Distinguishing Majorana and Kondo modes in a quantum dot-topological quantum wire setup.

Transcription

1 Distinguishing Majorana and Kondo modes in a quantum dot-topological quantum wire setup. Luis G. Dias da Silva (1) (1) Instituto de Física, Universidade de São Paulo - IFUSP in collaboration with: David Ruiz-Tijerina (1) Edson Vernek (2,3) J. Carlos Egues (3) (2) Instituto de Física, Universidade Federal de Uberlândia (3) Instituto de Física de São Carlos, USP

2 What are Majorana fermions?

3 Majorana Fermions Majorana solution: Representarions of Dirac matrices with only imaginary non-zero elements while still satisfying Real solutions: A Dirac fermion can be written in terms of two Majoranas fermions or

4 Where do we find Majorana fermions?

5 Majorana fermions in condensed matter? Fractional Quantum Hall liquids (ν=5/2): non-abelian anyons. Two-channel Kondo non-fermi-liquid state. Quantum spin systems. Moore and Read, Nucl. Phys. B (1991). Maldacena, Ludwig, Nucl. Phys. B (1997). Zhang, Hewson, Bulla, Solid State Comm (1999). Kitaev, Ann. Phys (2003). Interface of topological insulators with BCS superconductors Fu and Kane, Phys. Rev. Lett (2008). Spin-polarized ( spinless ) p-wave superconductors. Read and Green, Phys. Rev. B (2000). Kitaev, Phys. Usp (2001). Motivation: entanglement of particles with non-abelian statistics= topologically protected quantum computation.

6 1D p-wave superconductor (Kitaev model) P-wave pairing term (spinless fermions) Energy spectrum: Gapped (E + -E - >0): trivial Gappless modes (E=0) : or Gapped: topological ( 0) J. Alicea, Rep. Prog. Phys. 75, (2012)

7 Majorana states in the Kitaev model. Gapped: trivial. Special case: Gapped: topological. Special case: Topological regime: Majorana fermions (e=µ=0!!!) at the edges of the chain!

8 Can the Kitaev model be realized experimentally?

9 How to realize a p-wave SC: Quantum wires. Theory: Lutchyn et al. PRL, 105, (2010); Oreg et al. PRL,105, (2010); Experiment: V. Mourik et al. Science (2012) Experiment: Majorana is found at the ends of a quantum wire

10 Emergent Majorana modes in p-wave superconductors Lutchyn, et al., PRL 105, (2010); Oreg et al., ibid µ t V Z <µ< µ t + V Z Physical realization: = ++ + gap Strong Rashba SO (e.g., in InSb nanowires) Magnetic field (to split the Rashba bands & make the system spinless ) Superconductivity (by proximity)

11 Experiments on InSb nanowires a 300 mk 100 Signatures appear for: S N nd voltage on gate 2 at 175 mt near 5 V (yellow dotted lines). not equally visible everywhere). we compare voltage sweeps on. Temperature is 50 mk. [Note dependence. di/dv versus V at e) and are taken at different 50, and 300 mk). di/dv outside FWHM of 20 mev is measured G [e2 /h] 0 V (µv) N -100 Zero-bia peak in tunneling spectroscopy S 60 mk G (e2/h) di/dv (2e2/h) G (e2/h) 0.3 of EZ (highlighted by orange dotted lines). We have followed these resonances up to high bias voltages in (20) and identified them as Andreev states bound within the gap of the bulk NbTiN superconducting electrodes (~2 mev). In contrast, the ZBP sticks to zero energy over a range Mourik et al., Science 336, (2012) 100 of DB ~ 300 mt centered around ~250 mt. Again 1.2 Deng et al., Nano Lett. 12, 6414 (2012) at ~400 mt, we observe two peaks located at symmetric, finite biases. Das et al., Nature Phys. 8, 887 (2012) To identify the origin of these ZBPs, we need B (mt) Prada et al., Phys. Rev. B 10086, (2012) 40 to consider various options including the Kondo effect, Andreev bound states, weak antilocal0 20 Churchill et al., Phys. Rev. B 87, (2013) V (µv) NATURE PHYSICS DOI: /NPHYS2479 V (µv) SD SD 100 ARTIC ization, and reflectionless tunneling versus a conjecture of Majorana bound states. ZBPs due 100 a bound c to the Kondo effect (24) or Andreev states c Type I b B = 120 mt to s-wave superconductors (25) can occur at B = 100 mt A finite B; however, with changing B, these peaks VLG VGG VRG Au LG then split and move to finite energy. A Kondo B=9 Au RG resonance moves with 2EZ (24), which is easy to1.6 GG Au dismiss as the origin for our ZBP because of the 2 SiO large g factor in InSb. (Note that even a Kondo effect from an impurity with g = 2 would be dis cernible.) Reflectionless tunneling is an benhancetype II ment of Andreev reflection by time-reversed paths in a diffusive normal region (26). AsAuin B = 65 mt LG the case of weak antilocalization, the resulting1.2 Au ZBP is maximal at B = 0 and disappears when RG GG B = 2 Au B is increased; see also (20). We thus conclude 2 SiO that the above options for a ZBP do not provide natural explanations for our observations. We AI Au d VSD (µv) are not aware of any mechanism that could exv(x) plain our observations, besides the conjecture of a Majorana. B = 2EFmT B = 0 mt 0.8 To further investigate the zero-biasness of AI VSD + VAC 200 nm our peak, we measured gate voltage depend25 nm nm 0 nm RG ences. Figure 3A shows a color panel with volt- 200LG Cold ground V (µv) GG age sweeps on gate 2. The main observation is VsdSD(µV ) the occurrence of two oppositefigure types of behav24 A suspended Al InAs nanowire on goldofpedestals above p-type Thetop-type silicon a globald4). gate a, (GG) coated with Figure Low-bias conductance as a function applied magnetic fieldsilicon. parallel the wire axisserves (type as II device, Colour plot. b,c,15c G (e2/h) 150 mt Large enough magnetic field (topological phase) Not too big (that it kills the induced superconductivity) Perpendicular to Rashba SO 50 nm g reproducibility and the abs. We indicate the gap edges een dashed lines (highlighted A pair of resonances crosses AR d G (e2/h) D 0.1 Downloaded from on May slope equal to ~3 mev/t (indicated by sloped yellow dotted lines). Traces in (A) are extracted from (B). NATURE PHYSICS DOI: /NPHYS2479 e 1. us V

12 Experiments on InSb nanowires Skepticism: Tunneling spectroscopy probes the BULK too Solution*: Local probing of the wire ends Possible origins of the zero-bias peak: Localization due to disorder Andreev reflection Kondo effect Nadj Perje et al., Science346, (2014) Fig. 1. Topological superconductivity and Majorana fermions in ferromagnetic atomic chains on a superconductor. (A) SchematicoftheproposalforMQPrealizationanddetection:Aferromagnetic atomic chain is placed on the surface of strongly spin-orbit coupled superconductor and studied using STM. (B) Band structure of a linear suspended Fe chain before introducing spin-orbit coupling or superconductivity.the majority spin-up (red) and minority spin-down (blue) d-bands labeled by azimuthal angular momentum m are split by the exchange interaction J (degeneracy of each band is noted by the number of arrows). a, interatomicdistance.(c) Regimesfortrivial andtopological superconducting

13 Are there ways to make sure one is measuring a Majorana-related resonance?

14 Majorana physics in a non interacting QD R A(ω)= 2 Γ Im(G (ω)) dd lead lead V V QD t 0 Topological superconductor Coupling to a QD allows local probing of the MZM. H = " d d d + d d + H leads + H dot-leads λ, Γ 0.0, , , , , , D " F = " d = V 2 D ω Liu and Baranger, Phys. Rev. B 84, (R) (2011) D

15 Majorana physics in a non interacting QD lead V lead V QD t 0 Topological superconductor Vernek, et al. Phys. Rev. B 89, (2014) Ruiz-Tijerina, et al. Phys. Rev. B 91, (2015)

16 Better way to measure? Liu and Baranger, Phys Rev B (2011). Vernek et al., Phys Rev B (2014). U=0!!! Connect a quantum dot + metallic leads at the end of the nanowire. Measure conductance through the dot 0.5 e 2 /h = signature of the Majorana mode for U=0 What happens for the (common) case of non-zero U??? D. A. Ruiz-Tijerina et al. Phys Rev B (2015).

17 Majoranas + interaction Kondo impurity + Majorana edge states (NRG) R. Zitko, Phys. Rev. B 83, (2011). R. Zitko, P. Simon, Phys. Rev. B 84, (2011). Quantum dot + Kitaev (NRG) M. Lee, et al., Phys. Rev. B 87, (2013). Chirla et al., Phys. Rev. B 90, (2014). Ruiz-Tijerina et al., Phys. Rev. B 91, (2015). Quantum dot + Kitaev (DMRG) Korytár and Schmitteckert, JPCM (2014). Cheng et al., Phys. Rev. X 4, (2014). Interacting Kitaev model (DMRG) Stoudenmire et al., Phys. Rev. B (2011). Thomale et al., Phys. Rev. B (R) (2013).

18 Interacting quantum dot lead Chain edge (a) " 0, + U lead QD... lead lead Quantum fixed wire: adjusted H wire = H TB (µ, t, V Z )+H Rashba ( )+H SC ( ) QD " 0, " F =0 Quantum dot: H dot = X " 0,s n 0,s + Un 0," n 0,# s=",# QD-wire coupling: H dot-wire = t 0 X s=",# h i c 0,s c 1,s + c 1,s c 0,s Topological phase for V Z > p µ Rainis et al., Phys. Rev. B 87, (2013)

19 Iterative Green s function method = N-2 N-1 N-2 g N-2 G N-1 G N-2 lead QD 1 lead G 1

20 Iterative Green s function method = N-2 N-1 N-2 g N-2 G N-1 G N-2 = =0 in the QD and NLs " 0, + U Because of the Coulomb interaction, the NL QD part displays many body correlations (Kondo physics). " F =0 lead QD 1 " 0, We use an approximation based on the Hubbard I method* to obtain the Green s function. lead G 1 J. Hubbard, Proc. Roy. Soc. (London) A276, 238 (1963)

21 Iterative Green s function method " d" =0 " d# =0 " d" = 6.25 " d# = 6.25 " d" =6.25 " d# =6.25 Particle hole symmetry Ruiz-Tijerina, et al. Phys. Rev. B 91, (2015)

22 Iterative Green s function method Ruiz-Tijerina, et al. Phys. Rev. B 91, (2015)

23 Twowire Hubbard The in signature either bands case. appear of This the Majorana in is the a consequ interac zer Shortcomings of the andhi εof Zeeman the approximation ε quantum field V wire Z in the appears wire, in whic thez dot + U (the double occupancya RUIZ-TIJERINA, VERNEK, DIAS DA SILVA, statesand from the ρ spin-up (Fig. EGUES 4), level as an in the additiona QD. to f H? 0.5 field (in V Z,thespin-uplevelintheQD unitsof? 1/πƔ) pinned tor F noninteracting case, this resonance are no a The signature of the Majorana zer to v gate voltage [Figs. 4(a), 4(c), wire andin4( e of the quantum wire appears in thez ρ results for a spinless model Zeeman presen states ρ (Fig. 4), as an additional fth with Ref. [27]. The 0.5 signature from the 0.5 (inunitsof1/πɣ) pinned to case for ε noninteracting 0,s 0 [Figs. 4(b) andf w field V case, this resonance Z f, features are observed in ρ gate voltage [Figs. 4(a), 4(c), (apart Thefro si v and4( bands). However, for ε results for a spinless model 0,s = of U/2[ the qu ρ presen for ε with 0,s < 0, with ε Ref. [27]. The 0.5 0,s Ɣ (notth U states ρ signature appears with a reduced amplitude w m 0.5 (in( FIG. 4. Spin-down local density case of states for εof 0,s the QD 0 [Figs. for the 4(b) wire and negative energies. For V features are observed in Z < 0.4 µev f in nonintera in the topological phase for the same parameters of Fig. 3. ρ Note (apart fro ρ can in fact be completely suppressed 1t. outside QD parameters of the the reduced are Kondo Ɣ = amplitude 1 µev regime gate volt and and t 0 the = shift bands). towardhowever, negative energies for ε 0,s of = the U/2[ e blockade in the dot. Again, results we remfo U central peak in panel (d) for finite Ufor and ε 0,s ε 0, with ε 0,s Ɣ (not of the 0,s < 0. a Hubbard I approximation with Ref th m appears with a reduced amplitude ( FIG. 4. Spin-down local density describe of statesthe of the ground QD for state theof wire case the syst for in negative energies. For V Z < 0.4 µev erical results for in the V (dot) topological Z = 0 phase for the same parameters of Fig. 3. Note featuresρ can in fact be completely suppressed the reduced amplitude and the shift toward negative energies ofbands). the eh w the spin-up and spin-down local DOS blockade in the C. dot. Numerical Again, results we remfo central peak in panel (d) for finite U and ε tively, with the wire in the topological Ruiz-Tijerina, of the 0,s < 0. for ε a et al. Hubbard Phys. Rev. I Bapproximation 0,s < 91, (2015) The only difference between the th al density of states of the QD for the wire e, with t = 10 mev, E SO = 50 µev, V Z = The Hubbard I approximation captures the Majorana physics It doesn t capture the Kondo correlations What if there is a strong Kondo-Majorana interplay? Particle hole symmetry

24 Kondo effect: Anderson model U E F G(T)/ρ(0) ε 0 /Γ=-4 ε 0 /Γ=-3 ε 0 /Γ=-2 ε o ρ(t) T/T K NRG calculations: scaling with T K T.A. Costi, et al., J Phys Cond Mat (1994).

25 Kondo effect: Anderson model Γ=πV 2 ρ(e F ) U E F G(T)/ρ(0) ε 0 /Γ=-4 ε 0 /Γ=-3 ε 0 /Γ=-2 ε o Γ ρ(t) T/T K NRG calculations: scaling with T K T.A. Costi, et al., J Phys Cond Mat (1994).

26 Kondo effect: Anderson model Γ=πV 2 ρ(e F ) U T K E F G(T)/ρ(0) ε 0 /Γ=-4 ε 0 /Γ=-3 ε 0 /Γ=-2 ε o Γ ρ(t) T/T K NRG calculations: scaling with T K T.A. Costi, et al., J Phys Cond Mat (1994).

27 Wilson s NRG: Discretizing the band 1. Slice up the conduction band: log-spaced energy intervals (parameter Λ>1). ρ(ε) 2. Within each energy slice, only one state is taken (still an infinite Hilbert space! 3. Map into a tight binding chain: each site will represent an energy scale. 4. All energy scales are included in an (infinite) chain! t n ~Λ -n/2

28 Renormalization Procedure Iterative numerical procedure. Keep low energy states at each energy scale. Get the spectrum! ξ N... H N γ n ~ ξ n Λ -n/2 H N+1

29 NRG: local density of states ε d +U t γ 1 γ 2 γ 3... ε d γ n ~Λ -n/2 Spectral density: Single-particle peaks at ε d and ε d +U. Many-body peak at the Fermi energy: Kondo resonance (width ~T K ). DM-NRG: very good resolution at low ω. Γ ε d ~T K Γ ε d + U

30 An effective low energy model lead lead QD t 0 Topological superconductor Lee et al., Phys. Rev. B 87, (2013) Effective model: MZM couples directly to the QD spin down (VZ > 0). H e = H dot + H leads + H dot-leads + d # d # µ t V Z <µ< µ t + V Z H dot = X " 0 H leads = X ~k (" d,v (dot) Z )n 0 + Un 0" n 0# " k c ~ k c ~k = ++ + gap H dot-leads = X ~k V~k d c ~k + H. c. E(k)/t For a positive Zeeman splitting V Z, the wire couples only to the QD spin-dn. ka

31 An effective low energy model lead lead QD MZM Lee et al., Phys. Rev. B87, (2013) Effective model: MZM couples directly to the QD spin down (VZ > 0). H e = H dot + H leads + H dot-leads + d # d # µ t V Z <µ< µ t + V Z H dot = X " 0 H leads = X ~k (" d,v (dot) Z )n 0 + Un 0" n 0# " k c ~ k c ~k = ++ + gap H dot-leads = X ~k V~k d c ~k + H. c. E(k)/t For a positive Zeeman splitting V Z, the wire couples only to the QD spin-dn. ka

32 An effective low energy model E. model Full model With the right choice of λ, we reproduce the numerical results for a given t0.

33 Effective model zero energy mode creates a fermion in state α number operator (=0,1) Majorana operators Quantum dot (V Z : Zeeman ; U: e-e interaction) Metallic leads Coupling to the metallic leads Coupling to one Majorana NRG: spectral function and conductance D. A. Ruiz-Tijerina et al. Phys Rev B (2015).

34 The numerical renormalization group D D 2 D.. 3 D " F =0 QD MZM 3 D 2 D D D

35 NRG formulation: quantum numbers Majorana operators Fermion operators OK! not a good QN! However: OK! Build blocks such as: etc,

36 Majorana-Kondo coexistence INTERACTION EFFECTS ON INTERACTION A MAJORANAEFFECTS ZERO MODE ON A MAJORANA... " d" =0 " d# =0 " d" = 6.25 " d# = 6.25 T K T M " d" =6.25 " d# =6.25 In agreement with: FIG. 6. NRGFIG. calculations 7. NRG calculations of the zero-temperature offig. the7. zero-temperature NRG spin-up calculations local spin-down of the zero-temp FI density of states local at density the QD of site, statesinathe theabsence QD localsite, density ofina the magnetic ofabsence statesfield atof the a magnetic QD site, in the local abse [V (dot) Z = 0]. field The[V interacting (dot) Z = 0]. (noninteracting) The field (noninteracting) case [V (dot) Z is presented = 0]. The case ininteracting the is presented (noninteracting) Zeem Lee et al., Phys. Rev. B 87, (2013) Cheng et al., Phys. Rev. X 4, (2014)

37 I. SEPARATING THE KONDO-MAJORANA GROUND to a recent STATE entangled Majorana-Kondo coexistence he results of Figs. 6 and 7 have established that the DOS the ns interacting throughqd near particle-hole symmetry features c. ed Kondo V. Asa and Majorana signatures. According to a recent y [33], the QD spin-down channel is strongly entangled hized the by Majorana the mode and the lead electrons through nref.[30], conservation of the parity Pˆ defined in Sec. V. Asa sequence, 0.5 peak is strongly renormalized by the end on the -lead hybridization Ɣ.ThiswasdemonstratedinRef.[30], re the Majorana energy scale was shown to depend on the ibits ridization Kondo as λ/ Ɣ. he QD spin-up channel, on the other hand, exhibits Kondo processes relations which arise through virtual spin-flip processes spin-down een the lead electrons and the QD spin-up and spin-down ls. gests The persistence that, of the Kondo effect suggests that, pite its entanglement with the Majorana mode, the spinn degree of freedom of the singly occupied QD takes part, the spintakes processes. part It follows that the Kondo temperature the hese rature the th of the zero-bias peak in ρ must be renormalized by Majorana-QD coupling λ. malized n Fig. 9 weby present the dependence of the Kondo (T K )and jorana (T M )energyscalesonλ, asextractednumerically o the (T K density )andof states. The Kondo temperature was ulated as the width at half-maximum of the zero-bias kumerically in ρ.asfort M,theprocesswassomewhatsubtler rature requires some wasclarification. eonsider zero-bias the top curve (squares) in Fig. 9(a),whereλ Ɣ. his case the Kondo temperature is T K 10 2 Ɣ,andthe jorana hatsubtler scale is T M 10 7 Ɣ.Theformerisobtainedfrom (not shown), but for this value of λ it can also be seen in ρ, hown ereλin Fig. Ɣ. 9(b):Withε presented in a logarithmic scale, positive-energy half of the Kondo peak looks simply as a FIG. 9. (Color online) Extracting the Majorana energy scale T M from the spin-down density of states, for three values of λ. The spin-down DOS is shown close to the Fermi level in panel (a). In panel (b) it is shown with the energy axis in logarithmic scale. For T M T K

38 Experimentally distinguishing between Majorana and Kondo QD conductance vs. gate voltage (Vg) QD V g =0 p h symm. V g The Kondo effect is killed Majorana contribution is robust [Vernek et al., PRB 89, (2014)] Lee et al., Phys. Rev. B 87, (2013)

39 Experimentally distinguishing between Majorana and Kondo The Kondo contribution killed by a magnetic field. The Majorana contribution is robust. TM unchanged. QD Conductance vs. magnetic field VZ (dot) V (dot) Z / Universality can be observed experimentally

40 Summary QD DOS: MZM leaks into the interacting QD. T K Results consistent with a Majorana Kondo many body ground state To distinguish Majorana and Kondo: Gate voltage Magnetic field (universality) Phys. Rev. B, 91, (2015) " 0, = 6.25 T M Financial support by: