3. physics of josephson junctions: the voltage stateโฌยฆย ยท 2017) ohmic dependence with sharp...
TRANSCRIPT
AS-Chap. 3 - 1
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Chapter 3
Physics of Josephson Junctions:
The Voltage State
AS-Chap. 3 - 2
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For bias currents ๐ผ > ๐ผsm
Finite junction voltage
Phase difference ๐ evolves in time: ๐๐
๐๐กโ ๐
Finite voltage state of the junction corresponds to a dynamic state
Only part of the total current is carried by the Josephson current additional resistive channel capacitive channel noise channel
Key questions
How does the phase dynamics look like?
Current-voltage characteristics for ๐ผ > ๐ผsm?
What is the influence of the resistive damping ?
3. Physics of Josephson junctions: The voltage state
AS-Chap. 3 - 3
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At finite temperature ๐ > 0
Finite density of โnormalโ electrons Quasiparticles Zero-voltage state: No quasiparticle current For ๐ > 0 Quasiparticle current = Normal current ๐ผN Resistive state
High temperatures close to ๐c
For ๐ โฒ ๐๐ and 2ฮ ๐ โช ๐๐ต๐: (almost) all Cooper pairs are broken up Ohmic current-voltage characteristic (IVC)
๐ผN = ๐บN๐, where ๐บN โก1
๐ Nis the normal conductance
Large voltage ๐ > ๐g =ฮ1+ฮ2
๐
External circuit provides energy to break up Cooper pairs Ohmic IVC
For ๐ โช ๐c and |V| < Vg
Vanishing quasiparticle density No normal current
3.1 The basic equation of the lumped Josephson junction3.1.1 The normal current: Junction resistance
AS-Chap. 3 - 4
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For ๐ โช ๐c and V < ๐g
IVC depends on sweep direction and on bias type (current/voltage) Hysteretic behavior Current bias ๐ผ = ๐ผs + ๐ผN = ๐๐๐๐ ๐ก. Circuit model
Equivalent conductance GN at T = 0:
Voltage state
Bias current ๐ผ ๐ผs ๐ก = ๐ผc sin๐ ๐ก is time dependent IN is time dependent
Junction voltage ๐ =๐ผN
๐บNis time
dependent IVC shows time-averaged voltage ๐
Current-voltage characteristic
3.1.1 The normal current: Junction resistance
AS-Chap. 3 - 5
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Finite temperature Sub-gap resistance ๐ sg ๐ for ๐ < ๐g ๐ sg ๐ determined by amount of thermally excited quasiparticles
๐ ๐ Density of excited quasiparticles
for T > 0 we get
Nonlinear conductance ๐บN ๐, ๐
Characteristic voltage (๐ผc๐ N-product)
Note: There may be a frequency dependence of the normal channel Normal channel depends on junction type
3.1.1 The normal current: Junction resistance
AS-Chap. 3 - 6
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If ๐๐
๐๐กโ 0 Finite displacement current
Additional current channel
With ๐ = ๐ฟc๐๐ผs
๐๐ก, ๐ผN = ๐๐บN, ๐ผD = ๐ถ
๐๐
๐๐ก, ๐ฟs =
๐ฟc
cos ๐ ๐ฟc and ๐บN ๐, ๐ =
1
๐ N
Josephson inductance
For planar tunnel junction
๐ถ โ junction capacitance
3.1.2 The displacement current: Junction capacitance
AS-Chap. 3 - 7
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Equivalent parallel ๐ฟ๐ ๐ถ circuit ๐ฟc, ๐ N, C Three characteristic frequencies
Plasma frequency
๐๐ โ๐ฝc
๐ถ๐ด, where ๐ถ๐ด โก
๐ถ
๐ดis the specific junction capacitance
๐ < ๐p ๐ผD < ๐ผs
Inductive ๐ฟc/๐ N time constant
Inverse relaxation time in the normal+supercurrent system ๐c follows from ๐c (2nd Josephson equation) ๐ผN < ๐ผc for ๐ < ๐c or ๐ < ๐c
Capacitive ๐ N๐ถ time constant
๐ผD < ๐ผN for ๐ <1
๐๐ ๐ถ
๐c = ๐ผc๐ N
Characteristic frequencies
3.1.3 Characteristic times and frequencies
AS-Chap. 3 - 8
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Quality factor
Limiting cases
๐ฝ๐ถ โช 1 Small capacitance and/or small resistance Small ๐ N๐ถ time constants (๐๐ ๐ถ๐p โช 1)
Highly damped (overdamped) junctions
๐ฝ๐ถ โซ 1 Large capacitance and/or large resistance Large ๐ N๐ถ time constants (๐๐ ๐ถ๐p โซ 1)
Weakly damped (underdamped) junctions
Stewart-McCumber parameter and quality factor
Stewart-McCumber parameter
(๐ compares the decay of oscillation amplitudes to the oscillation period)
3.1.3 Characteristic times and frequencies
AS-Chap. 3 - 9
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Fluctuation/noise Langevin method: include random source fluctuating noise current type of fluctuations: thermal noise, shot noise, 1/f noise
Thermal Noise
Johnson-Nyquist formula for thermal noise (๐B๐ โซ ๐๐, โ๐):
relative noise intensity (thermal energy/Josephson coupling energy):
๐ผ๐ โก thermal noise current๐ = 4.2 K ๐ผ๐ โ 0.15 ฮผA
(current noise power spectral density)
(voltage noise power spectral density)
3.1.4 The fluctuation current
AS-Chap. 3 - 10
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Shot Noise
Schottky formula for shot noise (๐๐ โซ ๐B๐ ๐ > 0.5 mV@ 4.2 K):
Random fluctuations due to the discreteness of charge carriers Poisson process Poissonian distribution Strength of fluctuations variance ๐ฅ๐ผ2 โก ๐ผ โ ๐ผ 2
Variance depends on frequency Use noise power:
includes equilibriumfluctuations (white noise)
1/f noise
Dominant at low frequencies Physical nature often unclear Josephson junctions: dominant below about 1 Hz - 1 kHz Not considered here
3.1.4 The fluctuation current
AS-Chap. 3 - 11
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Kirchhoffโs law: ๐ผ = ๐ผs + ๐ผN + ๐ผD + ๐ผ๐น
Voltage-phase relation: ๐๐
๐๐ก=
2๐๐
โ
Basic equation of a Josephson junction
Nonlinear differential equation with nonlinear coefficients
Complex behavior, numerical solution
Use approximations (simple models)
Super-current
Normalcurrent
Displace-ment
current
Noise current
3.1.5 The basic junction equation
AS-Chap. 3 - 12
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Differential equation in dimesionless or energy formulation
โก ๐ โก ๐F(๐ก)
with
Tilted washboard potential
Resistively and Capacitively ShuntedJunction (RCSJ) model
Mechanical analogGauge invariant phase difference โ Particle with mass M and damping ๐ in potential U:
Approximation ๐บ๐ ๐ โก ๐บ = ๐ โ1 = const.๐ = Junction normal resistance
3.2 The resistively and capacitively shunted junction (RCSJ) model
AS-Chap. 3 - 13
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Normalized time:
Stewart-McCumber parameter:
Plasma frequency
Neglect damping, zero driving and small amplitudes (sin๐ โ ๐)
Solution:
Plasma frequency = Oscillation frequency around potential minimum
Finite tunneling probability:Macroscopic quantum tunneling (MQT)
Escape by thermal activation Thermally activated phase slips
Motion of โphase particleโ ๐ in the tilted washboard potential
3.2 The resistively and capacitively shunted junction (RCSJ) model
AS-Chap. 3 - 14
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) The pendulum analog
Plane mechanical pendulum in uniform gravitational field Mass ๐, length โ, deflection angle ๐ Torque ๐ท parallel to rotation axis Restoring torque: ๐๐โ sin ๐
Equation of motion ๐ท = ๐ฉ ๐ + ๐ค ๐ + ๐๐โ sin ๐
ฮ = ๐โ2 Moment of inertiaฮ Damping constant
๐ผ โ ๐ท๐ผc โ ๐๐โ๐ท0
2๐๐ โ ๐ค
C๐ท0
2๐โ ๐ฉ
๐ โ ๐
For ๐ท = 0 Oscillations around equilibrium with
๐ =๐
โโ Plasma frequency ๐p =
2๐๐ผc
๐ท0๐ถ
Finite torque (๐ท > 0) Finite ๐0 Finite, but constant ๐0 Zero-voltage stateLarge torque (deflection > 90ยฐ) Rotation of the pendulum Finite-voltage state
Voltage ๐โ Angular velocity of the pendulum
Analogies
3.2 The resistively and capacitively shunted junction (RCSJ) model
AS-Chap. 3 - 15
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Underdamped junction
๐ฝ๐ถ =2๐๐ผ๐๐
2๐ถ
โโซ 1
Capacitance & resistance large๐ large, ๐ smallHysteretic IVC
Overdamped junction
๐ฝ๐ถ =2๐๐ผ๐๐
2๐ถ
โโช 1
Capacitance & resistance small๐ small, ๐ largeNon-hysteretic IVC
(Once the phase is moving, the potential has to be tilt back almost into the horizontal position to stop ist motion)
3.2.1 Under- and overdamped Josephson junctions
(Phase particle will retrap immediately at ๐ผc because of large damping)
AS-Chap. 3 - 16
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โApplied Superconductivityโ One central question is
โHow to extract information about the junction experimentally?โ
3.3 Response to driving sources
Typical strategy
Drive junction with a probe signal and measure response
Examples for probe signals
Currents (magnetic fields) Voltages (electric fields) DC or AC Josephson junctions AC means microwaves!
Prototypical experiment
Measure junction IVC Typically done with current bias
Motivation
AS-Chap. 3 - 17
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) Time averaged voltage:2๐
Total current must be constant (neglecting the fluctuation source):
where:
๐ผ > ๐ผc Part of the current must flow as ๐ผN or ๐ผD
Finite junction voltage ๐ > 0 Time varying ๐ผs ๐ผN + ๐ผD varies in time Time varying voltage, complicated non-sinusoidal oscillations of ๐ผs,
Oscillating voltage has to be calculated self-consistently Oscillation frequency ๐ = ๐ ๐ท0
๐ = Oscillation period
3.3.1 Response to a dc current source
AS-Chap. 3 - 18
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For ๐ผ โณ ๐ผc
Highly non-sinusoidal oscillations Long oscillation period
๐ โ1
๐is small
I/Ic = 1.05
I/Ic = 1.1
I/Ic = 1.5
I/Ic = 3.0
Analogy to pendulum
3.3.1 Response to a dc current source
For ๐ผ โซ ๐ผc
Almost all current flows as normal current Junction voltage is nearly constant Almost sinusoidal Josephson current
oscillations Time averaged Josephson current almost
zero Linear/Ohmic IVC
AS-Chap. 3 - 19
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) Strong damping
๐ฝ๐ถ โช 1 & neglecting noise current
๐ < 1 Only supercurrent, ๐ = sinโ1 ๐ is a solution, zero junction voltage๐ > 1 Finite voltage, temporal evolution of the phase
Integration using
gives
Periodic function with period
Setting ๐0 = 0 and using ๐ =๐ก
๐c
tanโ1 ๐ tan ๐ฅ + ๐is ๐-periodic
3.3.1 Response to a dc current source
AS-Chap. 3 - 20
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with
We get for ๐ > 1
and
3.3.1 Response to a dc current source
AS-Chap. 3 - 21
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๐๐ ๐ถ =1
๐ N๐ถis very small
Large C is effectively shunting oscillating part of junction voltage ๐ ๐ก โ ๐ Time evolution of the phase
Almost sinusoidal oscillation of Josephson current
Down to ๐ โโ๐๐ ๐ถ
๐โช ๐c = ๐ผc๐ N
Corresponding current โช ๐ผc Hysteretic IVC
Weak damping
๐ฝ๐ถ โซ 1 & neglecting noise current
Ohmic result valid for ๐ N = ๐๐๐๐ ๐ก.
Real junction IVC determined by voltage dependence of ๐ N = ๐ N V
3.3.1 Response to a dc current source
AS-Chap. 3 - 22
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๐ฝ๐ถ โ 1
Numerically solve IVC General trend
Increasing ๐ฝ๐ถ โ Increasing hysteresis
Hysteresis characterized by retrapping current ๐ผr
๐ผr โ washboard potential tilt whereEnergy dissipated in advancing to next minimum = Work done by drive current
Analytical calculation possible for ๐ฝ๐ถ โซ 1 (exercise class)
Intermediate damping
Numericalcalculation
3.3.1 Response to a dc current source
๐ผ r/๐ผc
AS-Chap. 3 - 23
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Phase evolves linearly in time:
Josephson current ๐ผ๐ oscillates sinusoidally Time average of ๐ผs is zero
๐ผD = 0 since๐๐dc
๐๐ก= 0
Total current carried by normal current ๐ผ =๐dc
๐ N
RCSJ model Ohmic IVCGeneral case ๐ = ๐ N ๐ Nonlinear IVC
3.3.1 Response to a dc voltage source
AS-Chap. 3 - 24
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) Response to an ac voltage source
Strong damping ๐ฝ๐ถ โช 1
Integrating the voltage-phase relation:
Current-phase relation:
Superposition of linearly increasing and sinusoidally varying phase
Supercurrent ๐ผs(๐ก) and ac voltage ๐1 have different frequencies Origin Nonlinear current-phase relation
3.3.2 Response to ac driving sources
AS-Chap. 3 - 25
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Fourier-Bessel series identity:๐ฅ๐ ๐ = ๐th order Bessel function of the first kind
and:
Ac driven junction ๐ฅ = ๐1๐ก, ๐ =2๐
๐ท0๐1and ๐ = ๐0 +๐dc๐ก = ๐0 +
2๐
๐ท0๐dc๐ก
Frequency ๐dc couples to multiples of the driving frequency
Imaginary part
Some maths for the analysis of the time-dependent Josephson current
3.3.2 Response to ac driving sources
AS-Chap. 3 - 26
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Ac voltage results in dc supercurrent if ๐dc โ ๐๐1 ๐ก + ๐0 is time independent
Amplitude of average dc current for a specific step number ๐
๐dc โ ๐๐
๐dc โ ๐๐1 ๐ก + ๐0 is time dependent Sum of sinusoidally varying terms Time average is zero Vanishing dc component
3.3.2 Response to ac driving sources
Shapiro steps
AS-Chap. 3 - 27
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Ohmic dependence with sharp current spikes at ๐dc = ๐๐ Current spike amplitude depends on ac voltage amplitude
๐th step Phase locking of the junction to the ๐th harmonic
Example: ๐1/2๐ = 10 GHz
Constant dc current at Vdc = 0 and ๐๐ = ๐๐1๐ท0
2๐โ ๐ ร 20 ฮผV
3.3.2 Response to ac driving sources
AS-Chap. 3 - 28
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Difficult to solve Qualitative discussion with washboard potential
Increase ๐ผdc at constant ๐ผ1 Zero-voltage state for ๐ผdc + ๐ผ1 โค ๐ผc, finite voltage state for ๐ผdc + ๐ผ1 > ๐ผc Complicated dynamics!
๐๐ = ๐๐1ฮฆ0
2๐Motion of phase particle synchronized by ac driving
Simplifying assumption During each ac cycle the phase particle
moves down ๐ minima
Resulting phase change ๐ = ๐2๐
๐= ๐๐1
Average dc voltage ๐ = ๐๐ท0
2๐๐1 โก ๐๐
Exact analysis Synchronization of phase dynamics with external ac source for a certain bias current interval
Strong damping ๐ฝ๐ถ โช 1 (experimentally relevant)
Kirchhoffโs law (neglecting ID) ๐ผ๐ sin ๐ +1
๐ N
๐ท0
2๐
๐๐
๐๐ก= ๐ผdc + ๐ผ1 sin๐1๐ก
3.3.2 Response to ac driving sources
Response to an ac current source
Steps
AS-Chap. 3 - 29
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Experimental IVCs obtained for an underdamped and overdamped Niobium Josephson junction under microwave irradiation
3.3.2 Response to ac driving sources
AS-Chap. 3 - 30
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) Superconducting tunnel junction Highly nonlinear ๐ ๐
Sharp step at ๐g =2๐ฅ
๐
Use quasiparticle (QP) tunneling current ๐ผqp ๐
Include effect of ac source on QP tunneling
Bessel function identity for ๐1-term Sum of terms Splitting of qp-levels into many levels ๐ธqp ยฑ ๐โ๐1Modified density of states!
Tunneling current
Sharp increase of the ๐ผqp ๐ at ๐ = ๐g is broken up into many steps of smaller current
amplitude at ๐๐ = ๐g ยฑ๐โ๐1
๐
Model of Tien and Gordon: Ac driving shifts levels in electrode up and down
QP energy: ๐ธqp + ๐๐1 cos๐1๐ก
QM phase factor
3.3.4 Photon-assisted tunneling
AS-Chap. 3 - 31
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Example QP IVC of a Nb SIS Josephson junction without & with microwave irradiation Frequency ๐1 2๐ = 230 GHz corresponding to โ๐1 ๐ โ 950 ฮผV
Shapiro steps
Appear at ๐๐ = ๐โ
2๐๐1
Amplitude ๐ฝ๐2๐๐1
โ๐1
Sharp steps
QP steps
Appear at ๐๐ = ๐โ
๐๐1
Amplitude ๐ฝ๐๐๐1
โ๐1
Broadended steps(depending on ๐ผqp ๐ )
๐1 2๐ = 230 GHz
3.3.4 Photon-assisted tunneling
AS-Chap. 3 - 32
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) Thermal fluctuations with correlation function:
Larger fluctuations
Increase probability for escape out of potential well Escape at rates Gnยฑ1
Escape to next minimum Phase change of 2๐
๐ผ > 0 Gn+1 > Gn-1๐๐
๐๐ก> 0
Langevin equation for RCSJ model
Equivalent to Fokker-Planck equation:
Normalized force
Normalized momentum
3.4 Additional topic: Effect of thermal fluctuations
Small fluctuations Phase fluctuations around equilibrium
AS-Chap. 3 - 33
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) ๐(๐ฃ, ๐, ๐ก) Probability density of finding system at (๐ฃ, ๐) at time ๐ก
Static solution (๐๐
๐๐ก= 0)
with:
Boltzmann distribution (๐บ = ๐ธโ๐น๐ฅ is total energy, ๐ธ is free energy)
Constant probability to find system in nth metastable state
statistical average of variable ๐
3.4 Additional topic: Effect of thermal fluctuations
Small fluctuations
AS-Chap. 3 - 34
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Attempt frequency ๐A
Weak damping (๐ฝ๐ถ = ๐๐๐๐ ๐ถ โซ 1) ๐ผ = 0 ๐A = ๐p (Oscillation frequency in the potential well)
๐ผ โช ๐ผc ๐A ยป ๐p
3.4 Additional topic: Effect of thermal fluctuations
๐ can change in time
for ฮ๐+1 โซ ฮ๐โ1 and ๐A
ฮ๐+1โซ 1 ๐A = Attempt frequency
Amount of phase slippage
Large fluctuations
Strong damping (๐ฝ๐ถ = ๐๐๐๐ ๐ถ โช 1)
๐p โ ๐c (Frequency of an overdamped oscillator)
(underdamped junction)
(overdamped junction)
AS-Chap. 3 - 35
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For ๐ธJ0 โซ ๐B๐ Small escape probability โ exp โ๐0 ๐ผ
๐B๐at each attempt
Barrier height: 2๐ธJ0 for ๐ผ = 0
0 for ๐ผ โ ๐ผc
Escape probability ๐A/2๐ for ๐ผ โ ๐ผcAfter escape Junction switches to IRN
Experiment
Measure distribution of escape current ๐ผM Width ๐ฟ๐ผ and mean reduction ๐ฅ๐ผc = ๐ผc โ ๐ผM Use approximation for ๐0 ๐ผ and escape rate
๐A/2๐ exp โ๐0 ๐ผ
๐B๐
Considerable reduction of Ic when kBT > 0.05 EJ0
3.4.1 Underdamped junctions: Critical current reduction by premature switching
Provides experimental information on real or effective temperature!
AS-Chap. 3 - 36
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) Calculate voltage ๐ induced by thermally activated phase slips as a function of current
Important parameter:
3.4.2 Overdamped junctions: The Ambegaokar-Halperin theory
AS-Chap. 3 - 37
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Finite amount of phase slippage Nonvanishing voltage for ๐ผ โ 0 Phase slip resistance for strong damping (๐ฝ๐ถ โช 1), for U0 = 2EJ0:
๐ธJ0
๐B๐โซ 1 Approximate Bessel function
or
attempt frequency
Attempt frequency is characteristic frequency ๐c
Plasma frequency has to be replaced by frequency of overdamped oscillator:
Washboard potential Phase diffuses over barrier Activated nonlinear resistance
Modified Bessel function
3.4.2 Overdamped junctions: The Ambegaokar-Halperin theory
Amgegaokar-Halperin theory
AS-Chap. 3 - 38
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epitaxial YBa2Cu3O7 film on SrTiO3 bicrystalline substrate
R. Gross et al., Phys. Rev. Lett. 64, 228 (1990)Nature 322, 818 (1988)
Example: YBa2Cu3O7 grain boundary Josephson junctions Strong effect of thermal fluctuations due to high operation temperature
3.4.2 Overdamped junctions: The Ambegaokar-Halperin theory
AS-Chap. 3 - 39
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Determination of ๐ผc ๐ close to ๐c
Overdamped YBa2Cu3O7 grain boundary Josephson junction
R. Gross et al., Phys. Rev. Lett. 64, 228 (1990)
thermally activated
phase slippage
3.4.2 Overdamped junctions: The Ambegaokar-Halperin theory
AS-Chap. 3 - 40
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3.5 Voltage state of extended Josephson junctions
So far
Junction treated as lumped element circuit element Spatial extension neglected
Spatially extended junctions
Specific geometry as as in Chapter 2 Insulating barrier in ๐ฆ๐ง-plane In-plane ๐ต field in ๐ฆ-direction Thick electrodes โซ ๐L1,2 Magnetic thickness ๐ก๐ต = ๐ + ๐๐ฟ,1 + ๐๐ฟ,2 Bias current in ๐ฅ-direction
Phase gradient along ๐ง-direction
๐๐ ๐ง,๐ก
๐๐ง=
2๐
๐ท0๐ก๐ต๐ต๐ฆ ๐ง, ๐ก
Expected effects
Voltage state ๐ธ-field and time-dependence become important Short junction and long junction case
AS-Chap. 3 - 41
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) Neglect self-fields (short junctions)
๐ฉ = ๐ฉ๐๐ฑ
Junction voltage ๐ = Applied voltage ๐0 Gauge invariant phase difference:
Josephson vortices moving in ๐ง-direction with velocity
3.5.1 Negligible Screening Effects
AS-Chap. 3 - 42
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) Long junctions (๐ณ โซ ๐๐)
Effect of Josephson currents has to be taken into account Magnetic flux density = External + Self-generated field
with B = ๐0H and D = ํ0E:in contrast to static case,
now E/t 0
with ๐ธ๐ฅ = โ๐/๐, ๐ฝ๐ฅ = โ๐ฝ๐ sin ๐ and ๐๐/๐๐ก = 2๐๐/ฮฆ0:
consider 1D junction extending in z-direction, B = By, current flow in x-direction
(Josephson penetration depth)
(propagation velocity)
3.5.2 The time dependent Sine-Gordon equation
AS-Chap. 3 - 43
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Time dependentSine-Gordon equation
๐ = velocity of TEM mode in the junction transmission line
Example: ํ โ 5 โ 10, 2๐L
๐โ 50 โ 100 ๐ โ 0.1๐
Reduced wavelength For f = 10 GHz Free space: 3 cm, in junction: 1 mm
with the Swihart velocity
Other form of time-dependent Sine-Gordon equation
3.5.2 The time dependent Sine-Gordon equation
AS-Chap. 3 - 44
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Mechanical analogue
Chain of mechanical pendula attached to a twistablerubber ribbon
Restoring torque ๐J2 ๐
2๐
๐๐ง2
Short junction w/o magnetic field 2/z2 = 0 Rigid connection of pendula Corresponds to single pendulum
Time-dependent Sine-Gordon equation:
3.5.2 The time dependent Sine-Gordon equation
AS-Chap. 3 - 45
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) Simple case 1D junction (W โช ฮปJ), short and long junctions
Short junctions (L โช ฮปJ) @ low damping
Neglect z-variation of ๐
Equivalent to RCSJ model for ๐บN = 0, ๐ผ = 0Small amplitudes Plasma oscillations(Oscillation of ๐ around minimum of washboard potential)
Long junctions (L โซ ฮปJ)
Solution for infinitely long junction Soliton or fluxon
๐ = ๐ at ๐ง = ๐ง0 + ๐ฃ๐ง๐กgoes from 0 to 2๐ for โโ โ ๐ง โ โ Fluxon (antifluxon: โ โ ๐ง โ โโ)
3.5.3 Solutions of the time dependent SG equation
AS-Chap. 3 - 46
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Pendulum analog Local 360ยฐ twist of
rubber ribbon
Applied current Lorentz forceMotion of phase twist (fluxon)
Fluxon as particle Lorentz contraction for vz โ c
Local change of phase difference Voltage
Moving fluxon = Voltage pulse
Other solutions: Fluxon-fluxon collisions, โฆ
๐ = ๐ at ๐ง = ๐ง0 + ๐ฃ๐ง๐กgoes from 0 to 2๐ for โโ โ ๐ง โ โ Fluxon (antifluxon: โ โ ๐ง โ โโ)
3.5.3 Solutions of the time dependent SG equation
AS-Chap. 3 - 47
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Linearized Sine-Gordon equation
๐1 = Small deviation Approximation
Substitution (keeping only linear terms):
๐0 solves time independent SGE ๐2๐0
๐๐ง2= ๐J
โ2sin ๐0
๐0 slowly varying ๐0 โ ๐๐๐๐ ๐ก.
3.5.3 Solutions of the time dependent SG equation
Josephson plasma waves
AS-Chap. 3 - 48
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๐ < ๐p,J
Wave vector k imaginary No proagating solution๐ > ๐p,J
Mode propagation Pendulum analogue Deflect one pendulum Relax Wave like excitation๐ = ๐p,J
Infinite wavelength Josephson plasma wave Analogy to plasma frequency in a metal Typically junctions ๐p,J โ 10 GHz
For very large ๐J or very small I
Neglectsin ๐
๐J2 term Linear wave equation Plane waves with velcoity ๐
=๐p2
4๐2cos๐0
3.5.3 Solutions of the time dependent SG equation
Solution:
Dispersion relation ฯ(k) :
Josephson plasma frequency
(small amplitude plasma waves)
Plane waves
AS-Chap. 3 - 49
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) Interaction of fluxons or plasma waves with oscillating Josephson current
Rich variety of interesting resonance phenomena Require presence of ๐ฉ๐๐ฑ
Steps in IVC (junction upconverts dc drive)
For ๐ตex > 0
Spatially modulated Josephson currrent density moves at ๐ฃ๐ง = ๐ ๐ต๐ฆ๐ก๐ต Josephson current can excite Josephson plasma waves
On resonance, em waves couple strongly to Josephson current if ๐ = ๐๐
Eck peak at frequency:
Corresponding junction voltage:
3.5.4 Resonance phenomena
Flux-flow steps and Eck peak
AS-Chap. 3 - 50
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Alternative point of view
Lorentz force Josephson vortices move at ๐ฃ๐ง =๐
๐ต๐ฆ๐ก๐ต
Increase driving force Increase ๐ฃ๐ง Maximum possible speed is ๐ฃ๐ง = ๐ Further increase of ๐ผ does not increase ๐) Flux flow step in IVC
๐ffs = ๐๐ต๐ฆ๐ก๐ต = ๐๐ท
๐ฟ=
๐p
2๐
๐J
๐ฟ๐ท0
๐ท
๐ท0
Corresponds to Eck voltage
Traveling current wave only excites traveling em wave of same direction Low damping, short junctions Em wave is reflected at open end Eck peak only observed in long junctions at medium damping when
the backward wave is damped
3.5.4 Resonance phenomena
AS-Chap. 3 - 51
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Standing em waves in junction โcavityโ at ๐๐ = 2๐๐๐ = 2๐ ๐
2๐ฟ๐ =
๐ ๐
๐ฟ๐
Fiske steps at voltages for L ยป 100 ฮผmfirst Fiske step ยป 10 GHz(few 10s of ฮผV)
Interpretation
Wave length of Josephson current density is 2๐
๐
Resonance condition ๐ฟ = ๐
2๐๐๐ =
๐
2๐ โ ๐๐ฟ = ๐๐ or ๐ท = ๐
๐ท0
2
where maximum Josephson current of short junction vanishes Standing wave pattern of em wave and Josephson current match Steps in IVC
Influence of dissipation
Damping of standing wave pattern by dissipative effects Broadening of Fiske steps Observation only for small and medium damping
๐/2-cavity
๐๐ = ๐๐๐ฃph ๐ฟ
๐ฃph = Phase velocity
3.5.4 Resonance phenomena
Fiske steps
AS-Chap. 3 - 52
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) Fiske steps at small damping and/or small magnetic field
Eck peak at medium damping and/or medium magnetic field
For ๐ โ ๐Eck and ๐ โ ๐๐ ๐ผ๐ = ๐ผc sin ๐0๐ก + ๐๐ง + ๐0 โ 0 ๐ผ = ๐ผN ๐ = ๐/๐ N ๐
3.5.4 Resonance phenomena
AS-Chap. 3 - 53
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Motion of trapped flux due to Lorentz force (w/o magnetic field) Junction of length L, moving back and forth
Phase change of 4๐ in period ๐ =2๐ฟ
๐ฃz
At large bias currents (๐ฃ๐ง โ ๐)
๐zfs = ๐โ
2๐=4๐
๐
โ
2๐=
4๐
2๐ฟ/ ๐
โ
2๐=
โ
2๐
๐
๐ฟ=๐p
๐
๐J๐ฟ๐ท0
For ๐ fluxons
๐๐,zfs = ๐๐๐ง๐๐ ๐๐,zf๐ = 2 ร Fiske voltage ๐๐ (fluxon has to move back and forth)
๐ffs = ๐๐,zfs for ๐ท = ๐๐ท0 (introduce n fluxons = generate n flux quanta)
Example:IVCs of annular Nb/insulator/PbJosephson junction containing adifferent number of trapped fluxons
3.5.4 Resonance phenomena
Zero field steps
AS-Chap. 3 - 54
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Voltage state: (Josephson + normal + displacement + fluctuation) current = total current
RCSJ-model (๐บ๐ ๐ = ๐๐๐๐ ๐ก.)
Motion of phase particle in the tilted washboard potential
Equivalent LCR resonator, characteristic frequencies:
Equation of motion for phase difference ๐:
Quality factor:
๐๐
๐๐ก=2๐๐
โ
Summary (Voltage state of short junctions)
๐ฝ๐ถ = Stewart-McCumber parameter
AS-Chap. 3 - 55
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IVC for strong damping and ๐ฝ๐ถ โช 1
Driving with ๐ ๐ก = ๐dc + ๐1 cos๐1๐ก
Shapiro steps at ๐๐ = ๐๐ท0
2๐๐1
with amplitudes ๐ผs ๐ = ๐ผ๐ ๐ฅ๐2๐๐1
๐ท0๐1
Photon assisted tunneling
Voltage steps at ๐๐ = ๐๐ท0
๐๐1 due to nonlinear
QP resistance
Summary (Voltage state of short junctions)
Effect of thermal fluctuations
Phase-slips at rate ฮ๐+1 =๐A
2๐exp โ
๐0
kB๐
Finite phase-slip resistance ๐ p even below ๐ผc
Premature switching
AS-Chap. 3 - 56
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) โข voltage state of extended junctionsw/o self-field:
โข with self-field: time dependent Sine-Gordon equation
characteristic velocity of TEM mode in the junction transmission line
Summary
characteristic screening length
Prominent solutions: plasma oscillations and solitons
nonlinear interactions of these excitations with Josephson current: flux-flow steps, Fiske steps, zero-field steps