3. physics of josephson junctions: the voltage state · mass m, length ℓ, deflection angle µ,...
TRANSCRIPT
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 ?
AS-Chap. 3 - 3
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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 current- for 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
AS-Chap. 3 - 4
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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:
3.1.1 The Normal Current: Junction Resistance
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
AS-Chap. 3 - 5
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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
3.1.1 The Normal Current: Junction Resistance
AS-Chap. 3 - 6
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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
AS-Chap. 3 - 7
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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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3) quality factor:
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
AS-Chap. 3 - 9
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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)
AS-Chap. 3 - 10
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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
AS-Chap. 3 - 11
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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
AS-Chap. 3 - 12
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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
AS-Chap. 3 - 13
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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
AS-Chap. 3 - 14
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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
AS-Chap. 3 - 15
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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
AS-Chap. 3 - 16
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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
AS-Chap. 3 - 17
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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
AS-Chap. 3 - 18
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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
3.3.1 Response to a dc Current Source
AS-Chap. 3 - 19
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with
we get for i > 1:
and
3.3.1 Response to a dc Current Source
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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
3.3.1 Response to a dc Current Source
Current-Voltage Characteristics:
Weak damping: ¯c À 1, neglecting noise current:
AS-Chap. 3 - 21
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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:
3.3.1 Response to a dc Current Source
AS-Chap. 3 - 22
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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
3.3.1 Response to a dc Current Source
AS-Chap. 3 - 23
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Summary
plasma frequency:
Lc/RN time constant:
RNC time constant:
Stewart-McCumber parameter:
AS-Chap. 3 - 24
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time averaged voltage:
strong damping: ¯c ¿ 1, neglecting noise current:
Summary
AS-Chap. 3 - 25
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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
AS-Chap. 3 - 27
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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
3.3.2 Response to ac Driving Sources
some mathematics for the analysis of the time-dependent Josephson current:
AS-Chap. 3 - 28
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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
3.3.2 Response to ac Driving Sources
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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
3.3.2 Response to ac Driving Sources
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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)
3.3.2 Response to ac Driving Sources
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experimental IVCs obtained for an underdamped and overdamped Niobium Josephson junction under microwave radiation
3.3.2 Response to ac Driving Sources
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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:
AS-Chap. 3 - 33
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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
3.4 Additional Topic: Effect of Thermal Fluctuations
AS-Chap. 3 - 36
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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)
3.4 Additional Topic: Effect of Thermal Fluctuations
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:
AS-Chap. 3 - 41
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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:
3.4.2 Overdamped Junctions: The Ambegaokar-Halperin Theory
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
AS-Chap. 3 - 42
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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
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high temperature superconducting grain boundary Josephson junctions
3.4.2 Overdamped Junctions: The Ambegaokar-Halperin Theory
La2-xCexCuO4 on SrTiO3
optical lithography
x500
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Vgrain boundary
ab
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B. Welter, Ph.D. Thesis, TU München (2007)
AS-Chap. 3 - 44
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determination of IC(T) close to Tc
overdamped YBa2Cu3O7 grain boundary Josephson junction
3.4.2 Overdamped Junctions: The Ambegaokar-Halperin Theory
R. Gross et al., Phys. Rev. Lett. 64, 228 (1990)
thermally activated
phase slippage
AS-Chap. 3 - 45
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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):
3.4.2 Overdamped Junctions: The Ambegaokar-Halperin Theory
S. Schuster, R. Gross, et al., Rev. B 48, 16172 (1993)
AS-Chap. 3 - 47
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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
AS-Chap. 3 - 48
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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)
AS-Chap. 3 - 49
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time dependentSine-Gordon equation
3.5.2 The Time Dependent Sine-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
3.5.2 The Time Dependent Sine-Gordon Equation
time-dependent Sine-Gordon equation:
AS-Chap. 3 - 52
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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
AS-Chap. 3 - 53
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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, …
3.5.3 Solutions of the Time Dependent SG Equation
AS-Chap. 3 - 54
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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)
3.5.3 Solutions of the Time Dependent SG Equation
solution:
dispersion relation !(k):
Josephson plasma frequency:
(small amplitude plasma waves)
AS-Chap. 3 - 55
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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
3.5.3 Solutions of the Time Dependent SG Equation
AS-Chap. 3 - 56
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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
3.5.4 Resonance Phenomena
other point of view:
AS-Chap. 3 - 58
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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
3.5.4 Resonance Phenomena
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)
3.5.4 Resonance Phenomena
AS-Chap. 3 - 60
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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
3.5.4 Resonance Phenomena
AS-Chap. 3 - 61
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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
AS-Chap. 3 - 62
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IVC: (strong damping, 𝛽𝐶 ≪ 1)
driving with Shapiro steps at
amplitudes:
Summary (voltage state of short junctions)
AS-Chap. 3 - 63
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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)
AS-Chap. 3 - 64
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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
AS-Chap. 3 - 65
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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
3.6.1 Quantum Consequences of the small Junction Capacitance
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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
3.6.1 Quantum Consequences of the small Junction Capacitance
AS-Chap. 3 - 68
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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
3.6.1 Quantum Consequences of the small Junction Capacitance
commutation rules for the operators:
uncertainty relation:
AS-Chap. 3 - 69
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• Hamiltonian in flux basis 𝜙 =ℏ
2𝑒𝜑 =
Φ0
2𝜋𝜑 :
3.6.1 Quantum Consequences of the small Junction Capacitance
commutator: 𝜙, 𝑄 = 𝑖ℏ Q and 𝜙 are canonically conjugate (cf. x and p),
deviations from “classical” description: secondary quantum macroscopic effects
AS-Chap. 3 - 70
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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)
AS-Chap. 3 - 71
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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
3.6.2 Limiting Cases: The Phase and Charge Regime
• the phase regime: ℏ𝜔𝑝 ≪ 𝐸𝐽0 , 𝐸𝑐 ≪ 𝐸𝐽0 (phase 𝜑 is good quantum number)
Mathieu equation:
AS-Chap. 3 - 72
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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
3.6.2 Limiting Cases: The Phase and Charge Regime
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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
3.6.2 Limiting Cases: The Phase and Charge Regime
• 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
3.6.2 Limiting Cases: The Phase and Charge Regime
• the charge regime: ℏ𝜔𝑝 ≫ 𝐸𝐽0 , 𝐸𝑐 ≫ 𝐸𝐽0 (charge Q is good quantum number)
AS-Chap. 3 - 75
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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
3.6.4 Coherent Charge and Phase States
coherent superposition of charge states: experiment by Nakamura, Pashkin, Tsai
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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
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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
3.6.4 Coherent Charge and Phase 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
3.6.6 Macroscopic Quantum Tunneling
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
3.6.6 Macroscopic Quantum Tunneling
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)