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

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

Andrei Golov:

3He-A: order parameter

vs

vs

l dp-wave, spin triplet Cooper pairs

Two anisotropy axes:

l - direction of orbital momentum

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

Continuous vorticity: large length scale Discrete degeneracy: domain walls

l

nm

γ

β

αOrder 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)

Groundstates, vortices, domain walls:

(slab geometry, small H and vs)

=0 vs=0

>0

vs>0

>0

Topological defects (textures)

Azimuthal component of superflow

Two-quantum vortex

Vortex and wall can be either dipole-locked or unlocked

Rcore~ 0.2D

(lz, dz)=

Domain walls

Vortices in bulk 3He-A(Equilibrium phase diagram, Helsinki data)

LV2similar to CUV except d = l(narrow range of small )

dl-wall

l-wall

ATC-vortex (dl)

ATC-vortex (l)

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

HF=2-4 G

vc

H

vd~1 mm/s

Hd≈25 G

vc∝D-1vc∝H

vc∼vd

Models for vc (intrinsic processes)

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

Instability of v-aligned l-texture: at

or

Rcore2m

v3

c

ħ

mm/s12 3

=Dm

v

ħc

(Feynman 1955, et al…)

lz=+1dz=-1

lz=+1dz=+1

lz=-1dz=-1

lz=-1dz=+1

d-wall

dl-walll-walllz=+1dz=-1

lz=+1dz=+1

lz=-1dz=+1

lz=-1dz=-1

orGroundstate (choice of four)

Multidomain texture(metastable)

(obtained by cooling at H=0 while rotating)

(obtained by cooling while stationary)

Also possible:

d-walls only dl-walls only

(obtained by cooling at H=0 while rotating)

(obtained by cooling while stationary)

lz=+1dz=+1

lz=-1dz=-1

lz=+1dz=+1

lz=+1dz=-1

Fredericksz transition (flow driven 2nd order textural transition)

0.0 0.5 1.00.0

0.5

1.0

v c / v

FH / H

F

vF = FR

Orienting forces: - Boundaries favour l perpendicular to walls (“uniform texture”, UT)- Magnetic field H favours l (via d) in plane with walls (“planar”, PT)- Superflow favours l tends to be parallel to vs (“azimuthal”, AT)

Theory (Fetter 1977):

vF

122

HF

Hv vF ~ D-1 HF ~ D-1

2 walls

Ways of preparing textures

vortices

uniform

rotation

azimuthal

domain walls

rotation

planar uniform

H

Initialpreparation

Uniform l-texture: cooling through Tc while rotating:

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: two dl-walls on demand

Rotating at > vcR introduces vortices

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

Rotating torsional oscillator

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

Rotation produces continuous counterflow v = vn - vs

H

Vs Vs Vs

Normal Texture

Azimuthal Texture

Textures with defects

Because s < s we can distinguish:

0 rvs = 0

vn= rv

0 r

vn= rvvs = 0

Principles of vortex detection

Superfluid circulation Nκ :vs(R) = Nκ(2πR)-1N vortices

Rotating normal component :vn(R) = R

Rotation

If counterflow | vn - vs | exceeds vF ,texture tips azimuthally

TO detection of counterflow

Main observables

F

c

1. Hysteresis due to vc > 0 2. Hysteresis due to pinning

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

-4

-3

-2

-1

0

BtoAD = 0.26 mm

R (

mH

z)

(rad/s)

1)2)3)4)

vs or vs

Hysteresis due to pinning

no pinning

weak, vp< vc

Horizontal scale set by c = vc /R

Vertical scale set by trap = vp /R

trap

c

c

2c max

vs

strong, vp> vc

?

Strong pinning: trap = c

Because trap can’t exceed c

(otherwise antivortex nucleates)

Uniform texture, positive rotation (H = 0)

Four fitting parameters:

F c R-Rc

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

F

c

0.88 0.90 0.92 0.94 0.96 0.98 1.000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

D = 0.26 mm

v c (m

m/s

)

T / Tc

D = 0.44 mm

vc = cR

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

Critical velocity vs. core radius

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

10-1 100 101 102 103 104 105100

101

102

103

104

105

106

vc = / (2R

core)

3He-AH = 0

3He-AH >H

d

3He-B

4He

v c /

(mm

-1)

Rcore

(nm)

10-1 100 101 102 103 104 105100

101

102

103

104

105

106

vc = / (2R

core)

3He-AH >H

d

3He-B

4He

v c /

(mm

-1)

Rcore

(nm)

Uniform texture, weak pinning

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.012

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

c

- = 0.06 rad/s

(vc

-= 0.3 mm/s)

-

saturated = -0.0092 rad/s

(N-

trapped = 11)

c

+ = 0.10 rad/s

(vc

+= 0.5 mm/s)

+

saturated = 0.0046 rad/s

(N+

trapped = 5)

D = 0.26 mm

pers (

rad

/s)

prep

(rad/s)

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

Nu

mb

er

of t

wo

-qu

an

ta v

ort

ice

s

Uniform texture, weak pinning

-0.10 -0.05 0.00 0.05 0.10-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

0.005

Good texture

N = (2/)R

2

R = 5 mm

pers (

rad

/s)

prep

(rad/s)

-3

-1

1

3

5

7

9

11

Nu

mb

er

of q

ua

nta

of c

ircu

latio

n, N

Handful of pinned vortices

0.00 0.01 0.02 0.03 0.04 0.05

-4.0

-3.0

-2.0

-1.0

0.0

Re

son

an

t fre

qu

en

cy s

hift

(m

Hz)

(rad/s)

(rad/s) before sweep:

0.000 -0.002 -0.005 -0.007 -0.009 -0.011

2 4 6 80

1

2

3

4

5

6

7

8

Nu

mb

er

of

Eve

nts

Quantum number N

D=0.44mm

When no pinned vortices leftCan tell the orientation of l-texture

0.00 0.01 0.02 0.03 0.04-5

-4

-3

-2

-1

0

Fre

qu

en

cy S

hift

(m

Hz)

|| (rad/s)

Positive Negative

One MH vortex with one quantum of circulation

Negative rotation: strange behaviour

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020Order:1) 2) 3) 4) 5)

pers (

rad

/s)

prep

(rad/s)

(only for D = 0.44mm)

0.00 0.02 0.04 0.06 0.08 0.10 0.12

0

2

4

6

8

10

12

B (

mH

z)

(rad/s)

Acceleration Deceleration

Vc

Vc1Vc2

D (mm) V+c V-c V-c1 V-c2 Vc(walls) (mm/s)0.26 0.5 0.3 -- -- 0.20.44 0.3 0.2 0.2 0.5 0.2

No hysteresis!

c

F

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

two domain walls v

c, 0.26 mm

vc, 0.44 mm

v c (m

m/s

)

H (Gauss)

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

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

uniform azimuthal v

c, 0.26 mm

vc, 0.44 mm

v c (m

m/s

)

H (Gauss)

What difference will two dl-walls make?Critical velocity:

Just two dl-walls: pinning in field

0.000

0.005

0.010

0.015

0.020

0.0250 5 10 15 20 25

Magnetic Field (Gauss)

pers (

rad/

s)

Good texture

N2

0

= 2R2pers

/20

0

5

10

15

20

25

N2

0 [num

ber

of

two

-qua

ntu

m v

ortic

es]

0 5 10 15 20 250.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

vc (vortex nucleation)

vF (Freedericksz tr.)

F

HF

(r

ad/s

)

Magnetic Field (Gauss)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8 V

elo

city

[ v

=

R] (

mm

/s)

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.AT

UT PT

Vortices

D=0.26mm

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0 NtoA, D = 0.44 mm BtoA, D = 0.44 mm

v c (m

m/s

)

H (Gauss)

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

v c (m

m/s

)

H (Gauss)

Theory: bulk l-wall

Theory: bulk dl-wall(Kopu et al, PRB 2000)

D = 0.44 mm

D = 0.26 mm

With many walls in magnetic field: vc

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

v c (m

m/s

)

H (Gauss)

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

Trapped vorticity

-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06

0

2

4

6

trap

Uniform

(rad/s)

0

2

4

6

trap

trap

NtoA

B

(m

Hz)

0

2 trap

trap

BtoA

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

0 10 20 30 40 50 60 700.000

0.005

0.010

0.015

0.020

0.025

N to A

tra

p (ra

d/s

)Time (hours)

0

10

20

30

40

50

60

Nu

mbe

r o

f ci

rcla

tion

qu

an

ta

vs(R) = Nκ0(2πR)-1, trap = vs/R

vs

Pinning by networks of walls

-0.4 -0.2 0.0 0.2 0.4

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

c= 0.03 rad/s

(vc= 0.15 mm/s)

trap

= 0.03 rad/s

trap

(ra

d/s)

prep

(rad/s)

NtoA, 0.44 mm BtoA, 0.44 mm NtoA, 0.26 mm BtoA, 0.26 mm

Strong pinning: single parameter vc :

c = vc /R trap = vc/R

Web of domain walls

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 or PrOs4Sb12,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]

Trapping of vorticity by defects of order parameter is intrinsic pinning

3-wall junctions might play a role of pinning centres

dl-walll-walld-wall

can carry vorticity

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

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

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

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

Energy of domain walls

0 20 40 60 80 1000

20

40

60

80

100

120

140

160

180

dl-wall

d-wall

l-wall

Fre

e E

nerg

y F D

2 / s||

Slab Thickness D/D

D=0.26mm D=0.44mm

Web of domain walls

dl-walll-walld-wall

can carry vorticity

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

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

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

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

Edl = El = Ed

Edl << El Ed

(expected for D >> ξd = 10 μm )

dl

d

l

dll

d

What if only dl-walls?

To be metastable, need pinning on surface roughness

dl-wall

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

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

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

No metastability in long cylinder

Then vortices could be trapped too

Summary

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.

Unpinning by Magnus force Annihilation with antivortex

In experiment, vp = min (vM, vc)(i.e. the critical velocity is capped by vc)

Unpinning mechanisms

FM

v

v > vMv > vc

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 = sD0v equals pinning force Fp )

Weak pinning, vM < vc Strong pinning, vM > vc

Model of strong pinning

All vortices are pinned forever

Maximum pers is limited to c

due to the creation of antivortices

vc

vc

vc

vp

vp

vc*

*

*

1) No pinning vp = 0

Cryostat rotatingW > 2Wc

2) Weak pinning 0 < vp < vc

3) Strong pinning vc < vp

Cryostat stationaryW = 0

vn

vs

n

Two models of critical state

1. Pinning force on a vortex Fp equals Magnus force FM= (sD0) v

2. Counterflow velocity v equals vc (nucleation of antivortices)

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 .

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

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

vp=Fp/ s0 D

strong, vp> vc

trap

c

c

2c max

no pinning

weak, vp< vc

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

-0.4 -0.2 0.0 0.2 0.4-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

trap

= 0.005 rad/s

trap

= 0.03 rad/s

c= 0.10 rad/s (v

c=0.5 mm/s)

c= 0.03 rad/s (v

c=0.15 mm/s)

N, 0.26mm N, 0.40mm

B, 0.26mm B, 0.40mm

- cooled

tr

ap (

rad

/s)

max

(rad/s)

Trapping by different textures

domain walls

rotation

planar uniform

rotation

azimuthal

domain walls

rotation

planar

0 5 10 15 20 250.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16 HF

HD= 15 G = 1.5 mT

vD= 0.7 mm/s

C

(ra

d/s

)

Magnetic Field (Gauss)

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

-4

-3

-2

-1

0

BtoAD = 0.26 mm

R (

mH

z)

(rad/s)

1)2)3)4)

-0.10 -0.05 0.00 0.05 0.10

-4

-3

-2

-1

0

trap

trap

R (

mH

z)

(rad/s)

1 2 3 4

0 10 20 30 40 50 60 700.000

0.005

0.010

0.015

0.020

0.025

N to A

pers (

rad

/s)

Time (hours)

0

10

20

30

40

50

60

Nu

mb

er

of

qu

an

ta v

ort

ice

s

In textures with domain walls:

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

Trapped vorticity

vs(R) = Nκ0(2πR)-1

trap = vs/R

vs

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

HF=2-4 G

vc

H

vc~1 mm/s

Hd≈25 G

vc∝D-1vc∝H

Theory for vc (intrinsic nucleation)

Alternative theory

vc ~ D-1 :

Why? Not quite aligned texture!

(numerical simulations for v = 3 vF)

-0.4 -0.2 0.0 0.2 0.4-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

Wc-=0.038 rad/s(v

c=0.19 mm/s)

W = 0.055rad/s(v

c= 0.28 mm/s)

Uniform 0.26 mm Model 0.26 mm Uniform 0.44 mm Model 0.44 mm

+

0 = 0.0047 rad/s

(N-

trap = 11, v

p=0.024 mm/s)

-

0= -0.0092 rad/s

(N-

trap= 22, v

p= 0.047 mm/s)

c

- = 0.055 rad/s

(vc

-= 0.28 mm/s)

c

+ = 0.095 rad/s

(vc

+= 0.48 mm/s)

tr

ap (

rad/

s)

prep

(rad/s)

However, these are also possible:

lz=+1dz=-1

lz=+1dz=+1

lz=-1dz=-1

lz=-1dz=+1

d-wall

dl-walll-wall

lz=+1dz=+1

lz=-1dz=+1

unlocked walls presentdl-walls only

or

-0.4 -0.2 0.0 0.2 0.4

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

c= 0.03 rad/s

(vc= 0.15 mm/s)

trap

= 0.03 rad/s

trap

(ra

d/s)

prep

(rad/s)

NtoA, 0.44 mm BtoA, 0.44 mm NtoA, 0.26 mm BtoA, 0.26 mm

-0.4 -0.2 0.0 0.2 0.4

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

trap

= 0.005 rad/s

c = 0.10 rad/s

(vc = 0.5 mm/s)

trap

(ra

d/s)

prep

(rad/s)

Uniform texture, D = 0.26 mm

Models of critical state

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

Horizontal scale set by c = vc /R

Vertical scale set by trap = vp /R vs

?

-0.4 -0.2 0.0 0.2 0.4

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

c= 0.03 rad/s

(vc= 0.15 mm/s)

trap

= 0.03 rad/s

trap

(ra

d/s)

prep

(rad/s)

NtoA, 0.44 mm BtoA, 0.44 mm NtoA, 0.26 mm BtoA, 0.26 mm

-0.4 -0.2 0.0 0.2 0.4

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

trap

= 0.005 rad/s

c = 0.10 rad/s

(vc = 0.5 mm/s)

trap

(ra

d/s)

prep

(rad/s)

Uniform texture, D = 0.26 mm

Hysteretic “remnant magnetization”

Horizontal scale set by c = vc /R

Vertical scale set by trap = vp /R vs

?

(p.t.o.)

What sets the critical state of trapped vortices?