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

Superconductivity in Cu:Bi2Se3

model order parameters and surface states Sungkit Yip Institute of Physics Academia Sinica, Taipei, Taiwan

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

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

Superconductor: can be topological non-trivial due to superconducting order parameter even though normal state band structure trivial Examples:

      | ) ( | ) (

y x y x

ik k ik k

          

y x y x

ik k ik k

~(2D TI)

          

y x z z y x

ik k k k ik k

~(3D TI)

3He-B

Balian-Werthamer (c.f. s-wave, topologically trivial; no surface states) planar state

Cu:Bi2Se3 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 )

d

D3

P U C E D d   ) 3 , 2 , (

2 3 3

band inversion at k=0  point Model for k ~ 0:

x y x x y y z z N

s k s k A k B M H    ) ...)( ( ..) ( ...) (       

 z

P1

 z

P2

Fu and Berg 10: parity operator:

z



parity operator:

x



h

D

k k

E   

Dirac like Hamiltonian 2m E |k|

Boundary condition 1 2 1 2 1 2 1 2 x o | x o | x o | x o

| 1     

z



2 TI vac Surface bound state as Dirac cone if

) sgn( 

z

mv

Positive energy branch along

k n v   ˆ

E

n ˆ

Superconducting state: Fu and Berg 10: local pairs, investigated phase diagram with local interaction

        2 1 | , 2 1 | ,.... 2 1 | , 1 1 |

(6 total) Also noted, different symmetries:

g

A

1 u

A

1 u

A

2 u

E     2 1 | 2 1 |

Fully gapped, topological superconducting state

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

     2 2 | 1 1 |

“intrasite opposite spin”

     2 1 | 2 1 |      2 1 | 2 1 |

“interorbital odd parity”

     2 1 | 2 1 |

Hao and Lee 11 :

     2 1 | 2 1 |

Hsieh and Fu 12:

) sgn( 

z

mv ) sgn( 

z

mv

Topological SC 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)

• dd parity: pseudospin triplet

          

y x z z y x

id d d d id d

) (k d  

spin-vector k dependent

• dd in k

h

D6

Yip and Garg 93 complete basis set, up to invariant functions even

• dd

h

D6

d

D3

x P = Parity

A1u and E u: what linear combination?

d

D3

I: Construct pseudospin basis II: project k

• k
• k

P T

| | | | ) (      k k k k k

x

        

pseudospins:

        1 1

) , , (

z y x

  

like an axial vector (pseudospin) (1) (2)   up   down

 

acting on a 2D Hilbert space

(1) P and T

• rbital

spin

• thers defined by P and T

(on half of fermi surface) k

• k
• k

P T

(2)

 

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

Projection: 1 x

 ) / (  m

see later (planar)

     2 2 | 1 1 |

s-wave, A1g

     2 1 | 2 1 |

A1u

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

u

A

1

     2 1 | 2 1 |

depends on

) sgn(

z

mv

• cf. Balian Werthamer state

z 

k d   //

surface state: Dirac cone

Surface states (single band picture):

• dd parity

in

k

• ut

k

• ut

in

d d   

Choose quantization axes along

• ut

in

d d   

in

d 

• ut

d 

in

d 

• ut

d 

 phase difference

   

positive branch along ( if glancing incidence) E  gap edge

) (k d  

          

y x z z y x

id d d d id d

Normal state Superconducting state E > 0 branch:

k n v   ˆ

u

A

1

in z

k n v mv    ˆ ) sgn(

u

E

if TI follows from

v < 0 single band

• r

two band

) sgn( 

z

mv ) sgn( 

z

mv

single band

) sgn( 

z

mv

two band

u

A

1

v < 0 single band

• r

two band

) sgn( 

z

mv ) sgn( 

z

mv

single band

) sgn( 

z

mv

two band

u

A

1 F

k k  | |

// F

k k  | |

//

determined by superconducting order parameter determined by normal state topology

Q: Anomalous dispersion an indicator of TI ? NO: only d(k) on the Fermi surface matters

z x y y x y z z x N

s k s k v k v k C m H    ) ...)( ( ..) ( ...) (

2

       

• pp signs

2

Ck m m  

k on Fermi surface can change from anomalous to ordinary for increasing C or ||

     2 1 | 2 1 |      2 1 | 2 1 |

Yamakage et al 12

u

A

1 u

E

Summary: pseudospin basis bulk order parameters

• rbital/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: m = 0, line nodes on equator

,  y x

d

z z

k d 

full two band:  m 

  m    m

TI SC W=1

2 2

    m   m    m

trivial insulator continuous at m=0 if 

m ’’ 

     2 1 | 2 1 |

 ) / (  m

' ' 1g

A

but can be smoothly connected if include

     2 2 | 1 1 |