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Superconductive Electronics Lecture Overview • • • • • Superconductors - basic principles Josephson junctions Rapid Single-Flux Quantum (RSFQ) circuits Reversible parametric quantron Superconducting quantum computers Principles of Superconductivity • Fermions & Bosons • Coherent bosonic systems • Cooper pairs & BCS theory Particle Exchange • Consider a quantum state of two identical particles in single-particle states x and y, respectively. – Amplitude given by wavefunction (x,y) • Imagine any physical process (descibed by a unitary matrix U) whose effect is just to exchange the locations of the two particles. – Because two such swaps gives the identical quantum state, UU=1 (identity operation), – One swap U must multiply the state vector by 1. – There are only two square roots of 1: Namely, 1 and 1. • Now, what happens if x=y? Fermions & Bosons • Fermions are simply those particles such that, when they are swapped, the state vector is multiplied by 1. – (y,x)= (x,y), so if x=y then (y,x) = (x,x) = (x,x) – But (x,x)(x,x) unless (x,x)=0, • so, there is always 0 probability for two fermions to be in the same state x. (Pauli exclusion principle.) – Examples of some fundamental fermions: • Electrons, Quarks, Neutrinos • Bosons are those particles that when swapped, multiply the state vector by 1. – The quantum statistics of Bosons turns out actually to give a statistical preference for them to occupy the same state. – Examples of some fundamental bosons: • Photons, W bosons, Gluons Compound Bosons • Note that exchanging two identical pairs of fermions multiplies the state vector by (1)2 = 1. • Two identical systems that each contain an even number of fermions behave like bosons. • If they contain an odd number of fermions, they behave like fermions. – Protons, Neutrons (3 quarks each) are fermions – Atoms w. an even number of neutrons are bosons • n protons + n electrons + 2k neutrons = even # of fermions = boson Coherent Bosonic Condensates • Large numbers of bosons can occupy the same quantum state and form a large, many-particle system having a definite quantum state. • Three (more or less) familiar examples: – Laser beams - Bosonic condensates of photons. – Supercurrents - Bosonic condensates of “Cooper pairs” of conduction electrons. – Bose-Einstein condensates - E.g. In 1995 Cornell & Wieman cooled large numbers of 87Rb atoms (37 protons + 50 neutrons + 37 electrons = boson) to a single quantum state at a temperature of ~20 nK. History of Superconductivity • Discovered by Kammerlingh-Onnes in 1911: In solid mercury below 4.2 K resistance is 0! – Superconducting loop currents can persist for years. • Meissner & Oschenfeld discovered in 1933 that superconductors exclude magnetic fields. – Induced countercurrent sets up an opposing field. • Electron-atom interactions shown to be involved in 1950. • Bardeen, Cooper, & Schrieffer proposed a working theory of superconductivity in 1957 – BCS theory. Electron-Lattice Interactions • Electron moving through lattice exerts an attractive force on nearby + ions. – Causes a lattice deformation & local concentration of + charge. • Positively charged “phonon” (quantum of lattice distortion) propagates as particle/wave in “wake” of electron. – Later, phonon may be absorbed by a 2nd electron. Cooper Pairs • Two electrons exert a net attractive force on each other due to the exchange of + phonons to which they are both attracted. – Repulsive below some distance. – Typical separation: ~1 m • Binding energy of pair = ~3kBTc – Tc is critical superconducting temperature • Note that phonon exchange doesn’t change total momentum of pair. Multiple overlapping pairs • The lowest-energy state is when each electron is paired with the maximum number of neighbors. • Most favored when all pairs have same total momentum. - Wavefunctions in phase • As a result, each electron’s momentum is “locked” to its neighbors. – All of the pairs move together. • 3kBTc energy to break a given Cooper pair. – This energy not thermally available if T<<Tc. Josephson Junctions Insulator (thin) • Structure very simple: – Thin insulator between two superconductors. • Current-controlled switch: – Cooper pair wavefunctions tunnel ballistically I through the barrier. ~10Å Superconductive metal • below critical current Ic • Hysteretic I-V curve: – After current exceeds Ic, resistance stays “high” • Till I drops back to 0. Ic Device has built-in “memory” 1.5 ps switching speed V Leftovers from Last Lecture • Most superconducting devices require very low (<5K temperatures). – However, “high-temperature” superconductors were discovered in the 1980’s • Tc ranging from ~90-130 K (compare 0°C = ~273 K). • Electron pairing mechanism not well understood – High-temperature Josephson junctions have also been proposed • 77K, liquid-N temp. deemed feasible (Braginski 1991) • Discussion of BCS mechanism was very oversimplified – see van Duzer & Turner for details Microstrip Transmission Lines • Nice features: – – – – Short (ps) waveforms Near c speeds Low attenuation & dispersion Dense layout with low crosstalk • JJs can be impedance-matched w. TLs – avoids wave reflection off of junction – permits ballistic wave transfer – 10 can be obtained, w. V < 3 mV • Resistive state P = V2/R < 1 W. • 100 Mjunctions 100 W? Overview of JJ Logics • Voltage-state logics – IBM project in 1970s • primarily dealt with magnetically-coupled gates – Resistor-junction logic families (Japan, 1980s) • RCJL (Resistor-Coupled Josephson Logic) • 4JL (four-junction logic) • MVTL (Modified Variable Threshold Logic) – also used inductors & magnetic coupling – These technologies not found to be practical... • Better: Single-Flux-Quantum (SFQ) logics – Encode bits using single quanta of magnetic flux! V (t ) dt = 0 : h/2qe = ~2 mV·ps A Simple Element: Current Latch • Bias current Ib slightly less than JJ critical Ic • Incoming current pulse Iin(t) – pushes JJ current over Ic, JJ switches to “off” state • Part of bias current shunted into output TL • JJ hysteresis means Iout is latched in high state Ib < I c current Iout Iin Iout (Ic) Iin time How to turn JJ back on? Overdamped Josephson Junctions • Place resistor in parallel w. JJ – Brings junction current back below Ic when input pulse goes away – Restores junction back to “on” state waiting for another pulse – Iout becomes another pulse similar to input pulse • Switching speeds up to 770 GHz have been measured! current – Voltage-state JJ logics were limited to 1 GHz • Were not competitive with modern CMOS Iin time Iout Problems w. superconductors • Typical logic gates complex, hard to understand – Simpler gates might yet be discovered • Low temperatures increase total free-energy loss for a given signal energy dissipated – E.g. T=5 K: 60x worse than @ 300 K • Superconducting effect may go away in nanoscale wires (10 nm or less) – Cooper pairs too big to fit – Seems true for metal-based superconductors – But, other nanoscale structures may take over! • Superconductivity has been shown @<20K in carbon nanotubes (Sheng, Tang et al. ‘01) • Low temperatures imply lower maximum clock frequencies, by Margolus-Levitin bound. – E.g., 5 K circuits limited to a 300 GHz average frequency of nbops