Numerical simulation evidence of dynamical transverse Meissner effect and moving Bose glass phase

5 Pages·0.76 MB·English

by  E. Olive

Numerical simulation evidence of dynamical transverse Meissner effect and moving
Bose glass phase
E. Olive,1 J.C. Soret,1 P. Le Doussal,2 and T. Giamarchi3
1LEMA, CNRS-UMR 6157, CEA, Universit´e Franc¸ois Rabelais, Parc de Grandmont 37200 Tours, France
3 2LPTENS, CNRS-UMR 8549, ENS, Universit´e Paris-Sud, 24 rue Lhomond 75231 Cedex 05 Paris, France
0 3DPMC, University of Geneva, 24 Quai Ernest Ansermet, 1211 Geneva 4, Switzerland
0 (Dated: 2nd February2008)
2
Wepresent3Dnumericalsimulationresultsofmovingvortexlatticesinpresenceof1Dcorrelated
n
disorder at zero temperature. Our results with field tilting confirm the theoritical predictions of a
a
moving Bose glass phase, characterized by transverse pinning and dynamical transverse Meissner
J
effect, the moving flux lines being localized along the correlated disorder direction. Beyond a
4
critical transverse field,vortex lines exhibit along all their length a ”kink”structureresulting from
2
an effective static ”tin roof” pinningpotential in thetransverse direction.

]
n
o Depinninganddynamicalorderingofperiodiclattices, atureinarandompinninglandscapeofcolumnardefects.
c suchasvortexlattices(VL)drivenbyanexternalforcein Theelasticallymovingvortexlinesconfirmstheexistence
- presence ofstatic point disorderhas been widely studied of MBoG displaying DTME. Above a critical angle, the
r
p these last years. A large velocity expansion [1] showed vortexlinesexhibita”kink”structurealongtheirlength.
u
thatthemotionreducestheeffectsofdisorder. Inthisap- This results from the existence of an effective static pin-
s
. proximationthe effects ofdisorderon movingVL results ninglandscapewhichisalmostperiodicinthetransverse
t
a only in an additional ”shaking” temperature Tsh 1/v direction to the vortex flow. Finite size effects are dis-
m with expectation of moving crystalline VL at lar∼ge ve- cussed within a simple model of a single flux line in a
- locities. It was however shown in [2] that, due to the static periodic potential. In particular, it confirms the
d
componentsofthedisorderanddisplacementstransverse existence of a moving ”kink” structure observed in our
n
to the motion, a static disorder persists even at large simulation.
o
c velocity, and leads to a moving glass phase (MG). The WemodelastackofNz Josephson-coupledparallelsu-
[ motion of particles in the MG occurs along rough static perconductingplanes ofthicknessdwithinterlayerspac-

channels. Depending on the velocity the channels can ings. Eachlayerinthe(x,y)planecontainsN pancake
1 v
v be coupled or decoupled due to dislocations, thus giving vorticesinteractingwitharandompinningbackgroundof
1 two generic moving glasses. For weak disorder or large 1D correlated disorder, namely N columnar pins paral-
p
7 velocityind=3,channelsareelasticallycoupledleading leltothez direction. AtT =0theoverdampedequation
4
to a topologically ordered structure: the moving Bragg of motion of a pancake i in position r (z) reads
1 i
glass(MBG)[2]. Forlargerdisorderorind=2channels
0
dr
3 aredecoupledbydislocationsgivingamovingtransverse η i = Uvv(ρ ,z ) Uvp(ρ )+FL+Ftilt(z)
i ij ij i ip
0 glass (MTG) or smectic [2, 3] . The predicted channel dt − ∇ − ∇
j6=i p
/ structure and dynamical ordering was confirmed by nu- X X
t
a merical [4, 5, 6, 7] and further analytical [8] studies and where ρij and zij are the components of rij = ri rj
m −
observed experimentally [9]. in cylindrical coordinates, ρip is the in-plane distance
- between the pancake i and a pinning site in the same
d Within the MG model [2], the situation for correlated
layer at r , and is the 2D gradient operator act-
n disorder such as heavy-ion columnar tracks has been re- p ∇i
o ing in the (x,y) plane. The viscosity coefficient is η,
cently investigated [10]. For all velocities at T = 0 or
c FL = FLxˆ is the Lorentz driving force due to an ap-
v: for weak velocities at T > 0 it was predicted a moving plied current and Ftilt(z) is the surface force due to
Bose glass phase (MBoG) characterized by a transverse
i the field tilting away from the z axis in the y direc-
X critical force and a diverging tilt modulus due to local-
tion. This force acts as a torque on each flux line, i.e.
r ization effect arising from the columns. This novel fea- Ftilt(z =0)= Ftilt(z =N s)=FtiltyˆandFtilt(z)=0
a z
ture specific to correlated disorder results in a vanishing −
for pancakes in the bulk. The tilting force modulus is
tilt response below a critical transverse field. This so-
defined by Ftilt = ǫ2φ H /4π = 8πǫ2ǫ λ2 H /√3a2H ,
called dynamical transverse Meissner effect (DTME) is 0 y 0 ab y 0 z
where ǫ = (φ /4πλ )2, λ is the in-plane magnetic
0 0 ab ab
a crucial consequence of the MG theory; its observation
penetration depth, a is the average vortex distance,
0
should provide an unambiguous signature of the MBoG
H is the transverse field component, and ǫ is the
y
phase. However, up to now there is no experimental nor
anisotropy parameter. The intra-plane vortex-vortex re-
numerical simulation evidence for this phase.
pulsiveinteractionis givenbyamodified Besselfunction
This Letter reports 3D numerical simulation of field Uvv(ρ ) = 2ǫ dK (ρ /λ ). Following [11], the inter-
ij 0 0 ij ab
tiltinginamovinglayeredvortexsystematzerotemper- planeattractiveinteractionbetweenpancakesinadjacent
2
layers of altitude z and (z+s) reads Uvv(ρ ,z =s)=
ij ij
(2sǫ /π)[1+ln(λ /s)][(ρ /2r )2 1] for ρ 2r and
0 ab ij g − ij ≤ g 10
Uvv(ρ ,z = s) = (2sǫ /π)[1 + ln(λ /s)][ρ /r 2] 1.2
ij ij 0 ab ij g
−
otherwise; in this expression r =ξ /ǫ, where ξ is the 1
g ab ab
in-planecoherencelength. Finally,theattractivepinning 8 0.8 Z
potentialisgivenbyUvp(ρip)= αApe−(ρip/Rp)2,where 0.6
− 0.4
A =(ǫ d/2)ln[1+(R2/2ξ2 )] as proposed in [12] and α
p 0 p ab 0.2
is a tunable parameter. We consider periodic boundary 6
0
conditionsof(Lx,Ly)sizesinthe(x,y)planewhileopen tanθΒ 0 1 2Y3 4 5 N=19
boundariesaretakeninthez direction. Alldetailsabout z
4
our method for computing the Bessel potential with pe-
riodic conditions can be found in [13]. Molecular Dy- Nz=29
namics simulation is used for Nv = 30 vortex lines in a 2 Nz=39
rectangularbasiccell(Lx,Ly)=(5,6√3/2)λab,andfora Nz=49
number of layers varying from Nz =19 up to Nz =149. Nz=149
The number of columnar pins is set to Np = 30. We 0
0 0.2 0.4 0.6
consider the London limit κ = λ /ξ = 90, with an
ab ab H /H
average vortex distance a =λ , and d=2.83 10−3λ , y z
0 ab ab
s = 8.33 10−3λ , R = 0.22 λ , ǫ = 0.01, η = 1.
ab p ab
Thesevaluesarecoherentwithstronganisotropylayered
Figure 1: Average vortex line inclination tanθB = By/Bz
superconductors with B 1000 G. The choice of the
z ∼ versus the field inclination tanθH = Hy/Hz for several sys-
second order Runge-Kutta algorithm time iteration step tem sizes Nz = 19,29,39,49,149 (filled circles). The linear
δt is dictated by the dominant force FL, and we take responseofthepinningfreevortexsystemisshownwithopen
δt =7 10−2t , where t =ηξ /FL. We choose the tun- circles. Sizeeffectsshowthatthepartialtransversescreening
0 0 ab
able pinning parameter α = 1/25 so that the maximum observed for Hy/Hz 0.50 tends to be total in the infinite
≺
tphinenuinngitffoorrccee idsefiFnmvepadxb∼y tFh0e/B5ewssheelrientFer0ac=tio2nǫ.0dF/iλnaabllyis, tsihzeeclirmitiitcaNlztr→ans∞ve,rssheofiweilndgHthyce/Hexzis∼ten0c.5e0o.fItnhseerDt:TpMroEjebcetlioown
inthe(y,z)planeofthe30vortexlinescomposedofNz =149
the driving force applied along a principal vortex lattice
layers illustrating the MBoG phase (thick lines) and DTME,
directionxissettoFL 2.8F0. Thiscorrespondstothe compared to the vortex lines of the non-disordered system
∼
fully elastic flow limit since FL 25FL where FL is the (thin obliquelines). In theMBoG phase, the transversefield
∼ c c
critical Lorentz force along x. penetratesonlyatthesurfaceandvortexlinesinthebulkare
aligned with the columnar defect direction z. In the pinning
The ”experimental” procedure is the folllowing: we
freesituation,thevortexlinesformstraightlinesinanoblique
start in 2D with the (x,y) plane by randomly throw-
direction defined by the field inclination and Josephson cou-
ing N pancakes and N gaussian pins, and relaxation
v p pling. Notethatthewholevortexlatticeismovingelastically
with zero Lorentz force yields a vortex structure with in the x direction, i.e. perpendicularly to theimage.
dislocations. The Lorentz force is then slowly increased
far in the elastic phase up to FL 2.8F with steps
0
∼
δFL 3.5 10−3F0 every 3.6 104t0. The successive dy- straightvortexlinesmovingelasticallyalongthexdirec-
∼
namical regimes observed in our 2D vortex system are tion through a random columnar pin landscape. Then,
the following: (i) pinned regime where all vortices have starting from this situation, we now slowly tilt the mag-
zerovelocity;(ii)plasticchannelsflowingthroughpinned neticfieldHawayfromz inthey directionbyincreasing
regionsandwherethemotionisperiodic(washboardfre- the H component transversally to the vortex flow. For
y
quency) as seen in [5, 6]; (iii) plastic turbulent flow with several system sizes, Fig. 1 displays the vortex line re-
no stationnary vortices as seen in [6, 7]; (iv) weakly de- sponse to low field tilting, i.e. the average vortex line
coupledchannelsnamedMTGsincequasilongrangelon- inclination tanθ = B /B versus the field inclination
B y z
gitudinal order exists in each channel; (v) elastic phase tanθ = H /H , compared with the linear response of
H y z
where all vortices have the same average velocity and thepinningfreevortexsystem. Itshowsapartialscreen-
named MBG since no dislocation appears and the mo- ing of the transverse component H of the field, which
y
tionoccursthroughroughstaticchannelseveninthehigh tends to be total for infinite vortex line length since the
drivingphaseFL 25FLstudiedbelow. Our2Dnumer- slopes tend towards zero when N . This trans-
∼ c z → ∞
ical results are therefore in agreement with the moving verse screening is followed by a transition at a critical
glass theory developed in [2]. Finally, note that they are field Hc/H 0.50, above which the averagevortex line
y z ∼
consistent with the phase diagram region corresponding inclination recovers the one obtained without any pin-
to the intermediate pinning of Ref 7. ning. Therefore, Fig. 1 shows the existence of DTME
Now starting from this situation, we extend the vor- which is the very signature of the MBoG phase as pre-
ticesandthepinsinthez directioninordertoobtain3D dicted in [10]. As a good illustrationof DTME, we show
3
20
1.2
15 0.8
P7
Z (a)
tanθΒ10 P5 0.4
P3 0
5 0 1 2 Y 3 4 5
-0.2
(b)
dZ/dY
0 -0.4
0 0.2 0.4 0.6 0.8 1 1.2 0 1 2 Y 3 4 5
H /H
y z
Figure2: Plateauxandhysteresisobservedabovethetransi- Figure 3: (a) Projection in the (y,z) plane of the 30 mov-
tion for Nz =49. The origin of each plateau is connected to ing vortex line composed of Nz = 149 layers and obtained
a given number of ”pinned” regions passed through by each for Hy/Hz 0.53 while increasing the transverse field. In
∼
vortex line. For details about the plateaux labels and the grey are materialised the ”pinned” regions which define an
”pinned” regions defining the effective static pinning land- effective pinning landscape which is static and almost peri-
scape,seeFig.3andtextbelow. Finitesizeeffectsshowthat odic. (b)Derivativedz/dyforallthe30vortexlinesshownin
theplateauxandhysteresisdisappearintheinfinitesizelimit (a). The vortex line modulations along their length become
Nz . Plateaux andhysteresismaythereforebeobserved clearly visible. Taking into account the minima and inflex-
→∞
in thin real samples. ion points of the derivative allows the determination of the
”pinned”regions.
in the insert of Fig. 1, the vortex line configuration of
the MBoG comparedwith the non-disorderedlattice ob- tex lines with respect to z. The small ones are termed
tained for the same field tilting. One clearly sees that ”pinned” regions and are linked by ”unpinned” regions
below the transition the transverse field enters at the where the vortex inclination is bigger. The materiali-
surface without penetrating into the bulk, in opposition sation of the ”pinned” regions is shown in grey in Fig.
to the pinning free vortex system. So below the critical 3(a). We visualize inthis waya static pinning landscape
transverse field, the vortex lines remain parallel to the in the transverse direction y with effective columnar pin
columnardefectseventhoughtheyaremovingatenough width correspondingto twice the input pinning ”radius”
large velocity. R . Each plateau of Fig. 2 can be thereby labelled with
p
We shall now examine the situation above the crit- anoddnumbercorrespondingtothenumberof”pinned”
ical transverse field. Far above Hc/H and whatever regions passed through by each vortex line. In this way,
y z
the vortex line size is, the average vortex line inclina- whileincreasingthetransversefieldabovethetransition,
tion recovers the one obtained without any pinning (not the vortex lines firstly pass through 3 pinned regions for
shown). A more interesting case happens in the inter- a given range of H defining the ”P3” plateau, then the
y
mediate transverse field range above the transition and system jumps to the following plateau ”P5” where the
shownfor N =49 layersin Fig.2. Increasingthe trans- vortex lines pass through 5 pinned regions, and so on.
z
verse field above its critical value Hc/H 0.50 gives The same process holds while decreasing the transverse
y z ∼
rise to plateaux in the tanθ vs H /H curve. A con- field generating thereby hysteresis. Finally, the central
B y z
sequence of these plateaux is that decreasing the trans- pointdisplayedinFig.3(a)isthatthevortexlineswhich
verse field generates hysteresis in the vortex inclination. are moving in the x direction perpendicularly to the fig-
A typical vortex line configuration above the transition ure plane see an effective static tin roof potential in the
is shown for N = 149 layers in Fig. 3(a). A modulated transverse direction y. This is a direct consequence of
z
structure z(y) appears along their length. This modu- the existence of static pinned channels of motion [2, 10],
lation is more obvious looking at the derivative dz/dy and clearly shows that the action of the disorder in the
shown for all the vortex lines in Fig. 3(b). By using the moving system cannot be simply reduced [1] to a tem-
minima and maxima of the derivative we may roughly perature. Notethatthiseffectivepotentialisnotstrictly
define two parts of small and big inclinations in the vor- periodic because of the channel roughness.
4
We now turn to the discussion of the finite size effects
in our simulation by considering the tilt response of a
20
single static elastic vortex line in a tin roof pinning po- 20
tential at T = 0. Such a simple model allows to get
tanθ P7
a valuable insight into the transverse properties of the
10
moving vortex lattice in presence of correlated disorder, 15
e.g. the TME and the kink vortex line structure. The
energy of a line of length L is given by: 0 P5
0 0.4 0.8 1.2
tanθ 10 f*
L c du 2
E(u)= dz +V (u) +f(u(L) u(0)),(1)
eff
Z0 2(cid:18)dz(cid:19) ! − P3
5
where u(z) is the one-dimensional displacement field in
the y direction, c = ǫ2ǫ is the elastic constant, f H
0 y
∝
is a surface force, and V (u) = g 1 cos(4πu/√3a )
eff 0
−
withgbeingaconstant. Thereexist(cid:2)sagroundstateuL,f(cid:3) 00 0.2 0.4 0.6 0.8 1 1.2
which minimizes E(u). We calculate the tilt response of
f*
the vortexline definedby tanθ =(u (L) u (0))/L.
L,f L,f
−
Inthe limit L ,we finda criticalforcefc =4√gc/π
→∞
below which uL=∞,f 0, giving tanθ = 0 and hence Figure4: Plateauxandhysteresisobservedwithinthesimple
≡
TME. For f fc the vortex line is tilted and its ground model(Eq. 1) fora finitelength L=0.4 a0 and g/c=18.75.
≻
stateu (z)areexpressedintermsofJacobianellip- Blackfilledcirclesrepresentthemetastablestatesfollowedby
L=∞,f
tic functions which generate kink vortex line structures the elastic line when the surface force f is slowly increased,
then decreased. The white filled circles represent the true
asobservedinoursimulation(seeFig.3(a)). Thetiltre-
equilibrium states. Insert: Eq. 1 results in the infinite size
sponse is shown in the insert of Fig. 4 together with the
limit L showing the existence of a critical dimension-
pure elastic response, i.e. g = 0. In the latter case, the less forc→e f∞∗ = √3/2π2. g/c and the disappearance of the
c
groundstatesarelinearfunctionsandhencethevortexis plateaux and hysteresis.
a straightline of slope tanθ =f/c. The kink vortex line p
structure obviously rely only on the periodic potential.
Note that f∗ = √3f/8πc is the equivalent quantity of
H /H . For a finite size line and f = 0, the minimum
y z
energysolutionisu 0,i.e. thelineispinned. Iff
L,f=0
≡
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