Last lecture (#4): J vortex. J tr

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1 Last lecture (#4): We completed te discussion of te B-T pase diagram of type- and type- superconductors. n contrast to type-, te type- state as finite resistance unless vortices are pinned by defects. B f vortex Homogeneous state Meissner effect type- sc nomogeneities at isolated points vortices type- sc nomogeneities at weak links osepson effect, SQUDs tr 1

2 Lecture 5: Weak links and te osepson pase relation osepson critical current DC and AC osepson effect wit voltage source (current source given in an appendix) Gauge-invariant pase Quantum interference for weak links Te DC SQUD Applications of SQUDS Oter applications of osepson penomena: frequency mixers and voltage standards Literature: Waldram cs 6 & 18

3 Weak Links " 1 i " e ψ 1 SC 1 V d SC " i " e Bulk superconductors 1 & are separated by a very tin region of normal metal or insulator. A simple model of a 1D superconducting weak link, d<<ξ, and strong link, d>>ξ, is given in te appendix. Here we consider a more general penomenological approac Pase difference ϕ θ 1 θ evolves as (lecture ) " / t ev / 3

4 n GL free energy density and in expression for current we replace "#$ by $ 1 " $ d % exp(i& ) " 1 Te current troug te link is ten of te form sin ϕ n a weak link (i.e., d<<ξ for 1D sc link) te current is periodic in ϕ wit period π. Te current is consistent wit a free energy term of te form ΔF -F 0 cos ϕ, were F 0 ћ /(e). Proof: Power "#F / "t F 0 sin$ "$ / "t V as required. 4

5 Voltage-Biased osepson Weak Links Te osepson Current-Pase Equation Consider te resistively sunted junction (RS): V s n Te total current wit bias voltage V is V s + n sin + R Since V (ћ/e) ϕ / t, we can rewrite in terms of te pase ϕ alone Tis is a strange circuit equation unlike any known in conventional circuit teory and it leads to remarkable -V caracteristics. is known as te osepson critical current of te weak link and is a constant tat depends on te microscopic details of te junctions. Typical values of are in te range 10-6 A to 10 - A. R " sin " + er t O 5

6 Te osepson Current-Voltage Relation From te pase-voltage relation V (ћ/e) ϕ / t, we can write ϕ as an integral over V(t) " " 0 + e t # V(t )dt 0 were ϕ 0 is a constant. Tus, te current-voltage form of te osepson equation becomes $ sin " 0 + e & % t ' # V(t )dt ) + V 0 ( R Consider first te case V 0. Ten n 0 and s sin 0 Current flows witout an applied voltage, i.e., s is indeed a supercurrent flowing troug te weak link. Tis is te DC-osepson effect. 6

7 AC-osepson Effect Te main surprise comes wen we apply a finite voltage. Consider first a DC voltage V. Tis leads to a time-dependent pase ev t 0 + and tus to an oscillatory component in te current V ev sin ( $ + # t) +, were # " V / 0 R 0 f " # V $ 0 ( x 10 8 Hz/1µV) V Remarkably, a DC applied voltage drives an oscillating DC supercurrent at a frequency tat is (1/φ 0 ) per unit of voltage applied. Tis is te AC-osepson Effect. Te DC -V caracteristic of a RS weak link is given on te rigt. is te osepson frequency. s n V/R V 7

8 Combined DC and AC Applied Voltages Now include bot a DC and an AC voltage so tat " " + t ev 0 / sin( were and /. ev Substituting V and ϕ in sinϕ + V/R, we find after some manipulations, using well known armonic expansions* * sin( " sin x) ) & * + ' $ " ",+ ( % V0 V + + cos R R $ # %$ odd sin [# (" )sin( x), t 0 + ( V 0 V0 + V cos( t) + " t) ) t] side band frequencies cos( " sin x) $ # %$ even (" )cos( x) 8

9 Sapiro Spikes Te AC voltage generates a current response at te sideband frequencies ω 0 +νω. Te DC part of te current is just V 0 /R unless te te osepson frequency matces a multiple of te AC frequency, ω 0 + νω 0. n tat case we generate a DC supercurrent s ν (ω /ω ). Tis is known as te inverse AC osepson effect. Even toug tere is a quantum interpretation te potons supply te energy needed to lift a pair across te junction te effect is really more subtle. n particular te DC tends to zero as te power is increased (since ν 0 for all ν). n V 0 /R n te DC -V caracteristic te supercurrent appears V 0 at te so-called Sapiro spikes as sown rigt. "V 0 # e f $ 0 ultra sarp spikes (parts in 10 9 ) 9

10 Gauge nvariant Pase So far we ave ignored te effect of te coupling of te cange to te vector potential. Tis coupling requires tat we look for a gauge invariant form of te pase ϕ. Recall tat to obtain a gauge invariant current we required (lecture ) # " # + ea By integration we arrive at a gauge invariant generalization of te pase ϕ & " (# 1 $ # ) % (# 1 $ # ) $ ( e ' ) + - A, ds * 1 Tis as major consequences for a wide weak link and for two weak links in an applied field. Here we consider two weak links used in te design of a SQUD. 10

11 Macroscopic Quantum nterference Between Two Weak Links: Matter Field nterferometer a a tot (1) flux φ () b b Pase cange ϕ 1 from (1) to () is given in two ways $ $ 1 1 (pat (pat a) b) $ $ a b e # e # A " ds pat a A " ds pat b ϕ a is pase cange across junction a, & ϕ b is pase cange across junction b Since ϕ 1 (pat a) ϕ 1 (pat b), we get " a # " b # e & A $ ds # e % #' % % 0 11

12 a a tot (1) flux φ () b b Define ϕ a + ϕ b ϕ ave, so tat " a " ave # e " b " ave + e Te total current a + b is ten tot sin( ave ) cos' # ( e * ") + " & $ " 0 % sin ave sin( ave + e ") 1

13 Te critical osepson current for te pair of links is c oscillates wit φ wit period equal to te flux quantum φ 0. c c cos( " 0 ) φ 0 Analogy to interference from a pair of Young slits, but now for matter waves instead of ligt waves. Superconducting Quantum nterference Device or SQUD: ig-sensitivity measurements of magnetic fields, voltages and currents in te ft, fv and fa ranges, respectively. φ 13

14 SQUD Applications Te device is igly sensitive: under ideal conditions one can measure a cange of 10-6 φ 0 / Hz. SQUDs are used as precision magnetometers in te examples below: Scanning SQUD microscopy for exotic experiments on ig-t c s (more later ) Magnetoencepalograpy measures tiny magnetic fields (ft range) created by active areas in te brain Picture credits:. R. Kirtley; 4-D Magnetic properties measurement system for susceptibility measurements etc. can detect moments down to ~10-13 Am 14 Neuroimaging; Quantum Design

15 Oter Applications of te osepson Effect Te exactness of te osepson frequency-voltage relation ω ev/ as led to te adoption of osepson junction arrays as te primary voltage standard: an incident field is tuned to matc te Sapiro steps of te array. Frequency can be measured igly accurately, and te Sapiro steps are extremely sarp, giving a relative voltage uncertainty of 1 part in Tis is one out of several quantum standards tat ave revolutionized metrology, te quantum Hall effect resistance standard being anoter prominent example. Some oter applications include microwave detectors and frequency mixers exploiting te strong nonlinearity and te sideband generation of te weak link, respectively. A lot of researc effort is presently going into developing superconducting transistors and quantum computers using superconducting qubits based on circulating currents and enclosed flux in weak link circuits or non-analytic anyons in exotic pairing states (more later ). 15

16 Appendix 1: Sort Quasi-1D Superconducting Weak Link ψ(0) 1 Assume β α : 0 d for a weak link, 0<x<d<<ξ, te first GL equation conditions " 1 # x # i (1 # e d Te current is proportional to $ sin m (" *" #) d For a strong link, 0 < x < d >> ξ, te first GL equation gives ψ (1 - ψ ) 0 ψ exp(-iϕx/d) ) " ( d) e # i # " '' + "(1 " ) 0 reduces to ψ 0 ψ a + bx so tat from boundary x so tat $ m( " *" #) (instead of sin ) d d 16

17 Appendix : Simplified Treatment of Combined DC and AC Applied Voltages S sin [ t + sin ( 0 B + Te term (cos A) B is proportional to cos( 1 A 0 t) sin( sin[( ) t] + Tus, we will get a DC osepson effect wen + t) 0 V e [ sin A cos B + cos A sin B] B 1 t)] sin[( becomes DC if if ϕ 0 0 " 0 ) t] 0 ev 17

18 Appendix 3: Current Biased Weak Links n practice, it is usually te current rater tan te voltage tat is controlled in a weak link. We need to invert te osepson pase equation. Suppose first tat we apply a DC current 0 to te RS. Ten from p. 5 te pase ϕ is given by # 0 " sin er # t f 0, te pase reaces an equilibrium value given by ϕ / t 0, i.e., 0 sin ϕ equil. Note tat ϕ / t 0 means V 0. f 0 > suc an equilibrium is not possible and ϕ keeps canging wit time. 0 : equilibrium ϕ ϕ 0 > : rolling, rolling, rolling 18

19 t can be sown tat for a given 0 te pase ϕ satisfies tan " V 0 t 0 tan + R were V 0 is te mean voltage across te link defined by V / R 0 Tis gives te current-bias -V caracteristic sown on te rigt. V 0 Te -V caracteristic under current bias and under voltage bias are terefore quite different. Generally, te weak link equations ave to be integrated numerically. 19