2. physics of josephson junctions: the zero voltage state ... · s nd © r-r-(2001 -• 2013)...

Chapter 2

Physics of Josephson Junctions:

The Zero Voltage State

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AS-Chap. 2 - 2

• small spatial dimensions: gauge invariant phase diff. & current density are uniform variations of supercurrent density on length scale larger than lL

≈ 10 nm – 1 ¹m

Josephson junction: ns strongly reduced: lL→ lJ ≈ 10 - 100 ¹m

• 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

2.1 Basic Properties of Lumped Josephson Junctions

2.1.1 The Lumped Josephson Junction

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AS-Chap. 2 - 3

„0“-junction „¼“-junction

• 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):

2.1.1 The Lumped Josephson Junction

2.1.2 The Josephson Coupling Energy

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AS-Chap. 2 - 4

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

2.1.2 The Josephson Coupling Energy

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AS-Chap. 2 - 5

–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 stable minima n, unstable maxima n

states for different n: equivalent

~

2.1.3 The Superconducting State

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AS-Chap. 2 - 6

• properties of the washboard potential

• close to Ic: a ≡ 1 - I/Ic << 1, we get the approximations:

• washboard potential extremely useful in describing junction dynamics for I > Ic

0 for I/Ic 1

2.1.3 The Superconducting State

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AS-Chap. 2 - 7

Ls is negative for

• Josephson inductance:

(for V > 0: oscillating Josephson current)

• energy storage in JJ nonlinear reactance

• for small variations around Is = Ic sin: JJ equivalent to inductance

with Josephson inductance

no minima for I > Ic

2.1.4 The Josephson Inductance

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AS-Chap. 2 - 10

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

2.1.4 Junction Fabrication

S. Nagasawa et al., Physica C: Superconductivity Volumes 426–431, Part 2, pp. 1525–1532 (2005)

http://www.sciencedirect.com/science/journal/09214534









http://www.sciencedirect.com/science/journal/09214534/426/supp/P2



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AS-Chap. 2 - 11

• plane mechanical pendulum in uniform gravitational field:

phase difference : angle of pendulum with respect to equilibrium supercurrent Is: torque voltage V: angular velocity of pendulum

• particle moving in tilted washboard potential coordinate x / velocity v / d/dt / V

2.1.5 Mechanical Analogs

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AS-Chap. 2 - 12

• 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

2.2 Short Josephson Junctions

• JJ at finite voltage temporal interference oscillation of Josephson current

• JJ at finite phase gradient spatial interference magnetic field dependence of Josephson current

• relevant length scale for transition from short to long Josephson junction:

Josephson penetration depth

density in weak coupling region

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AS-Chap. 2 - 14

• 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 > lL

• magnetic field: Be=(0,By,0)

• magnetic thickness: tB = d+lL1+lL2

• 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

2.2.1 Quantum Interference Effects - Short JJ in an Applied Field

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AS-Chap. 2 - 15

gauge invariant phase gradient in the bulk superconductor:

gauge invariant phase difference across the barrier:

1 2 3 4

1 & 3 are differences acrosss the junction:

2 & 4 differences in the bulk, supercurrent equation for 𝛻𝜃:


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AS-Chap. 2 - 16

• integration of A around closed contour enclosed flux F

• integration of Js excludes insulating barrier incomplete contour C‘:

• substitution:

• 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 (ÀlL) Js exponentially small therefore:

• total flux enclosed by the loop:


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AS-Chap. 2 - 17

nota bene: if t1<¸1 and t2<¸2:

• similar argument for P and Q separated by dy in y-direction

then

• 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


magnetic field component parallel to junction plane induces phase gradient

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AS-Chap. 2 - 18

1. Josephson equation:

2. Josephson equation:

• JJ with spatially homogeneous Js and : lumped element


tilted washboard potential

Josephson inductance

short JJ: no self-field ↔ long JJ: self-field Josephson penetration depth

• short junction in external field

field induces gradient of phase difference

Brief Summary

• extended junctions: Js = Js(r) and = (r)

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AS-Chap. 2 - 19

• how does depend on the applied field Be = (0,By,0)?

• integration in in y-direction:

• integral: complex, multiplication by ei0 does not change magnitude • maximum Josephson current: magnitude

magnetic field dependence of Ism

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

2.2.2 The Fraunhofer Diffraction Pattern

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AS-Chap. 2 - 20

• maximum current density integrated along y-direction

• spatially homogeneous maximum current density Jc(y,z) Fraunhofer diffraction pattern:

• experiment: study of the homogeneity of the supercurrent flow in JJ


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AS-Chap. 2 - 21

• intepretation of the shape of Ism(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


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AS-Chap. 2 - 22

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

local Josephson current density in negative and positive x-direction, but no net total current


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AS-Chap. 2 - 23

• closed current loop • no penetration of applied field

into electrodes • Josephson vortex • no normal core • vortex core in barrier region

Josephson vortex


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AS-Chap. 2 - 24

• arbitrary direction of the applied field within barrier plane:


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AS-Chap. 2 - 25

• 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 |Ism(k)| between 0 and k

information on Jc(y,z) on small length scale high fields spatial resolution ≈ 1/By:

measurement of Ism(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:

2.2.3 Determination of the Maximum Josephson Current Density

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AS-Chap. 2 - 26

• Ism(F)-dependence without side lobes:

• junction shape: should approache Gauss curve for homogeneous Jc(y,z)

integrated current density in y-direction: Gaussian profile

2.2.3 Determination of the Maximum Josephson Current Density

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AS-Chap. 2 - 27

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 ↔ Ism(By)

backtransformation gives Pi (spatial resolution given by # of diffraction orders) ↔ phase is lost, backtransformation of intensity (Ic

m)2 (By)

autocorrelation function of supercurrent distribution

2.2.3 Determination of the maximum Josephson current density

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AS-Chap. 2 - 28

• 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 Ism(B) spatial resolution 0.01 £ junction width

• statistical information in envelope of |Ism

(By)|2: e.g. inhomogeneities with probability p(a) / 1/a (p £ length scale a = const.) |Is

m(By)|2 / 1/By („spatial 1/f noise“)


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AS-Chap. 2 - 29

• random distribution of filaments with width a:

envelop const. up to k = 2¼/a, then ≈ 1/By

2

“spatial shot noise”

• analysis of AC gives statistical information on current density inhomogeneities

• YBa2Cu3O7-± grain boundary JJ slope of envelop is about -0.65 |Is

m(By)|

2 1/By1.3

p(a) / 1/a1.5

small scale inhomogeneities are more probable


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AS-Chap. 2 - 30

2.2.4 Additional Topic: Direct Imaging of Supercurrent Distribution

scanning of JJ by focused electron / laser beam measure change dIs

m(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)

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AS-Chap. 2 - 31

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

2.2.5 Additional Topic: Short JJ - Energy Considerations

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AS-Chap. 2 - 32

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)

2.2.5 Additional Topic: Short JJ - Energy Considerations

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AS-Chap. 2 - 33

• 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:

2.2.6 The Motion of Josephson Vortices

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AS-Chap. 2 - 34

• 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

2.2.6 The Motion of Josephson Vortices

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AS-Chap. 2 - 35

• spatial distribution of is(z) = ∫ Js(y,z)dz

2.2. Summary – Short Josephson Junctions

• short: lateral junction dimensions small compared to Josephson penetration depth

• effect of magnetic field parallel to junction electrodes phase gradient Josephson current density

• Josephson vortex

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AS-Chap. 2 - 36

2.2. Summary – Short Josephson Junctions

• magnetic field dependence of maximum Josephson current:

flux though junction

Fraunhofer diffraction pattern analogy to optics (single slit diffaction)

• motion of Josephson vortices: motion of single vortex across junction results in phase change of 2𝝅

no inverse Fourier transform (missing phase)

autocorrelation function

constant motion of vortices constant junction voltage / vortex rate

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AS-Chap. 2 - 37

2.3 Long Josephson Junctions

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

Josephson penetration depth:

stationary Sine-Gordon equation (SSGE) (nonlinear differential equation)

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AS-Chap. 2 - 38

two-dimensional stationary Sine-Gordon equation:


• relation between London and Josephson penetration depth

with

insert into expression for Josephson penetration depth

lJ corresponds to London penetration depth of weak coupling region with reduced superelectron density ns

*

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AS-Chap. 2 - 39

• small linearization: sin ≈


additional topic: analytical solutions of the SSGE

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 < J

c or By(z=0) ≤ µ0 Jc¸J

• for a small junction L ¿ ¸J :

short junction result

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AS-Chap. 2 - 40

particular solution of the SSGE:

2.3.2 The Josephson Vortex

general solution: particular + homogeneous solution important case: junction of infinite length, d/dz vanishes at z = §1

particular solution ́ complete solution

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AS-Chap. 2 - 41

• decay length for Js and By: ¸J

• maximum of Js • integration: total current = 0, total flux = ©0

Josephson vortex in a long JJ

2.3.2 The Josephson Vortex

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AS-Chap. 2 - 43

energy stored in long JJ:

insulator

superconductor

stored energy for vortex solution, thick electrodes:

using and integration over y and x:

now we use the vortex solution

2.3.2 The Josephson Vortex additional topic: Energy of the Josephson Vortex Solution

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AS-Chap. 2 - 44

energy per unit length of vortex:

magnetic flux density Bc1 for first vortex entrance:

Bc1 ≈ magnetic flux density of a single flux quantum distributed over an area tB ¢ ¸J

EVortex > 0 we need: external field and /or current to supply energy

2.3.2 The Josephson Vortex additional topic: Energy of the Josephson Vortex Solution

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AS-Chap. 2 - 45

• 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 !!

2.3.3 Junction Types and Boundary Conditions

• three basic types of junction geometries: overlap junction

in-line junction grain boundary junctions

J. Mannhart, J. Bosch, R. Gross, R. P. Huebener Phys. Lett. A 121, 241 (1987)

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AS-Chap. 2 - 46

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

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AS-Chap. 2 - 47

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 - Bz

el has negligible impact since W << L Fel

z = BelWtB << Fely = BelLtB

- finite inline admixture s = W/L ¿ 1


HTS bicrystal grain boundary junction

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AS-Chap. 2 - 57

overlap junction: highest zero field Ism ( junction area Ai = LW)

inline junction: Ism 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 Is

m with field


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AS-Chap. 2 - 59

Josephson penetration depth (short – long junctions)


nonlinear inductance

washboard potential

in-plane magnetic field By

spatial oscillations of the Jospehson current density:

Summary (short junctions)

integral Josephson current FT of ic(z):

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AS-Chap. 2 - 60

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

Summary (long junctions)

2. physics of josephson junctions: the zero voltage state ... · s nd © r-r-(2001 -• 2013)...

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