Metastability and hysteretic vortex pinning near the order-disorder transition in NbSe$_2$: An interplay between plastic and elastic energy barriers?

1.3 MB·

by  M. Marziali Bermúdez

Metastability and hysteretic vortex pinning near the order-disorder transition in
NbSe : An interplay between plastic and elastic energy barriers?
2
M. Marziali Bermu´dez,1,2 E. R. Louden,3 M. R. Eskildsen,3 C. D. Dewhurst,4 V. Bekeris,1,2 and G. Pasquini1,2,∗
1Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Argentina.
2Instituto de F´ısica de Buenos Aires, Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas, Argentina.
3Deptartment of Physics, University of Notre Dame, Notre Dame, IN 46556, USA.
4Institut Laue-Langevin Avenue des Martyrs BP156, 38042 Grenoble Cedex 9, France.
(Dated: February 1, 2017)
Westudiedthermalanddynamichistoryeffectsinthevortexlattice(VL)neartheorder-disorder
7 transition in clean NbSe single crystals. Comparing the evolution of the effective vortex pinning
2
1 andthebulkVLstructure,weobservedmetastablesuperheatedandsupercooledVLconfigurations
0
that coexist with a hysteretic effective pinning response due to thermal cycling of the system. A
2
novelscenario,governedbytheinterplaybetween(lower)elasticand(higher)plasticenergybarriers,

n isproposedasanexplanationforourobservations: Plasticbarriers, whichpreventtheannihilation
a orcreationoftopologicaldefects,requiredynamicassistancetobeovercomeandtoachieveastable
J VL at each temperature. Conversely, thermal hysteresis in the pining response is ascribed to low

0 energy barriers, which inhibit rearrangement within a single VL correlation volume and are easily
3 overcome as the relative strength of competing interactions changes with temperature.

] PACSnumbers: 74.25.Uv.,61.05.fg,64.60.Cn,64.60.My
n
o
c I. INTRODUCTION tive pinning. This ordered phase is characterized by the
-
r weak elastic decay of spatial correlations over large dis-
p
tances, coexisting with a glassy behavior. With increas-
u A wide variety of complex systems can be described
ing field and/or temperature, the system undergoes an
s as collectively interacting particles on random poten-
t. tial landscapes. Among others, vortices in type- ODT to a disordered glass11,16 with correlation lengths
a II superconductors,1 charge density waves,2 electron ζ oftheorderofthemeaninter-vortexdistance, a0. The
m crystals,3 colloids,4 and skyrmions5 belong to this cat- suddenincreaseintheeffectivepinningattheODTgen-
- egory. History effects related to plasticity, glassy behav- erates the well-known Peak Effect (PE) anomaly.
d
n ior and metastable configurations in the vicinity of an In very low pinning materials, such as clean NbSe2
o order-disorder transition (ODT) are common features in single crystals, the BG phase is stable in a large re-
c thesesystems.6 Inthiscontext, vortexmatterisaproto- gion of the vortex field-temperature phase diagram.17
[ typemodelwherevortex-vortexinteractions,favoringan Here, the ODT takes place near the superconducting

ordered vortex lattice (VL), compete with both thermal phase boundary.18 Intermediate responses observed in
1
v fluctuations and pinning interactions that tend to disor- transport experiments in the region of the PE, indicat-
8 der the system, resulting in multiple vortex phases and ing an intermediate degree of pinning, had been origi-
9 complexdynamics. Inthelastdecades,alargeamountof nally ascribed to a contamination produced by the in-
7 workhasbeendedicatedtothestudyofvortexmatterin coming disordered vortices trough the sample edges.13
8 the presence of strong pinning, both in conventional and However, the existence of a narrow transitional region
0
non-conventional superconducting materials. Such sys- between the BG and the disordered phase, in the vicin-
.
1 tems are promising for practical transport applications.7 ity of the PE, has been theoretically proposed19 and ex-
0 On the other hand, from a fundamental point of view,8 perimentally confirmed. Recently, by joint small-angle
7
interesting physics arises when the inter-vortex interac- neutron scattering (SANS) and in-situ AC susceptibility
1
tion plays a major role, competing with a weak vortex- measurements,wewereabletoshowthattheapplication
:
v pinning interaction.8,9 of a shaking AC field in the transitional region drives
Xi In a variety of superconducting materials, character- the VL to bulk robust configurations with intermediate
ized by the presence of randomly distributed weak pin- degrees of disorder.22 Numerical simulations15 suggested
r
a ning centers, the stable vortex phase at low tempera- that these intermediate configurations could arise from
ture and low magnetic fields is an ordered Bragg Glass stationary fluctuating dynamic states, similar to those
(BG) phase without dislocations.9,10 Yet, when the sys- observed in colloidal systems.20,21 Stable VL configu-
tem is cooled from the normal state in an external mag- rations with intermediate dislocation density have also
netic field (field-cooled [FC]), energy barriers trap the been observed in recent scanning tunneling spectroscopy
VLindisorderedmetastableconfigurationswheretheVL (STS) experiments carried out in the transitional region
is more strongly pinned.11 By applying high transport of irradiated23 and Co-doped24 NbSe2 single crystals.
current densities11–13 or large oscillatory shaking mag- Inourpreviousexperiment,in-situlinearACmeasure-
netic fields,14,15 the system overcomes energy barriers mentsrevealedthatconfigurationswithintermediatede-
andreachestheorderedBGphase,whichhaslowereffec- grees of disorder, created through different dynamic his-
2
tories, display an intermediate effective pinning.22 The stronger pinning, saturating to χ(cid:48) = −1/4π at perfect
observed correlation between stronger effective pinning AC shielding (λ /D (cid:28)1).
R
and shorter spatial correlation lengths (higher disloca- Thehighesth allowedintheCampbellregimeisre-
AC
tion densities) suggested that dynamic history effects in stricted by the condition that the mean vortex displace-
the linear AC response are a consequence of changes in mentuinducedbytheACfieldbesmallerthanthechar-
the bulk VL dislocation density. On the other hand, acteristic pinning potential size, estimated to be of the
hystereticeffectivepinningobservedaftercoolingand/or order of the superconducting coherence length ξ. The
warming the system without any dynamic assistance are displacement is usually largest near the edge of a sam-
poorly understood up to now.14,25–27 Here, we report a ple. Whenh ||H,themaximumvortexdisplacementin
AC
systematic study of the effective pinning and structural asampleofsizeD(perpendiculartothemagneticfield)is
characterizationoftheVLaftervariousthermalhistories roughly u ∼ D h /H. Therefore, fields h < H ξ/D
AC AC
inNbSe singlecrystals. Ourresultssupportanovelsce- guaranteeu<ξ throughoutthesample,anecessarycon-
2
nariotodescribethermalhistoryeffectsinvortexmatter dition to perform linear non-invasive AC susceptibility
and the controversial relationship between the VL struc- measurements. The validity of the Campbell regime can
ture and the effective pinning. be confirmed by testing the linearity of the frequency
This paper is organized as follows: In section II, an independent AC response and the smallness of the dissi-
introductiontotheexperimental techniquesanditsrela- pative component χ(cid:48)(cid:48).31
tionship with the underlying VL properties is presented.
In section III, some relevant technical details regarding
experiments and samples are given. In section IV, re- B. SANS in vortex matter
sults corresponding to the AC susceptibility VL pinning
measurementsandSANSstructuralobservationsarepre- SANS is a non invasive technique suitable for prob-
sented and discussed. Section V summarizes the conclu- ing magnetic structures at nanometric scales. Here, the
sions. incident neutrons’ magnetic moment causes them to be
diffracted by the magnetic field modulation due to the
VL. Diffracted neutrons are collected using a position-
sensitive detector, thus obtaining the scattered intensity
II. EXPERIMENTAL TECHNIQUES AND
UNDERLYING PHYSICS distribution on the detector plane.
For a perfectly regular hexagonal lattice with lattice
parameter a , reflections will take place only for scat-
A. Linear AC susceptibility and the Campbell 0
tering vectors Q matching any of the reciprocal lattice
regime 0
vectors; thus first order Bragg peaks (BPs) will corre-
Inordertoinvestigatethespontaneousmodificationof spond to Q0 = a04√π3. The scattering process is elastic
andthescatteringangleθ ,measuredrelativetothefield
VLpinningproperties,anon-invasiveexperimentaltech- 0
direction, is given by Bragg’s law:
nique is imperative. An ideal way to measure the VL re-
sponseinweakpinningtype-IIsuperconductorsisbyAC Q
susceptibility restrictedtothe linearCampbellregime:28 sinθ0 = 2K0, (2)
The application of a very small AC field h , superim-
AC
posed to the DC field H forces vortices to perform small whereK =2π/λNistheneutronwavevectorcorrespond-
(harmonic) oscillations inside their effective pinning po- ing to a wavelength λN. Since Q0 (cid:28) K, the scattering
tential wells without modifying their spatial configura- angle is small and sinθ0 ≈θ0.
tion. These oscillations propagate through the sample For an imperfect lattice with dislocations, spatial cor-
due to the vortex-vortex repulsion, with a characteris- relations decay exponentially beyond a characteristic
tic penetration depth λ = λ +iλ . In the Campbell length ζ associated with the spatial scale at which dislo-
AC R I
regime λ (cid:28)λ and cations appear.9 The anisotropy of the elastic constants
I R
is reflected in different ζ for i = x,y,z. Due to finite
i
φ B correlation lengths, the Bragg reflections are broadened
λ2 =λ2 + 0 , (1)
R L 4παL in reciprocal space, with ∆Qi ∝ ζi−1.10 Consequently,
Braggpeakswillappearbroaderinthedetectorplaneas
where λ is the London penetration depth, φ =h/2e(cid:39) wellasforanglesaroundtheBraggcondition, θ =θ . In
L 0 0
2068 T nm2 is the flux quantum and α is the curva- addition, the intensity distribution is further smeared by
L
ture of the effective mean pinning potential, also known the instrumental resolution. Due to the poorer in-plane
as the Labusch constant. Since the real component of resolution, only the longitudinal characteristic length in
the AC susceptibility χ(cid:48) is determined by the experi- the direction of the flux lines ζ can be reliably deter-
z
mentalgeometryandthedimensionlessparameterλ /D, mined from the measured intensity distribution I(θ), as
R
withDacharacteristicsampledimension,30 χ(cid:48) canbedi- long as ζ does not exceed resolution limits. This char-
z
rectlyrelatedtotheeffectivepinningpotentialinamean acteristic length is directly related to the mean distance
field approximation.29 Lower values of χ(cid:48) correspond to between edge dislocations.33
3
Thetotalscatteredintensitydependsonthedensityof temperature regulation within ∆T (cid:54) 2 mK down to
scatterersandtheirindividualstrength,whichispropor- T ∼4.3 K.
min
tional to the square of the individual vortex form factor, In order to study thermal history effects throughout a
F(Q).32 An approximate expression for F(Q) can be ob- widerangeoftemperatures,experimentswereperformed
tained from the solution to the London equation by in- at a moderate DC field, H = 1 kOe. For all suscepti-
troducing an ad-hoc cut-off due to the finite vortex core bilitymeasurements,ACfieldamplitudeswererestricted
size. This leads to35 to h < 10 mOe. Linear response and low dissipation,
AC
which are characteristic features of the linear Campbell
exp(−c Q2ξ2)
regime, were verified. For some sequences, susceptibility
F(Q)∝ , (3)
1+Q2λ2 measurementswereinterruptedatcertaintargettemper-
L
atures to apply a shaking AC field. Here, 1000 cycles
where ξ is the superconducting coherence length and c of a larger sinusoidal field with amplitude (cid:62) 5 Oe and
is a heuristic factor. The modified London model has frequency 1 kHz was applied before resuming linear sus-
been successfully applied to fit experimental data corre- ceptibility measurements.
sponding to d-wave or multi-gap superconductors using
The SANS experiments were conducted using the D33
c between 1/4 and 2.35,36 instrument at the Institut Laue-Langevin (ILL).39 Ex-
Since λL and ξ increase with temperature, the mag- perimental parameters were chosen for the best com-
netic field modulation smears out and thus F(Q) → 0 promisebetweenminimizingthebackground, optimizing
as T → Tc. It is therefore not feasible to measure full the resolution, and maximizing the scattered intensity:
rocking curves to obtain the intensity distribution I(θ) H (cid:39)5kOe (Q (cid:39)0.011 ˚A−1), λ (cid:39)7 ˚A (±10%). Based
0 N
at temperatures near the ODT. However, when spatial on the wavelength spread and beam collimation param-
correlation is broken by dislocations, the integrated in- eters (12.8 m collimation, 30 mm source aperture, 5 mm
tensityisconserved.10,32 Therefore, theproductbetween samplepinhole),theangularresolutionalongtherocking
the width ∆θ and the maximum intensity Imax = I(θ0) anglewasestimatedtobe∆θres ∼0.086◦ (FWHM).The
of a BP remains proportional to F2. As a result, dif- applied magnetic field was parallel to crystal c axis and
ferences in ζz between configurations at equal T and H nearly parallel to the beam. A special sample holder,
can be assessed by comparing their corresponding Imax. equipped with a set of coils, allowed measuring the AC
Moreover, using independent data for λL(T) and ξ(T), susceptibility response in situ.
it is possible to calculate the expected temperature de-
pendence of F(Q), and thereby trace the evolution of ζ
z
with temperature by the deviation of I (T) from the
max IV. RESULTS AND DISCUSSION
expected F2-like dependence.
A. History effects in the effective pinning
III. EXPERIMENTAL DETAILS
Linear AC susceptibility curves measured after differ-
ent thermal and dynamic histories are shown in Figure
A. Samples
1. As reported in previous works, in a field-cooled cool-
ing process (FCC, black curve), the VL remains trapped
The study has been performed in clean NbSe2 single in a strongly pinned configuration and χ(cid:48)(T) displays no
crystalswithT ∼7.2K,growninBellLabsasdescribed
c PE. After shaking the VL at T well below the onset
sh
elsewhere.37 The phase diagram for a number of crystals of the PE (T ), an unambiguous rise in χ(cid:48) (red curve)
po
wascharacterizedusinga7-TMPMSXL(QuantumDe-
signals an increment in λ and thus a softening of the
AC
sign)andreportedpreviously.38Ourcrystalsonlyshowed
effective pinning. In addition, a strong PE is observed
minor, insignificant variations compared to other litera-
uponwarmingtheshakenVL.Similardynamicassistance
ture results. Most of the linear AC susceptibility mea-
can also be provided by a transport current or a sudden
surementsweredoneusingasingle(1×1×0.2)mm3sam-
increase/decrease of the magnetic field (e.g., zero-field-
ple, whereas SANS observations and test in-situ AC sus- cooled configurations). Recent SANS22 and STS23,24
ceptibility measurements were performed using a larger observations, in agreement with pioneer experiments,40
(5.6×4.7×0.2) mm3 crystal.
corroborated that the decrease of the effective pinning
after applying a shaking field, or an equivalent dynami-
cal assistance, is correlated with a bulk ordering of the
B. Experimental setups and technical details VLwheremetastablesupercooledVLdislocationsarere-
moved.
Linear AC susceptibility experiments were performed Certainthermalprotocolscanalsoreduceeffectivepin-
using a home-made susceptometer based on the mutual ning without any dynamic assistance.14 After cooling
inductance technique where both the AC and DC fields the system down to sufficiently low temperatures, χ(cid:48)(T)
are parallel to the c axis of the sample. The suscep- curves measured during subsequent warming processes
tometer was installed in a bath He cryostat that allowed (FCW, green curves in Figure 1) display the PE and a
4
0 .0 B. History effects in the VL structure
0 .4 D
-0 .2 PE’c Thermal-history-dependent effective pinning has been
0 .0 [n ascribed to spontaneous VL ordering promoted by the
.u.]-0 .4 H = 1 k O e -0 .4 orm.] fsapsetcetrtsotrpeningntihnegnifnogrcoefsvaosrttehxe-vsyosrtteemxinistecroaoclteidoninswthitehBrGe-
.m D c phase.14,25 Following the observed connection between
pc4 ’ [e-0 .6 4 F C W 5T m in [K ]6 B G 7 ’P E tlohceateiffoencstiavfeteprindnyinnagmriecdaulcatsisoinstaanncde,thspeornetmanoevoaulsofandnisi--
-0 .8 (T m in = 4 .4 5 K ) hilation of dislocations might also be expected to occur
duringtheproposedthermalordering. Astructuralchar-
F C C
F C W
T s h T p o T p acterization through SANS experiments was performed
-1 .0 in order to assess the role of dislocations in the observed
6 .4 6 .6 6 .8 7 .0 hysteretic pinning.
T [K ]
If dislocations were removed as a result of thermal cy-
cling, larger correlation lengths ζ due to a reduction of
FIG.1: In-phaselinearACsusceptibilityχ(cid:48)(T)recordeddur-
the VL dislocation density should be observed as nar-
ing a FCC process (black curve), after shaking the system at
rower rocking curves (RCs). In Figure 2, we show RCs
T < T (red curve), and during various FCW from differ-
sh po
entminimumtemperaturesT (greenandbluecurves). Ar- recordedatthebasetemperature(3K,panela)andata
min
rows indicate the direction of temperature variation; vertical higher temperature (4.65 K, panel b). RCs acquired af-
dotted lines indicate relevant temperatures (see text). FCW
processes display lower effective pinning than FCC without
any dynamic assistance. Inset: Normalized height of the PE
(Eq. 4, see text) measured in FCW processes as a function
ofthelowesttemperaturereached,T . Thedottedlineisa
min 2 .0
guide to the eye. H = 5 k O e ( a )

1 .5 T = 3 K
.] B G
clear reduction of effective pinning with respect to the [a.u1 .0 F C C DD (cid:1) = 0 .0 8 1 (4 ) d e g
originalFCCprocess. Towhatextentpinningissoftened ) (cid:1)= 0 .2 3 (2 ) d e g
q
depends on the lowest temperature reached Tmin. For a I (
clearer comparison among the various thermal histories, 0 .5
we define the height of the PE as
0 .0
∆χ(cid:48) =χ(cid:48)(T )−χ(cid:48)(T ), (4)
PE po p 0 .8
( b )
where T is the local maximum of χ(cid:48)(T) (matching the 0 .6 T = 4 .6 5 K
po .] B G D (cid:1) = 0 .0 8 3 (3 ) d e g
onset of the PE) and Tp, the local minimum (maximum .u
heseffetiegcohtftivFfeoigrpuairnVenL1in,sg∆h)aχwk(cid:48)PehEne,ninntothrhmeeaPBlErizaegidsgbGoyblasteshrsevpechdoa.rsreeI,nsipstoshnhedoiwinnng- qI () [a0 .4 FF CC WC DD (cid:1)(cid:1) == 00 ..22 42 66 ((98 )) dd ee gg
as a function of T . The PE occurs (i.e., ∆χ(cid:48) > 0) 0 .2
min PE
only after the sample was cooled below a threshold tem-
perature (∼ 6.5 K in the example shown in Figure 1).
0 .0
Once the PE develops, ∆χ(cid:48) increases with decreasing
PE
T . The curves are independent of cooling and warm-
min
-0 .5 0 -0 .2 5 0 .0 0 0 .2 5 0 .5 0
ing rates and whether linear susceptibility was measured
during the cooling process or not. FCW curves warmed (cid:1) - (cid:1)0 [d e g ]
from the lowest accessible temperatures in our experi-
ments display marginally different heights and, for all
FIG. 2: Rocking curves from SANS measurements after dif-
Tmin, ∆χ(cid:48)PE(Tmin) < ∆χ(cid:48)PE(BG). Hence, the effective ferent thermal and dynamic histories recorded at the base
pinning reduction due to thermally cycling the system temperature 3 K (a) and at 4.65 K (b): FCC (black open
approaches a limit as T → 0, without displaying as squares),VLshakenat5.6K(redopencircles)andFCWfrom
min
strong a reduction as that observed after dynamically 3K(blueopentriangles). ContinuouslinesareLorentzianfits
assisting the system. with full width at half maximum ∆θ.
5
ter shaking the VL at 5.6 K, in the BG phase (red open at least for low temperatures. NbSe is a multi-band
2
circles), display resolution-limited Bragg Peaks at both superconductor41 withsix-foldstarshapedvortexcore;42
temperatures, indicating correlation lengths much larger therefore, modelspredictingF2(T)basedontheLondon
than our resolution. Their corresponding full width at theory should be used with caution. Still, using c=0.12
half maximum (FWHM) was estimated from Lorentzian and estimating the vortex core size as
fitstothedata, obtaining∆θ =0.081(4)◦ and0.083(3)◦, (cid:115)
φ
respectively. These values are slightly smaller and imply ξ(T)= 0 , (5)
a better resolution limit (ζz (cid:46) 40 µm) than the rough 2πHc2(T)
estimate ∆θ (cid:39)0.087◦ (Sec. III). Conversely, the FCC
res themodifiedLondonmodel(Eq. 3andredlineinFigure
RCs (black open squares) are clearly wider than ∆θ ,
res 3)isingoodagreementwiththeexperimentalI (T)of
indicatingaVLwithahighdensityofdislocations. Here, max
the resolution-limited BPs. Here, H (T) was estimated
positional order is lost beyond a characteristic length c2
fromtheonsetofthediamagneticresponse,38 andλ (T)
ζ = 4.9(2) µm.22 A FCW RC, recorded after warm- L
z wasextractedfromdirectexperimentaldata.43Thesmall
ing a FCC configuration from the base temperature to
value of c may be a consequence of overestimating ξ in
4.65 K, is also shown in Figure 2b (blue open triangles).
Eq. 5orduetoalessereffectofthefinitecoresizeinthis
We found this FCW RC to be near identical to the FCC
system. Thedeviationofexperimentaldatafromtheex-
RC, with ζ = 4.5(2) µm. This result indicates no sub-
z pecteddecayaroundT ∼6K(stillintheorderedphase)
stantial modification of ζ during the cooling-warming
z may be due to a decrease of the elastic BG correlation
process up to this temperature.
length below experimental resolution.10
AsexplainedinSec. IIB,measuringfullRCsathigher
ThesamemodelandparameterswereusedfortheFCC
temperatures was not feasible. Instead, I (T) was ac-
max data, with the expected F2(T) scaled to match I at
max
quired at several temperatures for the various VL con-
the lowest temperature (black line in Figure 3). As with
figurations (Figure 3). The fact that FCC and BG RCs
theBG,thereisnoapparentdeviationfromtheexpected
recorded at each temperature have the same integrated
decay up to T ∼ 6 K, suggesting no substantial modifi-
intensity within the typical precision of the SANS mea-
cation of ζ nor of the mean dislocation distance22 up to
z
surements (10%) strongly supports using I as probe
max this temperature. This result is consistent with the fact
for changes in ζ . As long as ζ is beyond the exper-
z z that FCC RCs recorded at both temperatures displayed
imental resolution, the maximum BP intensity scales
similar ∆θ, and extends to higher temperatures. More-
as the integrated intensity. Therefore, for the ordered
over, in spite of the large decrease in the effective pin-
Bragg Glass BPs (red dots), I (T) is expected to de-
max ning during warming processes (Figure 1), I (T) cor-
cay as the square of the single-vortex form factor F2(T), max
responding to FCC and FCW configurations show only
minordifferencesuptothehighesttemperatureatwhich
we were able to acquire neutron data. Therefore, corre-
lation lengths of a bulk dislocated metastable FCC VL
B G are not substantially modified after a cooling/warming
1 F C W T p o process in the BG phase. This behavior is qualitatively
] different to that observed in Nb single crystals, where a
its gradual decrease of ζ during FC processes in the BG
n (a ) F C C z
. u phase has been reported.26 Our results suggest that the
b0 .1 spontaneous thermal ordering does not involve signifi-
r (b )
[a cantannihilationofVLdislocations,incontrastwiththe
ax strong change in the dislocation density obtained after
Im an assisted dynamic ordering.
H = 5 k O e
0 .0 1
C. Towards a consistent description
3 4 5 6
T [K ]
Hysteretic responses in glasses are ascribed to energy
barriersthatmaytrapthesysteminmetastableconfigu-
FIG. 3: Maximum BP intensity, I (T), recorded during
max rations. Our observations suggest that there are mainly
cooling/warming processes after different dynamic histories:
two different types of energy barriers: 1) those that are
VLshakenintheBGphase(reddots),FCC(blacksquares),
and FCW (blue triangles). Continuous lines show the ex- easilyovercomeandvanishastherelativeweightbetween
pected F2(T)-like dependence from Eq. 3, using an effec- competinginteractionschangeswithtemperature,and2)
tive vortex core size (see text). Arrows indicate the direc- those exceeding the available thermal energy and there-
tionoftemperatureevolutioninthethreeprocesses(labelled fore requiring dynamical assistance to annihilate topo-
FCC, FCW, and BG) and the intensities corresponding to logical defects and drive the VL to the ordered Bragg
RCs shown in Figure 2 (labelled a and b). A vertical dashed glass phase at low temperature. These two kinds of en-
line marks T .
po ergybarriersexplainourresultsandrevealaninteresting
6
Eachintermediatecoolingresponsedependsontheshak-
-0 .4 H = 1 k O e ing temperature.14 After cooling the system to the same
T , all warming curves reveal an important reduction
min
of the effective pinning with respect to the correspond-
-0 .5 (3 ) (3 ) (3 )
ing cooling process, as in FCC-FCW cycles. Moreover,
.]
.u BG (2 ) eachwarmingresponseisintermediatebetweentheFCW
.m-0 .6 (1 ) and the BG response, and correlated with the shaking
c ’ [e FC W (2 ) (1 ) tTehmepreefroarteu,rtehetsheartesduelttesrimndiniceadteeathchatdtihselorecarteimonaindsensosimtye.
p4-0 .7 (2 ) (1 ) memory of the original dislocation density after ther-
mally cycling the system. This dislocation density mem-
F C C
-0 .8 T p o oryisinagreementwithrecentdirectSTSobservationsof
supercooled configurations,27 in which dislocations were
6 .8 0 6 .8 5 6 .9 0 6 .9 5 still present upon cooling the system to low tempera-
T [K ]
tures. The fact that pinning is reduced while disloca-
tions remain supports our proposing two kinds of energy
FIG. 4: AC response measured during cooling (open sym- barriers.
bols) and immediately warming (full symbols) VL configu-
It is particularly interesting whether both kinds of
rations with intermediate disorder. Each of these configura-
barriers play an important role in the PE. The spon-
tions (dark yellow, orange and green symbols) was prepared
by shaking the VL at a different temperature in the transi- taneous VL disordering responsible for the increase in
tionalregion,nearT (representedbyarrowsindottedlines). the effective pinning in the transitional region (sudden
po
Arrows in continuous lines indicate the direction of temper- drop in χ(cid:48)(T) above T ) has been a subject of contro-
po
ature evolution and numbers in parenthesis, temporal order. versy for decades. The first simple explanation, given by
Continuous lines reproduce FCC (black), FCW (blue) and Pippard,44 was based on the accommodation of vortices
BG (red, shaken well below Tpo) curves for reference. Each totherandompinningcenterdistribution: AsH isap-
c2
warmingresponseisstillintermediatebetweenFCWandBG
proached,therigidityoftheflux-linelatticedropstozero
responses.
faster than the pinning strength. This allows the flux
lattice to adapt more easily to the underlying quenched
disorder, so that the effective pinning is increased. Sub-
underlying physics. In a metastable VL, the size of the sequent models by Larkin and Ovchinnikov45 provided
correlated volumes, determined by the dislocation den- quantitative relations between the elastic VL modulus,
sity, are not drastically affected by the thermal history. pinning forces, and the effective critical current density.
However,atlowtemperature,elasticvortex-vortexinter- These models lost attention years later, when the role
actions prevail over vortex-pinning interactions, produc- of VL dislocations in increasing the effective pinning at
ing short-range vortex accommodation inside each cor- the ODT became evident. This new observation sug-
relation volume. In this local reorganization, the VL gested that, near the depinning transition, plastic creep
overcomes lower energy barriers, leading to spontaneous and plastic dynamics largely boost the effective critical
local ordering. This subtle accommodation can drasti- current density and reduce VL mobility.46 Still, our re-
callyaffecttheVLeffectivepinningand, thus, linearAC sults show that the dislocation density only partially de-
response, but would not significantly affect the neutron termines the effective pinning. May both descriptions of
scattering pattern. the PE hold under different conditions?
Results presented in Secs. IVA and IVB support the In order to clarify the spontaneous thermal disorder-
proposed scenario based on observations in extreme sit- ing above T , we conducted a systematic study of his-
po
uations: themostordered(BG)andthemostdisordered tory effects in the AC response. The linear AC response
(FC) VL configurations. To further test the consistency recorded in various thermal cycles, each starting from
of our picture, we analyzed the thermal cycling response an ordered VL, is shown in Figure 5. After interrupting
of VL configurations initially stabilized with various in- the warming process (full red dots) in the transitional
termediate dislocation densities. These configurations region, all cooling curves show an intermediate degree
werepreparedbyshakingthesystematvarioustempera- of pinning, correlated with the maximum temperature
tures near T , within the transitional region.22 The dis- reached. In Figure 5a, cooling processes from two dif-
po
location density is expected to increase as the shaking ferent upper temperatures (open pink and purple dots)
temperatureapproachesT .14 Figure4showstheACre- areshownasexamples. Strikingly,thesubsequentwarm-
p
sponses corresponding to cooling-warming processes for ing processes (full pink and purple dots) recover the re-
various VL configurations obtained (green, orange, and versible BG response without any dynamic assistance.
dark yellow symbols). We compare these results with Identical responses suggest that the system evolves in
the thermally cycled response of an initially ordered VL such a way that the spontaneous disordering produced
without dislocations, shaken well below T (red curve), in the PE region is recovered by a spontaneous ordering
po
andamoredisorderedFCWVL,notshaken(bluecurve). during the cooling process. A similar memory effect has
7

peratures. Consequently, the memory of the dislocation
densitywillbelostifthesystemiswarmedbeyondsome
-0 .4 B G (1 ) characteristic temperature T∗. The number of sponta-
(3 ) neous dislocations (and disclinations) is expected to in-
(3 )
crease when approaching the disordered phase,24 and a
-0 .5 (2 )
.] completely amorphous VL would lie above the spinodal
.u (2 ) line.47 Panel 5b shows the evolution of the AC response
.m-0 .6 startingfromanorderedconfigurationobtainedbyshak-
’ [e ing the VL in the BG phase (red) and thermally cycling
c the sample reaching temperatures well above T : near
p po
4-0 .7 the maximum effective pinning temperature Tp (open
H = 1 k O e green symbols) and above T (open cyan symbols). It
p

can be seen that the BG response is no longer recovered
F C C ( a )
-0 .8 duringthesecondwarmingprocesses(fullgreenandcyan
-0 .4 (1 ) symbols), and the memory is lost. Within the context of
B G theproposedpicture,ourresultssupportanovelscenario
to understand the PE: A dynamic assistance would gen-
-0 .5 (3 ) erate VL dislocations immediately above T , breaking
po
.] (3 ) metastableorderedconfigurationswhichwouldotherwise
.u (2 ) remain up to T∗. In contrast, above T∗, plastic energy
.m-0 .6 barriers are lower and thermal energy alone is sufficient
c ’ [e (2 ) to overcome them.
p
4-0 .7
V. CONCLUSIONS
F C C ( b )
-0 .8 T *
We have studied thermal and dynamic history effects
6 .8 8 6 .9 2 6 .9 6
in the VL in clean NbSe , inspecting the evolution of ef-
2
T [K ] fective pinning by means of AC susceptibility techniques
and observing the VL bulk structure in SANS experi-
FIG. 5: AC response measured while thermally cycling an ments. The reported results suggest a novel scenario to
initiallyorderedVL.Straightarrowsindicatethedirectionof describetheorder-disordertransitionandthePeakEffect
temperature evolution and red curved arrows mark turning anomaly. Additionally, the proposed framework helps
points. Ineachsequence,(1)afresh BGwaswarmedtoTmax to clarify the controversial relationship between the VL
(red symbols), (2) cooled to 4.5 K (open symbols), and (3) structure and effective pinning. In the proposed picture,
warmedagain(fullsymbols). ThereisatemperatureT∗(ver-
theinterplaybetweentwodifferenttypesofenergybarri-
tical dashed line) such that, if T < T∗, the BG response
max ers is the origin of the history and memory effects in the
is recovered (panel a), but memory is lost when T > T∗
max effective pinning potential. On one hand, high plastic
(panel b).
barriers impeding the annihilation or creation of topo-
logicaldefects, thatwouldotherwisereducethetotalVL
energy, require dynamic assistance to be overcome. On
beenreportedinpreviousSANSexperiments.22Therewe the other hand, low elastic barriers prevent the VL from
showed that a low-temperature RC measured after ther- accommodating the pinning landscape at the local level.
mally cycling an ordered VL through the transitional re- Thesebarriersareeasilyovercomeandvanishastherela-
gion remained beyond the resolution limit. This implies tiveweightbetweencompetinginteractionschangeswith
thatsolelywarmingthesystemuptothetransitionalre- temperature. This subtle elastic arrangement would fa-
giondoesnotcreateasignificantnumberofdislocations, vor the elastic vortex-vortex interaction at low temper-
ifanyatall. Inaddition,superheatedconfigurationsfree atures and the vortex-pinning interaction in the transi-
of dislocations above Tpo have been recently observed in tional region. This constitutes the main ingredient for
STS experiments.27 Altogether, these experiments evi- the spontaneous PE and the observed thermal hysteresis
dencethatthePEcannotbeaconsequenceoftheprolif- in the linear AC response.
eration of dislocations exclusively. In particular, the ef- In our scenario, thermal history does not drastically
fective pinning increase displayed right above Tpo would affect the size of the correlated volumes, which is mainly
onlyinvolvetheelasticaccommodationofvorticestothe determined by the mean dislocation density, but clearly
random quenched disorder, which can be easily undone affects vortex arrangements within the correlated vol-
by cooling the system. umes. This provides an explanation for different pin-
Thermal history effects associated to the creation of ning intensities in configurations showing similar low-
dislocationsarelikelytobecomeapparentathighertem- temperature SANS rocking curves. Quenched disloca-
8
tionsinfieldcoolingprotocolsareresponsibleforthewide cessed through invasive techniques. This explains the
RCs recorded in SANS experiments22 and supercooled disagreement between observations by linear AC suscep-
disordered VLs observed by STS at low temperature.27 tibilityorultrafasttransporttechniquesandconventional
Highenergybarriersalsopreventthecreationofdisloca- transport experiments, in which VL self-reordering and
tionsinmetastablesuperheatedorderedconfigurationsin edge contamination are also present.13
the transitional region (between the PE onset tempera- Finally, we mention that partial spontaneous removal
tureTpo andacharacteristictemperatureT∗ (cid:46)Tp),pre- of dislocations in the cooling process of a disordered VL
serving narrow RCs and memory in the effective pinning cannot be ruled out. The FC VL at low T displays bulk
afterthermallycyclingthesystembelowT∗. BeyondT∗, orientationalorderandthemeandislocationdistance,es-
plastic energy barriers vanish, favoring the spontaneous timated from the width of the RCs, is much larger than
creation of topological defects. In STS experiments per- a . Therefore, it is expected that successive thermal cy-
0
formed in doped NbSe2 crystals, a topological transition cles in an adequate range of temperatures could eventu-
has been identified at T ∼ Tp, above which stable VL ally help the progressive removal of dislocations without
configurations are highly disordered with proliferation dynamic assistance. Direct observations in the full tem-
of VL dislocations and disclinations.27 The vanishing of perature range and a systematic study of different ther-
plastic barriers above Tp is possibly associated with this malcycles may enable thecorroborationof theproposed
topological transition. Although the proposed picture is picture.
consistentwiththeexperimentalevidenceandphysically
sound, a formal theoretical treatment is needed. For in-
stance, hysteretic Campbell response in the presence of
strong pinning centers has been recently demonstrated
Acknowledgements
through a simple model.48 A similar approach for VLs
in weak pinning landscapes should help to better under-
stand the connection between thermal history leading to MMB,VB,andGPweresupportedbyCONICETun-
metastable VL configurations and hysteretic Campbell der Grant PIP 536 and Universidad de Buenos Aires;
responses. ERL and MRE were supported by the U.S. Department
An important remark: When vortex displacements in- of Energy, Office of Basic Energy Sciences, under Award
volve larger distances, dislocations are created, annihi- No. DE-FG02-10ER46783. We are grateful to G. Cuello
lated, and/or redistributed, erasing metastable configu- and G. Garbarino for their collaboration in submitting
rations and its related memory effects. Hence, the pro- the experimental proposal and their assistance with lo-
posed picture and its underlying physics cannot be ac- gistics,andtoG.NievaforprovidingtheNbSe samples.
2
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