Rotating 3HeA in a slab geometry: vortex pinning and persistent currents in ltextures with defects

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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
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3He-A order parameter
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)(mjinj)dµ Velo
city of flow depends on 3 d.o.f. vs
-h(2m3)-1(??cosß?a)
Continuous vorticity large length scale
Discrete degeneracy domain walls
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Groundstates, vortices, domain walls (slab
geometry, small H and vs)
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Topological defects (textures)
Two-quantum vortex
Azimuthal component of superflow
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Vortices in bulk 3He-A(Equilibrium phase
diagram, Helsinki data)
LV2 similar to CUV except d l (narrow range of
small ?)
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Models for vc (intrinsic processes)
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 lt H lt 25 G Rcore ?H
? H-1 ? vc ? H ? H gt 25 G Rcore ?d 10 µm
? vc1 mm/s
(Feynman 1955, et al)
or
When l is aligned with v (Bhattacharyya, Ho,
Mermin 1977) Instability of v-aligned l-texture
at
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or
Groundstate (choice of four)
Multidomain texture (metastable)
(obtained by cooling at H0 while rotating)
(obtained by cooling while stationary)
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Also possible
dl-walls only
d-walls only
(obtained by cooling at H0 while rotating)
(obtained by cooling while stationary)
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Fredericksz transition (flow driven 2nd order
textural transition)
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)
vF ?FR
vF D-1 HF D-1
Theory (Fetter 1977)
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Ways of preparing textures
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, ? gt ?F, H 0 makes azimuthal
textures
Applying H gt HF at ? 0 makes planar texture,
then ? gt ?F two dl-walls on demand
Rotating at ? gt vcR introduces vortices Value
of vc and type of vortices depend on texture
(with or without domain walls)
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Rotating torsional oscillator
Disk-shaped cavity, D 0.26 mm or 0.44 mm, R5.0
mm
The shifts in resonant frequency vR 650 Hz and
bandwidth vB 10 mHz tell about texture
Because ?s?? lt ?s? we can distinguish
Normal Texture
Azimuthal Texture
Textures with defects
Rotation produces continuous counterflow v vn -
vs
Vs
Vs
Vs
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Principles of vortex detection
If counterflow vn - vs exceeds vF , texture
tips azimuthally
TO detection of counterflow
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Main observables
2. Hysteresis due to pinning
1. Hysteresis due to vc gt 0
vs
or
vs
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Hysteresis due to pinning
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Uniform texture, positive rotation (H 0)
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 with vch(2m3ac)-1
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Critical velocity vs. core radius
Adapted from U. Parts et al., Europhys. Lett. 31,
449 (1995)
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Uniform texture, weak pinning
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Uniform texture, weak pinning
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Handful of pinned vortices
D0.44mm
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When no pinned vortices leftCan tell the
orientation of l-texture
One MH vortex with one quantum of circulation
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Negative rotation strange behaviour
(only for D 0.44mm)
?c
Vc2
Vc1
No hysteresis!
Vc
?F
D (mm) Vc V-c V-c1 V-c2 Vc(walls) (mm/s) 0.26
0.5 0.3 -- -- 0.2 0.44 0.3 0.2 0.2 0.5 0.2
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What difference will two dl-walls
make? Critical velocity
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Just two dl-walls pinning in field
Three times as much vorticity pinned on a domain
wall at H25 G than in uniform texture at
H0. 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
D0.26mm
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With many walls in magnetic field vc
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Trapped vorticity
vs
vs(R) N?0(2pR)-1, ?trap vs/R
In textures with domain walls total circulation
of 50 ?0 of both directions can be trapped
after stopping rotation
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Pinning by networks of walls
Strong pinning single parameter vc ?c vc /R
?trap vc/R
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Web of domain walls
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 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
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Energy of domain walls
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Web of domain walls
l
dl
Edl El Ed Edl ltlt El Ed (expected
for D gtgt ?d 10 µm )
d
l
dl
d
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What if only dl-walls?
(lz1, dz1) -- (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
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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.
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Unpinning mechanisms
to remove an existing vortex (vM) or to create an
antivortex (vc)?
Pinning potential is quantified by Magnus
velocity vM Fp /?s?0D (such that Magnus force
on a vortex FM ?sD?0v equals pinning force Fp )
Weak pinning, vM lt vc
Strong pinning, vM gt vc
Annihilation with antivortex
Unpinning by Magnus force
In experiment, vp min (vM, vc) (i.e. the
critical velocity is capped by vc)
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Model of strong pinning
All vortices are pinned forever
Maximum ?pers is limited to ?c due to the
creation of antivortices
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Two models of critical state

• Pinning force on a vortex Fp equals Magnus force
  FM (?sD?0) 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 gt vc existing vortices can
move when vn-vs gt vp
If vclt vp (strong pinning), v vc If vc gt
vp (weak pinning), v vp vpFp/ ?s?0 D
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 vclt
vp .
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Trapping by different textures
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domain walls
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Trapped vorticity
In textures with domain walls total circulation
of 50 ?0 of both directions can be trapped
after stopping rotation
vs(R) N?0(2pR)-1 ?trap vs/R
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Theory for vc (intrinsic nucleation)

• Hydrodynamic instability at vch(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 lt H lt 25 G ac ?H ?H-1 ? vc ? H
• ? H gt 25 G ac ?d 10 µm ? vc1 mm/s

Alternative theory
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vc D-1 Why? Not quite aligned texture!
(numerical simulations for v 3 vF)
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However, these are also possible
or
unlocked walls present
dl-walls only
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Models of critical state
Horizontal scale set by ?c vc /R Vertical
scale set by ?trap vp /R
Strong pinning (vM gt vc) Single parameter, vc
?c vc /R ?trap vc/R
Weak pinning (vp lt vc) Two parameters, vc and vM
?c vc /R ?trap vp/R
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Hysteretic remnant magnetization
Horizontal scale set by ?c vc /R Vertical
scale set by ?trap vp /R
What sets the critical state of trapped vortices?