson current should flow in opposite direction
- t → -t, Js → - Js
• the time evolution of the macroscopic wave functions is  exp [iq(t)] = exp [iwt]
- if we reverse time, we have
t → -t
• if the Josephson effect stays unchanged under time reversal, we have to demand
satisfied only by sin-terms
Chap. 1 - 49
1.3.1 Josephson Equations
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
second Josephson equation:
• time derivative of the gauge invariant phase difference:
• substitution of the energy-phase relation
gives:
• supercurrent density across the junction is continuous (Js (1) = Js(2)):
(term in parentheses = electric field)
voltage –
phase relation
2. Josephson equation:
voltage drop across barrier
Chap. 1 - 50
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.3.1 Josephson Equations
• for a constant voltage across the junction:
• Is is oscillating at the Josephson frequency n = V/F0:
 voltage controlled oscillator
• applications:
- Josephson voltage standard
- microwave sources
Chap. 1 - 51
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.3.2 Josephson Tunneling
• what determines the maximum Josephson
current density Jc ?
- insulating tunneling barrier, thickness d
• calculation by wave matching method:
• energy-phase relation:
E0 = kinetic energy
 time depedent macroscopic wave function:
• wave function within barrier with height V0 > E0:
only elastic processes
 time evolution is the same outside and inside barrier
 consider only time independent part
 time independent Schrödinger(-like) equation for region of constant potential
Chap. 1 - 52
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
1.3.2 Josephson Tunneling
• assumption:
homogeneous barrier and supercurrent flow  1d problem
• solutions:
- in superconductors
- in insulator: sum of decaying and growing exponentials
- characteristic decay constant:
barrier properties
• coefficients A and B are determined by the boundary conditions at x = ± d/2:
n1,2 , q1,2: Cooper pair density and wave function phase at the boundaries x = ± d/2
Chap. 1 - 53
1.3.2 Josephson Tunneling
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
• solving vor A and B:
• supercurrent density:
• substituting the coefficients A and B:
current-phase relation
 maximum Josephson current density:
• real junctions:
V0 ≈ few meV  1/κ < 1 nm, d few nm  κd À 1, then:
• maximum Josephson current decays exponentially with increasing thickness d:
Chap. 1 - 54
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
Summary
Macroscopic wave function ª:
describes ensemble of macroscopic number of superconducting pairs
|ª|2 describes density of superconducting pairs
Current density in a superconductor:
Gauge invariant phase gradient:
Phenomenological London equations:
flux/fluxoid quantization:
Chap. 1 - 55
Summary
R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013)
Josephson equations:
maximum Josephson current density Jc:
wave matching method
Chap. 1 - 56