Studying Topological Insulators. Roni Ilan UC Berkeley

Transcription

1 Studying Topological Insulators via time reversal symmetry breaking and proximity effect Roni Ilan UC Berkeley Joel Moore, Jens Bardarson, Jerome Cayssol, Heung-Sun Sim

2 Topological phases Insulating phases with metallic surface states. Bulk is insulating but non-trivial! Properties/existence of surface states protected by bulk properties. Topological invariants of the bulk are immune to continuous deformations.

3 Probes of topological properties Many features are attributed to surface states as mirrors of topological properties of the bulk How can we observe this in experiment?

4 Topological phases and symmetry Some topological phases require no symmetry: Quantum Hall effect Bulk topological = xy = Chern number invariant Chiral edge modes protected due to spatial separation

5 Topological phases and symmetry TI surface states are protected by time reversal symmetry Quantum Spin Hall effect (2DTI) An opportunity to study surface states by breaking that symmetry

6 Surface states of 2 and 3D TI s 2D 3D Spin-momentum locking Dirac fermions Figure from Y. Ando J. Phys. Soc. Jpn. 82, (2013)

7 Time reversal symmetry Half integral spin T 2 = 1 [H, T] =0 i T i ht i = ht 2 T i = h T i = ht i Absence of backscattering h V T i =0

8 Experimental detection: 2DTI CdTe HgTe CdTe I. Trivial II-IV. Non trivial! L=20 microns (L,W) =(1,1) micron (L,W) = (1,0.5) micron! [Konig et al 07]

9 Experimental detection: 3DTI Bi2X3 Figure from Moore and Hasan, Annu. Rev. Condens. Matter Phys Experiments by Xia et al. 2008, 2009 and Hsieh et al. 2009

10 Interesting questions 2D TI: Surface states are 1D. G =2e 2 /h Helical states.!! Evidence of luttinger liquid physics? Impurity physics? Exact solutions for non-equilibrium transport?

11 Interesting questions 3D TI: Surface states are 2D. Dirac Fermions.! Transport? Topological effects? Majorana fermion physics?

12 Luttinger liquid physics on the edge of a 2DTI Does 2e 2 h mean edge is non-interacting? No! contact contact QSHE G = g e2 h QHE contact contact G = e2 h [Maslov and Stone 95, Safi and Schulz 95,97; Furusaki and Nagaosa 96; Oreg and Finkel stein 96; Alekseev et al. 96; Egger and Grabert Wen 90, Milliken et al 96]

13 Luttinger liquid physics on the edge of a 2DTI To observe interaction effects we must have backscattering. Need to break time reversal symmetry: Local Magnetic impurity Local breaking of TR Global magnetic field + impurity [Maciejko et al 09; Tanaka et al 11; Beri and Cooper 12; Soori et al. 12; Timm. 12, Lezmy Oreg and Berkooz, 12]

14 Goal To find a controlled way to induce local backscattering Use the special properties of 2DTI to obtain exact solution for noneq. conductance

15 designing a tunable impurity Point contact in the QHE Point contact in the QSHE?

16 magnetic field Breaks time reversal symmetry Rashba spin orbit coupling H = i~v F x i~ 2 x} y H = i~v F x + µ B g e 2 ~ B ~ E 2 =(vp h z ) 2 + h 2? E h z E g =2h? v! v = p 2 + v 2 h z p E g =2h? = tan 1 ( /v F )

17 tunable impurity H = i~v F x + h z i~ =0 6= 0 =0 2 x} y E g =2 h/v F Rashba SO controllable by gate [Nitta et al 97;Hinz et al 06]

18 Non interacting electrons (x) = 8 >< Re iprx + r L e ip Lx x<0 a + + e >: ip +x + a e ip x 0 <x<d t R e ip Rx x>d E/E R R (h /v 2 F ) M/E 0 h/(~v F /d)

19 Interacting edge H 0 = iv( x " x # ) H int = u 2 " " # # + u 4 h ( " ") 2 +( # #) 2 i H BS = L (0) R + h.c "= R #= L Helicity reduces the number of degrees of freedom by locking the spin to the momentum

20 Interacting edge Maps to a spinless Luttinger Liquid H = v 4 g ((@ x R) 2 +(@ x L )) 2 H BS = cos( R L ) H BS = 1 P d ne 2 n in R(0) e in L(0) + h.c. d` =(1 n2 g) 1/4 <g<1 One relevant term G(T ) e2 h h 1 T BT 2(1 g) i T B /T! 0 G(T ) e2 h 2(1/g 1) T T T B /T!1 B T B 1/(1 g) [Kane and Fisher 92, Kane and Teo 09, and many others]

21 Spinless luttinger liquid with impurity R L o (x, t) = 1 p 2 ( L (x, t) R( x, t)) e (x, t) = 1 p 2 ( L (x, t)+ R ( x, t)) R(x, t) o (x + t) L(x, t) o ( x + t) H SG = 1 8 g R 1 0 [ 2 +(@ x o ) 2 ]+ cos o (0) Integrable model, can be solved exactly [Fendley, Ludwig, Saleur 95; Fendley, Lesage, Saleur 96]

22 Integrability [Ghoshal and Zamolodchikov 94; Fandley Ludwig and Saleur 95; Fendley Lesage and Saleur 96] Infinite set of conserved quantities with impurity H SG = 1 8 g R 1 0 [ 2 +(@ x o ) 2 ]+ cos o (0) find quasiparticle basis Additive energies One body S matrices Two body S matrices in the bulk H = v P i p i Particles scatter one-by-one Particles scatter elastically off each other Does not mean scattering is trivial! Permute quantum numbers, phase shifts f j = 1 1+e µ j /T + j j ( ) TBA equations for

23 EXACT SOLUTION AT g =1 1/m I(V,T B,T)= T (m 1) 2 Z cosh 2 [ d 1+e (m 1)V/2T ln(t B /T )] ln + ( ) 1+e (m 1)V/2T +( ) 1+e (m 1)V/2T +(1) 1+e (m 1)V/2T +(1) G(V,T B,T)= di BS dv = G(V/2T,T B/T ) G(V/2T,0) = g Include contact correction V = 1 1 I(W )+W g [Fendley Lesage and Saleur 96, Koutouza, Siano and Saleur 01]

24 Conductance vs. temperature G(T ) e2 h h 1 T BT 2(1 g) i g =1 1 m G/(e 2 /h) m V/T = T B /T G(T ) e2 h 2(1/g 1) T T B

25 VOLTAGE DEPENDENCE With FL contact G/G m G/G m T B /T = V/T 0.2 T B /T = V/T W/O FL contact (shifted down)

26 Conclusion A setup for transport experiment on the QSHE edge: controlled TR symmetry breaking + controlled SO coupling = point contact Interacting edge theory is integrable model: Exact solutions for conductance with FL leads From experiment: estimate LL parameter Confirm integrability Confirm effects of leads

27 Perfect transmission and Majorana fermions in 3DTI Detect a perfectly transmitted mode and signature of Majorana fermions

28 Problem with current setups Large number of modes and finite chemical potential Large contribution to transport from bulk states

29 Dirac fermion on a curved surface- TI nanowires x (x, y + W )=e i (x, y) [Ando and Suzuura 2002, Ran, Vishwanath, and Lee 2008, Rosenberg, Guo, and Franz 2010, Ostrovsky, Gornyi, and Mirlin 2010, Bardarson and Moore 2013]

30 Dirac fermion on a curved surface- TI nanowires x (x, y + W )=e i +i2 / 0 (x, y) odd number of modes = 0 /2 T 2 = 1 S T = S Perfectly transmitted mode [Ando and Suzuura 2002, Ran, Vishwanath, and Lee 2008, Rosenberg, Guo, and Franz 2010, Ostrovsky, Gornyi, and Mirlin 2010, Bardarson and Moore 2013]

31 Proximity effect - Josephson Junctions Whereas bulk transport tends to obscure the observation of surface state transport in the normal state, the supercurrent is found to be carried mainly by the surface states [Veldhorst et al. Nature Mater. 2012] Majorana states = [Fu and Kane, PRL 2008] [ Alicea Rep Prog Phys (2012), Beenakker, Annu. Rev. Cond. Mat. Phys. 2013, Grosfeld and Stern 2011, Potter and Fu 2013, Weider et al 2013]

32 3DTI and Majorana fermions L Short JJ [Beenakker 2006, Titov and Beenakker 2006] E n ( )= q 1 n sin 2 ( /2) = + =1 E =0 Zero energy Majorana state requires a perfectly transmitted mode

33 Josephson junction on a 3DTI nanowire L/W 1 Hosts a PTM = 0 /2 Hosts Majorana fermions = [Ando and Suzuura 2002, Ran, Vishwanath, and Lee 2008, Rosenberg, Guo, and Franz 2010, Ostrovsky, Gornyi, and Mirlin 2010, Bardarson and Moore 2013, Fu and Kane PRL 2008, Cook and Franz 2011, Cook at al 2012]

34 Current Phase relation - clean wire E n ( )= q 1 n sin 2 ( /2) [Titov and Beenakker 2006] n = k 2 n k 2 n cos 2 (k n L)+(µ/~v) 2 sin 2 (k n L) k n = p (µ/~v) 2 q 2 n I / X n /@ =0 = 0 /2 µw/~v [RI, Jens Bardarson, H.S. Sim, Joel Moore,2013]

35 Current Phase relation - clean wire E( )=± cos( /2) =0 p p Parity anomaly = 2 c = 1 i 2 1 = 0 /2 c = 1 + i 2 µw/~v [RI, Jens Bardarson, H.S. Sim, Joel Moore,2013]

36 Current Phase relation - clean wire Parity anomaly E( )=± cos( /2) p p =0 1 = 2 c = 1 i 2 = 0 /2 c = 1 + i 2 µw/~v [RI, Jens Bardarson, H.S. Sim, Joel Moore,2013]

37 Current Phase relation - disorder hv (r)v (r 0 )i = g ~v 2 2 D e r r0 /2 2 D [Bardarson et al. PRL 2007] = 0 /2 µw/~v = 20 [RI, Jens Bardarson, H.S. Sim, Joel Moore,2013]

38 Critical current - disorder µ =0 µ D /~v =0.5 g Ic [e / h] / 0 [RI, Jens Bardarson, H.S. Sim, Joel Moore,2013]

39 Critical current - disorder g =2 1.0 µ D / hv =2.0 Ic [e / h] / 0 0 [RI, Jens Bardarson, H.S. Sim, Joel Moore,2013]

40 Conclusion Josephson junctions on 3DTI nanowires are a great place to look for signatures of surface physics: PTM, Majorana fermions At finite chemical potential, enhanced by disorder (and at finite temperature) Not constrained by parity conservation

41 Thanks Paul Fendley (UVA) David Goldhaber Gordon Group (Stanford) Kam Moler Group (Stanford) Fernando De Juan (Berkeley)

42 Effects of temperature E( ) p p I c ( ) e /2~ e /2~(1 (T/ ) 2 ) T 30mK 1K 1/2 / 0

43 Some numbers q E = ± (v p + v F h/v ) h2 /v 2 B = 10mT v F = m/s m/s E g mK Temperature in experiment ~30mK (taken from Konig et al)

44 Non interacting electrons H = i~v F x i~ 2 x} y (x 1 )=T x1,x 0 (x 0 ) T x1,x 0 = P x e i R x 1 x 0 dx (v F z + y ) v 2 F + 2 [E+ i 2 (@ x ) y] R(x) =(vf ) 1/4 e i (x) cos i sin L(x) =(vf ) 1/4 e i (x) i sin cos (x) = Z x 0 E p v 2 F + (x) 2. 2 (x) = tan 1 ( /v F ) (x 0 )= vf v 1/2 e i x (x + 0 )

45 Non interacting electrons H = i~v F x + h z i~ 2 x} y (x 1 )=T x1,x 0 (x 0 ) T x1,x 0 = P x e i R x 1 x 0 dx (v F z + y ) v 2 F + 2 [E+h z + i 2 (@ x ) y] (x 0 )= vf v 1/2 e i x (x + 0 )

46 Experimental feasibility Short junction limit Mode suppression L L/W 1 Flux through wire > 0 /2 Distance from Dirac point µ D /~v Disorder strength g W 400nm B 0.2T g 1 n cm 2 D 10nm [Sacepe et al. Nature Commun 2011, Kong et al. Nano Letters 2010, Beidenkophf et al. 2011,,Williams et al. PRL 2012, Tian et al. Sci. Rep. 2012, Veldhorst et al Nature Mater. 2012, Kim et al Nature Phys. 2012, Cho et al. Nature Commun ]

47 What s the big deal about Majorana fermions? = 2 c = 1 i 2 1 c = 1 + i 2 Ground state degeneracy Topological protection Potential for the perfect qubit Non Abelian statistics