Kondo effect of magnetic impurities on nanotubes

1.1 MB·English

by  P. P. Baruselli

ff
Kondo e ect of magnetic impurities on nanotubes
P.P.Barusellia,b,A.Smogunovb,c,d,e,M.Fabrizioa,c,f,E.Tosattia,c,f
aSISSA,ViaBeirut2/4,Trieste34014,Italy
bCNR-IOMDemocritos,ViaBeirut2/4,Trieste34014,Italy
cICTP,StradaCostiera11,Trieste34014,Italy
dVoronezhStateUniversity,UniversitySquare1,Voronezh394006,Russia
epresentaddress:CEASaclay,France
fINFM,DemocritosUnita´diTrieste,ViaBeirut2/4,Trieste34014,Italy
2Abstract
1
0Theeffectofmagneticimpuritiesontheballisticconductanceofnanocontactsis,assuggestedinrecentwork,amenabletoabinitio
2study[1]. Ourmethodproceedsviaaconventionaldensityfunctionalcalculationofspinandsymmetrydependentelectronscat-
teringphaseshifts,followedbythesubsequentnumericalrenormalizationgroupsolutionofAndersonmodels–whoseingredients
n
aandparametersarechosensoastoreproducethesephaseshifts. WeapplythismethodtoinvestigatetheKondozerobiasanomalies
Jthatwouldbecausedintheballisticconductanceofperfectmetallic(4,4)and(8,8)singlewallcarbonnanotubes,ideallyconnected

6toleadsatthetwoends,byexternallyadsorbedCoandFeadatoms. Thedifferentspinandelectronicstructureoftheseimpurities
1arepredictedtoleadtoavarietyofKondotemperatures,generallywellbelow10K,andtointerferencebetweenchannelsleading

toFano-likeconductanceminimaatzerobias.
]
elKeywords: Kondoeffect,phaseshifts,carbonnanotubes,magneticimpurities,zero-biasanomalies
-
r
t
s
.1. Introduction adjusted to yield, within the Hartree-Fock approximation, the
t
a same channel- and spin-dependent impurity scattering phase
m Thezero-biasanomaliesobservedinSTSconductancespec- shiftsasthosethatwecalculateabinitiobyDFT–whoseinput
troscopythroughadsorbedmagneticimpuritiesandtosomeex-
- information is therefore put to maximal use. For the last step,
dtent in metal break junctions have recently revived interest in
solvingtheAndersonmodels,weemployedastandardnumer-
ntheKondoeffect. Addressingthesesystemstheoreticallyposes
icalrenormalizationgroup(NRG)scheme. Whileothergroups
coseveral problems. In the first place, and unlike quantum dots, have dealt with the overall Kondo problem in different ways
[abinitioelectronicstructurecalculationssuchasdensityfunc- [2,3], wefindour“DFT+NRG”routeextremelyinstructive,
tional theory (DFT) are essential to establish a quantitatively
andworthexploringinmorecomplexsituationsthanthesimple
1meaningful starting point. Which among the impurity-related
v Au-Ni-AucontactstudiedinRef[1]. Inthepresentapplication
levelsandresonancesdrivethespinpolarization, whatistheir
3 weconsiderasinglewallcarbonnanotube(SWNT)asourlin-
multiplicity,theirhybridization,etc. areallquestionsthatneed
0 earconductingsystem,andasingleexternallyadsorbedtransi-
3an ab initio calculation. Next, this information must be trans-
tionmetalatom,eitherCoorFe,asthemagneticimpurity. To
3latedintosomemanageablemanybodyHamiltonian,possibly
beginwith,themetallicnanotubehastwoconductingchannels
1.withoutthelossofthebrutequantitativeinformationprovided insteadofonlyoneasAu. Themagneticatomsinturnhavein
0byDFT.Finally,themanybodyHamiltonian(s)mustbesolved,
principle a richer multiplicity of magnetic levels than Ni. We
2toextractKondoparametersandthepredictedconductancefea-
wishtoexplorewhatthisrichnessmightbring.
1
tures near zero bias, possibly with their behavior with param-
:
veters such as nanocontact geometry, temperature and external
Xifield, to be eventually compared with experiment. One ap- 2. Systemsandsymmetries
proach in this direction was recently taken by our group [1].
r
aGivenananocontactbetweentwoleads,oneidentifies,withthe We considered alternatively Co or Fe impurities on either
help of symmetry, the impinging and outgoing channels that (4,4) or (8,8) metallic SWNTs (see fig. 1). If z is the SWNT
carry current across the impurity. From the matching symme- axis,itselectronicstatesofcanbeclassifiedaccordingtoparity
tryselectedlocaldensitiesofstatesattheimpurity,oneidenti- withrespecttoxyplanereflection(e−o,even-odd)andxzplane
fiestheimportantimpurityorbitalswiththeirdifferentmagnetic reflection(s−a,symmetricandantisymmetric). DFTcalcula-
splittings and hybridizations. This leads to formulate multi- tions(seesection3)predictthattheexternallyadsorbedimpu-
orbital Anderson models, which contain a multiplicity of pa- ritiesshouldhaveminimumenergywhenatthehollowsite(see
rameters to be adjusted. In our scheme the parameters are fig. 2), that is above the center of a carbon hexagon. Assum-
ing that geometry, the impurity electronic states can be clas-
Emailaddress:[email protected](P.P.Baruselli) sified according to the same parity numbers as those of clean
PreprintsubmittedtoPhysicaE January17,2012
(cid:88)
SWNTs. Weareinterestedinparticularin3d and4simpurity H = J σ(cid:126) ·σ(cid:126) (2)
Hund i j
orbitals,whoseparitiesareshownintab. 1. i<j,j=1,4
1(cid:88)
σ(cid:126) = d†(cid:126)s d (3)
3. Abinitioelectronicstructure i 2 iα αβ iβ
αβ
(cid:88)
Wecarriedoutstandarddensity-functionaltheory(DFT)cal- HAnd = (cid:15) c†c −tc†c+V(c†d+dc†)+(cid:15)d†d+Un↑n↓(4)
i ik ik ik i i i i i i i i i i i i i i
culations, allowing for full relaxation of all atomic positions
k
in a unit cell, which comprised 80 and 160 carbon atoms for
whered† createsanelectronontheimpurityorbitalwithsym-
the (4,4) and (8,8) tubes respectively plus one Co or Fe ad- i
sorbed impurity. Calculations used the standard plane-wave metryi,c† createsanelectroninakconductionstatewithsym-
ik
package Quantum-ESPRESSO [4] within the generalized gra- metryi,(cid:126)sarethePaulimatrices,tiisapotentialscatteringterm
dient approximation (GGA) to exchange-correlation function- duetotothechargedensityoftheimpurity, Vi isthecoupling
als in parametrization of Perdew, Burke and Ernzerhof. The oftheimpurityorbitalwiththeconductionelectronstates,(cid:15)i is
planewavecut-offswere30Ryand300Ryforthewavefunc- thebareenergyoftheimpurityorbital,UiistheHubbardrepul-
tions and for the charge density, respectively. Integration over sionontheorbital, J <0isaglobalHundexchangeparameter
theone-dimensionalBrillouinzonewasaccomplishedusing8 (favouring high spin for the isolated impurity), and the single
k-pointsandasmearingparameterof10mRy. Whennecessary particleenergiesofconductionelectrons(cid:15)ik aresuchastogive
to test the sensitivity of DFT results to correlation effects, we aconstantdensityofstates,withexactlythesamevalueasthat
extended to “GGA+U” with a reasonably small Hubbard “U” ofthecleanSWNT(perspindirection)ascomputedbyDFT:
[4]–butgenerallythestraightDFTresultwasused. (cid:88) 1
WefoundthatCobehavesasaS = 1 impurityonboth(4,4) δ((cid:15)−(cid:15)ik)=ρ(cid:39) , i=es,eo,as,ao (5)
2 12eV
and (8,8) SWNTs, its d orbital driving the spin polarization. k
xz
TheCoatomswitchesfromthe3d74s2configurationoftheiso- Inpractice,onlyconductionbandscoupledtoamagneticor-
lated atom to a slightly surprising low-spin 3d94s0 one when bital are retained in our NRG procedure (see section 6). This
adsorbedonthenanotube. FebehavesasaS = 1impurityon leavesuswithasinglebandcoupledtoasingleimpuritylevelin
the (8,8) tube, similarly switching from the high-spin 3d64s2 thecaseofCo(onceorbitald isignored),andwithtwobands,
xy
oftheisolatedatomtoalow-spin3d84s0 intheadsorbedstate. eachcoupledtooneimpuritylevel,inthecaseofFe/(8,8)(once
Here the pair of orbitals d and d is magnetically polarized orbitald isignored).ThecaseofFe/(4,4)ismoreinvolvedand
xz xy z2
(seefig. 3andtab. 2),aresultingoodagreementwithprevious wewillpresentlynotdealwithit.
calculations[5,6]. Orbitalsd forCoandd forFearepartly
xy z2
empty,andfallneartheFermienergyinstraightDFT:butthey
5. JoiningupDFTandmanybody
promptlymovebelowE whenevenasmallU isswitchedon.
F
Weconcludethattheseorbitalsarenotgoingtobeinvolvedin Hamiltonian eq. 1 is easily solved in the (unrestricted)
Kondobehaviourandcanbeneglectedtoafirstapproximation Hartree-Fockapproximation[7],breakingspinrotationalsym-
inordertokeepthemany-bodymodelsimple. Thebehaviorof metry. This leads to a phase shift in conduction electrons of
Fe/(4,4) is complicated. The s orbital is partly filled, and d symmetryi(i=es,eo,as,ao)attheFermienergy
z2
is magnetically polarized besides the (dxz,dxy) pair, so here Fe Γ
shouldbehaveasaS = 23 impurity. δσi =φi+arctan(cid:15)σi (6)
As in previous work [1] we implemented DFT computa- i
tionofthe(spin-polarized)mean-fieldballistcconductanceand, whereφi = arctanπρti (cid:39) 0isthephaseshiftcausedbytheim-
more importantly, of the impurity-related spin- and channel- purity charge scattering. This is numerically found to be neg-
selected phase shifts suffered by the SWNT conduction elec- ligible, so we shall ignore it from now on. The peak of the
tronsasafunctionofenergy. Anexampleisshowninfig. 4for impurityDOSisfoundtobeat
Coonthe(4,4)SWNT. J
(cid:15)σ =(cid:15) +U(cid:104)n−σ(cid:105)−σ (m −m) (7)
i i i i 2 tot i
4. GeneralizedAndersonmodel where
1 Γσ
(cid:104)nσ(cid:105)= arctan i (8)
TheKondomodelisusuallyunderstoodbymeansofamany- i π (cid:15)σ
bodyAndersonHamiltonian[7]. Inourcaseweneedtoextend i
it in principle to the four SWNT conduction bands, each hy- istheaverageoccupationofup/downorbital
bridized with some impurity orbital among the 3d and 4s, of m =(cid:104)n↑(cid:105)−(cid:104)n↓(cid:105) (9)
same symmetry. These impurity orbitals in turn are mutually i i i
coupledbyanintra-atomicferromagneticHundexchangeterm, isthemagnetizationofeachorbitaland
(cid:88) (cid:88)n
H = HiAnd+HHund (1) mtot = mi (10)
i=es,ea,os,oa i=1
2
isthetotalmagnetizationoftheatom. Asin[1],wechooseto SinceinDFTtheyare(cid:39) 0,theycanbesafelyneglected. Sum-
reproduce the same phase shifts at the Fermi energy for each mingup,bothCo/(4,4)andCo/(8,8)shouldexhibita(zerotem-
symmetry, and the same peaks in the density of states of the peratureandzerobias)conductanceG ∼ G ,whereasFe/(8,8)
0
impurity orbitals as those computed by DFT. This allows to should haveG ∼ 0. These results remain valid so long as ei-
uniquely fix (cid:15), U and Γ as long as just one magnetic orbital thertemperatureand/orbiasremainwellbelowT . However,
i i i K
isconsideredineq. 1–thatisthecaseofCo(i = os). When itturnsoutthatKondotemperaturesT arequitelow(seetab.
k
morethanoneorbitalisinvolved,suchasinFe,(orinCoifor- 3), which might make this effect hard to observe in a real ex-
bitald weretobetakenintoaccount)weneedtofixJaswell. periment. Interestingly, a much higher Kondo temperature of
xy
WecanextractJfromtheDFTcalculatedexchangesplittingof about 15 K has been quoted for Co/graphene[11]. While the
filledorbitals,accordingto reasonsforthisdifferencebetweengrapheneandnanotubesare
presentlybeinginvestigated,itshouldbenotedthatseveralfac-
(cid:15)↑−(cid:15)↓ = Jm (11) tors differ, including symmetry, and heavy doping in real, de-
f f 2 tot
positedgraphene.
Since different d orbitals have slightly different splittings, we Finally, we can qualitatively address the predicted bias-
just took an average value as deduced form different orbitals. dependent lineshape of the Kondo conductance anomaly.
ThisyieldsJ ∼1eVinCoandJ ∼1.2eVinFe. Through the Keldysh technique for non-equilibrium Green-
functions it is possible to compute the finite-bias conductance
[12],oncetheimpurityGreenfunctionG((cid:15))iscalculatedfrom
i
6. ResultsofNRGcalculations NRG[13]:
We solved the Anderson Hamiltonian by means of NRG gs,a =1−Γ(cid:61)Gi((cid:15)) (17)
[8, 9], which allows to compute all the needed static and dy-
Forsimplicity,wehavetaken
namic quantities we need in an almost exact, albeit numeri-
cal, way. We extracted the conduction electrons phase shifts Γ /Γ
G((cid:15))= k (18)
fromthesingleparticleenergiesatthezeroenergyfixedpoint, i (cid:15)+iΓ
k
andtheKondotemperaturefromtheimpurityGreenfunctionat
where
imaginaryfrequency:
wπ
k T = Γ =0.342Γ (19)
1 1 (cid:88)|(cid:104)GS|d|n(cid:105)|2 b K 4 K k
G(i(cid:15))= = (12)
i(cid:15)−(cid:15)i−Σ(i(cid:15))+iΓi Zpart n i(cid:15)−(cid:15)n ThisgivesrisetoaFanolineshape[14]
(Zpart is the partition function and GS the ground state). The g = (q+v)2 , v≡(cid:15)/Γ (20)
Kondotemperatureisgivenby s,a (q2+1)(v2+1) k
πwZΓ withq = 0,soforeachband(s−a)thelineshapeispredicted
TK = 4k (13) to be a symmetric antilorentzian, with a width proportional to
b
theKondotemperature. Smallsourcesofasymmetrywillarise
wherew = 0.4128istheWilsoncoefficientandZ isthequasi- froma)thepotentialscatteringt whichweignoredinEq. 4;b)
i
particleresidue fromtheinterferencewithorbitalsbelongingtothesameband
a or s, but with different symmetry e/o; and c) from particle-
∂Σ(i(cid:15))
Z−1 =1− (14) hole asymmetries in eq. 18. However, we estimate that the
∂(i(cid:15))
asymmetryparameterqshouldgenerallyremainbelow0.1. In
Alternatively,anapproximateformula[10],validforoneimpu- Co/(4,4) and Co/(8,8), only gs contributes to the lineshape, ga
ritycoupledtoonechannel, beingalmostone–andmoreoverindependentfromenergyon
the Kondo energy scale. In Fe/(8,8), both g and g have an
(cid:114) s a
UΓ antilorentzianshape, althoughwithvery differentwidths. The
TK ∼0.4107 2i ieπ(cid:15)i((cid:15)i+Ui)/2ΓiUi (15) totallineshapeisjusttheirsum(seefig. 5).
couldbeused,withsimilarresults.
The zero-bias conductance is given, in terms of the final 7. Conclusions
phaseshifts,by
We implemented our recently devised DFT+NRG scheme
G [1]tocalculatetheKondoeffectcausedbyCoandFeadsorbed
g≡ =cos2(δ −δ )+cos2(δ −δ )≡g +g (16)
G es os ea oa s a impuritiesontheconductanceof(4,4)and(8,8)nanotubes. On
0
the methodological side, the present calculation represents a
where G ≡ 2e2 is the quantum of conductance. Note that in goodpedagogicalillustrationofourtechnique. Forthesystems
0 h
the clean tube G = 2G . Phase shifts are only computed for chosen, the predicted anomalies are symmetric antilorentzian
0
Kondo channels, and are found to be always (cid:39) π/2. For the dips, reducing total zero bias conductance to zero for Fe, and
non-Kondochannels,theycanbedirectlyextractedfromDFT. byafactor1/2forCo.Whiletherearenodatatocomparewith,
3
s a
e d ,d ,s d
z2 x2−y2 xy
o d d
xz yz
Table1: Symmetriesofdandsorbitalswithrespecttothexyplane(evene-
oddo)andthexzplane(symmetrics-antisymmetrica).
Imp. SWNT Orb. (cid:15) (eV) U(eV) Γ(eV) T (K)
d K
Impurity MagneticOrbital Symmetry Spin
Co (4,4) d -1.62 2.17 0.082 ∼1
Co d os 1 xz
xz 2 (8,8) dxz -1.83 2.11 0.054 ∼1
(d ) ea 0
xy Fe (8,8) d -1.24 2.01 0.060 ∼10−4
Fe ddxz eoas 121 dxxyz -1.38 2.13 0.043 ∼10−3
xy 2
(dz2,s) es 0 Table3: RecapitulativetableofimportantquantitiesofourAndersonmodels.
KondotemperatureissolowinFeduetothereducedbroadeningΓandtothe
Table2:MagneticorbitalsasfoundfromDFTcalculations,theirsymmetry,and HundcouplingJthatcouplesthetwoimpurityorbitals.
spintheycarry(S =1/2foreachmagneticorbital).Orbitalsinparenthesesare
notKondoorbitals,sodonotcontributetothetotalspinoftheimpurityandare
ignoredinthemany-bodymodel,butstillparticipatesintransport.
thispredictionshouldinprinciplebeamenabletoexperimental
check. However, we note that our calculated Kondo tempera-
tures are very small, which might constitute and experimental
challenge.
ThisworkwassponsoredunderPRIN/COFIN,andalsoun-
derFANAS/AFRI.WeareindebtedtoA.Ferretti,P.Lucignano,
R. Mazzarello, and L. De Leo for early collaboration and dis-
cussions.
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Figure1:Aschematicviewofclean(4,4)and(8,8)SWNTswiththeirsymme-
tries.
4
Co (4,4)
S
O
D
-2.5 -2 -1.5 -1 -0.5 0 0.5 1
Energy(eV)
Fe (8,8)
S
O
D
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5
Energy(eV)
Figure3:Projecteddensityofstatesofimpurity3dand4sorbitals.Above:Co
on(4,4)SWNT;below:Feon(8,8)SWNT.
Figure2: Aschematicviewof(4,4)and(8,8)SWNTswithanimpurityad-
sorbedinthehollowposition.
5
Co (4,4)
1.5
1
2
d) 0.5
a
Phase shift (r -0 .05 eeeassudu ductance g 1 .15
ead on
-1 osu C
osd 0.5
oau
oad
-1.5
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
0
Energy (eV) -1 -0.5 0 0.5 1
Bias potential(mV)
Co (4,4)
2
1 Symmetric band
Antisymmetric band
Total
0.8 1.5
g
e
mission 0.6 nductanc 1
ns Co
a 0.4
Tr 0.5
su
0.2 sd
au
ad 0
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
0 Bias potential ( V)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Figure5:Predictedzero-biasanomalyforCoimpurityonboth(4,4)and(8,8)
Energy (eV)
SWNTs(above)andforFeon(8,8)SWNT(below)(g≡G/G0,G0=2e2/h).
Figure4:Above:conductanceasafunctionofenergyofconductionelectrons
forCoon(4,4)SWNT,foreachsymmetrys−aandspindirectionu−d(up-
down);below:phaseshiftofconductionelectronsfordifferentsymmetriess−a,
e−oandspindirectionsu−d.
6