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On the Kondo effect in carbon nanotubes at half halfing

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a r X i v :c o n d -m a t /0407193v 1 [c o n d -m a t .m e s -h a l l ] 8 J u l 2004On the Kondo effect in carbon nanotubes at half halfingB.Babi´c ,T.Kontos,andC.Sch¨o nenberger ∗Institut f¨u r Physik,Universit¨a t Basel,Klingelbergstr.82,CH-4056Basel,Switzerland (Dated:February 2,2008)Abstract In a single state of a quantum dot the Kondo effect arises due to the spin-degeneracy,which is present if the dot is occupied with one electron (N =1).The eigenstates of a carbon nanotube quantum dot possess an additional orbital degeneracy leading to a four-fold shell pattern.This additional degeneracy increases the possibility for the Kondo effect to appear.We revisit the Kondo problem in metallic carbon nanotubes by linear and non-linear transport measurement in this regime,in which the four-fold pattern is present.We have analyzed the ground state of CNTs,which were grown by chemical vapor deposition,at fillings N =1,N =2,and N =3.Of particular interest is the half-filled shell,i.e.N =2.In this case,the ground state is either a paired electron state or a state for which the singlet and triplet states are effectively degenerate,allowing in the latter case for the appearance of the Kondo effect.We deduce numbers for the effective missmatch δof the levels from perfect degeneracy and the exchange energy J .While δ∼0.1−0.2(in units of level spacing)is in agreement with previous work,the exchange term is found to be surprisingly small:J 0.02.In addition we report on the observation of gaps,which in one case is seen at N =3and in another is present over an extended sequence of levels.PACS numbers:73.63.Fg,73.63.Kv,73.23.Hk,72.15.Qm,81.07.De,85.35.KtI.INTRODUCTIONIn the past decade,transport measurements have emerged as a primary tool for exploring the electrical properties of structures on the nanometer scale.Due to their unique electronic bandstructure,much attention has been focused on carbon nanotubes(CNTs).1For metallic single wall carbon nanotubes(SWNTs)just two spin degenerate one-dimensional(1d)modes should govern their transport properties at low energies,which makes them interesting model systems to explore the physics in reduced dimensions.2Due to thefinite length,given by the lithographically fabricated contacts on opposite sides of the CNT(two-terminal device with source and drain contacts),the one-dimensional CNT is turned into a quantum dot3at low temperatures(typically at 10K),i.e.into a zero-dimensional object with a discrete level spectrum.The confinement is formed by the finite back-reflection at the edges of the contacts.The level spacingδE is determined by the contact separation L and is inversely proportional to it.This particle in the box-model holds provided the level-broadeningΓand the temperature are both smaller thanδE.Γdescribes the life-time broadening proportional to the coupling strength to the leads.Until now,three transport regimes have been identified in SWNTs:A)single-electron tunneling,4which is dominated by the on-site Coulomb repulsion expressed by the energy term U=e2/CΣ,where CΣis the total capacitance;B)the regime of correlated transport, in which higher-order tunneling processes,are appreciable,leading to the emergence of the Kondo effect;5,6and C)the open SWNT for which Coulomb interaction may be neglected and the residual gate-dependence of G can be described as in a tunable Fabry-Perot resonator.7 A)holds for low,B)for intermediate and C)for high transparent contacts.The Kondo-effect,which occurs at intermediate contact transparency,can be seen as the Holly Grail of many electron physics.It hasfirst been observed in quantum dots by Goldhaber-Gordon et al.5and in CNTs by Nyg˚ard et al.8In contrast to the Coulomb Blockade(CB)regime, which only probes the electrons confined on a QD,the Kondo effect incorporates delocalized electrons in the leads coherently.The presence of a degenerate ground state in the quantum dot(for example,a singly occupied level with two-fold degeneracy due to the spin degree of freedom)forms the basis for the Kondo effect.A multitude of coherent second-and higher-order elastic tunneling processes between the Fermi seas of the leads and the quantum-dot state are enabled,leading to the appearance of a narrow peak in the density of states(DOS)right at the Fermi level(the Kondo resonance)at sufficiently low temperatures.Its width is given by the Kondo temperature T K which measures the binding energy of this many-electron state.The peak in the DOS enhances the probability for electrons to tunnel from source to drain.As a consequence the zero-temperature linear-conductance saturates at the quantized conductance G0=2e2/h(unitary limit),provided the device is coupled symmetrically to source and drain.As in atoms,eigenstates in quantum dots(QDs)may be degenerate due to symmetries and together with the spin degeneracy and Pauli principle lead to the formation of elec-tronic shells.Indeed,striking shell patterns have been observed in QDs.9,10The eigenstates (Bloch-states)at the Fermi energy of graphene(two-dimensional sheet of graphite)is two fold degenerate.The two wave functions correspond to the two carbon sublattices(the unit-cell is composed of two C-atoms).This degeneracy is preserved in CNTs and should therefore lead to two degenerate orbitals in afinite-length nanotube in0d.Together with the spin degeneracy,the shells are expected to be four-fold degenerate.This shell pattern has recently been observed by Buitelaar and coworkers11in multi-wall carbon nanotubes (MWNTs)and by Liang et al.in SWNTs.13Within one shell the ground-state spin was shown to follow the sequence S=0→1/2→0→1/2in the former work,whereas a pos-sible triplet ground state for two added electrons was suggested by the latter authors,i.e. the sequence S=0→1/2→1→1/2.Here we focus on CVD-grown metallic carbon nanotubes(CNTs).12,14We willfirst ex-amine the four-fold shell pattern in great detail and demonstrate that the half-filled ground state(i.e.2electrons added to an empty shell)is either a paired electron with S=0or the six possible two-electron states are effectively degenerate due to a level broadening exceeding the orbital mismatch and exchange energy.We furthermore have discovered striking gaps in several samples.This anomaly is at present not understood.II.EXPERIMENTALSingle wall carbon nanotubes(SWNTs)have been grown from patterned catalyst islands by the chemical vapor deposition method on Si/SiO2substrates.15The degenerately doped silicon,terminated by a400nm thick SiO2layer,is used as a back-gate to modulate the electrochemical potential of the SWNT electrically contacted with a source and drain ter-minal.The contacts are patterned by electron-beam lithography(EBL)using polymethyl-methacrylate(PMMA)as resist,followed by metallization with palladium and lift-off.14,16 Once the samples are made,semiconducting and metallic SWNTs are distinguished by the dependence of their electrical conductance G on the gate voltage V g measured at room temperature(T≈300K).14In the rest of the paper we report on measurements performed on metallic SWNTs with relatively low-ohmic contacts,such that co-tunneling and Kondo physics is observable.The electrical characterization of the devices has been performed at low temperature in a3He system at300mK.We measure the electrical current I with a low noise current to voltage amplifier as a function of source-drain(V sd)and gate(V g)voltage and determine the differential conductance G d:=∂I/∂V sd numerically.Finally,the collected data G d(V sd,V g) are represented in a two-dimensional grey-scale representation in which the grey-scale corre-sponds to the magnitude of G d.The linear-response conductance G:=I/V sd with V sd→0 is measured at a small butfinite source-drain voltage of40µV.III.RESULTS AND DISCUSSIONIn this section,we will focusfirst on one set of measurements which we will analyze in great detail.This set of data is shown in Fig.1.Fig.1a shows the linear-response conductance G as a function of gate voltage V g.Fig.1b and1c display the corresponding grey-scale plots of the differential conductance G d in zero magneticfield and B=5T,respectively.White corresponds to low and black to high conductance.The observed patterns correspond to a quantum dot with a relatively strong coupling to the contacts.Signatures of the latter are high conductance‘ridges’,observed at zero bias (V sd≈0)and B=0,caused by the the Kondo effect.This effect is a many-electron effect and requires a relatively high tunneling coupling to the leads in order to be appreciable at temperatures where the measurements take place.As required,5,6wefind that G increases if the temperature is lowered below≈4K to saturate at the lowest temperature close to the unitary limit of G=2e2/h.The characteristic energy scale,i.e.the Kondo temperature T K, has been deduced from the temperature dependence of G in ridge3(not shown)and found to be T K≈2K.The conductance enhancement due to the Kondo effect is observed at zero source-drain voltage if B=0.In a magneticfield,however,the conductance enhancement isreduced and a splitting of the peak conductance tofinite source-drain voltages is expected.5,18 This splitting is visible in Fig.1c which was measured in a perpendicular magneticfield of 5T.That the linear-response conductance G is suppressed in a magneticfield is clearly seen in Fig.1a in which the solid(dashed)curve correspond to B=0(B=5T).Because the many-electron effects(Kondo effect)are suppressed in magneticfield,we can use the linear-response conductance measurement in magneticfield(dashed curve in Fig.1a)to assign the charge states of the quantum dot with reference to the single-electron tunneling picture.A transition from a ground state with N electrons in the dot to one with N+1gives rise to a peak in the conductance,whereas G is suppressed in between. This pattern is nicely seen in the dashed curve of Fig.1a,in which transitions have been labelled.Evidently,these conductance peaks form a repeating pattern in clusters of four. This pattern is the generic shell pattern of an ideal CNT quantum dot.11,13It is caused by the four-fold degeneracy of0d-eigenstates,two of which stem from spin and the other two from the so-called K−K′orbital-degeneracy of graphene.1The four-fold pattern can be regarded as a measure of the quality of the SWNTs.It is not observed in all SWNTs and even if observed it is not usually present over the whole gate voltage range.But it can repeat over several shells,not just two as shown in Fig.1.The degeneracy may be lifted by disorder and by the contacts which may couple differently to the two orbital states.As has been pointed out be Oreg et al.,the four fold pattern may be absent even in a‘perfect’SWNT because the two orbital states can respond differently to the electrostatic gate-field if inhomogeneous.19Let us continue to analyze our data in terms of the constant-interaction model.20In order to assign the states only two parameters are needed:the single-electron charging energy U:=e2/CΣ(which can be expressed by the total capacitance CΣand is assumed to be a constant in this model)and the single-electron level spacingδE.Note,thatδE measures the energy difference between afilled shell to a state with one additional electron.This is sketched in the inset of Fig.1a.To add an electron one has to provide an‘addition energy’composed of charging energy U plus level-spacingδE,the latter only if the electron must be added into a new shell.Therefore,the addition energy∆E equalsδE+U for thefirst electron in a shell,whereas it amounts to‘only’U for the following three added electrons belonging to the same shell.Since∆E is proportional to the gate-voltage difference between adjacent conductance peaks(the conversion factor equals eC g/CΣ),the labelling of statesin terms of charge N in Fig.1a should be understandable.N=0mod4corresponds to a ground state with afilled shell.Due to the large addition energy,the conductance is strongly suppressed for afilled shell,giving rise to the diamond-like white areas(denoted by A,B and C)in the grey-scale plots.Adding electrons to thefilled shell one by one(peaks in G, dashed curve of Fig.1a),we reach the state N=4which corresponds again to afilled shell.In the following,the ground states will be labelled by N=0...3for thefirst quartet and N=0′...3′for the second,where N=4=0′.Relying on the constant interaction model, the ratio between the average level spacing and charging energy amounts toδE/U≈1in our data.It is seen,however,that U is constant to a good approximation,but thatδE varies.For the respective diamonds A,B and C,the level spacingδE amounts to≈7, 5,and3meV,respectively.Theoretically,the level-spacing of an ideal SWNT is given by δE=hv F/2L,where v F=8×105m/s is the Fermi velocity and L the length of the tube that determines the1d cavity.1Taking the nanotube length L measured from the edges of the contacts,which for this sample amounts to L∼300nm,the equation predicts a level-spacing ofδE∼5.5meV in good agreement with the experimental values of3−7meV.The data in Fig.1yields for the charging energy U=5.3±0.5meV and a gate-conversion factor ofα:=C g/CΣof0.08.Focusing on the high-conductance Kondo ridges at zero bias voltage,we see in Fig.1b a ridge at charge states N=1and N=3,whereas G is suppressed at half-filling,i.e.at N=2.The situation is different for the second quartet,where Kondo ridges are observed for all three states N=1,2and3.This phenomenon was reported before by Liang et al.13 Whereas a spin-1/2Kondo effect is expected for N=1(one electron)and N=3(one hole), the situation at half-filling,i.e.at N=2is less obvious.The observed Kondo-effect was assigned to a spin-1triplet state in Ref.13.In the following we re-examine this assignment. To do so,we have to go beyond the‘free’electron model and consider among other things the exchange interaction.There are three additional parameters:First,it has been pointed out that the orbital degeneracy need not to be exact.11,13The orbital mismatch is denoted byδ.With regard to on-site charging energy the doubly occupancy of one orbital is a bit higher in energy as compared to placing each of the two electrons in a separate orbital.This parameters has been introduced by Oreg et al.19and is denoted byδU.Finally,placing the two electrons in different orbitals gives rise to a spin-dependent exchange energy term, which according to Hund’s rule favors the triplet state,i.e.the state with spin S=1.Thisparameter is denoted by J.These parameters have been extracted,both for MWNTs11 and SWNTs13and the analysis of our data confirm the previously obtained values.The importance of the parameters in descending order isδ,J andδU as the least important one. The former work by Buitelaar et al.reportsδ≈0.2and J<0.09and the latter work by Liang et al.reportsδ≈0.3,J≈0.1andδU<0.1(all numbers are measured in units of level spacingδE).We neglectδU because it is small and typically much smaller than the bare level broadeningΓ,which-as we will emphasize-matters as well.Since the Kondo effect is the dynamic screening of the local spin by exchange with a sea of electrons,it is tempting to assign the Kondo ridge for N=2to a spin-1(triplet)ground state.However, with regard to the just mentioned parameters,this appears to be unlikely,because J<δ, favoring a spin-0ground state.19Let usfirst contrast the possible states at N=1and N=2for the case of degenerate orbitals(δ=0)and without exchange(J=0),shown in Fig.2a,with the case of afinite level mismatch and afinite exchange energy,shown in Fig.2b.In thefirst(maybe too naive)model of Fig.2a,the degeneracy equals4at N=1and clearly Kondo physics can emerge.At N=2the degeneracy is even larger,amounting to6and second-order elastic spin-flip processes are energetically allowed so that Kondo physics can emerge as well.Here, two states are paired-electron states and the other four may be labelled as one singlet state with spin S=0and three triplet states with S=1,denoted as S and T states.The Kondo effect may be expected to be even enhanced in this case due to the larger number of states.21This scenario corresponds to the Kondo effect for which the singlet and triplet states are degenerate.This has been realized experimentally in semiconducting quantum dots by tuning the states with either a magnetic or an electricfield.21,22,23Once we go over to the more realistic model shown in Fig.2b,assuming that exchange and level mismatch are non-zero and of comparable magnitude,the N=1states remains‘normal’in the sense that only the lowest lying orbital need to be considered.At half-filling,i.e.at N=2,there are however two possibilities:19if exchange dominates(J>δ),the ground state is the spin triplet(T)state,whereas if the opposite holds,the ground state is a paired electron(PE) residing on the lowest orbital state.The energy difference between the T and PE states is given by∆=δ−J.24Although the Kondo effect is(in principle)possible for the triplet, there is no Kondo effect possible for paired electrons.Note,that unlike previous discussions, there are three cases to consider at half-filling.Two may give rise to Kondo and one doesnot.To have an abbreviation at hand we denote the three N=2states with ST(degenerate ground state),PE(paired electron ground state)and T(triplet ground state).As mentioned above,the Kondo ridge at N=2has been assigned to the triplet state.13Though this is tempting atfirst sight,there is no necessity.In fact,this assignment is unlikely,first because J is measured to be small(usually smaller thanδ)and secondly,the Kondo temperature T K for triplet Kondo is predicted to be much smaller than T K for S=1/2-Kondo25(an estimate will follow below).We now look at the excitation spectra for N=1,3and N=2in zerofield.This is shown in Fig.3(a-c),where(a)and(b)correspond to oddfilling(N=1,3)and(b)to half-filling (N=2).The energy∆of thefirst excited state atfixed N relative to the ground state is given by the level mismatchδ,both for N=1and N=3,i.e.∆1,3=δ.This is illustrated in the respective insets on the right.Thefirst excited states show up as a conductance peak atfinite V sd,corresponding to the excitation energy.This is a so-called inelastic co-tunneling process.We obtain from the measurement∆1=0.92meV and∆3=0.85meV. Hence,the level mismatch is given by0.89±0.4meV.For the N=2case we have to distinguish two possibilities:if J>δ,the ground state is the triplet(T)and the excited state the paired electron(PE)state,yielding∆2=J−δ.If on the other hand J<δ,the states are reversed,yielding∆2=δ−J.In general,∆2=|δ−J|.From the experiment (Fig.1c)we deduce∆2=0.88meV.We stress that we measure on one and the same shell so that we can use the parameterδ,measured for the N=1,3case,also for the N=2 paring the numbers,leaves open two possibilities:either the exchange parameter is quite small,i.e.J∼0(taking the possible errors into account J 0.1meV),or it is quite large J 1.65meV.If the latter would be true,it would be a remarkable coincidence that wefind|δ−J|∼δwith J≈2δ.Moreover,the ratio J/δE>0.3would be quite remarkable with regard to previous measurements.On the other hand,comparable values for J have theoretically been predicted,however only for small diameter tubes.For example, J/δE was estimated to be≥0.22and≥0.44for a(10,10)and(5,5)tube,respectively.19 However,the diameter d of CVD-grown NTs is known to vary substantial and in particular wefind that d 2nm,14from which one would theoretically predict an exchange parameter of order J/δE∼0.1,which disagrees with thefinding above.If J were indeed as large as 2δ,and therefore J>δ,the triplet state would be the ground state at half-filling,i.e.at N=2and N=2′.The Kondo effect at half-filling must then be assigned to the S=1Kondo effect.To explain the absence of the Kondo effect for N=2and its presence for N=2′,one would have to argue that the Kondo temperature is smaller than300mK at N=2,whereas it is larger at N=2′.Pustilnik et al.26showed that the Kondo temperature for the triplet state is smaller than T K,1/2for the spin-1/2case.More precisely,T K,1 T S=1Kcan be estimated according to k B T K,1= k B T K,1/2 2/δE.The average width of the zero-bias resonances at N=1,3in the left quartet is measured to be0.35meV,yielding as a prediction T K,1≈0.25K.In the right quartet the same procedure yields for N=1′,3′a mean width of0.8meV,from which one predicts T K,1≈1.5K.Hence,the comparison with the measuring temperature does not exclude S=1Kondo,as T K,1 0.3K in the left quartet and T K,1>0.3K in the right one.However,the ratio T K,1/T K,1/2measured in the right quartet is inconsistent with theory which predicts T K,1/2/δE.The former is evaluated to0.8...1.6,whereas the latter is at most0.3.In simple terms,the appearance of the two resonances at N=1′and N=2′with essentially one and the same width(1.1and0.9meV, respectively),makes triplet-Kondo quite unlikely.In magneticfield the states further split due to the Zeeman energy given by±gµB B/2, whereµB is the Bohr magneton and g the so-called g-factor.g has been measured in related electrical measurements on carbon nanotube quantum dots and found to agree with the free electron value of g=2.3,11,27Due to the Zeeman-splitting the excitation spectrum changes.At N=1and for small magneticfields(with regard to the level mismatch),the spin-1/2Kondo resonance is expected to split,11evolving into inelastic co-tunneling with an excitation energy given by∆Z=gµB B.Because of the relatively large width of the zero-bias resonances,this shift is hardly visible for small magneticfields in the experiment.That is why we have chosen a relatively largefield of5T.Thisfield yields for the Zemann excitation energy0.58meV,taking g=2.Note,however,that there is a second excited state given by the level mismatchδ≈0.9meV.If we analyze the non-linear differential conductance as a function of V sd,we see two excitation lines(one at positive and one at negative bias),which are markedly broadened,suggesting an overlap of two excitation features,see Fig.3a(grey overlaid graph).The onset of the excitation peaks agrees with the Zeeman energy(arrows from below).This analysis is of particular importance in the N=2case,because it allows us to distinguish the PE from the T ground state unambiguously,see Fig.4.If the ground state is the paired-electron state,thefirst excitation occurs at energy∆2=δ−∆Z−J ≈δ−∆Z(because J≈0)and the second lies atδ.In contrast,if the ground state isthe spin-1triplet state thefirst two excited states have energy∆=J+∆Z−δ≈δ+∆Z and J+∆Z≈2δ+∆Z(because J≈2δin this case).This is shown(approximately to scale)in the illustrations of Fig.4b and c,respectively.Based on thefield-dependence of the excitation spectrum we can predict the position of the excitation peaks for the two cases. In the measurement,shown in Fig.4a,the upper black arrows point to expected excitations if the ground state is the paired-electron(PE)state,whereas the upwards pointing open arrows correspond to the expected excitations for the triplet(T)ground state.It is obvious that the agreement with the PE state is much better.The excitation peaks at zerofield do not move out to larger energies expected for the T ground state,but rather shrink.In particular there are clear low-energy shoulders visible which agree quite reasonably with the expected lowest energy excitation energy for the PE ground state.Taking all arguments together,this makes a convincing case for the ground state at half-filling,i.e.for N=2,which is the paired-electron state.Moreover,the exchange energy must be very small.How do we then have to explain the pronounced Kondo ridge at N=2′, visible in Fig.1b?As we have pointed out when discussing Fig.2,there are two cases at half-filling that allow for Kondo:spin-1Kondo in case of the triplet state or the degenerate ground state,i.e.the ST-state.Based on our previous discussion the former can be excluded, so that the only remaining possibility requires degenerate orbitals.We know that the orbitals are not exactly degenerate.The level mismatch,as deduced from the N=1−3states, amounts toδ≈0.9meV,which is quite appreciable.Due to the relative large width of the zero-bias conductance peaks at N=1′...3′we are not able to deduce the level mismatch on the second shell along the same lines as before for thefirst shell.Though it is possible thatδis smaller in the second shell,it is unlikely zero.We emphasize,however,that it is crucial to compare the level mismatch with the level width,due to the tunneling couplings Γs andΓd to the respective contacts.Ifδ<Γ,whereΓ:=Γs+Γd,the two orbital states cannot be distinguished and are in effect degenerate.We know from other measurements on carbon nanotubes thatΓmay vary a lot with gate voltage.Our picture of the half-filled state is correct,if we can show thatΓis smaller thanδwithin thefirst shell,but larger within the second.There are several ways to deduceΓ.One possibility is to deduce it from the width of the excitation features atfixed N,another one is to analyze the transitions atfinite bias at the edge of the Coulomb-blockade(CB)diamonds.For the left shell the excitation spectra for states N=2and N=3(see Fig.3)yieldΓ≈0.7meV,whereasa cut at V g=3.1V,corresponding to the transition0↔1,yieldsΓ≈0.9meV.Because we cannot resolve excitation features in the right shell,we have to rely on transitions at the edge of CB-diamonds.We deduceΓ≈3meV at0′↔1′andΓ≈1.9meV at3′↔4′. ClearlyΓ δfor the left shell andΓ>δfor the right one in support of our statement.To conclude this part,we can say that the Kondo effect at N=2′is not a triplet Kondo,but arises becauseΓis larger than the level mismatch,resulting in a ground state in which the paired-electron,the singlet and triplet states are effectively degenerate.Our data is only consistent with a very small exchange exchange term of J/δE 0.02.Such a small value can only be reconciled with theory19if either the tube has a large diameter of order∼10nm or the interaction is locally screened,possibly by the presence of other nanotubes forming a bundle.Examination of the measured data shows that for the Kondo resonances labelled with1′and3′in Fig.1b,the positions of the maximum conductance are situated at non-zero bias. This is shown in Fig.5.This phenomenon has been observed in semiconducting quantum dots and was termed the anomalous Kondo effect by Simmel et al.28It was suggested by these authors that the effect is due to asymmetric and energy-dependent coupling strengthsΓs and Γd to the two reservoirs.The effect has thereafter been confirmed theoretically in a single-impurity Anderson model.29The authors show that the peak conductance is shifted provided thatΓs=Γd,but an energy dependence ofΓs,d is not required.We stress here,however,that the Anderson model introduces an additional model-dependent asymmetry in that U→∞, which is not realized in a real quantum dot.At half-filling,there is particle-hole symmetry where electrons(holes)can be exchanged via both the bare state at energyǫ0and the one atǫ0+U.In this case,no shift of the Kondo peak is expected even ifΓs=Γd.Extrapo-lating G(T)to the unitary limit G(0)at zero temperature(not shown)using the standard expression tofit the Kondo effect,i.e.G(T)=G(0)/ 1+(21/s−1)(T/T K)2 2,30we obtainfor the ridge at charge state N=3a zero temperature conductance of G(0)=1.68e2/h, out of which theΓratio is estimated to be≈2.Hence,there is an asymmetry of magnitude comparable to Ref.28.Our statement,that the shift of the Kondo peak tofinite bias is absent at half-filling is beautifully reflected in the data of Fig.5.Due to the four-fold symmetry, half-filling corresponds to charge state N=2′and indeed,this peak has its maximum at V sd=0.The other two resonances are shifted oppositely,one to V sd>0(N=1′)and the other to V sd<0(N=3′).The shift amounts to0.22meV.These shifts are comparable in。

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Carbon Nanotubes as Molecular Biosensors

Carbon Nanotubes as Molecular Biosensors

Carbon Nanotubes as MolecularBiosensorsCarbon nanotubes (CNTs) are cylindrical structures made up of carbon atoms that are one-atom thick and can have a diameter of just a few nanometers. The unique physical and chemical properties of CNTs make them a promising candidate for a wide range of applications, including molecular biosensing.Molecular biosensors are devices that can detect and quantify the presence of specific biomolecules such as proteins, DNA, and RNA. These biomolecules are often indicative of disease states or environmental pollutants, and their detection can provide valuable insight into the underlying biological and chemical processes involved.To function as biosensors, CNTs are often modified with specific biomolecules, such as antibodies or aptamers, which bind to the target molecule of interest. The binding of the target molecule induces a change in the electrical or optical properties of the CNTs, which can be measured and used to detect the presence and concentration of the target molecule.One of the key advantages of CNT-based biosensors is their high sensitivity and selectivity. Due to their small size and high surface area, CNTs can detect biomolecules at very low concentrations, even down to the single-molecule level. Additionally, the surface chemistry of CNTs can be tailored to enhance their binding affinity for specific biomolecules, further improving their selectivity.Another advantage of CNT-based biosensors is their compatibility with various detection platforms. CNTs can be integrated with a range of transducers, such as field-effect transistors, optical sensors, or electrochemical sensors, to amplify and detect the binding-induced changes in electrical or optical signals.CNT-based biosensors have already shown promise in a range of applications, including early detection of cancer biomarkers, monitoring of bacterial infections, and detection of environmental pollutants such as heavy metals and pesticides.However, there are also some challenges and limitations to the use of CNTs as biosensors. One important consideration is their potential toxicity and biocompatibility. While CNTs are generally considered safe, there have been concerns about their potential accumulation and effects on living organisms. Therefore, extensive studies and regulations are necessary to ensure the safe use of CNTs in molecular biosensing applications.In conclusion, CNTs are emerging as a promising platform for molecular biosensing, with their high sensitivity, selectivity, and compatibility with various detection platforms. Further research and development in this field could lead to the development of CNT-based biosensors with improved performance and specificity, enabling their broader applications in disease diagnosis, environmental monitoring, and other areas.。

Kondo effect and STM spectra through ferromagnetic nanoclusters

Kondo effect and STM spectra through ferromagnetic nanoclusters

a r X i v :c o n d -m a t /0112258v 2 [c o n d -m a t .s t r -e l ] 15 A p r 2002Kondo effect and STM spectra through ferromagnetic nanoclustersGregory A.Fiete 1,Gergely Zarand 1,Bertrand I.Halperin 1and Yuval Oreg 21Department of Physics,Harvard University,Cambridge,Massachusetts 012382Weizmann Institute of Science,P.O.Box 26Rehovot 76100,Israel(Dated:February 1,2008)Motivated by recent scanning tunneling microscope (STM)experiments on cobalt clusters ad-sorbed on single wall metallic nanotubes [Odom et al.,Science 290,1549(2000)],we study theo-retically the size dependence of STM spectra and spin-flip scattering of electrons from finite size ferromagnetic clusters adsorbed on metallic surfaces.We study two models of nanometer size ferro-magnets:(i)An itinerant model with delocalized s,p and d electrons and (ii)a local moment model with both localized d-level spins and delocalized cluster electrons.The effective exchange coupling between the spin of the cluster and the conduction electrons of the metallic substrate depends on the specific details of the single particle density of states on the cluster.The calculated Kondo coupling is inversely proportional to the total spin of the ferromagnetic cluster in both models and thus the Kondo temperature is rapidly suppressed as the size of the cluster increases.Mesoscopic fluctuations in the charging energies and magnetization of nanoclusters can lead to large fluctuations in the Kondo temperatures and a very asymmetric voltage dependence of the STM spectra.We compare our results to the experiments.PACS numbers:73.22.-f,72.10.Fk,73.63.Fg I.INTRODUCTION Kondo physics[1]has seen a revival in recent years due to an exciting series of new experiments on mesoscopic[2,3]and nanoscale[4,5,6,7]systems that has enabled a more thorough and controlled study of the basic problem of a local moment interacting with a sea of conduction electrons.It is now possible,for example in quantum dots,to tune the Kondo temperature,T K ,with externally applied gate voltages[2,3,6]or,with the scanning tunneling microscope (STM),to study the Kondo effect at single magnetic impurities on metallic surfaces.[4,5]The superb spatial resolution of the STM permits unprecedented direct local spectroscopic detail of Kondo impurities.Recent low temperature(∼4K)STM experiments[7]have probed the spectroscopy of isolated nanometer and sub-nanometer ferromagnetic (Co)clusters on metallic single walled carbon nanotubes.[8]The tunneling spectrum of these small ferromagnetic clusters exhibited several interesting features–most notably Kondo effect in sub-nanometer diam-eter clusters and well resolved discrete level spacing in nanometer size clusters.Both sub-nanometer and nanometer size clusters exhibited a Coulomb charging gap in the spectrum near zero bias.The Kondo effect in the sub-nanometer size clusters appeared in the spectrum as a Fano-like sharp peak [9,10]with a half-width ∼15meV (T K ∼80K)at 4K.This peak was not present at 100K,presumably due to thermal destruction of the Kondo resonance.The goal of this paper is to present a theoretical framework to study the spin-flip scattering of conduction electrons from finite size ferromagnetic clusters adsorbed on metallic surfaces.In particular,we are interested in understanding some of the size-dependent features observed in the tunneling spectrum.We focus on how the Kondo Effect,the level spacing and the level widths depend on the size of the cluster.Cluster ferromagnetism may be described by two alternate models:(i)In the experiments of Odom et al.[7]magnetism of the cluster is most likely itinerant in nature as bulk Co is an itinerant band ferromagnet.The basic physics of itinerant clusters is captured in the model of Refs.11,12(see Sec.II).(ii)In some other cases,however,including the case of many semiconducting ferromagnets [13]and rare earth materials,ferromagnetism is better described in terms of local moments that couple ferromagnetically or antiferromagnetically to the cluster conduction electrons (holes).This motivated us to introduce another exactly solvable model,where d-electrons produce highly localized magnetic moments.While the excitation spectrum of the two models turns out to be very similar,theway they couple to the metallic substrate is rather different.The most appropriate model to use in a given physical situation depends on the material that composes the nanoparticle and whether it is more appropriate to think of the magnetism of the particle as due to localized moments or itinerant electrons that,although free to move about,are nevertheless polarized strongly enough to give a net spin to the cluster.We make several assumptions during our computations:We assume the clusters to have between,say,eight and forty atoms and that they are large enough that bulk properties,such as the magnetization per atom,are not changed substantially due to their finite size.Furthermore,we assume that the single particle states on the island are of extended character,and therefore the single particle level spacing decreases as ∼1/N A with an increasing number of cluster atoms,N A .We also assume that the ferromagnetic spin splitting ∆s of the extended states on the island is much larger than the single particle level spacing and is approximately independent of the size of the cluster.Thusthe total spin of the cluster is rather large and is roughly proportional to∼N A≫1.Depending on the strength of the electron tunneling between the metal and the cluster we may distinguish two regimes:weak and strong.For weak tunneling we derive an effective Kondo Hamiltonian in second order perturbation theory in the tunneling.In this regime wefind that at very low energy scales the cluster behaves most similarly to a quantum dot with a single unpaired electron.However,ferromagnetic interactions induce a large spin on the cluster and at the same time reduce the value of the effective exchange coupling between the spin of the cluster and the electrons in the metal.The sign of the exchange coupling depends on the specific details of the band structure on the nanoparticle.For a Co cluster,in particular,the itinerant model gives a tendency for antiferromagnetic coupling,and possibly produces a Kondo effect.However,the Kondo temperature T K,decreases very fast with increasing cluster sizes,and therefore the Kondo effect can be observed only for very small clusters.The effective couplings we obtain,J eff,turn out to be in general M×M matrices describing spin-flip scattering among the M conduction band orbital modes that couple to the cluster(M=4for metallic nanotubes[14]).Therefore we expect that a series of Kondo effects takes place at Kondo temperatures corresponding to the antiferromagnetic eigenvalues of J eff.The total spin of the impurity,S T gets successively screened at each Kondo temperature as the temperature is lowered:S T→S T−1/2→S T−1....For a nanotube this compensation can never be complete[15]for any spin greater than2since there are only four channels available for screening.[14]The degeneracy of the remaining unscreened spin will ultimately be lifted by magnetic anisotropy induced by spin-orbit coupling.[16]To make closer contact with the experiments of Odom et al.[7]we compute the STM tunneling spectrum using the itinerant model.The simultaneous observation of the Coulomb gap and the discrete levels,whose widths almost equal the level spacing,suggest that the clusters investigated are in the intermediate tunneling regime.Nevertheless, carrying out perturbative calculations in the weak tunneling limit,we are able to reproduce the essential features of the STM spectrum above the the Coulomb charging energy.Wefind that smallfluctuations in the charging energies can give rise to very asymmetrical STM spectra.From the STM spectra of the clusters at higher voltages,we can crudely estimate the Kondo temperature,T K,of a small Co cluster of the size that exhibited a Kondo effect in Ref.7.However,our estimates tend to give a T K that is too small.Wefind that typical mesoscopicfluctuations in the charging energies of a ferromagnetic cluster can considerably increase the Kondo temperature,however,they do not account for the difference between our theoretical estimates and the experimentally observed T K.[7]Nevertheless,the fact that some small Co clusters exhibit Kondo effect while others do not,indicates that these mesoscopicfluctuations may indeed play an important role in determining T K: Since T K depends exponentially on J eff,relatively small changes in the latter can induce large variations of T K. This paper is organized as follows.In Sec.II we introduce the itinerant model of nanoscale ferromagnets.In Sec.III we study the spin-flip scattering of conduction electrons from an itinerant cluster using second order perturbation theory in the cluster-substrate tunneling,and we derive a general expression for the matrix of Kondo couplings,J eff. In Sec.IV we make contact to the experiments of Odom et al.[7]by calculating in a simple approximation the STM tunneling spectrum of a ferromagnetic Co cluster on a metallic substrate in the limit of weak tunneling.In Sec.V we introduce a new local moment model for ferromagnetism and show how to compute the effective interaction J effin this case.In Sec.VI we discuss in detail the connection between our work and recent experiments.Finally,in Sec.VII we present our conclusions.II.ITINERANT FERROMAGNETIC CLUSTER MODELAn itinerant meanfield model of a magnetic nanoparticle wasfirst proposed by Canali and MacDonald[11]and later extended by Kleffet al.[12]to explain the dense spectrum in tunneling measurements of nanometer size Co particles.[17]The basic Hamiltonian isˆH= j,σǫj c†jσc jσ−1N A S· S+E Ccluster2 α,α′c†jα σαα′c jα′(2)is the total spin of the cluster.The second term in Eq.(1)corresponds to ferromagnetic exchange on the cluster: It gives rise to a spontaneous polarization of the cluster by making spin alignment of different levels energetically favorable.Finally the last term describes the charging of the cluster with E C the Coulomb charging energy,ˆN= j,σˆn jσthe total number of electrons on the cluster,and n g a dimensionless gate voltage that determines the number of electrons in the ground state.[18]Throughout this paper we are neglectingfluctuations in the level spacing.In a real nanoparticle,the level spacing between the j th and j+1th energy level can be written asδj= δj +ηj,where δj is the level spacing corresponding to the bulk energy-dependent density of states, δj ∼1/̺(ǫj),andηj( ηj =0)is afluctuating part due to shape irregularities in the nanoparticle.In this paper we takeηj≡0,corresponding to an infinitely strong level repulsion. The advantage of this model is that S z,S2and n j are conserved quantities and therefore the Hamiltonian in Eq.(1)can be diagonalized exactly.In the more general situation a magnetic anisotropy term should be added to the Hamiltonian.Nevertheless,for the experimentally investigated clusters this anisotropy is estimated to be much less then the width of the Kondo resonance observed,and therefore it can be neglected.[19]The ground state can be explicitly constructed:[12]|S0,S z=S0 N G=n↑j=1c†j↑n↓ j=1c†j↓|vac (3)with S0and N the spin and particle number in the ground state.The remaining states within the ground state multiplet may be explicitly constructed using the lowering operator:|S0,S z N G= (2S0)!(S0−S z)!(S−)(S0−S z)|S0,S0 N G.(4)In itinerant ferromagnets there is an approximate rigid band splitting between spin up and spin down electron density of states.[20]Throughout this paper,we will assume that our clusters are large enough that the band splitting energy,∆s,is well approximated by the bulk value of the specific material we are considering.In our model ∆s can be defined as the energy difference between the highest occupied spin up level,ǫA,and the highest occupied spin down level,ǫI:∆s≡ǫA−ǫI,and from band structure calculations it is known to be typically of the order of a few electron volts[20](see Fig.1). It can be determined by demanding that the ground state of the ferromagnetic cluster be stable tofluctuations of energy level occupations with constant particle number,and is related to the interaction parameter U as[11]U=N A2S08∆s2S08∆sδE−,↑≈¯ǫ+E−C− 1N A d0+3S0 ,(10)δE−,↓≈¯ǫ+E−C+ 1N A d0+1S0 ,(11)where the corrections scale as O(1/N2A),and E±C denote the charging energies in the limit of vanishing ferromagnetic coupling and an infinitely dense single particle spectrum:E±C≡E C(12 k,k′Jµν S·a†µkα σαα′aνk′α′(16)where Jµνis a Hermitian matrix of spin-flip exchange couplings among the various orbital channels,indexed byµand ν.(In a one-dimensional system,such as a carbon nanotube,µlabels left-going and right-going modes of different symmetries.In an isotropic two or three dimensional hostµlabels angular momenta about the cluster.)The total spin of the cluster isˆS and the a†µkα(aµkα)are creation(annihilation)operators for conduction electrons of the metal in orbital channelµ,wavenumber k and spinα.Afinite size cluster in an isotropic host will generally scatter electrons among all orbital channels,but in practice Jµνcan be truncated to include afinite number of channels,the largest angular momentum being l∼LIn the limit of weak tunneling,Jµνcan be calculated in second order perturbation theory in the tunneling.The Hamiltonian we consider isˆH=ˆH0+ˆV,whereˆH0=ˆH metal+ˆH cluster and whereˆH metal= µ,k,σǫµ,k a†µkσaµkσis the Hamiltonian of the free conduction electrons of the metal andˆV= µ,σ,j,k(V j,kµc†jσaµkσ+h.c.),(17) whereV j,k µ≡N pn=1V nϕ∗j( R n)ψµ,k( R n),(18)and N p is the number of points of electrical contact between the cluster and the metal.Theϕj( R n)are the wave functions of the j th levels on the cluster at the points of contact R n and theψµ,k( R n)are the wave functions of the metal at R n.The tunneling amplitude V n can be estimated with a knowledge of the wave functions near the cluster-metal contact points and the work functions of the cluster and the metal.We can determine Jµνby equatingf|ˆH effKondo|i = n f|ˆV|n n|ˆV|i2√S0+1/2 j≤I V j,k fµ∗V j,k iν 1δE−,↓+∆s+ǫI−ǫj(21)Jµνe=−1δE+,↑−ǫA+1+ǫj−1S0Aj=I+1V j,k fµ∗V j,k iν 1δE−,↓+ǫA−ǫj ,(23)with J µνs having only non-negative eigenvalues.[23]This is a result of the Pauli principle,because only electrons (holes)in states with spin antiparallel to that of the cluster may hop onto singly occupied levels in second order perturbation theory,and thereby reduce their kinetic energy.Eqs.(21),(22),and (23)constitute one of the central results of the paper.They describe how J µνdepends on the density of states of the singly occupied levels on the cluster,the excitation energies to the N ±1manifold of states and the tunneling amplitudes to the various levels of the cluster.In general,there is a competition among the three terms of Eq.(20),and thus the sign of eigenvalues of J µνdepends on the specific structure of the single particle density of states on the cluster,and other mesoscopic parameters such as δE ±,σ.In the absence of ferromagnetic interactions on the cluster,U =0,however,the ferromagnetic contributions J µνe and J µνd identically vanish,and the effective interaction is always antiferromagnetic,provided there is an odd number of electrons on the cluster.The itinerant model has been used independently in Ref.22to describe spin S =1metallic nanoclusters with similar results.In that case the ferromagnetic interaction is weak,U <δ,the island is far from a ferromagnetic unstability,and the S =1ground state of the island results rather from two single particle states of the island having energies accidentally closer than U <δ.Then the ferromagnetic contribution of the doubly occupied and unoccupied levels is usually negligible,and the effective exchange interaction is dominated by the antiferromagnetic contributions of the singly occupied levels.[22]The off-diagonal elements of J µνare a sum of random numbers since the ϕ∗j ( R n )ψµ,k ( R n )of Eq.(18)have random phase for differentj and µ’s.Therefore the size of the off-diagonal elements of J µνwill be down by a factor of ∼ 2S 0∼ ∆s compared to the diagonal ones with µ=ν,and the matrix Jµνwill be dominated by the diagonal elements if the number of scattering channels M is much less than 2S 0.B.Weak Tunneling at a Single Point of ContactIt is instructive and also useful to specialize the results of Sec.IIIA to the important case of a ferromagnetic cluster that makes contact with a metallic substrate in only a single point.Then we can choose a basis where J µνconsists of a single non-zero element as all but one conduction channel will decouple from the cluster (that is,a linear combination of the host orbital modes can be found such that only one orbital mode in the new basis has non-zero amplitude at the impurity location).With the origin taken as the single point of contact,V j,k µof Eq.(18)reduces to V j ≡V ϕ∗j (0)ψk (0).Eq.(23),e.g.,then becomesJ s = j |V j |2δE +,↓−ǫI +1+ǫj +1S 0 ∆s 0dξ̺(ǫI +ξ) 1δE −,↓+∆s −ξ ,(25)with ̺(ǫ)the single particle density of states on the cluster,and |V j |2 j the average hybridization strength.The ferromagnetic couplings J d and J e can be expressed by similar integrals.For S 0≫1/2these integrals can be combined in a straightforward calculation where the integrand1S 0 P ∞−∞∆s ̺(ξ)dξδE −,↑ +̺(ǫA )ln δE +,↑rise to a ferromagnetic contribution.The last two terms of Eq.(26)are mesoscopicfluctuations that depend on the specific charging energies of the nanocluster and the density of states at the top of the minority and majority bands.Other mesoscopicfluctuations come from variations in the tunneling matrix elements and in the level spacing,which are ignored in our model.These mesoscopic corrections become more pronounced for smaller cluster sizes and leadto strongfluctuations around J eff.For Co,the single particle density of states has a maximum within the spin-polarized part of the spectrum,ǫI<ξ<ǫA,resulting in a J effthat tends to be antiferromagnetic.Numerical evaluation of Eq.(26)using the actual density of states for Cobalt[20]shows that the ferromagnetic contributions to J efffrom J d and J e reduce the dominantantiferromagnetic contribution from J s by roughly50-60percent.Fluctuations ofδE±,σin small clusters by a factor of2can lead to sample to samplefluctuations of∼10%in J efffor Co.This ultimately leads to largefluctuations inthe Kondo temperature,T K,from cluster to cluster,since J effappears in the exponent of the expression for T K. Eq.(26)also describes how J effscales with cluster size.From the normalization of the cluster wave functions wefind |V j| 2∼1N A∝1E C∝1λF ,the matrix Jµνis approximately of size∼l2×l2for a cluster in bulk or∼l×l for a cluster in contact witha two-dimensional electron gas.For a nanotube at most4orbital channels can scatter low energy electrons.[14]As we discussed earlier in this case,in principle at least,one may observe a series of Kondo effects.The cluster displays an under-screened Kondo effect,where the spin of the impurity is large and several channels with different coupling constants try to screen it.The channel with the largest coupling to the impurity screens half a spinfirst and ceases to interact with the impurity.Then the channel with the largest coupling from the remaining channels screens half a spin and so on.In practice,however,the various Kondo temperatures are exponentially separated,and the very small Kondo temperatures are likely to be much smaller then the spin-anisotropy energy of the cluster.Therefore in a realistic situation one can only observe one,or possibly two Kondo effects.[25]IV.STM TUNNELING SPECTRA AND KONDO EFFECTA.STM Tunneling Spectra of Ferromagnetic ClustersIn the simplest model of STM spectra the tip is treated as a point source of electrons and the resulting signal is proportional to the local density of states(LDOS)at the tunneling position r,[26]dIπIm[G R( r,ω=eV)].(27)Here G R( r,ω)is the retarded Green’s function of the substrate below the tip,and V is the voltage difference between the STM tip and the substrate.Below we consider only tunneling directly into the cluster,ignoring any Fano-type interference effects.[9,10]Let usfirst focus on the high-energy part of the spectrum atω>T K,where the Kondo-type strong correlations are ing standard manipulations we can express the retarded Green’s function asG R( r,ω)≈1(ω−(ˆE−E G))−M R(ω) EE′ E′|ˆϕ†( r)|G + G|ˆϕ†( r)|E 1dV∝ j,σ,±|ϕj( r)|2Γj±,σwe ignore(i)the amplitude modulation,(ii)fluctuations in the level spacing of the cluster beyond that given by band structure calculations[20](as we discussed in Sec.II)and(iii)fluctuations in the hybridizationΓj0≈Γ0.These effects could be included with some effort,but they would not be expected to change our conclusions.In principle,contributions from phonon scattering or electron-electron interactions also contribute to the relaxation rateΓj±,σ;however,in the experiments of Ref.7these can probably be neglected with respect to the single particle relaxation channel considered here.To compute the tunneling spectrum of an actual cluster with N A atoms we generated a discrete set of levelsǫj with a level spacing corresponding to the single particle density of states in Co.[20]As discussed above,the non-uniform density of states is important to obtain a predominantly antiferromagnetic coupling,but is also important to obtain a stable ground state with a partially polarized band.The STM spectrum for low bias voltages is governed by the level structure nearǫA andǫI since these are the only states probed in an STM measurement at low voltage bias.The majority and minority level spacing at these energies can be estimated from band structure calculations asδA=5.55 eV/N A,δI=1.43eV/N A and S0=0.855N A.[20]For Co,∆s=ǫA−ǫI≈2eV.For the sake of simplicity we set¯ǫ=(ǫI+ǫA)/2=0in our computations,effectively absorbing it into n g.This corresponds to a specific choice of contact potential between the lead and the cluster,and does not influence the overall features of the spectrum.Eq.(30)assumes the cluster is in its ground state before the electron from the STM tunnels into it.Thus,in the numerical calculation it is important to ensure that the parameters of the model give a ground state with definite S0 and ing Eq.(1)and Eq.(5)and keeping track of the1/N A corrections which are all put into d0,one can derive the conditions for stability of the ground state,which we assume to have N s≈1.71N A singly occupied levels.By considering thefluctuations in the ground state spin shown in Fig.1B and C one can show that to guarantee stability, d0must satisfy the conditions:−N A(S+1) δA−∆s<d0<3.21eVN Aand E−C>2.48eVN Aat the Fermi level with relative weight∼T KJ eff̺0e−1/J eff̺0(32) where D is the bandwidth of the conduction electrons of the metallic host and̺0is the density of states of the hostat the impurity site.For a ferromagnetic cluster,however,this formula is incorrect,since the Kondo Hamiltonian provides an appropriate description of the cluster dynamics only below the characteristic energy of inelastic excitations,E inel≡min{E C,δA,I}.As shown for example in the two-level system model,[27]above this energy scalefluctuations to various excited states destroy the coherent spin processes leading to the Kondo effect,and in the renormalizationgroup approach in the regime above E inel the Kondo coupling remains unrenormalized.Therefore,for a single point of contact we estimate the Kondo temperature asT K∼min{E C,δA,I}∞−∞∆s̺(ξ)dξδE−,↑ +̺(ǫA)ln δE+,↑2πS0 Pfluctuation effects typically change the value of J effaltogether by10−15%.It seems unlikely suchfluctuations could bring the theoretical estimate of T K into the experimentally observed range.Thus,there is a discrepancy between experiment and theory.There may be several explanations for this disagreement that we discuss in detail in Sec.VI of the paper.Eq.(34),on the other hand,is in qualitative agreement with the experiments in that it predicts an extremely rapid decrease of T K with increasing cluster size.Since for a single point of contactΓ0∼1/N A,S0∼N A,and̺(ξ)∼N A, the dimensionless exchange coupling scales as J eff̺0∼1/N A.We also verified numerically that the scaling J eff∝1N A ˆS c·ˆS d+E C(S d+1)∆s+O(1/N A),(37)where∆s is now the band splitting of the conduction band.Considering the type of particle-hole excitations shown in Fig.1and using Eq.(37),onefinds that the excitation spectrum is very similar to that of the itinerant model.In particular,wefind that for chargefluctuationsδE±,σ∼E C/2,and spinfluctuations have a gap∼δA,δI.putation of the exchange couplingThe coupling constants ˜J µνof the local moment model depend on the sign of the exchange coupling,J ,of Eq.(35).Let us first focus on the case J >0and S d >S c .For the local moment model, f |ˆH effKondo |i =12S T ˜J µν.To evaluate the RHS of Eq.(19)we must again evaluate contributions from the conduction electron states with double,single and no occupation:˜J µν=˜J µνd +˜J µνs +˜J µνe .(38)For the sake of simplicity,let us consider the contribution of singly occupied levels,˜J µνs .Various matrix elements of the type N +1j S T +1/2,S T −1/2|c †j ↓|S T ,S T N arise in course of the evaluation of ˜Jµν,and in contrast to the itinerant cluster model,the intermediate states of the local moment model (with J >0and S d >S c )have an increase in totalspin on the cluster in this case.To evaluate N +1j S T +1/2,S T −1/2|c †j ↓|S T ,S T N we first expand |S T ,S T N S c ,S d and |S T +1/2,S T −1/2 N +1S c −1/2,S d using Eq.(36)to obtainN +1j S T +1/2,S T −1/2|c †j ↓|S T ,S T N =S c S z c =−S c S c ,S d ;S z c ,S T −S z c |S T ,S TS c −1/2,S d ;S z c −1/2,S T −S z c|S T +1/2,S T −1/2 N +1j S c −1/2,S z c −1/2|c †j ↓|S c ,S z c N .(39)What remains to be computed in Eq.(39)is the matrix element N +1j S c −1/2,S z c −1/2|c †j ↓|S c ,S z c N .To determinethis,we use the states of Eq.(4)with S c replacing S 0.The overlap is computed by first directly evaluating N +1j S c −1/2,S c −1/2|c †j ↓|S c ,S c N and then applying the Wigner-Eckhart Theorem for general S z c .This yields N +1j S c −1/2,S z c −1/2|c †j ↓|S c ,S z c N = 2S c .This can then be substituted into Eq.39which finally givesN +1j S T +1/2,S T −1/2|c †j ↓|S T ,S T N =S c S z c =−S c2S c S c ,S d ;S z c ,S T −S z c |S T ,S T S c −1/2,S d ;S z c −1/2,S T −S z c |S T +1/2,S T −1/2 (40)The results of the evaluation of all the matrix elements on the right hand side of Eq.(19)for the local moment model as well as the expression equivalent to Eq.(20)are somewhat lengthy so we relegated them to Appendix B.We evaluated them numerically for S T =S d −S c (antiferromagnetic J and S d >S c )and found that the single particle contribution to the final exchange coupling differs in an overall sign from the itinerant model result of Sec.III,and it is ferromagnetic.On the other hand,if J is ferromagnetic,then S T =S d +S c and the matrix elements above agree in sign with those of Sec.III.This can qualitatively be understood as follows:Suppose the total spin,S T ,of the cluster points upward.Then by assumption S d >S c the spin of the d-levels,S d ,also points up.If the internal interaction,J ,between S d and S c is antiferromagnetic,delocalized electrons of the cluster with spin down will partially screen the local spin,S d ,so that the singly occupied states tend to have have spin down.A substrate conduction electron that hops on a singly occupied state must have,therefore,spin up that is parallel to the total spin,resulting in a ferromagnetic contribution to the effective interaction between the total cluster spin and the substrate.On the other hand,due to the antiferromagnetic interaction with the local spin,hopping to empty states with spin down have an energy smaller than those with spin up (parallel to the total spin),and give rise to an antiferromagnetic contribution to the effective interaction between the cluster spin and the substrate.The case J <0can be understood along the same lines.Similar to the itinerant case,the signs of ˜J µνd and ˜J µνe are always opposite to that of ˜J µνs regardless of the sign of J .Therefore,there is in general a competition between these terms,and the sign of the final coupling depends on specific band structure features.For an antiferromagnetic coupling,J >0,in the limit where ∆s ,E C ≫δ,and S c ,S d ,S T ≫1we can obtain the following simple estimate for a single point of contact,analogous to Eq.(34):˜J eff̺0∼− |V j |2 (ξ−ǫI )(ǫA −ξ)−̺(ǫI )ln δE +,↑δE −,↑ .(41)。

Properties of Carbon Nanotubes

Properties of Carbon Nanotubes

Properties of Carbon NanotubesCarbon nanotubes (CNTs) are cylindrical materials made from pure carbon, composed of graphene sheets rolled into tubes. They are one of the most extensively studied nanostructures with unique mechanical, electrical, and thermal properties. These characteristics make them ideal candidates for a wide range of applications such as drug delivery systems, electronics, energy storage, and nanocomposites. In this article, we will explore some of the properties that make carbon nanotubes so unique.1. Mechanical propertiesOne of the most significant properties of CNTs is their remarkable mechanical strength. CNTs are strong and stiff, and a single-walled carbon nanotube (SWCNT) can withstand strains of up to 10% without breaking. This remarkable strength is due to the strong covalent bonds between carbon atoms and the sp2 hybridization of carbon atoms within the graphene sheets. These features result in the tubes' exceptional stiffness, with an elastic modulus of up to 1 TPa, making them the stiffest known materials.2. Electrical propertiesCNTs possess unique electrical properties due to their one-dimensional structure, which behaves like a one-dimensional quantum wire. SWCNTs are either metallic or semiconducting, depending on their diameter and chirality. This feature makes them ideal for building electronic circuits with high performance and miniaturization attributes. Additionally, CNTs have outstanding electrical conductivity, outperforming copper by a factor of 100-1000 times. The unique combination of semiconducting and metallic properties of SWCNTs has made them an ideal material for building high-performance transistors.3. Thermal propertiesCNTs exhibit excellent thermal conductivity that can surpass copper's conductivity, making them an ideal candidate for high-temperature applications such as heat exchangers, electronics, and aerospace. The thermal conductivity of CNTs is dependenton their length, diameter, and chirality. The unique thermal properties of CNTs can be attributed to their one-dimensional structure.4. Optical propertiesThe unique one-dimensional structure of CNTs results in unusual optical properties such as strong light absorption in the visible and near-infrared regions of the electromagnetic spectrum. Additionally, CNTs have a high surface area that can be used to improve the efficiency of dye-sensitized solar cells.ConclusionCarbon nanotubes exhibit exceptional properties that make them ideal for a wide range of applications. Their remarkable mechanical strength, electrical conductivity, thermal properties, and optical properties make them ideal for use in a wide range of applications such as electronics, aerospace, nanocomposites, and energy storage. The future of carbon nanotubes looks very promising with exciting advances in their synthesis, characterization, and practical applications. As we continue to unravel the secrets of the carbon nanotube, we can only expect their wide-ranging applications to grow even further in the future.。