Position-dependent effect of non-magnetic impurities on superconducting properties of nanowires

2.2 MB·

by  L. -F. Zhang

epl draft
Position-dependent effect of non-magnetic impurities on super-
conducting properties of nanowires
4L.-F. Zhang, L. Covaci and F. M. Peeters
1
0Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium
2

c
e
D PACS 74.78.Na–Superconductivity,Mesoscopic and nanoscale systems
PACS 61.72.-y–Scattering by defects

1 PACS 73.63.-b–Electronic transport, nanoscale materials
3
Abstract. - Anderson’s theorem states that non-magnetic impurities do not change the bulk
]
propertiesofconventionalsuperconductors. However,asthedimensionalityisreduced,theeffectof
n
o impuritiesbecomesmoresignificant. Hereweinvestigatesuperconductingnanowireswithdiameter
c comparabletotheFermiwavelengthλF (whichislessthanthesuperconductingcoherencelength)
- by using a microscopic description based on the Bogoliubov-de Gennes method. We find that:
r
p 1) impurities strongly affect the superconducting properties, 2) the effect is impurity position-
u dependent, and 3) it exhibits opposite behavior for resonant and off-resonant wire widths. We
s showthatthisisduetotheinterplaybetweentheshaperesonancesoftheorderparameterandthe
.
t sub-bandenergyspectruminducedbythelateralquantumconfinement. Theseeffectscanbeused
a
to manipulate the Josephson current, filter electrons by subband and investigate the symmetries
m
of thesuperconducting subbandgaps.
-
d
n
o
c
[
Introduction. – The effect of impurities on super- crosssectionofthe wireisshorterthanthe superconduct-
2
conductivity have intrigued scientists for several decades. ing coherence length, ξ (also known as the healing length
v
Adding impurities to a superconductor can not only pro- of the OP), superconductivity weakens there, thus affect-
9
1videanusefultooltoinvestigatethesuperconductingstate ing transport properties of the wires. Quantum phase
3butcanalsobeusedtooptimizetheperformanceofcertain slips,leadingtothelossofphasecoherence,areoneofthe
4devices. For example, impurities are used to distinguish main reasons for the appearance of normal regions giving
.
1between various symmetries of the superconducting state rise to a finite resistance below Tc in nanowires [13–19].
0[1,2], to identify topologicalsuperconductors [3] or to pin Phase slip junctions can be fabricated in order to take
4vortices in order to enhance the superconducting critical advantage of this mechanism [20]. Normal regions can
1
current [4]. Strong disorder is often used to study the also be caused by interactions with photons and this can
:
vsuperconducting-insulatortransitionsandthe localization be used to build single-photon detectors [21,22]. Second,
i
XoftheCooperpairs[5–10]. Itiswellknownthatmagnetic surfaceeffectsinnanoscalesuperconductorsresultintopo-
impurities,whicharepairbreakers,suppresssuperconduc- logical superconducting states [23]. Majorana fermions,
r
ativity. On the other hand, as Anderson’s theorem points whicharetheirownanti-particles,werepredictedasquasi-
out[11],forbulkconventionalsuperconductors,smallcon- particles in such p-wave nanowires [24,25]. Third, quan-
centrations of non-magnetic impurities do not change the tum confinement results in quantum size effects [26–28],
thermodynamic properties of the system, such as the su- quantum-size cascades [29], facilitates the appearance of
perconductingcriticaltemperatureT ,andthatimpurities new Andreev-states [30] and give rise to unconventional
c
haveonlylocaleffects[1]onthesuperconductingorderpa- vortex states [31] induced by the wavelikeinhomogeneous
rameter (OP) and the density of states. spatial distribution of the OP [27].
On the other hand, quasi-one-dimensional supercon- Up to now, the theoretical study of superconducting
ductingnanowiresshowdifferentpropertiesfromthebulk nanowires was limited to defect-free cases in the clean
andcouldhavewide potentialapplicationsin the nearfu- limit orto systemswith weaklinks, whichweretreatedin
ture [12]. First, if in some regions, the diameter of the the one-dimensional limit. However, experiments such as
p-1
L.-F. Zhang et al.
ScanningTunnelingMicroscopy(STM)arenowadaysable For simplicity, we consider a two-dimensional problem
tomeasurethelocaldensityofstates(LDOS)withatomic and introduce a long unit cell whose area is S = L L
x y
resolution [32–36] and consequently the finite width of where L is the length of the unit cell and L is the wire
x y
nanowires can no longer be neglected. It is known that width. Periodic boundary conditions are set in the x di-
in low dimensional superconductors phase fluctuations rectionandL is setto be longenoughsuchthatphysical
x
should play an important role [37–39]. The calculation properties are L -independent. Due to quantum confine-
x
presented here is done at zero temperature and at mean- mentinthetransversedirectiony,wesetDirichletbound-
−→ −→
field level, therefore it cannot describe fluctuations. Nev- aryconditionatthesurface(i.e. u (r)=v (r)=0, r ∈
n n
ertheless, by calculating the modifications induced by the ∂S).
impurityontheJosephsoncurrentweareabletoinferthe In order to solve more efficiently the self-consistent
phase robustness of the condensate, therefore indirectly BdG equations (1)-(3), we expand u (v ) by using a two
n n
provide information about the enhancement of fluctua- step procedure. First, in order to get the eigenstates of
tions and the appearance of phase slips. the single-electron Schr¨odinger equation K0φl = Elφl we
In this Letter, we investigate the effect of a nonmag- Fourier expand φ :
l
netic impurity on superconducting nanowires with diam-
2 πjy
eter comparable to the Fermi wavelength, λF. The bulk φl(x,y)= cj,kexp(ikx)sin , (4)
coherence length at zero temperature, ξ0, is much larger sLxLy j>0,k (cid:18) Ly (cid:19)
X
than λ . For such systems, quantum confinement ef-
F where c are the coefficients of the expansion and wave
fectsdominateandadescriptionbasedonthemicroscopic j,k
vector k =2πm/L , m∈Z. Here we use a large number
Bogoliubov-de Gennes (BdG) equations is required. The x
ofbasisfunctionsinordertoensuretheaccuracyofthere-
impurity,whichinducespotentialscatteringonsingleelec-
sults. Next,weexpandu (v )intermsofφ (x,y). Inthis
trons,notonlyaffectslocalpropertiessuchastheOPand n n l
step, only states with energies E < E +ε are included.
theLDOS,butalsohasglobaleffectsonthecriticalsuper- l F
In our calculation, ε is taken to be 30E and we find that
current. Quantum confinement leads to inhomogeneous c
any larger cut-off does not modify the results. We have
superconductivity andtherefore the effect of the impurity
checked our results, for a specific choice of parameters,
strongly depends on its transverse location.
against a more computationally intensive finite-difference
Theletterisorganizedasfollows: firstwebrieflypresent
method approach.
ournumericalmethodandthepropertiesofthenanowires
We choose to study NbSe2 nanowires as an example
inthecleanlimit,thenweshowtheeffectofnon-magnetic
because of the availability of high-quality nanowires with
impurities on the profile of the order parameter and the
2 − 25nm in diameter [40,41]. Also, theoretical calcu-
local density of states. At the end we investigate the im-
lations are in good agreement with experiments, espe-
purity effect on the Josephson critical current.
cially, for vortex lines [42–44]. In addition, the lower
Theoretical approach. – We start from the well- Fermi energy makes the calculations feasible. The pa-
known BdG equations: rameters of bulk NbSe2 are the following: m = 2me,
E = 40meV, E = 3meV and coupling constant g is
−→ −→ −→ −→ F c
[K0−EF]un(r)+∆(r)vn(r)=Enun(r), (1) set so that the bulk gap ∆0 = 1.2meV, which yields
−→ ∗ −→ ∗ −→ −→
∆(r) un(r)−[K0 −EF]vn(r)=Envn(r), (2) Tc ≈ 8.22K, ξ0 = 14.7nm and kFξ0 = 21.23. In
where K0 = −(~∇)2/2m + U(−→r) is the kinetic energy nanowires, the mean electron density ne is kept to the
value obtained when L ,L → ∞ by using an effective
x y
withU beingthepotentialbarrierinducedbytheimpurity E ,wheren = 2 |u |2f(E )+|v |2[1−f(E )] .
and E the Fermi energy, u (v ) are electron(hole)-like F e S n n n n n
F n n All the calculations are performed at zero temperature.
quasiparticle eigen-wavefunctions, En are the quasiparti- P R (cid:8) (cid:9)
cle eigen-energies. The impurity potential is modeled by NumericalResults.– WefirstshowinFig.1impor-
−→ −→
a symmetrical Gaussian function, U(r) =U0exp[−(r − tant superconducting properties of clean nanowires, i.e.
−→r0)2/2σ2] where U0 is the amplitude, −→r0 is the location of with U ≡ 0. The spatially averaged OP, ∆¯, shows quan-
the impurity and σ is the width of the Gaussian. tumsizeoscillationsasafunctionofthewidth,Ly,asseen
The pair potential is determined self-consistently from fromFig.1(a). Theresonantenhancementsappearalmost
the eigen-wavefunctions and eigen-energies: regularly with a period of half the Fermi wavelength, i.e.
λ /2. Theyareduetothefactthatthebottomoftherel-
F
−→ −→ ∗ −→
∆(r)=g u (r)v (r)[1−2f(E )], (3) evant single-electron subbands passes through the Fermi
n n n
EnX<Ec surface, which results in a significant increase in the den-
sity of states, i.e. in the number of electrons which can
where g is the coupling constant, E is the cutoff energy,
c form Cooper-pairs. The different energy spectra for the
and f(E ) = [1 + exp(E /k T)]−1 is the Fermi distri-
n n B resonance and off-resonance cases are the key to under-
bution function, where T is the temperature. The lo-
stand the behavior of the nanowires. For completeness,
−→
cal density of states is calculated as usual: N(r,E) =
we show in Fig. 1(b)-(h) more information for the res-
[δ(E −E)|u (−→r)|2+δ(E +E)|v (−→r)|2].
n n n n onance case, with L = 3.4nm, intermediate case, with
n y
P
p-2
Effect of impurities on superconducting nanowires
(a) (b) j=1 j=2 j=3
1:L=3.4nm 2:L=3.8nm
y y
4 α γ E1 E2 E3
V) 2 12 V) α* OS
me me 2 3:Ly=5.4nm D
∆ ( 1 3 ∆|| ( β β* α′β′ LDOS impurity at side impurity at center
0 0
4 6L (nm)10 12 y (nm) E1
y
(g) E2
eV) 4 E 4 4 j=3 E E3
m c c
E ( 2 j=1 j=2 2 j=1 j=2 2
(c) (e) ←j=1 j=2
0 0 0 Fig. 2: (Color online) Effect of the impurities on the elec-
−1.6−1 0 1 1.6 −1.6−1k (nm−1)1 1.6 −1.6−1 0 1 1.6 tronic structurewhen the nanowire is in normal state. (upper
LDOS 4 (d)αγ:: yy==20..5855nnmm 3 (f)α*: y=0.98nm 2 (h) β′: y=2.7nm→ ppaonsietli)onDO(bSluaessaolifdunlcintieo)naonfdenatercgeyntfoerreadnpiomsiptuiornity(raetdssiodleidd
d 2
malize 2 β: y=1.7nm 1 β*: y=1.9nm 1 α′: y=1.46nm lairnee,fosrhiUfte=d u0pafsorrecflearreitnyc)e,s.res(ploewcteirveplayn.eTlsh)eSbclhaecmkastoilcidcolirnrees-
or sponding LDOS(x,y) at energy E1, E2 and E3, respectively.
N 0 0 0
−4 −2 0 2 4 −4 −2 0 2 4 −4 −2 0 2 4 The open circles indicate impurities’ position.
bias energy (meV)
′
Fig. 1: (Color online) Properties of clean NbSe2 nanowires. proportional to |∆(y)|, i.e. the LDOS at α (local maxi-
(a)Spatially averaged ∆¯ asafunction of Ly. Theopen circles mum) is larger than the one at β′ (local minimum). Be-
1−3 indicate the resonance case (Ly = 3.4nm), intermediate sides the sub-gap shift seen in the energy spectrum, an-
case (Ly = 3.8nm) and off-resonance case (Ly = 5.4nm), re- other important difference between the two cases is that
spectively. (b) OP |∆(y)| for the three cases. The αβγ are for resonance there is always a large number of low mo-
defined as the positions where the OP has a local maximum
mentum quasi-particles [see Fig. 1(c)] which are involved
∗ ∗
/ minimum for resonance, the α β are for intermediate and
inpairing,whereastherearenoquasi-particlesatk =0in
theα′β′ areforoff-resonance. Thevalueofthesepositionsare
the off-resonance case [see Fig. 1(e)]. Note that subband
shown in the inset of the panels (d),(f) and (h), respectively.
j =3inoff-resonancesitsjustoutsideofE andgenerates
(c) and (d) energy spectrum and the corresponding LDOS at c
a largeasymmetric peak in the LDOS [see Fig. 1(f)]. The
positions α, β and γ for resonance. (e) and (f) the same but
fortheintermediatecaseandpositionsα∗ andβ∗. (g)and(h) intermediate case[see Fig.1(e) and (f)] mixes the charac-
the same but for off-resonance and positions α′ and β′. Note teristicfrombothresonanceandoff-resonancecaseswhere
that in panel (d) and (f) the LDOS for position β and β∗ are it has the intermediate gap difference between subbands
shifted for clarity. j = 1 and j = 2 and the intermediate k-value for most
quasi-particles states with subband j =2.
Next,weconsideranimpurityinthenanowire. Theim-
L = 3.8nm, and off-resonance case, with L = 5.4nm purity is strong enough to suppress completely the wave-
y y
[45]. The energy spectrum for resonance [see Fig. 1(c)] functions of the electronic states locally. Meanwhile, the
shows two energy gaps: a smaller gap for subband j = 1 spread of the impurity is smaller than the width of the
and a larger one for j = 2. Most quasiparticle states are nanowire. The effects of the impurity on the electronic
just below the cutoff energy E for subband j = 2. This structures of normal state are presented in Fig. 2. The
c
results in an enhancement of the OP, which shows two scattering due to the single impurity does not change
pronounced peaks and is strongly inhomogeneous in the the quasione-dimensionalcharacteristicseenin the DOS.
y direction [see Fig. 1(b)]. We consider the spatial posi- However, it results in additional small oscillations when
′ ′
tions αβγ and αβ as given in the insets of Fig. 1(d)(h) comparing with the clean case with U = 0. In addi-
where the OP has either a maximum or a minimum. The tion, the oscillations show different pattern for impurity
corresponding LDOS at positions α and β are shown in at sided position and at centered position. It indicates
Fig. 1(d). We notice that there are two gaps at α and the influence of the impurity depending on its position
only one gap at β. The smaller sub-gap at α and the gap and the quantum number of the electronic state, i.e. j
at β comes from the j = 1 subband. The main gap at α andk. Due to the sine-shapedtransverseelectronicwave-
comes mostly from the j = 2 subband which is indicated functions,theimpurityatcenterpositionaffectsthestates
by the ratio of the LDOS peaks at the main gap and at withsubbandj =1morethanstateswithsubbandj =2.
the sub-gap. Incontrast,the off-resonancecaseshowsthe Thus, the DOS of the impurity at center position shows
same energy gap for j = 1 and j = 2 [see Fig. 1(g)] and stronger oscillations than the one of the impurity at side
the LDOS [see Fig. 1(h)] is more conventional: only one position between the peaks of j = 1 and j = 2. For the
superconducting gap and the amplitude of the LDOS is same reason, it shows opposite behavior in DOS between
p-3
L.-F. Zhang et al.
(a) Resonance, impurity at α (b)Resonance, impurity at β LDOS

24 yyy===αβγ yy==αβ || (meV)∆ y (nm) ((ba)) 14
0 0
−40 −20x (nm)0 20 −40 −20 x (nm)0 20 −30 −20 −10 0 1 0
y (nm) ((dc)) 24 y (nm) ((dc)) 104
0 −30 −20 −10 0 10
−30 −25 −20 −15 −10 −5 0 5
x (nm) 1.2 (e) 0.4
(e) 1 m)
nm) 0.8 y (n 0.2
y ( 0.6 (f)
(f) 0.4 0
0.2 −30 −20 −10 0 10
x (nm)
−30 −25 −20 −15 −10 −5 0
x (nm)
Fig. 4: (Color online) Contour plots of LDOS at the ener-
gies’ (sub-)gap in the presence of an impurity. (a-d) are for
Fig. 3: (Color online) OP in thepresence of an impurity. (a)
the resonance case with an impurity sitting at α at sub-gap
and(c)show|∆(x)|aty=α,β,γ(definedinFig.1)andcontour
(a) E/meV = 2.32 and gap (b) E/meV = 2.93 and with an
plotof|∆(x,y)|foranimpuritysittingatαinresonancecase,
impurity sitting at β at sub-gap (c) E/meV = 2.36 and gap
respectively. (b)and(d)arethesameas(a)and(c)butforthe
(d)E/meV =2.93. (e) and (f) are for off-resonance case with
impuritysittingat thecenterβ. (e)and(f)showcontourplot ′ ′
′ ′ the impurity at α and at β at gap E/meV = 0.68 and 0.65,
of|∆(x,y)|foranimpuritysittingatα andβ inoff-resonance
respectively. Theopencirclesindicateimpurities’profilewhere
case, respectively. The white open circles indicate impurities’
U =0.1U0.
profile where U =0.1U0.
j =1playarolesothatscatteringontheimpurityresults
the peak of j = 2 and j = 3. Furthermore, the impurity
in oscillations of the wave-functions. On the other hand,
leads to fast oscillations in the LDOS at energies E1 and
the impurity at α affects mostly subband j = 2 but all
E3 [see Fig. 2 (lower panels)] because these states have
states with j =2 have low k so that the OP recovers fast
largewave-number,i.e. k. In contrast,the impurity leads
andhaslongerwave-lengthoscillations. The localdensity
to oscillations with long wave-length in the LDOS at en-
of states in Figs. 4(a-d) show evidence that supports this
ergy E2, due to the states with small k-value dominating
explanation. For the off-resonancecase [see Figs. 3 (e-f)],
near E2.
due to the combination of the j = 1 and j = 2 subbands
Nowwe moveonto the superconducting stateandcon-
which now have only high k quasi-particles, the impurity
sideranimpurityatposition(0,α)and(0,β)forresonance
alwaysinducesstrongoscillationsintheOP.Notethatthe
′ ′
case and at (0,α) and (0,β ) when off-resonance. We set
OP is affected more strongly when the impurity sits at a
U0 =20EF andσ =0.02nmsothattheimpuritystrongly local maximum. The LDOS shown in Figs. 4 (e-f) show
suppresses the local OP and the spread of the potential
the same patterns as the OP.
(full width at 1/10th of maximum) is 0.86nm, which is
Finally,westudytheeffectoftheimpurityonthetrans-
shorterthanthewidthofthenanowire. WeshowinFig.3
port properties such as the Josephson current. In order
the profile of the amplitude of the OP in the presence of
to do so, we set a junction link of length L = 30nm
j
the impurity. For the resonance, [see Figs. 3(c)(d)], the
at the center of the nanowire where ∆ is obtained self-
impurity suppresses the OP over the whole width which
consistently. Outside the link, we fix the phase of the or-
is more pronounced when in α than in β. Meanwhile, the
der parameter and impose a phase difference δθ between
asymmetrical impurity, at α, results in a local enhance- thetwosidesofthelink,i.e. ∆(x<−15nm)=|∆|ei0 and
ment at position (0,γ), as seen from Fig. 3(a). For the ∆(x > 15nm) = |∆|eiθ. Then, the supercurrent induced
impurity sitting in the center, β, the OP oscillates along
by the phase difference δθ can be calculated as follows:
the wire over a longer distance althoughthe impurity sits
atalocalminima ofthe OPwhichintuitivelyshouldhave e~
J~(~r)= {f(E )u∗(~r)▽u (~r)
a smaller effect. As seen from Fig. 3(b), the OP oscilla- 2mi n n n
tions extend to x = ±50nm, which is much farther than EnX<Ec
∗
the extent of the oscillations for the impurity at α. The + [(1−f(En)]vn(~r)▽vn(~r)−h.c.}.
reason is that the centered β impurity strongly affects
the j = 1 subband quasi-particles due to the sine-shaped PleasenotethatJ~satisfiesthecontinuitycondition∇·J~=
transversewave-function,leavingthej =2subbandunaf- 0 in the link due to the self-consistent ∆ [46,47]. Outside
fected. Moreover, only high k quasi-particles in subband the link, J~ is discontinuous due to the fixed phase of the
p-4
Effect of impurities on superconducting nanowires
20 10 ScanningGateMicroscopy(SGM)inordertolocallymimic
(a)Resonance (b)Off−resonance
15 the effect of the impurity and suppress the order parame-
I (pA)105 clean ata αt β clean at α′ 5I (pA) toenre.cBanymmaopniotuotritnhgetshyemcmhaetnrgyeoifnththeeorJdoesreppharsaomnectuerrreanntd,
at β′
the distribution of the supercurrentalongthe y direction.
0 0
0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1
θ/π θ/π J (mA/m) Conclusions. – In conclusion, we studied the effect

(c) 20 of non-magnetic impurities on narrow superconducting
nm) 0 nanowires in the clean limit, in which quantum confine-
y ( (d) 5 mentplaysanimportantrole. ByapplyingBdGtheoryto
0 the case ofNbSe2, we uncoveredseveralregimesin which
the impurity affectsthe superconductingpropertiesofthe
5 nanowire in different ways. First, depending whether the
(e)
nanowireisintheresonantoroff-resonantregime,theOP
nm) 0 willshowsloworfastoscillationsawayfromthe impurity,
y ( respectively. This is due to the different nature of the
5
(f) quasi-particles involved in the formation of the Cooper
0 pairs, i.e. small or large momentum. Additionally, the

−15 −10 −5 0 5 10 15 impurity has a strong position-dependent effect on the
x (nm)
Josephson critical current with opposite behavior in the
resonant and off-resonance cases. In the resonant case an
Fig. 5: (Color online) Effects of an impurity on Josephson
impurity at the center of the wire will strongly suppress
current where the junction length is Lj =30nm. (a) Current-
the current while for the off-resonance case the current is
phase relation in the resonance case for clean limit, with im-
slightly suppressedfor animpurity sitting atanoff-center
purity at α and at β. (b) Results in the off-resonance case
for clean limit, impurity at α′ and at β′. (c)-(f) The spatial location.
distributions of current for an impurity at α, β for resonance Inexperiments,thecrystalstructureofthematerial,the
caseandat α′,β′ foroff-resonancecase withphasedifferences surfaceroughnessofthespecimenandthepropertiesofthe
where currents reach their critical Josephson currents. The substrate will modify the Fermi level, the band structure,
color indicates the amplitude of the current and the stream- the electronic wave functions or the electron mean free
lines indicate the direction of the current. The open circles path. All these factors could broaden the single-electron
indicate theimpurity profile for which U =0.1U0. levels and modify the specific scattering patterns of the
OP and electronic structures (LDOS). However, the su-
perconducting shape resonances are robust, as shown in
orderparameterbut these areascanbe treatedascurrent
more realistic theoretical models [48]. In this case, quan-
sources.
tum confinement dominates and the position dependence
Figs. 5(a) and (b) show the J −θ relation for resonant
of the amplitude of the electronic wave functions and OP
and off-resonant cases, respectively. For resonance, the
arerobust. Asaresult,ourconclusionsabouttheposition-
current increases monotonically with the phase for the
dependent impurity effect (especially the effect on the
impurity at α but it shows a sine-shaped curve for the
Josephson current) will not be significantly altered, al-
impurityatβ. WefindthatthecriticalJosephsoncurrent
though specific details might change.
issuppresseddramaticallyfor the impurity atβ while the
impurity at position α has little effect. The explanation We believe that these effects could be used to investi-
is that when the current is small, only the lowest quasi- gatethenatureofthesuperconductingcondensateandthe
particlestatesareinvolved. andalmostallsuchstatesare scatteringofthevarioussubbandsontheimpurity. Alsoin
fromthej =1branchandasaresultthecurrentdistribu- realistic nanowires, which contain impurities, one should
tionj(y)∝sin(πy/L )isasinthecleanlimit. Thiscanbe seeastrongimpurityeffectintheresonantcase. Although
y
seenfromFig.5(d)forxfarawayfromtheimpurity. Thus, the mean-field calculation presented here cannot describe
the impurity blocking effect on the current for the impu- fluctuations, by showing that the Josephson current can
rity at the center β has a larger effect than the off-center be strongly suppressed in the resonant case, one can in-
position α. In contrast, for the off resonance case, the fer that fluctuations will become more important. Our
current with the impurity at α′ is more suppressed. The calculations could also provide a basis for a phenomeno-
reasonisthesamebut,here,thej =1andj =2branches logicaltoymodelofa1DdisorderedJosephsonarraywith
contribute both to the current, and as a consequence the position impurity dependent junction parameters.
currentdistributionisj(y)∝|sin(2πy/L )|+|sin(πy/L )|
y y
and has a maximum at the α′ position. This can also be ∗∗∗
seen from Figs. 5(f) far away from the impurity. This po-
sition dependent effect could be used to investigate the This work was supported by the Flemish Science Foun-
nature of the superconducting condensate, e.g. by using dation (FWO-Vlaanderen) and the Methusalem funding
p-5
L.-F. Zhang et al.
of the Flemish Government. [25] Das A. , Ronen Y., Most Y., Oreg Y., Heiblum M.
and Shtrikman H., Nat. Phys., 8 (2012) 887
[26] ShanenkoA.A.,CroitoruM.D.andPeetersF.M.,
Phys. Rev. B, 75 (2007) 014519
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p-6
Effect of impurities on superconducting nanowires
Supporting Online Materials
We present results for NbSe2 nanowires for the reso-
nance case (L = 5.8nm) and off-resonance case (L =
y y
7.5nm), where there are three peaks in ∆(y) now.
First, we show in Fig. S1 the electronic properties in
the clean limit for both cases. They are similar to the
results we presented in the main paper. The difference
is that, for the resonance case here, the energy spectrum
shows a subgap for both subbands j = 1 and j = 2 and
the main gap for the subband j = 3. Only the quasi-
particlesofthesubbandj =3havelowmomentumk ≈0.
Note that the folding of the subband j = 3 around k = 0
resultsinadouble peakinthe LDOS[seeFig.S1(d)]. For
the off-resonance case, all three subbands mix and have
onlyonegap. Furthermore,noquasiparticlestate haslow
momentum.
Next,weconsideranimpuritywiththesameparameters
as in the main paper at positions (0,α), (0,β) and (0,γ)
′ ′ ′
for resonance case and positions (0,α), (0,β ) and (0,γ )
for off-resonance case. These are defined in Fig. S1. In
Fig.S2, we show the spatialdistributionof the amplitude
of the order parameter in the presence of the impurity.
In Fig. S3 and Fig. S4, we show the corresponding LDOS
for resonance and off-resonance cases, respectively. All
the results are similar to the ones presented in the main
paper. The effect of the impurity depends on its position
and shows opposite behaviors for the resonant and off-
resonant cases. These are due to the differences in the
subband energy spectrum.
Finally, we show the Josephson current for the reso-
nance case in Fig. S5 and off-resonance case in Fig. S6.
All other parameters are the same as in the main paper
suchas the length of the junction link L =30nm. Again
j
allresultsaresimilartotheonesshowninthemainpaper
but the impurity has now less overall effect on the total
current. The reasons are that (1) The relative size of the
impuritybecomessmallerwhenthewidthofthenanowire
increases. (2) More subband quasiparticles take part in
the current transport, which results in the current distri-
bution along the lateral direction being more uniform.
Inconclusion,alltheadditionalresultsshownheresup-
porttheexplanationspresentedinthemainpaper. Mean-
while,thisshowsthatthe effectoftheimpurities decrease
withincreaseofthe widthofthe nanowire. Finally, asex-
pected all the global properties converge to the bulk ones
and Anderson’s theorem is recovered.
p-7
L.-F. Zhang et al.
L (nm) k (nm−1) k (nm−1)
y
4 6 8 10 12 −1.6−1 0 1 1.8−1.6−1 0 1 1.8
3
(a) (e)
(meV)2 3 bulk meV)4 Ec j=4 Ec 4
∆ 1 E (2 j=1 j=3 j=2 2
j=3
4
0 (c) j=1 j=2 0
0 α γ 1.5
3.7 (b) (d) (f)
S α: y=1.0nm β′: y=2.62nm
V) 3 3 LDO2 γ: y=2.9nm 1
me 2 ed γ′: y=3.75nm
∆ ( 1 β ←γ′ 4 maliz1 β: y=2.0nm α′: y=1.46nm 0.5
α′ ←β′ or
N
0 0 0
0 2 4 6 8 −4 −2 0 2 4 −4 −2 0 2 4
y (nm) bias energy (meV) bias energy (meV)
Fig. S1: Properties of the clean NbSe2 nanowires. (a) Spatially averaged ∆¯ as a function of Ly. The open circles 3 and 4
indicate the resonance case (Ly = 5.8nm) and off-resonance case (Ly = 7.5nm), respectively. (b) Order parameter |∆(y)| for
resonance and off-resonance cases. The αβγ are defined as the positions where the order parameter has a local maximum /
′ ′ ′
minimum for resonance and the αβ γ for off-resonance. The value of these positions are shown in the inset of the panels (d)
and(f). (c) and (d)energy spectrum andthecorresponding LDOSat positions α,β andγ for resonance. (e) and(f) thesame
butfor off-resonance and positions α′, β′ and γ′.

(a) 4

3
)
m (b)
n 2
(

y
1
(c)
−35 −25 −15 −5 0 5 |∆| (meV)

1.5
(d)

1
)
m (e)
n
(

y 0.5
(f)
0
−30 −20 −10 0 10
x (nm)
Fig.S2: Contourplotsoftheorderparameterinthepresenceofanimpurity. (a-c)|∆(x,y)|foranimpuritysittingatα,β and
′ ′ ′
γ (defined in Fig. S1) in resonance case. (d-f) are the same as (a-c) but for the impurity sitting at α, β and γ in resonance
case. The white open circles indicate impurities’ profile where U =0.1U0.
p-8
Effect of impurities on superconducting nanowires
LDOS
1
(a)
0.5
)
m
n 0
(

y 2
(b)
1
0
−30 −20 −10 0 10
1
(c)
0.5
)
m
n 0
(
y 2
(d)
1
0
−30 −20 −10 0 10
1
(e)
0.5
)
m
n 0
(

y 2
(f)
1
0
−30 −20 −10 0 10
x (nm)
Fig. S3: Contour plots of the LDOS for the resonance case with an impurity sitting at α (a-b), β (c-d) and γ (e-f). The bias
energyforpanels(a,c,e) isat sub-gapE/meV =2.17 andforpanels(b,d,f) isat gapE/meV =2.73. Theopencircles indicate
impurities’ profile where U =0.1U0.
LDOS
0.5
(a)
0.4
m) (b) 0.3
n
(

y 0.2

(c) 0.1
0

−30 −20 −10 0 10
x (nm)
′ ′ ′
Fig. S4: Contour plots of the LDOS for the off-resonance case with an impurity sitting at α (a), β (b) and γ (c) at gap
E/meV =0.76. The open circles indicate impurities’ profilewhere U =0.1U0.
p-9
L.-F. Zhang et al.
30
(a) Resonance
20I

(
clean p
A
at α )
10
at β
at γ
0
0 0.2 0.4 0.6 0.8 1
θ/π J (mA/m)
(b)
25
20
) (c)
m
15
n
(

y
10

(d)
5
0

−15 −10 −5 0 5 10 15
x (nm)
Fig. S5: Josephson current for the resonance case. (a) Current-phase relation for clean limit, with impurity at α, β and γ.
(b)-(d)Thespatialdistributionsofcurrentforanimpurityatα,β andγ forphasedifferenceswherecurrentsreachtheircritical
Josephson currents. The color indicates the amplitude of the current and thestreamlines indicate thedirection of the current.
The open circles indicate the impurityprofile for which U =0.1U0.
p-10