Vortex Liquid Crystals in Anisotropic Type II ... - Purdue Physics
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
r |Ψ|
ξ
r In the high temperature superconductors, κ ≡ λξ ≈ 100 λ: screening currents and magnetic field ξ: the normal “core”
Vortex Lattice Melting
Circular Cross Sections:
Lattice
Liquid
Raise T or B
Cross Section of a Vortex
ISOT ROP IC
CircularP rof ile
AN ISOT ROP IC
m
m=
m 2
mc λ2 = 4πe 2 ns → − → − λ2∇2 B − B = Φoδ2(r) → − Φo K (r/λ) B = 2πλ 2 o
m=
EllipticalP rof ile
m
mx my mz 2
mx c = λ2 (B||zb) x 4πe2 ns → − → − λ2∇2 B − B = Φoδ2(r) → − Φo K (r/λ) B = 2πλ 2 o
Anisotropic Vortex Lattice Melting Elliptical Cross Sections:
Anisotropic Interacting “Molecules” → Liquid Crystalline Phases Abrikosov Liquid Crystals?
Smectic-A Liquid-Like
Nematic
Solid-Like
Lattice
Raise T or B
Raise T or B
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 symmetry unbroken translational symmetry
Nematic:
2-fold rotational symmetry unbroken translational symmetry
Smectic:
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 Entangled Flux Liquid Chain States Hexatic Smectic-C Driven Smectic
Abrikosov (1957) Nelson (1998) Ivlev, Kopnin (1990) Fisher (1980) Efetov (1979) Balents, Nelson (1995) Balents, Marchetti, Radzihovsky (1998)
Our Assumption:
¤ Explicitly broken rotational symmetry
Instability of Ordered Phase: Lindemann Criterion for Melting
u = (ux , uy ) 2 < u2 >=< u2 x > + < uy > ≥
Typically, lattice melts for c ≈ .1 Houghton, Pelcovits, Sudbo (1989) Extended to Anisotropy: uy ux
< u2 x > ≥
1 c2 a 2 x 2
< u2 y > ≥
1 c2 a 2 y 2
Look for one to be exceeded well before the other.
c 2 a2
Method 2 Calculate < u2 x > and < uy > ¤ based on elasticity theory of the ordered state ¤ using k-dependent elastic constants ¤ from Ginzburg-Landau theory
F =
1 (2π)3
R
¯·~ d~k~ u·C u
where ~ u = (ux , uy ) = vortex displacement
¯= C
Ã
c11 (~k)kx2 + ce66 ky2 + ce44 (~k)kz2 c11 (~k)kx ky
c11 (~k)kx ky c11 (~k)ky2 + ch66 kx2 + ch44 (~k)kz2 ... for B||ab
Elastic constants are known for uniaxial superconductors:
m ¯ = ANISOTROPY: γ 4 =
mab mc
mab mab mc
= ( λλabc )2 = ( ξξabc )2
!
Elastic Constants TILT MODULI h − q)= ce44 (→
− ch44 (→ q)=
B 2 1−b 4π 2bκ2 2
B 1−b 4π 2bκ2
1 m2λ +(qx2 +qy2 )+γ −2 qz2
h³
i
+
B2 5 ln 4π 2bκ2
m2λ +(γ −4 qx2 +qy2 +γ −2 qz2 ) m2λ +(qx2 +γ 4 qy2 +γ 2 qz2 )
BULK MODULI
´³
¡ ¢ e κ + 1−b 2 ´i
1 m2λ +qx2 +qy2 +γ −2 qz2
2
→ − − − q ) = ce,o q ) = ch11 (→ ce11 (→ 44 ( q )
SHEAR MODULI c66 =
Φo B (8πλab )2
=
2
B 2 2 (1−b) γ 8bκ2 4π
ce66 = γ 6 c66 ch66 = γ −2 c66
where: m2λ =
e κ= b≡
Magnetic Field
1−b 2bκ2
q
1+γ −4 κ2 +2bκ2 γ −2 qz2 1+bκ2 +2bκ2 γ −2 qz2
B ab (T ) Bc2
=
B ab (T =0)(1−t) Bc2
Input parameters: b, γ, κ =
λab ξab
−4
¡
γ ln e κ+ +B 4π 2bκ2
1−b 2
¢
Elastic Constants Vanish at Bc2 B B (T=0) c2
"All Core"
Vortices
Meissner
Tc
Bcc2 = ab = Bc2
T
φo 2 (T ) 2πξab φo 2πξc (T )ξab (T )
ξ(T ) =
ξ(T ) √ 1−t
t = T /Tc
⇒
ab = Bc2
φo (1 2πξc (0)ξab (0)
− t)
Which way will it melt?
¤ Elastic constants scale on short length scales. ¤ Scaling breaks down at long length scales, c1 1, c4 4 →
Lindemann criterion < u2x >≥ 12 c2 a2x
B2 4π
Lindemann ellipse follows eccentricity of the lattice Fluctuations are less eccentric
< u2y >≥ 21 c2 a2y 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: Ã 2
¯= C
c11 (~k)kx + ce66 ky2 + ce44 (~k)kz2 c11 (~k)kx ky
c11 (~k)kx ky c11 (~k)ky2 + ch66 kx2 + ch44 (~k)kz2 + ∆
Using Lawrence-Doniach model,
∆=
√ 2 8 πBc2 (b−b2 )ξc γ 2 −8ξc2 /s2 e s 3 β A κ2
where βA ≈ 1.16 Pinning vanishes exponentially as the Bc2 (T ) line is approached.
→ →
Pinning favors smectic-C
Anisotropy favors smectic-A
!
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
ab = 842T Hc2
= 55
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...
In the absence of pinning: Parameters:
mc mab
= 10
κ=
λab ξab
= 100
Lindemann parameter: c = .2 0.1
B/Hc2(0)
0.08
0.06
0.04
0.02
0 0.2
0.4
0.6
T/Tc
0.8
1
Stronger anisotropy: mc mab
= 100
κ=
λab ξab
= 100
Lindemann parameter c = .2
0.1
B/Hc2(0)
0.08
0.06
0.04
0.02
0 0.2
0.4
0.6
T/Tc
0.8
1
Theoretical Phase Diagram
B H (T=0) c2
-A
tic
tic
ma
Ne
ec
Sm Lattice Meissner
Tc
T
Long Range Order? Smectic + spontaneously broken rotational symmetry Has at most quasi long-range order Smectic order parameter: ρ = |ρ|eiθ
F =
1 2
R
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 =
1 2
R
dr[α0 (∇x θ)2 + β 0 (∇y θ)2 + β 00 (∇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 melting T > 0:
2D bosons 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
x
ρ =0 x
z
ρ =0 y ρ =0 z
(FL=0) 3D Superconductivity
Smectic-A ρ =0 x ρ =0 y
(FL || liquid-like)
ρ =0 z
(FL=0)
ρ =0 x
( FL || liquid-like)
2D Superconductivity between smectic layers
Smectic-C
ρ =0 y ρ =0 z
(FL=0)
2D Superconductivity between smectic layers
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
Solid-Like Correlations
Where to look for the Smectic-A 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
E. W. Carlson A. H. Castro Netro D. K. Campbell Boston University
cond-mat/0209175
Vortex
B
λ
r |Ψ|
ξ
r In the high temperature superconductors, κ ≡ λξ ≈ 100 λ: screening currents and magnetic field ξ: the normal “core”
Vortex Lattice Melting
Circular Cross Sections:
Lattice
Liquid
Raise T or B
Cross Section of a Vortex
ISOT ROP IC
CircularP rof ile
AN ISOT ROP IC
m
m=
m 2
mc λ2 = 4πe 2 ns → − → − λ2∇2 B − B = Φoδ2(r) → − Φo K (r/λ) B = 2πλ 2 o
m=
EllipticalP rof ile
m
mx my mz 2
mx c = λ2 (B||zb) x 4πe2 ns → − → − λ2∇2 B − B = Φoδ2(r) → − Φo K (r/λ) B = 2πλ 2 o
Anisotropic Vortex Lattice Melting Elliptical Cross Sections:
Anisotropic Interacting “Molecules” → Liquid Crystalline Phases Abrikosov Liquid Crystals?
Smectic-A Liquid-Like
Nematic
Solid-Like
Lattice
Raise T or B
Raise T or B
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 symmetry unbroken translational symmetry
Nematic:
2-fold rotational symmetry unbroken translational symmetry
Smectic:
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 Entangled Flux Liquid Chain States Hexatic Smectic-C Driven Smectic
Abrikosov (1957) Nelson (1998) Ivlev, Kopnin (1990) Fisher (1980) Efetov (1979) Balents, Nelson (1995) Balents, Marchetti, Radzihovsky (1998)
Our Assumption:
¤ Explicitly broken rotational symmetry
Instability of Ordered Phase: Lindemann Criterion for Melting
u = (ux , uy ) 2 < u2 >=< u2 x > + < uy > ≥
Typically, lattice melts for c ≈ .1 Houghton, Pelcovits, Sudbo (1989) Extended to Anisotropy: uy ux
< u2 x > ≥
1 c2 a 2 x 2
< u2 y > ≥
1 c2 a 2 y 2
Look for one to be exceeded well before the other.
c 2 a2
Method 2 Calculate < u2 x > and < uy > ¤ based on elasticity theory of the ordered state ¤ using k-dependent elastic constants ¤ from Ginzburg-Landau theory
F =
1 (2π)3
R
¯·~ d~k~ u·C u
where ~ u = (ux , uy ) = vortex displacement
¯= C
Ã
c11 (~k)kx2 + ce66 ky2 + ce44 (~k)kz2 c11 (~k)kx ky
c11 (~k)kx ky c11 (~k)ky2 + ch66 kx2 + ch44 (~k)kz2 ... for B||ab
Elastic constants are known for uniaxial superconductors:
m ¯ = ANISOTROPY: γ 4 =
mab mc
mab mab mc
= ( λλabc )2 = ( ξξabc )2
!
Elastic Constants TILT MODULI h − q)= ce44 (→
− ch44 (→ q)=
B 2 1−b 4π 2bκ2 2
B 1−b 4π 2bκ2
1 m2λ +(qx2 +qy2 )+γ −2 qz2
h³
i
+
B2 5 ln 4π 2bκ2
m2λ +(γ −4 qx2 +qy2 +γ −2 qz2 ) m2λ +(qx2 +γ 4 qy2 +γ 2 qz2 )
BULK MODULI
´³
¡ ¢ e κ + 1−b 2 ´i
1 m2λ +qx2 +qy2 +γ −2 qz2
2
→ − − − q ) = ce,o q ) = ch11 (→ ce11 (→ 44 ( q )
SHEAR MODULI c66 =
Φo B (8πλab )2
=
2
B 2 2 (1−b) γ 8bκ2 4π
ce66 = γ 6 c66 ch66 = γ −2 c66
where: m2λ =
e κ= b≡
Magnetic Field
1−b 2bκ2
q
1+γ −4 κ2 +2bκ2 γ −2 qz2 1+bκ2 +2bκ2 γ −2 qz2
B ab (T ) Bc2
=
B ab (T =0)(1−t) Bc2
Input parameters: b, γ, κ =
λab ξab
−4
¡
γ ln e κ+ +B 4π 2bκ2
1−b 2
¢
Elastic Constants Vanish at Bc2 B B (T=0) c2
"All Core"
Vortices
Meissner
Tc
Bcc2 = ab = Bc2
T
φo 2 (T ) 2πξab φo 2πξc (T )ξab (T )
ξ(T ) =
ξ(T ) √ 1−t
t = T /Tc
⇒
ab = Bc2
φo (1 2πξc (0)ξab (0)
− t)
Which way will it melt?
¤ Elastic constants scale on short length scales. ¤ Scaling breaks down at long length scales, c1 1, c4 4 →
Lindemann criterion < u2x >≥ 12 c2 a2x
B2 4π
Lindemann ellipse follows eccentricity of the lattice Fluctuations are less eccentric
< u2y >≥ 21 c2 a2y 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: Ã 2
¯= C
c11 (~k)kx + ce66 ky2 + ce44 (~k)kz2 c11 (~k)kx ky
c11 (~k)kx ky c11 (~k)ky2 + ch66 kx2 + ch44 (~k)kz2 + ∆
Using Lawrence-Doniach model,
∆=
√ 2 8 πBc2 (b−b2 )ξc γ 2 −8ξc2 /s2 e s 3 β A κ2
where βA ≈ 1.16 Pinning vanishes exponentially as the Bc2 (T ) line is approached.
→ →
Pinning favors smectic-C
Anisotropy favors smectic-A
!
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
ab = 842T Hc2
= 55
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...
In the absence of pinning: Parameters:
mc mab
= 10
κ=
λab ξab
= 100
Lindemann parameter: c = .2 0.1
B/Hc2(0)
0.08
0.06
0.04
0.02
0 0.2
0.4
0.6
T/Tc
0.8
1
Stronger anisotropy: mc mab
= 100
κ=
λab ξab
= 100
Lindemann parameter c = .2
0.1
B/Hc2(0)
0.08
0.06
0.04
0.02
0 0.2
0.4
0.6
T/Tc
0.8
1
Theoretical Phase Diagram
B H (T=0) c2
-A
tic
tic
ma
Ne
ec
Sm Lattice Meissner
Tc
T
Long Range Order? Smectic + spontaneously broken rotational symmetry Has at most quasi long-range order Smectic order parameter: ρ = |ρ|eiθ
F =
1 2
R
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 =
1 2
R
dr[α0 (∇x θ)2 + β 0 (∇y θ)2 + β 00 (∇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 melting T > 0:
2D bosons 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
x
ρ =0 x
z
ρ =0 y ρ =0 z
(FL=0) 3D Superconductivity
Smectic-A ρ =0 x ρ =0 y
(FL || liquid-like)
ρ =0 z
(FL=0)
ρ =0 x
( FL || liquid-like)
2D Superconductivity between smectic layers
Smectic-C
ρ =0 y ρ =0 z
(FL=0)
2D Superconductivity between smectic layers
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
Solid-Like Correlations
Where to look for the Smectic-A 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
