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Journal Logo 00(2015)1–8 5 A scattering model of 1D quantum wire regular polygons 1 0 2 n Cristian Estarellasa, Llorenç Serraa,b, ∗ a J aInstitutdeF´ısicaInterdisciplina`riaideSistemesComplexosIFISC(CSIC-UIB),E-07122PalmadeMallorca,Spain 0 bDepartamentdeF´ısica,UniversitatdelesIllesBalears,E-07122PalmadeMallorca,Spain 3 ] l l a h Abstract - s Wecalculatethequantumstatesofregularpolygonsmadeof1Dquantumwirestreatingeachpolygonvertexasascatterer. The e vertexscatteringmatrixisanalyticallyobtainedfromthemodelofacircularbendofagivenangleofa2Dnanowire.Inthesingle m modelimitthespectrumisclassifiedindoubletsofvanishingcirculation,twofoldsplitbythesmallvertexreflection,andsinglets . withcirculationdegeneracy. Simpleanalyticexpressions oftheenergyeigenvalues aregiven. Itisshown howeachpolygon is t a characterizedbyaspecificspectrum. m Keywords: quantumwires,scatteringtheory - d n 1. Introduction o c [ Nanowiresarealong-lastingtopicofinterestinnanosciencefortwomainreasons.Oneis,undoubtedly,theiruse aselectronwaveguidesindevicesandtechnologicalapplications. Theotheristhepossibilityofusingnanowiresto 1 artificiallycontrolquantumpropertiesatthenanoscale,thusimprovingourfundamentalunderstandingandprediction v capabilities. Innanowiresmadewith2DelectrongasesofsemiconductorslikeGaAsquantumbehaviormanifestsat 8 0 amesoscopicscale,beyondtheatomisticdescription,anditcanbedescribedwitheffectivemassmodelsandsmooth 7 potentials[1,2]. 7 The quantum states on nanowire bends attracted much interest some years ago [3, 4, 5, 6, 7, 8, 9, 10]. It was 0 shownthatboundstatesformonthebend,andthetransmissionandreflectionpropertiesasafunctionoftheenergy . 1 andwidth ofthewire werethoroughlyinvestigated. Inthesingle-modelimitofa 2D wirea bendcan bedescribed 0 withaneffective1Dmodelcontaininganenergy-dependentpotential[6]. Inthecaseofacircularbend,thesituation 5 becomessimplerandanalyticalapproximationsusinga squarewell whosedepthand lengthare fixedbythe radius 1 : andangleofthebendweresuggested[5]. v Motivatedbythe abovementionedinterestonnanowirebendswestudyin thisworkpolygonalstructuresmade i X of 1D quantum wires (Fig. 1). We focus on the single-mode limit and describe each vertex as a scatterer. The r vertexscatteringmatrixistakenfromthemodelofcircularbendsin2Dnanowires. Wefindthatthespectrumfully a characterizesthe polygonstructure. Two kindsof states are present: a) doublets, with a small energysplitting due toreflectionandwithavanishingcirculationalongtheperimeterofthepolygon;andb)singlets,withanunderlying degeneracybywhichthecirculationcantakearbitraryvalueswithinagivenrange.Theenergysplittingofthedoublets depends on the reflection probability of the vertex, while the number of singlets between consecutive doublets (in Correspondingauthor. ∗ Emailaddress:[email protected](LlorençSerra) 1 C.EstarellasandL.Serra/ 00(2015)1–8 2 a) 3 ℓ ... 1 x 2 b) i+1 a(i) r a(li) b(ri) i−1 b(i) i l Figure1. a)Sketchoftheregularpolygonsmadeof1Dnanowires.Thevertexlabelingandthedefinitionofalongitudinalcoordinatexisindicated onlyforthetriangle.b)Scatteringamplitudesasdefinedforagivenvertexi. energy)canbeusedtoinferthenumberofvertices. Theenergyscale,settingtheseparationbetweenstates,isfixed bythelengthofthenanowireformingthesidesofthepolygon. We finally remark that, although our approachis on physicalmodeling of nanostructures, there is a connection withthemoremathematicallyorientedstudyofquantumgraphs[11].Indeed,fromthisperspectiveourresultscanbe viewedasaparticularapplicationtothecaseofregularpolygonsofthemoregeneralquestionofreconstructingthe graphtopologyfromtheknowledgeofthespectrum. 2. Theoreticalmodelandmethod We consider a regular polygonwith N vertices, made of 1D quantum wires of length ℓ (Fig. 1). The distance v along the perimeter is measured by variable x, with origin arbitrarily taken on the first vertex. The wave function betweenverticesiandi+1isasuperpositionofleft-andright-wardpropagatingplanewaves ψ(x) = a(ri)e−ip(x−xi)+b(ri)eip(x−xi) = a(li+1)eip(x−xi+1)+b(li+1)e−ip(x−xi+1) , (1) where the first and secondequalities take the referencepointon verticesi and i+1, respectively. The a(i) and b(i) l/r l/r coefficientsarethecharacteristicinputandoutputscatteringamplitudesdefinedinFig.1b. Weareassumingasingle propagatingmodeofwavenumberp(seeAppendix A)andthesetofvertexpositionsis x,i=1,...,N . i v { } Thei-thvertexscatteringequationrelatesoutputandinputamplitudesas b(i) r t a(i) l = l , (2)  b(ri)  t r ! a(ri)  where t and r are the usual transmission and reflection complex coefficients. The scattering matrix in Eq. (2) is summarizing the physical effect of the vertex by means of two complex quantities, t and r. As they are required inputsinourmodel,wewilltakethesevaluesfromtheknownscatteringmatricesofcircularbendsin2Dnanowires. Actually,asdiscussedinAppendix A,inthesingle-modelimitthereisananalyticaldescriptionofthescatteringby acircularbendintermsofaneffectivesquarewelldependingontheradiusandangleofthebend[5]. Thereisarelationbetweeninputandoutputamplitudesofsuccessivevertices, b(i) = a(i 1)e ipℓ , (3) l r− − b(i) = a(i+1)e ipℓ . (4) r l − Thephasee ipℓ inEqs.(3)and(4)appearsduetotheassumptionoftwodifferentreferencepointswhenconsidering − thescatteringprocessesfromtwoconsecutivevertices. 2 C.EstarellasandL.Serra/ 00(2015)1–8 3 (cid:0)(cid:3)(cid:4)(cid:5) (cid:15) (cid:0)(cid:3)(cid:4)(cid:16) (cid:2) (cid:14) (cid:1) (cid:0)(cid:3)(cid:4)(cid:12) (cid:0)(cid:3)(cid:4)(cid:17) (cid:2) (cid:1) (cid:12)(cid:13)(cid:1)(cid:1) (cid:12)(cid:13)(cid:1)(cid:12) (cid:12)(cid:13)(cid:2)(cid:1) (cid:12)(cid:13)(cid:2)(cid:12) (cid:12)(cid:13)(cid:1)(cid:12) (cid:12)(cid:13)(cid:2)(cid:1) (cid:12)(cid:13)(cid:2)(cid:12) (cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11) (cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11) Figure2. Energydependenceofthe measureinarbitraryunits(black)andthecirculation inunitsof~p/m(gray-red). Theunitofenergyis ~2/md2. Thewavenumber pandtheFwireparameterdaredefinedinSec.2. Asindicated,eCachpanelcorrespondstoapolygonwithadifferent numberofverticesNvbutthesamearmlengthℓ =30d. Thefigureshowsthepresenceofdoubletswithvanishingcirculationandsingletswith maximalpositivecirculation 0. Noticethatattheenergiesofthesingletsthecirculationcantakevaluesfrom 0 to+ 0. Otherparameters: R=1.3d. C −C C Wedefinethecirculation ~p a(i)2 b(i)2 N ℓ, (5) C≡ m (cid:18)(cid:12) l (cid:12) −(cid:12) l (cid:12) (cid:19) v (cid:12) (cid:12) (cid:12) (cid:12) independentonthechoiceofvertexiduetofluxconse(cid:12)rva(cid:12)tion.(cid:12)Th(cid:12)ecirculationisalsothesameifonechoosestheright coefficients(a(i) and b(i)) instead of the leftonesin Eq. (5). Physically, the circulationis measuringthe probability r r currentflowingalongthepolygoninaparticularstate,characterizedbyasetofaandbcoefficientsandawavenumber p(correspondingtoanenergyE). InAppendix Bwedefineanenergy-dependentmeasure (E),suchthatitvanishesfortheeigenenergiesofthe F polygon. Inpractice,wescannumericallythevaluesof (E)inordertodeterminetheeigenenergieswithinagiven F energyinterval. 3. Results Figure2 showstheenergydependenceof and forpolygonalstructuresfrom3 to 6 verticesfora represen- F C tative set of parameters. As mentionedabove, the energiesfor which vanishes are the physicaleigenenergiesof the polygon. We considered a representative energy interval above thFe threshold ε = ~2π2/(2md2). The energy 1 ε correspondsto the first transverse mode of the 2D nanowire of width d from which our 1D model derives (see 1 Appendix A). From Fig. 2 we notice that the polygonspectra are characterized by a sequence of doublets, with small energy splittingsandwithvanishingcirculation . Inadditiontothedoublets,thereareavaryingnumberofsingle-energy modes(singlets) with finite lying in beCtween doublets. For N = 3, for instance, doublets around5.02 and 5.13 v C withtwointermediatesingletsareseenintheupperleftpanel. Remarkably,doubletsatthesameenergiesarepresent in polygons with odd and even number of vertices. Polygons with an even number of vertices, however, possess additionaldoubletsatintermediateenergies. Singlets in Fig. 2 are characterized by having N ~p/m . Notice, however, that different values of v 0 C ≈ ≡ C thecirculationarepossibleforeachsingletenergy.Indeed,withthealgorithmofAppendix Bwefindthatapositive circulation isobtainedifrequiringa(i) =1(foranarbitraryi)andnegativecirculation ifrequiringa(i) =1.This C0 l −C0 r indicatesthatforeachsingletthecirculationcanactuallytakeanyvaluewithintherange < < byadequately 0 0 −C C C superposingthesolutionsforpositiveandnegativecirculations. Wehaveexplicitlycheckedthispossibilityfromthe 3 C.EstarellasandL.Serra/ 00(2015)1–8 4 Table1.Differenttypesofmodesinvanishing-circulationdoublets.Thepairsformingdoubletsare(I,II)and(III,IV). type left-right successivevertices equation I a =a a(i) =a(i+1) t+r=e ipℓ l r − II a = a a(i) =a(i+1) t r=e ipℓ l r − − − III a =a a(i) = a(i+1) t+r= e ipℓ l r − − − IV a = a a(i) = a(i+1) t r= e ipℓ l r − − − − − numericalsolutions. Thisbehaviormanifeststheconnectionbetweenthe symmetrybreakingandthe -degeneracy C ofthesinglets. A qualitative difference is seen when comparingleft and right panels of Fig. 2. Polygonswith an odd number of verticeshave N 1 singletsin between vanishing-circulationdoublets, while polygonswith an evennumberof v − verticesonlyhaveN /2 1. Thisexplicitlyshowsthatbyknowingthespectrumonemaycharacterizethepolygonal v − structure. Forinstance,aspectrumconsistingofasequenceofmodedoubletswiththreeintermediatesingletswould correspondtoanoctagon. Forthespectrawithanevennumberofintermediatesingletsaconfusionmightarisebetweenthepolygonswith N and2N vertices. Forinstance, boththetriangleandthehexagoninFig. 2havetwo intermediatesinglets. Still, v v onemay differentiatethe two situationswith the discussion on doubletsof nextsubsection. In essence, forodd-N v polygons all doublets are similar in that each polygon side contains an even number of half wavelengths. On the contrary,foreven-N polygonsconsecutivedoubletsalternatefromaneventoanoddnumberofhalfwavelengths.As v shownbelow,thisdifferencecanbeseencountingthenumberofdensitymaximaonasideofthepolygon. 3.1. Doublets Thestates with vanishingcirculationare characterizedby having a = b on eachside ofthe polygon. Besides | | | | vanishingcirculation,doubletshaveadensitythatremainsinvariantbyatranslationfromonepolygonsidetothenext one(Figs.3and4). Thishappensbecausethescatteringamplitudesofsuccessiveverticesfulfilleithera(i) =a(i+1) or a(i) = a(i+1). Atthesametime, foragivenvertexthecoefficientsalsofulfilla = a . Therefore,thesevanishing- l r − ± circulationstatescanbeclassifiedinfourtypesassummarizedinTab.1. Table1alsogivesthesecularequationdeterminingthewavenumber pandtheenergyoftheeigenmodeforeach typeofmode. Notice thatsolutionsof typesI andIIIare associatedto stateswith maximaldensityoneachvertex, while types II and IV have minimal density on the vertex, cf. Figs. 3 and 4. The split pair forming a doublet are thereforecomposedbysolutionsoftypes(I,II)and(III,IV).ItisalsoworthstressingthatsolutionsoftypeIIIandIV arenotallowedinpolygonswithanoddnumberofsides. Thereasonismosteasilyunderstoodassumingr 0and t 1. Inthiscase, thesecularequationfortypesIIIandIVsimplifiesto eipl = 1; thatisℓ = (2n+1)λ/2≈andan ≈ − odd numberof half wavelengthsshould fit in a polygonside. For odd valuesof N this is not compatiblewith the v additionalconditionthatthefullperimetershouldcontainandintegernumbern ofwavelengths,N ℓ=nλ. ′ v ′ SimpleanalyticalapproximationsforthedoubletsplittingscanbeobtainedfromthesecularequationsinTab.1 assumingt 1andr r eiφ. Thisleadstotheconditions(n=1,2,3...) 0 ≈ ≈ pℓ=2nπ r sinφ, (I,II), pℓ=(2n+∓1)0π r sinφ, (III,IV). (6) 0 ∓ For both kinds of pairs the splitting in wave numbers is ∆p = 2r sinφ/ℓ, with an associated energy splitting 0 ∆ = (~2/m)p∆p. Equation (6) clearly shows that in (I,II) doublets, the solution of type I decreases its energydue toreflectionwhilethatoftypeIIincreases,andanalogouslyfor(III,IV)doublets. WerecallthattypeIandIIIsolu- tionsarethosewithamaximaldensityoneachvertex. Finally,we noticethatastheenergyincreasesthereflection coefficientisreduced,andsothedoubletsplittingisalsoreduced,ashintedalsoinFig.2. Summarizing,theenergiesofthezero-circulationdoubletsareapproximately ~2π2n2 n=1,2,3,...(evenN ), E =ε + ∆, v (7) n± 1 2mℓ2 ± ( n=2,4,6,...(oddNv). 4 C.EstarellasandL.Serra/ 00(2015)1–8 5 Physically, our result of Eq. (7) says that the arm length ℓ determines the separation between successive doublets, whilethereflectionpropertiesofthevertexdeterminethesplitting∆ofthepairformingadoublet. 3.2. Singlets Themostremarkablefeatureofsingletsisthatthedensityisnotequivalentondifferentsidesofthepolygon. As mentionedabove,foreachsingletdifferentsolutionswithcirculationrangingfrom to+ ,where N ~p/m, 0 0 0 v −C C C ≈ arepossible.TheintermediatepanelsofFigs.3(energiesof5.049and5.086)andinFig4(5.042and5.097)showthe densitiescorrespondingtosingletsforthetriangleandsquare.Continuumanddashedlinesareforstateswithmaximal andvanishingcirculation,respectively. We notice thatforstates with nonvanishing , the densityfor positiveand C negativecirculationscoincide. For nonvanishingcirculation the density doesnotvanish at any point, but oscillates aroundafinitemeanvalue.Inallsinglets,differentbehaviorsareseenonthesidesofthepolygon. Thesingletenergiesandtheirunderlying -degeneracycanbeeasilyunderstoodinthelimitofvanishingreflec- C tion. Indeed,whenr 0andt 1thesolutionstoEqs.(2)and(3)decoupleforpositiveandnegativecirculations. Theinputcoefficients≈forvertex≈kreadinthiscase a(k) = αeipℓ(k 1) , ( <0), r − C (8) a(k) = βe ipℓ(k 1) , ( >0), l − − C whereαandβarearbitraryindependentconstantsthatcanbefixedbynormalization. Amaximalcirculationstateis obtainedwheneitherαorβvanish.Arbitrarysuperpositionscanbeformedforα,0andβ,0,includingavanishing circulationstateforα=β. ThecyclicconditioninEq.(8),eipℓNv =1,leadstotheenergies ~2π2(2n)2 E =ε + , (n=1,2,3,...). (9) n 1 2mℓ2N2 v Theseenergiescoincidewithadoubletcondition[Eq.(7)for∆=0]eachtimenisanintegernumberoftimesN . It v canbeeasilyverifiedthatEq.(9)impliesthatforodd-N polygonsthereareN 1singletsbetweentwosuccessive v v − doubletsandthatforeven-N polygonsthereareN /2 1,inagreementwiththeresultsofFig.2 v v − 4. Conclusions Amodelofclosedpolygonsmadeof1Dquantumwireshasbeenpresentedwhereeachvertexisdescribedasa scatterer. The vertex transmission and reflection coefficients have been described with the model of circular bends in 2D wires. The polygon spectra are characterized by a sequence of doublets, with a small energy splitting, and a typicalnumberof singlets lyingin betweendoublets. Odd-N polygonshave N 1 singlets while even N have v v v − N /2 1singletsinbetweentwodoublets. Thedoubletsplittingsarecausedbythereflectiononvertices. Doublets v − have a vanishing circulation and singlets can have circulations ranging from a negative to positive characteristic values.Approximateanalyticalexpressionsforthesingletanddoubletenergieshavebeengivenandthecorresponding densitiesdiscussed. Ourresultsexplicitlyshowhowthepolygoncharacteristicscanbeinferredfromthespectrum. Acknowledgement C.E. gratefully acknowledges a SURF@IFISC fellowship. This work was funded by MINECO-Spain (grant FIS2011-23526),CAIB-Spain(Conselleriad’Educacio´,CulturaiUniversitats)andFEDER. Appendix A. Scatteringbya2Dcircularbend We considera 2D nanowirewith a circularbend of angle2θ and radiusR, as sketchedin Fig. A.5. Notice that θ,definedashalfthebendangle,isrelatedinourmodeltothenumberofpolygonverticesbyθ = π/N . Thelateral v extensionofthenanowireisd, settingtheenergyofthefirsttransversemodetoε = ~2π2/(2md2). FollowingRef. 1 [5],itispossibletoderiveananalyticalexpressionforthereflectionandtransmissioncoefficientsofthecircularbend inthesingle-modelimit. 5 C.EstarellasandL.Serra/ 00(2015)1–8 6 5.019 5.021 (cid:18)(cid:19)(cid:18)(cid:21) (cid:18)(cid:19)(cid:18)(cid:20) (cid:18)(cid:19)(cid:18)(cid:18) 5.049 5.086 (cid:18)(cid:19)(cid:18)(cid:21) (cid:25)(cid:26)(cid:27)(cid:28) (cid:23)(cid:24) (cid:22) (cid:18)(cid:19)(cid:18)(cid:18) 5.126 5.132 (cid:18)(cid:19)(cid:18)(cid:21) (cid:18)(cid:19)(cid:18)(cid:20) (cid:18)(cid:19)(cid:18)(cid:18) (cid:18) (cid:30)(cid:18) (cid:31)(cid:18) (cid:18)(cid:18) (cid:30)(cid:18) (cid:31)(cid:18) (cid:18) (cid:29) (cid:29) Figure3. DensitiesforthetrianglemodesofFig.2. Alldistancesaremeasuredinunitsofd(definedinAppendix A).Upperandlowerpanels areforvanishing-circulationdoubletsoftype(I,II).Intermediatepanelsaresingletswithmaximal(solid)andvanishing(dashed)circulation. The energyofthemode(asinFig.2)isindicatedineachpanel.Thepositionofverticesisgivenbytheverticallines.RestoftheparametersasinFig. 2. Defininga = R¯θ,whereR¯ = √R(R+d)isanaverageradius,the2Dscatteringprobleminthesinglemodelimit isapproximatedbya1Dquantumwellofwidth2aanddepthV = ~2/(8mR¯2). Theanalyticalexpressionsofthet 0 andrcomplexcoefficientsaregivenintermsoftwowavenumbers(E−>ε ) 1 2m p = (E ε ), (A.1) r~2 − 1 2m q = (E ε V ). (A.2) r~2 − 1− 0 Specifically,theyread 4pq t = , (A.3) (p+q)2e2i(p q)a (p q)2e2i(p+q)a − − − 2i(p q)(p+q)sin(2qa) r = − . (A.4) −(p+q)2e2i(p q)a (p q)2e2i(p+q)a − − − Noticethat,asexpected,forlargeenergiesitis p qandEqs.(A.3)and(A.4)thenyieldt 1andr 0. Wealso stress that, due to symmetry, the transmission and≈reflection coefficients for left and right i≈ncidences≈are identical, t =t,r =r,leadingtothescatteringmatrixofEq.(2). ′ ′ Appendix B. Algorithm Equations(2),(3)and(4)aresetupasahomogoneouslinearsystem a M =0, (B.1) b ! 6 C.EstarellasandL.Serra/ 00(2015)1–8 7 0.02 5.016 5.022 0.01 0.00 5.042 5.097 y nsit 0.01 de 0.00 5.066 5.071 0.01 0.00 0 30 60 90 1200 30 60 90 120 x x Figure4.SameasFig.3forthesquaremodes.Upperpanelsarethefirstpair[type(I,II)],whilelowerpanelsarethesecondpair[type(III,IV)]in theresultsofFig.2forthesquare.Thetwocentralpanelsaresingletswithmaximal(solid)andvanishing(dashed)circulation. d 2θ y R x R+d FigureA.5.Circularbendina2Dnanowire. wherewedefinethevector aT a(1)...a(Nv)a(1)...a(Nv) , (B.2) ≡ l l r r (cid:16) (cid:17) withananalogousdefinitionforb. NontrivialsolutionsofEq.(B.1)occuronlyforenergiessuchthatthedeterminant ofMvanishes,det(M)=0. Inpractice,wearbitrarilypickoneofthecomponentsofa,saya(β),andfixitto1,transformingEq.(B.1)intothe α inhomogeneousproblem a M =R, (B.3) ′ b ! where RT (0...010...0...0), (B.4) ≡ withthevalue1onthepositionofthechosencomponenta(β). 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