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Continuous topological defects in 3 He-A in a slab

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Andrei Golov:

Trapping of vortices by a network of topological defects in superfluid 3He-A

Continuous topological defects in 3He-A in a slab

Models for the critical velocity and pinning (critical states).

Vortex nucleation and pinning (intrinsic and extrinsic):

- Uniform texture: intrinsic nucleation and weak extrinsic pinning

- Texture with domain walls: intrinsic nucleation and strong universal pinning

Speculations about the networks of domain walls

P.M.Walmsley, D.J.Cousins, A.I.Golov Phys. Rev. Lett. 91, 225301 (2003) Critical velocity of continuous vortex nucleation in a slab of superfluid 3He-A

P.M.Walmsley, I.J.White, A.I.Golov Phys. Rev. Lett. 93, 195301 (2004) Intrinsic pinning of vorticity by domain walls of l-texture in superfluid 3He-A

l

γ

d

vs

l

vs

β

n

m

α

p-wave, spin triplet Cooper pairs

Two anisotropy axes:

l - direction of orbital momentum

d-spin quantization axis (s.d)=0

Order parameter: 6 d.o.f.:

Aμj=∆(T)(mj+inj)dµ

Velocity of flow depends on 3 d.o.f.: vs = -ħ(2m3)-1(∇γ+cosβ∇α)

Continuous vorticity: large length scale

Discrete degeneracy: domain walls

=0

vs=0

vs

>0

>0

>0

Groundstates, vortices, domain walls: (slab geometry, small H and vs)

Domain walls

(lz, dz)=

Rcore~ 0.2D

Vortex and wall can be either

dipole-locked or unlocked

Two-quantum vortex

Azimuthal component of superflow

l-wall

ATC-vortex (l)

dl-wall

ATC-vortex (dl)

LV2

similar to CUV except d = l

(narrow range of small )

Models for vc (intrinsic processes)

vc

H

vd~1 mm/s

ħ

~

v

vc∼vd

ħ

c

2m

Rcore

~

v

1

mm/s

=

3

c

x

2

m

vc∝H

3

D

vc∝D-1

HF=2-4 G

Hd≈25 G

When l is free to rotate:

Hydrodynamic instability at

Soft core radius Rcore vs. D and H :

♦H= 0 :Rcore ∼D→vc∝ D-1

♦2-4 G < H< 25 G:Rcore ∼ξH ∝ H-1→vc ∝H

♦H> 25 G :Rcore ∼ξd = 10 μm →vc∼1 mm/s

(Feynman 1955, et al…)

or

When l is aligned with v (Bhattacharyya, Ho, Mermin 1977):

Instability of v-aligned l-texture: at

lz=+1

dz=+1

lz=-1

dz=+1

lz=+1

dz=+1

lz=+1

dz=-1

lz=-1

dz=-1

lz=-1

dz=+1

lz=+1

dz=-1

lz=-1

dz=-1

dl-wall

l-wall

d-wall

Groundstate

(choice of four)

or

Multidomain texture

(metastable)

(obtained by cooling

at H=0 while rotating)

(obtained by cooling

while stationary)

lz=+1

dz=+1

lz=+1

dz=+1

lz=+1

dz=-1

lz=-1

dz=-1

Also possible:

dl-walls only

d-walls only

(obtained by cooling

at H=0 while rotating)

(obtained by cooling

while stationary)

2

2

æ

ö

æ

ö

v

H

ç

÷

ç

÷

+

=

1

ç

÷

ç

÷

2 walls

vF

HF

è

ø

è

ø

Orienting forces:

- Boundaries favourl perpendicular to walls (“uniform texture”, UT)

- Magnetic field Hfavours l (via d) in plane with walls (“planar”, PT)

- Superflow favoursl tends to be parallel to vs (“azimuthal”, AT)

vF =FR

vF ~D-1HF ~D-1

Theory (Fetter 1977):

uniform

uniform

rotation

rotation

H

azimuthal

domain walls

vortices

planar

Uniform l-texture: cooling through Tc while rotating:

Initial

preparation

NtoA (moderate density of domain walls): cooling through Tc at = 0

BtoA (high density of domain walls): warming from B-phase at = 0

Applying rotation, > F,H = 0: makes azimuthal textures

Applying H > HF at 0: makes planar texture,

then > F: twodl-walls on demand

Rotating at > vcR introduces vortices

Value of vc and type of vortices depend on texture (with or without domain walls)

H

vn=r

vs = 0

v

vn=r

r

0

v

vs = 0

r

0

Disk-shaped cavity, D = 0.26 mm or 0.44 mm, R=5.0 mm

The shifts in resonant frequency vR ~ 650 Hz and bandwidth vB ~ 10 mHz tell about texture

Because s < swe can distinguish:

Normal Texture

Azimuthal Texture

Textures with defects

Rotation produces continuous counterflowv= vn - vs

Vs

Vs

Vs

Superfluid circulation Nκ :

vs(R) = Nκ(2πR)-1

N vortices

Rotating normal component :

vn(R) = R

Rotation

If counterflow | vn - vs | exceeds vF ,

texture tips azimuthally

TO detection of counterflow

WF

Wc

2.Hysteresis due to pinning

1.Hysteresis due to vc > 0

vs

or

vs

?

strong, vp> vc

vs

trap

weak, vp< vc

c

Horizontal scale set by c = vc/R

Vertical scale set by trap = vp/R

no pinning

max

c

2c

Strong pinning: trap = c

Because trapcan’t exceedc

(otherwise antivortex nucleates)

WF

Wc

Four fitting parameters:

WF Wc R-Rc Dn

D = 0.26 mm: R - Rc = 0.30 ± 0.10 mm

D = 0.44 mm: R - Rc = 0.35 ± 0.10 mm

Vortices nucleate at ~ D from edge

vc=cR

vc = 4vF ~ D-1, in agreement withvc∼ħ(2m3ac)-1

Critical velocity vs. core radius

Adapted from U. Parts et al., Europhys. Lett. 31, 449 (1995)

D=0.44mm

One MH vortex with one quantum of circulation

(only for D = 0.44mm)

c

Vc2

Vc1

No hysteresis!

Vc

F

D (mm)V+cV-cV-c1V-c2Vc(walls) (mm/s)

0.26 0.50.3 -- --0.2

0.44 0.30.20.20.50.2

Bulk dl-wall (theory: Kopu et al. Phys. Rev. B (2000))

What difference will two dl-walls make?

Critical velocity:

Just two dl-walls: pinning in field

Three times as much vorticity pinned on a domain wall at H=25 G than in uniform texture at H=0.

Other possible factors:

- Pinning in field might be stronger (vortex core shrinks with field).

- Different types of vortices in weak and strong fields.

Vortices

AT

UT

PT

D=0.26mm

D = 0.44 mm

Theory: bulk dl-wall

(Kopu et al, PRB 2000)

D = 0.26 mm

Theory: bulk l-wall

NtoA after rotation in field H >Hd: l–walls

With many walls in magnetic field: vc

vs

vs(R) = Nκ0(2πR)-1, trap = vs/R

In textures with domain walls: total circulation of ~ 50 0 of both directions can be trapped after stopping rotation

Strong pinning: single parameter vc :

c = vc/R trap = vc/R

++ (lz=+1, dz=+1)

+- (lz=+1, dz=-1)

-+ (lz=-1, dz=+1)

-- (lz=-1, dz=-1)

dl-wall

l-wall

d-wall

can carry vorticity

3-wall junctions might play a role of pinning centres

Trapping of vorticity by defects of order parameter is intrinsic pinning

vs.pinning due to extrinsic inhomogeneities (grain boundaries or roughness of container walls)

Intrinsic pinning in chiral superconductors

In chiral superconductors, such as Sr2RuO4, UPt3 orPrOs4Sb12,vortices can be trapped by domain walls between differently oriented ground states [Sigrist, Agterberg 1999, Matsunaga et al. 2004]

Anomalously slow creep and strong pining of vortices are observed as well as history

dependent density of domain walls (zero-field vs field-cooled) [Dumont, Mota 2002]

D=0.26mm

D=0.44mm

++ (lz=+1, dz=+1)

+- (lz=+1, dz=-1)

-+ (lz=-1, dz=+1)

-- (lz=-1, dz=-1)

dl-wall

l-wall

d-wall

can carry vorticity

l

dl

Edl= El = Ed

Edl<< El» Ed

(expected for D >> ξd = 10 μm)

d

l

dl

d

Then vortices could be trapped too

++ (lz=+1, dz=+1)

-- (lz=-1, dz=-1)

dl-wall

To be metastable, need pinning on surface roughness

E.g. the backbone of vortex sheet in Helsinki experiments

No metastability in long cylinder

In 3He-A, we studied dynamics of continuous vortices in different l-textures.

Critical velocity for nucleation of different vortices observed and explained as intrinsic processes (hydrodynamic instability).

Strong pinning of vorticity by multidomain textures is observed. The amount of trapped vorticity is fairly universal.

General features of vortex nucleation and pinning are understood. However, some mysteries remain.

The 2-dimensional 4-state mosaic looks like a rich and tractable system. We have some experimental insight into it. Theoretical input is in demand.

v > vc

v > vM

v

FM

to remove an existing vortex (vM) or to create an antivortex (vc)?

Pinning potential is quantified by “Magnus velocity” vM= Fp /s0D

(such that Magnus force on a vortex FM = sD0vequalspinning force Fp)

Weak pinning, vM < vc

Strong pinning, vM > vc

Annihilation with antivortex

Unpinning by Magnus force

In experiment, vp = min (vM, vc)

(i.e. the critical velocity is capped by vc)

All vortices are pinned forever

Maximum pers is limited to c

due to the creation of antivortices

strong, vp> vc

trap

weak, vp< vc

c

no pinning

max

c

2c

- Pinning force on a vortex Fp equals Magnus force FM= (sD0) v
- Counterflow velocity v equals vc (nucleation of antivortices)

two critical parameters: vc and vp (because Magnus force ~ vs):

(anti)vortices can nucleate anywhere when |vn-vs| > vc

existing vortices can move when |vn-vs| > vp

If vc< vp (strong pinning), |v| = vc

If vc > vp (weak pinning), |v| = vp vp=Fp/ s0D

In superconductors, vp (Bean-Levingston barrier) is small but flux lines can not nucleate in volume,

hence superconductors are normally in the pinning-limited regime |v| = vp even though vc< vp .

uniform

rotation

rotation

rotation

azimuthal

domain walls

planar

planar

domain walls

Trapped vorticity

vs

In textures with domain walls:

total circulation of ~ 50 0 of both directions can be trapped after stopping rotation

vs(R) = Nκ0(2πR)-1

trap = vs/R

vc

vc~1 mm/s

vc∝H

vc∝D-1

H

Hd≈25 G

HF=2-4 G

ħ

ħ

=

=

v

1

mm/s

~

v

c

x

c

2

m

2m

D

3

D

3

Hydrodynamic instability at vc∼ħ(2m3ac)-1 (Feynman)

(when l is free to rotate)

Soft core radius ac can be manipulated

by varying either:

slab thickness D

♦H= 0 :ac ∼D→vc∝D-1

or magnetic field H

♦2-4 G < H< 25 G:ac ∼ξH ∝H-1→vc ∝H

♦H> 25 G :ac ∼ξd = 10 μm →vc∼1 mm/s

Alternative theory

(numerical simulations for v = 3 vF)

lz=+1

dz=+1

lz=+1

dz=-1

lz=+1

dz=+1

lz=-1

dz=-1

lz=-1

dz=+1

lz=-1

dz=+1

dl-wall

l-wall

d-wall

or

dl-walls only

unlocked walls present

Models of critical state

?

vs

Horizontal scale set by c = vc/R

Vertical scale set by trap = vp/R

Strong pinning (vM>vc):

Single parameter, vc :

c = vc/R trap = vc/R

Weak pinning (vp<vc):

Two parameters, vc and vM:

c = vc/R trap = vp/R

Hysteretic “remnant magnetization”

(p.t.o.)

?

vs

Horizontal scale set by c = vc/R

Vertical scale set by trap = vp/R

What sets the critical state of trapped vortices?