Chapter 3. The Voltage State

Transcription

1 AS-Chap. 3-1 Chapter 3 Physics of Josephson Junctions: The Voltage State

2 3. Physics of Josephson junctions: The voltage state AS-Chap. 3-2 For bias currents I > I s m Finite junction voltage Phase difference φ evolves in time: dφ dt V 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 I > I m s? What is the influence of the resistive damping?

3 3.1 The basic equation of the lumped Josephson junction The normal current: Junction resistance AS-Chap. 3-3 At finite temperature T > 0 Finite density of normal electrons Quasiparticles Zero-voltage state: No quasiparticle current For V > 0 Quasiparticle current = Normal current I N Resistive state High temperatures close to T c For T T c and 2Δ T k B T: (almost) all Cooper pairs are broken up Ohmic current-voltage characteristic (IVC) I N = G N V, where G N 1 R N is the normal conductance Large voltage V > V g = Δ 1+Δ 2 e External circuit provides energy to break up Cooper pairs Ohmic IVC For T T c and V < V g Vanishing quasiparticle density No normal current

4 AS-Chap The normal current: Junction resistance Current-voltage characteristic Voltage state Bias current I I s t = I c sin φ t is time dependent I N is time dependent Junction voltage V = I N G N is time dependent IVC shows time-averaged voltage V For T T c and V < V g IVC depends on sweep direction and on bias type (current/voltage) Hysteretic behavior Current bias I = I s + I N = const. Circuit model Equivalent conductance G N at T = 0:

5 3.1.1 The normal current: Junction resistance AS-Chap. 3-5 Finite temperature Sub-gap resistance R sg T for V < V g R sg T determined by amount of thermally excited quasiparticles n T Density of excited quasiparticles for T > 0 we get Nonlinear conductance G N V, T Characteristic voltage (I c R N -product) Note: There may be a frequency dependence of the normal channel Normal channel depends on junction type

6 AS-Chap The displacement current: Junction capacitance If dv dt 0 Finite displacement current C junction capacitance For planar tunnel junction Additional current channel With V = L c di s dt, I N = VG N, I D = C dv dt, L s = L c L cos φ c and G N V, T = 1 R N Josephson inductance

7 3.1.3 Characteristic times and frequencies Characteristic frequencies Equivalent parallel LRC circuit L c, R N, C Three characteristic frequencies Plasma frequency ω p J c, where C C A C is the specific junction capacitance A A ω < ω p I D < I s Inductive L c /R N time constant V c = I c R N Inverse relaxation time in the normal+supercurrent system ω c follows from V c (2 nd Josephson equation) I N < I c for V < V c or ω < ω c Capacitive R N C time constant I D < I N for ω < 1 τ RC AS-Chap. 3-7

8 3.1.3 Characteristic times and frequencies AS-Chap. 3-8 Stewart-McCumber parameter and quality factor Stewart-McCumber parameter Quality factor (Q compares the decay of oscillation amplitudes to the oscillation period) Limiting cases β C 1 Small capacitance and/or small resistance Small R N C time constants (τ RC ω p 1) Highly damped (overdamped) junctions β C 1 Large capacitance and/or large resistance Large R N C time constants (τ RC ω p 1) Weakly damped (underdamped) junctions

9 3.1.4 The fluctuation current AS-Chap. 3-9 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 (k B T ev, ħω): (current noise power spectral density) (voltage noise power spectral density) relative noise intensity (thermal energy/josephson coupling energy): I T thermal noise current T = 4.2 K I T 0.15 μa

10 3.1.4 The fluctuation current AS-Chap Shot Noise Schottky formula for shot noise (ev k B T V > K): Random fluctuations due to the discreteness of charge carriers Poisson process Poissonian distribution Strength of fluctuations variance ΔI 2 I I 2 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

11 3.1.5 The basic junction equation AS-Chap Kirchhoff s law: I = I s + I N + I D + I F Voltage-phase relation: dφ = 2eV dt ħ Basic equation of a Josephson junction Normal current Supercurrent Displacement current Noise current Nonlinear differential equation with nonlinear coefficients Complex behavior, numerical solution Use approximations (simple models)

12 3.2 The resistively and capacitively shunted junction (RCSJ) model Resistively and Capacitively Shunted Junction (RCSJ) model Approximation G N V G = R 1 = const. R = Junction normal resistance Differential equation in dimesionless or energy formulation i i F (t) Mechanical analog Gauge invariant phase difference Particle with mass M and damping η in potential U: with Tilted washboard potential AS-Chap. 3-12

13 3.2 The resistively and capacitively shunted junction (RCSJ) model Finite tunneling probability: Macroscopic quantum tunneling (MQT) Escape by thermal activation Thermally activated phase slips Normalized time: Stewart-McCumber parameter: Motion of phase particle φ in the tilted washboard potential Plasma frequency Neglect damping, zero driving and small amplitudes (sin φ φ) Solution: Plasma frequency = Oscillation frequency around potential minimum AS-Chap. 3-13

14 3.2 The resistively and capacitively shunted junction (RCSJ) model The pendulum analog Plane mechanical pendulum in uniform gravitational field Mass m, length l, deflection angle θ Torque D parallel to rotation axis Restoring torque: mgl sin θ Equation of motion D = Θ θ + Γ θ + mgl sin θ Θ = ml 2 Moment of inertia Γ Damping constant Analogies I I c Φ 0 2πR CΦ 0 2π φ D mgl Γ Θ θ For D = 0 Oscillations around equilibrium with ω = g l Plasma frequency ω p = Finite torque (D > 0) Finite θ 0 Finite, but constant φ 0 Zero-voltage state Large torque (deflection > 90 ) Rotation of the pendulum Finite-voltage state Voltage V Angular velocity of the pendulum 2πI c Φ 0 C AS-Chap. 3-14

15 3.2.1 Under- and overdamped Josephson junctions AS-Chap Overdamped junction β C = 2eI cr 2 C 1 ħ Capacitance & resistance small M small, η large Non-hysteretic IVC (Phase particle will retrap immediately at I c because of large damping) Underdamped junction β C = 2eI cr 2 C 1 ħ Capacitance & resistance large M large, η small Hysteretic IVC (Once the phase is moving, the potential has to be tilt back almost into the horizontal position to stop ist motion)

16 AS-Chap Response to driving sources Motivation Applied Superconductivity One central question is How to extract information about the junction experimentally? 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

17 3.3.1 Response to a dc current source AS-Chap Time averaged voltage: 2π T = Oscillation period Total current must be constant (neglecting the fluctuation source): where: I > I c Part of the current must flow as I N or I D Finite junction voltage V > 0 Time varying I s I N + I D varies in time Time varying voltage, complicated non-sinusoidal oscillations of I s, Oscillating voltage has to be calculated self-consistently Oscillation frequency f = V Φ 0

18 AS-Chap Response to a dc current source For I I c Highly non-sinusoidal oscillations Long oscillation period V 1 T is small For I I c 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 Analogy to pendulum I/I c = 3.0 I/I c = 1.5 I/I c = 1.1 I/I c = 1.05

19 3.3.1 Response to a dc current source Strong damping β C 1 & neglecting noise current i < 1 Only supercurrent, φ = sin 1 i is a solution, zero junction voltage i > 1 Finite voltage, temporal evolution of the phase Integration using gives Setting τ 0 = 0 and using τ = t τ c Periodic function with period tan 1 a tan x + b is π-periodic AS-Chap. 3-19

20 3.3.1 Response to a dc current source AS-Chap with and We get for i > 1

21 3.3.1 Response to a dc current source Weak damping β C 1 & neglecting noise current ω RC = 1 is very small R N C Large C is effectively shunting oscillating part of junction voltage V t V Time evolution of the phase Almost sinusoidal oscillation of Josephson current Down to V ħω RC V e c = I c R N Corresponding current I c Hysteretic IVC Ohmic result valid for R N = const. Real junction IVC determined by voltage dependence of R N = R N V AS-Chap. 3-21

22 AS-Chap Response to a dc current source Intermediate damping β C 1 Numerically solve IVC General trend Increasing β C Increasing hysteresis Hysteresis characterized by retrapping current I r I r washboard potential tilt where Energy dissipated in advancing to next minimum = Work done by drive current Analytical calculation possible for β C 1 (exercise class) I r /I c Numerical calculation

23 3.3.1 Response to a dc voltage source AS-Chap Phase evolves linearly in time: Josephson current I s oscillates sinusoidally Time average of I s is zero I D = 0 since dv dc = 0 dt Total current carried by normal current I = V dc R N RCSJ model Ohmic IVC General case R = R N V Nonlinear IVC

24 3.3.2 Response to ac driving sources AS-Chap Response to an ac voltage source Strong damping β C 1 Integrating the voltage-phase relation: Current-phase relation: Superposition of linearly increasing and sinusoidally varying phase Supercurrent I s (t) and ac voltage V 1 have different frequencies Origin Nonlinear current-phase relation

25 3.3.2 Response to ac driving sources AS-Chap Some maths for the analysis of the time-dependent Josephson current Fourier-Bessel series identity: J n b = n th order Bessel function of the first kind and: Imaginary part Ac driven junction x = ω 1 t, b = 2π and a = φ Φ 0 ω 0 + ω dc t = φ 0 + 2π V 1 Φ dc t 0 Frequency ω dc couples to multiples of the driving frequency

26 3.3.2 Response to ac driving sources AS-Chap Shapiro steps Ac voltage results in dc supercurrent if ω dc nω 1 t + φ 0 is time independent Amplitude of average dc current for a specific step number n V dc V n ω dc nω 1 t + φ 0 is time dependent Sum of sinusoidally varying terms Time average is zero Vanishing dc component

27 3.3.2 Response to ac driving sources Ohmic dependence with sharp current spikes at V dc = V n Current spike amplitude depends on ac voltage amplitude n th step Phase locking of the junction to the n th harmonic Example: ω 1 /2π = 10 GHz Constant dc current at V dc = 0 and V n = nω 1 Φ 0 2π n 20 μv AS-Chap. 3-27

28 3.3.2 Response to ac driving sources Response to an ac current source Strong damping β C 1 (experimentally relevant) Kirchhoff s law (neglecting I D ) I c sin φ + 1 Φ 0 dφ = I R N 2π dt dc + I 1 sin ω 1 t Difficult to solve Qualitative discussion with washboard potential Increase I dc at constant I 1 Zero-voltage state for I dc + I 1 I c, finite voltage state for I dc + I 1 > I c Complicated dynamics! V n = nω 1 Φ 0 2π Motion of phase particle synchronized by ac driving Simplifying assumption During each ac cycle the phase particle moves down n minima Resulting phase change φ = n 2π T = nω 1 Average dc voltage V = n Φ 0 2π ω 1 V n Exact analysis Synchronization of phase dynamics with external ac source for a certain bias current interval Steps AS-Chap. 3-28

29 3.3.2 Response to ac driving sources AS-Chap Experimental IVCs obtained for an underdamped and overdamped Niobium Josephson junction under microwave irradiation

30 3.3.4 Photon-assisted tunneling Superconducting tunnel junction Highly nonlinear R V Sharp step at V g = 2Δ e Use quasiparticle (QP) tunneling current I qp V Include effect of ac source on QP tunneling Model of Tien and Gordon: Ac driving shifts levels in electrode up and down QP energy: E qp + ev 1 cos ω 1 t QM phase factor Bessel function identity for V 1 -term Sum of terms Splitting of qp-levels into many levels E qp ± nħω 1 Modified density of states! Tunneling current Sharp increase of the I qp V at V = V g is broken up into many steps of smaller current amplitude at V n = V g ± nħω 1 e AS-Chap. 3-30

31 3.3.4 Photon-assisted tunneling AS-Chap Example QP IVC of a Nb SIS Josephson junction without & with microwave irradiation Frequency ω 1 2π = 230 GHz corresponding to ħω 1 e 950 μv ω 1 2π = 230 GHz QP steps Appear at V n = n ħ e ω 1 Amplitude J n ev 1 ħω 1 Broadended steps (depending on I qp V ) Shapiro steps Appear at V n = n ħ 2e ω 1 Amplitude J n 2eV 1 ħω 1 Sharp steps

32 3.4 Additional topic: Effect of thermal fluctuations Thermal fluctuations with correlation function: Small fluctuations Phase fluctuations around equilibrium Larger fluctuations Increase probability for escape out of potential well Escape at rates G n±1 Escape to next minimum Phase change of 2π I > 0 G n+1 > G n-1 dφ > 0 dt Langevin equation for RCSJ model Equivalent to Fokker-Planck equation: Normalized force Normalized momentum AS-Chap. 3-32

33 3.4 Additional topic: Effect of thermal fluctuations AS-Chap σ(v, φ, t) Probability density of finding system at (v, φ) at time t statistical average of variable X Small fluctuations Static solution ( dσ dt = 0) with: Boltzmann distribution (G = E Fx is total energy, E is free energy) Constant probability to find system in n th metastable state

34 3.4 Additional topic: Effect of thermal fluctuations AS-Chap Large fluctuations p can change in time Amount of phase slippage for Γ n+1 Γ n 1 and ω A Γ n+1 1 Kramers approximation ω A = Attempt frequency Attempt frequency ω A Weak damping (β C = ω c τ RC 1) I = 0 ω A = ω p (Oscillation frequency in the potential well) I I c ω A» ω p Strong damping (β C = ω c τ RC 1) ω p ω c (Frequency of an overdamped oscillator) (underdamped junction) (overdamped junction)

35 3.4.1 Underdamped junctions: Critical current reduction by premature switching AS-Chap For E J0 k B T Small escape probability exp U 0 I k B T at each attempt Barrier height: 2E J0 for I = 0 0 for I I c Experiment Escape probability ω A /2π for I I c After escape Junction switches to IR N Measure distribution of escape current I M Width δi and mean reduction ΔI c = I c I M Use approximation for U 0 I and escape rate ω A /2π exp U 0 I k B T Considerable reduction of I c when k B T > 0.05 E J0 Provides experimental information on real or effective temperature!

36 AS-Chap Overdamped junctions: The Ambegaokar-Halperin theory Calculate voltage V induced by thermally activated phase slips as a function of current Important parameter:

37 3.4.2 Overdamped junctions: The Ambegaokar-Halperin theory Amgegaokar-Halperin theory Finite amount of phase slippage Nonvanishing voltage for I 0 Phase slip resistance for strong damping (β C 1), for U 0 = 2E J0 : E J0 k B T Modified Bessel function 1 Approximate Bessel function attempt frequency or Attempt frequency is characteristic frequency ω c Plasma frequency has to be replaced by frequency of overdamped oscillator: Washboard potential Phase diffuses over barrier Activated nonlinear resistance AS-Chap. 3-37

38 3.4.2 Overdamped junctions: The Ambegaokar-Halperin theory Example: YBa 2 Cu 3 O 7 grain boundary Josephson junctions Strong effect of thermal fluctuations due to high operation temperature epitaxial YBa 2 Cu 3 O 7 film on SrTiO 3 bicrystalline substrate R. Gross et al., Phys. Rev. Lett. 64, 228 (1990) Nature 322, 818 (1988) AS-Chap. 3-38

39 AS-Chap Overdamped junctions: The Ambegaokar-Halperin theory Overdamped YBa 2 Cu 3 O 7 grain boundary Josephson junction thermally activated phase slippage Determination of I c T close to T c R. Gross et al., Phys. Rev. Lett. 64, 228 (1990)

40 3.5 Voltage state of extended Josephson junctions AS-Chap 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 yz-plane In-plane B field in y-direction Thick electrodes λ L1,2 Magnetic thickness t B = d + λ L,1 + λ L,2 Bias current in x-direction Phase gradient along z-direction φ z,t = 2π t z Φ B B y z, t 0 Expected effects Voltage state E-field and time-dependence become important Short junction and long junction case

41 3.5.1 Negligible Screening Effects Neglect self-fields (short junctions) B = B ex Junction voltage V = Applied voltage V 0 Gauge invariant phase difference: Josephson vortices moving in z-direction with velocity AS-Chap. 3-41

42 3.5.2 The time dependent Sine-Gordon equation Long junctions (L λ J ) Effect of Josephson currents has to be taken into account Magnetic flux density = External + Self-generated field with B = μ 0 H and D = ε 0 E: in contrast to static case, now E/ t 0 consider 1D junction extending in z-direction, B = B y, current flow in x-direction with E x = V/d, J x = J c sin φ and φ/ t = 2πV/Φ 0 : (Josephson penetration depth) (propagation velocity) AS-Chap. 3-42

43 3.5.2 The time dependent Sine-Gordon equation AS-Chap Time dependent Sine-Gordon equation with the Swihart velocity c = velocity of TEM mode in the junction transmission line Example: ε 5 10, 2λ L d c 0.1c Reduced wavelength For f = 10 GHz Free space: 3 cm, in junction: 1 mm Other form of time-dependent Sine-Gordon equation

44 3.5.2 The time dependent Sine-Gordon equation AS-Chap Time-dependent Sine-Gordon equation: Mechanical analogue Chain of mechanical pendula attached to a twistable rubber ribbon Restoring torque λ J 2 2 φ z 2 Short junction w/o magnetic field 2 / z 2 = 0 Rigid connection of pendula Corresponds to single pendulum

45 3.5.3 Solutions of the time dependent SG equation AS-Chap Simple case 1D junction (W λ J ), short and long junctions Short junctions (L λ J low damping Neglect z-variation of φ Equivalent to RCSJ model for G N = 0, I = 0 Small amplitudes Plasma oscillations (Oscillation of φ around minimum of washboard potential) Long junctions (L λ J ) Solution for infinitely long junction Soliton or fluxon φ = π at z = z 0 + v z t goes from 0 to 2π for z Fluxon (antifluxon: z )

46 3.5.3 Solutions of the time dependent SG equation AS-Chap φ = π at z = z 0 + v z t goes from 0 to 2π for z Fluxon (antifluxon: z ) Pendulum analog Local 360 twist of rubber ribbon Applied current Lorentz force Motion of phase twist (fluxon) Fluxon as particle Lorentz contraction for v z c Local change of phase difference Voltage Moving fluxon = Voltage pulse Other solutions: Fluxon-fluxon collisions, breathers, bound states,

47 3.5.3 Solutions of the time dependent SG equation AS-Chap Josephson plasma waves Linearized Sine-Gordon equation φ 1 = Small deviation Approximation Substitution (keeping only linear terms): φ 0 solves time independent SGE 2 φ 0 z 2 = λ J 2 sin φ 0 φ 0 slowly varying φ 0 const.

48 3.5.3 Solutions of the time dependent SG equation AS-Chap Solution: (small amplitude plasma waves) Dispersion relation ω(k) : = ω p 2 Josephson plasma frequency 4π ω < ω 2 cos φ 0 p,j Wave vector k imaginary No propagating solution ω > ω p,j Mode propagation Pendulum analogue Deflect one pendulum Relax Wave like excitation ω = ω p,j Infinite wavelength Josephson plasma wave Analogy to plasma frequency in a metal Typically junctions ω p,j 10 GHz Plane waves For very large λ J or very small I sin φ Neglect λ2 term Linear wave equation Plane waves with velocity c J

49 3.5.4 Resonance phenomena AS-Chap Interaction of fluxons or plasma waves with oscillating Josephson current Rich variety of interesting resonance phenomena Require presence of B ex Steps in IVC (junction upconverts dc drive) Flux-flow steps and Eck peak For B ex > 0 Spatially modulated Josephson currrent density moves at v z = Josephson current can excite Josephson plasma waves V B y t B On resonance, em waves couple strongly to Josephson current if c = v z Corresponding junction voltage: Eck peak at frequency:

50 3.5.4 Resonance phenomena AS-Chap 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 Alternative point of view Lorentz force Josephson vortices move at v z = V B y t B Increase driving force Increase v z Maximum possible speed is v z = c Further increase of I does not increase V) Flux-flow step in IVC V ffs = cb y t B = c Φ = ω p L 2π Corresponds to Eck voltage λ J L Φ 0 Φ Φ 0

51 3.5.4 Resonance phenomena Fiske steps Standing em waves in junction cavity at ω n = 2πf n = 2π c Fiske steps at voltages Interpretation 2L π n = c n L Wave length of Josephson current density is 2π k Resonance condition L = c n = λ n kl = nπ or Φ = n Φ 0 2f n 2 2 where maximum Josephson current of short junction vanishes Standing wave pattern of em wave and Josephson current match Steps in IVC λ/2-cavity ω n = nπv ph L v ph = Phase velocity for L» 100 μm first Fiske step» 10 GHz (few 10s of μv) Influence of dissipation Damping of standing wave pattern by dissipative effects Broadening of Fiske steps Observation only for small and medium damping AS-Chap. 3-51

52 3.5.4 Resonance phenomena AS-Chap Fiske steps at small damping and/or small magnetic field Eck peak at medium damping and/or medium magnetic field For V V Eck and V V n I s = I c sin ω 0 t + kz + φ 0 0 I = I N V = V/R N V

53 3.5.4 Resonance phenomena AS-Chap Zero field steps Motion of trapped flux due to Lorentz force (w/o magnetic field) Junction of length L, moving back and forth Phase change of 4π in period T = 2L v z At large bias currents (v z c) V zfs = φ ħ 2e = 4π ħ T 2e = 4π ħ 2L/ c 2e = h c 2e L = ω p λ J π L Φ 0 For n fluxons V n,zfs = nv zfs V n,zfs = 2 Fiske voltage V n (fluxon has to move back and forth) V ffs = V n,zfs for Φ = nφ 0 (introduce n fluxons = generate n flux quanta) Example: IVCs of annular Nb/insulator/Pb Josephson junction containing a different number of trapped fluxons

54 Summary (Voltage state of short junctions) Voltage state: (Josephson + normal + displacement + fluctuation) current = total current dφ dt = 2eV ħ Equation of motion for phase difference φ: RCSJ-model (G N V = const.) Motion of phase particle in the tilted washboard potential Equivalent LCR resonator, characteristic frequencies: Quality factor: β C = Stewart-McCumber parameter AS-Chap. 3-54

55 AS-Chap Summary (Voltage state of short junctions) IVC for strong damping and β C 1 Driving with V t = V dc + V 1 cos ω 1 t Shapiro steps at V n = n Φ 0 2π ω 1 with amplitudes I s n = I c J n 2πV 1 Φ 0 ω 1 Photon assisted tunneling Voltage steps at V n = n Φ 0 π ω 1 due to nonlinear QP resistance Effect of thermal fluctuations Phase-slips at rate Γ n+1 = ω A 2π exp U 0 k B T Finite phase-slip resistance R p even below I c Premature switching

56 Summary AS-Chap voltage state of extended junctions w/o self-field: with self-field: time dependent Sine-Gordon equation characteristic velocity of TEM mode in the junction transmission line 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

57 Solitons AS-Chap A soliton is a self-reinforcing solitary wave (a wave packet or pulse) that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium.

58 Solitons AS-Chap d classic soliton in the water canal, discovered in 1834 by J.S.Russel : Described by Kortweg de Friz (KdF) equation similar to SG equation.

59 Solitons AS-Chap d solitons on the shallow water surface: Usually described by cnoidal wave solution of KdF equation.

60 Solitons AS-Chap d Falaco soliton in the water pool: Two vortices are linked together with a turbulent channel deep in the water and moving as the whole.

61 Solitons AS-Chap Morning Glory Clouds, Australia, coastline: