Chapter 3 Physics of Josephson Junctions: The Voltage State
AS-Chap. 3-2 3. Physics of Josephson Junctions: The Voltage State for I > I sm : - 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 questions: - how does the phase dynamics look like? - current-voltage characteristics for I > I sm? - what is the influence of the resistive damping?
AS-Chap. 3-3 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 I N resistive state high temperature close to T c : - for T T c : 2 (T) k B T: (almost) all Cooper pairs are broken up, Ohmic IVC: G N = 1/R N : normal conductance large voltage V > V g = ( 1 + 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
AS-Chap. 3-4 3.1.1 The Normal Current: Junction Resistance for T T c and V < V g : voltage state: I s (t) = I c sinj(t) is time dependent I N is time dependent junction voltage V = I N /G N is time dependent IVC: time averaged voltage equivalent conductance G N at T = 0: IVC depends on sweep direction (and external source) hysteretic behavior current source: I = I s + I N = const. circuit model
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 frequency dependence of normal channel - normal channel depends on junction type
3.1.2 The Displacement Current: Junction Capacitance AS-Chap. 3-6 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 = CdV/dt and L s = L C /cosj L c, G N (V,T) = 1/R N : Josephson inductance
3.1.3 Characteristic Times and Frequencies AS-Chap. 3-7 equivalent circuit: L c, R N, C 3 characteristic frequencies plasma frequency: scales (J c /C A ) 0.5, C A = C/A: specific junction capacitance for! <! p : I D < I s L c /R N time constant: inverse relaxation time in system of normal and supercurrent! c follows from V c (2 nd Josephson eq.) characteristic frequency I N < I c for V < V c or! <! c = R N /L c R N C time constant: I D < I N for! < 1/ RC Stewart-McCumber parameter: (related to quality factor of LCR circuit)
3.1.3 Characteristic Times and Frequencies AS-Chap. 3-8 quality factor: parallel LRC circuit Q compares decay of amplitude of oscillations to oscillation period c 1: small capacitance and/or small resistance small R N C time constants ( RC! p 1) highly damped or overdamped junctions c À 1: large capacitance and/or large resistance large R N C time constants ( RC! p À 1) weakly damped or underdamped junctions
3.1.4 The Fluctuation Current AS-Chap. 3-9 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 (k B T À ev, ~!): (current noise power spectral density) (voltage noise power spectral density) relative noise intensity (thermal energy/josephson coupling energy): I T : thermal noise current for T = 4.2 K: I T ' 0.15 ¹A
3.1.4 The Fluctuation Current AS-Chap. 3-10 Shot Noise: Schottky formula for shot noise (ev À k B T, 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
3.1.5 The Basic Junction Equation AS-Chap. 3-11 Kirchhoff s law: & voltage-phase relation: I = I s + I N + I D + I F dj/dt = 2eV/~ basic equation of Josephson junction nonlinear differential equation with nonlinear coefficients complex behavior, numerical solution use approximations (simple models)
3.2 The Resistively and Capacitively Shunted Junction Model Resistively and Capacitively Shunted Junction model (RCSJ): approximation: G N (V) = G = 1/R = const. Josephson junction: L s = L c /cosj with L c = ħ/2ei c R = 1/G: junction normal resistance nonlinear differential equation: i i F (t) compare motion of gauge invariant phase difference to that of particle with mass M and damping in potential U: with tilted washboard potential AS-Chap. 3-12
3.2 The RCSJ Model finite tunneling probability: macroscopic quantum tunneling (MQT) escape by thermal activation thermally activated phase slips tilt washboard potential normalized time: Stewart-McCumber parameter: plasma frequency: neglect damping, zero driving and small amplitudes (sinj ' j): solution: motion of φ in tilted washboard potential plasma frequency oscillation frequency around potential minimum AS-Chap. 3-13
3.2 The RCSJ Model The pendulum analogue: physical pendulum: mass m, length l, deflection angle µ, torque D parallel to rotation axis restoring torque: l m g sin equation of motion: = ml 2 : moment of inertia : damping constant analogy to Josephson junction: I $ D I c $ mgl F 0 /2¼R $ CF 0 /2¼ $ gauge invariant phase difference φ $ angle µ for D = 0: oscillations around equilibrium with! = (g/l) 1/2 $ plasma frequency! p = (2¼I c /F 0 C) 1/2 finite torque finite µ 0 finite j 0 large torque (deflection > 90 ) rotation of the pendulum finite voltage state AS-Chap. 3-14
3.2.1 Underdamped and Overdamped Josephson Junctions overdamped junction: c = 2eI c R 2 C/~ 1 capacitance & resistance small M small, large non-hysteretic IVC underdamped junction: c = 2eI c R 2 C/~ À 1 capacitance & resistance large M 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) AS-Chap. 3-15
3.3 Response to Driving Sources 3.3.1 Response to a dc Current Source time averaged voltage: 2¼ T: oscillation period total current must be constant (neglecting the fluctuation source): where: for I > I c : part of the current must flow as I N or I D finite junction voltage V > 0 V > 0 time varying I s I N + I D is varying in time time varying voltage, complicated non-sinusoidal oscillations of I s oscillating voltage has to be calculated self-consistently oscillation frequency: f = hvi/f 0 AS-Chap. 3-16
AS-Chap. 3-17 3.3.1 Response to a dc Current Source for I I c : highly non-sinusoidal oscillations long oscillation period hvi / 1/T : small for I >> I c : almost all current flows as normal current junction voltage is about constant oscillations of Josephson current are almost sinusoidal time averaged Josephson current almost zero linear/ohmic IVC I/I c = 3.0 I/I c = 1.5 I/I c = 1.1 analogy to pendulum I/I c = 1.05
3.3.1 Response to a dc Current Source Current-Voltage Characteristics: strong damping: c 1, neglecting noise current: for i < 1: only supercurrent, j = sin -1 i is a solution, junction voltage = zero for i > 1: finite voltage, temporal evolution of the phase integration using: gives setting 0 = 0 and using = t/ c periodic function with period: AS-Chap. 3-18
3.3.1 Response to a dc Current Source AS-Chap. 3-19 with and we get for i > 1:
3.3.1 Response to a dc Current Source AS-Chap. 3-20 Current-Voltage Characteristics: Weak damping: c À 1, neglecting noise current:! RC = 1/R N C 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 I c hysteretic IVC
3.3.1 Response to a dc Current Source AS-Chap. 3-21 additional topic: intermediate damping: c» 1 numerically solve IVC increasing McCumber parameter increasing hysteresis decreasing return-current I R I R given by tilt of washboard where: energy dissipated in advancing to next minimum = work done by drive current calculation of I R for c À 1: for I I R : 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 resistive state: minimum junction kinetic energy must be positive E 2E J0 limit I = I R E = 2E J0 energy dissipation: work done by the current source: then: valid for c À 1
AS-Chap. 3-23 Summary plasma frequency: L c /R N time constant: R N C time constant: Stewart-McCumber parameter:
AS-Chap. 3-24 Summary time averaged voltage: strong damping: c 1, neglecting noise current:
3.3.2 Response to a dc Voltage Source AS-Chap. 3-25 phase evolves linearly in time: Josephson current I s oscillates sinusoidally time average of I s is zero I D = 0 since dv/dt = 0 total current carried by normal current IVC: - RCSJ model: ohmic IVC - in more general R = R N (V): nonlinear IVC
3.3.2 Response to ac Driving Sources AS-Chap. 3-26 response to an ac voltage source weak damping, c 1 integrating the voltage-phase relation: current-phase relation: superposition of linearly increasing! dc t = (2¼V dc /F 0 )t and sinusoidally varying phase current I s (t) and ac voltage V 1 have different frequencies origin: nonlinear current-phase relation
3.3.2 Response to ac Driving Sources some mathematics for the analysis of the time-dependent Josephson current: n th order Bessel function of first kind Fourier-Bessel series identity: and: imaginary part with x = w 1 t, b = 2¼V 1 /F 0 w 1 and frequency! dc couples to multiples of the driving frequency AS-Chap. 3-27
3.3.2 Response to ac Driving Sources AS-Chap. 3-28 Shapiro steps: ac voltage results in dc supercurrent if [ ] is time independent amplitude of average dc current for a specific n (step number): for V dc V n : [ ] is time dependent sum of sinusoidally varying terms time average is zero vanishing dc component
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: w 1 /2¼ = 10 GHz constant dc current at V dc = 0 and V n = n ω 1 Φ 0 /2π n 20 ¹V AS-Chap. 3-29
3.3.2 Response to ac Driving Sources response to an ac current source, strong damping, c 1 (experimentally relevant) Kirchhoff s law (neglecting I D ): 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 voltage state for I dc + I 1 > I c complicated dynamics for V n = n w 1 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) AS-Chap. 3-30
3.3.2 Response to ac Driving Sources AS-Chap. 3-31 experimental IVCs obtained for an underdamped and overdamped Niobium Josephson junction under microwave radiation
3.3.4 Photon-Assisted Tunneling superconducting tunnel junction: highly nonlinear R(V), sharp step at V g = 2 /e use 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 cosw 1 t qm phase factor: Bessel function identity for V 1 -term: sum of terms splitting of qp-levels in many levels E qp n~w 1 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 V g n~! 1 /e AS-Chap. 3-32
3.3.4 Photon-Assisted Tunneling qp IVC of a niobium SIS Josephson junction without and with microwave irradiation frequency 2¼w 1 = 230 GHz corresponding to ~w 1 /e ' 950 ¹V Shapiro steps: qp steps: (note that qp steps have no constant voltage and different amplitude) AS-Chap. 3-33
3.4 Additional Topic: Effect of Thermal Fluctuations 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: AS-Chap. 3-34
3.4 Additional Topic: Effect of Thermal Fluctuations AS-Chap. 3-35 normalized momentum: ¾(v,j,t): probability density of finding system at (v,j) at t statistical average of variable X small fluctuations static solution: with: Boltzmann distribution (G = E Fx: total energy, E: free energy) constant probability to find system in n th metastable state: large fluctuations: p can change in time: amount of phase slippage for n+1 À n-1 and! A / n+1 À 1:! A : attempt frequency
3.4 Additional Topic: Effect of Thermal Fluctuations AS-Chap. 3-36 attempt frequency! A : for I = 0:! A plasma frequency! p (oscillation frequency in potential well) for I < I c :! A»! p strong damping ( C =! c RC 1):! p! c (frequency of overdamped oscillator)
3.4.1 Underdamped Junctions: I c Reduction by Premature Switching 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, barrier height 0 for I I c escape probability! A /2¼ for I I c after escape: junction switches to IR N experiment: one measures distribution of escape current I M width ±I and mean reduction h I c i = I c - hi M i use approximation for U 0 (I) and escape rate /! A /2¼ exp(-u 0 (I)/k B T): considerable reduction of I c when k B T > 0.05 E J0 AS-Chap. 3-39
AS-Chap. 3-40 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:
3.4.2 Overdamped Junctions: The Ambegaokar-Halperin Theory Amgegaokar, Halperin: finite amount of phase slippage nonvanishing voltage for I 0 phase slip resistance for strong damping ( c 1), for U 0 = 2E J0 : E J0 /k B T À 1: approximate Bessel function attempt frequency or 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. 3-41
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. 3-42
3.4.2 Overdamped Junctions: The Ambegaokar-Halperin Theory high temperature superconducting grain boundary Josephson junctions grain boundary I V x500 a b V I a b B. Welter, Ph.D. Thesis, TU München (2007) La 2-x Ce x CuO 4 on SrTiO 3 optical lithography AS-Chap. 3-43
AS-Chap. 3-44 3.4.2 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)
3.4.2 Overdamped Junctions: The Ambegaokar-Halperin Theory thermally activated phase slippage rounding of IVC at I I c rounding decreases with increasing 0 ( increasing U 0 ) analytical IVC for strong damping ( c 1): (Ambegaokar, Halperin) for small junctions (L < J): close to T c : measurement of R P at const T I c (B): S. Schuster, R. Gross, et al., Rev. B 48, 16172 (1993) AS-Chap. 3-45
AS-Chap. 3-47 3.5 Voltage State of Extended Josephson Junctions 3.5.1 Negligible Screening Effects neglect self-fields B = B ex (valid for short junctions) junction voltage V = applied voltage V 0 gauge invariant phase difference: Josephson vortices are moving in z-direction with velocity
3.5.2 The Time Dependent Sine-Gordon Equation Long junctions (L >> l J ): effect of Josephson currents has to be taken into account magnetic flux density = sum of external and self-generated field with B = ¹ 0 H and D = ² 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. 3-48
AS-Chap. 3-49 3.5.2 The Time Dependent Sine-Gordon Equation time dependent Sine-Gordon equation with the Swihart velocity: velocity of TEM mode in the junction transmission line example: 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 other form of time-dependent Sine-Gordon equation
AS-Chap. 3-50 3.5.2 The Time Dependent Sine-Gordon Equation time-dependent Sine-Gordon equation: mechanical analogue: chain of mechanical pendula attached to a twistable rubber ribbon Note: short junction w/o magnetic field: 2 j/ z 2 = 0 rigid connection of pendula corresponds to single pendulum
AS-Chap. 3-52 3.5.3 Solutions of the Time Dependent SG Equation 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 = 0 small amplitudes plasma oscillations (oscillation of j around minimum of washboard potential) (ii) long junctions (L À J), solitons solution for infinitely long junction soliton or fluxon: j = ¼ at z = z 0 +v z t goes from 0 to 2¼ for z going from -1 to 1 fluxon (antifluxon: 1 to -1)
3.5.3 Solutions of the Time Dependent SG Equation j = ¼ at z = z 0 +v z t goes from 0 to 2¼ for z from -1 to 1 fluxon (antifluxon: 1 to -1) 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, AS-Chap. 3-53
AS-Chap. 3-54 3.5.3 Solutions of the Time Dependent SG Equation linearized Sine-Gordon equation: Josephson plasma waves let: φ 1 : small deviation approximation: substitution (keeping only linear terms): j 0 solves time independent SG, 2 φ 0 z 2 = sin φ 0/λ J : approximate: j 0 = const (j 0 slowly varying) solution: dispersion relation!(k): Josephson plasma frequency: (small amplitude plasma waves)
AS-Chap. 3-55 3.5.3 Solutions of the Time Dependent SG Equation for! >! p,j : mode propagation pendulum 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
3.5.4 Resonance Phenomena interaction of fluxons/plasma waves with oscillating Josephson current rich variety of interesting resonance phenomena (i) flux-flow steps and the Eck peak for B ext > 0: - spatially modulated Josephson currrent density moves at v z = V/B y t b Josephson current can excite Josephson plasma waves resonance: em waves couple strongly to Josephson current if c = v z corresponding junction voltage: Eck peak at frequency: 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. 3-56
AS-Chap. 3-57 3.5.4 Resonance Phenomena other point of view: - Lorentz force Josephson vortices move at - increase driving force increase v z - maximum possible speed: v z = c step in IVC: flux flow step corresponds to Eck voltage
AS-Chap. 3-58 3.5.4 Resonance Phenomena (ii) Fiske steps standing em waves in junction cavity @ Fiske steps at voltages: for L» 100 ¹m first 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
3.5.4 Resonance Phenomena Fiske steps at small damping and/or small magnetic field Eck peak at medium damping and/or medium magnetic field for voltages V Eck or V n : I s 0 I = I N (V) = V / R N (V) AS-Chap. 3-59
3.5.4 Resonance Phenomena (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/v z, phase change: 4¼ at large bias currents: n fluxons: V n,zfs = n V zfs V n,zfs = 2 Fiske voltage V n (fluxon has to move back and forth) V ffs = V n,zfs for F = n F 0 IVCs of annular Nb/insulator/Pb Josephson junction containing a different number of trapped fluxons Vortex-Cherenkov radiation lecture notes AS-Chap. 3-60
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 circuit: LCR resonator, characteristic frequencies: quality factor: AS-Chap. 3-61
AS-Chap. 3-62 Summary (voltage state of short junctions) IVC: (strong damping, β C 1) driving with Shapiro steps at amplitudes:
AS-Chap. 3-63 Summary (voltage state of long junctions) 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
3.6 Full Quantum Treatment of Josephson Junctions Secondary Quantum Macroscopic Effects AS-Chap. 3-64 so far: Josephson junction was treated classically φ and dφ/dt [ Q = CV = C ħ 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) dφ 2 Q 2 kinetic energy K φ, associated with = 1 dt 2C 2 CV2 (equivalent circuit: charge stored on junction capacitance) 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 simultaneously more precise quantum theory is needed secondary quantum macroscopic effects
3.6.1 Quantum Consequences of the small Junction Capacitance validity of classical treatment: E consider an isolated, low-damping junction, I = 0 cosine potential, depth 2E J0 approx. close to minimum: harmonic oscillator, frequency ω p, level spacing ħω p 2E J0 ħω p ħω p /2 j classical treatment valid as long as E J0 ħω p enter quantum regime by decreasing junction area E J0 E C 1/2 1 (level spacing potential depth) AS-Chap. 3-65
AS-Chap. 3-66 3.6.1 Quantum Consequences of the small Junction Capacitance numbers: (i) area A = 10 µm 2 : - barrier: d = 1 nm, ² = 10 - J c = 100 A/cm 2 E J0 = 3 x 10-21 J - C = ²² 0 A/d ' 9 x 10-13 F E C 1.6 x 10-26 J classical junction (ii) area A ' 0.02 µm 2 : - C ' 1 ff E C E J0 we also need T 100 mk for k B T << E C
AS-Chap. 3-67 3.6.1 Quantum Consequences of the small Junction Capacitance consider a strongly underdamped junction (negligible damping) with dφ/dt 0 kinetic energy: energy due to extra charge Q on one junction electrode due to V total energy: U φ 1 cos φ: K φ φ 2 : potential energy kinetic energy consider E as junction Hamiltonian, rewrite kinetic energy: K = p 2 /2M p = ħ 2e Q
AS-Chap. 3-68 3.6.1 Quantum Consequences of the small Junction Capacitance canonical quantization, operator replacement: with N = Q/2e: # of Cooper pairs: we get the Hamiltonian: describes only Cooper pairs E C = e 2 /2C: charging energy for a single electron charge commutation rules for the operators: N Q/2e: deviation of # of CP in electrodes from equilibrium uncertainty relation:
AS-Chap. 3-69 3.6.1 Quantum Consequences of the small Junction Capacitance Hamiltonian in flux basis φ = ħ 2e φ = Φ 0 2π φ : commutator: φ, Q = iħ Q and φ are canonically conjugate (cf. x and p), deviations from classical description: secondary quantum macroscopic effects
3.6.2 Limiting Cases: The Phase and Charge Regime the phase regime: ħω p E J0, E c E J0 (phase φ is good quantum number) lowest energy levels: localized near bottom of potential wells at φ n = 2π n Taylor series for U φ harmonic oscillator, frequency! p, eigenenergies: E n = ħω p n + 1 2 ground state: narrowly peaked wave function at φ = φ n small phase fluctuations Δφ large fluctuations of Q on electrodes since ΔQ Δφ 2e small E C pairs can easily fluctuate, large ΔQ negligible Δφ classical treatment of phase dynamics is good approximation AS-Chap. 3-70
3.6.2 Limiting Cases: The Phase and Charge Regime the phase regime: ħω p E J0, E c E J0 (phase φ is good quantum number) Hamiltonian: define: a = E E J0 /E C, b = E J0 /2E C and z = φ/2 Mathieu equation: general solution: known from periodic potential problem in solid state physics energy bands Bloch waves: q: charge/pair number variable, q is continuous (cf. charge on capacitor) Ψ is not 2¼-periodic 1-dimensional problem numerical solution AS-Chap. 3-71
3.6.2 Limiting Cases: The Phase and Charge Regime variational approach for approximate ground state trial function for E C E J0 : (choose ¾ to find) minimum energy: first order in E J0 E min 0 for E C E J0 E C E J0 = 0. 1 E min = 0. 1 E J0 tunneling coupling exp 2E J0 E ħω p very small since ħω p E J0 tunneling splitting of low lying states is exponentially small AS-Chap. 3-72
AS-Chap. 3-73 3.6.2 Limiting Cases: The Phase and Charge Regime the charge regime: ħω p E J0, E c E J0 (charge Q is good quantum number) dφ 2 kinetic energy E c dt is 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 Hamiltonian: appropriate trial function: α 1 approximate ground state energy second order in E J0
3.6.2 Limiting Cases: The Phase and Charge Regime the charge regime: ħω p E J0, E c E J0 (charge Q is good quantum number) E C E J0 = 2. 5 E min = 0. 95 E J0 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 AS-Chap. 3-74
Brief Summary AS-Chap. 3-75 classical description valid for: level spacing potential depth the phase regime: ħω p E J0, E c E J0 small phase fluctuations large charge fluctuations the charge regime: ħω p E J0, E c E J0 large phase fluctuations small charge fluctuations
3.6.3 Coulomb and Flux Blockade Coulomb blockade in normal metal tunnel junctions: voltage: V charge: Q = CV, energy: E = Q 2 /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 V CB = V c = e/2c Coulomb blockade thermal fluctuations: E C = e2 > k 2C BT C < e2 (small thermal fluctuations) 2k B T numbers: C 1 ff @ 1 K, for d = 1 nm and e = 5 A 0.02 µm 2 quantum fluctuations: ΔE Δt ħ: finite tunnel resistance τ RC = RC (decay of charge fluctuations) Δt = 2πRC, ΔE = e 2 /2C R h e 2 = R K = 24.6 kω Coulomb blockade in superconducting tunnel junction: for Q2 > k 2C BT, ev (Q = 2e) no flow of Cooper pairs threshold voltage: V V CB = V c = 2e = e 2C C Coulomb blockade charge is fixed, phase is completely smeared out (small quantum fluctuations) AS-Chap. 3-76
3.6.3 Coulomb and Flux Blockade AS-Chap. 3-77 phase or flux blockade in a Josephson junction: current: I flux: Φ = L I, energy: E = Φ 2 /2L phase is blocked due to large E J0 = Φ 0 I c /2π - I c takes the role of V CB - phase change of 2¼ equivalent to flux change change of Φ 0 flux blockade I I FB = I c = Φ 0/2π L c fluctuations, we need: E J0 k B T and cf. charge blockade V V CB = V c = e C analogy: I V, 2e Φ 0 2π, C L ΔE Δt ħ: with Δt = 2πL/R and ΔE = 2E J0 R h (2e) 2 = 1 4 R K
AS-Chap. 3-78 3.6.4 Coherent Charge and Phase States coherent charge states island charge continuously changed by gate Cooper pair box for independent charge states: parabola: E = Q n 2e 2 /2C Σ for E J0 > 0: interaction of n and n + 1 at the crossing points Q = n + 1 2 2e coherent superposition states: splitting of charge energy at crossing points level anti-crossing splitting magnitude Josephson coupling energy E J0
3.6.4 Coherent Charge and Phase States AS-Chap. 3-79 average charge on the island as a function of the applied gate voltage coherent superposition of charge states: experiment by Nakamura, Pashkin, Tsai
3.6.4 Coherent Charge and Phase States AS-Chap. 3-80 coherent superposition states: splitting of charge energy at crossing points level anti-crossing splitting magnitude Josephson coupling energy
3.6.4 Coherent Charge and Phase States coherent phase states interaction of two adjacent phase states e.g. rf-squid magnetic energy of flux φ = Φ 0 2π in the ring φ Φ ext = Φ 0 /2 tunnel coupling: experimental evidence for quantum coherent superposition of states AS-Chap. 3-81
3.6.5 Quantum Fluctuations AS-Chap. 3-82 violation of conservation of energy on small time scales, obey ΔE Δt ħ creation of virtual excitations include Langevin force I F (with adequate statistical properties) fluctuation-dissipation theorem: E ω, T : energy of a quantum oscillator vacuum fluctuations occupation probability of oscillator (Planck distribution) transition from thermal Johnson-Nyquist (ħω, ev k B T) noise to quantum noise: classical limit (ħω, ev k B T): quantum limit (ħω, ev k B T):
3.6.6 Macroscopic Quantum Tunneling AS-Chap. 3-83 escape of the phase particle from minimum of washboard potential by tunneling macroscopic: phase difference is tunneling (collective state) states easily distinguishable competing process: thermal activation low temperatures neglect damping dc-bias: term ħiφ/2e in Hamiltonian curvature at potential minimum: i = I/I c (classical) small oscillation frequency: (attempt frequency)
3.6.6 Macroscopic Quantum Tunneling AS-Chap. 3-84 quantum mechanically: tunnel coupling of bound states to outgoing waves continuum of states but: 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 E 0 = ħω A /2 mass effective barrier height
3.6.6 Macroscopic Quantum Tunneling AS-Chap. 3-85 constant barrier height: escape rate: increasing bias current: very small for small j U 0 decreases with i becomes measurable temperature T * where Γ tunnel = Γ TA exp U 0 k B T for I 0: for I > 0: for ω p 10 11 s 1 T *» 100 mk
AS-Chap. 3-86 3.6.6 Macroscopic Quantum Tunneling additional topic: effect of damping coupling of the system to the environment (heat bath): see e.g. Caldeira & Leggett damping suppresses MQT crossover temperature: for À 1:! R! A lower T * quantum junction: lightly damped classical junction: moderately damped phase diffusion by MQT see lecture notes
Brief Summary AS-Chap. 3-87 Coulomb/flux blockade coherent charge and phase states macroscopic quantum tunneling effects of dissipation
AS-Chap. 3-88 Summary (secondary quantum macroscopic effects) classical description only in the phase regime (large junctions): E C E J0 for E c E J0 : quantum description (negligible damping): phase difference j and Cooper pair number N = Q/2e are canonically conjugate variables: phase regime: charge regime: charge regime at T = 0: tunneling only for flux regime at T = 0: flux motion only for Δφ 0 and ΔN ΔN 0 and Δφ Coulomb blockade: V CB e/c flux blockade: I FB Φ 0 /2πL c at I < I c : escape out of the washboard by: thermal activation tunneling (macroscopic quantum tunneling) crossover between TA and tunneling: