Disorder induced power-law response of a superconducting vortex on a plane

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by  N. Shapira

Disorder induced power-law response of a superconducting vortex on a plane
N. Shapira,1 Y. Lamhot,1 O. Shpielberg,1 Y. Kafri,1 B. J. Ramshaw,2
D. A. Bonn,2 Ruixing Liang,2 W. N. Hardy,2 and O. M. Auslaender1
1Department of Physics, Technion - Israel Institute of Technology, Haifa, 32000, Israel
2Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada
(Dated: January 7, 2014)
We report drive-response experiments on individual superconducting vortices on a plane, a real-
ization for a 1+1-dimensional directed polymer in random media. For this we use magnetic force
microscopy (MFM) to image and manipulate individual vortices trapped on a twin boundary in
YBCO near optimal doping. We find that when we drag a vortex with the magnetic tip it moves
4 in a series of jumps. As theory suggests the jump-size distribution does not depend on the applied
1
force and is consistent with power-law behavior. The measured power is much larger than widely
0
accepted theoretical calculations.
2

n
a While the dynamics of driven systems in ordered me- can make the optimal path for an isolated vortex elab-
J diaarewellunderstood,disordergivesrisetomuchmore orate. Despite this, DPRM theory provides many pre-

6 elaborate behavior. Particularly interesting are phenom- dictions for disorder-averaged quantities [18]. For exam-

enaarisingfromtheinterplaybetweendisorderandelas- ple, the thermal and disorder averaged offset distance
]
n ticity [1, 2] such as the conformations of polyelectrolytes from the field axis zˆ (∆) scales like a power-law given
o [3] (e.g. polypeptides and DNA [4]), kinetic roughening by the wandering exponent ζ(d⊥): (cid:104)∆(cid:105) ≡ δR ∼ Lζ(d⊥)
c
ofdriveninterfaces(e.g. wettinginpaper[5,6],magnetic for L (cid:29) a (L is the sample thickness, a is a sample-
- z z
r and ferroelectric domain wall motion [7–10], the growth dependent lower-cutoff), which is a universal number.
p
ofbacterialcolonyedges[5]),non-equilibriumeffectsthat Theoretically ζ(d ) describes a wide variety of systems
u ⊥
occur in randomly stirred fluids [11] and more. Super- [13] but there are only a few measurements of it [7–
s
. conducting vortices, in materials in which they behave 10, 19–21]. While a power-law also describes classical
t
ma like elastic strings, are among the most important ex- random walks (δR ∼ L21) disorder both enhances wan-
amples of such systems [12]. Despite a dearth of direct dering (ζ(d ) > 1) and stretches the distribution of off-
⊥ 2
d- experimental proof, these quantized whirlpools of super- set distances W(∆) from gaussian to W(∆)∼∆−αtheory
n current are considered textbook examples of the theory (αtheory > 0), significantly increasing the prevalence of
o of directed polymers in random media (DPRM) [13–15], trajectories with large excursions [18].
c a foundation model for systems where disorder and elas-
[ The power-law form of W(∆) implies the absence of a
ticitycompete. Thismodel,thatyieldsmanyresultsthat
characteristic length-scale and the existence of a signifi-
1 are considered generic and universal, provides the back-
cant number of vortex trajectories with a wide variety of
v drop for our experiment.
5 ∆’sandwithfree-energiesalmostaslowasthatoftheop-
9 Here we concentrate on vortices that are trapped on timalvortexpath. Sincethesetrajectoriesconstitutethe
9 a twin boundary (TB), a planar defect in YBa2Cu3O7−δ low-energy excitations of the system they are important
0 (YBCO)[16]. Wecoolthesamplethroughthesupercon- for thermodynamics and response functions [18]. While
.
1 ducting transition temperature Tc in the presence of an in thermal equilibrium the system has time to find these
0 external magnetic field H(cid:126) =Hzˆ, which directs the curve metastablestatesitisnotclearwhathappensoutofequi-
4
along which vortices cross the sample. Figure 1a depicts librium, although one can expect that near equilibrium
1
avortexawayfromaTB(VinFig.1a)thatisfreetome- these trajectories remain important.
:
v ander in the d = 2 directions perpendicular to H(cid:126). For
i ⊥ Inthisworkweexperimentallycharacterizethetrajec-
X a vortex trapped on a TB (TBV in Fig. 1a) the mean-
tories of individual vortices confined to move on a TB.
dering is limited to a plane, i.e. d =1. We concentrate
r ⊥ Unlikemostpreviousworkweusealocalprobe(magnetic
a on TB-vortices both because the reduced dimensionality
force microscopy, MFM) to measure individual vortices.
makes data analysis simpler and, more importantly, be-
TheheartofMFMisasharpmagnetictipsituatedatthe
cause, unlike DPRM in higher dimensions, 1+1-DPRM
end of a cantilever driven to oscillate in the z-direction
is a tractable model [17].
normal to the sample surface at a resonant frequency f.
The path of a vortex across a sample is determined by A force F(cid:126) = F xˆ+F yˆ+F zˆ acting on the tip shifts f
x y z
the competition between elasticity and disorder: while by ∆f = f −f ≈ −f /(2k)∂F /∂z (f is the natural
0 0 z 0
meandering allows a vortex to lower the energy of the resonant frequency, k is the cantilever spring constant, z
system by locating its core near defects, the associated is the tip-sample distance) [22]. For an image we record
stretching is limited by finite line tension κ [12]. As ∆f while rastering the tip at constant z. In addition we
a result the unavoidable random disorder in a sample usethetip-vortexinteractiontoperturbvorticesindivid-
2
]m[ x
a. b. c. 6.3 d. 01 5 0 5- 01- 8-
7.3 6-
89..33 4-
V TBV 123...4444 𝚫𝒇 0242- ym []
4.4 6
5.4 𝟒 𝝁𝒎 m n7175.2511=pitZ ,K93.4=T @ csa.dwb-fd-662_CS\52-10-2102\:Y8
𝑻 ≈ 𝟏𝟓𝑲
𝚫𝒇
e.
𝐻
twin boundary 𝑯 ≈ 𝟐𝒎𝑻
𝟓 𝝁𝒎 𝟐 𝝁𝒎
𝑻 ≈ 𝟒.𝟑𝟖𝑲 𝑻 ≈ 𝟏𝟓𝑲
FIG. 1. Vortices on and off a twin boundary (TB). (a) Illustration of vortices in a superconductor. The left vortex (V) can
meander in d = 2 dimensions perpendicular to an external magnetic field H(cid:126) while the vortex trapped on the TB (TBV), a
⊥
commonplanardefect,canmeanderonlyontheplane,i.e. ind =1dimensions. (b)Polarized-lightmicroscopyphotoofour
⊥
80µm-thick sample revealing two TBs (white arrows). (c) MFM scan of vortices (black discs) that form a high density chain
along a TB and an Abrikosov lattice around it (z ≈1.15µm, ∆f spans 0.93Hz). (d) MFM scan at 0≤H ≤10µT. Vortices
(blue discs) accumulate on a TB and exhibit 1+1-dimensional physics (z ≈0.28µm, ∆f spans 1.6Hz). (e) Many vortices in
the chain in (d) are isolated because their separation is much larger than λ ≈120nm (here z≈0.4µm, ∆f spans 0.6Hz).
ab
ually [23]. Such perturbations show up as abrupt shifts pattern consisted of line-scans in which the tip moved
of the signal from a vortex, which we dub ”jumps”. back and forth (Fwd/Bwd) at 125nm along the x-axis
sec
ThesampleweusedisanearlyoptimallydopedYBCO paralleltotheTB.Aftereachline-scanwereducedz and
single crystal (T ≈ 91K [24]) grown from flux in a steppedthetipinthey-direction. Sincetheforcethetip
c
BaZrO crucible for high purity and crystallinity [25]. exerts on a vortex depends on both z and the tip-vortex
3 (cid:112)
The L = 80µm-thick platelet-shaped sample has faces lateral distance ρ = (x−xv)2+(y−yv)2 (xvxˆ+yvyˆ
parallel to the crystal ab-plane and contains two parallel is the vortex position in the scan, see [30]), a complete
TBs(Fig.1b). FieldcoolingwasdonewithH(cid:126) =Hzˆpar- scan series gives the response of a TB-vortex to a wide
allel to the crystal c-axis and along the TB plane with range of forces along the TB, Fx.
the tip magnetized for attractive tip-vortex interactions. Figure 2a shows typical line-scans for an almost static
Figure 1c is an MFM scan of vortex arrays on a TB vortex. ∆f becomes increasingly negative as the tip ap-
and around it for H ≈ 2mT. Such a highly ordered proaches the vortex due to the increasing tip-vortex at-
Abrikosovlattice[26,27]atsuchalowfieldatteststothe traction until it passes the minimal ρ in the line-scan.
scarcity of strong defects other than the TB. Figure 1d From there ∆f increases until the interaction becomes
isanMFMscanofaTBat0≤H ≤10µT. Inthisnear- negligible again. The line-scans in Fig. 2a show one
zero field almost all of the vortices were trapped by the of the first jumps for this particular vortex - a shift
TBs, further attesting to the high quality of the sample in ∆f(x) at x = x . We associate this shift with
jump
andinagreementwithearlyexperimentsshowingthatin a tip-induced abrupt change in the position of the up-
YBCO TBs are strong vortex traps [28]. Despite their per part of the vortex. We determine the jump length
relative high density, many of the TB-vortices can be ∆ = |x −x∗| from the first position after the
jump jump
considered isolated since their nearest-neighbor distance jump satisfying ∆f(x∗) = ∆f(x ) [31]. In addition,
jump
is much larger than the penetration depth λab ≈120nm we calculate the value of Fx before each jump using an
[29] (Fig. 1e). approximation for the magnetic field from a single vor-
We tested how strongly vortices are trapped by a TB tex and a model for the tip [30]. Figure 2b shows typical
byperforminglow-height(andhencestronglateralforce, line-scans for a moving vortex. While the signal in the
up to 20 pN) scans. However, even for our lowest passes central region of the line-scan contains numerous sharp
acrosstheTBandevenforT ≈0.85Tc weneverobserved changes, the envelope resembles a stretched version of
a vortex dislodging from the TB. This experimentally the signal expected from a static vortex at the same z.
verifiesthatfortherangeofforcesweappliedTB-vortices This indicates that in the central region the top of the
behave as one-dimensional (1D) objects in an effective vortex moves with the tip in a series of jumps. The ob-
d=1+1 geometry. served asymmetry between the Fwd and Bwd line-scans
Next,weperformedaseriesofrasterscansoveraniso- are typical for a moving vortex and suggest that the sys-
latedTB-vortex(Fig.1e)inordertoperturbit. Thescan tem is not in thermal equilibrium.
3
𝑧≈370 𝑛𝑚 𝑧≈370 𝑛𝑚
a. 15
Fwd 0.5 𝜇𝑚 Bwd 0.5 𝜇𝑚
10 Probability

𝑧𝐻 𝛥𝑗𝑢𝑚𝑝
Δ𝑓 Fwd Bwd 𝒛𝑯𝟔 5 0.2
𝟏. 0
𝑥∗ 𝑥𝑗𝑢𝑚𝑝
0
𝑥-axis parallel to TB [𝜇𝑚]
b.
𝑧≈150 𝑛𝑚 𝑧≈150 𝑛𝑚 400 Δ
𝑗𝑢𝑚𝑝
Fwd 0.5 𝜇𝑚 Bwd 0.5 𝜇𝑚
0
𝑧𝐻 Fwd Bwd 𝒛𝑯𝟑 200
𝑓 -2 𝟒.
Δ
-4 -1 0 1 100
𝑥-axis parallel to TB [𝜇𝑚]
200
300
FIG. 2. Manipulation scans of TB-vortices at T =15K. (a)
Forward (Fwd) and backward (Bwd) line-scans (taken along
the dashed lines from the scans in the insets) containing a
tip-induced vortex jump of size ∆ = |x∗ −x | that
jump jump FIG. 3. Histograms binning all measured jump lengths
we associate with an abrupt change in the position of the
(∆ )fordifferentrangesoftheforceexertedalongtheTB
upper part of thevortex. (b) Fwd andBwd line-scans taken jump
by the tip (F ). Inset: Normalized distributions of ∆
alongthedashedlinesfromthescansintheinsets. Numerous x jump
forthedifferentF ranges. Allthedistributionscollapseonto
vortexjumpswithavarietyof∆ ’sareapparent. Thedif- x
jump each other revealing the independence of ∆ from F .
ference between the overall shapes of the Fwd and Bwd line- jump x
scans suggests that non-equilibrium effects may be involved.
Insets: The scans from which the line-scans in (a) and (b)
were taken. The scan height and the span of ∆f is indicated lowercutoffa . Thesaturationisastrongindicationthat
x
for each panel. The horizontal double arrows indicate the W˜(∆ ) is a power-law for ∆ > a = 49±3nm
jump jump 0
back or forth scan direction along the TB (the x-axis) and with the power given by α = 2.75±0.06 (80% con-
meas
thelargeverticalarrowsindicatethedirectionwestepthetip
fidence level). We emphasize that we determined ∆
after each back and forth cycle (the y-axis). jump
directly and without theoretical assumptions and that
W˜(∆ ), α anda arenotsensitivetoseveralim-
jump meas 0
portant sources of systematic error (the independence of
Figure 3 shows histograms containing all jumps of W˜(∆ ) on F implies that it is not sensitive to sys-
jump x
two vortices chosen for their large separation from their tematic errors in force estimates, the scale invariance of
neighborsandeachother(enoughtosafelyconsidertheir power-laws implies that α is insensitive to errors in
meas
disorder environments independent). The histograms length calibration).
separate the jumps into three ranges of F . When we
x Accordingtothefluctuation-susceptibilityrelation[18]
compare the distribution of ∆ within each F range
jump x the statistics of the jump length (∆ ) gives informa-
jump
we find that the distributions collapse onto each other.
tiononthepropertiesoftherare, large-scale, low-energy
Moreover, we find the same collapse when we consider
excitationsofthesystemcharacterizedby∆. Onemight
jumps from each vortex separately [32]. This shows that
worry that when driven out of equilibrium short jumps
for the range of forces we applied the distribution of
will occur more readily than the long jumps required to
∆ does not depend on F and justifies lumping all
jump x reachoneofthemorefavorablepathsfartheraway. How-
the jumps together regardless of the force.
ever, the properties of the accessible vortex trajectories
Our main result is the force-independent distribution ensure that such behavior is unlikely [18].
W˜(∆jump) for both vortices together (Fig. 4). The most The independence of W˜(∆jump) on Fx (Fig. 3), which
significant feature of W˜(∆jump) is a long tail indicat- atfirstglancemayseempuzzling,isattributedbyDPRM
ing that disorder is important - it is in complete dis- theory[18]tostatistical tilt symmetry. Thissymmetryis
agreement with the gaussian distribution expected for a a manifestation of the absence of correlations in the dis-
system where disorder is irrelevant [18]. Another impor- order which means that for sufficiently large force [18],
tant feature is the saturation of αfit obtained from best as in this experiment [33], each time we tilt a vortex it
fits of W˜(∆ ) to a power-law for different values of a samples a new random environment and is equivalent to
jump
4
estimate for the cutoff along z, i.e. a =(a2κ)/(k T)≈
500 Jump count z 0 B
exp −𝛽Δ2 fit 4.5µm (cid:28) L = 80µm, consistent with the experiment
𝑗𝑢𝑚𝑝
Δ−1.5 fit being in the thick sample regime.
100 Δ𝑗−𝑢2𝑚.7𝑝5 fit To conclude, we have used the interaction between a
𝑗𝑢𝑚𝑝
magnetic tip and superconducting vortices on a TB to
tnuo 𝛼𝑓𝑖𝑡 𝑎𝑥 study the behavior of individual directed 1D objects.
c
pmuJ10 3 𝛼𝑚𝑒𝑎𝑠 Tbehtiwsepernoveildaestsicaintyidaenadl sdeitsuorpdefor,r wsthuidchyinisgutbhiequinittoeurpslainy
nature. After experimentally showing that vortices on a
2
TB behave as 1D objects in an effective 1+1 random
1 medium we proceeded to pull them one at a time along
10 50 100 150 𝑎𝑥 [𝑛𝑚]
the TB and measured the distribution of jump lengths
15 30 100 300 W˜(∆ ). We find that W˜(∆ ) is independent of
Jump size Δ𝑗𝑢𝑚𝑝 [𝑛𝑚] jump jump
the force applied by the tip and is the same for two
FIG.4. Measuredvortexjumplengths(∆ )andfitstothe widely separated vortices, confirming the predicted sta-
jump
data. Althoughthedataisconsistentwithapower-lawdistri- tisticaltiltsymmetryinthesystem. Ourcentralresultis
butiontheexponentweobtaindoesnotmatchαtheory =3/2 thepower-lawformofW˜(∆ )thatsuggeststhateven
jump
predicted for a system in equilibrium. Inset: Values of a
out of equilibrium excitations do not have a characteris-
power-law exponent α obtained by fitting the data in the
fit tic length-scale beyond the sample-specific lower cutoff
main panel for different values of the lower cutoff a . α
x fit
a . The direct measurement of a provides a new char-
saturates(arrow)forax >a0 =49±3,aclearindicationthat 0 0
α =2.75±0.06 is the best-fit exponent for the distribu- acterization of the local disorder strength D around the
meas
tion. TB, complementing other measures such as the critical
current [39, 40].
We thank Thierry Giamarchi, who encouraged us to
an un-tilted vortex experiencing a new disorder realiza- focus on vortex motion along the TB, as well as Ana-
tion. The observed statistical tilt symmetry implies that toli Polkovnikov and Daniel Podolsky for comments and
theoretically we could have obtained disorder-averaged GadKorenforhelpwithcharacterization. N.S.acknowl-
quantitiesfrommeasurementsofjustonevortex. Indeed, edgessupportfromtheGutwirthFellowshipandPosnan-
when we examine the force-independent distributions of skyResearchFundinHighTemperatureSuperconductiv-
∆jump for each vortex separately [34] we find that the ity. O.M.A. is supported by an Alon Fellowship and as a
distributions are statistically similar. This observed self- HorevFellowissupportedbytheTaubFoundation. The
averaging corroborates the statistical tilt symmetry and projecthasreceivedfundingfromtheEuropeanCommu-
means that the measured distribution of jump lengths is nitys Seventh Framework Programme (FP7/2007-2013)
indeed equivalent to the distribution of rare, large-scale, under Grant Agreement n◦ 268294.
low-energy excitations; i.e. W˜(∆ )=W(∆).
jump
While DPRM predicts the power-law behavior of
W(∆), the value we extract disagrees with the theoret-
ical value: α = d +2−ζ−1(d ) [18]. The value
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