Conserved spin and orbital phase along carbon nanotubes connected with multiple ferromagnetic contacts
C. Feuillet-Palma, T. Delattre, P. Morfin, J.-M. Berroir, G. Fève, D.C. Glattli, B. Placais, A. Cottet, T. Kontos
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Conserved spin and orbital phase along carbon nanotubes connected with multipleferromagnetic contacts
C. Feuillet-Palma , , T. Delattre , , P. Morfin , , J.-M. Berroir , ,G. F`eve , , D.C. Glattli , , ,B. Pla¸cais , , A. Cottet , and T. Kontos , ∗ Ecole Normale Sup´erieure, Laboratoire Pierre Aigrain,24, rue Lhomond, 75231 Paris Cedex 05, France CNRS UMR 8551, Laboratoire associ´e aux universit´es Pierre et Marie Curie et Denis Diderot, France Service de physique de l’´etat Condens´e, CEA, 91192 Gif-sur-Yvette, France. (Dated: September 8, 2018)We report on spin dependent transport measurements in carbon nanotubes based multi-terminalcircuits. We observe a gate-controlled spin signal in non-local voltages and an anomalous conduc-tance spin signal, which reveal that both the spin and the orbital phase can be conserved alongcarbon nanotubes with multiple ferromagnetic contacts. This paves the way for spintronics devicesexploiting both these quantum mechanical degrees of freedom on the same footing.
PACS numbers: 73.23.-b,73.63.Fg
I. INTRODUCTION
The scattering imbalance between up and down spinsat the interface between a non-magnetic metal and a fer-romagnetic metal is at the heart of the principle of themagnetic tunnel junctions or multilayers celebrated inthe field of spintronics . Although these devices use thequantum mechanical spin degree of freedom and electrontunneling, they do not exploit a crucial degree of free-dom involved in quantum mechanics: the phase of theelectronic wave function. In most of the devices studiedso far, this aspect has not been developed owing to theclassical-like motion of charge carriers in the conductorsused .Quantum wires or molecules have emerged recentlyas a promising means to convey spin information .In these systems, the electronic gas is confined in twoor three directions in space, making quantum effects apriori prominent. In this context, most of the stud-ies have been carried out in two terminal devices, i.e.with two ferromagnetic contacts. The need for inte-gration and more complex architectures for manipulat-ing spin information brings on the question of whathappens when a spin active nanoscale conductor is con-nected to more than two reservoirs. Multi-terminal trans-port has been central in (spin independent) mesoscopicphysics, in particular with the observation of non-localelectric signals due to the delocalization of electronicwave functions . Can this quantum mechanical non-locality survive and ultimately be exploited in spintronicsdevices combining nanoscale conductors and ferromag-nets ?In this article, we address this question through multi-terminal spin dependent transport measurements in sin-gle wall carbon nanotubes (SWNTs) with ferromagnetic ∗ To whom correspondence should be addressed: [email protected]
FIG. 1: a. Schematics diagram of the devices studied in thearticle. b. Non-local voltage V for sample I as a function ofthe external magnetic field H for side gate voltages V SG = − . V , V SG = − . V and a back gate voltage V BG = − . V . and non-magnetic contacts. Non-local voltage and con-ductance measurements reveal that the spin as well asthe orbital phase are conserved along the whole activepart of our SWNTs. We observe a non-local spin field ef-fect transistor -like action which is a natural consequenceof quantum interference in a few channel conductor. Inspite of the inherent complexity of the spectrum of ourdevices, we can account well for our findings using a sim-ple theory based on a scattering approach. These re-sults bridge between mesoscopic physics and spintron-ics. They open an avenue for nanospintronics devicesexploiting both the spin and the orbital phase degrees offreedom, which could provide new means to manipulatethe electronic spin, because the orbital phase of the car-riers can easily be coupled to the local electric field innanoscale conductors.The principle of non-local transport measurements isto use a multi-terminal structure with two terminalsplaying the roles of source and drain, and the othersthe role of non-local voltage probes. Since the pio-neering work by Johnson and Silsbee in metals , non-local spin dependent voltages have been studied in vari-ous multichannel diffusive circuits based on semiconduct-ing heterostructures , metallic islands and graphene .These signals are well captured using a classical bipartiteresistors network, with two branches corresponding to op-posite spin directions . The non-local spin signal stemsfrom the imbalance between the up spin and down spinbranches of the network, which reflects the imbalancesbetween e.g. the two spin populations. Importantly, thisinterpretation is valid only when one can neglect quan-tum mechanical non-local signals which arise from thedelocalization of the carrier’s wave function.Coherence effects induce only corrections to transportat low temperatures in metals or semiconducting het-erostructures which involve many conducting channels .In contrast, coherence becomes essential in understand-ing transport in molecules or quantum wires where quan-tum mechanics primarily controls conduction. The stud-ies of non-local spin transport in the coherent regime havebeen elusive so far. Here, we use the high versatility ofSingle Wall carbon NanoTubes (SWNT) to achieve therequired devices and to explore these phenomena. We ob-serve a gate-controlled spin signal in non-local voltagesand an anomalous spin conductance which are specific tothe coherent regime. II. EXPERIMENTAL SETUP
We use the measurement scheme represented on figure1a. Our devices are made out of a SWNT connected to 4electrodes labelled 1,2,3 and 4 from the left to the right,with 2 and 3 ferromagnetic
N iP d electrodes and 1 and4 non-magnetic
P d electrodes. In addition, the device iscapacitively coupled to a back-gate electrode with voltage V BG and two side gate electrodes with respective voltages V SG and V SG , acting mainly on sections 12 and 34 ofthe nanotube respectively. Throughout the paper, thetemperature is set to 4 . K .A SEM picture of a typical device is shown in figure2a. We use chemical vapor deposition with a standardmethane process to fabricate our SWNTs on a Si sub-strate. We localize the SWNTs with respect to Au align-ment markers by Scanning Electron Microscopy (SEM)or Atomic Force Microscopy (AFM). We fabricate thecontacts and gates using standard e-beam lithographyand thin film deposition techniques. We deposit thenormal and the ferromagnetic contacts in one fabrica-tion step using shadow evaporation techniques. The FIG. 2: a. SEM picture of sample I. The NiPd electrodes are high-lighted in blue and the Pd stripes are highlighted in yellow.The SWNT is highlighted in purple. The orange scale baris 1 µm . b. MFM characterization of the NiPd electrodes atroom temperature on a test device similar to sample I withoutthe SWNT. The black scale bar is 1 µm . central ferromagnetic electrodes consist of a 30nm-thick N i . P d . layer below a 70nm-thick P d layer. Thenormal contacts consist of 70nm of Pd. Such a methodallows to achieve two probe resistances as low as 30kOhmbetween the normal and the ferromagnetic reservoirs. Inaddition to the highly doped Si substrate with 500 nmSiO which is used as a global backgate, we fabricatetwo side gates whose voltages V SG and V SG are usedto modulate transport in our devices. Each nanotubesection defined in this manner has a length ranging from300 nm to about 600 nm .Our measurements are carried out applying an AC biasvoltage V sd of about 200 to 300 µV between the normalelectrode 1 and the ferromagnetic electrode 2, at a typi-cal frequency of 77 . Hz . This generates a finite non localvoltage V between the ferromagnetic electrode 3 andthe normal electrode 4. We also measure simultaneouslythe conductance G = dI /dV sd . Note that a finite V has already been observed in similar but non-magneticdevices due to coherent propagation of electrons in theSWNT and lifting of the K/K’ degeneracy . Here, wefocus on the specific effects due to ferromagnetic leads.A spin contrast is obtained by comparing the electric sig-nals in the parallel (P) configuration (magnetizations ofelectrodes 2 and 3 pointing in the same direction) andin the (AP) configuration (magnetizations pointing inopposite directions). A finite magnetic field is appliedin plane parallel to the easy axis of the ferromagneticelectrodes (for samples I and III), which is transverseas shown by MFM characterization carried out at roomtemperature (see figure 2b). The observed magnetic con-trast shows the presence of large transverse domains ofa typical size of 1 µm . Due to the different widths ofrespectively 150 nm and 250 nm , the coercive fields ofthe two ferromagnetic electrodes are different. Gener-ally, this leads to a sharp switching at about 50 mT forone of the electrodes. For the lower field switching, itturns out to be more difficult to obtain systematicallyswitchings as sharp as those of sample I. The P and APconfiguration can be obtained selectively by sweeping theexternal magnetic field. We determine M V = V P − V AP and M G = 100(1 − G AP /G P ). III. GATE CONTROLLED NON-LOCAL SPINSIGNAL.
The magnetic field dependence of the non-local volt-age V of sample I is shown in figure 1b. Upon increaseand decrease of the external magnetic field H , the char-acteristic hysteretic switching of a spin valve is observed.We observe sharp switchings which show that the exter-nal field is well aligned with the magnetic anisotropy ofboth electrodes in this case. Upon increasing H (see redline in figure 1b), we obtain the AP configuration for H ∈ [10 mT ; 50 mT ], and the P configuration otherwise.For the particular gate voltage set used in figure 1b, V changes from V P = − . µV to V AP = − . µV uponswitching from the P to the AP configuration, leadingto a finite M V . Unlike the majority of our samples,this spin signal is superimposed to an intrinsic back-ground here (see discussion in section IV). A finite
M V has already reported in multichannel incoherent diffusiveconductors . One of the main results of the presentwork is the observation of a gate control of M V , as a con-sequence of quantum interferences, which contrasts withthese previous works. Such a fact not only sheds lighton the peculiar nature of spin injection in coherent fewchannel conductors, but also allows to rule out non-spininjection effects related to stray fields for example as wewill see in section IV.As soon as a metallic electrode is deposited on thetop of a SWNT, a scattering region is created below thecontact, which partially decouples the two sides of thenanotube defined by the electrode. The multi-dot natureof our devices appears on figure 3a, where V P is rep-resented in a greyscale plot as a function of V SG and V SG for sample I. We observe white horizontal and ver-tical stripes, rather regularly spaced, which correspondto negative anti-resonances in V P . Such a ”tartan” pat-tern is very much alike the stability diagram of a doublequantum dot in the electrostatically decoupled regime .The stripes correspond to discrete energy levels ”engi-neered” by defining the 3 different sections of the nan-otube with the 4 electrodes. The fact that horizontal aswell as vertical stripes are observed shows that the side gate electrodes control essentially independently differentparts of the device, which carry different energy levels.Our devices can be seen as a series of three Fabry-Perotelectronic interferometers with local gate control. Thenature of the coupling between these interferometers isa crucial question for the development of orbitally phasecoherent spintronics. V SG2 (V) V S G ( V )
10 20 -0.1 V P ( m V) aC V P ( m V ) V ( m V ) V P34 ( m V ) SG1BG bc FIG. 3: a. Greyscale plot showing the ”tartan” pattern ofthe non local voltage V P of sample I in the P configuration. b. ”Minor hysteresis loop” for the non-local voltage V ofsample I for V SG = − . V , V SG = − . V and V BG = − . V . c. V P and MV as a function of V SG . In purple,the prediction from the multi-terminal scattering theory ofref. 27 with the parameters described in the appendix. One can measure the spin signals by placing the sys-tem in the remanent state of magnetization either in theP or in the AP configuration (for samples with a suffi-ciently high stability). This is done by imposing to thedevice a ”minor loop” which is represented in figure 3b.In such a cycle, the magnetic field is swept in such a wayas to reverse selectively one magnetization without re-versing the other. Depending on how the external fieldis swept back to zero, one can reach either the P or theAP configuration. Then, for each of these configurations,we measure in a single shot the gate dependence of V .This method has been used to obtain M V in figure 2c.The existence of quasi bound-states inside the nanotubeinduces variations of both V and M V as a function ofthe gate voltages. This effect can be observed when V SG is swept, V SG being kept constant, for example (see fig.3c). The interference fringes observed correspond to the”tartan” pattern of figure 3a. For V SG = − . V and V BG = − . V , V P and V AP evolve almost in paral-lel as a function of V SG . This results in a weakly gatedependent M V with a constant positive sign as shownin figure 3c. We find that V P and V AP can be of op-posite sign, as well as of the same sign depending onthe values of V SG and V SG . In carbon nanotubes, thisphenomenon originates both from transverse and longi-tudinal size quantization.In the spectroscopy of our devices, Coulomb blockadeeffects are generally absent (see e.g. Fig. 6b). Thismotivates a comparison between our data and the non-interacting scattering model of ref. 27 (see appendix fordetails). This model uses four scattering channels, toaccount for the up/down spins and the K and K’ or-bitals of carbon nanotubes. For simplicity, we assumethat the spin and K/K’ degrees of freedom are conservedalong the whole device. Between two consecutive con-tacts i ∈ { , , } and j = i + 1, electrons acquire a”winding” quantum mechanical phase δ ij . The effect ofeach metal/nanotube contact is described with a scatter-ing matrix which depends on the contact transmissionprobability. In the case of a ferromagnetic contact, wealso take into account the spin-polarisation of the trans-mission probabilities and the Spin Dependence of Inter-facial Phase Shifts . This scattering model is fullycoherent, i.e. the phase of the electronic wave functionis conserved even when electrons pass in the nanotubesections below the ferromagnetic contacts 2 and 3. Theresults of the scattering theory of ref. 27, shown in ma-genta in figure 3c, are in qualitative agreement with ourdata. The variations of V P are well accounted for as wellas the sign and the order of magnitude of M V . Impor-tantly, the coherent model of ref. 27 involves resonanceloops which are extended on several sections of the nan-otube, e.g. between leads 1 and 3.The gate modulations of V P as well as the gate depen-dence of M V is a natural consequence of delocalization ofthe electronic wave function in our devices. Similarly tooptics, the multiple reflections at the contacts give rise to(electronic) interference which lead to gate modulationsof the physical signals. It is important to note that theorigin of this gate modulation is not related to the spin-orbit interaction which lead to energy splittings of the or-der of 0 . meV in SWNTs . This fact will become evenclearer in section V where we identify the energy scaleresponsible of the modulations as the single particle levelspacing of one of the NT section (namely section 12). Fi-nally, it is important to note that here, contrary to themultichannel diffusive case, coherence naturally couplesthe spin and the charge of carriers. Therefore, a non-local measurement does not ”separate spin and chargetransport” as is often stated in the coherent few chan-nel case. Rather, it gives a new path for manipulatingspin information with electric fields at low temperature. IV. BACKGROUND MAGNETORESISTANCEAND STRAY FIELD EFFECTS
As one can see in figure 1b, there is a finite backgroundfor the non-local voltage as a function of the magnetic -150 -100 -50 0 50 100 150 -0.20.0 V
SG2 = -6.0 V------------><------------ V
SG2 = -2.65 V------------><------------ V ( µ V ) H(mT)
FIG. 4:Non-local voltage V for sample I as a function of the externalmagnetic field H for V SG = − . V and − . V . The curvesfor V SG = − . V has been shifted up to make them coincidewith those for V SG = − . V at zero field. field, superimposed to the hysteresis. This might ques-tion the effect of the stray fields on the observed signals.Note, however, that no background is observed in figure5a and c, a behavior which is common to the majorityof our samples. This makes the device essentially insen-sitive to stray fields for the majority of samples stud-ied. In order to rule out the stray field effects for sam-ple I, we present in figure 4 hysteresis loops for two ofthe different gate voltages, namely V SG = − . V and V SG = − . V . The curves for V SG = − . V havebeen shifted up to make them coincide with those for V SG = − . V at zero field. As one can see on this fig-ure, while the backgrounds are almost exactly the same(up to small gate shifts), the MV’s clearly differ. There-fore, the observed gate dependence of the MV for sampleI cannot be attributed to stray field effects. V. ANOMALOUS NON-LOCALMAGNETORESISTANCE
In the multichannel diffusive incoherent regime, a hys-teretic non-local voltage can arise, but one can show thatthe intrinsic locality of charge transport makes it very dif-ficult for the conductance G = dI /dV sd to depend onthe relative magnetic configuration of the ferromagneticelectrodes . This contrasts with our devices as shownin fig. 5b and d where M G = 0 is obtained. In order toshow that the spin signals observed in G and V arisefrom a property of the device as a whole, it is crucial tomeasure G and V simultaneously. In figure 5a and b(resp. 5c and d), the magnetic field dependences of thenon-local voltage and the conductance of sample II (resp.III) are shown for different gate voltages. A hysteresis isobserved simultaneously for both quantities upon cyclingthe magnetic field. For the measurements of figure 5a andb, contrarily to the two other samples presented in thisarticle, the external magnetic field has been applied inplane, perpendicular to the magnetic anisotropy of theferromagnetic electrodes. In such a situation, the motionof the magnetic domains often displays a complex behav- FIG. 5: a. Non-local voltage V for sample II as a functionof the external magnetic field H for V SG = 0 . V , V SG =0 . V and V BG = 4 . V or V BG = 3 . V . b. Similar plotfor G of sample II. c. Non-local voltage V for sample IIIas a function of H for V SG = − . V , V BG = − . V and V SG = − . V or V SG = − . V . d. Similar plot for G of sample III for V SG = − . V , V BG = − . V and V SG = 4 . V or V SG = − . V . ior which is revealed by the complex switching features ofboth V and G in figure 5a and b. Because of their com-plexity, these features show that the hysteretic behaviorsof V and G have strong correlations. As expected, weobtain more regular switchings if the magnetic field isapplied along the easy axis anisotropy, as shown in fig-ures 1b, 5c and 5d. As highlighted by the vertical dashedgreen lines, both the shape and the sign of the spin sig-nals are again strongly correlated, which confirms thatthey have the same physical origin i.e. the change in therelative magnetic configuration of the two ferromagneticcontacts. Due to quantum interferences, M G naturallydepends on V SG (see figure 5b and 5d) . Therefore,we observe a spin field effect transistor action which isnon-local with respect to the position of both the ferro-magnetic electrodes and the gates. Note that in figure5a and 5c, we observe a negative M V . This behavior isspecific to the coherent regime and can be reproducedwith the model of ref. 27.The dependence of G and its hysteretic part M G =100(1 − G AP /G P ) on the side gate voltages further re-veals how the spin signals are affected by non-local quan-tum interferences. Figure 6a displays the colorscale plotof G as a function of V SG and V SG for sample III. Asindicated by the tilted red stripes, interference fringes areobserved in the conductance. The modulations in G arecontrolled essentially by a single winding phase, namely δ , which can be tuned via V SG or V SG . As shownin figure 6b, the colorscale plot of the normalized G asa function of V sd and V SG displays the characteristic V SG2 (V) V s d ( m V ) normG P - -5.0-10.00.05.0-13.25 -12.75 -12.25 V SG1 (V) V S G ( V ) a bc G P(e /h) G P ( e / h ) G P ( e / h ) no r m G P M G ( % ) hv F /L FIG. 6: a. Colorscale plot of G P as a function of V SG and V SG for sample III. b. Colorscale plot of the normalized G of sample III as a function of V sd and V SG . c. Simultaneousvariations of MG and G P as a function of V SG for sampleIII. Fabry-Perot pattern with a level spacing of about 5 meV ,consistent with the lithographically defined length L ofabout 300 nm and a Fermi velocity of 8 × m/s . Thesimultaneous measurement of G and M G as a functionof V SG is shown in figure 6c. Here, we have measured M G versus V SG by recording a full hysteresis cycle foreach set of gate voltages. As shown on the bottom panel, G P oscillates from 0 .
008 to 0 . × e /h when V SG isswept. Oscillations of about 30% are also found in M G .The solid magenta lines correspond to the result of thescattering theory at T = 4 . K . We find a very goodagreement with our experimental findings. From the the-oretical fit of figure 6c, we conclude that M G varies dueto changes in δ but also δ and δ . First, the gateelectrode 2, which is nearby section 34 of the device, alsoacts on δ and δ thanks to the long range nature ofCoulomb interaction ( an effect described e.g. by β = 0in the appendix). Second, the M G signal is also affectedby V SG due to the spatial extension of the carriers wavefunction over the whole device. The non-local transistor-like action shown in figure 6c is therefore non-local elec-trostatically and quantum mechanically. Importantly, infigure 6c, the position of the maxima in G P do not co-incide with those of M G as highlighted by the verticalorange dashed lines. This reveals that G AP oscillates ina similar fashion as G P but with a different phase . Thephase shift between G P and G AP clearly illustrates thatthe phase of the carriers is conserved upon scattering be-low the ferromagnetic contact 2. Indeed, this effect canonly be explained by invoking coherent electronic wavefunctions which extend from contact 1 to 3 at least, andgive rise to spin-dependent resonance effects sensitive tothe magnetic configuration of both leads 2 and 3. Thetheoretical curve of figure 6c reproduces accurately thiseffect. We conclude that, in our devices, both the spinand the orbital phase are conserved over the whole ac-tive part of the nanotube, even below the ferromagneticcontacts . VI. CONCLUSION
In this work, we have studied various non-localtransport phenomena in single wall carbon nanotubesconnnected to two ferromagnetic and two normal elec-trodes. These multiterminal spintronics devices exploitactively both the spin and the orbital phase degrees offreedom on the same footing, in spite of the use of ferro-magnetic elements. These findings could have interestingimplications for the manipulation of the electronic spinin nanoscale conductors.
VII. APPENDIX : MODELING OF OURDEVICES
Throughout the paper, we use the theory of reference27 to explain our experimental findings. Each of our de-vice is characterized by the set { T K , T K ′ , T , P , ϕ R , ∆ ϕ R , ϕ T , ∆ ϕ T , T , P , ϕ R , ∆ ϕ R ,ϕ T , ∆ ϕ T , T K , T K ′ , C Q , C Q , C Q } , where T K [ K ′ ] is the transmission probability from the normal electrode1(4) to the nanotube for the K[K’] orbital, T is thetransmission probability between the two nanotube sec-tions adjacent to contact 2(3), P is the correspond-ing tunnel spin polarization, ϕ T [ R ]2 , (3) is the spin averagedscattering phase for an electron transmission below thecontact 2(3) [an electron reflection against 2(3)], and∆ ϕ R , (3) , ∆ ϕ T , (3) are the Spin Dependence of InterfacialPhase Shifts (SDIPS) at contact 2(3) (see ref. 27). Dueto the unitarity of the contact scattering matrices, thetransmission from lead 2(3) to the nanotube is set by theabove parameters. The results of the scattering theoryat T = 4 . K shown in magenta correspond to the set { . , . , . , . , π, , , , . , . , π, , , , . , . , , . , . } in figure 3c and to the set { . , . , . , . ,π, . π, . π, . π, . , . , π, . π, . π, , . , . , , . , . } in figure 6c. The capacitances are in aF units. For the second case, contrarily to the caseof sample I, we had to include a finite SDIPS at theferromagnetic contacts to enhance the amplitude of theoscillations in M G .We emphasize that the above parameters are subjectto several constraints which minimize substantially theallowed phase space for our fitting procedure. The capac-itances can be estimated from the resonance patterns inthe G and V greyscale plots (see section V). The trans-mission probabilities can be estimated from the measure-ment of the two probe conductance of each section of thedevice at room temperature. The values of G , M G , V and M V measured at low temperature constraint fur-ther the transmission probabilities but also the scatter-ing phases and the tunnel spin polarizations. Note thatfor sample III, we find a very small value of T combinedwith a high value of P , and ∆ ϕ T , (3) . These parametersare necessary to obtain the high M G and very low G P observed in figure 6c. The observed zero bias anomaly in G is a possible signature of electron-electron interactions.This effect is compensated in figure 4b by normalizing G by its average V sd -dependence over all the gate voltagespresented in the figure.To describe the influence of the gate voltages on thecircuit, we introduce the relation δ ij = πC Qij ( α ij V SG + β ij V SG ) /e , e being the elementary charge, C Qij =2 e L ij /hv F being the quantum capacitances of each nan-otube section, the dimensionless couplings α ij and β ij being determined by the full electrostatic problem of ourdevices. FIG. 7: (Figure file too big-see published version for the figure)
Elec-trostatic diagram of our devices. We assume here only nearestneighbor electrostatic coupling.
In determining the gate dependence of the theoreti-cally expected signals, it is important to supplement thescattering theory with a self-consistent determination ofthe electrostatic potential of the circuit. We use a coarse-grained version of the Poisson equation which we solveself-consistently in order to determine the different sidegate actions. Our calculation proceeds along the linesof ref. . We start with the full electrostatic matrix ca-pacitance of our devices which can be derived from theelectrostatic diagram of figure 7. We use a nearest neigh-bor scheme. The total capacitance matrix C T OT reads: C T OT = C L − C L C R − C R C F L − C F L − C F L
00 0 0 2 C F R − C F R − C F R C G − C G C G − C G − C L − C F L − C G C Σ − C m
00 0 − C F L − C F R − C m C Σ − C m − C R − C F R − C G − C m C Σ (1)with C Σ = C L + C F L + C m + C G , C Σ = C m + C F R + C F L + C m , C Σ = C R + C F R + C m + C G In principle, we should determine self-consistently theelectrostatic potentials on each section of the nanotubeusing the full scattering matrix of the problem. Thiswould go beyond the scope of this work. For the sakeof simplicity, we assign a constant value for the electro-chemical capacitance of each section. This assumptionis reasonable in our case because the high coupling ofthe SWNT to the normal electrodes reduces the energydependence of the electrochemical capacitance. The self- consistent equation for the electrostatic potential of eachNT section reads : ( C Σ + 2 C Q ) δU NT − C m δU NT = C G δV SG − C m δU NT + ( C Σ + 2 C Q ) δU NT − C m δU NT = 0 − C m δU NT + ( C Σ + 2 C Q ) δU NT = C G δV SG (2)Finally, we get, α β α β α β = ( C µ C µ − C m ) C G C µ C µ C µ − C µ C m − C µ C m C m C m C G C µ C µ C µ − C µ C m − C µ C m C µ C m C G C µ C µ C µ − C µ C m − C µ C m C µ C m C G C µ C µ C µ − C µ C m − C µ C m C m C m C G C µ C µ C µ − C µ C m − C µ C m ( C µ C µ − C m ) C G C µ C µ C µ − C µ C m − C µ C m (3)For a realistic set of capacitances { C L , C F L , C R , C F R , C m , C m , C G , C G , C Q , C Q , C Q } of about { aF, aF, aF, aF, aF, aF, aF, aF, aF, aF, aF } , we find α ′ s and β ′ s which are ingood agreement with the observed slopes in the differenttartan patterns. For example, for the above parameters,we get the following coupling matrix : . . . . . . (4)For each fitting procedure, one has to adjust the valuesof the α ′ s and the β ′ s in order to account for the gatedependence of the observed signals. We use values whichare consistent with the above determination. Note that we have omitted the influence of the back gate voltagehere since it is set to a constant value in our measure-ments. Acknowledgments
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