Current noise in topological Josephson junctions

Transcription

1 Current noise in topological Josephson junctions Julia S. Meyer with Driss Badiane, Leonid Glazman, and Manuel Houzet INT Seattle Workshop on Quantum Noise May 29, 2013

2 Motivation active search for Majorana fermions in superconducting hybrid structures proposed signatures: zero-energy bound state (visible in tunneling DoS) fractional AC Josephson effect? study properties of a biased topological Josephson junction 2

3 Topological Phases BEST KNOWN EXAMPLE: QH system NOVEL ASPECTS: time-reversal symmetry strong spin-orbit interaction band inversion Example: HgTe/CdTe quantum wells Koenig et al., Science 421, 766 (2007) ) E(k) (ev) ) E(k) (ev) [also InAs/GaSb Knez, Du & Sullivan, PRL 107, (2011)] k (A 1 ) X-L Qi & S-C Zhang, RMP 83, 1057 (2011) k (A 1 ) 3

4 2D Topological Insulator (QSH): Helical Edge States ) E(k) (ev) effective low-energy Hamiltonian: (Dirac fermions) k (A 1 ) Gap Vacuum TI Vacuum Gap > 0 Gap < 0 Gap > 0 Position 4

5 2D Topological Insulator (QSH): Helical Edge States ) E(k) (ev) effective low-energy Hamiltonian: (Dirac fermions) k (A 1 ) R 14,23 / Ω G = 0.01 e 2 /h T = 30 mk R 14,23 / kω T = 1.8 K T = 0.03 K G = 2 e 2 /h G = 0.3 e 2 (V g V thr ) / V /h 10 4 G = 2 e 2 /h (V g V thr ) / V Koenig et al

6 1D Topological Superconductor: Majorana Fermions topological superconductor: 1D spinless p-wave edge state = Majorana fermion Oreg, Refael & von Oppen, PRL (2010) Lutchyn, Sau & Das Sarma, PRL (2010) Alicea, Rep. Prog. Phys. (2012) (b) E Kitaev, Phys. Usp. (2001) E so µ Possible realizations: 2D topological insulator (QSH) + conventional superconductor nanowire with strong spin-orbit coupling & Zeeman field + conventional superconductor proximity effect topological superconductivity 6 k

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8 Fractional AC Josephson Effect? Setup V Superconductor Superconductor 2D Topological Insulator 8

9 Fractional AC Josephson Effect? Setup V Superconductor Superconductor 9

10 Andreev / Majorana Bound States Superconductor φ 1 Superconductor φ 2 Hamiltonian: x x L φ x x L M x x L 10

11 Andreev / Majorana Bound States Superconductor φ 1 Superconductor φ 2 Ε φ 1.0 M = 0: bound state backscattering? potential barrier magnetic barrier spin-flip scattering Ε φ φ φ bound state where D transmission probability For comparison: Ε φ φ S-N-S with potential barrier 11

12 DC Josephson Effect Josephson current carried by ABS: Ε φ φ 1.0 equilibrium Josephson current: where ε < is the energy of the ABS below the Fermi level 2π-periodic 12

13 DC Josephson Effect Josephson current carried by ABS: Ε φ φ 1.0 non-equilibrium Josephson current: δφ 2eV dc t in the absence of inelastic processes fractional Josephson effect at frequency ev dc = ω J /2? only even Shapiro steps? Kitaev, Ann. Phys. (2003) Fu & Kane, PRB (2009) for ev dc = kω (instead of 2eV dc = kω in S-N-S) 13

14 Outline Andreev Bound States: S-N-S vs S-TI-S Josephson Junctions Fractional AC Josephson effect? Effective Model: Bound State Dynamics Characteristic Time Scales Average current & Noise Conclusions I Scattering Theory Noise: comparison with the effective model Additional signatures MAR features Conclusions II NOTE: δϕ = ϕ = χ V = V dc 14

15 Main Result 2 characteristic time scales: Ø lifetime of the Majorana bound state τ s due to dynamic coupling to the continuum Ø phase adjustment time τ R due to the resistive environment provided by the circuit observable signatures: even/odd effect in Shapiro steps for τ s À τ R more robust: peak in finite-frequency noise S(ω) at ω = ev dc (half of the usual Josephson frequency) 15

16 Effective Model 1. switching probability: Landau-Zener 2. bound state dynamics: Markov process 16

17 Switching Probability: Landau-Zener 2 Π 4 Π 6 Π 8 Π 10 Π 12 Π 14 Π 16 Π 18 Π 20 Π 22 Π 24 Π 26 Π 28 Π 30 Π at finite bias: ϕ(t) = 2e s t V(t ) dt dynamical coupling between the bound state and the continuum occupation of the bound state may switch 17

18 Switching Probability: Landau-Zener 2 Π 4 Π 6 Π 8 Π 10 Π 12 Π 14 Π 16 Π 18 Π 20 Π 22 Π 24 Π 26 Π 28 Π 30 Π x Hamiltonian : φ x x L x L consider weak backscattering: R = 1 D 1 where R ~ (ML/v) 2 Ε Χ 1 M x x L 2 Π 2 Π Χ t 1 18

19 Switching Probability at R 1 Ε Χ 1 consider only states with energy near Δ: ϕ 1 vp Δ 2 Π 2 Π Χ t 1 apply unitary transformation & diagonalize particle-hole space + bound state similar to Shiba state created by a magnetic impurity in a superconductor Shiba 1968, Rusinov

20 Switching Probability at R 1 + time-dependent phase: ϕ = 2eV dc t use adiabatic basis Ε Χ two-band version of transition from discrete state to continuum Demkov-Osherov (1967) Π 10 Π 5 Π 10 0 Π 10 Π 5 3 Π 10 Χ t switching probability 20

21 Switching Probability at R 1 Ε Χ adiabaticity parameter: 3 Π 10 Π 5 Π Π 10 Π 5 3 Π 10 Χ t switching probability time scale for the transition: 21

22 Switching Probability at R s Λ 23 s s 0.93 Λ 54 2 Λ3 e ΛR 32 ev dc M. Houzet, JSM, D.M. Badiane & L.I. Glazman, arxiv:

23 Bound State Dynamics: Markov Process 2 Π 4 Π 6 Π 8 Π 10 Π 12 Π 14 Π 16 Π 18 Π 20 Π 22 Π 24 Π 26 Π 28 Π 30 Π n t 0 Ε t I t Χ t 2Π Χ t 2Π Χ t 2Π 23

24 Bound State Dynamics: Markov Process P probability of fermionic state to be occupied Q = 1 P probability of fermionic state to be empty s switching probability per half-period 1 s s 4n¼ 4 Π (4n+2)¼ 6 Π s 1 s Χ t current: where similar models have been used for Landau-Zener transitions near avoided crossings in conventional junctions Averin et al. 1995, Pikulin et al. 2011, Sau et al

25 Realistic Circuit: Time Scales P 2(n+k) = P 2n +(1 s) 2k (P 2n P 2n), P 2(n+k)+1 = P 2n+1 +(1 s)2k+1 (P 2n P 2n ). with P n 1 = [1 ( 1) n s/(2 s)]/2 Markov process defines characteristic time scale τ s = 2π / [ev dc ln(1 s)] ¼ 2π / (sev dc ) for s 1 resistive environment provided by the circuit sets time scale τ R for adjusting the phase across the junction 25

26 RSJ Model consider a voltage-biased Josephson junction in series with an external resistance R where V (t) =V dc + V ac cos (Ωt) V(t) rewrite I S (t) =±I 0 sin (ϕ(t)/2) ) and introduce the varia with ϕ(t) =kωt + 2eV ac sin (Ωt)+φ(t) Ω for 2eV dc ¼ kω, one finds 26

27 RSJ Model characteristic time scale: R =1/(eRI k) with I k = I 0 J k/2 (ev ac /Ω) τ (k) if τ (k) R τ s, parity conserved during formation of step: only even Shapiro steps! even-odd effect shape of the step: I dc = k δv k R 1 θ 1 2 RIk 1 δv k 2 RIk δv k where δv k = V dc kω/(2e) and k even 27

28 τ s τ R : Average Current non-adiabatic processes important average current: determined by long-time probabilities independent of initial state & 4¼-periodic 2¼-periodic strongly suppressed for s 1 NOTE: for V ac V dc & Ω δ, dependence of s on ac bias negligible Shapiro steps suppressed 28

29 τ s τ R : Noise determined by correlator where n i = Int[ϕ i /2¼] 29

30 Noise DC Bias S(ω) = 4sI2 J π(2 s) (ev dc ) 3 4 cos 2 πω [ω 2 (ev dc ) 2 ] 2 4 cos 2 πω 2eV dc 2eV dc + s2 1 s finite-frequency noise: peak at! = ± ev dc for! ¼ ± ev dc & s 1 peak width characteristic time scale τ s similar results for finite-size Majorana nanowires equivalently: fractional AC effect in transient regime Pikulin & Nazarov 2011 San-José, Prada & Aguado

31 Noise DC+AC Bias noise peak shifted to lower (more accessible?) frequencies under microwave irradiation: 31

32 Conclusions I signatures of Majorana fermions are subtle Non-equilibrium properties of S-TI-S junctions: interplay between 2 time scales: Ø lifetime of the Majorana bound state τ s Ø phase adjustment time τ R even/odd effect in Shapiro steps for τ s À τ R clear signatures of fractional Josephson effect in finite-frequency noise D.M. Badiane, M. Houzet & JSM, PRL 107, (2011) M. Houzet, JSM, D.M. Badiane & L.I. Glazman, arxiv:

33 Alternative Approach: Scattering Theory 33

34 I-V characteristics: Multiple Andreev Reflections Superconductor Superconductor electrons/holes gain/lose energy ev dc when traversing the barrier only quasi-particles with energy ε > Δ can escape into reservoir 2eV dc Octavio, Blonder, Tinkham & Klapwijk

35 Scattering Formalism Superconductor Superconductor 35

36 Scattering Formalism Superconductor Superconductor Andreev reflection scattering matrix Andreev reflection a Ε 1 e h e h Andreev reflection: scattering matrix for electrons: Ε + applied voltage: scattering matrix for holes: 36

37 Scattering Formalism Superconductor Superconductor incoming electron from the left e h e h derive recursion relations for coefficients A n, B n obtain expression for current: where D Averin & A Bardas, PRL (1995) 37

38 Noise compute S(ω) = e2 2h ±ω p dɛ dɛ J(ɛ)J(ɛ ) { [ ( δ(ɛ ɛ ± ω +2peV ) n (f(ɛ)[1 f(ɛ )] + f(ɛ )[1 f(ɛ)]) (9) [ ] 2 A n+pa n + a 2(n+p) a 2nĀ n+pā n (1 + a 2(n+p) a 2n)Bn+pB n + (1 + a 2(n+p)+1 a 2n+1 ) ( Cn+p C n ) 2 ) D n+p D n n +2δ(ɛ ɛ [ ] 2 ] ± ω +(2p 1)eV ) A n+p C n + a 2(n+p) a 2n+1Ā n+p C n (1 + a 2(n+p) a 2n+1 )B n+p D n n +(f(ɛ)f(ɛ ) + [1 f(ɛ)][1 f(ɛ )]) [ δ(ɛ + ɛ [ ± ω +2peV ) Ap n B n B p na n a 2(p n)a 2n (Āp n B n B )] 2 p nā n n +δ(ɛ + ɛ 2 ± ω +2(p +1)eV ) (1 a 2(p n)+1 a 2n+1 )(D p nc n C p nd n ) n +2δ(ɛ + ɛ [ ± ω +(2p +1)eV ) Ap n D n a 2(p n)a 2n+1Āp nd n ( 1 a 2(p n) a ) 2n+1 Bp n C ] 2 ]} n. n 38

39 1(V 0) 0andI 1 (D 0) 0 n ac Josephson effect? Results: Noise noise:probeof4π-periodicity current noise: {δî(t), δî(t+τ)}, I(t), and the bar deg. as a consequence of ak due to dynamical coupling between the bound state peak at ω = ±ev dc signature of the fractional Josephson effect Badiane, Houzet & JSM

40 Comparison: Effective Model vs Numerical Results effective model in the limit R 1 and τ R 1 : S(ω) = 4sI2 J π(2 s) (ev dc ) 3 4 cos 2 πω [ω 2 (ev dc ) 2 ] 2 4 cos 2 πω 2eV dc 2eV dc + s2 1 s fit numerical results with s as a free parameter SΩ 4eI Ω MAR numerical calculation Fit D = 0.9 & ev dc = 0.025Δ D = 0.9 & ev dc = 0.05Δ SΩ 4eI Ω MAR numerical calculation Fit 40

41 Comparison: Effective Model vs Numerical Results compare with s(λ) (Demkov-Osherov): 1.0 s , D 0.85, D 0.90, D Λ R 3 2 ev dc 41

42 Additional Signatures: DC Current I 0 / (G N /e) I 0 / (G T /e!) D = 1.00 D = 0.99 D = 0.90 D = 0.70 D = 0.50 D = 0.30 D = 0.10 D = 0.01 G G T 2.e 2 N = De/h.D 2 /h For comparison: S-N-S I 0 / (G N /e) I 0 / (G T /e!) D = 1.00 D = 0.99 D = 0.90 D = 0.70 D = 0.50 D = 0.30 D = 0.10 D = 0.01 G T = 2.e 2 /h.d G N = 2De 2 /h ev/! ev = 2Δ ev/! midgap resonance Transparent junction (D = 1): I S-TI-S = 1/2 I S-N-S Tunnel junction (D 0): current onset at ev dc = Δ 42

43 Additional Signatures: DC Current G = e 2 /h subgap MAR structures at nev dc = Δ (vs nev dc = 2Δ for S-N-S) 0 n = 2 n = 3 Log( I/(G/e!) ) D = 0.90 D = 0.70 D = 0.50 D = 0.30 D = 0.10 D = !/eV 43

44 Conclusions II signatures of Majorana fermions are subtle Non-equilibrium properties of S-TI-S junctions: even/odd asymmetry in Shapiro steps for τ s τ R clear signatures of fractional Josephson effect in finite-frequency noise DC current subgap structures due to multiple Andreev reflections at ev dc = Δ/n tunneling limit: current onset at ev dc = Δ D.M. Badiane, M. Houzet & JSM, PRL 107, (2011) M. Houzet, JSM, D.M. Badiane & L.I. Glazman, arxiv:

45 Thank you! 45