Large gap electron-hole superfluidity and shape resonances in coupled graphene nanoribbons

1.6 MB·

by  M. Zarenia

Large gap electron-hole superfluidity and shape
resonances in coupled graphene nanoribbons
M. Zarenia1,*, A. Perali2, F. M. Peeters1, and D. Neilson2
1DepartmentofPhysics,UniversityofAntwerp,Groenenborgerlaan171,B-2020Antwerpen,Belgium
2DipartamentidiFisicaediFarmacia,Universita` diCamerino,62032Camerino,Italy
6 *[email protected]
1
0
2 ABSTRACT

n
a
Wepredictenhancedelectron-holesuperfluidityintwocoupledelectron-holearmchair-edgeterminatedgraphenenanoribbons
J
separatedbyathininsulatingbarrier. Incontrasttographenemonolayers, themultiplesubbandsofthenanoribbonsare
6
parabolic at low energy with a gap between the conduction and valence bands, and with lifted valley degeneracy. These
2
properties make screening of the electron-hole interaction much weaker than for coupled electron-hole monolayers, thus

] boostingthepairingstrengthandenhancingthesuperfluidproperties. Thepairingstrengthisfurtherboostedbythequasi-
n one-dimensional quantum confinement of the carriers, as well as by the large density of states near the bottom of each
o
subband. Thelattermagnifiesthesuperfluidshaperesonancescausedbythequantumconfinement. Severalsuperfluidpartial
c
condensatesarepresentforfinite-widthnanoribbonswithmultiplesubbands. Wefindthatsuperfluidityispredominatelyinthe
-
r strongly-coupledBECandBCS-BECcrossoverregimes,withlargesuperfluidgapsupto100meVandbeyond. Whenthegaps
p
exceedthesubbandspacing,thereissignificantmixingofthesubbands,aroundingoftheshaperesonances,andaresulting
u
reductionintheone-dimensionalnatureofthesystem.
s
.
t
a
m
Introduction
-
d
Superfluidityofspatiallyseparatedelectronsandholeswaspredictednearlyhalfacenturyago1butuptonowexperimental
n
o observationofthisexoticstatehasbeenelusiveatzeromagneticfield,notwithstandingmultipleattemptsonverydifferent
c systems. Thediscoveryofthewondermaterialgrapheneinconjunctionwiththelargebandgapinsulatorhexagonalboron
[
nitride(h-BN)hasraisednewhopesforrealizationofthisnewcollectivemanybodystate. Recently,superconductivityat

1 temperatures above liquid Heliumhas been reportedin doped monolayergraphene by fourgroups, amplifyinginterest in
v quantumcoherentphenomenaingraphene.2
2
Monolayergrapheneisanatomicallyflat,gaplesssemiconductorwithnearidenticalconductionandvalencebands.Spatially
4
separatedelectron-dopedandhole-dopedmonolayerscanbecompletelyinsulatedfromeachotherwithjustafewatomiclayers
9
6 ofh-BN.3,4Withsuchsmallspatialseparations,electron-holepairingbydirectCoulombattractionwouldbeexpectedtobe
0 strong.5–7 HoweverthelineardispersionofthemonolayergrapheneenergybandsresultsinverystrongCoulombscreeningof
1. theelectron-holepairingattraction,andthissuppressessuperfluidityincoupledelectron-holegraphenemonolayers.3,8 To
0 overcomethestrongscreening,Refs.9and10proposedusingcoupledelectron-holegraphenemultilayers. Usingmultilayers
6 takesadvantageofthenonlineardispersionoftheirenergybands,11,12andtheexistenceofagapbetweentheconductionand
1
valencebandswhenagatepotentialisapplied.
:
v Hereweproposeanewdesigntoboostelectron-holepairingandtheonsetofsuperfluidityusingnanoribbonsetchedin
Xi monolayergraphene. Monolayersheetsofgraphenearepromisingcandidatesforapplicationsintransparentconductivefilms,
electronicandopto-electronicdevices,actuators,sensors,composites,andmore. Howeveraseriouslimitationofgraphene
r
a monolayersisthatfield-effecttransistor(FET)devicesarenotpossiblebecausethemasslessnatureoftheelectronsprevents
electronconfinementingraphene. Quasi-one-dimensionalgraphenenanoribbonswithtuneablebandgapsresolvethisissue,
withimportantimplicationsforthefabricationofnovelandultrafastelectronicnanodevices. Forexample,FETdeviceswith
100GHzswitchingfrequencieshavebeenfabricatedusinggraphenenanoribbons.13 Thenanoribbonedgescanbeterminated
usingavarietyofdifferentatoms,whichopensupapplicationopportunities,inparticularfornanoribbonsinpolymerhostsfor
fabricationofnovelcompositematerials.14,15 Finally,graphenenanoribbonsareshowinggreatpromiseaselectrodematerials
forbatteriesandsupercapacitors.16
The electronic properties of graphene nanoribbons depend on the type of edge termination.17 We focus on armchair-
edgeterminatednanoribbonssince(i)theirsubbandsareparabolicaroundtheirminima(Fig.1(a)),(ii)thereisasizeable
semiconductor-like energy gap between the conduction and valence bands, and (iii) the valley degeneracy of monolayer
(a) (b)
4
3
V) 2
e
E (
1
0
-1
-1 -0.5 0 0.5 1 0 5 10 15
ky(π/3a) DOS (eV-1nm-1)
Figure1. (a)Lowestsingle-particleenergysubbandsε (k ),j=1,2,... inanarmchairgraphenenanoribbonofwidthW =2
j y
nm. (b)CorrespondingdensityofstatesDOS(E)innanoribbon. VanHovesingularitiesarevisibleatbottomofeachsubband.
(a)
grapheneislifted. Thesepropertiescombinetogreatlyreducethestrengthofscreeningoftheelectron-holepairinginteraction.
NotethatuniformarmchairgraphenenanoribbonsofwidthsW (cid:28)10nmhaverecentlybeenfabricated.18
Figure2showsthedeviceweareproposing. Itconsistsoftwoarmchair-edgeterminatedmonolayergraphenenanoribbons,
oneelectron-dopedandtheotherhole-doped,separatedbyafewatomiclayersofah-BNinsulatingbarrier. Thenanoribbons
areindependentlycontacted,andtopandbackmetalgatescontrolthecarrierdensities.
top gate
n
o
b
b
ori
n
a
n
e
n
e
h
p
a
gr
e
d grapnhaennoeribbon W h h-B N
back gate
Figure2. Proposeddevice. Upperelectron-dopedandlowerhole-dopedarmchair-edgeterminatedgraphenenanoribbonsof
widthsW separatedbyh-BNinsulatorofthicknessd. Topandbackgatescontrolelectronandholedensities. Gatesare
separatedfromnanoribbonsbyh-BNlayers. Nanoribbonsareindependentlycontacted.
Inadditiontoreducingtheeffectofscreening, electron-holepairingstrengthswillbefurtherboostedinourproposed
systembytheenhanceddensityofstatesneartheminimumofeachsubbandthatarisesfromthevanHovesingularitiesofthe
quasi-one-dimensionalnanoribbons,andalsobythequantumconfinementofthecarriersinthenanoribbons. Enhancementof
superconductinggapsandcriticaltemperatureinstripedsystemsduetoshaperesonancesandquantumconfinementatthe
nanoscalewaspredictedinRefs.19–21. Superconductivityhasbeenobservedinquasi-one-dimensionalsystemsincludingSn
andAlmetallicnanowiresandcarbonnanotubes,withenhancedtransitiontemperaturesascomparedwiththeirbulkvalues.22
Methods
Wetakethey-directionparalleltothenanoribbons,withthecarriersconfinedinthetransversex-direction.Figure1(b)showsthe
√ (cid:113)
single-particleenergysubbandsobtainedinthecontinuummodel,ε (k )=( 3ta /2) k2+k2, j=1,2,... foranarmchair
j y 0 y j
graphene nanoribbon of widthW =2 nm. The intralayer hopping energyt =2.7 eV 23 and the graphene lattice constant
2/8
1.6
1.4 d=2nm
1.2
1
V) d=3nm
e
( 0.8
x
a
m
∆
0.6 d=4nm
0.4
d=5nm
0.2
2 3 4 5 6
0
0 0.5 1 1.5 2 2.5 3
14 -2
n (10 cm )
Figure3. Maximumsuperfluidgap∆maxaveragedoverthesubbands. d isthicknessoftheinsulatingbarrierseparatingthe
nanoribbons. NanoribbonwidthisW =2nm. DensitiesatwhichE entersthebottomofanewsubbandε areindicatedby
F j
theverticallines.
√
(cid:2) (cid:3)
a =0.24nm. k =[jπ/W]− 4π/(3 3a ) isthequantizedwave-numberforthe j-subbandinthex-direction. Figure1(b)
0 j 0
showsthecorrespondingdensityofstatesDOS(E). ThevanHovesingularitiescoincidewiththebottomofeachsubband.
Notonlythefinitewidthofthenanoribbonsbutalsotheirmultipleoccupiedsubbandsmakethesystemonlyquasi-one-
dimensional. Inaddition,inthesuperfluidstatetheenergygapmixesclose-bysubbands. Thequasi-one-dimensionalitytogether
withthesubbandmixingwillsuppressorderparameterfluctuationsthatareresponsiblefordestroyingsuperfluidityinapure
one-dimensionalsystem. Forthesereasonswecancalculatepropertiesofthesuperfluidgroundstateusingmeanfieldtheory.
Recently Ref. 24 discussed a quasi-condensate of excitons in coupled electron-hole one-dimensional wires using the
weak-coupledBCSgapequationinthelowdensitylimitwithonlythelowestsubbandcontributingtothepairing,andwith
screeningneglected. Sinceonlyonechannelwasconsidered,therearenoshaperesonanceeffectsatfinitedensities. Also,
becauseoftheonedimensionality,fluctuationsoftheorderparametershouldbesevereandwouldstronglysuppresssuperfluidity.
Interestingly,Ref.24arguesthatevenintheone-channelcase,thefinitesizeofthenanoribbonswouldallowforshortrange
superfluidcorrelations. Inourcasethemanyavailablechannelsduetothemultiplesubbandsinvolvedinthepairingallowfora
suppressionofthecriticalfluctuationsanditshouldbestraightforwardtoobserveconventionallongrangesuperfluidity.
Ourcalculationsareforcoupledelectron-holearmchairgraphenenanoribbonsofequalwidthW andequal(two-dimensional)
electronandholedensitiesn=(r W)−1,wherer istheaverageinter-particlespacingalongthenanoribbon. Thesubbands
0 0
ε (k ), j=1,2,... areidenticalfortheelectronsandholes.
j y
Becauseofthemultiplesubbandstructure,thezerotemperaturemeanfieldequationsforthesuperfluidstateacquirean
additionalindexforthesubband j. Theequationsforthewave-vectordependentsuperfluidenergygaps∆ (k )forsubbands j
j y
become,
∆j(ky)=−L1y∑jjc(cid:48) ∑kkcy(cid:48) Fky,j,ky(cid:48),j(cid:48)Ve-h(ky−ky(cid:48))2∆Ej(cid:48)j(cid:48)((kky(cid:48)y(cid:48))) , (1)
where Ly is the nanoribbon length, Fky,j,ky(cid:48),j(cid:48) =[1+cos(θky,j−θky(cid:48),j(cid:48))]/2 is the form factor coming from the overlap of the
single-particlenanoribbonwavefunctions,withθ =tan−1(k /k ),V (q)istheeffectiveelectron-holepairinginteraction,
ky,j y j e-h
andE (k )=[(ε (k )−µ)2+∆ (k )2]1/2 isthesingle-particleenergydispersioninthesuperfluidstateforsubband j. The
j y j y j y
wave-vectork(cid:48) isboundedbytheBrillouinzoneboundary±k . Wetruncatethesumoverthesubbandindexat j(cid:48)= j ,where
y c c
j isthelowestsubbandwithaminimumabovethegraphenenanoribbonworkfunctionenergy,takentobe∼4.5eV.The
c
3/8
0.4
(a) 2.5 (b)
9 8 V) 0.2
d 5 7 6 e
0.35 140 2 ∆(k) (jy 0.1 m=e2
3
0
0.3 2
c (c)
0.3
(eV) 0.25 j=1 1.5eV) k) (eV)y 0.2 m=e3
max∆j E(F ∆(j 0.1
1
0.2 b 0
(d)
0.3
V)
e m=e
0.15 0.5 k) (y0.2 4
∆(j0.1
2 3 4 5 6
0.1 0
0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1
n (1014cm-2) ky(kF)
Figure4. (a)Maximumsuperfluidgap∆maxforsubbands j=1,2... asfunctionofdensityn. Thedottedlineshowsthetotal
j
FermienergyE . NanoribbonwidthsW =2nmandbarrierthicknessd=5nm. ThedensitiesatwhichE entersthebottom
F F
ofanewsubbandareindicatedbytheverticallines. Notethatthe∆maxarealloforderE . Rightpanels(b)-(d):
j F
momentum-dependentgaps∆ (k )forsubbands jatdensitiesmarkedbythearrowsinpanel(a)(atwhichµ =ε ,ε ,ε ).
j y 2 3 4
chemicalpotentialµ isfixedbythedensityequation,
2 jc kc
n = ∑∑v (k )2; (2)
j y
WL
y j ky
(cid:20) (cid:21)
1 ε (k )−µ
v (k )2 = 1− j y . (3)
j y
2 E (k )
j y
Thefactor2inEq.(2)isduetothespindegeneracy.
Inthecalculationsweneglectscreening,anapproximationthatisjustifiedaposterioriasfollows. Thescreeningisexpected
tobeweakwhenthesuperfluidityliesinthestrongly-coupledBECorBCS-BECcrossoverregimesbecausethesuperfluidgap
intheseregimesofpairingiscomparabletotheFermienergy,resultinginalargesmearingoftheFermisurface,andcompact
electron-holepairscomparedwiththeiraveragespacing. Thismakestheirmutualinteractionsdipolarandweak. IntheBECor
BCS-BECcrossoverregimeswecanthusexpectscreeningtobeweak,andsoweneglectit. Howeverforweak-coupledBCS
superfluidity,screeningisknowntobeastrongeffect,whichwouldmakeourunscreenedapproximationapooroneinthe
BCSregime. Theseeffectshavebeendocumentedintherelatedsystemofsuperfluidityincoupledelectron-holesheetswith
parabolicenergybanddispersion.9,25
ThusinEq.(1)weapproximatetheelectron-holepairingattractionV (q)bythebareCoulombinteraction. Forelectrons
e-h
andholesconfinedinnanoribbonsofwidthW,separatedbyaninsulatingh-BNbarrierofthicknessdanddielectricconstant
κ =3,weobtain,26
−2e2(cid:90) W(cid:90) W (cid:113)
V (q)= dx(cid:48)dxK (|q| (x−x(cid:48))2+d2). (4)
e-h κW2 0 0 0
FollowingFig.1ofRef.26,weexpectthatinterbandmatrixelementswillbesmallcomparedwithintrabandmatrixelements.
ThusinEq.(1)weneglectthecrosspairingterms,whereCooperpairswouldformwithcarriersfromdifferentsubbands.
Results
Figure3showsthemaximumsuperfluidgap∆maxasafunctionofthedensityn,averagedoverthemultiplesubbandsofthe
nanoribbons,calculatedusingEqs.1to4. ∆max isthemaximumvalueofthewave-vectordependent∆(k )averagedwith
y
4/8
respecttothesubbandindex j. ThenanoribbonwidthisW =2nm,anddisthethicknessoftheinsulatingbarrier. Thedensities
atwhichtheFermienergyentersthebottomofanewsubbandε , j=1,2,...,areindicatedbytheverticallines.
j
InFig.3wenoticealocalboostin∆max neartheminimumofeachsubbandforbarrierthicknessd=5nm. Thisboost
arisesfromshaperesonanceeffectsassociatedwiththevanHovesingularities(Fig.1(b))andthequantumsizeeffectsinthe
pairinginteraction. However,forthinnerbarriersd∼<4nm,wheretheelectron-holepairingbecomesprogressivelystronger,the
shaperesonanceeffectsaremaskedbyamixingofthesubbandscausedbythelargesuperfluidgap. As∆maxgrowslargerthan
thetypicalspacingbetweensubbands,thesystembecomesdecreasinglessone-dimensionalincharacter,thankstothemany
channelsavailablebothforCooperpairingandforformingthesuperfluidcondensate.
0.8
d=5nm
d=4nm
0.6 d=3nm
d=2nm
F
E 0.4
/m
0.2
0
0 0.5 1 1.5 2 2.5 3
n (1014cm-2)
Figure5. Ratioofchemicalpotentialµ toFermienergyE asfunctionofdensityn. d isthicknessoftheinsulatingbarrier
F
separatingthenanoribbons. NanoribbonwidthW =2nm. TheverticallinesshowthedensitiesatwhichE entersthebottom
F
ofasubband.
Figure4(a)showsthemaximumsuperfluidgap∆maxfortheseparatesubbands jasafunctionofnfornanoribbonwidth
j
W =2nmandbarrierthicknessd=5nm. Forcomparison,thetotalFermienergyE ofthenon-interactingnanoribbonsystem
F
atdensitynisalsoshown. TheverticallinesmarkthedensitiesatwhichE entersanewsubband. Forthelowestconduction
F
subbands,thereisanotablelocalboostin∆maxasE entersanewsubband. Thisboosttakestheformofashaperesonancein
j F
thesuperfluidgapsassociatedwithaparticularsubband. Howeverevenforthelowestsubbands,welosesomefinestructureof
theshaperesonancesbecauseofmixingbythegapofcloselyingsubbands. Overthedensityrangeshown,the∆maxremain
j
alwaysoforderE ,andhencetheylieinthestronglycoupledregime.
F
Figures4(b)-(d),showthemomentum-dependenceofthesubbandgaps∆ (k )fordensitiesatwhichthechemicalpotential
j y
µ entersanewlow-lyingsubband(markedinFig.4(a)bytheverticalarrows). Thepeaksin∆ (k )arebroadonthescaleof
j y
k =π/(2r ),theinverseoftheaverageinterparticlespacing,whichconfirmsthatweareintheBCS-BECcrossoverregimeof
F 0
compactelectron-holepairs. Inpanels(c)and(d)ofFig.4,themultiplepeaksof∆ (k )areassociatedwiththedifferentFermi
j y
energiesofthesubbands(k ) ,displayingaremainingfermioniccharacteroftheCooperpairingintheBCS-BECcrossover
F j
regime.
Figure5showsthechemicalpotentialµ asafunctionofdensitynfornanoribbonwidthW =2nmandbarrierthickness
d. µ isnormalizedtothecorrespondingFermienergyofthenon-interactingnanoribbonsystematdensityn. Thechemical
potentialisstronglyrenormalizedwithrespecttotheFermienergyoverthefullrangeofnandd shown. WhenE entersa
F
subband,µ hasadip. Thisisincontrasttothepeakseeninthesuperfluidgap,anditisashapeantiresonancecausedbythe
shape-resonance-generatedpeakinthegap. Asd isdecreasedandthepairingstrengthweakens,µ increasestowardsE and
F
theshape(anti)resonancesbecomesizeable,indicatingthatthesystemhasenteredtheBCS-BECcrossoverregime. Inthecase
ofd=2nm,µ (cid:28)E andtheshape(anti)resonancesarecompletelysmoothedout. Thisisaresultoflargesuperfluidgapsand
F
itsignalsthatthesystemisinthestrongpairingBECregime. Whenthedensityincreases,thesystemalwaysevolvestowards
theweakerpairingBCSregimeforallvaluesofd,withµ eventuallyarrivingatE .
F
TheaveragepairsizeoftheCooperpairsξ insubband jisdefinedastheexpectationvalueofthesquareoftherelative
j
coordinateoftheCooperpairswithrespecttothesquareoftheBCSwavefunctionprojectedinthesubband. Thisdefinition
5/8
6
d=5nm
5 BCS d=4nm
4
Crossover
)
0
r 3
( d=3nm
ξ
d=2nm
2
1
BEC
0
0 0.5 1 1.5 2 2.5 3
14 -2
n (10 cm )
Figure6. Paircorrelationlengthξ/r averagedoversubbandsasfunctionofn,thecarrierdensity. d isthicknessofthe
0
insulatingbarrierseparatingthenanoribbons. NanoribbonwidthW =2nm.
wasoriginallyintroducedinRef.27toinvestigatethedifferentregimesofpairinginhigh-T superconductivityincupratesasa
c
functionofdensity. IthasbeenextendedtoamultigapsuperconductorthroughouttheBCS-BECcrossoverinRef.28andtoa
multigapquasi-one-dimensionalsuperfluidofultracoldfermionsconfinedincigar-shapedtraps.29 Inwave-vectorspace,
(cid:34) (cid:35)1
ξ = ∑ky|∇ky(uj(ky)vj(ky))|2 2 , (5)
j ∑ky(uj(ky)vi(ky))2
whereu (k )2=1−v (k )2.
j y j y
Figure6showsthepaircorrelationlengthξ/r asafunctionofdensity.ξ/r isthepartialaveragepairsizeforeachsubband
0 0
averagedoverthesubbands. ThenanoribbonwidthW =2nm. Wedesignateξ/r <0.25astheBECregime,0.25<ξ/r <4
0 0
theBCS-BECcrossoverregime,andξ/r >4theBCSregime. Asdiscussed,ourapproximationofneglectingscreeningis
0
expectedtobeagoodonefordensitieslyingintheBCS-BECcrossoverandBECregimes. Asexpected,thedensityrangefor
thestrongly-coupledregimecontractswithincreasingbarrierthicknessd becausethepairingbecomesweaker.
Wehaveneglectedeffectsfromimpuritiesanddisorder. Weexpecttheseeffectstobesmallsincewhilethereisnodirect
informationonimpurityanddisordereffectsingraphenenanoribbons,butbasedonpropertiesofanalogouscoupledelectron-
holegraphenemonolayers,chargeimpuritiesconcentrationsupton <k /(πd)arenotexpectedtodestroysuperfluidity.30 At
i F
graphene-hBNinterfaces,thechargeimpuritydensityni∼>1010cm−2,31soford∼<5nm,theinequalityissatisfiedprovided
n∼>3×106cm−2. Thisdensityisordersofmagnitudelessthancurrentexperimentaldensities.
Conclusions
Thesuperfluidgapsinourcoupledelectron-holenanoribbonsystemsarelargeinabsolutevalueandcomparabletotheFermi
energy. Thequasi-one-dimensionalconfinementandthesuperfluidshaperesonancesduetoquantumsizeeffectsbothplay
an important role here. The van Hove singularities in the densities of states act non-linearly through the gap equation to
significantly enhance the magnitude of the superfluid gaps. In the range of nanoribbon densities and barrier separations
considered,wefindthattheelectron-holesuperfluidisforthemostpartinthestronglycoupledpairingregime,andsoCoulomb
screeningeffectsareexpectedtobeweak.
Whenthesuperfluidgapsarecomparabletothesubbandenergyseparations,thegapsmixthesubbandsandthisresultsina
roundingoftheshaperesonances. Thiseffectismostpronouncedforsmallseparationsbetweenthenanoribbonswherethe
6/8
electron-holecouplingisparticularlystrong. Forlargerseparations,theelectron-holecouplingisweakerandthesuperfluidgaps
aresmaller. Thisresultsinweakersubbandmixing. Whenthisisthecase,theshaperesonancesaresharperwhichstrengthens
thelocalamplificationofthegaps.
Inourquasi-one-dimensionalsystemthereisnodirectlinkbetweenthesuperfluidtransitiontemperatureandthesizeofthe
superfluidgapscalculatedwithinmeanfield. Inourproposeddevicethezerotemperaturesuperfluidgapsarecomparabletothe
Fermienergy,andcanbeoforderofhundredsofmeV.Thushightransitiontemperatureelectron-holesuperfluiditycouldbe
expected,withpropertiesthataretuneablebychangingthedensity. Thedeviceconfigurationsweproposeareexperimentally
realizablewithcurrentlyavailabletechnologies. Asuperlatticeformedofsuchnanoribbondevicescouldfurtherstabilizethe
electron-holesuperfluidphaseoverlargeareas.
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Acknowledgements
M.Z.acknowledgessupportbytheFlemishScienceFoundation(FWO-Vl),theUniversityResearchFund(BOF),andthe
EuropeanScienceFoundation(POLATOM).A.P.andD.N.acknowledgesupportbytheUniversityofCamerinoFARproject
CESEMN.TheauthorsthankthecolleaguesinvolvedintheMultiSuperInternationalNetwork(http://www.multisuper.org)for
exchangeofideasandsuggestionsforthiswork.
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