The Kondo-effect is a many-body phenomenon arising due to conduction
electrons scattering off a localized spin. Coherent spin-flip scattering off
such a quantum impurity correlates the conduction electrons and at low
temperature this leads to a zero-bias conductance anomaly. This has become a
common signature in bias-spectroscopy of single-electron transistors, observed
in GaAs quantum dots as well as in various single-molecule transistors. While
the zero-bias Kondo effect is well established it remains uncertain to what
extent Kondo correlations persist in non-equilibrium situations where inelastic
processes induce decoherence. Here we report on a pronounced conductance peak
observed at finite bias-voltage in a carbon nanotube quantum dot in the spin
singlet ground state. We explain this finite-bias conductance anomaly by a
nonequilibrium Kondo-effect involving excitations into a spin triplet state.
Excellent agreement between calculated and measured nonlinear conductance is
obtained, thus strongly supporting the correlated nature of this nonequilibrium
resonance.Comment: 21 pages, 5 figure

A Kogan

A Rosch

A. Rosch

AC Hewson

B Babić

C. M. Marcus

D Goldhaber-Gordon

DM Zumbühl

H Jeong

J Appelbaum

J Nygård

J Paaske

J Park

J. Nygård

J. Paaske

L Glazman

LH Yu

MN Kiselev

N. Mason

P Jarillo-Herrero

P. Wölfle

PW Anderson

S Sapmaz

S Sasaki

SM Cronenwett

T Ng

VN Golovach

W Liang

WG van der Wiel

Y Oreg

English

arXiv

Crossref

Non-equilibrium singlet–triplet Kondo effect in carbon nanotubes

Paaske, J.

Rosch, A

Wolfle, P

Marcus, CM

Nygård, J.

Copenhagen University Research Information System

u n i ve r s i t y  o f  co pe n h ag e n  Københavns UniversitetNon-equilibrium singlet-triplet kondo effect in carbon nanotubesPaaske, J.; Rosch, A; Wolfle, P; Marcus, CM; Nygård, J.Published in:Nature PhysicsDOI:10.1038/nphys340Publication date:2006Document versionEarly version, also known as pre-printCitation for published version (APA):Paaske, J., Rosch, A., Wolfle, P., Marcus, CM., & Nygård, J. (2006). Non-equilibrium singlet-triplet kondo effectin carbon nanotubes. Nature Physics, 2(7), 460-464. https://doi.org/10.1038/nphys340Download date: 02. Feb. 2020Nonequilibrium Singlet-Triplet Kondo Effect in Carbon NanotubesJ. Paaske, A. Rosch, P, Wo¨lfle, N. Mason, C. M. Marcus, J. Nyg˚ard.Supplementary Discussion1. Nanotube transport experimentsOur nanotubes were grown by chemical vapor deposition (CVD) on a doped silicon sub-strate with a 400 nm oxide cap layer. Ferric iron nitrate nanoparticles deposited from asolution in isopropyl alcohol acted as catalyst for the CVD process, which was carried out ina tube furnace by flowing methane and hydrogen over the sample at 900 ◦C1. This processyielded mostly individual single-wall nanotubes as determined by atomic force microscopy.The nanotubes were contacted by thermally evaporated metal electrodes (35 nm Au on4 nm Cr), spaced by 250 nm and patterned by electron beam lithography. The two-terminalconductance was measured using standard lock-in techniques with ∼ 5 µV ac excitationand voltage bias V applied to the source with the drain grounded through a low-impedancecurrent amplifier. The effective electron temperature can differ from the measured mixingchamber temperature in the dilution refrigerator. A base electron temperature of Tel ≈80 mK was estimated from the temperature dependence of device characteristics.In the range of gate voltage considered in this paper, the room temperature conductanceof the device is around 1.8 e2/h and independent of back-gate voltage Vg. At lower gatevoltages, a weak Vg dependence at room temperature is seen and can be attributed to asmall band gap in the metallic nanotube induced by perturbations.2 The coupling decreaseswith Vg and at low T the Coulomb blockade peaks disappear at Vg ∼ −2.5 V, i.e. around∆N = 40 electrons away from the present Vg range.312. Low energy two-particle states and effective Kondo-HamiltonianThe relevant single-particle states of the nanotube carry spin σ =↑, ↓ and orbital indexi = 1, 2, and the lowest lying two-particle states are denoted as follows:| s〉 =(| ↓, ↑〉 − | ↑, ↓〉)⊗ |1, 1〉/√2 (1)|−1〉 = | ↓, ↓〉 ⊗ (|1, 2〉 − |2, 1〉)/√2 (2)| 0〉 =(| ↓, ↑〉+ | ↑, ↓〉)⊗ (|1, 2〉 − |2, 1〉)/2 (3)| 1〉 = | ↑, ↑〉 ⊗ (|1, 2〉 − |2, 1〉)/√2 (4)| s′〉 =(| ↓, ↑〉 − | ↑, ↓〉)⊗ (|1, 2〉+ |2, 1〉)/2 (5)| s′′〉 =(| ↓, ↑〉 − | ↑, ↓〉)⊗ |2, 2〉/√2 (6)These states have energies Es, E−1,0,1 = Es + δ − J , Es′ = Es + δ and Es′′ = Es + 2δ, andsince J < δ, the singlet | s〉 is the ground-state. We include only the five lowest lying statesand neglect the highest lying singlet, | s′′〉, altogether. Within the Hilbert-space of these fivestates, the effective Kondo-Hamiltonian takes the form:H =∑k,σi=1,2α=L,R(εk − µα)c†αikσcαikσ +HintwhereHint =12νF∑k,k′,σ,σ′i,j=1,2α,α′=L,R{gijα′α[δij ~S + τ3ij~T + τ 1ij ~Pij]· ~τσ′σ+pijα′α[δij |s〉〈s|+ 12(τ+ij |s〉〈s′|+ τ−ij |s′〉〈s|)]δσ′σ+qijα′αδij∑m=−1,0,1,s′|m〉〈m| δσ′σ }c†α′ik′σ′cαjkσ (7)withgiiα′α = 2νF tiα′t∗iα/EC , g12α′α = (g21αα′)∗ = 2√2νF t1α′t∗2α/EC ,piiα′α = τ3iigiiα′α, p12α′α = (p21αα′)∗ = g12α′α,qijα′α = 0.(8)The vector of Pauli-matrices is denoted by ~τij and all terms in the interaction part have theform of spin, and orbital exchange, except for the two terms in the last line proportional2to δijδσ′σ which are pure potential scattering terms. Throughout, we use the conventionA± = Ax ± iAy, for any vector-operator ~A. The Hamiltonian is expressed in terms of thetwo-particle vector-operatorsS+ = (S−)† =√2 ( |1〉〈0|+ |0〉〈−1| ) ,Sz = |1〉〈1| − |−1〉〈−1|,P+12 = (P−21)† =√2 |1〉〈s|,P−12 = (P+21)† = −√2 |−1〉〈s|,P z12 = (Pz21)† = −|0〉〈s|,T+ = (T−)† =√2 (−|1〉〈s′|+ |s′〉〈−1| ) ,T z = |0〉〈s′|+ |s′〉〈0|,together with the scalar-operators|s〉〈s| = 13(P 2 − 12S2),|s〉〈s′| = (|s′〉〈s|)† = −13~P · ~T ,∑m=−1,0,1,s′|m〉〈m| = 13(T 2 + S2), (9)where ~P = ~P12 + ~P21. Defining also the operator M = [~T , ~P ]/3i = i(|s〉〈s′| − |s′〉〈s|), wenote that the ten operators {~S, ~T , ~P ,M} satisfy the commutation relations:[Si, Sj] = iεijkSk, [Si, Tj] = iεijkTk, [M,Ti] = iPi,[Ti, Tj] = iεijkSk, [Si, Pj] = iεijkPk, [M,Pi] = −iTi,[Pi, Pj] = iεijkSk, [Ti, Pj] = iδijM, [M,Si] = 0.(10)which identifies them as generators of the Lie-algebra SO(5). Notice that T 2+P 2+S2+M2 =4 is a Casimir-operator, i.e. a constant of motion. The Hamiltonian (7) is manifestlyinvariant under spatial rotations, but it will posses this abstract SO(5)-symmetry onlywhen δ = J = 0 and t1α = t2α, in which case one would expect to observe a conventionalzero-bias Kondo-peak, characterized by a Kondo-temperature which is enhanced comparedto the usual SU(2) Kondo-effect. Notice that an SO(5)-Kondo-effect has been discussedearlier in the context of a triple-quantum-dot system4.As pointed out for a double-dot system studied in Ref. 5, ~S and ~P generate SO(4),and it is the addition of the excited singlet |s′〉 which adds four new generators in the3present problem. In a double-dot system at even filling, the inter-dot tunneling breaksthe degeneracy between singlet and triplet states and an inelastic cotunneling channelis generally available. A finite-bias Kondo-resonance in such a double-dot system wassuggested in Ref. 6 and the current due to nonequilibrium cotunneling was calculated inRef. 7, but the combined nonequilibrium Kondo-effect has not yet been examined. Bysimply leaving out the excited singlet |s′〉, our present calculation could readily be appliedto this problem as well.3. Perturbative renormalization group equationsThe (one-loop) perturbative renormalization group (RG) equations satisfied by the fre-quency dependent couplings are established in much the same way as explained earlier inRef. 8. Here, they take the following form:∂giiα′α(ω)∂ lnD= −∑α′′{2giiα′α′′(α′′V/2)giiα′′α(α′′V/2)θ(D − |ω − α′′V/2|) (11)+12gi¯iα′α′′(α′′V/2)gi¯iα′′α(α′′V/2− τ 3iiδ)θ(D − |ω + τ 3iiδ − α′′V/2|)},∂g i¯iα′α(ω)∂ lnD= −12∑α′′{[3giiα′α′′(α′′V/2) + piiα′α′′(α′′V/2)− qiiα′α′′(α′′V/2)](12)×gi¯iα′′α(α′′V/2 + τ 3iiδ)θ(D − |ω − τ 3iiδ − α′′V/2|)+gi¯iα′α′′(α′′V/2)[3g i¯¯iα′′α(α′′V/2) + q i¯¯iα′′α(α′′V/2)− pi¯¯iα′′α(α′′V/2)]× θ(D − |ω − α′′V/2|)} ,∂piiα′α(ω)∂ lnD= −2τ 3ii∑α′′gi¯iα′α′′(α′′V/2)g i¯iα′′α(α′′V/2− τ 3iiδ)θ(D − |ω + τ 3iiδ − α′′V/2|), (13)∂pi¯iα′α(ω)∂ lnD=∂g i¯iα′α(ω)∂ lnD, (14)∂qiiα′α(ω)∂ lnD= −14∂piiα′α(ω)∂ lnD, (15)with the shorthand notation θx = θ(D − |x|), 1¯ = 2 and 2¯ = 1. For a given set of initialvalues (at scale D = D0), parametrized by the bare tunneling amplitudes tiα according to(8), these equations are readily solved numerically for arbitrary ω and D. Taking the limitof D → 0, we obtain the renormalized coupling functions used in the Golden Rule expressionfor the current.4 0  0 50100 g11−1g12−1=p12−1g22−1p11−1−p22−1−q11−1=q22−1ln(D/ TK) ln(D0/ T K)FIG. 1: Inverse couplings vs. bandwidth D. The inverse couplings all vanish at thesame energy-scale defining TK . The couplings shown here are based on the same bare pa-rameters as were used for the low temperature fit in the main paper: {tL1, tL2, tR1, tR2} ={0.032, 0.028, 0.108, 0.063}√EC/νF , J = 0 and D0 = 1 eV, which implies that TK ≈ 2 mK.As usual, when deriving the Kondo-model from an Anderson model, it is convenient tointroduce two angles, (cosφi, sinφi) = (tiL, tiR)/√t2iL + t2iR, and parameterize the exchangecouplings as{gijα′α, pijα′α, qijα′α} = {gij, pij, qij} cosφisinφi( cosφj sinφj )α′α(16)with initial conditions gij = 2(νF/EC)√(1 + τ 1ij)(t2iL + t2iR)(t2jL + t2jR), pii = τ3iigii, p12 =p21 = g12 and qij = 0. The L/R matrix-structure of the couplings is now exterior, in thesense that the RG-equations are identical for every (α′, α) component.For D À δ, V, T, ω, the RG-equations simplify to describe the flow of coupling constants(g, p, q)ij with no dependence on frequency, ω, nor lead-index, α = L,R. The differencein tunneling-strengths to the two orbitals still plays an important role, and enters via theinitial conditions. From the RG-equations it is clear that all the different couplings divergeat the same energy-scale and it is this scale which we henceforth refer to as the Kondo-temperature, TK (see Fig. 1). The numerical solution of these high-D RG-equations allowsus to study the dependence of TK on the three independent initial values g11, g22 and D0.5This reveals the following nearly perfect interpolation-formula for TK :TK ≈ D0e−1/(2(g11+g22)−0.84(g−111 +g−122 )−1), (17)where 0.84 is a good approximation to a universal number describing this particular Kondo-effect. For g11 = g22 this reduces to TK ≈ D0 exp(−0.28/g11). If we leave out the singletstate |s′〉, the RG-equations are modified accordingly and we find that the factor of 0.28 isreplaced by 0.36, as found earlier in a study of vertical quantum dots9(cf. also Refs. 10,11).Notice that in the standard Kondo-effect involving a zero-bias conductance peak, theKondo-temperature can be estimated directly from the width of the conductance peak. Inthe present problem, however, TK is of little physical significance insofar as the non-linearconductance is characterized mainly by the spin-relaxation rate Γ (see sec. 5) and thesubband-mismatch δ. Nevertheless, TK still encodes a scaling-property of the RG-solutionsand shows how our rather arbitrary choice of bare bandwidth D0 = 1 eV is linked tothe values of the bare couplings. In spite of the fact that the Kondo-temperature in thisproblem is enhanced over that of the standard SU(2) symmetric Kondo-model, our fittingparameters imply that TK ≈ 2 mK, which is much smaller than the value of 1 K found forthe neighboring Coulomb-blockade valley having an odd number of electrons on the tube(cf. Fig. 2b in main text). Estimating the effective coupling in the neighboring valley fromEq. (17) with g22 = 0, we find that this big difference in Kondo-temperature corresponds toa mere reduction of the total hybridization to orbital 1 (√t2L1 + t2R1) by roughly 30% whenchanging from N=1 to N=2.4. Nonequilibrium distribution functionsUsing the renormalized coupling-functions, we may calculate the transition rates betweenthe various two-particle states using the Golden Rule expressionWγ′γ(V, δ, T ) =2pi~∫ ∞−∞dω∑σ,σ′=↑,↓α,α′=L,R|gγ′;γα′,σ′;α,σ(ω)|2f(ω − µα)(1− f(ω + εγ − εγ′ − µα′)), (18)where f denotes the Fermi-function. The nonequilibrium distribution functions for the two-particle states, nγ, are then found by solving the steady-state quantum Boltzmann equation∑γ′Wγγ′nγ′ =∑γ′Wγ′γnγ, (19)6together with the constraint∑γ=s,−1,0,1,s′ nγ = 1, ensuring that the two electrons in thehalf-filled shell on the nanotube occupy exactly one of the five lowest two-particle states.These voltage-dependent distribution functions are subsequently plugged into the GoldenRule expression for the current.5. Spin-relaxationAs demonstrated in Ref. 12, a finite bias gives rise to Korringa-like spin-relaxation viainter-lead particle-hole excitations and for a single spin-1/2 the logarithmic singularities werefound to be cut off by the (V-dependent) spin-relaxation rates 1/T1,2. To be more precise,it is the broadening of the transverse dynamical susceptibility, 1/T2, which cuts off the log-renormalization of spin-flip exchange couplings, whereas the renormalization of non-spin-flip exchange couplings are contained by the broadening of the longitudinal susceptibility,1/T1. Diagrammatically, these rates arise from a combination of both self-energy, and vertexcorrections to the spin-susceptibility bubble and a simplification which simply omits thevertex corrections will lead to serious mistakes.In the present problem, the inter-orbital transition-rates (from excited to ground-state)are strongly enhanced by the large available phase-space. Ws,m (m = −1, 0, 1) and Ws,s′turn out to be larger than all other transition-rates by roughly an order of magnitude andthe self-energy broadenings or line-widths of the excited states are therefore much largerthan that of the ground-state. At least one of these large self-energy broadenings willcontribute to the cut-off in the log-renormalization of all exchange couplings except one.pii couples the ground-state to itself and the relevant self-energy broadening is very weak.For this particular coupling, vertex corrections will now play the dominant role and theysupply additional terms in the total broadening involving the large Ws,m and Ws,s′ . Alllog-singularities are therefore cut off by rates of the order of these dominant line-widths ofthe excited states.From the expression for the current (cf. main text Eq.(1)), it is clear that the conduc-tance peak at V ∼ δ is mainly determined by the inter-orbital exchange-couplings gi¯i andpi¯i. Furthermore, these couplings turn out to be larger than the others and altogether themain influence of spin-relaxation on the physical current is therefore via these inter-orbitalcouplings. In other words, broadening all log-renormalization by a single effective spin-relaxation rate estimated from an inter-orbital susceptibility is expected to produce only7minute errors.In terms of a generalized ss′-susceptibility, we determine the physical spin-relaxation-rateΓs,s′ as the sum of self-energy, and vertex corrections:Γss′ = Γs,s′v +12(∑γ′ 6=sWγ′,s +∑γ′ 6=s′Wγ′,s′), (20)where Γs,s′v is the contribution from vertex corrections. Just as the vertex-corrections tothe transverse spin-susceptibility for a single spin-1/2 include only non-spin-flip processes12,the correction Γs,s′v picks up only intra-orbital transition-rates and is therefore negligiblecompared to the dominant term Ws,s′ coming from the self-energy broadening of the excitedstate. Γsm (m = −1, 0, 1) is determined in a similar way and since these two different inter-orbital relaxation rates turn out to be very close in magnitude as well as in V-dependencewe define a single effective spin-relaxation rate as their average:Γ =12(Γss′ + Γs1). (21)The spin-relaxation mechanism is incorporated in the RG-equations by replacing θx byθ(D − |√x2 + T 2 + Γ2|) in Eqs. (11-15). Furthermore, all Fermi-functions occurring inthe transition-rates and the current are effectively smeared by Γ, by replacing the energy-conserving δ-functions appearing in the Golden rule expressions with Lorentzians of widthΓ (i.e. spin-susceptibilitites Im[χR]). For example, we evaluate the transition rate WST asfollows:WST (V, δ, T ) =2pi~∑σ,σ′=↑,↓α,α′=L,R∫ ∞−∞dω∫ ∞−∞dε |gs;1α′,σ′;α,σ(ω)|2f(ω − µα)(1− f(ω + ε− µα′))Im[χRST (ε)]≈ 2pi~∑σ,σ′=↑,↓α,α′=L,R∫ ∞−∞dω∫ Λ+δ−Λ+δdε(1µα − µα′ + ε∫ µαµα′−εdω′ |gs;1α′,σ′;α,σ(ω′)|2)×f(ω − µα)(1− f(ω + ε− µα′)) Γ/pi(ε− δ)2 + Γ2≈ 2pi~∑σ,σ′=↑,↓α,α′=L,R∫ µαµα′−δdω′ |gs;1α′,σ′;α,σ(ω′)|2∫ Λ−Λdε (1 +N(ε+ µα − µα′ + δ)) Γ/piε2 + Γ2,(22)where N is the Bose-function and Λ =√δ2 + J2 + V 2 + T 2 is an ultra-violet cut-off onthe spin-susceptibility ensuring convergence of the integral over ε. Notice that, in order tospeed up the numerical evaluation, we have replaced the square of the coupling-function8by its average over the window set by the Fermi-functions. The error introduced by thisapproximation is estimated to be subleading in the small parameter 1/ ln(δ/TK).We find that Γ(V = δ) ≈ 250 mK (varying between 220 mK and 360 mK when changingV over the measured range) and the data clearly sample the full crossover from low to hightemperatures with T lowestel ≈ 81 mK < Γ < 687 mK ≈ T highestel .1 Hafner, J. H., Cheung, C.-L., Oosterkamp, T. H. & Lieber, C. M. High-yield assembly ofindividual single-walled carbon nanotube tips for scanning probe microscopies. J. Phys. Chem.B 105, 743-746 (2001).2 Minot, E. D.,Yaish, Y., Sazonova, V. & McEuen, P. L. Determination of electron orbital mag-netic moments in carbon nanotubes. Nature 428, 536-539 (2004).3 Jarillo-Herrero, P. et al. Electronic Transport Spectroscopy of Carbon Nanotubes in a MagneticField. Phys. Rev. Lett. 94, 156802 (2005).4 Kuzmenko, T., Kikoin, K. & Avishai, Y. Dynamical Symmetries in Kondo Tunneling throughComplex Quantum Dots, Phys. Rev. Lett. 89, 156602 (2002).5 Kikoin, K. & Avishai, Y. Kondo tunneling through real and artificial molecules. Phys. Rev.Lett. 86, 2090-2093 (2001).6 Kiselev, M. N. ,Kikoin, K. & Molenkamp, L. W. Resonance kondo tunneling through a doublequantum dot at finite bias, Phys. Rev. B 68 155323 (2003).7 Golovach, V. N. & Loss, D. Transport through a double quantum dot in the sequential tunnelingand cotunneling regimes, Phys. Rev. B 69 245327 (2004).8 Rosch, A., Paaske, J., Kroha, J. & Wo¨lfle, P. The Kondo effect in non-equilibrium quantumdots: perturbative renormalization group. J. Phys. Soc. Jpn. 74, 118-126 (2005).9 Pustilnik, M., & Glazman, L. I. Conduction through a quantum dot near a singlet-triplettransition. Phys. Rev. Lett. 85 2993-2996 (2000).10 Eto, M. & Nazarov, Y. V. Multiparameter scaling of the Kondo effect in quantum dots with aneven number of electrons, Phys. Rev. B 66 153319 (2002).11 Golovach, V. N. & Loss, D. Kondo effect and singlet-triplet splitting in coupled quantum dotsin a magnetic field, Europhys. Lett. 62 83-89 (2003).12 Paaske, J., Rosch, A., Kroha, J. & Wo¨lfle, P. Nonequilibrium transport through a Kondo dot:decoherence effects. Phys. Rev. B 70, 155301 (2004).9

Non-equilibrium singlet-triplet kondo effect in carbon nanotubes

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Nonequilibrium Singlet-Triplet Kondo Effect in Carbon Nanotubes

Nonequilibrium Singlet-Triplet Kondo Effect in Carbon Nanotubes

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