Kondo impurities in nanotubes: the importance of being "in"

1.9 MB·English

by  P. P. Baruselli

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
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