Table Of ContentNonlocal transistor based on pure crossed Andreev reflection in a
EuO-graphene/superconductor hybrid structure
Yee Sin Ang,1,2,3 L. K. Ang,1,2 C. Zhang,3 and Zhongshui Ma4,5
1Engineering Product Development, Singapore University of Technology and Design, Singapore 487372
2SUTD-MIT International Design Center, Singapore University of Technology and Design, Singapore 487372
3School of Physics, University of Wollongong, Wollongong, NSW 2522, Australia
4School of Physics, Peking University, Beijing 100871, China
6 5Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
1
0 Westudy theinterbandtransport in a superconductingdevicecomposed of graphene with EuO-
2 induced exchange interaction. We show that pure crossed Andreev reflection can be generated
exclusivelywithouttheparasiticlocalAndreevreflectionandelasticcotunnellingoverawiderange
n
ofbiasandFermilevelsinanEuO-graphene/superconductor/EuO-graphenedevice. Thepurenon-
a
local conductance exhibits rapid on/off switching and oscillatory behavior when the Fermi levels
J
in the normal and the superconducting leads are varied. The oscillation reflects the quasiparticle
3
propagationinthesuperconductingleadandcanbeusedasatooltoprobethesubgapquasiparticle
1
mode in superconducting graphene, which is inaccessible from the current-voltage characteristics.
Ourresults suggest that thedevicecan beused as a highly tunable transistor that operates purely
]
l in the non-local and spin-polarized transport regime.
l
a
h PACSnumbers: 74.50+r,74.25F-,74.45+c,72.80Vp
-
s
e
Introduction - Andreev reflection (AR) is the ex- (a)
m
citation of a hole in a normal/superconductor inter-
at. faacCeowohpeenr tpwaoiropinpotshitee-sspuipnerecleocntdrounctsoar1r.e coIunplead ninotro- ER EECC
-m momale/tsruy,ptewrcooenldeuctcrtoonr/sncoarnmcaolu(pNle/Slo/cNa)llythirneet-hteersmaminealngoer-- (cid:1831)(cid:3007)(cid:481)(cid:2869) (cid:1857)(cid:1848)AAR (cid:3400) (cid:1831)(cid:3007)(cid:481)(cid:2870)CAR
d S
n mal lead or non-locally in different normal leads to form (cid:3400)
o a Cooper pair in the superconductor. The local cou- (b)
c pling producesthe ‘usual’Andreev reflection(AR) while
[ the non-local coupling produces the exotic crossed An- EuO Superconductor EuO
1 dreev reflection (CAR)2. The reverse process of CAR in (cid:1848)(cid:3034)(cid:481)(cid:2869) CAR (cid:1848)(cid:3034)(cid:481)(cid:2870)
v N/S/N device has been proposed as the basis of Cooper ER
1 pairsplitterthatgeneratesentangledelectronpairincon- Graphene
Substrate
1 densed matter environment3–7. High Cooper pair split-
(cid:1857)(cid:1848) (cid:1835)
1 tingefficiencyof∼90%hasbeenexperimentallyachieved (cid:1877) (cid:1876)
3
in a carbon nanotube-based N/S/N device8. Moreover, (cid:1876)(cid:3404)(cid:882) (cid:1876)(cid:3404)(cid:1856)
0 FIG. 1. (a) Mechanism of pCAR in a gapped
the pairing symmetry of a superconductor can also be
. and spin-split dispersion; (b) Schematic of the EuO-
1 probed by CAR signal9. Unfortunately, the generation
0 G/superconductor/EuO-G device. The incident energy is re-
ofCAR-dominatedtransportis challengingsince itisin-
6 lated to thebias by E =eV.
evitably plagued by electron elastic co-tunnelling (EC)
1
and local AR10.
:
v
Generating pure CAR (pCAR) using energy band provided that the Fermi levels are placed within one su-
i
X topology was first proposed in a graphene bipolar perconducting gap with respect to the conduction and
r transistor11. By precisely tuning the Fermi levels and valence band extrema.
a
the bias voltage,EC and local AR excitations are forced ThestringentconditionofhavingpreciselyfixedFermi
to lie exactly on the Dirac points. Due to the vanishing levels can be circumvented, for example, by engineering
quasiparticle density, EC and AR are completely elim- thevalley-helicity16ofzigzaggraphenenanoribbon17and
inated. Despite its conceptual simplicity, the experi- byshiftingthevalley-spinsplitting18 inthevalenceband
mental realization is difficult since precisely fixed Fermi of MoS 19. In systems with tunable bandgap such as
2
levels and bias are required. A significant improvement silicene and bilayer graphene20,21, the existence of 1D
can be achieved by using a gapped energy dispersion12. topologicallyprotectededgestate22 providesanotherop-
AR and EC are completely blockedby the whole contin- portunity to create widely tunable pCAR. Remarkably,
uum ofthe bandgapinsteadof asingle Diracpoint, thus the suppressed intervalley scattering forces a further re-
lifting the constraint on the bias voltage. pCAR medi- moval of the normal electron reflection (ER)23. Beyond
ated by bandgap blocking can occur in semiconductor12, supercoductivity, tunable pCAR has been predicted24
silicene13, MoS 14 and quantum spin hall insulator15, in the topological exciton condensate in 3D topological
2
2
insulator25. Thisoffersanexcitingalternativecondensed 90 (a) 0.8
matter platform to generate entangled electrons. 45 0.6
Theoreticalconcept -Weproposeadifferentstrategyto ◦φ() 0 0.4
−45 0.2
achievewidelytunablepCARinthiswork. Weshowthat −90 0
theinterbandtransport inagappedandspin-splitenergy 4950 (b) 0.8
danisdpeFresriomni lceavnelssu.sTtahiencpoCncAeRptoisveilrluastwraidteedrianngFeigo.f1b(aia)s. ◦φ()−450 000...246
Consider the case where the Fermi level of the incident −901. 5 2 2.5 3 0
d/ξ
side, E , is placed between the two conduction spin-
F,1
subband edges and that of the transmitted side, EF,2, is 90 (c) (d) 1.0
placedbetween the two valence spin-subband edges. For 0.8
45
anincidentelectronresidinginthelowerconductionspin-
0.6
subband, no opposite-spin electrons are available below ◦φ() 0
the Fermi level for local AR. In the transmitted region, 0.4
spin conservation forbids the electron from tunnelling −45 0.2
into the opposite-spin valence subband. As a result, the
only permissible processes are ER and the much sought- −900 1 2 3 4 5 0 1 2 3 4 5 0
E/∆sc E/∆sc
after pCAR. The conditions of having precisely tuned
bias and Fermi levels are both relaxed. To demonstrate
FIG. 2. TCAR as a function of superconductor width d for
this, we consider an Europium oxide-graphene (EuO-G) (a) (EF,1,EF,2) = (30,−40) meV; and (b) (EF,1,EF,2) =
ferromagnetic hybrid-structure26 [1(b)]. First-principle (60,−60) meV with E/∆sc = 0.9 (ξ = ~vF/π∆sc). (c) and
calculations predicted that EuO strongly spin-polarizes (d) shows the TCAR as a function of incident energy E/∆sc
the π-orbitals of graphene27,28 and induces a large ex- at the same Fermi levels as (a) and (b), respectively, with
change splitting. A sizable spin-dependent bandgap, d=2.5ξ (∆sc =1 meV and µS =200 meV).
which crucially blocks the local AR and EC excita-
tions, is present. We found that the non-local conduc-
tance in EuO-G/S/EuO-G exhibits fast on-off switch- perconducting gap is ∆ (x) = ∆ for 0 < x < d. We
sc sc
ing via normal leads gating. Furthermore, the non- take the phase in ∆ as zero. For ∆ (x) = 0, Eq. (1)
sc sc
local conductance exhibits an oscillatory behavior with canbedecoupledintoaspin-σelectronpartandaspin-σ¯
thesuperconductor-gatethatdirectlyreflectsthesubgap hole part. As k is a good quantum number, we write
y
superconducting Dirac quasiparticle propagation. We k± = −i∂/∂x±iky and solve Eq. (1) for the quasipar-
foundthataminimalsubthresholdswingof15.1mVand ticle eigenstates and the excitation energies. The trans-
alargeon-offratioof105 canbeachieved. Thisopensup port coefficients can then be straightforwardly obtained
thepossibilityofhighefficiencygraphene-basednon-local by matching the wavefunctions at x=0 and at x=d31.
transistor in whichalllocalandnon-entangledprocesses
Results & discussions - In the numerical calculation,
are suppressed.
we choose ∆ =1 meV which agrees with recent exper-
Model - In EuO/G, the K and K′ Dirac cones are imental valuesc32. For conciseness, we focus on the pCAR
mapped onto the Γ point due to the Brillouin zone
transportphenomenonoriginatingfromanincidentelec-
folding27,28. The low energy effective Hamiltonian can
tron residing in the σ = −1 conduction subband and
be written as29 Hk,σ = σhI +∆στz +~vσk·τ, where transmitted as a purely σ = +1 polarized valence hole.
σ = ±1 for spin-up and spin-down electrons, h is the
proximity-induced exchange interaction, k = (k ,k ) is Sincethereisalargecommon-gapof(∆++∆−−2h)=54
x y meV, only conduction hole is involved in the quasiparti-
the electron wavevector, τ = (τx,τy,τz) are the Pauli cle transport33. According to first-principle results27,29,
matrices,v and∆ arethespin-dependentFermiveloc-
σ σ the Fermi levels lie in the rangesof 18 meV <E <98
F,1
ity and bandgap, respectively. I is a 2×2 identity ma-
meVand−80meV<E <−36meV.Thiscorresponds
trix. The eigenenergy is ε (k) = η ∆2 +~v2k2+σh F,2
where k = |k| and η =σ±η 1 denotpes cσonducσtion and to a wide windows of ∆EF(c) = 80 meV and ∆EF(v) = 44
valence bands. The normalized eigenstate is ξ (k) = meV for conduction and valence bands respectively. We
ση first study the pCAR transmission probabilities, T ,
[(ε (k)−∆ )/2ε (k)]1/2 ~vσkx−iky,1 T, where T in Fig. 2. The angle of incidence of the electron iCsAdRe-
staσnηds for trσansposσeη. First-hpεrσinη(cki)p−le∆σcalciulation27 gives noted by φ. TCAR oscillates rapidly with d because of
thequasiparticleinterferenceinthesuperconductinggap
v = (1.4825 − 0.1455σ) × v , h = 31 meV and
σ F
[Fig. 2(a)]; transmissionpeaks occurs whenever the sub-
∆ = (58+9σ) meV. The Bogoliubov-de Gene (BdG)
σ
equation30 is given as gap superconducting quasiparticle wavevector matches
the resonance wavevector k = nπ/d. A significant dif-
0
(cid:18)Hk,∆σ(∗sxc)(x−)IEFI −(Hk∆,σ¯s(cx()x)−IEFI)(cid:19)(cid:18)uvσσ¯(cid:19)=Eσ(k)(cid:18)uvσσ¯(cid:19)foenraennccee ‘bsetrtiwpeeesn’ aFriegsa.lm2o(sat)vaerntdica2l(ba)ndiswtehlal-tsetphaerarteesd-
(1) in Fig. 2(a) (Fermi levels lie closer to the band edges)
where σ¯ = −σ, h(x) = h for 0 > x > d and the su- while inFig. 2(b) (Fermilevelslie fartherawayfromthe
3
5 1.5 able at certain incident angles for a given energy. With-
(a) (b)
out filtering out the local AR and EC processes via the
4
band topology, the tunnelling conductance is inevitably
/G03 1 mixed with local and non-entangled components. In the
R
GCA2 0.5 proposed device, (EF,1,EF,2) can be tuned without de-
stroying the pCAR. G as a function of (E ,E )
1 CAR F,1 F,2
is calculated in Figs. 3(c) and 3(d). The onset of G
CAR
00 0.5 1 1.5 0 2 4 6 80 for incident energy E is (18−E) meV and (−36+E)
d/ξ E/∆sc meVforE andE ,respectively. Beforetheseonsets,
F,1 F,2
0.8
(c) (d) GCAR is completely switched-off due to the depletion of
thechargecarriers. Remarkably,G risesverysharply
0.6 CAR
G0 postonset,suggestingapotentialinfaston-offswitching
/R0.4 application. To estimate the switching characteristic,we
A
GC define the Fermi level subthreshold swing as SS(EF,i)=
0.2 (dlog I /dE )−1 ≈(∆log G /∆E )−1 where
10 CAR F,i 10 CAR F,i
i = 1,2 denotes the two normal leads. We found that
0
0 50 100 −100 −50 0 in the linear-growth regime immediately after the on-
EF,1(meV) EF,2(meV)
set, SS(E ) is about 7.1 meV/dec (meV per decade).
F,1
FIG. 3. Non-local conductance GCAR. (a) d-dependence for For EF,2, the onset of GCAR is even sharper, yielding
(solid)(EF,1,EF,2)=(30,−40)meV,(dashed)(EF,1,EF,2)= SS(EF,2) ≈ 3.3 meV/dec. The gate-voltage subthresh-
(30,−60) meV and (dotted) (EF,1,EF,2) = (60,−60) meV old swing, SS(V ) = (dlog I /dV )−1, can be es-
g,i 10 CAR g,i
at E/∆sc = 0.9 (Data are offset vertically by 2G0 for clar- timated from experimental data35,36. We found that37
ity and E = 0.9∆sc); (b) E/∆sc-dependence with the same SS(V ) ≈ 60.5 mV/dec and SS(V ) ≈ 15.1 mV/dec.
EFeFr,m2i=lev−e4ls0,a−s5(0a,)−a7n0dmdeV= 2(s.5oξli;d,(cd)aEshFe,d1-daenpdenddoetntecde wliniteh, The rge,1markably small SS(Vg,2) shogw,2s an even steeper
on-off switching characteristic in comparison with state-
respectively);and(d)EF,2 dependencewithEF,1=30,40,60 of-the-art MoS -based transistor recently reported in38.
meV (solid, broken and dotted line, respectively). d = 2.5ξ 2
and E/∆sc = 0.9for (c) and (d). When EF,1 > 98 meV and This reveals the potential of the proposed device as a
EF,2 < −80 meV, GCAR decreases significantly due to the fast switching transistor that operates uniquely in the
onset of the competing local AR and EC processes. non-local and 100% spin-polarized transport regime.
We calculated G as a function of the supercon-
CAR
ducting graphene Fermi level, µ , in Figs. 4(a) to (c).
S
band edges) the resonance patterns are curved and are In general, GCAR exhibits oscillatory behavior with µS.
no longer well-separated. This contrasting behavior can Thefollowingsareobserved: (i)the oscillationfrequency
be seen in the eV-dependence of TCAR. Along a vertical is unaffected by (EF,1,EF,2) [Fig. 4(a)]; (ii) the fre-
cut at E/∆sc = 2 and E/∆sc = 5, four TCAR hotspots quency of GCAR oscillation is reduced by a smaller d
are clearly present in Fig. 2(d) instead of only two in [Fig. 4(b)]; and (iii) the oscillation frequency is un-
Fig. 2(c). Thefour-hotspotiscausedbythecurvedreso- affected by ∆sc but the amplitude is severely damped
nance pattern in Fig. 2(b) asit is composedof twopairs at larger ∆sc [Fig. 4(c)]. These oscillatory behaviors
of transmission resonance at constant d/ξ: one from the reflect the subgap (E < ∆sc) quasiparticle dynamics
central region of a resonance ‘stripe’ and one from the residing in the superconducting graphene. For E <
tail region of the preceding curved resonance ‘stripe’. ∆sc, the superconducting Dirac quasiparticle wavevec-
Thezero-temperaturenon-localconductance generated tor is composed of a propagating (real) term, kS =
by pCAR is given as34 GCAR/G0 = TCARcosφdφ. G0 (~vF)−2µ2S +q2, and an imaginary (damping) term,
is the ballistic normal conductance iRn σ = −1 channel. pκ = (∆sc/~vFkS)sin cos−1(E/∆sc) 33. µS-tuning di-
In Fig. 3(a), GCAR as a function of the superconduc- rectly modifies kS. W(cid:0)hen kS is tun(cid:1)ed across two suc-
tor width, d, is plotted. G oscillates rapidly with d cessive subgap standing-wave modes, a peak-valley-peak
CAR
due to the fast oscillation of T . Interestingly, G G -oscillationis produced. (E ,E ) do notplay a
CAR CAR CAR F,1 F,2
minima are near-zero only when the E ’s are close to directroleink . Hence, they do notalterthe oscillation
F S
the band edge. This is a direct consequence of the well- frequency[Fig. 4(a)]. Whendisdecreased,thedifference
separatedresonance‘stripes’asdiscussedinFig. 2(a). In between two successive standing-wavevectors becomes
Fig. 3(b),theG -resonancesoccuratE/∆ ≈2and larger as the quantized standing-wavevector k ∝ 1/d.
CAR sc 0
≈5. This is consistentwith the T hotspots observed The peak-to-peak transition thus requires a larger range
CAR
inFigs. 2(c)and(d). NotethatalthoughtheT peaks of µ to be scannedacross. This results in a reduced os-
CAR S
originates from the Fabry-P´erotinterference (FB) inside cillation frequency as seen in Fig. 4(b). ∆ affects only
sc
∆ , tunnelling current of solely pCAR is unachievable thedampingtermasκ∝∆ whenE/∆ →1. Increas-
sc sc sc
viaFBalonesinceG isanangular-averaged quantity. ing ∆ thus leads to a stronger damping which reduces
CAR sc
The selective enhancement of one transport process and theamplitudewithoutchangingitsoscillationfrequency.
the simultaneous suppression of the rest is only achiev- Physically, one can interpret the ∆ -dependence as fol-
sc
4
µS(meV) between ∆µ and ∆ can be explained by noting that
200 205 210 215 200 205 210 215 200 205 210 215 S sc
/G0 1(a) (b) (c) ×5 Fdigiss.in4t(hde)uannidt o(fe)ξ. ∝Si1n/c∆estcheanodschilelnactieonisfnreoqtufiexnecdy iins
R0.5
CA ×10 determined by the quantized standing-wavevectorwhich
G
0 is k0 ∝ d and d ∝ 1/∆sc, we have k0 ∝ ∆sc, which
106 (d) (e) 80 leads to the linear dependence. In the insets, we calcu-
I/IOnoff110045 I/Ionoff111000246 I/Ionoff111000024 5 10 246000∆µS lfioaaabnnxtsddeeeddarfvotuIernolanitvrt/hgaIaderot0iff∆oI=uaossncs~./∆avITFoσfffh/uiπanissn∆cctsdo0iiognhwnnfiirhfiaometcfrase∆dnt∆ths=lcey0wir=mdei0tdph1uo[FcmrdteiagedinVn.actte[4hF(soeiemfg)ua.]a.nl4llia(tW∆rdgo)σeef]
103 5 10 0 spin-dependentbandgapandfinitehinachievingefficient
2 4 6 8 10 2 4 6 8 10
∆sc(meV) ∆sc(meV) non-local current gating. Furthermore, strong Cooper
pairing does not lead to enhanced non-local transport.
FIG.4. µS dependenceofGCARwithE/∆sc =0.9. (a)Fermi Conclusion - In conclusion, we proposed widely tun-
levels (EF,1,EF,2) are (30,−40) meV (solid) and (60,−60) able pCAR in the interband transport of spin-split and
meV at d = 2ξ (dashed); (b) d = ξ (solid) and d = 3ξ
gapped dispersion in EuO-G/S/EuO-G. The proposed
(dashed); (c) ∆sc = 2 meV (solid) and ∆sc = 5 meV
(dashed) with d0 fixed at ~vF/∆˜sc where ∆˜sc = 1 meV. In device exhibits rapid on/off switching which can poten-
(b)and(c),(EF,1,EF,2)=(30,−40)meVareused. The∆sc- tially be used as a building block in CAR-based quan-
tum computing and spintronics. We emphasize that the
dependence of (d) Ion/Ioff and (e) ∆µs with E/∆sc = 0.9
and (EF,1,EF,2)=(18,−36) meV. The widths are d=ξ (◦), pCARmechanismproposedhereisfundamentallydiffer-
d=2ξ(⋄)andd=3ξ((cid:3)). Insetin(d)isthesameasthemain ent from the case of MoS2 with exchange interaction19.
plotexceptthatd=d0 (◦),d=2d0 (⋄) andd=3d0 ((cid:3)). In- Inourscheme,pCARisbasedontheinterband quasipar-
setin(e)showstheIon/Ioff atseveralsetsof(∆+,∆−,h),i.e. ticle transport between the conduction and the valence
(170,150,90) meV (∗), (210,190,10) meV (⋆) and (30,10,2) spin-split subbands via Fermi levels tuning. Due to the
meV(×). E/∆sc =0.9,d=d0 andtheFermilevelsarefixed inverted band topology between conduction and valence
at theband edges.
spin-split subbands, the elimination of the local AR and
EC branches can be straightforwardly achieved without
theneedtoshifttherelativeseparationbetweenthesub-
lowed: a larger ∆ leads to a shorter coherence length
sc bands via exchange interaction19. For EuO-G/S single-
ξ ∝1/∆ . Atafixedd ,the‘effective’barrierwidthbe-
sc 0 interface,localARiscompletelysuppressedintheregime
comes largerin the relativesense ofd /ξ. Therefore,the
0 studied here. As it is well-known that local AR gener-
CAR tunnelling current is heavily damped as the non-
ates Joule heating that undesirably lowers the cooling
local Cooper pairing of electrons has to overcome too
power of a normal/insulator/superconductor-basedelec-
many coherence lengths.
tronic refrigerator41–43, we expect EuO-G/S to exhibit
The G -oscillation offers an additional tunable pa-
CAR enhanced sub-Kelvin cooling performance. One major
rameter to control the transport by gating the super-
challenge to observe the rapid G -oscillation is the
conducting graphene. As µ -tuning cannot completely CAR
S fabricationofhighqualitysampleastheinterfacerough-
switch the G off and G oscillates periodically,
CAR CAR ness and Fermi level fluctuations can wash out the os-
wedefinetwoquantitiestocharacterizetheµ -switching
S cillation. Finally, we point out that the pCAR mecha-
effect: (i) the non-local current on-off ratio, I /I ≈
on off nism proposed here is universally applicable to systems
G(CmAaRx)/G(CmAinR)forsmallbiaswhereG(CmAaRx)andG(CmAinR)are with similar band topology such as YiG-graphene44 and
the maximum and minimum conductance determined at monolayer transition-metal dichalcogenides with mag-
the vicinity of µS ≈ 200 meV, respectively; and (ii) the netic doping45 orwithproximityto EuO46. These struc-
range of µS required for peak-to-valley switching, ∆µS. turesofferalternativeplatformstotestthevalidityofour
For small d, Ion/Ioff can be as high as 105 over a wide prediction.
range of ∆ [Fig. 4(d)]. Large ∆µ is desirable for ef-
sc S
ficient µ -switching so that the valley-to-peak transition We thank Shi-Jun Liang and Kelvin J. A. Ooi for
S
is robust against the Fermi level fluctuation induced by their fruitful discussions. ZSM is thankful for the
charge inhomogeneity and substrate39. The ∆µ is in support of NSFC (11274013) and NBRP of China
S
the undesirably small values of few meV at small ∆ (2012CB921300). This work was funded by the
sc
due to the rapid G -oscillation. Interestingly, ∆µ Singapore Ministry of Education (MOE) T2 grant
CAR S
increases linearly with ∆ [Fig. 4(e)] and can be im- (T2MOE1401),andtheAustralianResearchCouncilDis-
sc
proved to 70 meV at ∆ =10 meV. The linear relation covery Grant (DP140101501).
sc
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