Maeliss Ethis De Corny

Nicolas Chauvet

Guillaume Laurent

Mathieu Jeannin

Logi Olgeirsson

Aurélien Drezet

Serge Huant

Géraldine Dantelle

Gilles Nogues

Guillaume Bachelier
email: guillaume.bachelier@neel.cnrs.fr

Wave-Mixing Origin and Optimization in Single and Compact Aluminum Nanoantennas

Keywords:

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Introduction

gontrolling nd mplifying nonliner proesses t the nnometer sle is longEstnding quest for prtil pplitions inluding quntum teleommunition nd quntum omE putingF he gretest hllenge is to overome the intrinsilly low e0ieny of nonliner mterilsD whih is further redued upon downEsizing the tive medium to the nnosleF o this endD plsmoni nnostrutures hve een proposed s suwvelength resontors for enhning nonliner proesses suh s seond @rqA nd third @rqA hrmoni generE tionsF 15 he min dvntge of these strutures lies in the tunility of their resonnes s funtion of sizeD morphology or oupling on(gurtionF 69 hey lso strongly ene(t from their ility to on(ne the eletromgneti (eld elow the di'rtion limitD oosting their nonliner e0ieny ordingly thnks to the redued mode volumesF 10,11 sn this ontextD douly resonnt strutures re very promising sine they ene(t from the (eld enhnement t oth exittion nd emission stepsF 12 etD for optimized frequeny onversion yieldsD these plsmoni resonnes must e further sptilly modeEmthedF sn other wordsD the orreE sponding mplitude nd phse distriutions of the eletri (elds hve to e tilored in suh wy tht the generted nerE(eld nonliner urrents oherently rdite their hrmoni phoE tons in the frE(eldF uh strutures hve een reently hieved y oupling gold nnorod to Eshped ntenn with typil sizes down to ISH nmF 13 rereD we demonstrte tht these douleEresonne nd modeEmthing onditions n e rehed in single nd omE pt luminum ntenn for pplition purposes in the visile rngeF 1416 he net dvntge ompred to goldEsed nnostrutures is the lk of superimposed multiphoton luminesE ene kgroundD llowing diret redEout of pure rq signlsF hnks to this spei( P propertyD we show how the origin of the rq response n e unmiguously unveiled y quntittively ompring mesured nd simulted mps of the nonliner signlsD whih re otined y imging single ntenn under tightly foused emF Results and discussion sn order to infer the origin of the nonliner response of single luminum ntennsD we (rst investigted their spetrl response t (xed exittion wvelength of VSH nm y snning the monohromtor from RHH to TSH nm @see the experimentl setion for more detilsAF ynly one signl pek is oserved t RPS nmD iFeF t the hrmoni wvelengthF sts intensity is further shown to sle s the squred optil exittion powerX I(2ω) ∝ [I(ω)] 2 with orreltion oe0ient equl to r 2 = 0.996D unmiguously ssigning its origin to rq proessF his result signi(ntly di'ers from the se of gold nnontenns @frited y the sme proessA where the shrp pek t the hrmoni wvelength is superimposed to rodnd signl with mximum intensity t SSH nm @see pigure IAF his response is known to originte from the twoEphoton luminesene @vA of goldD whih is relted to its nd struture lose to the X nd L points providing lrge densities of sttesF 1721 iven higher order photoluminesene hs een reported in the litertureD smering the wek rq signl in n intense nd inoherent kgroundF 22 rereD the kgroundEfree signl of luminum n e diretly exploited nd quntittively ompred to simultionsF enother slient feture of luminum ntenns is their ility to sustin plsmoni resE onnes over the entire visile spetrumD in ontrst to gold nnostrutures where the resE onnes re hmpered ove the internd trnsition threshold @ner SPH nmA due to the strong sorption rte rising from the genertion of eletronEhole pirsF 14,15 his llows promoting the onept of oth douleEresonne nd modeEmthing onditions 13 in single nd ompt luminum nnontennF o evidene this peulir propertyD sorption spetr were simulted y pling in the viinity of the nnostruture pointEdipole soure ligned Q pigure IX rq intensity spetrum of single PPS nmElong luminum @lueA nd gold @redA ntenns illuminted t VSH nmF sn insetD the iw imge of single RPS nm luminium nnontenn @rightA nd typil topogrphi pro(le mesured with the proe of snning nerE(eld optil mirosope implemented in the setup @leftAF with the long ntenn xisF his llows exiting ll possile plsmon resonnes ssoited either with right or drk modes in the nnostruturesF 23 es shown in pigure P for PPS nm nd RPS nmElong ntennsD the spetrl distriution of the plsmoni resonnes n e tilored y djusting the ntenn spet rtioF st is therefore possile to reh resonnt onditions t the exittion wvelengthD t the emission wvelengthD or t oth wvelengthsD depending on the trgeted pplition or investigtion s shown lterF iven more interesting is the ility to hoose the symmetry of the plsmon modes for the exittion nd the emission steps in order to mximize the mode mthing t the nnosleF o give simple pitureD let us onsider the dipolr resonne tht is the most e0iently exited lolized surfe plsmon for suwvelength nnostruturesF st is ssoited with enhned eletri (eld mplitudes t eh ntenn pex with opposite surfe hrges @see pigure PAF essuming the rq urrents to e proportionl to the squred eletri (eld @this will e disussed in detil elowAD their sptil distriution hs here symmetri hrter with two loes shring the sme mplitude nd phse t the ntenn pexesF he plsmon mode mthing this ondition t the emission step is the qudripolr plsmon shown in R pigure PX A imulted sorption spetr for PPS nm @ornge urveA nd RPS nm @green urveA luminum nnontenns exited y n eletri point dipole in its viinityF lsmoni mode orders n re indited over eh resonneF ed nd lue dshed lines indite exE ittion nd detetion wvelengths used for ll mesurementsD respetivelyF A imulted sorption spetr for PPS nm snned long its long xis under foused em t the fundmentl frequeny @VSH nmAF snsetX Ph mps of the sttered eletri (eld norm t the fundmentl nd hrmoni frequenies otined inside the ntenn when the fous of the mirosope ojetive is entered on the ntennF hey evidene the n = 1 fundmentl nd n = 2 exited plsmon modesD respetivelyF he exittion on(gurtions for the sorption spetr re skethed y A the red dipole nd A foused emD respetivelyF pigure PF wore generllyD perfet mode mthing with the n th plsmon resonne exited t the fundmentl frequeny will e otined with the 2n th mode hving n opposite prity t the hrmoni frequenyF hese plsmon modes re often referred to s drk modes sine they hrdly ouple to plne wves propgting perpendiulrly to the sustrteF roweverD they do emit light t lrge ngles so tht hrmoni photons re e0iently olleted y lrge xe ojetivesF sn order to demonstrte these two mjor properties ! doule resonne nd mode mthE ing onditions ! Ph mps of the rq intensity sttered in the frE(eld were reorded y snning the smple @xEy plneA under the foused lser em @z xisAD see pigure QF he (rst onlusion tht n e esily drwn from these mesurements is tht the sptil distriution of the rq intensity depends on the ntenn lengthD s expeted from the tunility of the plsmon resonneF roweverD interpreting the rq dt is not strightforwrd here sine the nnostruture is lying on sustrteD is exited y strongly foused em ndD more S pigure QX rq exittion intensity mps in ountsGs exited t VSH nm for luminum nnonE tenns of inresing lengths from IPS to SPS nmF hite retngles re guides to the eyes inditing the position of orresponding nnostruturesF foth exittion nd detetion liner polriztions re set long the x xisF importntlyD is reorded while snning the smpleF his mens tht we do not imge given plsmon mode emitting the rq signl ut rther the overll e0ieny of hrmoni photon genertion nd olletion s funtion of the fous point positionF por exmpleD the PPS nm ntenn exhiits only one loe in ontrst with the twoEloe piture ssoited with the exited dipolr plsmon mode @see inset of pigure PAF hereforeD nerE(eld properties nd frE(eld mesurements ssoited with smple snning in re)etion mode must e disE entngledX the dipolr mode is mximlly exited t the enter of the PPS nm ntennD s shown in pigure P y the sorption urve otined y snning the nnontenn long its long xisD despite the ft tht it shows enhned eletri (elds t the ntenn endingsF etD the di'rtionElimited resolution provided y the immersion ojetive llows resolving nd di'erentiting the sptil rq distriution ssoited with eh single ntenn @one or two loes s shown in pigure QAF he seond outome of pigure Q is relted to the rq enhnement y plsmoni resoE nnesF hile nonEresonnt ntenns generte signls s low s SH photonsGsD RPS nmElong ntenns reh roughly PSH photonsGsD lrgely surpssed y the PPS nmElong ntenns with typilly IVHH photonsGs in the sme experimentl onditionsF sn terms of mgni(tionD this T orresponds to SEfold enhnement from non resonnt to simply resonnt @t the exittion stepA ntenns nd up to QTEfold enhnement in the douly resonnt regimeF his doule resonne ondition n lso e inferred from simulted nerE(eld mps suh s those shown in pigure P for the PPS nmElong ntennX pure dipole nd pure qudrupole re otined t the fundmentl nd hrmoni frequeny respetivelyD whtever the ntenn position with respet to the foused em @dt not shown due to their similrity with those of pigure PAF his is omplementry proof tht the plsmon modes re resonntly exitedD sine otherwise oherent superposition of modes hving distint node numers would hve een oservedF xote tht we did not ttempt to perfetly optimize the nnostruture morphology so tht the net enhnement might e even lrgerD demonstrting tht ompt luminum ntenns re idel ndidte for e0ient nd ompt rq oostersF eounting for the omplex redout of the optil mps @the frE(eld exittion nd the olletion lotions re hnged while snning the smpleA requires omplete simultion of the experimentl on(gurtionF pirstD the fousing nd olletion y optil elements is treted nlytilly @wtl R A y wvelet sumsF 24 his llows ounting for stndrd lenses s well s for lrge xe ojetivesD the ltter produing longitudinl eletri (eld omponents tht re fully tken into ount eyond the prxil pproximtionF eondD the e'et of immersion oil nd sustrte re desried in the wvevetor spe through presnel trnsmission nd re)etion oe0ientsF 24 hirdD full numeril simultion @piwD gomsol R RFQA is performed t eh position of the experimentl Ph mpsD from the exitE tion nd olletion proesses to the integrtion over the detetor surfeF 25 hereforeD due to the lrge omputtion resoure requirementsD the simultions hve een implemented on homeEmde luster in order to llow prmetri investigtion s funtion of wvelengthD polriztionD prtile morphology etF he tretment of the rq response n e found in efF PTDPUD lthough we dopt here di'erent strtegy to estlish the origin of the nonlinerityF snsted of nlyzing the frE(eld rdition digrms s funtion of the inident polriztionD we investigte the U Ph rtogrphy of the rq intensity distriution over single nnostrutureF reneD for eh reorded mpD two simultions hve een performed in order to ount for two distint ontriutions seprtelyX the rekdown of the entrosymmetry t the metl surfe nd the intensity grdients inside the ulk mterilF 28 he orresponding lol surfe nd nonlol ulk polriztionsD P surf,⊥ nd P bulk respetivelyD re given yX

P surf,⊥ (r, 2ω) = χ ⊥⊥⊥ E ⊥ (r, ω)E ⊥ (r, ω)n @IA P bulk (r, 2ω) = γ bulk ∇. [E(r, ω).E(r, ω)] , @PA
where the eletri (elds E(r, ω) re evluted inside the metl nd the polriztion vetor P surf,⊥ led to surfe urrents loted just outside the metlF 29 he susript ⊥ indites omponent long the unit vetor nD perpendiulr to the surfe nd pointing out of the metlF xote lso tht the ulk polriztion P bulk is entirely evluted inside the prtileY it does not t ross the prtile surfe nd therefore does not ontriute in this desription to the nonliner surfe urrentF pollowing ipe et lFD 29 the seond order suseptiility tensor elements ssoited with lol surfe nd nonlol ulk urrentsD χ ⊥⊥⊥ nd γ bulk respetivelyD re de(ned in our model yX 28

χ ⊥⊥⊥ = - a 4 [ r (ω) -1] e 0 mω 2 @QA γ bulk = - d 8 [ r (ω) -1] e 0 mω 2 @RA
where m nd e re the eletron mss nd hrge nd 0 r (ω) is the dieletri funtion of the metl 30 depending on the pulstion ωF a nd d re the dimensionl udnik nd pigure RX wesured nd simulted rq mps otined for single PPS nm luminum ntenn exited t VSH nm with n eletri (eld prllel to the ntenn xis @red rrowsAF he detetion polriztion is hnged etween pnels @A nd @A s indited y the lue rrowsF he rq mps simulted with χ ⊥⊥⊥ lol surfe nd γ bulk nonlol ulk ontriutions re leled urf nd fulkD respetivelyF he intensity pro(les long the dshed green lines in pnels @A nd @A re shown in pnels @A nd @dAF he simulted urves for the χ ⊥⊥⊥ lol surfe @red dshed lineA nd γ bulk nonlol ulk @violet dshed lineA ontriutions re divided y UW @see textA to mth the experimentl dt in pnel @AF he douly resonnt PPS nmElong ntenns were investigted (rst @see pigure RA with n W inident polriztion prllel to their long xisF foth detetion polriztions re mesured simultneously on two distint singleEphoton ounting modules @gwsA in order to ensure the ext position orreltion etween the two mpsF es evidened in pigures R@A nd R@AD the mesured nd simulted rq mps show good greement in terms of intensity distriution nd polriztion responsesF he mismth etween the mesured nd lulted signl levels re due to the optil element trnsmissions nd the sensitivity of the gwsF his ws heked y lirting our setup long the olletion pth @PUFQ7 trnsmissionA nd y tking into ount the dtsheets of the mirosope ojetive @VH7 trnsmissionA nd of the gws @ £ PS7 e0ieny t RSH nmAF purther onsidering the tul vlues of the udnik nd tern prmeters for luminum @see elowA leds to n expeted rtio of pproximtely UH etween simultions nd experimentsD to e ompred to the tul vlue of UW otined y dividing the respetive signls expressed in ountsGsF o e more quntittiveD the mesured nd lulted intensity pro(les long the long xis of the ntenn re shown in pigure R @A nd @dAF he orresponding widths nd polriztion rtios re in perfet greement up to the experimentl unertinties orresponding to the shot noise @see green error rsAX in oth sesD the rq signl olleted for polriztion prllel to the ntenn long xis is IFS times higher thn for rossed polriztionsF wore interesting is the omprison etween the reltive e0ienies of the χ ⊥⊥⊥ lol surfe nd γ bulk nonElol ulk ontriutions otined here using unitry udnik nd tern prmeters of the hydrodynmi nd leled ordingly in pigure RF sn ontrst to n intuitive eliefD oth ontriutions give the sme intensity pro(le entered on the ntennF pirstD s the fundmentl eletri (eld is mximl t the ntenn pexes @dipolr resonneA nd sine the χ ⊥⊥⊥ lol surfe ontriution is proportionl to the eletriE(eld squredD one might hve expeted one loe on eh side of the ntennF es evidened y the white retngles mimiking the ntennsD this is lerly not the se hereD even tking into ount the di'rtionElimited spotsF eondD the intensity pro(les re the sme for the χ ⊥⊥⊥ lol surfe nd γ bulk nonElol ulk ontriutions despite the ft tht the nonliner soures IH hve neither the sme lotions nor the sme intensity dependene @rememer tht the γ bulk nonElol ulk urrent is proportionl to the grdient of the eletri (eld squredAF his is pprently in greement with their inseprility for )t surfes s disussed in efF QPF pigure SX me s in pigure R for single RPS nm ntennF xote tht the intensity pro(les of pnel @dA hve een otined perpendiulrly to the ntenn long xisD s shown y the green dshed line in pnel @AD nd multiplied y ftor of two for etter visiilityF es disussed in efF QQD the γ bulk nonlol ontriution n e inluded in n e'etive surfe ontriutionD so tht the norml omponent of the polriztion sheet redsX

P ef f,⊥ (r, 2ω) = χ ⊥ E (r, ω)E (r, ω) + χ ⊥⊥⊥ E ⊥ (r, ω)E ⊥ (r, ω) + γ bulk (2ω) E(r, ω)E(r, ω) @SA
whih n e rest intoD 33

P ef f,⊥ (r, 2ω) = χ ⊥⊥⊥ -χ ⊥ E ⊥ (r, ω)E ⊥ (r, ω) + χ ⊥ + γ bulk (2ω) E(r, ω)E(r, ω) @TA II
his expression hs een used to show tht the only seprle quntities re χ ⊥⊥⊥ -χ ⊥ nd χ ⊥ +γ bulk / (2ω)D or in other word tht the three quntities χ ⊥⊥⊥ D χ ⊥ nd γ bulk nnot e extrted independently from optil experiments in the frE(eldF etD s disussed in efF QRD the existing models for the nonliner response of metls predit tht χ ⊥ = 0D so tht the polriztion redues to

P ef f,⊥ (r, 2ω) = χ ⊥⊥⊥ E ⊥ (r, ω)E ⊥ (r, ω) + γ bulk (2ω) E(r, ω)E(r, ω). @UA
es the two nonliner urrents re not linked to the eletri (eld in the sme wyD sptil deorreltion etween rq intensity distriutions for the χ ⊥⊥⊥ lol nd the γ bulk nonElol responses should our t lest in spei( on(gurtionsD llowing weighing their reltive ontriution y ompring mesured nd simulted intensity pro(lesF e hve therefore numerilly investigted the rq responses ssoited to χ ⊥⊥⊥ lol nd γ bulk nonElol ontriutions for vrious ntenn spet rtios in order to (nd prmeters where the sptil deorreltion is pronouned nd experimentlly ddressleF es shown in pigure S@AD this is hieved for RPS nmElong ntenns when the detetion polriztion is perpendiulr to the ntenn long xisD leding to di'erent sptil ptterns @see the supnel orresponding to the simultionsAF reneD our simultions unmiguously demonstrte tht the χ ⊥⊥⊥ lol surfe nd the γ bulk nonElol ulk ontriutions re seprle while imging resonnt nd nnoE sized strutures exited y tightly foused emF hereforeD they re not mergele into single e'etive nonliner surfe ontriution s it is the se for plnr surfes illuminted y plnr wvesF 32 xote tht the χ ⊥ lol surfe ontriution would lso produe rq intensity mp distint from those of χ ⊥⊥⊥ nd γ bulk in the rossed polriztion on(gurtion @dt not shownAF st is thus mergele with none of themF roweverD if χ ⊥ ws not zeroD it would e possile to reprodue the rq intensity distriution ssoited with the γ bulk nonlol ulk ontriution y omining the χ ⊥ nd χ ⊥⊥⊥ lol surfe souresD s expeted from iqution T using the pproprite vlues for the nonliner suseptiilitiesF IP hespite the wek mesured signl in the rossed polriztion on(gurtionD the rq intensity pttern lerly mthes the χ ⊥⊥⊥ lol surfe ontriution nd deprts from the γ bulk nonlol ulk oneD the ltter leding donutElike distriution in the simulted mpsF his suggests tht the χ ⊥⊥⊥ lol surfe urrent is the leding rq soure in eluminumF sn order to support this sttementD intensity pro(les were extrted from the mesurements nd simultions long two rossed diretions @see the green dshed lines in pigure SD di'erent from the ones of pigure RAF everl insights into the nonliner ehviors n e ginedF pirstD the signl rtio etween χ ⊥⊥⊥ lol surfe nd γ bulk nonlol ulk ontriutions does depend on the ntenn lengthD iFeF on the tul (eld distriutions nd grdients or the plsmon resonnes involvedF qenerlly spekingD it is therefore not strightforwrdD nor even orretD to ssign leding origin to the nonliner response s the reltive strength of χ ⊥⊥⊥ lol surfe nd γ bulk nonlol ulk ontriutions is strongly 'eted y the prtile morphology nd the exittionGolletion on(gurtionF etD s fr s the desription in terms of udnik nd tern prmeters is relevntD these oe0ients entering in the simultions re onstnt from one struture to nother nd extrtle from the experimentl dt s done in efF PVF rereD it is performed t the single nnostruture level nd tking into ount the e'et of fousing nd olletion y the mirosope ojetive together with the sustrte e'etF e lose inspetion of the simulted nd mesured intensity pro(les long the ntennD see pigure S@AD shows tht the experimentl dt nd the simulted signls for the χ ⊥⊥⊥ lol surfe ontriution nerly vnish t the enter of the luminum ntennD in ontrst to the γ bulk nonlol ulk ontriutionF his implies tht the ltter is lrgely overestimted with d = -1F his onlusion is further on(rmed in the rossed polriztion on(gurtion shown in pigure S@AD where the γ bulk nonlol ulk ontriution leds to donutElike intensity distriution drstilly di'erent from the experimentl dtF st is even lerer with the intensity pro(les tken perpendiulr to the ntenn in pigure S@dAD where the mesured dt perfetly follows the χ ⊥⊥⊥ lol surfe ontriutionsD wheres the γ bulk nonlol ulk response is y fr too lrge @t lest y ftor TFSA nd shows doule ell urve tht is not IQ oserved experimentllyF his disrepny with the hydrodynmi model hs lredy een pointed out for the seondEorder nonliner suseptiility of thin luminum (lmsF 34 king into ount the di'erene in nottions @ ftor of P ompred to iqF RAD the udnik nd tern prmeters were mesured to e |a| = 2.3 ± 0.72 nd |d| = 0.018 ± 0.004 F sf the χ ⊥⊥⊥ lol surfe ontriution is omprle to tht of goldD 28 the γ bulk nonlol ulk urrent is reltively PSH times smller leding to vnishingly smll sttered intensityF e hve lso evluted the χ ⊥ lol surfe ontriution using the udnik nd tern prmeter |b| = 0.0146 ± 0.004 of efF QR @dt not shownA nd otined less thn HFI photonsGsD fr elow the shot noise levelF purthermoreD neither the polriztion rtio of the simulted mps nor the intensity in the enter of the ntenn is omptile with the mesured dt of pigure SD so tht this ontriution n e ruled out in the present experimentsF his lerly supports our (ndings for single luminum nnostruturesD lthough no ler theoretil explntion hs een found to ount for this drsti devition from the hydrodynmi modelF 34 Conclusion sn onlusionD we hve demonstrted tht douly resonnt nd mode mthing onditions n e hieved for rq with single nd ompt luminum nnontenn y djusting its morphologil prmetersF ustntil rq enhnements re otined ompred to simE ply resonnt strutures @typil VEfold mgni(tionA nd to nonresonnt ntenns @nerly RH times lrger rq rteAF he Ph rq exittion mps otined y snning single nnontenn under the mirosope ojetive fous re quntittively ounted for in terms of intensity distriution nd polriztion response y (nite element method simultions tht model the ext experimentl on(gurtion @lrge xeD sustrteD re)etion modeD etF F F AF iven the signl mplitudes expressed in photonGs re found in greement when the trnsE mission losses nd detetor e0ienies re onsideredF sn seond stepD we demonstrte tht the rq soures rising from the χ ⊥⊥⊥ lol surfe nd the γ bulk nonlol ulk onE IR triutions n led to distint intensity distriutions while imging resonnt nnostrutures exited with tightly foused emF hey re therefore seprle provided tht χ ⊥ is ssumed to e zeroD in greement with the existing models for metlsF e further indentify the dominnt ontriution to the nonliner proessX in ontrst to gold nnostruturesD the symmetry reking indued y the surfe is the mjor rq soure in luminum nnontenE nsF reneD luminum nnostrutures n now e on(dently modeled nd used s uilding loks to oost the nonliner e0ieny t the nnosleF st inludes the design of novel hyrid strutures sed on nonliner nd nnosized monorystls emedded in plsmoni ntenns nd their optimiztion through quntittive @in terms of solute ount rtesA simE ultionsF he ltter will llow nswering the longEstnding question of the reltive e0ienies of plsmoniD dieletri nd semionduting mterils s rq genertors t the nnosle for potentil integrtionF heir extension to sum frequeny genertion @pqA is strightforE wrd in the ontext of the developed pproh nd is indeed lredy underwyF st opens new potentilities for ultrfst modultors with optimized designsD whih re furthermore investigle with our urrent experimentl setupF Experimental snvestigting nonliner proesses t the single prtile level requires di'rtionElimited spE til resolutionD high sensitivity nd low noise experimentl setupsF sn order to reh these spei(tionsD our setup hs een designed s followsF he lser soure is femtoseond osilltor iXpphire working t wvelength of VSH nm nd delivering IHH fs pulses t repetition rte of VH wrzF por the mesurementsD the verge input power is set t PRS µW to void dmging the mteril surfeD exept for pigure I for whih it ws redued to IPS µW F he fousing of the lser em is provided y n immersion oil mirosope ojetive @xIHHD numeril perture N A = 1.3AD providing sptil resolutions of QQH nd ITS nm t the fundmentl nd hrmoni frequeniesD respetively @λ/2N AAF xonliner signls IS re olleted in re)etion mode nd seprted from the residul fundmentl em y old mirrorF o overome hromti errtions from the mirosope ojetiveD utomted telesopes hve een set in to overlp the fousing plnes t the exittion nd olletion wvelengths with the surfe of the smpleF he ltter is mounted on set of losedE loop piezoeletri stges for lterl snning with nnometer ury under the foused emF goupled with trking lgorithmD the full omputer ontrol of the setup provides n utomted optimiztion of the mesured signls efore ny set of mesurementsD yielding extremely roust nd reproduile resultsF he olleted light is then spetrlly seleted y monohromtor with wekly dispersive grting @ISH gGmmA nd nlyzed through poE lrizing emEsplitter oupled to two gwsF en ultrEfst quisition rd is synhronized with the lser impulsions to disriminte etween the orrelted rq signl nd rndom noiseD reduing drk ounts ontriutions to few eventsGsF sndividul luminum nnostrutures re frited on glss sustrte y stndrd eleE tron em lithogrphyF e mirosope over slip dpted to our mirosope ojetive is lened in etone efore eing prepred y immersion in nitri id for QH s nd rinsed in deionized wterF en oxygen plsm lening is then performed for P minutes to remove residul orgni ontmintion of the surfeF he smple is susequently heted t IVH

  tern prmetersD 31 equl to I nd EI respetively in the hydrodynmi model pplied to the ondution eletronsF 29 ith these expressions nd tking into ount the rel experimentl onstrintsD the rq simulted intensities re expressed in terms of photonsGs nd n e diretly ompred to mesurementsF V es disussed in detil in efF QPD there re severl other ontriutions oming either from the prtile surfeD χ ⊥ nd χ ⊥ D or from the ulk β bulk E(r, ω) [∇.E(r, ω)] nd δ bulk . [E(r, ω).∇] E(r, ω)F eording to their mesurements on gold plnr sustrtesD their ssoited nonliner suseptiilities re t lest one order of mgnitude smller thn those onsidered in iqsF I nd PD leding to rq signls more thn two orders of mgnitude weker thn the leding ontriutions investigted hereF he rq intensities mesured t the single prtile level eing of the order of few hundreds of photonsGsD they do not llow extrting these extr ontriutions lying elow the shot noise in our experimentsF

  PRA xovotnyD vFY rehtD fF riniples of xnoEyptisF Cambridge U. Press, New York 2012D Pnd iditionF @PSA qloppeD eFY erlotD FY hupontEperrierD iFY iriD eFY onhrlD FY fhelierD qFY inentD FY erizetD yF fidimensionl nnoEoptomehnis nd topologil ktion in nonEonservtive rdition fore (eldF Nat. Nanotech. 2014D 9D WPHEWPTF @PTA futetD tFY fhelierD qFY ussierEentoineD sFY toninD gFY fenihouD iFY frevetD F pF snterferene etween eleted hipoles nd ytupoles in the yptil eondErrmoni qenertion from pheril qold xnoprtilesF Phys. Rev. Lett. 2010D 105D HUURHIF @PUA fhelierD qFY ussierEentoineD sFY fenihouD iFY toninD gFY frevetD F pF wultipolr seondEhrmoni genertion in nole metl nnoprtilesF JOSA B 2008D 25D WSSEWTHF @PVA fhelierD qFY futetD tFY ussierEentoineD sFY toninD gFY fenihouD iFY frevetD FEpF yrigin of the optil seondEhrmoni genertion in spheril gold nnoprtilesX lol surfe nd nonElol ulk ontriutionsF Phys. Rev. B 2010D 82D PQSRHQF @PWA ipeD tF iFY oD F gF FY pukuiD wFY tegemnD qF sF enlysis of PndEhmoni genertion t metlEsurfesF Phys. Rev. B 1980D 21D RQVWERRHPF @QHA kiD eF hFY hjurisiD eF fFY ilzrD tF wFY wjewskiD wF vF yptil properties of metlli (lms for vertilEvity optoeletroni deviesF Appl. Opt. 1998D 37D SPUIESPVQF @QIA udnikD tFY ternD iF eF eondEhrmoni rdition from metl surfesF Phys. Rev. B 1971D 4D RPURF @QPA ngD pF FY odriguezD pF tFY elersD F wFY ehorintD FY ipeD tF iFY uurnenD wF urfe nd ulk ontriutions to the seondEorder nonliner optil response of gold (lmF Phys. Rev. B 2009D 80D PQQRHPF @QQA ipeD tF iF Y wizrhi FY tegemnD F sF pundmentl di0ulty in using seond hrE moni genertion s stritly surfe proeF Phys. Rev. B 1987D 35D WHWIEWHWRF PH @QRA eplinD gF FY ogersD gF F ixperimentl exmple of isotropi surfe seondEhrmoni genertionX dEsputtered irEexposed luminum thin (lmsF Phys. Rev. B 2002D 65D IIF @QSA ghevlierD xFY onnefrudD FY wotteD tFEpFY runtD FY urriD uF epertureEsizeE ontrolled optil (er tips for highEresolution optil mirosopyF Rev. Sci. Instrum.For Table of Contents Use OnlyWave-mixing origin and optimization in single and compact aluminum nanoantennas.Maeliss Ethis de Corny, Nicolas Chauvet, Guillaume Laurent, Mathieu Jeannin, Logi Olgeirsson, Aurélien Drezet, Serge Huant, Géraldine Dantelle, Gilles Nogues, Guillaume Bachelier.The present Table of Content (TOC) Graphic illustrates the setup (reection mode) used to investigate the second harmonic generation (SHG) in single aluminum antennas ontained by e-beam lithography. The nonlinear signal recorded by scanning the sample under a tightly focused beam is quantitatively compared to numerical simulations taking into account the precise experimental conguration.

	mode @see insets of pigure IAF Acknowledgement he uthors knowledge the (nnil support of the egene xtionle de l eherhe @exA qrnt exEIREgiPTE HHHIEHIEsxD the niversité qrenole elpes ghire se @TA xoguezD gF urfe lsmons on wetl xnoprtilesX he sn)uene of hpe nd hysil invironmentF J. Phys. Chem. C 2007D 111D QVHTEQVIWF @UA figioniD FY rungD tFEFY rehtD fF xnontenns for visile nd infrred rditionF Rep. Prog. Phys. 2012D 75D HPRRHPF @VA xovotnyD vFY vn rulstD xF entenns for lightF Nat. Photon. 2011D 5D VQEWHF @ITA wetzgerD fFY quiD vFY puhsD tFY ploessD hFY rentshelD wFY qiessenD rF trong inhneE ment of eond rrmoni imission y lsmoni esonnes t the eond rrmoni velengthF Nano Lett. 2015D 15D QWIUEQWPPF @IUA hionneD tF eF xnoplsmonisX lsmons rok in metl ndsF Nat. Mat. 2013D 12D QVHEQVIF @2006D 77D HTQUHRF Graphical TOC Entry
	nd the vex vexip hh progrmF @WA wierD F eFY etwterD rF eF lsmonisX voliztion nd guiding of eletromgneti @IVA ngD FY hiD rFY ngD xF FY ghengD vF rFY qoD FY rungD vFY tingD F F
	woEhoton vuminesene nd eond rrmoni qenertion from qold wiroEltesF energy in metlGdieletri struturesF J. Appl. Phys. 2005D 98D HIIIHIF @IHA uurnenD wFY ytsD eF F xonliner plsmonisF Nat. Photon. 2012D 6D UQUEURVF References Sensors 2014D 14D IVQPVEIVQQTF
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	i0ient xonliner vight imission of ingle qold yptil entenns hriven y pewE @IIA gn(eldD fF uFY rusuD rFY vukknenD tFY fiD fF pFY uuittinenD wFY urunenD tFY iF iloring nd imging the plsmoni lol density of sttes in rystlline nnoprismsF
	trosymmetri nnodimersF Nano Lett. 2007D 7D IPSIEIPSSF gyle xerEsnfrred ulsesF Phys. Rev. Lett. 2009D 103D PSURHRF uurnenD wF vol (eld symmetry drives seondEhrmoni genertion in nonenE Nat. Mat. 2013D 12D RPTERQPF
	@PA wetzgerD fFY rentshelD wFY vippitzD wFY qiessenD rF hirdEhrmoni spetrosopy nd @PHA figioniD FY fridD hFY rungD tFEFY uernD tFY huoD vFY rehtD fFY pinzziD wFY gerulloD
	@IPA hygrjnD uFY ivierD FY voverD eFY wrtinD yF tF pF inhned seondEhrmoni qF hynmis of pourEhoton hotoluminesene in qold xnontennsF Nano Lett. modeling of the nonliner response of plsmoni nnontennsF Opt. Lett. 2012D 37D RURIERURQF genertion from doule resonnt plsmoni ntenneF Opt. Exp. 2012D 20D IPVTHEIPVTSF 2012D 12D PWRIEPWRUF
	• g @QA eouniD rFY xvrroEgiD wFY hmniD wFY idiropoulosD F F rFY rongD wF rFY @IQA gelernoD wFY uD FY fselliD wFY qroÃmnnD FY figioniD FY votelliD eFY he enE @PIA figioniD FY gelernoD wFY voiniD wFY qrniniD qFY fridD hFY wte(Eemp)iD
	for S minutes efore wwe SHkhGwwe WSHkh iElyer resist is spin oted onto yultonD F pFY wierD F eF wultiresonnt frodnd yptil entenns es i0ient gelisD gFY gerulloD qFY ysellmeD FY rehtD fFY huoD vFY giiD pFY pinzziD wF wode FY wte(Eemp)iD wFY huoD vFY rehtD fFY gerulloD qFY pinzziD wF hependene of the
	its surfeF ih lyer is ked t VH • g for S minutesF en dditionl IH nm thik lyer of luminum is deposited on the resist y eletronEgun evportion to help evuting the mthing in multiresonnt plsmoni nnontenns for enhned seond hrmoni genE twoEphoton photoluminesene yield of gold nnostrutures on the lser pulse durtionF unle xnosoures of eond rrmoni vightF Nano Lett. 2012D 12D RWWUESHHPF ertionF Nat. Nanotech. 2015D 10D RIPF Phys. Rev. B 2009D 80D HRSRIIF
	@RA qinzurgD FY ursvinD eFY onnefrudD FY wurphyD eFY ollrdD F tFY wierD F residul hrges during lithogrphyF efter lithogrphyD the luminum lyer is dissolved in @IRA unightD wF FY uingD xF FY viuD vF pFY iverittD rF yFY xordlnderD FY rlsD xF tF @PPA hengD rF hFY viD qF gFY hiD F pFY yuyngD wFY vnD FY ro(movD F eFY vyskD F eFY ytsD eF F xonlinerly oupled lolized plsmon resonnesX esonnt seondE strong se solutionF he resist is developed in wsfuXse nd (xed y se rinsingF e (nl eluminum for lsmonisF Acs Nano 2014D 8D VQREVRHF wF ize dependent ompetition etween seond hrmoni genertion nd twoEphoton hrmoni genertionF Phys. Rev. B 2012D 86D HVSRPPF QS nm thik luminum lyer is deposited onto the smple using eletronEgun evportionD efore xEmethylEPEpyrrolidone @xwA liftEo' is performed t VH • gF vterl dimensions of the @SA rentshelD wFY tiklD FY qiessenD rFY vippitzD wF untittive wodeling of the hird @ISA unightD wF FY viuD vF pFY ngD F wFY frownD vFY wukherjeeD FY uingD xF FY iverittD luminesene oserved in gold nnoprtilesF Nanotech. 2013D 24D VF
	ntenns @IHH nm in width nd IPS nm to SPS nm in lengthA re mesured using snning eletron mirosope @iwAD while the QS nm thikness is on(rmed y snning the tip of rF yFY xordlnderD FY rlsD xF tF eluminum lsmoni xnontennsF Nano Lett. @PQA xoguesD qFY werottoD FY fhelierD qFY veeD iF rFY ongD tF hF prition nd tuning rrmoni imission petrum of lsmoni xnontennsF Nano Lett. 2012D 12D QUUVE QUVPF 2012D 12D THHHETHHRF of plsmoni optil nnontenns round droplet epitxy quntum dots y thodoE
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	IT IU IV IW PI

PP


HAL is

A large genetic variability in tree architecture has been observed in biparental apple tree populations (Segura et al., 2006) QTLs related to architectural traits have been identified (Segura et al., 2008). These genotypic effets have not been integrated in the apple tree FSPM, MappleT (Costes et al. 2008).

             ( ) ( ) GDD eff RLE t RLE T t  
Main equations used to simulate genotypic effects

Results

Modelling phyllochron and probabilities of sylleptic branching

Model calibration and genome wide prediction

Conclusions

Parameter values (RLE Gdd , a syll , IN_length, Leaf_area) were estimated on 116 genotypes of the bi-parental ('Starkimson' x 'Granny Smith') population, observed in 2007.

Genotyping allowed detecting 3123 SNPs polymorphisms that were used for genome wide predictions.

Genome wide prediction was performed using Ridge Regression Blups (Endelman, 2011;Fodor et al. 2013).

Marker effects were evaluated using a mixed effect model assuming infintisemial effects.

1 2 n Y Y Y             … -1 1 -1 1 -1 1 -1 1 1 -1 1 1 1 -1 1 1 -1 1             … … … … … … 1 2 n g g g             … 2 ( 0

Simulations of phenotypic variability

The MAppleT model simulated large range of 1-year-old apple tree phenotypes (Fig. 2).

Correlations between simulated traits (trunk length, number of branches, number of leaves) were consistent with the correlations for the observed traits (Table 2), suggesting independant genetics controls. Four new parameters (Table 1) with major impact on the first year of growth were included in MAppleT.

The rate of leaf emergence (RLE GDD ) was assumed to depend on the thermal time

Model validation under independant climatic conditions

Trunk length

Observed phenotype

Simulated phenotype

This study corresponds to a first attempt to introduce genome-wide information in a FSPM for a fruit tree.

Even though futher improvements are required to better model genotype x environment interactions, it opens new possibilities for supporting plant breeding by in-silico evaluations of the impact of genotypic polymorphisms on integrative phenotypes.

High correlations and low RMSE between observations and simulations were found for the simulations of integrative traits in 2007 (Fig. 3).

Under environmental conditions not used for model calibration (year 2004), correlations between simulated and observed values were lower (Fig. 4).

The environmental effects presently included in MAppleT, i.e. temperature effect on the phyllochrone and sylleptic branching, are not sufficient to correctly simulate the GxE interactions.

  Using genome-wide predictions in a functional-structural plant model to simulate the genetic variation of 1-year-old apple tree development under contrasted soil water conditions Vincent Migault, Benoît Pallas, Jean-Michel Legave, Yann Guédon, Evelyne Costes To cite this version: Vincent Migault, Benoît Pallas, Jean-Michel Legave, Yann Guédon, Evelyne Costes. Using genomewide predictions in a functional-structural plant model to simulate the genetic variation of 1-year-old apple tree development under contrasted soil water conditions. IEEE International Conference on Functional-Structural Plant Growth Modeling, Simulation, Visualization and Applications (FSPMA 2016), Nov 2016, Qingdao, China. , poster abstracts (96), 2016 IEEE International Conference on Functional-Structural Plant Growth Modeling, Simulation, Visualization and Applications (FSPMA 2016). hal-01400054 Using genome-wide predictions in a functional-structural plant model to simulate the genetic variation of 1-year-old apple tree development Vincent Migault 1 , Benoît Pallas 1 , Jean Michel Legave 1 , Yann Guédon 2 , Evelyne Costes 1 1 INRA, UMR AGAP, 2 Cirad UMR AGAP & Virtual Plants Team. Avenue d'Agropolis, 34000 Montpellier. contact: pallas@supagro.inra.fr

Fig 1 .

 1 Fig 1. Architecture variability in the Starkimson x Granny F1 progeny

  of genotypes at each marker When running simulations, values of MAppleT parameters were estimated using the genetic information provided by the genome wide prediction model. The quality of simulations was assessed by comparing model simulations and observations of integrative plant variables (trunk length, number of sylleptics) in 2007 and under independant climatic conditions in 2004.

Fig 2 .

 2 Fig 2. Graphical outputs of MAppleT simulations for nine apple tree genotypes with contrasted architecture

(Fig 3 .

 3 Fig 3. Comparison of observed and simulated phenotypes in 2004.

Fig 4 .

 4 Fig 4. Comparison of observed and simulated phenotypes in 2004

Table 1 .

 1 New parameters of MAppleT

	( ) P t a syll	syll	1 days n	t i t n days	i RLE	( ) RLE t

Table 2 .

 2 Correlations between architectural traits for observations and simulations

		Nb_Leaves	Trunk length	Nb_Syll
	Nb_Leaves		0.34	0.26
	Trunk length	0.41		-0.09
	Nb_Syll	0.25	-0.03
	Correlations from observed data
	Correlations from simulated data
		Model validation on integrative traits

Fundings

This project is funded by the Agence Nationale de la Recherche (ANR) and by the Deutsche Froschungs-gemeinschaft (DFG).


Diego Moro-Melgar
email: diego.moro-melgar@obspm.fr

Alain Maestrini
email: alain.maestrini@obspm.fr

Jeanne Treuttel
email: jeanne.treuttel@obspm.fr.

Lina Gatilova
email: lina.gatilova@lpn.crns.fr.

Tomas Gonzalez
email: tomasg@usal.es

Beatriz Garcia Vasallo
email: bgvasallo@usal.es

Javier Mateos
email: javierm@usal.es.

T González

Monte Carlo Study of 2-D Capacitance Fringing Effects in GaAs Planar Schottky Diodes

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.



Abstract-Nanometer scale planar Schottky barrier diodes with realistic geometries have been studied by means of a twodimensional ensemble Monte Carlo simulator. The topology of the devices studied in this work is based in real planar GaAs Schottky barrier diodes used in THz applications, such as passive frequency mixing and multiplication, in which accurate models for the diode capacitance are required. The intrinsic capacitance of such small devices, which due to edge effects strongly deviates from the ideal value, has been calculated. In good agreement with the classical models, we have found that the edge capacitance is independent of the properties of the semiconductor beneath the contact and, as novel result, that the presence of surface charges at the semiconductor dielectric interface can reduce it almost 15%. We have finally provided a compact model for the total capacitance of diodes with arbitrary shape that could be easily implemented in design automation software such as ADS.

Index Terms-GaAs Planar Schottky diodes, Monte Carlo (MC) simulation, Edge Effect, Capacitance (C-V), JUICE-SWI.

I. INTRODUCTION

ast advancements have been done in the development of planar Schottky barrier diodes (PSBDs) for THz local oscillators and heterodyne receivers since this technology was introduced in the 90's, thus finally replacing the whiskercontacted Schottky diodes previously used for ultra-high frequency applications [START_REF] Maestrini | A 540-640-GHz highefficiency four-anode frequency tripler[END_REF]. Although several technologies are available for building THz sources and detectors, such as quantum cascade lasers (QCL), heterostructure barrier varactors (HBV) or hot electron bolometers (HEB), presently the Schottky technology is the most widespread used, including applications as frequency multipliers from several tens of GHz to few THz. The good performance of the PSBDbased circuits at room temperature have allowed to implement this technology in ground base and space-borne radio astronomy applications such as the Herschel Space Observatory [START_REF] Maestrini | A 540-640-GHz highefficiency four-anode frequency tripler[END_REF], [START_REF] Cherednichenko | Hot-electron bolometer terahertz mixers for the Herschel Space Observatory[END_REF] and the ALMA observatory [START_REF] Thomas | Application of substrate transfer to a 190 GHz frequency doubler and 380 GHz sub-harmomic mixer using MMIC foundry Schottky diodes[END_REF], [START_REF] Chattopadhyay | An all-solid-state broad-band frequency multiplier chain at 1500 GHz[END_REF]. Recently, we are carrying an intense research related to the Submillimeter Wavelength Instrument (SWI) proposed by the Spatial European Agency (ESA) in the frame of the future Jupiter ICe moons Explorer (JUICE) mission, working in two frequency channels: 540-640 GHz [START_REF] Treuttel | A 520-620-GHz Schottky Receiver Front-End for Planetary Science and Remote Sensing With 1070 K-1500 K DSB Noise Temperature at Room Temperature[END_REF] and 1080-1280 GHz [START_REF] Schlecht | Schottky diode based 1.2 THz receivers operating at roomtemperature and below for planetary atmospheric sounding[END_REF]. For the development of such ultra-high frequency multipliers it is necessary to have a precise description of the experimental values of the capacitance and resistance of the ultra-scaled GaAs PSBDs that are the core of the circuits. Indeed, excellent output power levels have been obtained during the last years by using fitted equations for the experimental C-V and I-V characteristics of the PSBDs [START_REF] Siles | A single-waveguide in-phase power-combined frequency doubler at 190 GHz[END_REF], [START_REF] Siles | A high-power 105-120 GHz broadband onchip power-combined frequency tripler[END_REF]. But such data are not often available, so that analytical models for the electrical characteristics of SBDs are typically used. The problem is that when reducing the size of the diodes for increasing the frequency of operation, the emergence of non-ideal phenomena can affect the accuracy of the models used in the circuit design process and thus dramatically reduce the final efficiency of the multiplication stage [START_REF] Chattopadhyay | An all-solid-state broad-band frequency multiplier chain at 1500 GHz[END_REF], [START_REF] Louhi | On the modeling and optimization of Schottky varactor frequency multipliers at submillimeter wavelengths[END_REF], [START_REF] Grajal | Performance evaluation of multiplication chains up to THz frequencies[END_REF]. The advanced physical models used at JPL (including not only fringing capacitances in the equivalent circuit of the PSBDs, but also carrier inertia, influence on the resistance of the doping-dependent mobility, etc., as explained in [START_REF] Schlecht | Schottky diode based 1.2 THz receivers operating at roomtemperature and below for planetary atmospheric sounding[END_REF] and [START_REF] Schlecht | Harmonic balance optimization of terahertz Schottky diode multipliers using an advanced device model[END_REF]) have allowed to reach output frequencies of 1.5 THz [START_REF] Chattopadhyay | An all-solid-state broad-band frequency multiplier chain at 1500 GHz[END_REF]. But just the use of the simple model for the edge capacitances proposed by Louhi et al. in [START_REF] Louhi | On the modeling and optimization of Schottky varactor frequency multipliers at submillimeter wavelengths[END_REF], jointly with an adequate modification of the value of the series resistance of the diodes (which has to be artificially increased), has allowed the increase of the output frequency of frequency multipliers up to 2.48-2.75 THz [START_REF] Maestrini | Design and characterization of a room temperature all-solid-state electronic source tunable from 2.48 to 2.75 THz[END_REF]. However, none of these models can accurately predict the output power of the circuits, and a good agreement between simulations and measurements can only be obtained if the models are adjusted once the experimental results are available. Therefore, the key point for the correct operation of the fabricated circuits is the experience of the designer in adequately tuning the values of the electrical parameters used in the models of the PSBDs. Since the most important parameter for optimizing the performance of frequency multipliers is the non-linearity of the C-V characteristic of the diodes, we will reassess the validity of the capacitance models used in the design of such applications, D. Moro-Melgar, A. Maestrini, Member, IEEE, J. Treuttel, L. Gatilova, T. González, Senior Member, IEEE, B. G. Vasallo and J. Mateos, Member, IEEE.

Monte Carlo Study of Two-Dimensional Capacitance Fringing Effects in GaAs Planar

Schottky Diodes V which are critical mainly when aiming at high-efficiency or high-power circuits at frequencies of the pump signal above 100 GHz. Also the I-V curve of the devices have to be correctly modeled, but this point is out of the scope of this paper.

We have to stress that the lumped-element circuit (LEC) model is valid in a certain the frequency range, since, even if using correct values for the capacitances and resistances of the PSBDs, such model fails when increasing the input frequency. For this sake, we refer to the results shown in [START_REF] Pardo | An assessment of available models for the design of Schottky-based multipliers up to THz frequencies[END_REF], where a review of the available models for the design and optimization of Schottky diode based multipliers is made. Pardo et al., using an harmonic balance circuit solver coupled to a lumpedelement circuit representation of the diode (LEC-HB), compare the results provided by drift-diffusion and hydrodynamic transport models with those obtained with a one-dimensional Monte Carlo (MC) simulator (also coupled with the HB solver, MC-HB). The MC simulator is considered as a reference when comparing the different models of Schottky diodes, since, by accounting for all the microscopic phenomena taking place within the devices, it provides a precise description of the semiclassical electron transport even under large signal or high frequency excitations. The main conclusion extracted in [START_REF] Pardo | An assessment of available models for the design of Schottky-based multipliers up to THz frequencies[END_REF] is that the operation of Schottkydiode based circuits up to very high frequencies (even above 1 THz) can be correctly described by means of simplified analytical LEC-HB simulators as long as correct values for the resistances and capacitances are used, which can be provided by means of MC simulations.

However, this conclusion is only valid as long as velocity saturation and carrier inertia phenomena are absent, i.e., at frequencies below a certain limit. Fortunately, the effect of velocity saturation can be avoided in experimental applications by reducing the bias and increasing the epilayer doping [START_REF] Grajal | 2D Design of Schottky Diodes[END_REF], thus allowing the LEC models to be valid for well-designed diodes at input frequencies even above 600 GHz (i.e. triplers approaching output frequencies of 2 THz). This result has been confirmed by dynamic simulations of the GaAs PSBDs studied in this work, carried out with our 2D-MC code, in which the influence of velocity saturation phenomena is not significant at input frequencies below 300 GHz. Above this limit, physical models accounting for these phenomena are necessary, as discussed in [START_REF] Schlecht | Harmonic balance optimization of terahertz Schottky diode multipliers using an advanced device model[END_REF].

In this context, the aim of the present work is to calculate, by means of MC simulations, the static capacitance of GaAs PSBDs to be used in the design of THz circuits using LEC-HB models, and also analyze the microscopic origin of the fringing capacitance. To this end we will use the 2D-MC simulator presented in [START_REF] Mateos | Effect of the T-gate on the performance of recessed HEMTs. A Monte Carlo analysis[END_REF], [START_REF] Mateos | Improved Monte Carlo algorithm for the simulation of δ-doped AlInAs/GaInAs HEMTs[END_REF]. From the calculated values of the capacitance we propose a simple compact analytical model for the C-V dependence, accounting for the influence of the surface charges on edge effects (EEs), which correctly describes the obtained results and can be readily included in commercial non-linear HB simulators. This analysis is especially important for high frequency applications, where the anode surface needs to be drastically reduced as the frequency increases and the available power is low, so that a precise design for an improved efficiency is highly demanded. The so-called "edge-effect" becomes important for such small PSBDs and can strongly modify the optimal conversion efficiency point. Previous studies of the EE have been performed by different authors [START_REF] Copeland | Diode edge effect on doping-profile measurements[END_REF], [START_REF] Louhi | The capacitance of a small circular Schottky diode for submillimeter wavelengths[END_REF], but always considering an ideal epilayer-dielectric interface in the proximities of the Schottky contact, i.e. disregarding the depletion region present at the semiconductor surface originated by trapped charges. The aim of this work is to shed light on the microscopic origin of the EE in the capacitance of PSBDs, including the contribution of surface charges. The influence of the epilayer structure and the surface charges at the semiconductor-dielectric interface on the EE capacitance will be quantified in order to provide a compact model that can be easily implemented in design automation software such as ADS, which will allow the precise design of THz MMICs based on GaAs PSBDs.

The paper has been structured as follows. In Section II the physical simulator based on the 2D MC method and the geometry of the PSBDs are introduced. In Section III the analytical model to characterize the charge variations in Schottky diodes derived from MC results is presented, as well as the influence of the surface potential on the depletion region generated by the Schottky contact. A geometrical analysis is also carried out in this section to identify the physical origin of the observed charge variations and associated capacitance. Our main conclusions are finally drawn in Section IV.

II. PHYSICAL MODEL

A. Monte Carlo Simulator

The present study has been performed by using a semiclassical ensemble MC simulator of carrier transport selfconsistently coupled with a 2D Poisson solver. Three nonparabolic spherical valleys (Γ, L, X) are used to model the conduction band of the GaAs semiconductor layers [START_REF] Fischetti | Monte Carlo simulation of transport in technologically significant semiconductors of the diamond and zinc-blende structures. I. Homogeneous transport[END_REF]. Ionized impurity, alloy, polar and nonpolar optical phonon, acoustic phonon and intervalley scattering mechanisms are taken into account, allowing the consideration of hot carrier effects in the proximities of flat-band in Schottky contacts [START_REF] González | Microscopic analysis of electron noise in GaAs Schottky barrier diodes[END_REF]. Fermi-Dirac statistics, using a self-consistent calculation of the Fermi level, are imposed for the occupancy of energy states by means of the rejection technique when selecting the final state after scattering events [START_REF] Mateos | Improved Monte Carlo algorithm for the simulation of δ-doped AlInAs/GaInAs HEMTs[END_REF]. This technique has already been successfully applied for the study of HEMTs [START_REF] Mateos | Monte Carlo simulator for the design optimization of low-noise HEMTs[END_REF]- [START_REF] Mateos | Design optimization of AlInAs-GalnAs HEMTs for low-noise applications[END_REF]. Regarding the contact models, both the Schottky and the ohmic contacts are simulated as in [START_REF] Mateos | Improved Monte Carlo algorithm for the simulation of δ-doped AlInAs/GaInAs HEMTs[END_REF], [START_REF] González | Microscopic analysis of electron noise in GaAs Schottky barrier diodes[END_REF]- [START_REF] Mateos | Design optimization of AlInAs-GalnAs HEMTs for low-noise applications[END_REF]. The Schottky contact is simulated as a perfect absorbing boundary, that is, all the carriers reaching the metal contact leave the structure and no carriers are injected from the metal into the semiconductor. This consideration leads to the modification of the Maxwellian velocity distribution of the electrons at some tens of nm from the Schottky interface to a perfect hemi-Maxwellian distribution at the interface [START_REF] Baccarani | Monte Carlo simulation of current transport in forward-biased Schottky-barrier diodes[END_REF]- [START_REF] Traversa | A generalized drift-diffusion model for rectifying Schottky contact simulation[END_REF]. Regarding the ohmic contact model, it imposes charge neutrality in the proximities of the electrode by injecting carriers with the appropriate thermal distribution (velocity-weighted hemimaxwellian) at the lattice temperature [START_REF] González | Microscopic analysis of electron noise in GaAs Schottky barrier diodes[END_REF].

In this work we focus on the determination of the junction capacitance as a function of the applied voltage (in reverse and forward bias below flat-band conditions) in 2D PSBDs. The junction capacitance in Schottky diodes is associated to bias-induced variations of the depletion region generated by the built-in voltage of the junction and the applied voltage. Additional (bias independent) depletion regions originated by the presence of surface charges at the semiconductor-dielectric interfaces can overlap the previous one, thus affecting the value of the junction capacitance. Such surface charges are accounted for in the simulator by means of the model used in [START_REF] Mateos | Effect of the T-gate on the performance of recessed HEMTs. A Monte Carlo analysis[END_REF], in which the value of the considered surface charge, , is related to that of a surface potential, V S , by:

𝜎 = -√2𝑞𝑁 𝐷 |𝑉 𝑆 |𝜀 𝑆𝐶 , (1)
where N D is the semiconductor doping, ε SC is the permittivity of the semiconductor and q is the electron charge. A frequent criterion to choose the V S value considers that it is the surface potential necessary to bring the Fermi level of the semiconductor near the middle of the bandgap [START_REF] Mateos | Improved Monte Carlo algorithm for the simulation of δ-doped AlInAs/GaInAs HEMTs[END_REF]. We will study the influence of the surface charges by simulating different Fermi-level pinning conditions, with values of V S ranging from 0 V to a maximum of -0.7 V, the half of the GaAs band gap.

To compute the DC capacitance of the PSBDs, we monitor the average number of electrons present inside the diode at every bias point. The intrinsic capacitance is then calculated from the charge variation from point to point as Q/V, neglecting the dielectric capacitance between contacts, whose value is much smaller, and the parasitic contributions of the accesses.

B. Simulated Structure

The MC simulated structures are based in real PSBDs fabricated using the E-beam photolithography LERMA-LPN process, presented in [START_REF] Treuttel | A 330 GHz frequency doubler using European MMIC Schottky process based on e-beam lithography[END_REF], [START_REF] Maestrini | Schottky diode-based terahertz frequency multipliers and mixers[END_REF]. Fig. 1(a) shows an image of a real PSBD used in a frequency doubler at 280 GHz, which is part of the local oscillator chain of the 600 GHz frequency receiver presented in [START_REF] Treuttel | A 520-620-GHz Schottky Receiver Front-End for Planetary Science and Remote Sensing With 1070 K-1500 K DSB Noise Temperature at Room Temperature[END_REF]. The red line indicates the transversal plane where the 2D MC simulated structure has been defined. Taking advantage of the symmetry of the anode, only half of the diode is considered in the simulation domain for reducing the computational requirements. The scheme of the simulated PSBDs is represented in Fig. 1(b), where the characteristic lengths are indicated. The GaAs layer structure consists of a highly doped n + substrate (with doping N S ) and a n epilayer (with low doping N E ). The Schottky contact is placed on the top of the epilayer, while the ohmic contact is deposited on the semiconductor substrate and isolated from the epilayer by etching and dielectric deposition (Si 3 N 4 ), which also passivates the global structure. The simulated surface charges σ are placed at the epilayer-dielectric and substrate-dielectric interfaces [in red in Fig. 1(b)] and their value (for a given V S ) is calculated following (1), according to the doping level of each semiconductor layer.

The simulated geometry resembles as closely as possible the fabricated PSBD. The substrate thickness W Sub and the ohmic contact length L OhmL are large enough to ensure a flat potential profile at the bottom of the structure. The length of the dielectric region L Diel that isolates the ohmic contact from the epilayer is similar to the epilayer thickness W EP , determined by the technological process [START_REF] Kern | Chemical Etching of Silicon, Germanium, Gallium Arsenide and Gallium Phosphide[END_REF], [START_REF] Mori | A New Etching Solution System, H 3 PO 4-H 2 O 2-H 2 O, for GaAs and Its Kinetics[END_REF]. The simulated Schottky anode length L SCH , and the epilayer length between contacts L EP , thickness W EP and doping level N E will be modified to study the influence of the epilayer geometry on the depletion region generated by the Schottky contact. Since this work is focused on the study of the backward junction capacitance of the PSBD, which is especially important in multiplying applications, the epilayer thickness in the simulated structures will be always large enough to avoid the penetration of the depletion region into the substrate layer in the applied bias range [START_REF] Grajal | Modeling and design aspects of millimeter-wave and submillimeterwave Schottky diode varactor frequency multipliers[END_REF].

According to these considerations, two different structures, presented in Table I, have been defined to carry out the study. The DiodeA structure is based on a PSBD used in the frequency mixer presented in [START_REF] Treuttel | A 520-620-GHz Schottky Receiver Front-End for Planetary Science and Remote Sensing With 1070 K-1500 K DSB Noise Temperature at Room Temperature[END_REF] but with a thicker epilayer to allow for higher reverse biasing, while the DiodeB structure is based on the frequency doubler of the same receiver. These structures present a very different Schottky anode size L SCH and a different epilayer doping level N E . The lengths L OhmL and W Subs are sufficiently large to avoid any artificial resistance coming from the simulation of the ohmic contact or the substrate. The geometry of the epilayer (W EP , L EP ) is defined according to its doping level to ensure that the depletion region does not reach the substrate layer or the vertical epilayer-dielectric interface placed at a distance L EP from the edge of the anode. Initially, a value of -0.5 V is considered for the surface potential V S ; then it will be modified in order to analyze its influence on the PSBD capacitance.

III. RESULTS

From the integration of the number of particles inside the diode obtained with the MC simulation for each bias point, we evaluate the variation with the bias of the total charge in the structure (and hence the capacitance), which can be associated to the variation of the depletion region generated by the Schottky contact. The charge variation has been analyzed in bias steps of 0.5 V when strongly reverse biasing the diode and 0.05 V when near flat band conditions. The C-V characteristics of nanometer scale GaAs PSBDs with realistic geometries (as shown in Table I) will then be obtained.

The simulations are also able to show the local contribution of the different regions of the device to the charge variation (and to the total capacitance of the diode) by subtracting the electron concentration in each cell of the mesh structure at two different bias points, as shown in Fig. 2. As clearly observed in the figure, the intrinsic capacitance of PSBDs deviates from the ideal value of a parallel-plate capacitor due to the presence of a depletion region not only below the anode but also around its edge. Moreover, the variations of the bias induced depletion region (and therefore the total junction capacitance) become strongly affected by the presence of surface charges when these are considered in the simulations, Figs. 2(b) and (d), mainly because of a decreased contribution of EEs.

The results of the 2D-MC model for the depleted charge (per unit length in the non-simulated dimension, thus in C/m) will be compared with the ideal value of the charge in the depletion region generated by the Schottky contact in the absence of EEs [START_REF] Sze | Physics of Semiconductor Devices[END_REF]:

𝑄 𝐼𝑑𝑒𝑎𝑙 (𝑉) = -𝐿 𝑆𝐶𝐻 𝑞𝑁 𝐸 𝑊(𝑉) = -𝐿 𝑆𝐶𝐻 √2𝑞𝜀 𝑠𝑐 𝑁 𝐸 (𝑉 𝐵 -𝑉) , (2)
where V B is the built-in voltage of the Schottky contact, V the applied bias and

𝑊(𝑉) = √ 2𝜀 𝑠𝑐 (𝑉 𝐵 -𝑉) 𝑞𝑁 𝐸 ( 3
)
the depth of the depletion region. In the MC simulation, the charge depleted by the bias voltage Q MC is calculated as the difference between the total charge in the diode for a given bias and that present under flatband conditions. The dependence of the depleted charge on the bias (below flatband) obtained with the 2D-MC simulations of DiodeA and DiodeB, normalized to the simulated anode length, is plotted in Fig. 3(a) when a surface potential V S =-0.5 V is considered. While in both diodes the expected √(𝑉 𝐵 -𝑉) dependence of the depleted charge is followed, there is a discrepancy when comparing the MC results with the ideal charge variation given by (2). A larger difference is observed for DiodeA, mainly because it has a smaller size (lower L SCH ). However, if the ideal depleted charge given by ( 2) is subtracted from the MC results, and we remove the normalization by the length L SCH , the resulting difference is practically the same for both diodes, as plotted in Fig. 3(b). Fig. 3(b) also shows that the additional depleted charge due to the EE presents a linear dependence on the bias (in reverse bias, far from flatband conditions), in good agreement with previous predictions [START_REF] Copeland | Diode edge effect on doping-profile measurements[END_REF], [START_REF] Louhi | The capacitance of a small circular Schottky diode for submillimeter wavelengths[END_REF]. This result indicates that the additional charge contribution originated from the 2D geometry of the diodes (EE) is independent of the anode size and doping level of the epilayer, result already obtained in analytical calculations of the electric potential distribution in metal-semiconductor junctions [START_REF] Gelmont | Capacitance-voltage characteristics of microwave Schottky diodes[END_REF], [START_REF] Wasserstrom | The Potential Due to a Charged Metallic Strip on a Semiconductor Surface[END_REF]. Indeed, we have performed simulations in a large variety of diodes and all of them follow this universal behavior. However, it is important to remark that the description of the response of PSBDs at very high frequencies (above the range of validity of the standard LEC approach) requires more than a single capacitance to account for complex non-harmonic effects related to velocity saturation, which do depend on the doping level of the epilayer [START_REF] Pardo | An assessment of available models for the design of Schottky-based multipliers up to THz frequencies[END_REF]. As observed in Fig. 2, this 2D EE observed with our MC model is strongly dependent on the presence or not of surface charges at the semiconductor-dielectric interface. We can therefore analyze the influence of the surface potential of the epilayer on the additional depleted charges associated to the 2D effects by plotting, Fig. 4, the difference between the MC results and the ideal depleted charge (Q MC -Q Ideal ) for different values of V S . As observed, it is possible to conclude that the linear tendency remains for any surface potential considered in the simulation, but with a slope that slightly depends on V S .

According to the previous observations, and in agreement with the classical models [START_REF] Copeland | Diode edge effect on doping-profile measurements[END_REF], [START_REF] Louhi | The capacitance of a small circular Schottky diode for submillimeter wavelengths[END_REF], we can propose a modification of (2) able to account for the 2D effects appearing in PSBDs by including an additional linear term, and considering a dependence of the EE parameter on the surface potential.

𝑄(𝑉) = -𝐿 𝑆𝐶𝐻 √2𝜀 𝑠𝑐 𝑞𝑁 𝐸 (𝑉 𝐵 -𝑉) + 𝛽(𝑉 𝑆 )𝜀 𝑠𝑐 (𝑉 -𝑉 𝐵 ), [START_REF] Chattopadhyay | An all-solid-state broad-band frequency multiplier chain at 1500 GHz[END_REF] where 3(b) and ( 4)] with β(V S ) the dimensionless EE parameter. This parameter was already defined in [START_REF] Copeland | Diode edge effect on doping-profile measurements[END_REF], [START_REF] Louhi | The capacitance of a small circular Schottky diode for submillimeter wavelengths[END_REF] as a phenomenological way of characterizing EEs in PSBDs, but it was taken as an universal constant. In fact, Eq. ( 4) highlights that the EEs do not depend neither on the anode size nor on the doping level of the epilayer. However, our MC simulations clearly show that the value of the EE parameter can be affected by the presence surface charges at the epilayer-dielectric interface. From the fitting of the MC results with (4) we have extracted the dependence of β on V S and plotted it in Fig. 5 for both simulated diodes. We can observe that, in spite of the differences in the geometry and doping levels, DiodeA and DiodeB show very similar values of the EE parameter. This result is coherent with the classic model firstly proposed in [START_REF] Copeland | Diode edge effect on doping-profile measurements[END_REF] and extended in [START_REF] Louhi | The capacitance of a small circular Schottky diode for submillimeter wavelengths[END_REF], in which a parameter called D 1 is proposed for characterizing the EE, with an universal value of 0.36, which coincides with our calculations in the absence of surface charges (since β 0 ≡D 1 =0.72). The decrease of β(V S ) with increasing negative values of V S observed in Fig. 5 is connected with the lower charge variations at the edge of the Schottky contact due to the depletion induced by the surface charges at the semiconductor-dielectric interfaces, as clearly illustrated in Figs. 2(b) and(d).

β(V S )•ε sc is the slope of the representation of Q MC -Q Ideal vs. V-V B [Figs.
We can then propose a second order polynomial approximation for β(V S ) that can be considered to be universal according to the MC results, since it is independent of the properties of the semiconductor beneath the contact, and so is the edge capacitance (as we will show next):

𝛽(𝑉 𝑆 ) = 0.720 + 0.216 • 𝑉 𝑆 + 0.105 • 𝑉 𝑆 2 , (5)
where V S <0 is defined in Volts.

Starting from the analytical expression for Q(V), given in (4), we can straightforwardly extend our analysis to the behavior of 2D EE in the capacitance of PSBDs, and provide a simple analytical expression to obtain the value of the capacitance per unit length of the PSBDs, C(V), as: Eq. ( 6) reveals a deviation from the ideal parallel-plate capacitance, C Ideal (V), due to the presence of a constant contribution to the total capacitance of the PSBD, C EE =β(V S ) sc , originated by the 2D EEs, and whose value is only dependent on the dielectric constant of the semiconductor and the parameter β(V S ).

In Fig. 6, the results of the MC simulations using three different values of V S are compared with the analytical expression [START_REF] Schlecht | Schottky diode based 1.2 THz receivers operating at roomtemperature and below for planetary atmospheric sounding[END_REF] [using the values of β(V S ) provided by eq. ( 5)] and the ideal parallel-plate capacitance C Ideal (V). Due to the EE, 1/C 2 is not a straight line anymore, but it is deviates from the ideal behavior. Such deviation is more pronounced in DiodeA as the anode size is reduced, since the ideal contribution decreases while the contribution of C EE remains constant. The implementation of this capacitance model in a non-linear HB simulator and its usefulness with respect previous models is a matter of a different analysis, but the conclusion obtained in [START_REF] Louhi | The capacitance of a small circular Schottky diode for submillimeter wavelengths[END_REF] has already been verified: the presence of EEs in small anodes leads to a more reactive impedance of the PSBD as the excited voltage signal enters in the inverse region, what leads to a reduction of the conversion efficiency of the multiplier as well as to a modification of the optimal bias.

In conclusion, ( 5) and ( 6) are able to correctly reproduce the results of the capacitance of PSBDs obtained with the MC simulations, so that they could even be used to extract the values of V S from the measurements of the C-V curve of any kind of PSBDs. However, this analysis is extremely difficult, since the parasitic contributions to the intrinsic capacitance of the PSBD should first be deembedded. Indeed, for practical design purposes, EE and parasitic contributions could both be included in [START_REF] Schlecht | Schottky diode based 1.2 THz receivers operating at roomtemperature and below for planetary atmospheric sounding[END_REF] just by modifying the value of β, so that this simple compact model for the PSBD capacitance can be easily implemented in commercial circuit simulators like ADS.

We must remark that our study of EEs has been performed with a 2D representation of a rectangular Schottky anode, assuming the diode is homogenous in the non-simulated dimension. This means that in a realistic case of a Schottky contact with a given geometry, to calculate the additional contribution of EEs to the absolute diode capacitance, C EE should be multiplied by the total length of the diode contour. Additionally, in the case of circular anodes a further correction to the value of the capacitance is necessary to account for the circular shape of the EE depletion region, consisting in a new term in (6) which involves a second EE parameter, D 2 [START_REF] Louhi | The capacitance of a small circular Schottky diode for submillimeter wavelengths[END_REF]. The value of D 2 (0.34 if no surface effect is considered) has in turn to be modified also by the effect of V S in a factor [β(V S )/β 0 ] 2 (the square exponent appears when integrating (6) in a circular geometry).

As a result, extending the model proposed in [START_REF] Louhi | The capacitance of a small circular Schottky diode for submillimeter wavelengths[END_REF] to the case of a general shape and considering surface effects, the absolute capacitance of a PSBD can be calculated as: which is the result of the addition of the ideal parallel-plate term (proportional to the area of the anode, A), the EE term (proportional to the length of the contour of the anode, L Contour , and independent of the epilayer doping) and a third term associated to the circular sections of the EE depletion region (independent of the anode geometry but dependent on the biasing and epilayer doping, as it is proportional to W). Notice that circular sectors are not only present in the depletion region of circular anodes, but also at the corners of rectangular ones, and their contribution to the total capacitance is not accounted for by the first two terms in [START_REF] Siles | A single-waveguide in-phase power-combined frequency doubler at 190 GHz[END_REF]. Therefore, the third term must be included even in the case of rectangular geometries. Indeed, such a term does not contain any dependence on characteristic parameters of a circular geometry. The third contribution to the capacitance is not present in our MC simulations, since the 2D approach implies homogeneity in the non-simulated dimension, and therefore absence of any circular section in the EE depletion region in such a direction.

𝐶 𝑇 (𝑉) =

IV. CONCLUSIONS

By means of 2D MC simulations of PSBDs, we have analyzed the deviations of the internal charge variation and the associate intrinsic capacitance with respect to the ideal 1D behavior. The 2D shape of the depletion region beneath the Schottky anode leads to an excess of depleted charge (EE) which increases linearly with respect to the reverse applied voltage, thus contributing with a constant term to the global capacitance of the PSBD (characterized by an EE parameter, β), which is proportional to the length of the contour of the anode but independent of the doping level of the epilayer. By modeling the surface charges at the epilayer-dielectric interface (characterized by the value of its surface potential V S ), we have evidenced a dependency of β on V S , leading to a reduction up to a 15% from its nominal value β 0 (with a dependence that can be approximated by a second order polynomial equation). The compact model provided by the analytical eqs. ( 5) and ( 7) can be easily implemented in LEC-HB simulators (with correct values of the series resistance obtained with a 2D current model) for an accurate prediction of the diode response in multiplying applications up to, at least, 300 GHz LO input signal [START_REF] Louhi | The capacitance of a small circular Schottky diode for submillimeter wavelengths[END_REF].

Fig. 1 .

 1 Fig. 1. (a) Image of a real PSBD fabricated by LERMA-LPN-CNRS and (b) scheme of the MC simulation domain based in the real devices.Table I. Physical and geometrical parameters of the two simulated PSBDs. Symbol DiodeA DiodeB L SCH (nm) 200 1500 W EP (nm) 180 350 L EP (nm) 230 460 L Diel (nm) 120 240 L BC (nm) 350 700 W Subs (nm) 350 500 L OhmL (nm) 300 500 N E (cm -3 ) 3•10 17 1•10 17 N S (cm -3 ) 5•10 18 5•10 18 V B (V) 0.745 0.695

Fig. 2 .

 2 Fig. 2. Map of the local contribution to the total capacitance per unit length (calculated as the variation of the charge per unit length between the bias points V=VB and V =-2.0 V, divided by the voltage difference) for both (a) and (b) DiodeA and (c) and (d) DiodeB. The MC simulations are performed considering (a) and (c) a null surface potential and (b) and (d) VS=-0.5 V. The depth of the depletion region calculated within the total depletion assumption, W, is shown, and the rectangular shaded region indicates the depletion region

Fig. 3 .

 3 Fig. 3. (a) Depleted charge per unit surface, obtained by normalizing by LSCH the charge per unit length obtained with 2D-MC simulations, compared with the ideal one (dashed lines) for DiodeA and DiodeB. (b) Excess charge per unit length depleted at the edges of the Schottky contact due to 2D phenomena calculated by subtracting the ideal charge given by eq. (2) from the 2D-MC result. A surface potential VS=-0.5 V has been considered in these simulations.

Fig. 4 .

 4 Fig. 4. Charge per unit length depleted by the Schottky contact in excess of the ideal value given by eq. (2) (QMC-QIdeal) for (a) DiodeA and (b) Diode B when considering VS= -0.1, -0.3, -0.5 and -0.7 V.

6 )Fig. 5 .

 65 Fig. 5. EE parameter variation when considering a surface potential sweep from 0 V to -0.7 V for DiodeA and DiodeB. An analytical approximation of β(VS) in the considered range is included.

Fig. 6 .

 6 Fig. 6. Comparison between the values of 1/C 2 in (a) DiodeA and (b) Diode B obtained with the MC simulations (symbols) and the analytical expression of eq. (6) (dashed lines) when considering VS=-0.1, -0.5 and -0.7 V. The solid black line represents the ideal parallel-plate capacitance of Schottky diodes.

This work was partially supported by the Labex ESEP (N° 2011-LABX-030) and the ANR program "investissements d'avenir" through the "initiative d'excellence" PSL* (convention ANR-10-IDEX-0001-02), the Dirección General de Investigación Científica y Técnica within the Ministerio de Economía y Competitividad under Project TEC2013-41640-R and by the Consejería de Educación de la Junta de Castilla y León under Project SA052U13.