Microscopic Current Dynamics in Nanoscale Junctions

0.79 MB·English

by  Na Sai

Microscopic Current Dynamics inNanoscaleJunctions
Na Sai,1,2 Neil Bushong,1 Ryan Hatcher,3 and Massimiliano Di Ventra1
1Department of Physics, University of California, San Diego, LaJolla, CA 92093-0319
2Department of Physics, The University of Texas, Austin, TX 78712
3Department of Physics, Vanderbilt University, Nashville, TN 37235
7
(Dated:January07,2007)
0
0 So far transport properties of nanoscale contacts have been mostly studied within the static scattering ap-
2 proach. Theelectrondynamicsandthetransientbehaviorofcurrentflow,however,remainpoorlyunderstood.
n Wepresentanumericalstudyofmicroscopiccurrentflowdynamicsinnanoscalequantumpointcontacts. We
a employanapproachthatcombinesamicrocanonicalpictureoftransportwithtime-dependentdensity-functional
J theory. Wecarryoutatomicandjelliummodelcalculationstoshowthatthetimeevolutionofthecurrentflow
6 exhibitsseveral noteworthy features, such asnonlaminarity andedge flow. Weattributethesefeaturestothe
2 interactionoftheelectronfluidwiththeioniclattice,totheexistenceofpressuregradientsinthefluid,andto
thetransientdynamicalformationofsurfacechargesatthenanocontact-electrodeinterfaces. Ourresultssug-

] gestthatquantumtransportsystemsexhibithydrodynamicalcharacteristicswhichresemblethoseofaclassical
ll liquid.
a
h
s- I. INTRODUCTION withabsorbingboundaryconditions.24Thismethod,however,
e is affected by the arbitrariness of how and where the ab-
m sorbing potentials are added, while the effect of the absorb-
Recent experimental progress has enabled imaging of co-
. ing potential on the conduction in nanojunctions is unclear.
t herent current flow dynamics in quantum point contacts
a Other methods have been developed based on the Landauer
m formed in semiconductor heterostructures.1,2,3,4 These ad- scatteringformalismoftransportthatemployopenboundary
vances in experimental techniques open the possibility that
- conditions.25,27,29,30
d current flow through atomic or molecular junctions will be
Morerecently,amicrocanonicalformalismthattreatselec-
n eventuallyimagedandcontrolled. Understandingthe micro-
tronictransportas adischargeacrossa nanocontactconnect-
o scopic electronic flow patterns can aid the design of novel
ing two largebut finite chargedelectrodeswas introduced.23
c electronic devices. However, very few theoretical studies of
[ TheformulationhasbeencombinedwithTDDFTtostudythe
current dynamics in nanoscale systems are currently avail-
dynamicalformationofquasi-steadycurrents,localchemical
1 able. Indeed, among the recent theoretical studies of trans-
potentials,28 and electron-ion interactions.26 This formalism
v port in nanoscale systems, much emphasis has been placed
wouldyieldtheexacttotalcurrentflowingfromoneelectrode
4 onthe steady-stateconductionproperties,5,6,7,8,9,10,11,12,13,14,15
3 totheotheriftheexactfunctionalwereknown,regardlessof
whereasthe transientbehaviorof the currentremainsan un-
6 whetherthesystemachievesasteadystate. Inpractice,inab-
exploredarea.
1 initiotransportcalculations,oneonlyusesapproximateforms
0 Electronic transport in nanoscale junctions is usually for- of the functionalsuch as the adiabatic local density approx-
7 mulated within the stationary scattering picture, such as the imation (ALDA). It has been shown recently that the elec-
0 one dueto Landauer,16 in which the conductionis treatedas tronic correlation effect beyond the ALDA gives rise to ad-
/
t a collection of scattering events. This stationary approach, ditionalresistanceinmolecularjunctions.35,36,37Thespurious
a
m widely used to study transport in mesoscopic and nanoscale self-interactionimplicitintheALDAfurthercomplicatescal-
systems, has led to considerable success in understanding culations of the conduction properties.39,40,41 The sensitivity
-
d current-relatedeffectsotherthanthe conductance,including, ofvarioustransportpropertiestothesuggestedcorrectionsre-
n e.g.,noise,localheating,andcurrent-inducedforces.17 Ithas mainsonlypartiallyunderstood.
o helped our understanding of the microscopic current distri- Thedynamicalestablishmentofaquasi-steadycurrenthas
c bution as well.18,19 Nevertheless, the stationary approachas- beeninvestigatedbyanumberofauthors.26,27,28,42Ithasbeen
:
v sumesthatthesystemisalreadyinasteadystate,leavingthe demonstrated that a quasi-steady current can establish itself
Xi questionsofhow a steadycurrentestablishesitself andwhat across a junction on a femtosecond time scale without the
other phenomena are related to the dynamical formation of presence of inelastic scattering.28 This is due to the geomet-
ar steadystatesinananojunctionunsolved. rical “squeezing” experienced by the electrons crossing the
Thedynamicalnatureofthecurrentflowisbetteraddressed nanojunction.23 The conductancecalculated from the micro-
in a time-dependent approach than in the stationary one. canonical formula was shown to be in good agreement with
Time-dependent or AC transport approaches have been pre- that obtained from the static DFT approach in prototypical
viously introducedin mesoscopicconductingsystems.20,21,22 atomicjunctions26,28aswellasinmolecularjunctions.43Nev-
Recently, an increasing amount of effort has been directed ertheless,astudyofthemicroscopicbehavioroftheelectron
toward developing ab-initio time- dependent approaches for flow,andinparticularofthecurrentflowmorphologyinnano-
nanoscalesystems.23,24,25,26,27,28,29,30,31,32,33 One such method junctions,isstilllacking.
was developedto study the AC conductanceusing the time- Inthispaper,wecarryoutreal-timenumericalsimulations
dependent density functional theory34(TDDFT) combined of current flow in metallic nanojunctions using the micro-
2
thickastheelectrodes.Thedistancebetweenthejelliumelec-
trodesis9.8A˚.Intheatomiccalculations,thejunctionisrep-
resented by two planar arrays of gold atoms sandwiching a
singlegoldatom. WeemployTDDFTandsolvetheeffective
single-particleSchro¨dingerequation
V(z) z
a
~2∇2
i~ψ˙(~r,t)= − +Veff(~r,t) ψ(~r,t), (1)
(cid:20) 2m (cid:21)
x
wheretheeffectivepotentialisgivenby
z
ρ(~r,t)
FisIGst.ud1i:ed(Cinoltohreopnalpineer.) SAktettch<of0,thaebniaasnoinjunthcetiofnormgeoomf eVtr(yz)th=at Veff(~r,t)=Vext(~r,t)+Z ~r−~r′d~r′+Vxc(~r,t). (2)
V0[H(z−za)−H(−z−za)]isappliedtothejunction(thecentral
constrictionisatz =0)suchthattheregions|z|>zabearapoten- ThetermVxc(~r,t)istheexchange-correlationpotentialcal-
tialoffsetfromthecentralconstriction,whereH(z)istheHeaviside
culatedwithintheadiabaticlocal-densityapproximation.The
stepfunction.
external (ionic) potential is modeled using pseudopotentials
fortheatomiccalculations,44whileinthejelliummodelitisa
localoperatorrelatedtotheuniformpositivebackgroundjel-
cAaLnDonAic.aWleforremstarilcistmth,ewfohrethrecowmeinegmdpilsocyusTsiDonDtFoTthweidthyinnamthie- lium density ρ0 via Vext(~r) = ~r−ρ0~r′d~r′, where ρ0 equals
calbehaviorofelectron/holechargesinthenanojunctionsun- to (4π3rs3)−1 inside the jellium aRnd zero outside, and rs is
derthelinearresponseregime,i.e.,inwhichthebiasissmall the Wigner-Seitz radius. In the jellium model, we choose
and the current-voltage characteristics are linear. The paper rs = 3aB (aB = Bohr radius) which gives a good repre-
isorganizedasfollows. InSec.IIwediscussmodeltransport sentation of bulk gold (see also discussion below). A “free-
systems and numerical methods. In Sec.III, we present and space”boundaryconditionisimplementedsuchthatthelong-
discusssimulationsofcurrentdynamicsinjelliumandatomic range potential is constructed only from the densities in the
junctions. Wealsoanalyzetheeffectsofhydrodynamicpres- supercell;45 that is, the system is not periodic. Additional
sure and electrode surface charges on the dynamics of the numerical details can be found in Appendix A. The single-
flow. In Sec.IV, we summarize the main conclusionsof our particletime-dependentcurrentdensityiscalculatedvia
work.
~
~j(~r,t)= ψ∗(~r,t)∇~ψ (~r,t)−∇ψ∗(~r,t)ψ (~r,t) ,
2mi n n n n
Xn h i
II. MODELANDMETHODS (3)
where ψ denotes individual Kohn-Sham single-particle
n
Thenanoscalejunctiongeometrystudiedinthispaperisil- states. Note thatin TDDFT the currentdensity is notneces-
lustratedinFig.1. Anarrowconstrictionseparatestwolarge sarilyexactevenifonecalculatesitwiththeexactfunctional
butfiniteelectrodes. Webeginthesimulationsbyapplyinga (viathecontinuityequation,onlythedivergenceofthecurrent
stepfunction-likeelectricbiasacrossthejunctionsuchthatthe density wouldbe exact). One wouldneed to resortto Time-
twoelectrodesbearequalandoppositepotentialsoffsetrela- DependentCurrentDensity FunctionalTheory (TDCDFT)46
tivetothepotentialatthecenterofthejunction.Thedistance toobtaintheexactcurrentdensity(withtheexactfunctional).
fromthebiasdiscontinuitytothecenterofthejunctionisz Nonetheless,duetothesmallviscosityoftheelectronliquid,
a
(seeFig.1).Thisbiasinducesachargeimbalancebetweenthe we have found that the current density one obtains by using
twosidesofthesystem. Att = 0,weremovethebiasanda TDDFTwithintheALDA,andtheoneobtainedbyTDCDFT
dischargethroughthenanojunctionensues. TheKohn-Sham within the Vignale-Kohn functional46 are qualitatively simi-
initial state therefore corresponds to the ground state of the lar.47
Hamiltonian in the presence of the bias. Here, we are inter- Even if the contact between the electrodeswere removed,
ested in the transientbehaviorduringthe phasein whichthe thecurrentbetweenthetwoelectrodeswouldnotcompletely
current is in the process of establishing a quasi-steady state vanish because of quantum tunneling. This bare tunneling
andimmediatelythereafter,i.e.,longbeforetheelectronsthat currentcan convenientlybe used to comparethe jellium and
havepassedthroughtheconstrictionhavehadchancetoreach theatomiccalculations. Thejellium edgesare placedathalf
thefarboundaryoftheelectrodes. the interplanarspacingof the lattice.48 This way, the jellium
To separate the effects of the electrons and of the atomic modelandthepseudopotentialcalculationsbothyieldtunnel-
lattice,wehavecarriedoutcalculationsusingajelliummodel ingcurrentdensitiesof∼ 0.05µA/A˚2 ata biasof0.2V.The
and an atomic model. In the jellium calculations, the elec- agreementindicatesthatthejelliummodelisagoodrepresen-
trodesarerepresentedbytwolargejelliumslabs2.8A˚ thick. tation of two large metal electrodes. This is consistent with
Thecontactisa rectangularjelliumblock2.8A˚ wideandas theresultsofpreviousdensity-functionalcalculations.49
3
FIG.2:(Coloronline)Currentfluxforaseriesoftimesinananoscalequantumpointcontactsysteminthejelliummodel.Theappliedbiasat
t<0is∆V =0.2V.Thefieldlinesineachpaneldepictthedirectionandamplitudeofthecurrentdensityvectors,whilethecolorsgiveextra
electron(red)orhole(blue)density. (a)t = 0.4fs;(b)t = 0.8fs;(c)t = 1.6fs. In(a),thesemicirclemarksthecontouralongwhichthe
radialcomponentofthecurrentdensityiscalculated(seetextandFig.3).
III. RESULTSANDDISCUSSION 0.3
t = 0.40 fs
]
2 t = 0.80 fs
A. Flowdynamicsthroughjelliummodeljunctions °A/A 0.25 t = 1.60 fs
µ 0.2
[
meInnsiaonnsanoofjuthnectlieoandssuacrheaussuaanllaytommuicchploairngtecrotnhtaanctt,htohseedoif- nsity 0.15
e
the central constriction. In addition, not far from the con- d
nt 0.1
tact, we expect the electron momentum to converge to the e
r
valuecharacteristicofthebulkleads. Therefore,themomen- cur 0.05
tum of an electron coming from the leads and entering the al
contact has to change considerably. This gives rise to resis- di 0
a
tance,andforatrulynanoscalejunction,thismomentummis- R
match is mainly responsible for the establishment of quasi- -0.05
-90 -60 -30 0 30 60 90
steadystates.23,28,50 Usingthe abovedynamicalapproachwe
Angle [Degrees]
can now study how this translates into microscopic current
flowthroughthenanocontactandintotheleadsbycalculating
FIG.3:(Coloronline)Timeseriesoftheradialamplitudeofthecur-
thecurrentdensityatdifferenttimes. rentdensitiesalongasemicircleofradius3A˚ centeredonthenano-
To begin, we follow the method described in the preced- junction,asafunctionofanglealongthecontour.
ingsectiontoimposeachargeimbalanceinthejelliummodel
system.Adiscussionoftheeffectofthelatticeontheflowdy-
namicswillbepresentedinthefollowingsection. Theinitial dominant flow is in the lateral direction, i.e., parallel to the
biasischosensuchthatthediscontinuityhappensattheedge facingsurfacesoftheelectrodes.
of the jellium slab near the central constriction(i.e., za ∼= 5 Toquantifytheevolutionoftheangulardistributionofthe
A˚).Theflowpatternisindependentofthelocationofthedis- electronflow,inFig.3weplotatimeseriesoftheradialcom-
continuityoncethecurrentstartstoflowthroughthecenterof ponentofthecurrentdensityalongasemicirclecontourcen-
thejunction. InFig.2,weplotthreesnapshotsofthecurrent tered on the junctionas a functionof the angle on the semi-
densitytoillustratetheevolutionoftheflow. Duetothebias circle(see Fig. 2 (a)). Onecan see thatinitiallya “wave”of
offset near the jellium edges, a dipolar layer forms on each excess charge approaches the nanocontact. Then, the radial
ofthetwocontact-electrodeinterfaces. Asaresult,theinitial currentdensitypeaksatverylargeangles(∼ ±75◦);i.e.,the
current flow is uniform on both sides as shown in Fig. 2(a). currentdensitynearthecontactisdominatedbytheflowalong
Verylittlecurrentflowsinthenanojunctionatthispoint,how- the electrode edges in the lateral direction. The peaks then
everthecurrentsteadilyrises. InFig.2(b),thecurrentdensity graduallymovetowardsthecentralaxis,andthecurrentden-
becomesconvergenttowardthecenterofthenanojunction.In- sityadjuststoamore“focused”patternasshowninFig.2(c).
terestingly,astheexcesschargefromtheleftelectrodereaches Wehavealsoexaminedajunctionconsistingofajelliumcir-
the contact, thereis a periodof adjustmentduringwhichthe cular“island”betweentheelectrodes. Wehavealsoobserved
4
FIG.4: (Coloronline)TimesequenceofelectroncurrentstreamlinesintheatomicjunctiondescribedinSec.II. Panels(a)-(d)correspond
tot=0.2,0.25,0.3,and0.35fs,respectively. Thedotsdenoteatomicsitesthatcorrespondstothe(001)facetsofthegoldFCClattice. The
appliedbiasatt<0is∆V =0.2V.
edgeflowinthiscaseaswell. Theedgeflowisnotaquantum the influence of the lattice on the flow, we carried out simu-
interferencepattern,andcannotbecomparedwiththefringes lations that included an ionic backgroundmodeled by pseu-
observedinthe2Delectrongasquantumpointcontacts.4 In- dopotentials. Theatomiccalculationswerecarriedoutfor2-
stead,wesuggestthattheflowpatterniscontrolledbyhydro- D gold nanojunctions and were initialized in the same way
dynamiceffectsandforcesduetosurfacecharges.Weanalyze asthejelliumcalculations. Thetheoreticalandexperimental
theseinSec.IIIC. conductanceareinverygoodagreementforthissystem.56We
Thestructureofthecurrentdensityisanalogoustoa clas- chose lattice arrangements that correspond to the (001) and
sical fluid flowing across a narrow constriction. This is not (111)facetsofthegoldFCClattice(thedotsinFig.4repre-
surprising because an inhomogeneous electron system can senttheatomicsitesforthe(001)configuration).
be indeed characterized by a set of hydrodynamical rela- Electric currentstreamlines at differenttimes in the simu-
tions expressed in terms of the particle density and velocity lation are plotted in Fig. 4(a) - (d). The streamlines are cal-
field.51,52,53,54 Morerecently, in particular,a hydrodynamical culated by integratingthe currentdensity field upstream and
approachwasproposedfornanoscaletransportsystems,53fur- downstream,d~r/ds=±~j(~r(s)). Themorphologyofthecur-
ther strengthening the analogy between the dynamics of the
rentflowintheatomicjunctionandinthejelliummodelisre-
electronliquidandtheoneofaclassicalliquid. Wenotethat markablysimilar,57indicatingthatthejelliummodelisagood
the present calculations do not take into account the physi-
representationofthegoldelectrodes. Nevertheless,anumber
cal viscosity of the electronliquid.54,55 An inviscid fluid can
ofnewfeaturesappearintheatomiccalculations.
thereforebeusedasamodelforthedynamicalbehaviorofthe
Fig.4(c)showsthatonceasteadyflowthroughthejunction
presentelectronliquid.
isestablished,thecurrentspreadsintoawedge-shapedregion
inside the electrodes. The flow morphology for each of the
two different lattice arrangements is similar except that the
B. Flowdynamicsthroughatomicjunctions flowspreadsoverabroaderwedge-shapedregioninthe(111)
lattice. Anothercommonfeatureintheatomiccalculationsis
The jelliummodelwas a convenientway to probethe mi- thepresenceofastagnantzoneatthecorneroftheelectrode
croscopiccurrentdynamicsinanelectrongas. Tounderstand boundary. Thereislittlecurrentflowintooroutofthiszone.
5
This is similar to a classical fluid where a stagnantzone can t = 0.00 fs (a)
sometimesbelocatedattheentranceorexitofachannel. -- --
+ +
One profound difference between the atomic and the jel-
liumcalculationsistheformationofeddiesevidentinthefor-
t = 0.20 fs
merbutnotinthelatter.Inthejelliumcalculationscarriedout
within the linear-response bias regime, the currentflux lines
are laminar. In contrast, in the atomic calculations, the ed- z
ρ
diesappearaslocalizedcircularflowthatcanbeobservedin t = 0.40 fs
Fig.4(d). Theeddiesdevelopinbothelectrodesandthesize
of the eddiesis comparableto the interatomic distance. The
eddiesarereminiscentofthevorticesthatforminaclassical t = 0.60 fs
fluid at higher Reynolds number when the fluid encounters + --
obstacles. As is well known, vortices can also occur when
velocity shear is present within a continuous fluid (Kelvin- -30 -15 0 15 30
Helmholtz instability). We suggest that the lattice ionic ob- °
z [A]
stacles and the boundaries separating the flow zone and the
stagnantzone facilitate the formationof the observededdies 16 (b) (c)
inoursimulations.
8
The formation of current vortices has been previously re-
psyosrtteemdisn, a2Dricbhalvliasrtiiectqyuoafntfluomwbpilaltitaerrdnss.5r8a,5n9gIinngthfersoemqureagnutulamr °x [A] 0
to chaotic is possible. While we cannot draw direct analo-
-8
gies with these open and mesoscopic transport systems, an
unstable and turbulent flow has been recently predicted in -16
nanoscale systems53. We have argued that the ALDA elec-
-16 -8 0 8 16
tron liquid in our simulations corresponds approximately to °
z [A]
aninviscidfluid. Wethereforesuggestthatinthepresenceof
a lattice, hydrodynamicalinstabilities, or turbulence can oc-
FIG.5: (a)Planaraveragedchargedensityfromt=0tot=0.6fs
curinnanotransportsystemsevenintheabsenceofaphysical
forajelliumjunction.Thechangeinthepeaksindicatesthedynamic
viscosity.
processinwhichexcesschargebuildsupatthesurfaces. Thesign
ofthesurfacechargesindicatesthatelectronchargesaccumulateson
therightandholechargesontheleft. (b)-(c)Excesschargeinthe
C. Hydrodynamicsandtheformationofsurfacecharges vicinity of the contact at t = 0 and t = 0.6 fs. Thick solid lines
indicate excess electrons, while thick dashed lines indicate excess
holes. Thin dashed lines mark the edges of the jellium electrodes
Tounderstandwhatdrivestheedgeflowalongtheelectrode
andthecontact.
surfaces,weexaminetheevolutionofthechargedistribution
near the surfaces of the junction. For this purpose, we ap-
ply a step-function bias such that the potential discontinuity
in each electrodeoccursat za ∼= 10A˚, whichcutsacrossthe caaplpfrooramcha.tionofthesesurfacechargesusingatime-dependent
electrode,asillustratedinFig.1. InFig.5(a),weplotatime
Wehavealreadyobservedthat,asthesurfacesoftheelec-
seriesofthex−yplane-averagedexcesschargedensityalong
trodes are populated by excess charges, a lateral flow starts
the z axis. At t = 0, two symmetricdipolarlayers formin-
to developalong the surfaces. This behavioris illustrated in
side theelectrodesasa resultofthebiasoffsets. Asthecur-
Fig. 3 where the radial currentflux at t = 0.8 fs shows two
rentstartstoflowthroughthecontactandgraduallyreachesa
pronouncedpeaks at very large angles. We attempt to inter-
quasi-steadystate,aglobalchargeredistributionbecomesap-
pret this behavior within the frameworkof an effectiveclas-
parent. The dipolarlayersgraduallyvanishand are replaced
sic hydrodynamic model of an inviscid charged fluid. The
bysurfacechargelayersthatformatthecontact-electrodein-
acceleration of the fluid is then given by Euler’s equation
terfaces.ThechargecontourplotsFig.5(b-c)furtherillustrate
theformationofsurfacechargesasaresultofcurrentflow. ∂~v/∂t+(~v·∇~)~v = −∇~P/men−e∇~ϕ/me,where~v isthe
The formation of surface charges around the central con- fluid velocity, n is the fluid particle density, me is the elec-
strictionisreminiscentoftheformationofresidual-resistivity tronmass, andϕ isthe electrostatic potential. Thefirst term
dipolesintroducedbyLandauer.16Ithasbeensuggestedthata ontherighthandsideistheaccelerationduetoagradientin
continuouscurrentflowarrivingatajunctionmustbeaccom- electronpressure. Thesecondtermistheaccelerationdueto
panied by self-consistently formed charges at the electrode the electric field of the excesschargeson the surfacesof the
surfaces.49Theeffectshouldbetakenintoaccounttocorrectly electrode. Theelectricfielddrivestheelectrons/holestoward
characterizetheelectrostaticpotentialandthenonequilibrium thesurfacestocancelouttheexcesscharges.
conductingpropertiesinatransportcalculation.61Inthiswork The inertial term in the above equation can be estimated
we providethe first numericaldemonstrationof the dynami- as |(~v · ∇~)~v| ∼ v2/L ∼ 106m2/s2 × L−1. Here, L de-
6
notes the characteristic length scale over which we expect a chargeshasnotbeenreportedinadiabaticquantumpointcon-
departure from uniform flow, so that ∇ ∼ L−1. Velocity tacts before. It would thus be interesting to develop experi-
of the flow in the simulation reaches v ∼ 103m/s ≪ v , mentaltechniquesto explorethe surface region of the quan-
F
where v ≈ 1.4 × 106m/s is the Fermi velocity of bulk tum point contact in a 2D electron gas and the charge accu-
F
gold. The hydrodynamicalpressure can be calculated from mulationthatweobserveinoursimulations.
thederivativeofthegroundstateenergywithrespectivetor ,
s
P/n = −rsǫ′(r ), where ǫ(r ) is the energy per particle.62
3 s s
Therefore, |∇~P|/m n ∼ 1v2|∇~n|/n. Here, we included IV. CONCLUSIONS
e 5 F
onlythegroundstateenergyofanoninteractingelectrongas,
ǫ0(rs)= 35ǫF.63Letδndenotethechangeoftheparticleden- Inthispaper,wehaveusedthemicrocanonicalapproach23
sity as a result of the currentor the formationof the surface tostudythetime-dependentcurrentflowmorphologyandthe
charges.Forthepresentjunctionsatthesebiases,wefindthat charge distribution in discharging nanojunctionsrepresented
typicallyδn/n.0.01.Then,theaccelerationduetothepres- by both a jellium model and pseudopotentials. We have
suregradientis showedthattheelectronflowinthenanojunctionsexhibitshy-
drodynamicfeaturesanalogousto a classical fluid. We have
|∇~P| 1v2 δn found that in the atomic case the current flow evolves into
|~a | = ∼ F
P m n 5 L n wedge-shaped pattern flanked by stagnant zones. The flow
e
m2 develops nonlaminar features including eddy currents. We
9 −1
. 3×10 ×L . (4) suggestthat the ionic lattice at the junctionplays the role of
s2
“obstacles” in the dynamicsof the electronliquid, with con-
Toestimatethemagnitudeoftheelectrostaticacceleration, sequentdevelopmentofthesefeatures. Wehavealsodemon-
we treat the layer of charge induced on the facing surfaces strated that excess surface charges accumulate dynamically
of the electrodes (as illustrated in Fig.5c) as an infinite uni- along the electrodes. In addition, we have observedthat for
formly charged wire. The electric field of the wire is given aperiodoftime,thereisstrongcurrentflowinthetransverse
by|∇~ϕ| = λ/2πε0L,whereλisthelineardensityofexcess direction. Byemployinganorder-of-magnitudeargumentwe
charge,Listhedistancetothewire,andε0isthevacuumper- suggestthatthisflow isdrivenbybothhydrodynamicforces
mittivity.Thelinearchargedensityiscalculatedbyaveraging due to the electron pressure, and electrostatic forces due to
the charge density difference eδn between the configuration the surface charge distributions. The latter forces dominate
withandwithoutthecurrentflowoverthelayerinwhichthe theinitialdynamicsinthejunctionsathand.
charge accumulated at the contact-electrode interfaces. For The present and a previous study28 demonstrate that
thesamejunctions,wefindλ∼0.016e/A˚. Theacceleration the microcanonicalapproachcombinedwith time-dependent
ofchargesduetothiselectricfieldcanbecalculatedas density-functional theory can be used to probe the transient
behaviorofthecurrentinnanojunctionssuchasatomic-scale
e λ pointcontactsormolecularjunctions.43Thepresentapproach
|~a | = (5)
el
me2πε0L supplementsexistingmethodsthatarebasedonthestaticscat-
m2 teringpictureandprovidesanothertooltostudyingrelatively
11 −1
∼ 5×10 ×L .
s2 unexplorednanoscaletransportphenomenafromfirst princi-
ples.
ThesecrudeestimatesimplythatoverthesamedistanceL
Theflowpatternsweobserveinmetallicnanojunctionscan
begeneralizedtoanumberofothersystems,suchasmolecu-
|(~v·∇~)~v|<|~a |<|~a |, (6)
P el larjunctions,althoughmanydetailsmayvary. Inviewofthe
recentadvancesinmicroscopicimagingtechniquesofcoher-
which suggests that the hydrodynamic pressure gradients
ent current flow in quantum point contacts in a 2D electron
dominateovertheinertiaofthefluid(theflowissubsonicand
gas,4 we hopethatnew experimentalwork exploringthe be-
compressible),whilethe maximumelectrostaticforcedueto
haviorofcurrentflowinatomiccontactsandmolecularjunc-
the surface chargesis comparable to or larger than the pres-
tionswillsoonemergetosupplementourstudies.
sure gradientforcebeforethe surfacechargehasbeenpassi-
vated. Therefore,itisplausiblethatthelateralflowobserved
inthesimulationisprimarilyofelectrostaticorigin. Indiffer-
Acknowledgments
entjunctiongeometriesorinadifferentconductanceregime,
theorderinginEq.6maybedifferent.
We have also carried out a similar simulation using a TheworkwassupportedbytheDepartmentofEnergygrant
parabolically- shaped constriction that resembles a quantum DE-FG02-05ER46204.
pointcontactinthe2Delectrongas.Atasimilarbiasasinthe
non-parabolicjunctions,wefindsimilarsurfacechargeaccu-
mulationalongtheboundariesoftheelectrodesinthevicinity APPENDIXA:NUMERICALDETAILS
of the contact. We believethe analysiswe haveprovidedon
the surface chargeformationappliesto this case as well. To We performed time-dependent density-functional calcula-
the best of our knowledge, the accumulation of the surface tions using the program socorro44 and an in-house pro-
7
gram which implements TDDFT within the jellium model. the calculations were performed using a real-space basis set
The gold ions were modeled by norm conserving Hamann where the space is uniformly discretized and the grid spac-
pseudopotentialswith6selectronsasvalenceelectrons.65We ingis0.7A˚.Theeigenvaluesvarylessthan3%bydecreasing
used the Perdew-Zunger (1981) LDA exchange-correlation thegridspacingby66%. Thetimeevolutionoperatorisrep-
functional.66 The atomic calculation employed a plane-wave resented using the Chebyshev method,67 with a time step of
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