Antiferomagnetism and triplet superconductivity in Bechgaard salts

Daniel Podolsky (Harvard and UC Berkeley)

Timofey Rostunov (Harvard)

Ehud Altman (Harvard)

Antoine Georges (Ecole Polytechnique)Eugene Demler (Harvard)

References: Phys. Rev. Lett. 93:246402 (2004) Phys. Rev. B 70:224503 (2004) cond-mat/0506548

Outline

• Introduction. Phase diagram of Bechgaard salts

• New experimental tests of triplet superconductivity

•Antiferomagnet to triplet superconductor transition in quasi 1d systems. SO(4) symmetry

• Implications of SO(4) symmetry for the phase diagram. Comparison to (TMTSF)2PF6

• Experimental test of SO(4) symmetry

Bechgaard salts

Stacked molecules form 1d chains Jerome, Science 252:1509 (1991)

Evidence for triplet superconductivity in Bechgaard salts

•Strong suppression of Tc by disorderChoi et al., PRB 25:6208 (1982)Tomic et al., J. Physique 44: C3-1075 (1982)Bouffod et al, J. Phys. C 15:2951 (1981)

•Superconductivity persists at fields exceeding the paramagnetic limit

Lee et al., PRL 78:3555 (1997)Oh and Naughton, cond-mat/0401611

•No suppression of electron spin susceptibility below Tc. NMR Knight shift study of 77S in (TMTSF)2PF6

Lee et al, PRL 88:17004 (2002)

P-wave superconductor without nodes

--

--

++

+

+

Order parameter

px

py

Specific heat in (TMTSF)2PF6

Garoche et al., J. Phys.-Lett. 43:L147 (1982)

Nuclear spin lattice relaxation rate

in (TMTSF)2PF6

Lee et al., PRB 68:92519 (2003)

For (TMTSF)2ClO4 similar behavior has been observed by Takigawa et.al. (1987)

Typically this would be attributed to nodal quasiparticles (nodal line)

This work: T3 behavior of 1/T1 due to spin waves

Spin waves in triplet superconductors

Spin wave:d-vector rotatesIn space

Dispersion of spin waves

Easy axis anisotropyFull spin symmetry

Spin anisotropy of the triplet superconducting order parameter

Spin anisotropy in the antiferromagnetic state:

Easy direction for the superconducting order parameter is along the b axis

For Bechgaard salts we estimate

Assuming the same anistropy in the superconducting state

Dumm et al. (2000)

Torrance et al. (1982)

Spin z axis points along the crystallographic b axis.

Experimental regime of parameters

Contribution of spin waves to 1/T1

Moriya relation: -- nuclear Larmor frequency

Creation or annihilation ofspin waves does not contribute to T1

-1

Scattering of spin wavescontributes to T1

-1

Contribution of spin waves to 1/T1

is the density of states for spin wave excitations. Using

For we can take where is the dimension

This result does not change when we include coherence factors

(1) (2)

Contribution of spin waves to 1/T1

• For small fields, T1-1 depends on the direction of the magnetic field

• When , we have T3 scaling of T1-1 in d=2

• When , we have exponential suppression of T1-1

These predictions of the spin-wave mechanism of nuclearspin relaxation can be checked in experiments

Spin-flop transition in the triplet superconducting state

S=1 Sz=0

S=1 Sx=0

S=1 Sy=0

At B=0 start with (easy axis). For this state doesnot benefit from the Zeeman energy.

For the order parameter flops into the xy plane.

This state can benefit from the Zeeman energy withoutsacrificing the pairing energy.

For Bechgaard salts we estimate

Field and direction dependent Knight shift in UPt3

Tau et al., PRL 80:3129 (1998)

Competition of antiferomagnetism and triplet superconductivity in Bechgaard salts

Coexistence of superconductivity and magnetismVuletic et al., EPJ B25:319 (2002)

Interacting electrons in 1d

L’ R’ L’ L’ L’ L’

g1 g2 g4 g4

R L R R L L R R

R’ R’

Interaction Hamiltonian

K

g1

SDW(CDW)

TSC(SS)

CDW CDW(SS)

SS(CDW)

SS

1/2 1 2

SDW/TSC transition at K=1.

This corresponds to

2g2 = g1

Phase diagram

Symmetries

Spin SO(3)S algebra

SO(3)S is a good symmetry of the system

Isospin SO(3)I symmetry

We always have charge U(1) symmetry

When K=1, U(1) is enhanced to SO(3)I because

SO(4)=SO(3)SxSO(3)I symmetry.Unification of antiferromagnetism and

triplet superconductivity.

transforms as a vector under spin and isospin rotations

isospinspin

Order parameter for antiferromagnetism:

Order parameter for triplet superconductivity:

SO(3)SxSO(4)I symmetry at incommensurate filling

Two separate SO(3) algebras

Isospin group SO(4)I= SO(3)RxSO(3)L

Umklapp scattering reduces SO(4)I to SO(3)I

Role of interchain hopping

Ginzburg-Landau free energy

SO(4) symmetry requires

SO(4) symmetric GL free energy

Weak coupling analysis

GL free energy. Phase diagram

Unitary TSC for . TSC order parameter

First order transitionbetween AF and TSC

r1

AFunitary TSC

r2

Unitary TSC and AF. Thermal fluctuations

AF

Unitary TSC

r2

r1

Extend spin SO(3) to SO(N). Do large N analysis in d=3

• First order transition between normal and triplet superconducting phases (analogous result for 3He: Bailin, Love, Moore (1997))

• Tricritical point on the normal/antiferromagnet boundary

Triplet superconductivity and antiferromagnetism.Phase diagram

First order transition becomes a coexistence region

AF TSC

NormalT

P

TSC

AF

Normal

T

“V”

Phase diagram ofBechgaard salts

Vuletic et al., EPJ B25:319 (2002)

Operator Charge Spin Momentum

0 1 2kF

2 1 0

2 0 2kF

operator rotates between AF and TSC orders

-mode should appear as a sharp resonance in the TSC phase

Energy of the mode softens at the first order transitionbetween superconducting and antiferromagnetic phases

Experimental test of quantum SO(4) symmetry

• New experimental tests of triplet pairing in Bechgaard salts: 1) NMR for T < 50mK and small fields. Expect strong suppression of 1/T1

2) Possible spin flop transtion for magnetic fields along the b axis and field strength around 0.5 kG 3) Microwave resonance in Bechgaard salts at . (For Sr2RuO4 expect such resonance at )

• SO(4) symmetry is generally present at the antiferromagnet to triplet superconductor transition in quasi-1d systems

• SO(4) symmetry helps to explain the phase diagram of (TMTSF)2PF6

• SO(4) symmetry implies the existence of a new collective mode, the resonance. The resonance should be observable using inelastic neutron scattering experiments (in the superconducting state)

Conclusions