Superconductivity in Cu:Bi 2 Se 3 model order parameters and surface - PowerPoint PPT Presentation

Superconductivity in Cu:Bi 2 Se 3 model order parameters and surface states Sungkit Yip Institute of Physics Academia Sinica, Taipei, Taiwan

Topological states (bulk) Surface /edge states Quantum Hall chiral E Quantum Spin Hall k // Topological Insulators

E 3D Dirac cone Bi2Se3, Bi2Te3, …. Spin-ARPES (Hsieh et al (Princeton))

Superconductor: can be topological non-trivial due to superconducting order parameter even though normal state band structure trivial Examples:       ( ) | ( ) | k ik k ik x y x y planar state ~(2D TI)     0 k ik   x y    0 k ik   x y 3 He-B     k ik k   x y z ~(3D TI)    k k ik   z x y Balian-Werthamer (c.f. s-wave, topologically trivial; no surface states)

Cu:Bi 2 Se 3 superconducting (Princeton) Tc varies with Cu concentration, up to ~ 4K Ando: suggest fully gapped

Cu intercalates electron doped chemical potential ~0.4 eV (>> Tc) Wray et al, Nature, 10

Tunneling experiments: -- controversial Osaka 11: NIST, MD

Q: Superconducting order parameter ? Topology of the superconducting state ? Surface states? what is the role of the topological insulator state?

Bi2Se3: normal state (H Zhang et al Nat. Phys. 09; C X Liu et al PRB 10 )   ( , 2 , 3 ) D d E C U P D 3 3 3 2 d

band inversion at k=0  point D  h Model for k ~ 0:           ( ...) ( ..) ( ...)( ) H M B k A k s k s 0 0 0 N z z y y x x y x   P 1 P 2 z z  parity operator: z Fu and Berg 10:  parity operator: x

E Dirac like Hamiltonian    E k k 2m |k|

vac ˆ n x o | x o | x o | x o 1 2 1 2 1 2 1 2 Boundary condition       1 | 0 z TI 2 E Surface bound state as Dirac cone if  sgn( ) 0 mv z   ˆ Positive energy branch along v n k

Superconducting state: Fu and Berg 10: local pairs, investigated phase diagram with local interaction (6 total)         | 1 1 , | 1 2 ,.... | 1 2 , | 1 2 Also noted, different symmetries: A 1 g Fully gapped, A 1 u topological superconducting state A 2 u   | 1 2 E u   | 1 2

surface bound states investigated, using this picture, by: Hao and Lee 11, Sato, Tanaka et al 12 Hsieh and Fu 12 Kamakage et al 12

Hao and Lee 11 :                | 1 2 | 1 2 | 1 2 | 1 2 | 1 1 | 2 2 “ intrasite opposite spin” “ interorbital odd parity”      | 1 2 | 1 2

Hsieh and Fu 12:      | 1 2 | 1 2 Topological SC  sgn( ) 0 mv z  sgn( ) 0 mv z mirror index

Superconductivity with strong spin-orbit 80’s: heavy fermions Anderson, Blount, Rice, …. if normal state time-reversal and parity symmetric each k: two degenerate states (pseudospin) Cooper pairing between pseudospin pairing wavefunction: even parity : pseudospin singlet need only specify momentum dependence (even) odd parity: pseudospin triplet       ( k ) spin-vector d d id d   x y z    k dependent d d id   z x y odd in k

Yip and Garg 93 D 6 h complete basis set, up to invariant even functions odd

D 6 x P = Parity h D 3 d

D 3 d A 1u and E u : what linear combination?

I: Construct pseudospin basis II: project   pseudospins: acting on a 2D Hilbert space k  (1) P T -k  -k    up          down 0 1            ( ) | | | | (2) k k k k k    1 0  x    ( , , ) like an axial vector (pseudospin) x y z

(1) P and T orbital spin (on half of fermi surface) others defined by P and T k  P T -k  -k 

  (2) axial vector (proper rotational properties) unitary transformation the rest by P and T

Projection: 1 x   ( / ) m see later (planar)

          | 1 2 | 1 2 | 1 1 | 2 2 s-wave, A 1g A 1u

normal state TI : normal state trivial (or single band) p-h p-h  mixing two band

     | 1 2 | 1 2 A 1 u sgn( ) depends on mv z  z   cf. Balian Werthamer state // d k surface state: Dirac cone

Surface states (single band picture):   odd parity ( k ) d k k out in       d id d   x y z    d d   d d id  in out z x y    Choose quantization axes along d d in out  d in    d out d in    phase difference   d (  if glancing incidence) E  gap edge out positive branch along

E > 0 branch: Normal state Superconducting state   ˆ if TI v n k    ˆ A E sgn( ) mv v n k 1 z in u u follows from

A 1 u v < 0    sgn( ) 0 sgn( ) 0 sgn( ) 0 mv mv mv z z z single band single band two band or two band

A 1 u v < 0    sgn( ) 0 sgn( ) 0 sgn( ) 0 mv mv mv z z z single band single band two band or two band k  | | k determined by superconducting order parameter // F k  determined by normal state topology | | k // F

Q: Anomalous dispersion an indicator of TI ? NO: only d(k) on the Fermi surface matters            2 ( ...) ( ..) ( ...)( ) H m C k v k v k s k s 0 0 N x z z y x y y x z opp signs   2 m m Ck k on Fermi surface 0 can change from anomalous to ordinary for increasing C or |  |

Yamakage et al 12 A 1 u      | 1 2 | 1 2 E u      | 1 2 | 1 2

Summary: pseudospin basis bulk order parameters orbital/spin  pseudospin basis anomalous dispersions related to peculiar d(k) the dispersion for k < k F purely property of the superconducting order parameter dispersion at k> k F : normal state property except duplication due to particle-hole

Single band picture: ,  d  0 m = 0, line nodes on equator d k x y z z full two band:   m  m        2 2 m trivial insulator TI m    m SC    continuous at m=0 m W=1 if 

  ( / ) m       ' ' | 1 2 | 1 2 A 1 g m  ’’ but can be smoothly connected if include      | 1 1 | 2 2