Ab initio study of spin-dependent transport in carbon nanotubes with iron and vanadium adatoms

0.49 MB·English

by  Joachim A. Fürst

Ab initio study of spin-dependent transport in carbon nanotubes with iron and
vanadium adatoms
Joachim A. Fu¨rst,1,2,∗ Mads Brandbyge,1 Antti-Pekka Jauho,1,3 and Kurt Stokbro4
1MIC – Department of Micro and Nanotechnology, NanoDTU,
Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark
2Atomistix A/S, c/o Niels Bohr Institute, 2100 Copenhagen, Denmark
3Laboratory of Physics, Helsinki University of Technology, P. O. Box 1100, FI-02015 HUT, Finland
4Department of Computer Science, Universitetsparken 1, DK-2100 Copenhagen Ø, Denmark
8 (Dated: February 3, 2008)
0
We present an ab initio study of spin dependent transport in armchair carbon nanotubes with
0
transition metal adsorbates, iron or vanadium. We neglect the effect of tube curvature and model
2
the nanotube by graphene with periodic boundary conditions. A density functional theory based

n nonequilibrium Green’s function method is used to compute the electronic structure and zero-bias
a conductance. The presence of the adsorbate causes a strong scattering of electrons of one spin
J
type only. The scattering is shown to be due to coupling of the two armchair band states to

5 the metal 3d orbitals with matching symmetry causing Fano resonances appearing as dips in the
2 transmissionfunction. Thespintype(majority/minority)beingscattereddependsontheadsorbate
and is explained in terms of d-state filling. The results are qualitatively reproduced using a simple
] tight-binding model, which is then used to investigate the dependence of the transmission on the
i
c nanotubewidth. Wefindadecreaseinthewidthofthetransmission dipasthetube-sizeincreases.
s
-
l PACSnumbers: 73.63.Fg,73.63.-b
r
t
m
I. INTRODUCTION SWNT’s canwork as spintronic devices demonstrating a
.
t spin polarisationclose to 90 % at the Fermi level as well
a
m Various carbon structures have attracted considerable as considerable magnetic moments. Interestingly, both
semiconducting and metallic SNWT’s showed high spin
- attention in the last decade or so, due to their po-
d tential use within future nanodevices1. Graphene, the polarisation. Kang et al.16 obtained similar conclusions
n for various iron nanowire configurations inside armchair
single-atom thick two-dimensional sheet of graphite is
o nanotubes16. On the other hand, work by Kishi et al.17
the basis for many carbon materials. Rolling up such
c shows that wires of iron and cobalt lose their magnetic
[ a sheet creates the one-dimensional single-walledcarbon
momentswhenadsorbedonarmchairSNWT’s. Fillingof
nanotube (SWNT) and wrapping up a sheet makes 0D
1 SWNT’s withtransitionmetals hasbeenrealisedexperi-
fullerenes. These materialshavea wide rangeofremark-
v mentallyseveralyearsago18,whichaddstothepotential
able electronic properties. Recently graphene ribbons –
7
of these systems. A few studies have been published on
9 cut graphene sheets – were used to make the first tran-
9 sistor carved in graphene2 which is stable at room tem- electron transport in this context. Iron-SWNT’s junc-
tions with C molecules19 and pristine SWNT’s20 have
3 perature. 60
. been proposed as magnetic tunnel junctions. First prin-
1 Not only pushing the limits within electronics, these
ciples transport calculations yield tunnel magnetoresis-
0 carbon materials may lay the ground for the emerging
tances of 11 % and40 %, respectively, for these systems.
8 field of spintronics3,4,5, where the functionality of the
A spin-polarised current has also been reported in theo-
0
device is based on the spin degree of freedom. While
: retical works on nanoribbons with substitutional boron
v graphene and SWNT’s are not inherently magnetic, in- atoms21, where spin-dependent scattering is found.
i troducing defects6,7,8, impurities or boundaries9,10 may
X
alter this fact. In recent years several studies have Motivated by the reported spin-dependent scattering,
r
a been carried out on SWNT/graphene - transition metal weseekheretoexplainthephysicalmechanismsfoundin
adatom systems. First principles calculations have the numerical calculations. We investigate the influence
been employed to determine equilibrium geometries and on the transport properties of a single iron or vanadium
magnetic properties: Fagan et al. reported electronic atomadsorbedonaSWNT.Byperformingabinitiospin-
structure studies on single iron11,12,13 and manganese polarised transport calculations we demonstrate a spin-
adatoms11,12onzigzagSWNT’sshowingatotalmagnetic dependent scattering. The spin type being scattered is
momentofthesystemsandinmostcasesamagnetisation shownto depend onthe type ofadsorbate. By analysing
ofthe tube itself. Asimilar earlierstudy with additional thePDOSofthe3dorbitalsoftheadsorbateswefindthe
transition metal elements on graphene was published by scatteringto be causedby coupling of the bandstates in
Duffy et al.14 reaching similar conclusions. Manganese the SWNT to these orbitals,resulting in Fano resonance
dimers, trimers and wires on zigzag SWNT’s12 exhibit phenomena. Usingasimpletight-bindingmodelwequal-
magnetic moments close to free manganese. Yang et itativelyreproducetheseresults,andalsoinvestigatethe
al.15haveproposedthatironandcobaltcoatedandfilled dependence of the transmission on tube size.
2
2
1.5
n
o
si
s
mi
s 1
n
a
r
T
Majority - DZP
Minority -DZP
0.5 Majority - SZ
FIG. 1: (Colour online) The unit cell of the system. The
Minority - SZ
sheet is cut in an armchair structure along the y-direction.
The adatom, iron or vanadium is placed in the center. The
shadedareasmarkedLandRaretheelectrodes,andC isthe 0
-2 -1 0 1 2
central region. Periodic boundary conditions are imposed in E-E (eV)
F
thetransverse direction (x).
FIG. 2: (Colour online) The transmission of the vanadium
adatomsystemusingasingle-zetabasissetforthewholesys-
tem versus using double-zetapolarised for theadatom.
The paper is organised as follows. Sec. II introduces
ourmodelsystemandthetechnicaldetailsofthecalcula-
tions. The ab initio results are presented in Sec. III and
theFanoresonanceisbrieflyintroducedalongwithtrans- riod of the cell in the transverse x direction is 8.52 ˚A,
mission eigenchannel analysis. In Sec. IV we present a which is then the smallest distance between adatoms
simple tight-binding model which is compared to the ab in neighbouring cells, and should be enough to prevent
initio results. Resultsforlargertubesaregivenbasedon significant adatom – adatom interactions. We employ
the simple model. Gamma point sampling (k = 0) in the periodic x direc-
tion. The system then corresponds to an (2,2) armchair
nanotube described by the approximate Graphene Sheet
Model26,thatis,neglectingcurvatureeffects. Wewillre-
II. SYSTEM
turntothedependenceoftheelectrontransmissionclose
to the Fermi energy on the GSM-armchair tube width
The system used for transport calculations is shown
(n,n) in Sec. IV. The mesh cutoff value defining the
in Fig. 1. The graphene sheet is cut in an armchair
realspace gridwas set to 175 Ry. A single-zeta (SZ) ba-
structure along the y-direction which results in a metal-
sis set was used. The difference between a SZ basis for
lic system. We employ periodic boundary conditions in
all atoms and increasing the vanadium basis to double-
the transverse x-direction. We have used the ab ini-
zeta polarised (DZP) can be seen on Fig. 2. The only
tio pseudopotential density functional theory (DFT) as
qualitative difference between using DZP and SZ is the
implemented in the SIESTA code22 to obtain the elec- additional dip (a single point in the graph) at E ≈ 0.25
tronic structure and relaxed atomic positions from spin-
eV in the SZ case. The reason for this not occuring for
polarisedDFT. We employ the GGA PBE pseudopoten-
DZP is merely a matter of resolution. The dips will be
tial for exchange-correlation23. Our spin transport cal-
addressed in the following two sections. In the remains
culation is based on the nonequilibrium Greens function
of the paper we have used a single-zeta basis.
method as implemented in the TranSIESTA24 code, ex-
tended to spin-polarised systems25. However, we only
consider here the zero bias limit and focus on electron
III. RESULTS
transmissioncloseto the Fermienergy. Accordingto the
TranSIESTAmethod24thesystemisdividedintoleftand
rightelectrodes,markedLandRonFig. 1,andacentral Thespinresolvedtransmissionsasafunctionofenergy
region marked C. A single iron or vanadium adatom is relative to the Fermi level, E , of the graphene sheet
F
placedinthemiddleofthecentralregion. Theelectrodes with iron and vanadium adatoms are shown in Fig. 3.
both contain 32 atoms while the central region consists The transmission of a pure sheet is 2 for each spin type,
of64sheetatoms. Theadatomsystemsarerelaxedusing sincetherearetwobandsintheenergywindoweachcon-
theCGmethodwithaforcetoleranceof0.01eV/˚A.The tributing withafully transmitting channelforeachspin.
carbonatoms werekeptfixedduring geometryoptimisa- It is seen that spin dependent scattering occurs due to
tion. Therelaxedpositionoftheadatomsisinthecenter the presence of the adatoms. In the case of iron the mi-
ofahexagonatadistanceof1.73˚Aand1.89˚Afromthe nority spin type is significantly suppressed around E ,
F
sheetplane for ironandvanadium, respectively. The pe- whereasthemajorityspinelectronstransmitcompletely.
3
Iron Vanadium
2
n
o1.5
si
s
mi 1
s
n
a
Tr0.5 Majority spin
Minority spin
-1 0 1 -1 0 1
Minority spin only: Majority spin only:
1
0.8
0.6
n
T
0.4
Channel 1
0.2 Channel 2
7
V) 6 yz
zx
1/e 5 x2-y2
( 4
S xy
O 3
D 2 3z2-r2
P
1
-1 0 1 -1 0 1
E-E (eV) E-E (eV)
F F
FIG.3: Top: Transmissionasafunctionofenergyforiron(left)orvanadium(right). Withina0.5eVrangeofE theminority
F
spin channels are suppressed in the case of iron whereas the majority channels are suppressed for vanadium. Middle: The
transmission of the two minority (majority) spin channels for the iron (vanadium) system. The sum of the two channels yield
to total transmission in each case. Bottom: The projected density of states (PDOS) of the 3d orbitals of the iron(vanadium)
adatom for minority(majority) spin.
For vanadium, likewise, scattering occurs for only one total spin of 4. This is again supported by the Mulliken
spinchannel,butinthis caseitis the majorityspinelec- analysis data as well as the projected density of states
trons which are scattered. The transmission of the two (PDOS). Comparing PDOS and the transmission there
bands for minority (iron) and majority (vanadium) spin is a strong correlation between adatom orbital energies
isshownonFig. 3,middlegraph. Weseethateachband andconductance dips. Thisis veryclearfor the twosep-
closes completely at certain energies. arated channel closings for vanadium at E ≈ E and
F
In the case of vanadium (3d34s2) we expect from E −EF ≈ −0.5 eV. The interference between a waves
involvingthe quasi-boundd-state on the adatomanddi-
Hund’s rules a total spin of 3 and the majority d-states
rectly transmitted band states yield a Fano (anti-) reso-
will be located around E , whereas the minority states
F
nance. The line shape of the resonance is given by the
are all empty and well above E . The Mulliken analysis
F Fano function27
indicates ahalffilling ofthe majorityspin3d and3d
yz zx
orbitals and a full filling of 3dx2−y2, 3dxy and 3d3z2−r2. (ǫ+q)2
Forminorityelectronsall3dorbitalsareempty. Thisisin f(ǫ)= , (1)
ǫ2+1
fullcorrespondencewiththe3dorbitalPDOSplotshown
in Fig. 3 bottom graph. In the case of iron (3d64s2) whereǫ=(E−E )/Γ. HereΓistheresonancewidth,E
R R
we expect from Hund’s rules that the majority states the resonance energy and q the asymmetry parameter.
are all filled and well below E , and now the partially Despite the overlapping of dips in our calculations their
F
filled minority states are located around E yielding a shapeappearssymmetricinthevicinityofzerotransmis-
F
4
FIG. 4: (Colour online) The real part of the (left-to-right) eigenchannel scattering states for the vanadium system, stemming
from majority spin electrons from both bands, at the energy of the first dip (-0.5 eV) (a),(b) and second dip (-0.1 eV) (c),(d)
in thetransmission spectrum. Onlythesolutions in thescattering region are shown. The size of theshapes indicates a cut-off
value for the wavefunction. White indicates a positive sign a blue a negative sign of the wavefunction. The involved orbitals
ofthevanadiumatom (markedwithdottedcircle) areseentobe(a) : 3dx2−y2 (andaminorpresenceof3d3z2−r2),(b): 3dxy,
(c): 3d and (d) : 3d .
yz zx
sion. Since the structure hasinversionsymmetryaround atoms have the same (opposite) sign in the symmetric
the adatom in the transport direction the parameter q (anti-symmetric) case. These solutions are indeed
mustberealandthusweexpecttohavezerotransmission matched by the vanadium atom orbitals, which is seen
dips as seen in Fig. 3 at the anti-resonancesǫ=−q28,29. on the figures by a match of signs (colours). From the
Averysharpresonancecanbeseenforvanadiumaround shape and sign in the plots the orbitals involved can be
−1eV which should also go to zero. However, it is re- identified for each band at both energies.
latedtothemostlocalizedd-orbital,d3d2−r2,andasmall
asymmetry in the numerical calculation and the resolu-
tioncauseitnottogoexactlytozero. SignaturesofFano
resonanceshavepreviouslybeenobservedexperimentally IV. A TIGHT-BINDING MODEL
intheconductanceofmulti-wallcarbonnanotubesatlow
temperature30. We will now rationalize the results for the channels
The anti-resonances are illustrated further by per- andtransmissionsintermsofthesimplestpossibletight-
forming an eigenchannelanalysis where scattering states binding model. We consider only the coupling of the
with well-defined transmissions (”eigenchannels”) are adatom d-orbitals with the 6 nearest π-orbitals. The
constructed31. We take vanadium as our example. starting point will be the two band states of the π-
The eigenchannel transmissions, T (ǫ), provide the electronsinthearmchairdirectionattheFermilevel,see
n
transmission of channel n at a given energy ǫ. Plotted Fig. 5. These are characterized by rotational symmetry
on Fig. 4 (a)(b) and (c)(d) are the real part of the around the tube axis and come in an odd/even version
majority spin scattering state solutions of both bands around the symmetry plane normal to x along the tube
for energies corresponding to the first dip (−0.5eV) and (y) and cutting through the adatom. They couple to
the second dip (−0.1eV), respectively. We see that the different d-orbitals on the adatom with the same sym-
wavefunction is nonvanishing only on the left side of the metry. Thus the symmetric/anti-symmetric band only
vanadium atom as expected since we have full reflection couple to the d-orbitals even/odd in x. To make a min-
inallcases. Theanti-symmetricandsymmetricsolutions imal model (with a minimum of parameters) we assume
are seen on Fig. 4 (a)(d) and (b)(c), respectively. Note that the d-orbitals have the same on-site energy E , and
d
that in the transverse direction the neighbouring C a coupling to the carbon π-orbitals described in units of
5
Fe (E = -0.4 eV, V = 1 eV) V (E = 0.5 eV, V = 2eV)
d pd d pd
Minority spin Majority spin
1 1
Symm.
Asymm.
0.8 0.8
n
o
missi0.6 0.6
ns0.4 0.4
a
Tr
0.2 0.2
0 0
xy
2 yz 6
S zx
DO x2-y2
al P 3z2-r2 4
bit1
Or 2
0 0
-1 0 1 -1 0 1
E-E (eV) E-E (eV)
F F
FIG. 6: (Colour online) The transmission through the
symmetric/anti-symmetric bands for Fe minority spin (left
panels) and V majority spin (right panels) calculated with
thesimpletight-bindingmodel. Lowerpanelsdisplaythepro-
jected density of states onto thed-orbitals.
FIG. 5: (Colour online) Top panel: The two π-band (p )
z
states (not normalized) in a hexagon where the arrow de-
notesthedirectionalongthearmchairtube(ydirection)with in an interval getting narrower with increasing n. The
φ = ei2π/3. Left/right panel correspond to symmetric/anti- latter is in agreement with the findings of Todorov and
symmetric solution on the A,B-dimers (x direction). Lower White33. They conclude that the larger the tube diame-
panels: Simpletight-bindingmodel. Thesignandrelativesize terthemorethescatteringduetoanimpurityisreduced.
of the hopping matrix elements between the ring π-orbitals This happens because the overlap with the wavefunc-
and the metal adatom d-orbitals when themetal-atom is sit-
tion (normalised around the tube circumference) with
uated 1.7˚A above themiddle of the ring.
the scattering potential goes down. The fact that the
conductance still drops to zero in our calculations is due
to the Fano resonance phenomena.
V =V takingV ≈0.5V 32. Thesizeandsignof
pd pdσ pdπ pdσ
the coupling of the different d-orbitals to the p orbitals
z
are illustrated in Fig. 5, left/right panels corresponding V. CONCLUSION
to coupling to the two types of bands. For both bands
it is seen that the d3z2−r2 orbitals do not couple at EF We have described spin-polarised zero bias transport
since the wavefunction values around the hexagon sums
calculations for armchair carbon nanotubes with ad-
up to zero. Therefore we do not expect significant con-
sorbed single iron or vanadium atoms. We find a signif-
tributions from this orbital, which will be very localised.
icant difference between transmissions of majority and
Weonlyconsiderthemajorityspind-statesforV,since minorityspin. Thepresenceofthemetaladatomscauses
they,asmentionedabove,willbelocatedaroundEF with spin-dependentclosingoftheconductionchannelsatcer-
Ed ≈0. Likewise, for iron we only consider the minority tain energies. The mechanism is due to Fano resonances
spin states. The model calculation with (Ed,Vpd) pa- related to the particular d-states close to the Fermi en-
rameters chosen to fit the data from the full ab initio ergy. Only d-orbitals with a symmetry matching the
calculationareshowninFig.6. It is seenthat the trans- symmetry of the Bloch band solutions take part in the
mission is more suppressed in the full calculation where scattering. The scattered spin type (minority or major-
the individuald-orbitalsareallowedtohavedifferenten- ityspin)canbe explainedbythe fillingofthe3dorbitals
ergies, but the fact that we see 4 dips corresponding to via Hund’s rule. Results of a simple tight-binding model
the four coupling d-orbitals is clear. show that increasing the width of the tube, and thus re-
With the simple model we can now explore what hap- ducing the concentrationofadatoms,results innarrower
pens when the diameter of (n,n)-tube is increased. This dips in the conduction, but still with complete closing.
is shown in Fig. 7 where it is seen that the transmis- The latteris explainedby the reflectionsymmetryofthe
sion still goes to zero at the anti-resonance points but system in the transport direction.
6
Acknowledgments
1
0.8
n
o0.6
si
s
mi
s
n
a0.4
r
T
n=5
0.2 n=10
n=15
n=20
0
-0.2 -0.1 0 0.1 0.2
E-E (eV) The authors would like to thank Jeremy Taylor for
F
useful discussions. Computational resources were pro-
vided by the Danish Center for Scientific Computations
FIG. 7: (Colour online) The dependence on tube-size, (n,n) (DCSC). APJ is grateful to the FiDiPro program of the
calculatedwiththesimpletight-bindingmodelatthefirstdip Finnish Academy for support during the final stages of
in theFecase. this work.
∗
Corresponding author: [email protected]
∗
Corresponding author: [email protected] 257203 (2003).
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