Vortex Liquid Crystals in Anisotropic Type II Superconductorserica/talks/vortexslides.pdf · Vortex...
Transcript of Vortex Liquid Crystals in Anisotropic Type II Superconductorserica/talks/vortexslides.pdf · Vortex...
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Vortex Liquid Crystals in
Anisotropic Type II
Superconductors
E. W. Carlson
A. H. Castro Netro
D. K. Campbell
Boston University
cond-mat/0209175
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Vortex
B
|Ψ|
λ
ξ
r
r
In the high temperature superconductors,
κ ≡ λξ ≈ 100
λ: screening currents and magnetic field
ξ: the normal “core”
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Vortex Lattice Melting
Circular Cross Sections:
Liquid
Raise T or B
Lattice
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Cross Section of a Vortex
ISOTROPIC
CircularProfile
m =
mm
m
λ2 = mc2
4πe2ns
λ2∇2−→B −−→B = Φoδ2(r)
−→B = Φo
2πλ2Ko(r/λ)
ANISOTROPIC
EllipticalProfile
m =
mx
my
mz
λ2x =mxc2
4πe2ns(B||z)
λ2∇2−→B −−→B = Φoδ2(r)
−→B = Φo
2πλ2Ko(r/λ)
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Anisotropic Vortex Lattice Melting
Elliptical Cross Sections:
Anisotropic Interacting “Molecules”
→ Liquid Crystalline Phases
Abrikosov Liquid Crystals?
Lattice NematicSmectic-A
Liquid-Like
Soli
d-L
ike
Raise T or B Raise T or B
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Symmetry of Phases
CRYSTALS: Break continuous rotational
and translational symmetries
of 3D space
LIQUIDS: Break none
LIQUID CRYSTALS: Break a subset
Hexatic: 6-fold rotational symmetryunbroken translational symmetry
Nematic: 2-fold rotational symmetryunbroken translational symmetry
Smectic: breaks translational symmetryin 1 or 2 directions
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Smectics
Smectic-A
Smectic-C
Full translational symmetry in at least one directionBroken translational symmetry in at least one direction(Broken rotational symmetry)
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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
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Instability of Ordered Phase:
Lindemann Criterion for Melting
xu = (u , u )y
< u2 >=< u2x > + < u2y > ≥ c2a2
Typically, lattice melts for c ≈ .1
Houghton, Pelcovits, Sudbo (1989)
Extended to Anisotropy:
ux
uy
< u2x > ≥ 12c2a2x
< u2y > ≥ 12c2a2y
Look for one to be exceeded
well before the other.
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Method
Calculate < u2x > and < u2y >
¤ 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 = (ux, uy) = vortex displacement
C =
(c11(~k)k2x + ce66k
2y + ce44(
~k)k2z c11(~k)kxky
c11(~k)kxky c11(~k)k2y + ch66k2x + ch44(
~k)k2z
)
... for B||ab
Elastic constants are known for uniaxial superconductors:
m =
mab
mab
mc
ANISOTROPY: γ4 = mab
mc= (λab
λc)2 = ( ξc
ξab)2
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Elastic Constants
TILT MODULI
ce44 (−→q ) = B2
4π1−b2bκ2
[1
m2λ+(q2x+q
2y )+γ
−2q2z
]+ B2
4π52bκ2 ln
(κ+ 1−b
2
)
ch44 (−→q ) = B2
4π1−b2bκ2
[(m2
λ+(γ−4q2x+q
2y+γ
−2q2z)m2
λ+(q2x+γ4q2y+γ
2q2z)
)(1
m2λ+q2x+q
2y+γ
−2q2z
)]+B2
4πγ−4
2bκ2 ln(κ+ 1−b
2
)
BULK MODULI
ce11 (−→q ) = ch11 (
−→q ) = ce,o44 (−→q )
SHEAR MODULI
c66 =ΦoB
(8πλab)2 =
B2
4πγ2 (1−b)
2
8bκ2
ce66 = γ6c66
ch66 = γ−2c66
where: Magnetic Field
m2λ =1−b2bκ2
κ =
√1+γ−4κ2+2bκ2γ−2q2z1+bκ2+2bκ2γ−2q2z
b≡ BBabc2(T )
= BBabc2(T=0)(1−t)
Input parameters: b, γ, κ = λab
ξab
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Elastic Constants Vanish at Bc2
Tc T
"All Core"
Meissner
Vortices
B
B (T=0)c2
Bcc2 =
φo
2πξ2ab(T )
Babc2 =
φo
2πξc(T )ξab(T )
ξ(T ) = ξ(T )√1−t ⇒ Bab
c2 =φo
2πξc(0)ξab(0)(1− t)
t = T/Tc
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Which way will it melt?
¤ Elastic constants scale on short length scales.¤ Scaling breaks down at long length scales, c11, c44→ B2
4π
Lindemann ellipsefollows eccentricityof the latticeLindemann criterion
< u2x >≥ 12c2a2x Fluctuations are
less eccentric
< u2y >≥ 12c2a2y
Anisotropy favorssmectic-A
Short wavelengths: Elastic constants softenLong wavelengths: Low energy cost
Both are important.
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YBCO with B||ab
We can compare to the uniaxial case experimentally.
Add pinning:
C =
(c11(~k)k2x + ce66k
2y + ce44(
~k)k2z c11(~k)kxky
c11(~k)kxky c11(~k)k2y + ch66k2x + ch44(
~k)k2z +∆
)
Using Lawrence-Doniach model, ∆ =8√πB2
c2(b−b2)ξcγ2
s3βAκ2 e−8ξ2c /s
2
where βA ≈ 1.16
Pinning vanishes exponentially as the Bc2(T ) line is approached.
→ Pinning favors smectic-C
→ Anisotropy favors smectic-A
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With Planar Pinning
Integrate < u2x > and < u2y > numerically to obtain melting curves
Compare to data on YBCO with B||ab
Parameters:Tc = 92.3K mc
mab= 59 κ = λab
ξab= 55 Hab
c2 = 842T
Lindemann parameter c = .19 (only free parameter)
88 89 90 91 92 93
T(K)
0
2
4
6
8
B(T)
Data is from Kwok et al., PRL 693370(1992)
Grigera et al. PRB (1998) find smectic-C in optimally doped YBCOwith B||ab
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But we are really interested in the case
without pinning.
Now consider the effect of anisotropy alone...
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In the absence of pinning:
Parameters: mc
mab= 10 κ = λab
ξab= 100
Lindemann parameter: c = .2
0.2 0.4 0.6 0.8 1T/Tc
0
0.02
0.04
0.06
0.08
0.1
B/H
c2(0
)
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Stronger anisotropy:
mc
mab= 100 κ = λab
ξab= 100
Lindemann parameter c = .2
0.2 0.4 0.6 0.8 1T/Tc
0
0.02
0.04
0.06
0.08
0.1
B/H
c2(0
)
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Theoretical Phase Diagram
c TT
Smectic-A
Nem
atic
Lattice
Meissner
c2H (T=0)
B
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Long Range Order?
Smectic + spontaneously broken rotational symmetry
Has at most quasi long-range order
Smectic order parameter: ρ = |ρ|eiθ
F = 12
∫dr[α(∇||θ)2 + β(∇2
⊥θ)2]
Smectic + explicitly broken rotational symmetry
Our assumption: mab 6= mc
Explicitly broken rotational symmetry
Costs energy to rotate the vortex smectic.
F = 12
∫dr[α′(∇xθ)2 + β′(∇yθ)2 + β′′(∇zθ)2]
Just like a 3D crystal.
3D smectic + explicitly broken rotational symmetry
→ Long Range Order
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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.
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C. Reichhardt and C. Olson
Numerical simulations on 2D vortices with anisotropic interactions
a b
c d
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C. Reichhardt and C. Olson
Numerical simulations on 2D vortices with anisotropic interactions
a b
c d
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Distinguishing the Smectics: Lorentz
between smectic layers2D Superconductivity
L(F =0)= 0
between smectic layers2D Superconductivity
3D Superconductivity
= 0
Smectic-A
Lattice = 0
z
y
x
ρ
ρ
ρ
ρ
L( F || liquid-like)
(F || liquid-like)L
L
ρ = 0
= 0
= 0
z
y
x
ρ (F =0)
ρ
ρ
ρ
zx
y
x
L(F =0)= 0
= 0
= 0
z
y
Smectic-C
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Experimental Signatures
Resistivity:
¤ Smectic may retain 2D superconductivity
¤ ρ = 0 along liquid-like smectic layers
¤ ρ 6= 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
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Where to look for the Smectic-A
So
lid-L
ike
Co
rrel
atio
ns 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-Ain the presence of a stripe nematic
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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