Science - USA (2020-05-01)

diffraction. The traces ofdV/dIwithina
field interval comprising two periods of the
fast mode are displayed in Fig. 1B. The large
peaks(Fig.1B,bluearrows)traceoutthescal-
loped boundary, whereas the weaker peaks
(Fig. 1B, red arrows) trace out the Fraunhofer
diffractionpattern.Inthecolormapsfortwo
large-area samples S2 and S6 (Fig. 1, C and D,
respectively), the fast mode is strikingly evi-
dent as the scalloped boundary surrounding
the entire dissipationless region, whereas the
slow mode is unresolved. We express the fast-
mode frequencyf 1 =1/DB 1 (whereDB 1 is the
period) in terms of the flux-penetration area
Af≡f 1 f 0.
The slow mode displaying the familiar
Fraunhofer diffraction pattern reflects the phase
of the bulk supercurrentJsb, which winds at a
frequencyf 2 = 1/DB 2 that is insensitive to the
crystal areaAphys(fig. S3). The conditions that
favor observation of either the slow mode (fig.
S5) or the fast mode (fig. S6) are described in
( 13 ), section 3.
Hereafter, we focus onf 1 to show that the
fast mode originates from an edge supercurrent
Jse. As shown in Fig. 2A,f 1 , expressed through

Af, scales asAf=h(B)Aphysacross five sam-

ples. The fractionh(B) expresses the degree

of flux penetration. In the Fig. 2A plot, the

black symbols and black dashed line refer to

the weak-field limit (B<1mT).Alreadyinthis

limit,Af=f 1 f 0 scales linearly withAphyswith

h(B~0)=0.35.

Inspection off 1 reveals that it increases

gradually withB. This chirp effect reflects in-

creasing flux penetration (on the scale of the

Pearl lengthL=2l^2 /d, wherelis the London

length). As indicated by the broad arrows and

the red symbols in Fig. 2A,Afin each sample

increases monotonically, reaching its physical

areaAphyswell beforeBreaches the critical

fieldBc. The plot off 1 versusBfor sample S1

in Fig. 2B shows that it saturates whenBex-

ceeds 6 mT. In all samples,h(B→Bc)→1 but

does not exceed 1. The partial screening im-

plies thatJsbis not confined to a monolayer,

but extends over the entire crystal volume.

AsshowninFig.2A,f 1 tracks the flux quanta

asAphysis increased ninefold at fixedBand

also asB→Bcat fixedAphys. Both trends sug-

gest fluxoid quantization within a closed loop

defined byJse.WeassumeJesflows along the

side wall (of depthd) encircling the crystal,

with a widthde(Fig. 2C), which we then es-

timated. A finitedeleads to a spread in the area

DAphys=deLpand a phase uncertaintydφ=2p

(deLp/f 0 )B, whereLpis the crystal perimeter.

Complete dephasing of the fast mode occurs

(at the dephasing fieldBd) whendφ→p;Bd=

0.6Bcto 0.8Bcis fixed by the experimental

resolution (Fig. 1, A and C, arrows; and fig. S9).

This yieldsde=f 0 /(2BdLp). From the observed

Bd= 7 mT in sample S1 (1.8 mT in sample S2),

we foundde<10nm(de~ 1/200 of the crystal

widthw). In the largest sample S6, the de-

phasing effect is especially clear (fig. S9).

Tomakethecaseforfluxoidquantization

[( 13 ), section 5], we assume that the edge con-

densate is described by a Ginzburg Landau

(GL) wave function (Y^e) distinct from that

describing the bulk (Y^b). The quantization of

fluxoids within an enclosed area causes the

edge superfluid velocityvsto vary asvs=

(2pℏ/m*Lp)(n–f/f 0 ), whereℏis Planck’s

constanthdivided by 2p,m* is the GL mass,

andn∈ℤ( 13 ). This leads to a set of free-energy

branchesDfn(f), each centered atf=nf 0

(Fig. 2D). At an intersection, the system jumps

between branches, leading to a sawtooth pro-

file forvs(f). The result is a characteristic

scalloped profile for the square of the wave

function amplitudeY^2 e≡jY^ej^2 which we write as

DY^2 e=Y^2 e¼Pðnf=f 0 Þ^2 ;

n

1

2

<f=f 0 <nþ

1

2



ð 1 Þ

with the prefactorP¼ð 2 pxÞ^2 =L^2 p, wherexis the

GL length [( 13 ), section 5].

In the classic Little Parks (LP) experiment

( 17 , 18 ), the relative change corresponding to

Eq. 1 is observed as a shiftdTc(f) very nearTc

(where the amplitudeYb¼jY^bj→0). Our

experiment, performed atT=Tc, falls in a

different regime; to drive both amplitudesYe

andYb→0, we appliedIclose toIc. The nar-

row widthdeoftheedgeGLwavefunctionY^e

renders it less susceptible thanY^bto field sup-

pression asIapproaches the boundaryIc(B).

Hence, the edgeJsecarries an increasing share

ofI. At the boundary,IceY^2 eacquires the pro-

file in Eq. 1,DIceDY^2 e(eq. S14).

Equation 1 predicts that the oscillation

amplitudeDIcdecreases steeply as 1=L^2 p. We

confirm that the observed decrease is con-

sistent with the prediction (fig. S8). The mod-

el also explains a striking observation. As

seen in samples S1, S2, and S6 in Fig. 1, the

fast-mode minima occur high above the hor-

izontal axis,I=0,whereastheslowmode

minima in S1 (also V2 in fig. S5) reach nearly

to zero. This occurs because the former arises

from a weak modulation of the amplitude

squaredDY^2 e, whereas the latter derives from

phase winding.

SCIENCEsciencemag.org 1 MAY 2020•VOL 368 ISSUE 6490 535

Fig. 1. Critical current maps in the Weyl superconductor MoTe 2 .Shown are color maps ofdV/dIversus
IandBtaken atT= 20 mK. (A) In sample S1, two oscillation modes are resolved. The fast mode, arising
from amplitude modulation of an edge supercurrent, is observed as the scalloped boundary of the low-
dissipation region. The slow mode, associated with the bulk supercurrent, displays the usual Fraunhofer
diffraction pattern. (B) Shown are 22 traces ofdV/dIversusI(shifted for clarity) taken in sample S1 in
steps of 30mT starting at 1.29 mT. Prominent peaks (blue arrows) track the fast mode, whereas the weak
peaks (red arrows) track the slow mode. (CandD) In large-area crystals [samples S2 and S6 in (C) and
(D), respectively], the fast mode is evident, whereas the slow mode is unresolved. (Insets) The Au contacts
evaporated on each crystal. Arrows in (A) and (C) indicate the dephasing fieldBd.

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