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SINGLE WALL NANOTUBES: ATOMIC LIKE BEHAVIOUR AND MICROSCOPIC APPROACH S. Bellucci 1 and P. Onorato 1 2 1INFN, Laboratori Nazionali di Frascati, P.O. Box 13, 00044 Frascati, Italy. 2Dipartimento di Scienze Fisiche, Università di Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy (Dated: February 2, 2008) 5 Recent experiments about the low temperature behaviour of a Single Wall Carbon Nanotube 0 (SWCNT)showedtypicalCoulombBlockade(CB)peaksinthezerobiasconductanceandallowedus 0 toinvestigatetheenergylevelsofinteractingelectrons. Otherexperimentsconfirmedthetheoretical 2 prediction about the crucial role which the long range nature of the Coulomb interaction plays in thecorrelated electronic transport through a SWCNTwith two intramolecular tunneling barriers. n Inordertoinvestigatetheeffectsonlowdimensionalelectronsystemsduetotherangeofelectron a electron repulsion, we introduce a model for the interaction which interpolates well between short J andlongrangeregimes. OurresultscouldbecomparedwithexperimentaldataobtainedinSWCNTs 4 and with those obtained for an ideal vertical Quantum Dot (QD). For a better understanding of some experimental results we also discuss how defects and doping ] l can break some symmetries of thebandstructureof a SWCNT. e - PACSnumbers: r t s . t I. INTRODUCTION energy, straightforwardly from the dispersion relation in a m a graphene sheet - In the last 20 years progresses in technology allowed d the construction of several new devices in the range of πm √3k √3k n εm,k = γ 1 4cos( )cos( )+4cos2( ) o nanometric dimensions. The known Moore prediction ± s − Nb 2 2 states that the silicon-data density on a chip doubles ev- (1) c [ ery 18 months. So we are going toward a new age when where Nb is the number of periods of the hexagonal lat- the devices in a computer will live in nanometer scale tice around the compact dimension (y) of the cylinder. 1 and will be ruled by the Quantum Mechanics laws. If the SWCNT is not excessively doped all the excita- v 1 QDs1,2, which could play a central role within the tionsofangularmomentumm=0(correspondingtothe 6 5 quantum computation as quantum bits (QBIT’s)3,4,5,6,7 transverse motion ky = m/R) cost a huge energy of or- 0 havebeenstudiedintensivelyinthelastyears8 thanksto der (0.3 γ 1 eV), so we may omit all transport bands ≈ 1 the advances in semiconductor technology9. except for the lowest one. From eq.(1) we obtain two 0 QDs are small devices, usually formed in semiconduc- linearly independent Fermi points K~ 2π with a 5 ± ≈ ±3√3 0 tor heterostructures, with perfectly defined shape and right- and a left-moving (r = R/L = ) branch around / dimensions8(<1µmindiameter). Theycontainfromone each Fermi point (see Fig.(1.c)). Th±ese branches are at toafewthousandelectronsand,becauseofthesmallvol- highly linear in the Fermi velocity vF 3γ/2 8 105 m ume available, the electron energies are quantized. The m/s up to energy scales E < D 1 eV≈. This l≈inea×r dis- - QDs are useful to study a wide range of physical phe- persion corresponds to that of a≈Luttinger Model for 1D d nomena: from atomic like behaviour10,11,12 to quantum electron liquids. Many experiments demonstrated a LL n chaos,to the QuantumHallEffect(QHE)whena strong behaviour22,23,24 inSWCNT25withameasurementofthe o transverse magnetic field acts on the device13,14,15,16. linear temperature dependence of the resistance above a c : Recently several scientists proposed a new carbon crossovertemperature Tc26. v based technology against the usual silicon one. In i X this sense the discovery of carbon nanotubes (CNs) in In nanometric devices, when the thermal energy k T B r 199117 opened a new field of research in the physics at is below the energy for adding an additional electron to a nanoscales18. the device (µ = E(N) E(N 1)), low bias (small N − − Nanotubes are very intriguing systems and many ex- V ) transport is characterized by a current carried by sd periments in the last decade have shown some of their successive discrete charging and discharging of the dot interesting properties19. An ideal SWCNT is a hexago- with just one electron. nal network of carbon atoms (graphene sheet) that has This phenomenon, known as single electron tunnel- beenrolleduptomakeacylinder. The uniqueelectronic ing (SET) or quantized charge transport, was observed propertiesofCNsareduetotheirdiameterandchiralan- in many experiments in vertical QDs at very small gle(helicity)parametrizedbyaroll-up(wrapping)vector temperature10,11,12. In this regime the ground state en- (n,m). This vector corresponds to the periodic bound- ergy determines strongly the conductance and the pe- ary conditions20,21 and gives us the dispersion relations riod in Coulomb Oscillations (COs). COs correspond of the one-dimensional bands, which link wavevector to to the peaks observed in conductance as a function of 2 gate potential (V ) and are crudely described by the ergyofSWCNT anddiscussthe rolewhichthe Coulomb g CB mechanism27: the N th conductance peak occurs interaction could play in a 1D system at small tempera- when15 αeV (N)=µ wh−ere α= Cg is the ratio of the tures (T =0.1 0.3 K). g N CΣ ÷ − gate capacitance to the total capacitance of the device. The peaks and their shape strongly depend on the tem- In this paper we analyze the effects of a long range perature as explained by the Beenakker formula for the electron electron interaction in order to determine the resonant tunneling conductance1,27 addition energy for models chosen by a vertical QD and a SWCNT in the Hartree Fock (HF) approximation. ∞ Vg µq InsectionIIweintroducemicroscopicmodelsandcor- G(Vg)=G0Xq=1 kBT sin−h(VkgB−Tµq) (2) rceusspinogndoinnagtHhaemorielttiocnailamnsodfoerlfSoWrtChNeiTnstearnadctiQonDpsoatlesnotfioa-l whichinterpolatesbetweenshortandlongrangetypein- here µ ,...,µ represent the positions of the peaks. 1 N teraction. In section III we show our results about the Many experiments showed the peaks in the conduc- SWCNTandQDsanddiscusstheeffects ofasymmetries tanceofQDsandafamousone11 alsoshowedatomic-like experimentally observed36. propertiesofaverticalQD.Therethe”addition energy”, neededtoplaceanextraelectroninasemiconductorQD, wasdefinedanalogouslyto the electronaffinityfor areal II. MICROSCOPIC APPROACH atom and was extracted from measurements. The typical addition energy ranges from 10 to 20meV, while the As showed in eq.(1) near each Fermi point we ob- disappearance of the COs happens above 50 K. ≈ − tain that the CN could be represented by a typical Lut- tinger Hamiltonian with linear branches depending on The transport in CNs often differs from the quantum k = k+αK (α = 1 labels the Fermi point). We in- CB theory for QDs, because of the one-dimensional na- F ± troduce the operators that create the electrons near one ture of the correlated electrons: so we could need a pe- culiar theory for resonant tunneling in LLs28,29. More ofthe Fermipoints (label α) belongingto one ofthe two recently a novel tunneling mechanism30 was introduced branches (label ζ corresponding to the sign of k), cα,k,s i.e. correlated sequential tunneling (CST), originating and c† . In terms of these operators the free and in- α,k,s from the finite range nature of the Coulomb interaction teraction Hamiltonians can be written as b inSWNTs,inorderto replaceconventionaluncorrelated b sequential tunneling. It dominates resonant transport H0 = vF |k|c†α,k,scα,k,s (3) at low temperatures and strong interactions and its prediction agree with experiments31. In the experiment a αX,k,s short nanotube segment was created with an addition Hint = Vk{,αp,is},s′(q)c†α1,k+q,sc†α2,p q,s′cα3,p,s′cα4,k,s(4). − energy largerthan the thermal energy at roomtempera- {Xαi}k,kX′,q,s,s′(cid:16) (cid:17) ture (T ), so that the SET can be observed also at T . R R The conductance was observed to follow a clear power The interaction Vζ{,αζ′i}(q) plays a central role in order law dependence with decreasing temperature, pointing to determine the properties of the electron liquid. at a LL behaviour in agreement with Ref.30. Aninterestingobservationaboutthe transportinCNs A very crucial question is the effective range of the concerns the long-range nature of the Coulomb inter- potentialanditspossiblescreeninginaCN.Ifwedenote action, which induces dipole-dipole correlations between by x the longitudinal direction of the tube and y the the tunneling events across the left and right barrier30. wrapped one, the single particle wave function for each The crucial question of the range of the interaction in electron reads CNs was investigated in different transport regimes e.g. inordertoexplaintheLLbehaviouroflargeMultiWall32 ϕ (x)=u (x,y)eiαKFxeζikx ζ,α ζ,α and doped33 CNs. √2πL where u (x,y) is the appropriatelinear combinationof HowevermanyexperimentsshowedCOs: e. g. in1997 ζ,α Bockrath and coworkers34 in a rope of CNs below about the sublattice states p=± andL the lengthofthe tube. So we can obtain a simple 1D interaction potential as 10 K observed dramatic peaks in the conductance as − follows: a function of the gate voltage according the theory of single-electron charging and resonant tunneling through 2πR tthhee rqoupaen3t4i.zeIdnetnheisrgryegliemveelsalosfotaheSWnaCnNotTubbeeshcaovmespaossianng U{ζα,ζi′}(x−x′) = Z0 dydy′ u∗ζα1(x,y)u∗ζ′α2(x′,y′) artificial atom and reveals its shell structure35(the data ×U0ζ,ζ′(x − x′,y−y′)uζ′α3(x′,y′)uζα4(x,y). were takenat 5mK). Recent measurements reportclean closed nanotube dots showing complete CB36, which en- These potentials only depend on x x′ and the 1D able us to deduce some properties from the addition en- fermion quantum numbers while Uζ,ζ′−(x x,y y ) is 0 − ′ − ′ 3 obtained from a linear combination of U(x x,y y + demonstratedthat verticalQDs havethe shape ofa disk ′ ′ pdδ ) sublattice interactions37. − − where the lateral confining potential has a cylindrical p, p′ Be−causeofthe screeningofthe interactioninCNs and symmetrywitharathersoftboundaryprofile. Underthis the divergence due to the long range Coulomb interac- hypothesisthequantumsingleparticlelevelsdependjust tion in 1Delectronsystems, it is customaryto introduce onn=n +n (theangularmomentumism=n n ) + + models, in order to describe the electron electron repul- − − − 1 sion. The usual model is the so called Luttinger model, ε =h¯ω (n+ ). n d where the electron electron repulsion is assumed to be 2 a constant in the space of momenta, corresponding to a The symmetry leads to sets of degenerate single-particle very short range 1D potential (Dirac delta). In order to states which form a shell structure: each shell (ε ) has n analyzetheeffectsoflongorshortrangeinteractions,we 2(n + 1) degenerate states so that the shells are com- introduce a model for the electron electron potential de- pletely filled for N =2,6,12,20,etc.electrons in the dot pending on a parameter r, which measures the range of (Magic Numbers). a non singularinteraction; it has as limits the veryshort The many body Hamiltonian corresponding to eq.(3) range potential (r 0, delta function) and the infinite and eq.(4) has the form → longrangeone(r ,constantinteraction). Sowecan →∞ cthoencvluerdyesthhoarttoruarnggeenoernaeltinotetrhaectiniofinnimtyodloenlgrarnagnegsefroonme Hˆ = ∞ εαnˆα+ 21 Vα,β,γ,δ cˆ†αcˆ†βcˆδcˆγ. (7) and eliminates the divergence of the Coulomb repulsion. Xα α,Xβ,γ,δ In this sense we suppose that our model is good for de- Here α (n,m,s) denotes the single particle state in scribing the interaction, if we do not take in account the ≡ the single particle energy level ε , cˆ creates a particle IR and the UV divergences. α †α in the state α and nˆ cˆ cˆ is the occupation number α ≡ †α α U (x x )=U e−|x−rx′| + r2 . (5) otipaelrqautoers.tioInntrheegafrodllionwginthgeseecleticotnrosnweeledcitsrcounssinthteeraecssteionn- r | − ′| 0 2r r2+|x−x′|2! in the dot (Vmn,,mn′′) and analyze in detail the screening of the effective potential. Theinteractionbetweentwodifferentelectronswithmo- menta k and q follows from the previous formula III. SINGLE WALL CARBON NANOTUBE: V(p= k q )= 1 Ldx L−xeipyU(y)dy . LOW TEMPERATURE BEHAVIOUR AND | − | L2 Z0 Z−x COULOMB BLOCKADE In the limit L we can calculate the Fourier trans- → ∞ Before proceeding with the calculations, we want to form of eq.(5) pointoutthat the realbandstructures ofmeasuredCNs 1 show some differences with respect to the ideal case dis- V (q q )=V πre rq q′ + . (6) r | − ′| 0 − | − | 1+r2 q q 2 cussed (eq.(3)): in order to clarify this point we shortly (cid:18) | − ′| (cid:19) discuss the model and the results of two recent experi- The scattering processes are usually classified accord- ments. ing to the different electrons involved and the coupling To begin with, we have to introduce a quantization strengths g are often taken as constants. This assump- due to the finite longitudinal size of the tube (L) in the tioncorrespondstothe usualLuttinger model,sowefol- dispersionrelationeq.(1). The longitudinal quantization low this historical scheme, in order to classify the in- introduces a parameter which also gives a thermal limit teractions. The backscattering gs,s′ involves electrons for the Atomic Like behaviour: in fact k wavevectors 1 in opposite branches with a large momentum transfer have to be taken as a continuum if KBT is as a critical (q ≈ 2kF) so g1s,s′ ≈ V(2kF). The forward scattering ivsalbueeloEwc(=orvnFeLhara)nEd.as a discrete set if the temperature occurs between electrons in opposite branches g with a c 2 small momentum transfer (q 2k ) so gs,s′ V(0) (or After the quantization we obtain shells with an 8-fold ≪ F 2 ≈ degeneracy due to σ (spin symmetry), α (K, K lattice bg2sr,asn′c≈h(Vg4()2πin/vLo)lv).esTthheepfaoirrwsa(rkd sckaFtt,epringkFin)athnedsgaivmees symRemceetnrty)e,xζpe(r(ikm−enKts)d,(oKno−tks)upLpuottritnsguecrhsyam−hmigehtrsyy)m. - ≈ ≈ g4k ≈ V(0)−V(p−k) and g4⊥ ≈ V(0). Further below, metry and different hypotheses were formulatedin order wherewediscussthe electronelectroninteraction,were- to explain this discrepancy. call the values of the g constants. According to Cobden and Nygard36 ”the sole orbital symmetryisatwo-foldone,correspondingtoaK-K’sub- Now we want to introduce the analogous model for banddegeneracy andresultingfrom theequivalenceof the a semiconductor ideal QD. Usually we describe the dot two atoms in the primitive cell of graphene structure”. like a 2D system with an harmonic confinement poten- Experimentally one can answer this question by observ- tialV(r)= 1m ω 2r2 accordingtomeasurements11 that ingthegroupingofthepeaksinplotsoftheconductance 2 ∗ d 4 lawsofourHamiltonianandfindtheNumberofelectrons (N), the Energy, the total linear momentum K =0 and the spin S,S =0. z The Fermi sea corresponds to the state where all the shells with energy below the Fermi energy (E = F v n h/L) are totally filled (N =4n ). This state will F F F F be our ground state Ψ (N ) and this is true also for in- 0 F teracting electrons in absence of correlation (i.e. in the HF approximation ). The effects ofcorrelationwillbe discussedin afurther article, here we have to explain the range of validity of our approximation. The first thing to consider is the interactionstrengthg V(0) V(2k ) comparedto the FIG.1: Thedispersionrelationandthequantizedlevels. The F ≈ − boxes in figure represent energy levels and can be filled by a kinetic energy vFh/L. The HF approximation is valid pair of electrons with opposite spins. a) The general case if g vFh/L. However also the temperature plays a ≪ without any degeneracy. b) The 4-fold degeneracy case with central role, in fact if the temperature increases we have δSM =0. c)The8-folddegeneracycase. d)Differencesinthe totakeintoaccountmoreexcitedstates(asortofthermal splitting dueto the comparison between δSM and ∆ε. cut-off corresponds to the energy k T) so if we are at a B verylow temperature we canassume the Fermi seastate as the ground state. versus the gate potential. Howeverin the experiment no four-fold groupingwas observedbecause degeneracywas At this point we are able to calculate the addition liftedby amixingbetweenstatesdue eithertodefects or energy (EA) following the Aufbau sequence explained to the contacts. in AppendNix A. EA has a maximum for some num- A different experiment38 displays conductance peaks N bers 4,8,12,..4n (n integer) due to the shell filling. The inclusters offour,indicating that there is a fourfold de- shells are filled sequentially and Hund’s rule determines generacy. In ref.38 two different shell filling models are whether a spin-down or a spin-up electron is added so put forward: the first one, when the subband mismatch that the singlet (S = 0) energy for a 4n+2 system is dominates, predicts that the spin in the SWCNT oscil- always greater than the triplet (S = 1) one. Obviously lates between S =0 and S =1/2. this is an effect of interaction and is quite different for the long and short range models. In order to take into account the strong asymmetries As we show in Fig.(2) the oscillations due to Hund’s measured experimentally we modify the dispersion rela- rulecorrespondtotheshortrangepotentialwhiletheat- tion. A first correction has to be introduced because of tenuation of these oscillations when the number of elec- the ”longitudinal incommensurability”: in general K is tronsinthe 1Dsystemincreasesis due to the long range not a multiple of π/L so K = (N +δN)π with δN < 1 L interaction. So we can draw the following conclusions: and the energy shift is ∆ε = vFhδLN. A second correc- The 4 folddegeneratemodelpredicts oscillationsin tion is due to the subband mismatch (δSM). The single the⋄additio−n energy due to the Hund’s rule quite similar electron energy levels are to the ones observed in QDs. The oscillations periodicity is 4 for this model (8 for π (1 p) ⋄ ε =h¯v l +pK + − δ (8) a system with two Fermi points) l,σ,p F SM | L | 2 Theoscillationsamplitudeisduetoanexchangeterm ⋄ (proportional to the short range interaction). where p= 1. Each cho±ice of parameters gives a different degener- The effect of a long range interaction is a damping ⋄ acy for the quantum levels: the 8-fold degeneracy corre- of the oscillations when the number of electrons in the sponds to δ = 0 and K = nπ; the 4-fold degeneracy system increases. SM L isfoundifweputjustδ =0andthe2folddegeneracy Themodelwith8-folddegeneracy(seeFig.(1))hastwo SM represents the general case (see Fig.(1)). basicsymmetries: k k,KL/π=NK andusuallythe →− interaction between the electrons with momenta near K andtheoneswithmomentanear Kisverysmallsothat − A. High band structure symmetry: damping in the wehavetwoindependent4-folddegenerateHamiltonians addition energy oscillations (nF =N/4<<NK). In the discussion which follows we analyze the effects of the range of the Coulomb interaction in a simplified B. Asymmetric model system with just two linear symmetric branches. In this model each shell is filled by 4 electrons with opposite Now we have to analyze models without symmetries momenta and spin. We can look for the conservation by using eq.(8) in the HF approximation. 5 2 1.75 1.5 1.25 Lh 2(cid:144) 1 e HG 0.75 0.5 0.25 3 4 5 6 7 8 9 10 FIG. 2: On the right we show analytical Aufbau results for V g the addition energy versus the number of electrons of a 4- fold degeneracy model corresponding to different values of FIG.3: Theasymmetricmodelcalculation fortheCOs(con- the range r (r =0 dark gray dashed line, r =0.4 black line, ductancevs. gatevoltage)atdifferenttemperaturesexpressed r=1grayline). Weshowhowthedampingintheoscillations in terms of ∆ calculated following the classical CB theory. isduetoalongrangeinteractionwhileitdoesnotappearfor Theoretical calculations show that the fine structure peaks a r = 0 model. Our predictions can be compared with the are appreciable just for very low temperatures. measured addition energy in theCobden Nygard experiment displayed on theleft. Starting from the Hamiltonian eq.(9) we are able to calculate the groundstates of the many electronsystem for Inorderto introduce the electronelectroninteraction, various N. we take in account just two (g and g ) of the many k ⊥ Our results can be compared to the experimental re- constants that we introduced in section II from eq.(6). sultswherestrongasymmetrieswerefound. InFig.(3)we For allowing to compare easily our results with those in show the peaks corresponding to an asymmetric model ref.39, as well as with experiments,we giveour results in to which we apply the classicaltheory of CB, i.e. eq.(2). terms of V and J, obtained as a linear combination of 0 Thefinestructurewith8periodicityisdestroyedbyther- the g constants. maleffects: itisappreciableatsmallertemperatures(we Following the usual method, in order to calculate the assumeaboutT 300 500mKforaNanotube’slength energy levels, we put s ≈ ÷ of about 100 300nm) and disappears at a temperature gk =V0−J ; g⊥ =V0 ; ∆= v2FLh, ; ∆ε=δNvFLh. Trio≈dic4iTtys a(Tpp≈÷ear1s..2÷So2,.0wKe sceoencrleufd.3e8)thwahterweejushstouald4npoet- be able to observe any small asymmetries effects if the Here ∆ε is the incommensurability shift (∆ε < 0.5∆). temperature increases. Fig.(3) shows how the Coulomb The single particle energieshave a differentstructure for peaks in the conductance disappear when the tempera- δ >2∆ε and δ <2∆ε, as we show in Fig.(1.d). SM SM From the experimental data38 we obtain ture increases. In a future article we will show how the end of the CB regime corresponds to the beginning of V0 U +δU +Jexp .42 and J Jexp 2δU .05 another one. ≈ ≈ ≈ − ≈ Nowwewantpointoutthelimitsoftheapproachused where we assume U = .22, δU = .05 and J = .15 in exp in the previous sections: the HF approximation ignores units of ∆. Under these conditions J is always less than the effects of correlation. This corresponds to the Fermi the level spacing. Liquidtheoryandgivesgoodresultsjustifwecanassume Now we can write the Hamiltonian of the nanotube depending on these parameters39 that the interaction is smaller than the kinetic energy (g ¯hv ). F ≪ H = ε nˆ (9) Thermal effects are quite important too. In fact the n,ζ,p n,ζ,p,s temperature appears in the Beennaker formula and is n,ζ,p,s X responsible of the disappearance of the Coulomb peaks. N(N +1) + V0 J δs,s‘nˆn,ζ,p,snˆn,ζ,p,s However,ifthetemperatureishigherforaSWCNT,itis 2 − n,ζ,p,sn‘,ζ‘,p‘,s the same Beennakerformula whichfails,because the HF X X calculated energy levels are very different from the real SinceJ islessthanthelevelspacingtheenergythatwe energy levels of the electron system. When T is above a need, in order to add one electron is ε, corresponding to critical value the lowestempty energy levelwith aninteractionenergy v h N(N 1) N 1 T = F (10) V0 2− −J 2− for odd N c LkB N(N 1) N we cannot consider the Fermi sea as the ground state V0 − J for even N. of the electron system in a SWCNT because some other 2 − 2 6 stateswiththesamelinearmomentumK =0anddiffer- ent kinetic energy are also available for the system. So fortemperaturesabovethe criticalvaluewehavetotake into account strong effects of correlation. C. Quantum Dots and Long Range Interactions As we did for a SWCNT we calculate the addition energy of a QD in the HF approximation. As we discussed in section II peaks and oscillations in themeasuredadditionenergycorrespondtotheonesofa shellstructureforatwo-dimensionalharmonicpotential. However, in experiments11, high values of the addition energyareobservedalsoforN =4,9,16,etc.corresponding to those values of the Number of electrons in the FIG. 4: In a),b),c) results from three different models of in- dot for which, respectively, the second, third and fourth teractingelectronsin QDsaredisplayed. Thesimple HFcal- culation in a) has to be corrected because it does not take shellsarehalffilledwithparallelspinsinaccordancewith intoaccount theclassical capacitive effect. In orderto deter- Hund’s rule. Half filled shells correspond to a maximum mine it we recall that the electrons in the dot give a charge spin state, which has a relatively low energy10,11. dropletofradiusRD notfixed,sothatwecanapproximateit Wecompareanon-interactingmodelwithmodelsthat with a disk of capacitance C =C0RD. The value of RD can include Coulomb interactions especially the exchange becalculated from classical equations, RD √N +1. So we term. In Fig.(4) we plot the calculated addition energy addtheclassical termtothedampedoscilla∝tions duetolong as a function of the number of electrons in the following range affected Coulomb exchange and obtain the b) and c) three different cases: plot in the figure. The non-interacting model gives us just the peaks ⋄ corresponding to the Magic Numbers. The model with constant parameters (g and g ) k ⊥ So we have to limit ourselves to describing an electron ⋄ corresponds to a short range interaction (Dirac δ). The system only when it is uncorrelated. This is true only CoulombexchangeterminHFgivestheHund’sruleand if the interaction strength is small (g << v h/L)40 and F allowsustoexplaintheoscillationsintheadditionenergy thetemperatureisbelowthecriticalvalueT (seeeq.10). c as we show in Fig.(4.a). The third model is the long range interaction one, ⋄ where we introduce two measured effects that are both duetothelongrangeCoulombinteraction. Thefirstone Acknowledgments is due to the classical capacitive effect shown in Fig.(4.b andc)asacontinuousline. The secondoneisduetothe long range correction to Coulomb exchange. This work was partially supported by the Italian Re- We could compare these results to the experiments searchMinistryMIUR,NationalInterestProgram,under (e.g. see ref11). grant COFIN 2002022534. IV. CONCLUSIONS APPENDIX A: AUFBAU AND ENERGIES FOR In this work we have analyzed some properties of a THE 4-FOLD DEGENERATE MODEL 1D (SWCNT) and 0D-2D (QDs) electron systems by in- troducingamodelofinteractioncapableofinterpolating We start by taking into account that each electron in from short to long range, in order to analyze the effects the k > k state interacts with all the electrons in the F ofthe longrangecomponentofelectronelectroninterac- | | filled shells with k below k so that we can have the F tions. following two (or four) Σ terms: HF Wehaveobtainedthatadampingintheoscillationsof the addition energy could be predicted for these models correspondingtothepresenceofalongrangeinteraction. Σ (k ,k)= kF Us,σ + −kF Us,σ The results can be compared to the experiments and we s,σ F k p k p | − | | − | discussed the limits for the experimental observation of Xp=1 pX=−1 ourpredictionsinSWCNT.Infactthe breakdownofthe FermiLiquidopensdifferentregimeswherethetunneling If we introduce n = Lk and consider the direct term F π F has to be considered as resonant tunneling in LLs28,29. of the interaction, we conclude that 7 where µN+1 =EN+1−EN and γ = JU2k0F Σ (k ,k) = 2V n F 0 F ↑↓ kF Σ (k ,k) = V 2n J (1 δ ) F 0 F p k p,0 ↑↑  − | − | −  p=X−kF   Obviously if we add an electron in the lowest The addition energy is also calculated as EA = eVner4gny empt2ykF+shLπelJl w.e obtain Σ(kF,kF + Lπ) ≈ µN+2−µN+1 so that N+1 0 F − p=Lπ p F(cid:16)or each shell we can(cid:17)consider the internal interaction P energy as kF w(k)=2V (3 J ) W(k )= w(p) 0 2k F − p=π XL so that the total energy of a system with n filled shells F reads kF−Lπ π E =2v n (n +1)+W(k )+4 Σ(p,p+ ) 4nF F F F F L p=0 EA = U (1 γ) EA =U (1+γ) X 4nF+1 0 − 4nF+2 0 ¯hv π EA = U (1 γ) EA =U (1+γ)+ F a. Energy levels 4nF+3 0 − 4nF+4 0 L (cid:18) (cid:19) When we add one by one the electrons to the 4n - F system we obtain π π µ = h¯v k + +Σ(k ,k + ) 4nF+1 F F L F F L (cid:16) π(cid:17) π µ = h¯v k + +Σ(k ,k + )+U (1 γ) 4nF+2 F F L F F L 0 − (cid:16) π(cid:17) π µ = h¯v k + +Σ(k ,k + )+2U 4nF+3 F F L F F L 0 (cid:16) π(cid:17) π µ = h¯v k + +Σ(k ,k + )+U (3 γ) 4nF+4 F F L F F L 0 − (cid:16) π(cid:17) π π µ = h¯v k +2 +Σ(k + ,k +2 ) 4nF+5 F F L F L F L (cid:16) (cid:17) 1 L.P. Kouwenhoven, C.M. Marcus, P.L. McEuen, S. 8 E.B. Foxman et al., Phys. Rev. B 47, 10020 (1993); R.C. Tarucha, R.M. Westervelt, and N.S. Wingreen, in Meso- Ashoori, Nature379, 413 (1996). scopic Electron Transport, (Kluwer, Dordrecht, The 9 T.J. Thornton, Rep.Prog. Phys. 57 (1994). Netherlands, 1997). 10 L.P. Kouwenhoven and C. Marcus, Physics World, June 2 P.L. McEuen, Science 278, 1729 (1997). 1999. 3 Spintronics and Quantum dots for Quantum Computing 11 S. Tarucha, D.G. Austing, T. 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