3. Physics of Josephson Junctions: The Voltage State · mass m, length ℓ, deflection angle µ,...

Chapter 3

Physics of Josephson Junctions:

The Voltage State

AS-Chap. 3 - 2

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• for I > Ism:

- finite junction voltage phase difference j evolves in time: dj/dt V

- finite voltage state of junction corresponds to dynamic state

• only part of the total current is carried by the Josephson current additional resistive channel,

capacitive channel, and noise

3. Physics of Josephson Junctions: The Voltage State

• questions:

- how does the phase dynamics look like?

- current-voltage characteristics for I > Ism ?

- what is the influence of the resistive damping ?

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3.1 The Basic Equation of the Lumped Josephson Junction

3.1.1 The Normal Current: Junction Resistance

• for T > 0: finite density of “normal” electrons quasiparticles- zero-voltage state: no quasiparticle currentfor V > 0: quasiparticle current ´ normal current IN resistive state

• high temperature close to Tc:- for T· Tc : 2¢(T) ¿ kBT: (almost) all Cooper pairs are broken up, Ohmic IVC:

GN = 1/RN: normal conductance

• large voltage V > Vg = (¢1 + ¢2)/e: external circuit provides energy to break up Cooper pairs Ohmic IVC

• for T¿ Tc and |V| < Vg: vanishing quasiparticle density no normal current

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• for T¿ Tc and |V| < Vg: IVC depends on sweep direction (and external source)hysteretic behaviorcurrent source: I = Is + IN = const.

circuit model

• equivalent conductance GN at T = 0:


voltage state: Is(t) = Ic sinj(t) is time dependent IN is time dependent junction voltage V = IN /GN is time dependent IVC: time averaged voltage

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3) • finite temperature: sub-gap resistance Rsg(T) for |V| < Vg

Rsg(T) determined by amount of thermally excited quasiparticles:

n(T): density of excited quasiparticles

for T > 0 we get:

nonlinear conductance GN(V,T)

• characteristic voltage (IcRN - product):

note: - there may be frequency dependence of normal channel- normal channel depends on junction type


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3.1.2 The Displacement Current: Junction Capacitance

• if dV/dt 0 finite displacement current

C: junction capacitance, for planar tunnel junction:

• additional current channel

with V = LcdIS /dt, IN = VGN, ID = CdV/dt and Ls = LC /cosj Lc, GN(V,T) = 1/RN:

Josephson inductance

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equivalent circuit: Lc, RN, C 3 characteristic frequencies

• plasma frequency:

scales (Jc/CA)0.5, CA = C/A: specific junction capacitancefor ! < !p: ID < Is

• Lc /RN time constant:

inverse relaxation time in system of normal and supercurrent!c follows from Vc (2nd Josephson eq.) characteristic frequencyIN < Ic for V < Vc or ! < !c = RN/Lc

• RNC time constant:

ID < IN for ! < 1/¿RC

• Stewart-McCumber parameter: (related to quality factor of LCR circuit)

3.1.3 Characteristic Times and Frequencies

AS-Chap. 3 - 8

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parallel LRC circuitQ compares decay of amplitude of oscillations to oscillation period

¯c¿ 1: small capacitance and/or small resistance

small RNC time constants (¿RC!p ¿1) highly damped or overdamped junctions

¯cÀ 1: large capacitance and/or large resistance

large RNC time constants (¿RC!p À 1) weakly damped or underdamped junctions

3.1.3 Characteristic Times and Frequencies

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fluctuation/noise Langevin method: include random source fluctuating noise current type of fluctuations: white noise, shot noise, 1/f noise

Thermal Noise:

Johnson-Nyquist formula for thermal noise (kBTÀ eV, ~!):

relative noise intensity (thermal energy/Josephson coupling energy):

IT: thermal noise currentfor T = 4.2 K: IT ' 0.15 ¹A

3.1.4 The Fluctuation Current

(current noise power spectral density)

(voltage noise power spectral density)

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3.1.4 The Fluctuation Current

Shot Noise:

Schottky formula for shot noise (eVÀ kBT, V > 0.5 mV @ 4.2 K):

random fluctuations, discreteness of charge carriers Poisson process Poissonian distribution strength of fluctuations: variance: variance depends on frequency use noise power:

includes equilibrium fluctuations (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

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Kirchhoff‘s law: I = Is + IN + ID + IF

& voltage-phase relation: dj/dt = 2eV/~

basic equation of Josephson junction

nonlinear differential equation with nonlinear coefficients

complex behavior, numerical solution

use approximations (simple models)

3.1.5 The Basic Junction Equation

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• Resistively and Capacitively Shunted Junction model (RCSJ): approximation: GN(V) = G = 1/R = const.

Josephson junction:Ls = Lc/cosj with Lc = ħ/2eIc

R = 1/G: junction normal resistance

nonlinear differential equation:

´ i ≡ iF(t)

• compare motion of gauge invariant phase difference to that of particle with mass M anddamping ´ in potential U:

with

tilted washboard potential

3.2 The Resistively and Capacitively Shunted Junction Model

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normalized time: Stewart-McCumber parameter:

plasma frequency: neglect damping, zero driving and small amplitudes (sinj ' j):

solution:

plasma frequency oscillation frequency around potential minimum

finite tunneling probability:macroscopic quantum tunneling (MQT)

escape by thermal activation thermally activated phase slips

3.2 The RCSJ Model

tilt washboardpotential

motion of 𝝋 in tiltedwashboard potential

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The pendulum analogue:physical pendulum: mass m, length ℓ, deflection angle µ, torque D parallel to rotation axisrestoring torque: ℓ m g sin£

equation of motion:

£ = mℓ2: moment of inertia¡: damping constant

analogy to Josephson junction: I$ DIc $ mgℓF0/2¼R $ ¡

CF0/2¼$£

gauge invariant phase difference 𝜑$ angle µ

for D = 0: oscillations around equilibrium with! = (g/ℓ)1/2 $ plasma frequency !p = (2¼Ic /F0C)1/2

finite torque finite µ0 finite j0

large torque (deflection > 90°) rotation of the pendulum finite voltage state

3.2 The RCSJ Model

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underdamped junction:

¯c = 2eIcR2C/~À 1

capacitance & resistance largeM large, ´ small hysteretic IVC

(once the phase is moving, the potential has to be tilt back almost into the horizontal position to stop ist motion)

overdamped junction:

¯c = 2eIcR2C/~¿ 1

capacitance & resistance smallM small, ´ large non-hysteretic IVC

3.2.1 Underdamped and Overdamped Josephson Junctions

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• time averaged voltage:2¼

• total current must be constant (neglecting the fluctuation source):

where:

• for I > Ic: part of the current must flow as IN or ID

finite junction voltage |V| > 0|V| > 0 time varying Is

IN + ID is varying in time time varying voltage, complicated non-sinusoidal oscillations of Is

oscillating voltage has to be calculated self-consistently oscillation frequency: f = hVi/F0

T: oscillation period

3.3 Response to Driving Sources 3.3.1 Response to a dc Current Source

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• for I Ic: highly non-sinusoidal oscillations

• long oscillation period

• hVi / 1/T : small

• for I >> Ic : almost all current flows as normal current

• junction voltage is about constant

• oscillations of Josephson current arealmost sinusoidal

• time averaged Josephson current almostzero

• linear/Ohmic IVC

I/Ic = 1.05

I/Ic = 1.1

I/Ic = 1.5

I/Ic = 3.0

3.3.1 Response to a dc Current Source

analogy to pendulum

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Current-Voltage Characteristics:

strong damping: ¯c ¿ 1, neglecting noise current:

for i < 1: only supercurrent, j = sin-1 i is a solution, junction voltage = zerofor i > 1: finite voltage, temporal evolution of the phase

integration using:

gives

periodic function with period:

setting ¿0 = 0 and using ¿ = t/¿c


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with

we get for i > 1:

and


AS-Chap. 3 - 20

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!RC = 1/RNC is very small large C is effectively shunting oscillating part of junction voltage time evolution of the phase:

almost sinusoidal oscillation of Josephson current:

down to

corresponding current¿ Ic hysteretic IVC


Current-Voltage Characteristics:

Weak damping: ¯c À 1, neglecting noise current:

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additional topic: intermediate damping: ¯c» 1 numerically solve IVC increasing McCumber parameter increasing hysteresis decreasing return-current IR

• IR given by tilt of washboard where:energy dissipated in advancing to next minimum =work done by drive current

• calculation of IR for ¯c À 1:for I¸ IR: normal current can be neglected junction energy:

energy dissipation within RCSJ model:


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resistive state: minimum junction kinetic energy must be positive E ≥ 2EJ0

limit I = IR E = 2EJ0

energy dissipation:

work done by the current source:

then:

valid for ¯cÀ 1


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Summary

plasma frequency:

Lc/RN time constant:

RNC time constant:

Stewart-McCumber parameter:

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time averaged voltage:

strong damping: ¯c ¿ 1, neglecting noise current:

Summary

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Josephson current Is oscillates sinusoidally time average of Is is zero ID = 0 since dV/dt = 0 total current carried by normal current IVC:

- RCSJ model: ohmic IVC- in more general R = RN(V): nonlinear IVC

3.3.2 Response to a dc Voltage Source

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• response to an ac voltage sourceweak damping, ¯c ≫ 1

integrating the voltage-phase relation:

current-phase relation:

superposition of linearly increasing !dct = (2¼Vdc /F0)t and sinusoidally varying phase

current Is(t) and ac voltage V1 have different frequencies origin: nonlinear current-phase relation

3.3.2 Response to ac Driving Sources

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Fourier-Bessel series identity:

nth order Bessel function of first kind

and:

with x = w1t, b = 2¼V1/F0w1 and

frequency !dc couples to multiples of the driving frequency

imaginary part


some mathematics for the analysis of the time-dependent Josephson current:

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Shapiro steps: ac voltage results in dc supercurrent if [ …] is time independent

amplitude of average dc current for a specific n (step number):

for Vdc Vn: […] is time dependent sum of sinusoidally varying terms time average is zero vanishing dc component


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Ohmic dependence with sharp current spikes at Vdc = Vn

current spike amplitude depends on ac voltage amplitude nth step: phase locking of the junction to the nth harmonic

example: w1/2¼ = 10 GHzconstant dc current at Vdc = 0 and Vn = n 𝜔1 × Φ0/2𝜋 ≃ n£20 ¹V


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• response to an ac current source, strong damping, ¯c¿ 1 (experimentally relevant)

Kirchhoff’s law (neglecting ID):

difficult to solve qualitative discussion with washboard potential:increase Idc at constant I1: zero-voltage state for Idc + I1 ≤ Ic

voltage state for Idc + I1 > Ic complicated dynamics for Vn = n¢w1¢©0/2¼: motion of phase particle synchronized by ac driving

assumption: during each ac-cycle phase the particle moves down n minima resulting phase change:

average dc voltage:

synchronization of phase dynamics with external ac source (for certain bias current interval)


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experimental IVCs obtained for an underdamped and overdamped Niobium Josephson junction under microwave radiation


AS-Chap. 3 - 32

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• superconducting tunnel junction: highly nonlinear R(V), sharp step at Vg = 2¢/e

use Iqp(V), include effect of ac source on qp-tunneling

• Bessel function identity for V1-term: sum of termssplitting of qp-levels in many levels Eqp§ n~w1

tunneling current:

sharp increase of the qp tunneling current at the gap voltage is broken up into many steps of smaller current amplitude at voltages Vg § n~!1/e

3.3.4 Photon-Assisted Tunneling

• Model of Tien and Gordon:ac driving shifts levels in electrode up and downqp-energy: Eqp + eV1cosw1tqm phase factor:

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qp IVC of a niobium SIS Josephson junction without and with microwave irradiation frequency 2¼w1 = 230 GHz corresponding to ~w1/e ' 950 ¹V

Shapiro steps: qp steps:

(note that qp steps have no constant voltage and different amplitude)

3.3.4 Photon-Assisted Tunneling

AS-Chap. 3 - 34

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• thermal fluctuations with correlation function:

• small fluctuations phase fluctuations around equilibrium

large fluctuations incease probability for escape out

of potential well

escape rates ¡n§1

escape to next minimum phase change of 2¼

for I > 0: ¡n+1 > ¡n-1 hdj/dti > 0

• Langevin equation for RCSJ model:

equivalent to Fokker-Planck equation:

normalized force:

3.4 Additional Topic: Effect of Thermal Fluctuations

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normalized momentum:

¾(v,j,t): probability density of finding system at (v,j) at t

small fluctuations static solution:

with:

Boltzmann distribution (G = E – Fx: total energy, E: free energy)

constant probability to find system in nth metastable state:

large fluctuations: p can change in time:

for ¡n+1 À ¡n-1 and !A/¡n+1 À 1: !A: attempt frequency

statistical average of variable X

amount of phase slippage


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attempt frequency !A :

for I = 0: !A plasma frequency !p (oscillation frequency in potential well)for I < Ic: !A » !p

strong damping (¯C = !c¿RC ¿ 1): !p !c (frequency of overdamped oscillator)


AS-Chap. 3 - 39

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for EJ0 >> kBT: small escape probability / exp(-U0(I)/kBT) at each attempt

barrier height: 2EJ0 for I = 0, barrier height 0 for I Ic

escape probability !A/2¼ for I Ic

after escape: junction switches to IRN

experiment: one measures distribution of escape current IM

width ±I and mean reduction h¢Ici = Ic - hIMi

use approximation for U0(I) and escape rate

/ !A/2¼ exp(-U0(I)/kBT):

considerable reduction of Ic when kBT > 0.05 EJ0

3.4.1 Underdamped Junctions: Ic Reduction by Premature Switching

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3.4.2 Overdamped Junctions: The Ambegaokar-Halperin Theory

calculate voltage 𝑉 induced by thermally activated phase slips as a function of current

important parameter:

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Amgegaokar, Halperin: finite amount of phase slippage nonvanishing voltage for I 0 phase slip resistance for strong damping (¯c ¿ 1), for U0 = 2EJ0:


EJ0/kBTÀ 1: approximate Bessel function

or

attempt frequency

attempt frequency is characteristic frequency !c

plasma frequency has to be replaced by frequency of overdamped oscillator:

washboard potential: phase diffuses over barrier activated nonlinear resistance

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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


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high temperature superconducting grain boundary Josephson junctions


La2-xCexCuO4 on SrTiO3

optical lithography

x500

I

I

V

Vgrain boundary

ab

a

b

B. Welter, Ph.D. Thesis, TU München (2007)

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determination of IC(T) close to Tc

overdamped YBa2Cu3O7 grain boundary Josephson junction


R. Gross et al., Phys. Rev. Lett. 64, 228 (1990)

thermally activated

phase slippage

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thermally activated phase slippage rounding of IVC at I ≃ Ic

rounding decreases with increasing °0 (´ increasing U0)

analytical IVC for strong damping (¯c ¿ 1): (Ambegaokar, Halperin)

for small junctions (L < ¸J):close to Tc: measurement of RP at const T Ic(B):


S. Schuster, R. Gross, et al., Rev. B 48, 16172 (1993)

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neglect self-fields B = Bex (valid for short junctions)junction voltage V = applied voltage V0

gauge invariant phase difference:

Josephson vortices are moving in z-direction with velocity

3.5 Voltage State of Extended Josephson Junctions3.5.1 Negligible Screening Effects

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Long junctions (L >> lJ): effect of Josephson currents has to be taken into accountmagnetic flux density = sum of external and self-generated field

with B = ¹0H and D = ²0E:in contrast to static case,

now E/t 0

3.5.2 The Time Dependent Sine-Gordon Equation

with 𝐸𝑥 = −𝑉/𝑑, 𝐽𝑥 = −𝐽𝑐 sin 𝜑 and 𝜕𝜑/𝜕𝑡 = 2𝜋𝑉/Φ0:

consider 1D junction extending in z-direction, B = By, current flow in x-direction

(Josephson penetration depth)

(propagation velocity)

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time dependentSine-Gordon equation


´ velocity of TEM mode in the junction transmission lineexample: e ≈ 5-10, 2¸L/d ≈ 50-100 Swihart velocity ≈ 0.1 · c

reduced wavelength: e.g. f = 10 GHz: free space: 3 cm, in junction: 1 mm

with the Swihart velocity:

other form of time-dependent Sine-Gordon equation

AS-Chap. 3 - 50

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mechanical analogue: chain of mechanical pendulaattached to a twistable rubber ribbon

Note: short junction w/o magnetic field: 2j/z2 = 0 rigid connection of pendula corresponds to single pendulum


time-dependent Sine-Gordon equation:

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• simple cases:1-dimensional junction (W¿ ¸J), short and long junctions

(i) short junctions (L¿ ¸J) @ low damping neglect z-variation of j:

equivalent to RCSJ model for G = 0, I = 0small amplitudes plasma oscillations(oscillation of j around minimum of washboard potential)

(ii) long junctions (LÀ ¸J), solitons solution for infinitely long junction soliton or fluxon:

j = ¼ at z = z0 +vztgoes from 0 to 2¼ for z going from -1 to1 fluxon (antifluxon: 1 to -1)

3.5.3 Solutions of the Time Dependent SG Equation

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j = ¼ at z = z0 +vztgoes from 0 to 2¼ for z from -1 to1 fluxon (antifluxon: 1 to -1)

pendulum analog:local 360° twist of rubber ribbon

applied current Lorentz forcemotion of phase twist (fluxon)

- fluxon as particle: Lorentz contraction for 𝑣𝑧 → 𝑐- local change of phase difference voltage

moving fluxon ´ voltage pulse- other solutions: fluxon-fluxon collisions, …


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• linearized Sine-Gordon equation: Josephson plasma waveslet:

𝜑1: small deviation approximation:

substitution (keeping only linear terms):

j0 solves time independent SG, 𝜕2𝜑0

𝜕𝑧2= sin 𝜑0/𝜆𝐽:

approximate: j0 = const(j0 slowly varying)


solution:

dispersion relation !(k):

Josephson plasma frequency:

(small amplitude plasma waves)

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for ! > !p,J: mode propagationpendulum analogue: deflect one pendulum relax wave like excitation

for ! = !p,J: infinite wavelength Josephson plasma wave (typically » 10 GHz)

• plane waves:

for very large ¸J or very small 𝐼: neglect sinj/¸J2 term

linear wave equation


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interaction of fluxons/plasma waves with oscillating Josephson current

rich variety of interesting resonance phenomena

(i) flux-flow steps and the Eck peak

for Bext > 0: - spatially modulated Josephson currrent densitymoves at 𝑣𝑧 = 𝑉/𝐵𝑦𝑡𝑏 Josephson current can excite Josephson plasma waves

resonance: em waves couple strongly to Josephson current if 𝒄 = 𝒗𝒛

3.5.4 Resonance Phenomena

Eck peak at frequency:

corresponding junction 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

AS-Chap. 3 - 57

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- increase driving force increase 𝑣𝑧- maximum possible speed: 𝑣𝑧 = 𝑐

step in IVC: flux flow step

corresponds to Eck voltage


other point of view:

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(ii) Fiske stepsstanding em waves in junction “cavity” @ Fiske steps at voltages:

for L» 100 ¹mfirst Fiske step » 10 GHz

wave length of Josephson current density 2¼/k/ B

resonance condition

here: maximum Josephson current of short junction vanishes

standing wave pattern of em wave and Josephson current match steps in IVC


damping of standing wave pattern by dissipative effects broadening of Fiske steps observation only for small and medium damping

AS-Chap. 3 - 59

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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 voltages VEck or Vn: 𝐼𝑠 ≃ 0 I = IN (V) = V / RN (V)


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(iii) zero field steps motion of trapped flux due to Lorentz force (w/o magnetic field) junction of length L, moving back and forth: T = 2L/vz, phase change: 4¼

at large bias currents:

n fluxons: 𝑉𝑛,𝑧𝑓𝑠 = 𝑛 ⋅ 𝑉𝑧𝑓𝑠𝑉𝑛,𝑧𝑓𝑠 = 2 × Fiske voltage Vn (fluxon has to move back and forth)

𝑉𝑓𝑓𝑠 = 𝑉𝑛,𝑧𝑓𝑠 for F = n F0

IVCs of annularNb/insulator/PbJosephson junction containing adifferent number of trapped fluxons

Vortex-Cherenkov radiation lecture notes


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voltage state: (Josephson + normal + displacement + fluctuation) current = total current

RCSJ-model (𝐺𝑁 𝑉 = 𝑐𝑜𝑛𝑠𝑡.):

motion of phase particle in the tilted washboard potential:

equivalent circuit: LCR resonator, characteristic frequencies:

equation of motion for phase difference 𝜑:

quality factor:

Summary (voltage state of short junctions)

𝑑𝜑

𝑑𝑡=2𝑒𝑉

ℏ

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IVC: (strong damping, 𝛽𝐶 ≪ 1)

driving with Shapiro steps at

amplitudes:

Summary (voltage state of short junctions)

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equation of motion: Sine-Gordon equation:

Swihart velocity (propagation velocity of em waves)

prominent solutions: plasma oscillations and solitons

nonlinear interactions of these excitations with Josephson current: flux-flow steps, Fiske steps, zero-field steps

Summary (voltage state of long junctions)

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• so far: Josephson junction was treated classically

𝝋 and 𝑑𝜑/𝑑𝑡 [∝ 𝑄 = 𝐶𝑉 = 𝐶ℏ

2𝑒

𝑑𝜑

𝑑𝑡] as purely classical variables

motion of 𝜑 is treated analogous to classical motion of particle in tilted washboard potential

classical energies:potential energy 𝑈(𝜑) , energy associated with Josephson coupling energy (equivalent circuit: flux stored in Josephson inductance)

kinetic energy 𝑲 𝝋 , associated with 𝑑𝜑

𝑑𝑡

2∝

𝑄2

2𝐶=

1

2𝐶𝑉2

(equivalent circuit: charge stored on junction capacitance)

3.6 Full Quantum Treatment of Josephson JunctionsSecondary Quantum Macroscopic Effects

• current-phase and voltage-phase relation have quantum origin (macroscopic quantum model)

primary quantum macroscopic effects

but so far: variables 𝑰, 𝑸, 𝑽, 𝝋 are assumed to be measurable simultaneouslymore precise quantum theory is needed secondary quantum macroscopic effects

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3.6.1 Quantum Consequences of the small Junction Capacitance

• validity of classical treatment:

E

j≃ ℏ𝜔𝑝/2

2E𝐽0

classical treatment valid as long as𝐸𝐽0

ℏ𝜔𝑝≃

𝐸𝐽0

𝐸𝐶

1/2≫ 1

(level spacing ¿ potential depth)

enter quantum regime by decreasing junction area

consider an isolated, low-damping junction, I = 0 cosine potential, depth 2EJ0

approx. close to minimum: harmonic oscillator, frequency 𝜔𝑝, level spacing ℏ𝜔𝑝

≃ ℏ𝜔𝑝

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numbers:

(i) area A = 10 µm2:

- barrier: d = 1 nm, ² = 10- Jc = 100 A/cm2

EJ0 = 3 x 10-21 J

- C = ²²0A/d' 9 x 10-13 F EC ≃ 1.6 x 10-26 J classical junction

(ii) area A ' 0.02 µm2:

- C' 1 fF EC ≃ EJ0

we also need T¿ 100 mK for kBT << EC


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• consider a strongly underdamped junction (negligible damping) with 𝑑𝜑/𝑑𝑡 ≠ 0

kinetic energy:

total energy:

𝑈 𝜑 ∝ 1 − cos𝜑: potential energy𝐾 𝜑 ∝ 𝜑2: kinetic energy

energy due to extra charge Q on one junction electrode due to V

• consider E as junction Hamiltonian, rewrite kinetic energy:

𝐾 = 𝑝2/2𝑀

𝑝 =ℏ

2𝑒𝑄


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canonical quantization, operator replacement:

with N = Q/2e: # of Cooper pairs:

we get the Hamiltonian:

describes only Cooper pairs𝐸𝐶 = 𝑒2/2𝐶: charging energy for a

single electron charge

𝑁 ≡ 𝑄/2𝑒: deviation of # of CP in electrodes from equilibrium


commutation rules for the operators:

uncertainty relation:

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• Hamiltonian in flux basis 𝜙 =ℏ

2𝑒𝜑 =

Φ0

2𝜋𝜑 :


commutator: 𝜙, 𝑄 = 𝑖ℏ Q and 𝜙 are canonically conjugate (cf. x and p),

deviations from “classical” description: secondary quantum macroscopic effects

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lowest energy levels: localized near bottom of potential wells at 𝜑𝑛 = 2𝜋 𝑛

Taylor series for 𝑈 𝜑 harmonic oscillator,

frequency !p, eigenenergies: 𝑬𝒏 = ℏ𝝎𝒑 𝒏 +𝟏

𝟐

ground state: narrowly peaked wave function at 𝜑 = 𝜑𝑛

small phase fluctuations Δ𝜑

large fluctuations of Q on electrodes since Δ𝑄 ⋅ Δ𝜑 ≥ 2𝑒

small EC pairs can easily fluctuate, large Δ𝑄

negligible Δ𝜑 ⇒ classical treatment of phase dynamics is good approximation

3.6.2 Limiting Cases: The Phase and Charge Regime

• the phase regime: ℏ𝜔𝑝 ≪ 𝐸𝐽0 , 𝐸𝑐 ≪ 𝐸𝐽0 (phase 𝜑 is good quantum number)

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Hamiltonian:

define: 𝑎 = 𝐸 − 𝐸𝐽0 /𝐸𝐶, 𝑏 = 𝐸𝐽0/2𝐸𝐶 and 𝑧 = 𝜑/2

known from periodic potential problem in solid state physics energy bands

general solution:

Bloch waves:

q: charge/pair number variable, q is continuous (cf. charge on capacitor) Ψ is not 2¼-periodic

1-dimensional problem numerical solution


• the phase regime: ℏ𝜔𝑝 ≪ 𝐸𝐽0 , 𝐸𝑐 ≪ 𝐸𝐽0 (phase 𝜑 is good quantum number)

Mathieu equation:

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variational approach for approximate ground state

trial function for 𝐸𝐶 ≪ 𝐸𝐽0:

(choose ¾ to find) minimum energy:

𝑬𝑪

𝑬𝑱𝟎= 𝟎. 𝟏

𝑬𝒎𝒊𝒏 = 𝟎. 𝟏 𝑬𝑱𝟎

tunneling coupling ∝ 𝐞𝐱𝐩 −𝟐𝑬𝑱𝟎−𝑬

ℏ𝝎𝒑 very small since ℏ𝜔𝑝 ≪ 𝐸𝐽0

tunneling splitting of low lying states is exponentially small

first order in EJ0

Emin ≃ 0 for EC ≪ EJ0


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kinetic energy ∝ 𝐸𝑐𝑑𝜑

𝑑𝑡

2is dominating, complete delocalization of phase

wave function should approach constant value, Ψ 𝜑 ≃ 𝑐𝑜𝑛𝑠𝑡. large phase fluctuations, small charge fluctuations: Δ𝑄 ⋅ Δ𝜑 ≥ 2𝑒 charge Q (corresponds to momentum) is good quantum number


• the charge regime: ℏ𝜔𝑝 ≫ 𝐸𝐽0 , 𝐸𝑐 ≫ 𝐸𝐽0 (charge Q is good quantum number)

appropriate trial function:

Hamiltonian:

approximate ground state energy

second order in EJ0

𝛼 ≪ 1

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𝑬𝑪𝑬𝑱𝟎

= 𝟐. 𝟓

𝑬𝒎𝒊𝒏 = 𝟎. 𝟗𝟓 𝑬𝑱𝟎

periodic potential is weak strong coupling between neighboring phase states broad bands compare to electrons moving in strong (phase regime) or weak (charge regime) periodic

potential of a crystal


• the charge regime: ℏ𝜔𝑝 ≫ 𝐸𝐽0 , 𝐸𝑐 ≫ 𝐸𝐽0 (charge Q is good quantum number)

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classical description valid for: level spacing ¿ potential depth

the phase regime: ℏ𝜔𝑝 ≪ 𝐸𝐽0 , 𝐸𝑐 ≪ 𝐸𝐽0

small phase fluctuations large charge fluctuations

the charge regime: ℏ𝜔𝑝 ≫ 𝐸𝐽0 , 𝐸𝑐 ≫ 𝐸𝐽0

large phase fluctuations small charge fluctuations

Brief Summary

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• Coulomb blockade in normal metal tunnel junctions:voltage: V charge: Q = CV, energy: E = Q2/2C

• single electron tunneling: charge on one electrode changes to 𝑄 − 𝑒 electrostatic energy: 𝐸′ = 𝑄 − 𝑒 2/2𝐶

tunneling only allowed for 𝐸′ ≤ 𝐸 we need 𝑄 ≥ 𝑒/2 or: 𝑽 ≥ 𝑽𝑪𝑩 = 𝑽𝒄 = 𝒆/𝟐𝑪 Coulomb blockade

• thermal fluctuations: 𝐸𝐶 =𝑒2

2𝐶> 𝑘𝐵𝑇 ⇒ 𝐶 <

𝑒2

2𝑘𝐵𝑇(small thermal fluctuations)

numbers: C ≈ 1 fF @ 1 K, for d = 1 nm and e = 5 A ≈ 0.02 µm2

• quantum fluctuations: Δ𝐸 ⋅ Δ𝑡 ≥ ℏ:finite tunnel resistance 𝜏𝑅𝐶 = 𝑅𝐶 (decay of charge fluctuations) Δ𝑡 = 2𝜋𝑅𝐶, Δ𝐸 = 𝑒2/2𝐶

𝑅 ≥ℎ

𝑒2= 𝑅𝐾 = 24.6 𝑘Ω (small quantum fluctuations)

• Coulomb blockade in superconducting tunnel junction:

for 𝑄2

2𝐶> 𝑘𝐵𝑇, 𝑒𝑉 (Q = 2e) no flow of Cooper pairs

threshold voltage: 𝑽 ≥ 𝑽𝑪𝑩 = 𝑽𝒄 =𝟐𝒆

𝟐𝑪=

𝒆

𝑪

• Coulomb blockade charge is fixed, phase is completely smeared out

3.6.3 Coulomb and Flux Blockade

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• phase or flux blockade in a Josephson junction:

current: I flux: Φ = 𝐿 𝐼, energy: 𝐸 = Φ2/2𝐿

phase is blocked due to large 𝐸𝐽0 = Φ0𝐼𝑐/2𝜋

- Ic takes the role of VCB

- phase change of 2¼ equivalent to flux change change of Φ0

flux blockade 𝑰 ≥ 𝑰𝑭𝑩 = 𝑰𝒄 =𝚽𝟎/𝟐𝝅

𝑳𝒄

cf. charge blockade 𝑽 ≥ 𝑽𝑪𝑩 = 𝑽𝒄 =𝒆

𝑪

analogy: 𝑰 ↔ 𝑽, 𝟐𝒆 ↔𝚽𝟎

𝟐𝝅, 𝑪 ↔ 𝑳

• fluctuations, we need:𝐸𝐽0 ≫ 𝑘𝐵𝑇

and

Δ𝐸 ⋅ Δ𝑡 ≥ ℏ: with Δt = 2𝜋𝐿/𝑅 and Δ𝐸 = 2𝐸𝐽0 → 𝑅 ≤ℎ

(2𝑒)2=

1

4𝑅𝐾

3.6.3 Coulomb and Flux Blockade

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coherent charge statesisland charge continuously changed by gate

for independent charge states:

parabola: 𝐸 = 𝑄 − 𝑛 ⋅ 2𝑒 2/2𝐶Σ

for 𝐸𝐽0 > 0: interaction of |𝑛⟩ and |𝑛 + 1⟩

at the crossing points 𝑄 = 𝑛 +1

2⋅ 2𝑒

Cooper pair box

3.6.4 Coherent Charge and Phase States

coherent superposition states: splitting of charge energy at crossing points ´ level anti-crossing splitting magnitude ∝ Josephson coupling energy 𝐸𝐽0

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average charge on the island as a function of the applied gate voltage


coherent superposition of charge states: experiment by Nakamura, Pashkin, Tsai

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coherent superposition states: splitting of charge energy at crossing points ´ level anti-crossing splitting magnitude Josephson coupling energy

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coherent phase states interaction of two adjacent phase states e.g. rf-SQUID

magnetic energy of flux 𝜙 =Φ0

2𝜋𝜑

in the ring

𝚽𝒆𝒙𝒕 = 𝚽𝟎/𝟐

tunnel coupling:experimental evidence for quantum coherent superposition of states


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violation of conservation of energy on small time scales, obey Δ𝐸 ⋅ Δ𝑡 ≥ ℏ

creation of virtual excitations include Langevin force IF (with adequate statistical properties) fluctuation-dissipation theorem:

𝐸 𝜔, 𝑇 : energy of a quantum oscillator

transition from “thermal” Johnson-Nyquist (ℏ𝜔, 𝑒𝑉 ≪ 𝑘𝐵𝑇) noise to quantum noise:

classical limit (ℏ𝜔, 𝑒𝑉 ≪ 𝑘𝐵𝑇):

quantum limit (ℏ𝜔, 𝑒𝑉 ≫ 𝑘𝐵𝑇):

3.6.5 Quantum Fluctuations

vacuumfluctuations

occupation probabilityof oscillator (Planck distribution)

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escape of the “phase particle” from minimum of washboard potential by tunnelingmacroscopic: phase difference is tunneling (collective state) states easily distinguishable

competing process: thermal activation low temperatures

neglect dampingdc-bias: term −ℏ𝐼𝜑/2𝑒 in Hamiltonian

curvature at potential minimum:

(classical) small oscillation frequency:

3.6.6 Macroscopic Quantum Tunneling

𝑖 = 𝐼/𝐼𝑐

(attempt frequency)

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quantum mechanically: tunnel coupling of bound states to outgoing waves continuum of statesbut: only states corresponding to quasi-bound states have high amplitude

in well states of width Γ = ℏ/𝜏 (¿ : lifetime for escape)

determination of wave functions wave matching method exponential prefactor within WKB approximation decay in barrier:

decay of wave function of particle with mass M and energy E


for 𝑈 𝜑 ≫ 𝐸0 = ℏ𝜔𝐴/2

mass effective barrier height

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constant barrier height: escape rate:

increasing bias current:

U0 decreases with i ¡ becomes measurable

temperature T* where Γ𝑡𝑢𝑛𝑛𝑒𝑙 = Γ𝑇𝐴 ≈ exp −𝑈0

𝑘𝐵𝑇

for 𝐼 > 0: for 𝜔𝑝 ≈ 1011 𝑠−1

T* » 100 mK

very small for small j


for 𝐼 ≃ 0:

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coupling of the system to the environment (heat bath): see e.g. Caldeira & Leggettdamping suppresses MQT

for ®À 1: !R ¿ ! A lower T*

quantum junction: lightly dampedclassical junction: moderately damped

phase diffusion by MQTsee lecture notes

3.6.6 Macroscopic Quantum Tunnelingadditional topic: effect of damping

crossover temperature:

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• Coulomb/flux blockade

• coherent charge and phase states

• macroscopic quantum tunneling

• effects of dissipation

Brief Summary

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classical description only in the phase regime (large junctions): 𝐸𝐶 ≪ 𝐸𝐽0

for 𝐸𝑐 ≫ 𝐸𝐽0: quantum description (negligible damping):

phase difference j and Cooper pair number N = Q/2e are canonically conjugate variables:

phase regime: Δ𝜑 → 0 and Δ𝑁 → ∞charge regime: Δ𝑁 → 0 and Δ𝜑 → ∞

charge regime at T = 0: Coulomb blockade:tunneling only for 𝑉𝐶𝐵 ≥ 𝑒/𝐶flux regime at T = 0: flux blockade:flux motion only for 𝐼𝐹𝐵 ≥ Φ0/2𝜋𝐿𝑐

at 𝐼 < 𝐼𝑐: escape out of the washboard by:thermal activationtunneling (macroscopic quantum tunneling)

crossover between TA and tunneling:

Summary (secondary quantum macroscopic effects)

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3. Physics of Josephson Junctions: The Voltage State · mass m, length ℓ, deflection angle µ,...

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