Nanoscale assembly of superconducting vortices with scanning tunnelling microscope tip

3.4 MB·

by  Jun-Yi Ge

Nanoscale assembly of superconducting vortices with scanning tunnelling microscope
tip
Jun-Yi Ge,1,∗ Vladimir N. Gladilin,1,2 Jacques Tempere,2 Cun Xue,1,3 Jozef T.
Devreese,2 Joris Van de Vondel,1 Youhe Zhou,4 and Victor V. Moshchalkov1,†
1INPAC–Institute for Nanoscale Physics and Chemistry,
KU Leuven, Celestijnenlaan 200D, B–3001 Leuven, Belgium
2TQC–Theory of Quantum and Complex Systems,
Universiteit Antwerpen, Universiteitsplein 1, B–2610 Antwerpen, Belgium
3School of Mechanics, Civil Engineering and Architecture,
7 Northwestern Polytechnical University, Xi’an 710071, China
1 4School of Aeronautics, Northwestern Polytechnical University, Xi’an 710071, P.R. China
0 (Dated: January 24, 2017)
2

n
a Vortices play a crucial role in determining the The scanning tunnelling microscope (STM) is one of
J
properties of superconductors as well as their ap- the most powerful techniques to image nanoscale topog-

3
plications. Therefore, characterization and ma- raphy and characterize the local physical properties [9–
2
nipulationofvortices, especiallyatthesinglevor- 12]. It has been widely used in the study of condensed

] tex level, is of great importance. Among many matter systems such as insulators [13], semiconductors
n
techniques to study single vortices, scanning tun- [14]andsuperconductors[15]. Alsothecapabilityofpre-
o
neling microscopy (STM) stands out as a power- cisepositioningmakesitapowerfultooltodesignvarious
c
- ful tool, due to its ability to detect the local elec- artificial nanostructures at atomic level [16–19]. In the
r
tronic states and high spatial resolution. How- study of superconductivity, the STM can probe the lo-
p
u ever, local control of superconductivity as well as cal electronic density of states and thus is used to map
s the manipulation of individual vortices with the the spatial distribution of superconducting gap and the
.
t STM tip is still lacking. Here we report a new vortex core state. The STM is the most suitable tech-
a
m function of the STM, namely to control the local nique to directly image vortices with atomic resolution,
pinning in a superconductor through the heating especially at high magnetic fields. A series of intricate
-
d effect. Such effect allows us to quench the super- phenomena related to vortex states have been revealed
n conducting state at nanoscale, and leads to the by the STM, such as the vortex lattice melting process
o
growth of vortex-clusters whose size can be con- [20], the anisotropy of the Fermi surface distribution of
c
trolled by the bias voltage. We also demonstrate the superconducting gap [21], order-disorder transition
[
theuseofanSTMtiptoassemblesinglequantum [22]. However, despite the importance of technical and

1
vortices into desired nanoscale configurations. scientificapplicationsbyutilizingvorticestodesignflux-
v
onic devices [23, 24], local control of superconductivity
6 Pair condensation of charge carriers from the normal
1 and the precise manipulation of vortices with the STM
to the superconducting state is associated with a gain of
3 is still lacking.
the free energy. However, in the presence of a magnetic
6
Here, we report a method of controlled quenching of
0 field, this gain is decreased due to the expulsion of the
a hot spot in a superconducting film by using the lo-
. external field by the superconductor. For type-II super-
1
cal heating effect of the tunnelling junction on vortex
0 conductors,thiscompetitionresultsinthepenetrationof
states, which is especially interesting at the nanoscale.
7 quantizedvorticeswhicharecharacterizedbytwolength
1 scales: the coherence length ξ, equivalent to the size of This method utilizes the heating effect of the tunneling
: junction with the power which can be well controlled by
v the vortex core, and the magnetic penetration depth λ,
tuning the bias voltage. We are also able to use this
i the decay length for the supercurrents encircling a vor-
X technique to precisely manipulate single quantum vor-
tex core [1]. Since the formation of the vortex core re-
r tices. Our results extend the pioneering work of Eigler
quires breaking of Cooper pairs, it is energetically favor-
a
and Schweizer [16] from manipulating single atoms to
ableforavortextobeplacedinanareawheresupercon-
manipulating single quantum vortices.
ductivity is already locally suppressed. Such positions
are called pinning centers and have various forms such
as atom vacancies [2], variations of sample composition Results
or/and thickness [3] and artificially patterned antidots Sample design and vortex assembling. The experi-
[4, 5]. The manipulation and control of pinning config- mentswereperformedbyimagingvorticesusingscanning
uration and its interaction with vortex are fundamental Hallprobemicroscopy(SHPM)whichcombinesthenon-
forcreatingnewfunctionaldeviceswithsuperconductors invasive Hall imaging technique and an STM (Figs. 1a,
[6–8]. b). TheSTMtip,typicallyusedonlytoprovidefeedback
Tip lift off; fast coolingdown to 
4.2 K; scan; 
Release tip at (0,‐8); 
v =‐0.5V; retract and 
bias
scan
172822 175707
2
a b d
Hall cross
STM tip V V+ I‐
bias
Hall cross
I+ V‐
STM tip 1.4
Conductive layer
Sample
c
)
T
e m
(
V vortex Bz
bias
e-
Au 35 nm 0
Ge 10 nm
e-
Pb 2e- 85 nm
SiO /Si
2
FIG.1. Introductiontotheoperationoftheheatingeffectgeneratedwithascanningtunnellingmicroscopetip.
(a) Schematic view of the scanning Hall probe microscope (SHPM). A scanning tunneling microscope (STM) tip is assembled
together with a Hall cross to make a sensor, which is aligned at a small angle (1 to 3 degrees) to the sample surface. (b)
An optical image of the Hall sensor. An SEM image of the Hall cross is shown in the inset. The longer (shorter) scale bar
corresponds to 20 (1) µm. (c) Schematic representation of the local heating effect by using the STM tip. The area close to
thetipisheatedupbythetunnellingcurrentwhiletheinsulatingGelayerandthesuperconductingPblayerarealsowarmed
up due to thermal transfer. Superconductivity is suppressed in a localized region where it is energetically favorable to place
vortices. (d) SHPM image of a vortex lattice observed after field cooling at H = 3.74×102 Am−1 from above T down to
c
T =4.2 K. (e) SHPM image after 5 seconds of tunnelling at bias voltage of 0.5 V and tunnelling current of 0.5 nA, and then
lifting up the tip for Hall probe imaging. A vortex cluster forms at the tip position due to the local quench of the hot spot.
to bring the Hall sensor in close proximity to the sample naturallyduringthesamplepreparationandcanbecon-
surface [25], is here used as a local heating source. In sidered as locations with a reduced electron mean free
the presence of tunnelling current, a pinning potential is path [26]. The vortex-vortex interaction can be weak-
created at the position of the tip due to the local sup- ened by choosing a superconducting film with relatively
pression of the superconductivity. As a result, nearby small Ginzburg-Landau parameter κ = λ/ξ. Therefore,
vortices will be attracted to this region. However, once a Pb film with thickness d = 85 nm, λ(0) = 94 nm and
the tunnelling disappears by retracting the tip to a cer- ξ(0)=52 nm is used (Supplementary Figure 1, 2 and 3,
tain distance (200 nm), the heating effect is stopped and and Supplementary Notes 1 and 2). In this case, once
theaccumulatedvorticeswillfinallyrelaxintoatriangu- vortices are forced together, the attraction between pin-
lar vortex lattice due to the repulsive interaction among ning centers and vortices overcomes the repulsive inter-
them. To avoid this and to retain the topological defects action between vortices, so that a specific vortex config-
afterquenching,asamplewithsufficientlystrongpinning uration, formed in the course of fast hot-spot quenching,
centers and weak vortex-vortex interactions is required. can be preserved.
The sample schematically shown in Fig. 1c is a mul- Figure 1d presents the vortex distribution SHPM im-
tilayered structure of Pb/Ge/Au deposited on a SiO /Si age after performing field-cooling (FC) from above T to
2 c
substrate (see Methods). The superconducting Pb film, 4.2 K. A disordered vortex lattice, homogeneously dis-
which is characterized by relatively weak vortex-vortex tributed in the scanned area, is observed. After apply-
interactions and a high density of quasi-homogeneously ing a pulse of tunnelling voltage, followed by the Hall
distributedpinningcenters,isanidealcandidatetostudy probe imaging, a vortex cluster is observed at the STM
the local heating effect. The pinning centers are created tip position. Since there is no multi-quantum vortex ob-
Increase T to7.4K; release tip at 
(0,‐8); fast coolingdown to T; 
retract and scan; 
Vbias=‐0.1V Vbias=‐0.2V Vbias=‐0.3V Vbias=‐0.4V Vbias=‐0.5V Vbias=‐0.6V
4.2K 4.2K 4.2K 4.2K 4.2K 4.2K
150940 171717 172220 165948 170516 171053
标准色
2 4 8 21 44
26
3
a V =0.1 V V =0.2 V V =0.3 V V =0.4 V V =0.5 V
bias bias bias bias bias
1.3
T)
m
(
Bz
0
b c d

0.8
r
vo r-tveoxr tcelxuster Vbias=0.1V mbe 40 1.2 Vbias=0.5 V
0.6 0
u
n
T) 0.4 ex T) 0.8
B (mz0.2 d vort 20 B (mz0.4
e
n
0.0 nfi 0 0.0
-2 -1 0 1 2 o 0.0 0.2 0.4 0.6 -4 -2 0 2 4
C
Scan (m) V (V) Scan (m)
bias
FIG. 2. Vortex clustering as a function of bias voltage. (a) Scanning Hall probe microscopy (SHPM) images observed
aftertunnelingpulseswithvariousbiasvoltagesV . Thesizeofthevortexclusterincreaseswithbiasvoltage. Theminimum
bias
cluster that could be observed contains two flux quanta at V =0.1 V. The scale bar equals 4 µm. (b) Field profiles across
bias
the center of an individual vortex (circles) and the smallest vortex cluster (squares), as indicated by the dashed lines in (a),
observedatV =0.1V.(c)Numberofvortices intheclusterasafunctionofthebiasvoltage(SupplementaryFigure4and
bias
Supplementary Note 3). The error bar corresponds to the number of vortices cut by the edges of the scanned SHPM images.
(d) Magnetic field profiles along different directions across the center of a vortex cluster observed at V = 0.5 V. All the
bias
profilesaresimilartoeachother,suggestingthatthevortexdistributionintheclusterisapproximatelyaxiallysymmetric. The
white circles mark the position of individual vortices at the periphery of the cluster (see Supplementary Figure 5).
servedinoursample,weassumethatthevortexclusteris vortex clustering can be seen. Figure 2b compares the
h=0.96673 mm
composedofsinglequantumvortices(seeSupplementary magnetic field profiles through the centers of the vortex
Note 4). BetwFe=en2t.h0e1vo Frt0ex cluster and the rest of in- cluster observed at Vbias = 0.1 V and a single-quantum
dividual vortices a vortex-free area is formed. Note that vortex. It is seen that the magnetic field at the center of
the positions of the vortices outside the cluster remain the cluster is doubled as compared to a single-quantum
unchanged. This confirms that the impact of heating is vortex, suggestingthatitcontainstwofluxquanta(Sup-
local. Thevortexclusterretainsthesamestructureeven plementary Note 4). The number of trapped vortices as
after a few hours relaxation, suggesting that it is a rel- a function of the bias voltage is shown in Fig. 2c. The
atively stable configuration. Similar results have been power dissipated in the sample is P ∝ V I, with I
bias
observedinthreedifferentsamples. Ithasbeenreported being the tunnelling current which is kept constant in
thattheSTMtunnellingjunctioncanreshapethetopog- ourexperiments. Correspondingly,thesizeoftheregion,
raphyofamaterial,generatingdefectsatthetipposition where superconductivity is suppressed by local heating,
[27]. Thispossibilityisruledoutsinceadisorderedlattice is expected to increase with bias voltage. This accounts
is again observed in the same region after a subsequent for the observed dependence in Fig. 2c.
FC procedure. The possibility of mechanically induced
One clear feature of the observed vortex clusters is
vortex attraction [28] is also ruled out since, in our case,
that, instead of forming a disordered Abrikosov vortex
there is no direct contact between the STM tip and the
lattice, clustered vortices are arranged into concentric
sample.
ring-like structures. As demonstrated in Fig. 2d, the
field profiles through the center of a vortex cluster are
Controlling the vortex cluster size. The size of the nearlysymmetricwithapeakattheclustercenter. Such
vortex clusters can be well controlled. Figure 2a shows a kindofring-shapedvortexclusters,predictedbyShapiro
seriesofimagesmeasuredafterapplyingthepulseoftun- et al [29], might be attributed to the Kibble-Zurek (KZ)
nelling current at the same position under various bias symmetry-breaking phase transition [30, 31]. When the
voltages. It is clear that the size of the vortex cluster tunnelling current is applied, the local area is heated up
increases with increasing bias voltage. The smallest vor- above T , leading to the formation of a normal domain
c
tex cluster is observed at V = 0.1 V, below which no with homogeneous distribution of magnetic flux in this
bias
4
a b experimentally with single vortex resolution. The possi-
4.2K 6.2K ble relation of our experiment to the KZ mechanism is
T=6.7 K
further supported by the theoretical simulations, where
vortex-antivortex pairs, as topological defects, appear
from quenching the hot spot (Supplementary Figure 6,
7 and 8, and Supplementary Note 5). However, due to
T=6.8 K
6.5K 6.7K the short annihilation time, the stabilization of KZ vor-
ticesandantivorticeshavenotbeenobservedexperimen-
tally. Further experimental evidence is needed to clarify
the mechanism, such as the relation of KZ vortices with
T=6.9 K
quenching rate.
nt
6.8K 6.9K ou As mentioned above, the vortex clusters are preserved
C
duetothenaturallyformedpinningcenters. Thepinning
T=7.0 K strength is weakened when approaching Tc [33]. There-
fore, we expect a competition between pinning and vor-
texlatticeelasticitywithvaryingtemperature. Thisalso
7.0K 7.1K provides an opportunity to study the relaxation of local
T=7.1 K non-equilibrium configuration in a macroscopic system.
Figure 3a displays the vortex cluster evolution with in-
creasing temperature. When the vortex cluster is cre-
ated at 4.2 K, it preserves the same geometry up to
0 1.48 1 2 3 4 T = 6.2 K (see Supplementary Figure 9), above which
B (mT) d (m)
z v-v the pinning strength is considerably weakened as com-
pared to the vortex-vortex interaction and the vortex
FIG. 3. Disintegration of vortex cluster with increas-
cluster starts to disintegrate. We notice that the dis-
ing temperature. (a) Scanning Hall probe microscopy
(SHPM) images observed after forming a vortex cluster at integration process starts from the interior of the vortex
4.2 K and then increasing the temperature gradually. The cluster, where vortex density is high, so that the vortex-
vortex cluster remains compact until T = 6.2 K. With fur- vortex interaction is strong. At intermediate tempera-
ther increasing temperature, the vortex cluster expands due tures (e.g., T = 6.8 K), vortices tend to form chains as
totheweakeningofpinningandfinallyahomogeneousvortex
highlighted by the dashed lines in Fig. 3a. Such vor-
lattice is observed at T =7.1 K. The Delaunay triangulation
tex chains might be associated with the one-dimensional
of the vortex pattern at T = 7.1 K is shown, where the ver-
temperature-induced vortex motion as observed in the
tices of each triangle represent the locations of the vortices.
Thesquares,circlesanddiamondsrepresentthevorticeswith vortex lattice melting process [20]. Only at high enough
five-, six- and seven-fold symmetries. The scale bar equals 4 temperature (e.g., T =7.1 K), vortex repulsion strongly
µm. (b) Histograms of the vortex nearest neighbor distance dominates and a homogeneous distribution of vortices,
obtained from the images in (a). The distribution at 7.1 K corresponding to a weakly distorted Abrikosov lattice, is
can be well simulated using the Gaussian curve.
formed. Additional information about the vortex clus-
terdisintegrationprocesscanbeobtainedbyconsidering
thenearestneighbordistance(NND)ofvortices. Aspre-
domain. After the heating is stopped by retracting the sented in Fig. 3b, the NND shows a broad distribution
STM tip, the recovery of the superconductivity starts at low temperatures. At T =7.1 K, it can be well fitted
when the temperature front deviates from the order pa- by the Gaussian distribution (solid line). The observed
rameter front. The temperature front accelerates while peak position at dv-v = 2.3 µm is consistent with the
the order parameter front, which presses the magnetic valueexpecte√dforanAbrikosovtriangularvortexlattice,
flux confined inside the shrinking normal region, decel- dv-v =(2Φ0/ 3H0)1/2 =2.27 µm.
erates, leading to an unstable normal domain ring area
with T <T and ψ =0. As a result, vortices nucleate in Theoretical simulations. The proposed scenario of
c
this normal domain ring at the perimeter of the heated vortex clustering is supported by the results of numeri-
area and the hot spot looses part of its magnetic flux. cal simulations, based on the time-dependent Ginzburg-
The order parameter front then accelerates briefly, be- Landau (TDGL) formalism and the local heating model
fore the above-described dynamics repeat [29, 32]. With (Supplementary Figure 10 and Supplementary Note 6).
further decreasing temperature, natural pinning in the The deviation of the local temperature T from its equi-
heated area becomes evident again, and the ring shaped librium value in the absence of tunnelling current, T , is
0
vortex cluster is preserved even when the temperature represented as T −T = αf(x,y) (see Fig. 4a), where
0
of the whole area is far below T . As far as we know, the coefficient α is proportional to the power dissipated
c
thering-shapedvortexclustershaveneverbeenobserved by the tunnelling current, while the spatial temperature
5
a b a b e ΦI,II=0.93Φ0
high IV
ure I II ΦIII=0.95Φ0
at
er /m III
z y mp 1
e c d
x T=T0+αf(x,y) lotw 0 0.6 V ortex-I V ortex-III0.6 Vortex-IV
c d e T) 0.4 V ortex-II 0.4
m
3.5 B (z0.2 0.2 0 0.29
0.0 0.0
-2 0 2 -2 0 2 Bz(mT)
Scan (m) Scan (m)
Hc2
α=0.2 α=0.3 α=0.5 B/μz0 FIG. 5. Manipulation of individual vortices with the
× scanning tunneling microscopy (STM) tip. (a) SHPM
210 image observed after field cooling from above Tc to T = 4.2
K at H = 4.3 Am−1. Three single flux quantum vortices
are obsZe0I,rII=v0e.d96 aμtmnaturZaIIIl=l1y.1f7o μrmmed pinning centers. (b) Image
-1 observed after moving vortex II with the STM tip along the
arrow in (a). Vortices I and II merged to form a vortex clus-
ter IV. (c, d), Magnetic field profiles through the centers of
FIG. 4. Simulations. (a) Temperature distribution at the
vortices I, II, III and vortex cluster VI. The solid lines are
scanning tunneling microscope tip position. (b) Vortex dis-
a fit with the monopole model. (e), A pattern V short for
tribution (upper panel) after field cooling down to the tem-
vortexisarrangedbyusingtheSTMtip.Thescalebarsequal
perature T = 0.58T at magnetic field H = 0.03H (T ).
0 c 0 c2 0 4µm for all the images.
The lower panel shows the distribution of the electron mean
free path, which models a set of quasi-random pinning cen-
ters. Themagneticfielddistributions,whichareformedinthe
course of a tunnelling current pulse (upper panel) and after magnetic flux. This pinning site attracts and traps vor-
switchingoffthepulse(lowerpanel)areshownfortheparam-
tices from the surrounding superconducting area, where
eter α=0.2 (c), 0.3 (d) and 0.5T (e). When the tunnelling
c the temperature is elevated and the mobility of vortices
currentison,localsuperconductivityisfullysuppressed. The
is increased due to their weaker interaction with pinning
scale bar equals 4 µm.
centers. The magnetic flux trapped within the normal
region increases with increasing α. When the tunneling
distribution f(x,y) is determined by material and ge- current is switched off and the local temperature relaxes
ometric parameters of the sample. Using this temper- toT0, thetrappedfluxtransformsintoacompactvortex
ature distribution to model the effect of a tunnelling cluster, which is stabilized by the pinning centers. Sim-
current pulse, the TDGL calculations are performed for ilarly to the experiment, the calculated vortex pattern
a 10 × 10 µm2 superconductor film with random pin- demonstrates the appearance of vortex-depleted regions
ning centers (Fig. 4b). Based on the used local heating around the vortex cluster, which become more evident
model and the material parameters, the quenching time when increasing the intensity of local heating and the
τ−1 = −T−1∂T/∂t| , corresponding to the thermal cluster size.
Q T=Tc
relaxation of the hot spot after switching off the tunnel-
ing current, is estimated to lie within the range of 1 to Manipulation of single-quantum vortices. The ob-
10psfortheinitialradiiofthenormaldomaininthehot tainedresultssuggestthatlocalheatingwithanSTMtip
spot from 1 to 3 µm. These quenching-time values are can providean efficient tool toarrange vortices insuper-
comparable to the Ginzburg-Landau characteristic time conductors. Finally, we demonstrate the ability of using
in our sample, tGL(T0) = 1 ps. The formation of vortex the STM tip to manipulate individual vortices (see also
clusters, resultingfromthesimulatedquenchingprocess, Supplementary Movie). This is an important extension
andtheirstabilizationduetovortexpinningtypicallyoc- of the pioneering work of Eigler [16] from manipulating
cursonthetimescale∼40to100ps. Thevortexcluster singleatomstomanipulatingsinglequantumvortices. To
formationtime,comparedwiththeshortquenchingtime, move a vortex, first, a tunneling junction is established
clearly places the quenching process in the KZ regime. by approaching the STM tip to the surface of the sam-
The simulation results are displayed in Figs. 4c to 4e ple at 1 µm away from the vortex center. A hot spot is
for different values of the coefficient α. The local heat- generated and the vortex is attracted to the location of
ing, caused by tunnelling current, leads to the formation the hot spot. Then the STM tip is retracted and moved
of a normal-state region in the vicinity of the STM tip. to the next position. By repeating the above process, we
The normal region can be considered as a big pinning are able to drag the vortex to any position in the super-
site, which is capable to accommodate a relatively large conductor. AsshowninFig.5a,asinglequantumvortex
6
(vortexII)canbemovedalongthepathindicatedbythe neticfielddistributionisrecordedusingthescanningHall
dashed arrow to merge with vortex I, forming a vortex probe microscope from Nanomagnetics in a lift-off mode
cluster (IV in Fig. 5b). We are also able to drag the (Supplementary Figure 12 and Supplementary Note 7).
vortex cluster as one object to a new position and then First, the Hall sensor approaches the sample under the
detach the two vortices by increasing the temperature control of a piezo until the tunneling current is estab-
(see Supplementary Figure 11). By using the local heat- lished. Then the whole sensor is retracted by 200 nm
ing method demonstrated above, a pattern V, short for and the magnetic field distribution is mapped. In all the
vortex, is arranged from Φ -vortices as shown in Fig. 5e. measurements, the magnetic field is applied perpendicu-
0
Compared with other techniques, such as magnetic force larly to the sample surface. WSxM software is used to
microscopy [6], that are used to manipulate individual process all SHPM images [35].
vortices, the big advantage of our technique is that no
magnetic field or transport current is needed. TDGL simulation. For a superconductor with pin-
ning centers, which originate from a local reduction of
Discussion the mean free path (cid:96), the TDGL equation for the or-
In conclusion, we have shown that a controlled heating der parameter ψ, normalized to 1, can be written in the
effect generated by the tunnelling current can be used to form [26]
locally suppress superconductivity, with this area play-
(cid:18) (cid:19)
∂ (cid:96)
ing the role of a pinning center. We are able to tune to +iϕ ψ= (∇−iA)2ψ
∂t (cid:96)
power of the nano heater simply by changing the bias m
voltage. Moreover, the tunnelling pulse applied through (cid:18)1−T2/T2 (cid:96) (cid:19)
+2ψ c − m|ψ|2 , (1)
the STM tip enables dragging and manipulation of in- 1−T2/T2 (cid:96)
0 c
dividual quantized vortices which may be used for the
where ϕ and A are the scalar and vector potentials, re-
development of new fluxon-based devices. Our results
spectively, and l is the mean free path value outside
demonstrate the utility of the local heating effect for m
the pinning centers. The relevant quantities are made
characterizing and manipulating vortices in a supercon- √
dimensionless by expressing lengths in units of 2ξ(T ),
ductor. This new approach provides a lot of information 0
time in units of π(cid:126)/[4k (T −T )]=2t (T ), magnetic
whichisofgreatinterestforstudyingothersystems,such B c 0 GL 0
fieldinunitsofΦ /[4πξ2(T )]=µ H (T )/2,andscalar
aslocalandnon-localphasetransitionsinsuperfluid,cos- 0 0 0 c2 0
potential in units of 2k (T −T )/(πe). Here, µ is the
mology and many-particle systems. B c 0 0
vacuum permeability, t is the Ginzburg-Landau time,
GL
H is the second critical field and ξ is the coherence
c2
length at l=l .
m
ThevectorpotentialA,forwhichwechoosethegauge
Methods
∇·A = 0, can be represented as A = A +A . A
Sample and experimental setup. The trilayer of 0 1 0
denotes the vector potential corresponding to the exter-
Au/Ge/Pb was prepared in two steps. First, to ensure a
nally applied magnetic field H , while A describes the
relatively small κ of the sample, a 85 nm thick Pb film 0 1
magnetic field H induced by the currents j, which flow
wasdepositedonaSiO /Sisubstrateusinganultrahigh 1
2
in the superconductor:
vacuum (3×10−8 Torr) electron beam evaporator cali-
brated with a quartz monitor. The substrate was cooled 1 (cid:90) j(r(cid:48))
A (r)= d3r(cid:48) . (2)
to 77 K by liquid nitrogen to ensure the homogeneous 1 2πκ2 |r−r(cid:48)|
growth of Pb. On top of Pb, a 10 nm thick Ge layer is
deposited to protect the sample surface from oxidation Here, κ = λ(T0)/ξ(T0) is the Ginzburg-Landau pa-
and also to avoid any proximity effect between Pb and rameter and λ is the penetration depth at (cid:96) =
thesubsequentlydepositedconductivelayer[34]. Second, (cid:96)m. √The current density is√expr√essed in units of
the sample was transferred into a sputtering machine, Φ0/[2 2πµ0λ(T0)2ξ(T0)] = 3 3/(2 2)jc(T0) with jc,
and a 35 nm thick Au layer was deposited to cover the the critical (depairing) current density of a thin wire or
whole sample, playing the role of a conducting layer for film. IntegrationinEq.(2)isperformedoverthevolume
the tunnelling current of the STM tip. The thickness of of the superconductor.
Auyieldsasmoothsurfacewitharoughnesslessthan0.2 In general, the total current density contains both the
nm. The critical temperature T =7.25 K is determined superconducting and normal components: j = j + j
c s n
using local ac susceptibility measurements. The pulse with
tunneling was applied by controlling the bias voltage be-
j =Im(ψ∗∇ψ)−A|ψ|2, (3)
tween the STM tip (Au) of a commercial Hall probe as s
shown in Fig. 1b and the sample, which is ground. In
(cid:18) (cid:19)
our experiments, the tunneling current is kept constant σ ∂A
j =− ∇ϕ+ , (4)
(∼ 0.5 nA) through a PID protocol control. The mag- n 2 ∂t
7
where σ is the normal-state conductivity, which is taken perconductivityatthenanoscale.NatureNanotechnology
as σ = 1/12 in our units [36]. The distribution of the 4, 142 (2009).
scalar potential ϕ is determined from the condition [13] Wong, D. et al. Characterization and manipulation of
individual defects in insulating hexagonal boron nitride
using scanning tunnelling microscopy. Nature Nanotech.
∇·j=0, (5)
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All authors contributed to the discussion and analysis.
V.V.M. coordinated the whole work.
Acknowledgements Competing financial interests
WeacknowledgethesupportfromtheMethusalemfund- The authors declare no competing financial interests.