Vortex Liquid Crystals in Anisotropic Type II Superconductors

Vortex Liquid Crystals in Anisotropic Type II Superconductors E. W. Carlson A. H. Castro Netro D. K. Campbell Boston University cond-mat/0209175

Vortex B λ Ψ r ξ In the high temperature superconductors, κ λ ξ 100 λ: screening currents and magnetic field ξ: the normal core r

Vortex Lattice Melting Circular Cross Sections: Lattice Liquid Raise T or B

Cross Section of a Vortex ISOT ROP IC m = m m m λ 2 = mc2 4πe 2 n s λ 2 2 B B = Φ o δ 2 (r) CircularP rof ile B = Φ o 2πλ 2K o(r/λ) ANISOT ROP IC m = m x m y m z λ 2 x = m xc 2 4πe 2 n s (B ẑ) λ 2 2 B B = Φ o δ 2 (r) EllipticalP rof ile B = Φ o 2πλ 2K o(r/λ)

Anisotropic Vortex Lattice Melting Elliptical Cross Sections: Anisotropic Interacting Molecules Liquid Crystalline Phases Abrikosov Liquid Crystals? Lattice Smectic-A Liquid-Like Nematic Solid-Like Raise T or B Raise T or B

Symmetry of Phases CRYSTALS: LIQUIDS: Break continuous rotational and translational symmetries of 3D space Break none LIQUID CRYSTALS: Break a subset Hexatic: Nematic: Smectic: 6-fold rotational symmetry unbroken translational symmetry 2-fold rotational symmetry unbroken translational symmetry breaks translational symmetry in 1 or 2 directions

Smectics Smectic-A Smectic-C Full translational symmetry in at least one direction Broken translational symmetry in at least one direction (Broken rotational symmetry)

Many Vortex Phases Abrikosov Lattice Abrikosov (1957) Entangled Flux Liquid Nelson (1998) Chain States Ivlev, Kopnin (1990) Hexatic Fisher (1980) Smectic-C Efetov (1979) Balents, Nelson (1995) Driven Smectic Balents, Marchetti, Radzihovsky (1998) Our Assumption: Explicitly broken rotational symmetry

Instability of Ordered Phase: Lindemann Criterion for Melting u = (u x, u y ) < u 2 >=< u 2 x > + < u 2 y > c 2 a 2 Typically, lattice melts for c.1 Houghton, Pelcovits, Sudbo (1989) Extended to Anisotropy: u y < u 2 u x x > 1 2 c2 a 2 x < u 2 y > 1 2 c2 a 2 y Look for one to be exceeded well before the other.

Method Calculate < u 2 x > and < u2 y > based on elasticity theory of the ordered state using k-dependent elastic constants from Ginzburg-Landau theory F = 1 (2π) 3 d k u C u where u = (u x, u y ) = vortex displacement ( c11 ( k)k 2 x + ce 66 k2 y + ce 44 ( k)k 2 z c 11 ( k)k x k y ) C = c 11 ( k)k x k y c 11 ( k)k 2 y + ch 66 k2 x + ch 44 ( k)k 2 z... for B ab Elastic constants are known for uniaxial superconductors: m = m ab m ab m c ANISOTROPY: γ 4 = m ab m c = ( λ ab λ c ) 2 = ( ξ c ξ ab ) 2

TILT MODULI c e 44 ( q ) = B2 1 b 4π c h 44 ( q ) = B2 1 b 4π Elastic Constants [ 1 2bκ 2 m 2 λ +(q2 x+qy 2 )+γ 2 qz 2 2bκ 2 [( m 2 ] + B2 4π + λ (γ 4 qx 2+q2 y +γ 2 qz 2 ) m 2+ λ (qx+γ 2 4 qy 2 +γ 2 qz 2 ) ) ( 5 ln ( κ ) + 1 b 2bκ 2 2 1 m 2 λ +q2 x+q 2 y +γ 2 q 2 z )] + B2 4π γ 4 ln ( κ ) + 1 b 2bκ 2 2 BULK MODULI c e 11 ( q ) = c h 11 ( q ) = c e,o 44 ( q ) SHEAR MODULI c 66 = Φ ob (8πλ ab ) 2 c e 66 = γ6 c 66 c h 66 = γ 2 c 66 = B2 4π γ2 (1 b) 2 8bκ 2 where: Magnetic Field m 2 λ = 1 b 2bκ 2 κ = b 1+γ 4 κ 2 +2bκ 2 γ 2 q 2 z 1+bκ 2 +2bκ 2 γ 2 q 2 z B B ab c2 (T ) = B ab c2 B (T =0)(1 t) Input parameters: b, γ, κ = λ ab ξ ab

Elastic Constants Vanish at B c2 B B (T=0) c2 "All Core" Vortices Meissner T c T B c c2 = B ab c2 = φ o 2πξ 2 (T ) ab φ o 2πξ c (T )ξ ab (T ) ξ(t ) = ξ(t ) 1 t B ab c2 = φ o 2πξ c (0)ξ ab (1 t) (0) t = T/T c

Which way will it melt? Elastic constants scale on short length scales. Scaling breaks down at long length scales, c 1 1, c 4 4 B2 4π Lindemann criterion Lindemann ellipse follows eccentricity of the lattice < u 2 x > 1 2 c2 a 2 x Fluctuations are less eccentric < u 2 y > 1 2 c2 a 2 y Anisotropy favors smectic-a Short wavelengths: Elastic constants soften Long wavelengths: Low energy cost Both are important.

YBCO with B ab We can compare to the uniaxial case experimentally. Add pinning: C = ( c11 ( k)k 2 x + c e 66 k2 y + c e 44 ( k)k 2 z c 11 ( k)k x k y c 11 ( k)k x k y c 11 ( k)ky 2 + c h 66 k2 x + c h 44 ( k)kz 2 + ) Using Lawrence-Doniach model, = 8 πb 2 c2 (b b2 )ξ c γ 2 s 3 β A κ 2 e 8ξ2 c /s 2 where β A 1.16 Pinning vanishes exponentially as the B c2 (T ) line is approached. Pinning favors smectic-c Anisotropy favors smectic-a

With Planar Pinning Integrate < u 2 x > and < u 2 y > numerically to obtain melting curves Compare to data on YBCO with B ab Parameters: T c = 92.3K m c m ab = 59 κ = λ ab ξ ab = 55 H ab c2 = 842T Lindemann parameter c =.19 (only free parameter) 8 B(T) 6 4 2 0 88 89 90 91 92 93 T(K) Data is from Kwok et al., PRL 69 3370(1992) Grigera et al. PRB (1998) find smectic-c in optimally doped YBCO with B ab

But we are really interested in the case without pinning. Now consider the effect of anisotropy alone...

Parameters: Lindemann parameter: c =.2 0.1 In the absence of pinning: m c m ab = 10 κ = λ ab ξ ab = 100 0.08 B/Hc2(0) 0.06 0.04 0.02 0 0.2 0.4 0.6 0.8 1 T/Tc

m c m ab = 100 κ = λ ab ξ ab = 100 Stronger anisotropy: Lindemann parameter c =.2 0.1 0.08 B/Hc2(0) 0.06 0.04 0.02 0 0.2 0.4 0.6 0.8 1 T/Tc

Theoretical Phase Diagram B H (T=0) c2 Nematic Smectic-A Lattice Meissner T c T

Long Range Order? Smectic + spontaneously broken rotational symmetry Has at most quasi long-range order Smectic order parameter: ρ = ρ e iθ F = 1 2 dr[α( θ) 2 + β( 2 θ)2 ] Smectic + explicitly broken rotational symmetry Our assumption: m ab m c Explicitly broken rotational symmetry Costs energy to rotate the vortex smectic. F = 1 2 dr[α ( x θ) 2 + β ( y θ) 2 + β ( z θ) 2 ] Just like a 3D crystal. 3D smectic + explicitly broken rotational symmetry Long Range Order

Other methods point to smectic-a 2D boson mapping: (Nelson, 1988) L τ 3D vortices 2D bosons T = 0 2D melting T > 0: (Ostlund Halperin, 1981) Short Burgers vector dislocations unbind first Smectic-A Quasi-Long-Range Order Our case: T = 0 Smectic can be long-range ordered.

C. Reichhardt and C. Olson Numerical simulations on 2D vortices with anisotropic interactions a b c d

C. Reichhardt and C. Olson Numerical simulations on 2D vortices with anisotropic interactions a b c d

Distinguishing the Smectics: Lorentz y Lattice ρ x = 0 z x ρ y = 0 ρ = 0 z (F =0) L 3D Superconductivity Smectic-A ρ x = 0 ρ y = 0 ρ z = 0 (F =0) L (F liquid-like) L 2D Superconductivity between smectic layers Smectic-C ρ x = 0 ρ y = 0 ρ = 0 z ( F liquid-like) L (F =0) L 2D Superconductivity between smectic layers

Experimental Signatures Resistivity: Smectic may retain 2D superconductivity ρ = 0 along liquid-like smectic layers ρ 0 along the density wave Structure Factor: (Neutron scattering or Bitter decoration) Liquid-like correlations in one direction Solid-like correlations in the other µsr: Signature at both melting temperatures

Where to look for the Smectic-A Solid-Like Correlations Liquid-Like Correlations Our Assumptions: Explicitly broken rotational symmetry No explicitly broken translational symmetry Cuprate Superconductors: Stripe Nematic Yields anisotropic superfluid stiffness Does not break translational symmetry B c: Look for vortex smectic-a in the presence of a stripe nematic

Conclusions Anisotropy favors intermediate melting (Lattice Smectic-A Nematic) - Lindemann criterion - 2D boston mapping - 2D numerical simulations (Reichhardt and Olson) Smectic-A has Long Range Order - 3D smectic + explicit symmetry breaking - 2D boson mapping Experimental Signatures: - Resistivity anisotropy - µsr - Bitter decoration - Neutron scattering