Vortex Liquid Crystals in Anisotropic Type II Superconductors

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

2 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

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

4 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)

5 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

6 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

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

8 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

9 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.

10 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

11 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 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

12 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

13 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.

14 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

15 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) T(K) Data is from Kwok et al., PRL (1992) Grigera et al. PRB (1998) find smectic-c in optimally doped YBCO with B ab

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

17 Parameters: Lindemann parameter: c = In the absence of pinning: m c m ab = 10 κ = λ ab ξ ab = B/Hc2(0) T/Tc

18 m c m ab = 100 κ = λ ab ξ ab = 100 Stronger anisotropy: Lindemann parameter c = B/Hc2(0) T/Tc

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

20 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

21 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.

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

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

24 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

25 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

26 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

27 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