Kondo impurities in nanotubes: the importance of being ”in” P. P. Baruselli,1,2 A. Smogunov,2,3,4,5 M. Fabrizio,1,2,3 and E. Tosatti1,2,3 1SISSA, Via Bonomea 265, Trieste 34136, Italy 2CNR-IOM, Democritos Unita´ di Trieste, Via Bonomea 265, Trieste 34136, Italy 3ICTP, Strada Costiera 11, Trieste 34014, Italy 4Voronezh State University, University Square 1, Voronezh 394006, Russia 5present address: CEA Saclay, France (Dated: January 17, 2012) Transition metal impurities will yield zero bias anomalies in the conductance of well contacted metallic carbon nanotubes, but Kondo temperatures and geometry dependences have not been anticipated so far. Applying the density functional plus numerical renormalization group approach of Lucignano et al. to Co and Fe impurities in (4,4) and (8,8) nanotubes, we discover a huge 2 difference of behaviour between outside versus inside adsorption of the impurity. The predicted 1 Kondo temperatures and zero bias anomalies, tiny outside the nanotube, turn large and strongly 0 radius dependent inside, owing to a change of symmetry of the magnetic orbital. Observation of 2 this Kondo effect should open the way to a host of future experiments. n a PACSnumbers: 73.63Rt,73.23.Ad,73.40.Cg J 6 1 Nanotubes provide a rich playground for a variety of teringphaseshifts. Asaspecificapplication,weexamine many body phenomena, in particular quantum trans- here Co and Fe impurities adsorbed on the outside sur- ] portbetweenmetalleads[1]. Dependingontransparency face of metallic armchair SWNTs. Results are at first l e of the electrical contact between the nanotube and the disappointing, predicting exceedingly small Kondo tem- - r leads, conduction may range from insulating with strong peratures, and tiny conductance anomalies that would t s Coulomb blockade for poor contacts [2], to free ballistic be hard to observe. When adsorbed inside the nanotube . transport with conductance close to 4e2/h when contact however, the same impurities should yield order of mag- t a transmission is close to one [3, 4]. Kondo effects in in- nitude larger Kondo temperatures, which moreover in- m trinsic nanotubes have been described, either for poor crease with decreasing nanotube radius. When inside, in - contacts [5, 6], and/or in connection with superconduct- fact, the impurity magnetic orbital symmetry switches d n ing leads [7], but none of the classic, extrinsic, single- from parallel to perpendicular to the tube axis, causing o atom impurity type. Here we focus on a high transmis- a dramatic increase of hybridization with the carbon π- c sion lead-nanotube-lead contacts, with a single magnetic orbitals, and a corresponding surge of Kondo energy. [ impurity adsorbed inside or outside a metallic nanotube Following Lucignano et al. [13] we first carry out a 1 segment – an extrinsic case. Conceptually, this should standardspin-polarizeddensityfunctionaltheory(DFT) v constituteareproduciblesystem,whoseconductancecan electronic structure calculation of the nanotube with 1 be precisely and predictably controlled by standard ex- 0 one impurity; the conduction π-electron phase shifts ex- ternal agents such as magnetic field, gate voltage and 3 tracted from that calculation are used to fix parameters 3 temperature. The ballistic conductance of the four nan- of an Anderson model; the model is solved by Numeri- . otube conduction channels will be altered by Kondo im- 1 calRenormalizationGroup(NRG)toobtainKondotem- purity screening, showing up as a zero bias anomaly [8], 0 peratures;finally,anonequilibriumGreenfunctiontech- 2 in a way and to an extent which is presently unknown. nique(NEGF)yieldstheconductancenearzerobias. We 1 Transportanomalieshavelongbeenreported[9]inatip- choose (4,4) and (8,8) single wall nanotubes (SWNT) v: impurity-deposited nanotube geometry, at tip-impurity- (Figs. 1, 2) with a Co or a Fe atom adsorbed at the Xi metal systems [10, 11]; and at molecular magnetic break hexagon center in a fully relaxed position alternatively junctions [12] – systems with very limited atomic and outside or inside the tube (details provided in Supple- r structural control. For an atomistically defined system a mental Material). like ours, we aim at theoretical predictions that are not The impurity projected density of electronic states is just generic – as is often the case in Kondo problems – shown in Fig. 3. butquantitativeandpreciseaboutKondotemperatures, conductance anomaly widths and lineshapes. For that DOS peaks mark the impurity d-states. Relative to purpose, we need to implement an ab-initio based pro- the impurity site, states are even (e) or odd (o) under tocol. According to the ”DFT+NRG” formulation by reflection across an xy plane (orthogonal to the tube), Lucignano et al. [13] that goal can be achieved by solv- and symmetric (s) or asymmetric (a) under reflection ingacustom-builtAndersonmodelwhoseparametersare across an xz plane slicing the tube lengthwise. Note determined by the first principles derived impurity scat- the ”magnetic” orbitals, where up and down spins are exchange-split below and above the Fermi level respec- 2 3dz2(es) 3dxz(os) 3dyz(oa) 3dx2-y2(es) 3dxy(ea) S 4s(es) O D -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 FIG.1. Sketchofthe(4,4)(left)and(8,8)(right)SWNTsin Energy(eV) theyzplane,withanimpurityadsorbedinthehollowposition (either inside or outside). S O D -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Energy(eV) FIG. 3. Symmetry resolved PDOS on the impurity atom for Co outside (a) and inside (b) the (4,4) SWNT. In the first case, orbital d is magnetic, while orbital d is weakly xz xy copolarized, and goes in the mixed-valence regime when the AIM is solved. In the second case, both orbitals are partly FIG. 2. Spatial distribution of dxy and dxz orbitals on the polarized; when the AIM is solved, orbital dxy goes to the (4,4) SWNT (different colours mean a change of sign of the Fermienergy,whileorbitaldxz isinthemixed-valenceregime. wavefunction).ForCo,d (wellhybridized)istherelevantor- Resultsarequalitativelythesameonthe(8,8)SWNT,where xy bital when inside, and dxz (poorly hybridized) when outside; howeverenergiesdifferencesbetweendxz anddxy orbitalsare for Fe, all orbitals are relevant. even smaller. When Fe is considered instead of Co, both orbitals are magnetic in all cases. tively. In Co/(4,4)–OUT there is a single magnetic or- impurity orbitals (one s and five d), a = 1,...,6, hence bital d with {o,s} symmetry indicating a S=1/2 state xz it is of the form for Co (3d94s0), and S=1 for Fe (3d84s0) on (8,8). Con- (cid:32) (cid:33) sider connecting the two ends of a nanotube segment to (cid:88) (cid:88) (cid:16) (cid:17) H = (cid:15) c† c + V c† d +H.c. metal leads, and passing a current. If the contacts are k ikσ ikσ ik,a ikσ aσ transparent, ballistic transport along the metallic nan- ikσ a (cid:88) otubewilltakeplacethroughthetwobandsatFermi(see + t c† c +H , (1) i,kk(cid:48) ikσ ik(cid:48)σ imp Fig. ??, Supplemental Material). Left- and right-moving ikk(cid:48)σ electronic states, φ and φ give rise in e and o combi- l √r where c† creates a spin σ electron in channel i with nations, φ = (φ ±φ )/ 2 to four channels with dis- ikσ e/o l r momentumk alongthetube,d† aspinσ electroninthe tinct symmetries {e/o,s/a} implying without impurities aσ orbitalaoftheimpurity. V isthehybridizationmatrix a conductance 4e2/h for perfectly transmitting contacts. ik,a elementbetweenconductionandimpurityorbitals,which A single impurity will cause each conduction channel to isfiniteonlyiftheysharethesamesymmetry,whilet scatterontotheimpurityorbital(s)ofsamesymmetry, if i,kk(cid:48) describes a local scalar potential felt by the conduction any, giving rise to a scattering phase shift. The (8×8) electronsbecauseofthetranslationalsymmetrybreaking unitary S-matrix is diagonal with eigenvalues e2iδµσ in caused by the impurity. H includes all terms that in- the {e/o,s/a} representation, where µ = es,ea,os,oa, imp volve only the impurity orbitals, which, since the orbital σ =↑, ↓. The transmission and reflection probabilities |t |2 = cos2(δ −δ ), |r |2 = sin2(δ −δ ), O(3) symmetry is fully removed by crystal field, can be ασ eασ oασ ασ eασ oασ written as α=s,a, also relate via the Friedel sum rule ∆ρ (E)= ασ π1dδαdσE(E) to the extra DOS ∆ρασ induced by the im- Himp =(cid:88)((cid:15)ana+Uana↑na↓) purity for each symmetry and spin. Phase shifts calcu- aσ latedbyDFTarethenusedtodeterminetheparameters (cid:88) + U n n +2J S ·S , (2) ab a b ab a b of an Anderson impurity model for the impurity. For a<b each channel, we introduce spin rotation angles defined as θ = 2(δ −δ ). The most general AIM should in- wheren =d† d ,n =(cid:80) n andJ <0,favoring µ µ↓ µ↑ aσ aσ aσ a σ aσ ab clude four scattering channels, i = es,ea,os,oa, and six a ferromagnetic correlation among the spin densities S a 3 of the different orbitals. The parameters of this Hamil- Kondo screening. The same conclusion should apply to tonian are fixed by requiring them to reproduce in the Fe/grapheneeventhoughthetwoorbitalsbecomedegen- mean field approximation the ab initio DFT shifts, in erate, because the Hund’s exchange forces the two elec- addition to orbital energies [13, 14]. The AIM hamilto- trons to occupy each a different orbital in a spin-triplet nian(1)foundinthiswayisstillnumericallyprohibitive. configuration. Since our ultimate goal is transport at low temperature and small bias, we can neglect orbitals that within DFT 103 hatainirnaoeTsdtnhehofeitoesthnrhcesaerFryrtueemucd,raimoaeeruileeotdbthftrilewtffyyrh,eooeorlcyeaumcnictnuctasgipiegvdinbeieenedemtotitwrochareigoeneenrsnymbeiditop=tieuacttylt0.so.hirFdepIbnoelitartuatnaCnhbelsdieos(.fiosaoneuprsetipsdbFrieodoitgxeah.,irmidsC2xea)ozs-. T(K)K1110010--10221 g (G)001..55012-20 -10Vb0(mV)10 20 g (G)001..01255-20 -10Vb0(mV)10 20 g (G)001..-012551000-500Vg (G)b0001(..-m5501210V05)000-5010000Vb0(mV5)001000 Its hybridization, 10-3 (cid:88) 10-4 Γos,xz =π Vo2sk,xzδ((cid:15)k−(cid:15)F), -0.25 -0.125 0 0.125 0.25 k the controlling parameter of the Kondo effect, is small. 102 When Co is inside on the contrary, the magnetic or- 10 1 bital is dxy, lying in the z = 0 plane, its radial lobes 10-1 (see Fig. 2) much more hybridized with the tube con- 10-2 dstbtoaiuodltceahtfiiaeoddlnndixdffzecdirhaonenaesnndistdnnedehoxlyisytsb.pardirluTdaeeiyhzmeataotaniogoyarnnbem.ritteiaavcIlje,norsarswtnahrldioetolctehfch.aicensrIgeycnhsobtbafaenoltFgtwhfieeeeieClindnno,sctoraaeuynnastddd-- T(K)K1111111100000000-1-------09876543 g (G)001..5501-2 -1Vb(0mV) 1 2 g (G)001..01255-2 -1Vb0(mV)1 2 g (G)001..01255-40 -20Vg (G)b00(01m..55012V-4)020-2040Vb(0mV)20 40 Fe, the tangential nature of the outside magnetic orbital -0.25 -0.125 0 0.125 0.25 impliesnostrongdependenceofhybridizationupontube radius. Conversely, the radial nature of the magnetic or- FIG.4. KondotemperaturesasafunctionofcurvatureforCo bital gives rise to a large radius dependence when the (a) and Fe (b); predixted zero bias anomalies are also shown imprity is inside, where it is better ”surrounded” espe- for each case. Dots show calculated values from tab. I, lines cially for smaller tube radius. The impurity-nearest car- are best fits assuming (independently for negative and posi- tivecurvatures)log(T )=a−b/|x|wherex=curvature. The boncouplingV leadstoahybridizationwidthfora(n,n) K shadedareashowstheregionoflowcurvature,whereoursin- nanotube, Γ∼πV2ρ/n∝1/n where ρ is the (radius in- gle band approximation breaks down, and additional terms, dependent) density of states at the Fermi energy. This suchasspin-orbitcoupling,mustbeincludedintheHamilto- increase of V with inverse radius explains the increased nian. As a consequence of considering higher subbands, the coupling of orbital d , for example, of Co on tube (4,4) Kondo temperature, that in our model goes exponentially to xz (0.087 eV) with respect to(8,8) (0.058 eV), only partly zero,cansaturateatafinitevalue(dottedlineinfig. (a),not compensated by a slight decrease of V due to the larger showninfig. (b);thesaturationvalueisguessed). Infig. (b), an arrow shows the range of temperatures where Fe behaves curvature. As a consequence, Kondo temperatures are like an underscreened impurity. predicted to decrease exponentially with increasing tube radius–solongashighersubbandscanbeneglected. For verylargetubes,oursinglesubbandmodelisinvalidand The simplified Anderson impurity models just ob- higher subbands must be taken into account. tained are solved by standard Numerical Renormaliza- It is worth here discussing, at least qualitatively, the tion Group (NRG) [16]; we adopt a two-band model, limit of zero curvature, graphene. For Co or Fe on whichtakesintoaccountchannelsosandea,andorbitals graphene the orbitals d and d are degenerate, and d and d . The approximate Kondo temperatures ob- xz xy xy xz occupied by three electrons only. This unstable SU(4) tainedinthismanner(detailsinSupplementalMaterial) symmetry can be broken by e.g., spin-orbit [15], or by a are given in Table I, column 6. We should warn here Jahn-Tellerdistortion,bothleadingtoanordinarySU(2) that Kondo temperature are by construction affected by Kondoeffect. Ineithercase,Co/nanotubeKondoisbasi- a large error, because of their intrinsic exponential de- cally different from Co/graphene. [15] For Fe/nanotube, pendence on parameters. With that caveat, we verify, as with two electron in two orbitals, the additional Hund’s already stated, that Kondo temperatures turn from very rule coupling, of order 1 eV , is larger than the crystal- small when impurities are outside the nanotube, to large field splitting, hybridization differences, and spin-orbit and radius dependent when inside. The impurity inside interaction – hence both orbitals should jointly undergo thenanotubeisthereforethegeometrywhichwepropose 4 metallic nanotubes. Impurity Nanotube Position Orbital Γ T (K) q K This work was supported by PRIN/COFIN Co (4,4) Out dxz 0.087 0.6 −0.03 20087NX9Y7. P. P. Baruselli would like to thank Co (8,8) Out dxz 0.058 0.1 −0.04 L. De Leo for providing the NRG code and for useful Co (8,8) In dxy 0.126 25 −0.10 discussions. Co (4,4) In d 0.380 600 −0.11 xy Fe (4,4) Out d 0.092 0.002 0.01 xz d 0.082 0.3 −0.02 xy Fe (8,8) Out d 0.062 10−7 0.06 xz [1] S. Ilani and P. L. McEuen, Annual Review dxy 0.044 10−8 −0.02 of Condensed Matter Physics 1, 1 (2010), Fe (8,8) In dxz 0.081 10−5 0.09 http://www.annualreviews.org/doi/pdf/10.1146/annurev- d 0.134 0.01 −0.01 conmatphys-070909-103928, URL http:// xy www.annualreviews.org/doi/abs/10.1146/ Fe (4,4) In d 0.126 0.05 0.35 xz annurev-conmatphys-070909-103928. d 0.396 30 −0.06 xy [2] S. J. Tans, M. H. Devoret, H. Dai, A. Thess, R. E. Smalley, L. J. Geerligs, and C. Dekker, Nature 386, 474 (1997), URL http://dx.doi.org/10.1038/386474a0. TABLEI.Kondoorbitalsforeachsystem,withbroadeningΓ [3] W. Liang, M. Bockrath, D. Bozovic, J. H. Hafner, and Kondo temperature T . K M. Tinkham, and H. Park, Nature 411, 665 (2001), ISSN 0028-0836, URL http://dx.doi.org/10.1038/ 35079517. for experimental verification. [4] B. Babi´c and C. Scho¨nenberger, Phys. Rev. B 70, The zero-bias conductance anomalies and lineshapes 195408 (2004), URL http://link.aps.org/doi/10. are calculated by the Keldysh method for non- 1103/PhysRevB.70.195408. equilibrium Green functions [17]. Approximating the [5] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, Kondo resonance by a Lorentzian, the conductance for D. Abusch-Magder, U. Meirav, and M. A. Kastner, Na- a single channel is a Fano resonance g(v) = (q+v)2 ture391,156(1998),ISSN0028-0836,URLhttp://dx. (q2+1)(v2+1) doi.org/10.1038/34373. where v ≡ VB/ΓK is the dimensionless bias potential, [6] J. Nygard, D. H. Cobden, and P. E. Lindelof, Nature q is the shape parameter, and total conductance is the 408, 342(2000), ISSN0028-0836, URLhttp://dx.doi. sum of even and odd channels. Results are shown in org/10.1038/35042545. Table I and Fig. 4. We generally predict in all cases a [7] A.Eichler,R.Deblock,M.Weiss,C.Karrasch,V.Meden, C. Sch¨onenberger, and H. Bouchiat, Phys. Rev. B conductanceminimumatzerobias(q (cid:39)0),exceptinthe 79,161407(2009),URLhttp://link.aps.org/doi/10. case of Co inside the (4,4) tube, where hybridization is 1103/PhysRevB.79.161407. so large that the Kondo effect gives way to a frank reso- [8] L. I. Glazman and M. E. Raikh, JETP Lett. 47, 352 nant level. Finally, the Fe impurity, having two different (1988),URLhttp://www.jetpletters.ac.ru/ps/1095/ Kondo temperatures can, especially when inside, behave article_16538.shtml. as an underscreened impurity in the range of tempera- [9] T. W. Odom, J.-L. Huang, C. L. Cheung, tures for which T (cid:28)T (cid:28)T . and C. M. Lieber, Science 290, 1549 (2000), K1 K2 http://www.sciencemag.org/content/290/5496/1549.full.pdf, In summary, we applied a DFT + NRG method to URL http://www.sciencemag.org/content/290/5496/ obtain first principle predictions of the Kondo effect in 1549.abstract. theconductanceofwellcontactedmetallicnanotubeseg- [10] N. Knorr, M. A. Schneider, L. Diekho¨ner, P. Wahl, ments hosting a single transition metal impurity atom. and K. Kern, Phys. Rev. Lett. 88, 096804 (2002), URL One first novelty is our claim of ab initio predicting http://link.aps.org/doi/10.1103/PhysRevLett.88. power, which has not been common in Kondo problems. 096804. Themainphysicalsurpriseisastrongdifferencebetween [11] L. Vitali, R. Ohmann, S. Stepanow, P. Gambardella, K. Tao, R. Huang, V. S. Stepanyuk, P. Bruno, outside and inside impurity adsorption; only when in- and K. Kern, Physical Review Letters 101, 216802 side, the Kondo temperatures are large and increase for (pages4)(2008),URLhttp://link.aps.org/abstract/ decreasing radius. The large radius limit, graphene, is PRL/v101/e216802. shown to be an intrinsically different case. Experimen- [12] J. J. Parks, A. R. Champagne, T. A. Costi, tally, it should be possible to insert an impurity inside W. W. Shum, A. N. Pasupathy, E. Neuscamman, a metallic nanotube segment, long enough that the fi- S. Flores-Torres, P. S. Cornaglia, A. A. Aligia, nitesizeleveldiscreetnessissmallerthanKondoenergies, C. A. Balseiro, et al., Science 328, 1370 (2010), http://www.sciencemag.org/content/328/5984/1370.full.pdf, andshortenoughforstrongcorrelationstobenegligible. URL http://www.sciencemag.org/content/328/5984/ ThepredictedKondotemperaturesandzerobiasanoma- 1370.abstract. lies for small radius nanotubes are very substantial and [13] P.Lucignano,R.Mazzarello,A.Smogunov,M.Fabrizio, should not only be measurable, but should open the way andE.Tosatti,NatMater8,563(2009),ISSN1476-1122, to a variety of new observations in a larger variety of URL http://dx.doi.org/10.1038/nmat2476. 5 [14] P. P. Baruselli, A. Smogunov, M. Fabrizio, and [18] P.Giannozzi,S.Baroni,N.Bonini,M.Calandra,R.Car, E. Tosatti, Physica E: Low-dimensional Systems and C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ- Nanostructures pp. – (2011), ISSN 1386-9477, URL cioni, I. Dabo, et al., Journal of Physics: Condensed http://www.sciencedirect.com/science/article/ Matter 21, 395502 (19pp) (2009), URL http://www. pii/S1386947711001639. quantum-espresso.org. [15] T. O. Wehling, A. V. Balatsky, M. I. Katsnelson, [19] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. A. I. Lichtenstein, and A. Rosch, Phys. Rev. B Lett.77, 3865(1996), URLhttp://link.aps.org/doi/ 81,115427(2010),URLhttp://link.aps.org/doi/10. 10.1103/PhysRevLett.77.3865. 1103/PhysRevB.81.115427. [20] P. P. Baruselli, A. Smogunov, M. Fabrizio, and [16] R. Bulla, T. A. Costi, and T. Pruschke, Rev. Mod. E. Tosatti, to be published. Phys. 80, 395 (2008), URL http://link.aps.org/doi/ [21] Y. Yagi, T. M. Briere, M. H. F. Sluiter, V. Ku- 10.1103/RevModPhys.80.395. mar, A. A. Farajian, and Y. Kawazoe, Phys. Rev. B [17] Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 69,075414(2004),URLhttp://link.aps.org/doi/10. 2512 (1992), URL http://link.aps.org/doi/10.1103/ 1103/PhysRevB.69.075414. PhysRevLett.68.2512.