Table Of ContentAb-initio investigation of structural, electronic, and optical properties of (5,0)
finite-length carbon nanotube
Mahdi Tarighi Ahmadpour,1 S. Javad Hashemifar,2 and Ali Rostamnejadi1
1Electroceram research center, Malek Ashtar University of Technology, Shahinshahr, Isfahan, Iran
2Department of Physics, Isfahan University of Technology, 84156-83111 Isfahan, Iran∗
(Dated: January 28, 2016)
We use density functional computations to study the size effects on the structural, electronic,
6
magnetic,andopticalpropertiesof(5,0) finitecarbon nanotubes(FCNT),with lengthintherange
1
0 of 4-44˚A. It is found that the structural and electronic properties of (5,0) FCNTs, in the ground
2 state,convergeatalengthofabout 30˚A,whiletheexcitedstatepropertiesexhibitlong-rangeedge
effects. We discuss that curvatureeffects govern theelectronic structure of short (5,0) FCNTs and
n
enhance energy gap of these systems, in contrast to the known trend in the periodic limit. It is
a
seen thatcompensation ofcurvatureeffectsin twospecial small sizes, may giverisetospontaneous
J
magnetization. Theobtainedcohesiveenergiesprovidesomeinsightsintotheeffectsofenvironment
7
on the growth of FCNTs. The second-order difference of the total energies reveals an important
2 magicsizeofabout15˚A.TheopticalanddynamicalmagneticresponsesoftheFCNTstopolarized
electromagneticpulsesarestudiedbytimedependentdensityfunctionaltheory. Theobtainedresults
]
l show that the static and dynamic magnetic properties mainly come from the edge carbon atoms.
l
a The optical absorption properties are described in term of local field effects and characterized by
h Casida linear response calculations.
-
s
e
I. INTRODUCTION finite sizeeffects onthe excitedstatespropertiesofthese
m
systems demands further theoretical investigations.
.
t Recently, finite-length carbonnanotubes (FCNTs) are The dependence ofelectronic andopticalpropertiesof
a
m of vast interest due to their potential applications in FCNTs on their geometric features is promising to de-
enhanced light emitting,1 spintronics designing,2,3 spin- sign flexible and sensitive optoelectronic and spintron-
-
d orbit interaction based optoelectronic,4 and heterojunc- ics devices. The specific aim of this study is to inves-
n tion nanoelectronic devices.5,6 It has been found that tigate length-dependent structural, electronic, magnetic
o the electronic properties of sub-nanometer length FC- andopticalpropertiesof(5,0)FCNTsbyusingstaticand
c NTsdiffersubstantiallyfromthoseofperiodicCNTs,be- time dependent first-principles calculations.
[
cause of the alternationofelectronic structure fromone-
1 dimensional to the zero-dimensional characteristic.7–9
v ThisfactimpliesthatenergystatesofFCNTsarestrictly II. METHODS
0 affected by the length10–12 and the edge structure of the
5
tube.13–15 For example, while periodic armchair CNTs The ground state calculations were performed within
3
represent a metallic character, finite armchair CNTs ex- the Kohn-Sham density functional theory (DFT) at
7
0 hibit an energy gap which decreases periodically toward the scalar relativistic limit, by using the local density
. zerobyincreasingthelength. InthecaseofzigzagCNTs (LDA)25 and local spin density approximations (LSDA)
1
the situation is different. On one hand, periodic zigzag implemented in the full-potential FHI-aims code.26 The
0
6 CNTs can be either metal or semiconductor, depending studied finite nanotubes were terminated by sufficient
1 ontheirchiralangle. Ontheotherhand,electronicstruc- number of hydrogen atoms to saturate their edge dan-
: ture of finite zigzag CNTs are dominated by some local- gling bonds. Full structural relaxations were applied
v
ized edge states, which are evidenced by scanning tun- to the systems down to the residual atomic forces of
i
X neling spectroscopy of graphite16 and may induce spin- less than 10−3 eV/˚A. The carbon and hydrogen atoms
r polarization in the system.15,17 were treated tightly in the calculations. The length of
a
Thermodynamic considerations suggest that (5,0) the tubes throughout this paper is described in unit of
CNT isthe narrowestfeasiblezigzagcarbonnanotube.18 a 0.2nm thick structural section along the tube axis
Becauseofitshighcurvature,theelectronicpropertiesof (Fig. 1a). The FCNTs are taggedby Si, where i denotes
thisnanotubeismorecomplicatedthanthelargerzigzag the number of sections of the system.
nanotubes.8 The well-known σ∗–π∗ rehybridization, as The optical absorption spectra were evaluated in the
a consequence of this high curvature, forces the zone- framework of time dependent DFT (TDDFT) and pseu-
folded semiconducting band gap of periodic (5,0) CNT dopotentialtechnique,implementedincomputerpackage
to close.19,20 While periodic structure of this tube is ex- Octopus.27TheTDDFTapproachhasalreadybeenused
tensivelystudied,20–24 thereisnosystematicstudyabout for successfulinterpretationofthe excitedstates proper-
finitesizeeffectsonthephysicalpropertiesofthissystem. ties of a (7,6) FCNT.28 We used norm-conserving pseu-
Moreover, first-principles studies on FCNTs are mainly dopotentials in the form of Troullier-Martins and time-
limitedtothegroundstatepropertiesandunderstanding dependent LDA (TD-LDA) kernel.29 It is already dis-
2
cussedthatTD-LDAisabletoprovidereliableresultsfor
finite systems.30 In Octopus,the ground state density is
evaluatedoveradiscretegridinrealspace. Theradiusof
overlappingspheresaroundeachatomandthegridspac-
ing were optimized to 3.5 and 0.16˚A, respectively. Elec-
tric dipole response (optical absorption) of the systems
are calculated by explicit real time-propagation of the
Kohn-Sham wave functions,31 in the presence of a weak
white uniformelectricpulse. To achieveafine spectrum,
nearly13000time-stepsaretakenforatotalpropagation (b)
time of20¯h/eV.The frequency responseis extractedvia
the Fourier transform of the time-dependent dipole mo-
1.46
ment. The Casida linear response approach was applied
h
forcharacterizationofthepredictedopticaltransitions.32 gt 1.44
n
In order to evaluate the spin-dipole response of the FC- e
l T
NTs, opposite magnetic pulses were applied to the ma- d 1.42
jority and minority spin channels.33 n L
o
B 1.40
III. STRUCTURAL PROPERTIES e 120
l
g
n 119
a β
anIdnotphtiiscawloprrkopweertiiensveosftifignaitteedletnhgethele(c5t,r0o)nciacrsbtornucntaunre- ond 113 α
otubes composedofupto 22sections. The atomicstruc- B 112
ture of S6, as a prototype, is depicted in Fig. 1a. The
longitudinal and transverse bonds are denoted by L and 111
T, respectively, and the bond angles are described by α
60
and β. After full structural relaxation,these parameters
are averagedover the central unit cell of the system and
d
then plotted in Fig. 1b. It is expected that by increas- n 40
o
ing the tube length, the edge effects vanish and central b
∆
unit cell acts as an infinite length periodic CNT. Ac- 20
cording to the figure, the L and T bond lengths, after
almost 16 sections, converge to the values of about 1.40 0
and 1.44˚A, respectively, close to those reported for pe- 0 5 10 15 20
riodic (5,0) CNT.20,23 For qualitative understanding of
Length (sections)
thehybridizationinthe systems,wecomparethesebond
lengths with those of the ideal graphene (1.42˚A) anddi-
amond (1.54˚A).34 FIG. 1. (a): Schematic diagram of a finite length (5,0) car-
The convergedvalue of the longitudinal bond (1.40˚A) bonnanotubewith6sections(S6). Thedashedrectangleen-
closes a tubesection. The longitudinal and transverse bonds
issmallerthanthecarbon-carbonbondinidealgraphene,
are indicated by L and T, while α and β shows two inves-
whiletheconvergedvalueofthetransversebond(1.44˚A)
tigated bond angles. (b): Variation of the L and T bond
hasslightlyincreasedtowardtheinteratomicbondlength lengths (˚A) and α and β bond angles (degree) versus the
in diamond. Hence, it seems that L bond has mainly a FCNT’s length. The corresponding values for periodic nan-
graphene like sp2 hybridization, while T bond may in- otube (infinite-length) are indicated with dashed lines. Dif-
volve a degree of sp3 contribution. ference(∆bond)betweentheaveragedLandTbondlengths
The trends of α and β bond angles also confirm the overthe entireof thetubes.
presence of a remarkable sp3 hybridization in the sys-
◦
tem. It is seen that α converges to a value of 111.2
(Fig. 1b), consistent with the sp3 bond angle (109.5◦), piedmolecularorbital(HOMO),bondlengths,andbond
◦
whileβ tendstoavalueofabout119.8 ,whichiscloseto angles of this system is compared with (7,0) and (9,0)
theidealsp2bondangle(120◦).34 Itshouldbenotedthat finite tubes with the same length (table I). The diame-
the converged values of the bond angles coincide with ter of (5,0), (7,0), and (9,0) CNTs are about 3.91, 5.48,
thecorrespondingvaluesinperiodic(5,0)CNT.20,23 The and7.05˚A,respectively. Inthe(9,0)tube,whichhasthe
appearance of sp3 hybridization in the systems comes largest diameter, the HOMO is seen to be localized on
from the significant curvature of the walls in such ul- theedgeatoms,whilethewallcurvatureofsmallertubes
trathin CNTs.35 In order to provide more insights into pushes this orbital toward the central part of the sys-
the curvature effects in (5,0) FCNTs, the highest occu- tem. Moreover,thebondlengthsandbondanglesofthis
3
TABLEI. Comparing thehighest occupied molecular orbital TABLE II. Effect of a further structural relaxation on the
(HOMO) and structural parameters of three finite zigzag totalenergy(∆Etot)andenergygap(∆gap)ofalongerfinite
CNTswiththesamelengthsanddifferentdiameters. L,T,α, CNT constructed from the relaxed structure of the shorter
andβ havethesame meaningasin Fig. 1andL andα are tube.
0 0
equilibrium bond length and bond angle in ideal graphene.34
S14 S16 S18 S20 S22
FCNT HOMO L/L0 T/L0 α/α0 β/α0 ∆Etot(Ry) 0.360 0.090 0.009 0.002 0.003
∆gap (Ry) 0.228 0.004 0.023 0.004 0.005
(5,0) 0.980 1.013 0.930 0.998
0.1 -8.0
(7,0) 0.993 1.000 0.962 1.000 V)
e
0 -8.5 y (
) g
V r
(9,0) 0.997 1.000 0.976 1.000 E (e2 -0.1 -9.0 e Ene
∆ -0.2 v
-9.5 si
∆ E he
-0.3 2 o
Cohesive -10.0 C
system are close to the corresponding parameters in the
ideal graphene (table I), revealing the dominance of sp2 -0.4
hybridization in this system. In the smaller finite tubes, 5 10 15 20
the curvature effects increase deviation of the bond pa- Length (sections)
rameters from the ideal graphene toward diamond, and
thus enhance sp3 hybridization in the systems. FIG. 2. Calculated cohesive energy (per carbon atom) and
In order to see the effects of passivation on structural second energy difference (∆ E) of the (5,0) FCNTs. The
2
properties of finite (5,0) CNTs, we tried to re-calculate dashed line shows thecohesive energy of the periodic tube.
someoftheFCNTsintheirpristinestructurewithnoad-
sorbedhydrogenatoms. The presence of dangling bonds
induces many difficulties in the self-consistent calcula- increasingthetubelength,thecohesiveenergyconverges
tion of these unpassivated systems. After considerable to the periodic limit (8.50 eV/atom). The absolute co-
efforts, we could only converge the self consistent elec- hesive energy decreases as the length increases, indicat-
tronic structure of S2, S3, and S4. It was found that, in ing more stability of shorter tubes. It is attributed to
contrasttothepassivatedCNTs,Lbondbecomeslonger the higher strength of C-H bond, compared with C-C
thanT bondinthese pristine FCNTs. Itmeans thatthe bond. The shorter FCNTs have higher fractional num-
unpassivatedFCNTshaveslightlylongerlengthandnar- ber of C-H bonds and hence are more stable. Calcu-
rowerdiameter,comparedwiththepassivatedones. This lated harmonic vibrationalfrequencies of S2 and S3 pro-
slight elongation may be attributed to the non-bonded vide further evidence for this statement; there is a very
electrons of the pristine FCNTs that induce an effective strong peak at about 3000cm−1 which is related to the
repulsive force between the two ends of the system. C-H bond. Barone et al. observed similar behavior in
We observe that the size dependency of the struc- graphene nanoribbons.36 They discussed that the stabil-
tural parameters, presented in Fig. 1b, becomes negli- ity of the passivated nanoribbons is proportional to the
gibly small for tubes longer than S16. Therefore, one molar fraction of hydrogen atom and therefore inversely
may omit the costly process of structural relaxation of proportionaltothe widthofthe system. Inourcase,the
longer(5,0)finite lengthcarbonnanotubesbyconstruct- cohesiveenergyofFCNTsisinverselyproportionaltothe
ingtheirstructurefromthe relaxedatomicconfiguration tube length.
of S16. In order to verify this statement, we constructed Onthe other hand,Rochefortet al. observedareverse
the S18 tube from the structural parameters of S16 and behavior,37 their absolute cohesive energy of CNTs per
then calculated its total energy and energy gap without atom increases as the length increases. This difference
and with a further structural relaxation. The same cal- may be ascribed to the different energy reference used
culations were performed for S20 and S22 tubes and the for calculation of cohesive energy in this work. Follow-
results were listed in table II. It is clearly seen that the ing some benchmark calculations, we realized that using
finitetubeslongerthanS16,constructedfromtheatomic a reference energy which includes C-H bond, like ben-
configuration of smaller tubes, in practice need not any zene or methane, give rises to a cohesive energy which
furtherstructuralrelaxationforinvestigatingtheirstruc- decreases (become more negative) by increasing length.
tural and electronic properties. Thesestatementsmaybehelpfultoidentify therequired
The cohesive energy of the systems as a function of experimentalsetupforsynthesisofshortorlongFCNTs.
the tube lengthisshowninFig.2. Itisexpectedthatby Itseemsthatinahighvacuumenvironment,longFCNTs
4
S4-H S4-L S9-H S9-L S13-H S13-L S16-H S16-L S22-H S22-L
4
0.6 V) Nonmagnetic
gy (e 0 Ferromagnetic
V) 0.4 Ener-4 Pseudopotential
e
p ( 0 π/a
a
G 0.2
0
2 4 6 8 10 12 14 16 18 20 22
Length (sections)
FIG. 3. Top: Isosurfaces of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital
(LUMO) of (5,0) FCNTs with different length. Bottom: Calculated energy gap of the finite tubes as a function of the tube
length. Inset: theelectronic band structureof a graphenenanoribbon with a width of four sections.
are more feasible while production of short FCNTs may curvatureeffects. Thezone-foldingapproachisapopular
be more practical in the presence of a proper hydrogen methodtoestimatetheelectronicstructureofananotube
gas flux. fromits graphene counterpart. This approachpredicts a
In the study of atomic clusters, a conventionalparam- semiconductinggapforinfinite (3n±1,0)zigzagCNTs,38
eter for addressingrelative stabilities is the second-order whichinallcaseslargerthan(5,0)CNT,agreeswiththe
difference of total energy (∆ E), defined as: accurate DFT based calculations. The metallic behav-
2
ior of the (5,0) CNT is attributed to the high curvature
∆2E(i)=Etot(i+1)+Etot(i−1)−2Etot(i) (1) of this narrow tube, which is not taken into account in
thezone-foldingapproach.35,39Thehighcurvatureofthis
where Etot(i) denotes to the relaxed total energy of the systemenhancestheinteratomicinteractionsandthusin-
cluster of size i. The positive values of this parameter creases the dispersion of valence and conduction bands.
determine the so called magic sizes of the system which Consequently, these bands overlap and form a metallic
exhibithigherstabilitywithrespecttotheadjacentsizes. system. We will discuss that the curvature effects may
The second-order energy difference of the given FCNTs havequite differentimpacts inthe zone-foldedelectronic
(Fig. 2) indicates that S7 and S8 are energetically the structure of the finite CNTs, which is derived from the
most favorable sizes, while S3, S5, S6, and S13 show the electronic structure of graphene nano-ribbons (GNRs).
leastrelativestabilities. Thisplotisexpectedtobecom-
parable with the envisaged mass spectra measurements Theelectronicbandstructureofthe GNRcorrespond-
on (5,0) FCNTs. ing to S4, is calculated and presented in the inset of
Fig. 3. It is seen that the valence and conduction bands
are degenerate and flat near edge of the Brillouin zone
IV. ELECTRONIC PROPERTIES
(BZ). The width of the degenerate region increases by
increasing the width of the GNR. Folding the GNR
The calculated energy gap of the finite CNTs, in the will discretize the allowed k-points of a (n,0) FCNT as:
non-magnetic state, as a function of their length is rep- kN = 2aπNn (N =0,±1,±2,···) where a is the GNR lat-
resented in Fig. 3. In order to understand this behavior, tice parameter.17 For a (5,0) tube, five k-points satisfy
one has to take into account the zone-folding and the the boundary condition, in which the last one falls on
5
thedegenerateregionandleavesazeroenergygapinthe
TABLE III. Calculated magnetic energy (∆E), spin moment
folded electronic structure of the systems. However, the
(µ), and spin density isosurfaces of some (5,0) FCNTs in the
obtainedresults(Fig.3)indicatethatallFCNTs,except
ferromagnetic state. The numbers in the parentheses show
S3andS4,exhibitanon-trivialenergygap. Thereasonis
magneticenergyofthesystemsintheantiferromagneticstate.
that the high curvature of these finite systems enhances
the interatomic interactions and consequently increases FCNT ∆E (meV) µ (µB) spin density
dispersionof the discrete energylevels to opena gapbe-
tween the highest occupied (HOMO) and the lowest un-
S3 -130 (0) 2.00
occupied (LUMO) levels. In other words, the curvature
effects act oppositely in the finite and periodic CNTs;
increasing band dispersion in a periodic CNT may give
rise to overlapping of the valence and conduction bands, S4 -127 (-6) 2.00
whereas increasing dispersion of discrete energy levels of
a finite CNT may enhance the HOMO-LUMO splitting.
S5 0 (2) 2.00
We use the difference between the average L and T
bond lengths (∆bond), as a proper measure of the cur-
vature of an FCNT. It is rationalized by the fact that
this difference decreases toward zero by increasing the
CNT diameter(table I).The results,presentedinFig.1, bution over the central sections of the FCNT. In con-
indicate that in the smallest FCNT (S2), L bond is on trast, LUMO exhibits persistent size dependent fluctua-
average longer than T bond. By increasing the FCNT tions even up to the size of S22 (44˚A). This observation
length, the L bond length falls quickly while the T bond provides more evidence for the resistant edge effects in
length rises sharply. It is clearly observed that within the excited state properties of (5,0) FCNTs.
S3 and S4 the longitudinal and transverse bond lengths Asitwasmentionedearlier,thecurvatureofS3andS4
are getting very close together. As a result, these sys- is compensated and hence the zero energy gap of these
tems involvethe lowestcurvature effects and hence their systems is mainly derived from the zone-folding of the
zeroenergygapinthenon-magneticstateagreeswiththe GNR sp2 band structure. In other investigated FCNTs,
zone-folding prediction. On the other hand, S2 exhibits the curvature effects play a significant role and induce a
the highest curvature effect and consequently a signifi- non-trivial sp3 hybridization in the systems. In the case
cantenergygapappearsinthissystem. Weobservethat of sp2 hybridization, the system has equivalent L and T
fromS4 toaboutS9,∆bondandconsequentlycurvature bonds, just as graphene, while contribution of sp3 hy-
aregrowing. Thistrendmayexplaintheincrementofen- bridization in the systems with high curvature increases
ergy gap between S4 and S9 (Fig. 3). For FCNTs longer the length of the transverse T bond, compared with the
thanS9,∆bondisalmostconvergedtotheperiodicCNT longitudinal L bond, and opens a gap in the systems.
valueandhencethecurvatureissaturatedinthisregion. This statement is justified by the behavior of ∆bond in
Therefore, the asymptotic decrease of the energy gap in Fig. 1.
this region should be explained by a different effect. ThezeroenergygapofS3andS4zigzagFCNTsshows
In larger FCNTs with lower curvature effect, similar thepotentialofspontaneousspinpolarizationinthesys-
asymptotic behavior, observed in a broad length range, tem. From a general point of view, the zigzag graphitic
isattributedtoquantumconfinementeffects.8Inthesim- materials are predicted to possess a ferromagnetic (FM)
ple picture of particle in a one-dimensional box, the dis- orantiferromagnetic(AF) spin-polarizedgroundstate.40
tance between quantum energy levels decreases asymp- Therefore, we studied the FM and AF states of the con-
toticallywithrespecttotheboxlength. Weobservethat sideredFCNTswithinspin-polarizedcalculations. Itwas
in the short (5,0) FCNTs, significant variation of cur- found that S3, S4, and S5 have FM and AF states (ta-
vature destroys the asymptotic behavior of the energy ble III), while all other FCNTs exhibit no spontaneous
gap,whileafter stabilizationofthe curvaturearoundS9, spin polarization. The calculated spin densities reveal
the asymptoticdecreaseofthequantumconfinementap- that the magnetic moment of S3, S4, and S5 is concen-
pears (Fig. 3). However, the results show the decrease tratedontheedgecarbonatoms. Thetotalspinmoment
of energy gapstops aroundS15 and convergesto a value ofthespin-polarizedsystemsisfoundtobe2µB,because
of about 0.2eV afterwards, whereas, the periodic (5,0) there are two allowed k-points (±4π/5a) in the degener-
CNT has a metallic character within DFT. Therefore, ate region of the corresponding GNR band structure.
long-rangeedgeeffectsareexpectedintheelectronicand The magnetic energy of S3, S4, and S5 is also pre-
optical spectra of (5,0) finite length CNTs. For better sented in table III. This parameter measures the differ-
understanding of this behavior, the HOMO and LUMO encebetweentheminimizedenergyofsysteminthenon-
isosurfaces of some investigated FCNTs are presented in magnetic (NM) and corresponding magnetic states. We
thetopofFig.3. Itisapparentthatthesizedependency observethatS3andS4preferaFMstatewhichis signif-
ofHOMOis limited to the tubes shorterthanS10,while icantly more stable than the NM and AF states. In the
for longer tubes this orbital exhibits a uniform distri- case of S5, the FM, AF, and NM state have very close
6
parallel perpendicular
∞
FIG. 5. Bond pattern of S10; red bonds are shorter than
1.40˚A, yellow bonds are between 1.40 and 1.44˚A, and blue
bondsare longer than 1.44˚A.
S22
V. OPTICAL SPECTRA
S14 As it was mentioned, we used time dependent DFT to
)
u. calculateabsorptionspectraofthedesiredfiniteCNTsin
a. thepresenceofapolarizedexternalelectromagneticfield.
(
n Thecalculatedabsorptioncrosssections,forparalleland
o
i perpendicular polarization, are presented in Fig. 4. In
ct S12
e the parallel polarization, the shorter tubes exhibit more
s
s sparse spectrum, originating from their discrete energy
s
o levels,while growingthe tube length, increasesthe num-
r
c
n ber of peaks and accumulates them together. It is due
o to the fact that by growing the tube length, the discrete
i
t
p energystatesaregettingmoredenseandgraduallytrans-
r
o
s S10 forming to one dimensional bands. We observe that de-
b
A creasingthetubelengthquenchestheabsorptionintensi-
ties. Thisquenchingisattributedtothelocalfieldeffects
(LFEs)or depolarizationfieldemergedfrommicroscopic
electric dipoles induced by the external field. Fortu-
nately,LFEsarewelltreatedintheTDDFTformalism.22
S7 It is argued that the strength of LFEs decreases rapidly
byincreasingtherelevantdimensionofthesystem.21Asa
result,itisseenthatgrowingthetubelengthsignificantly
S5
amplifies the absorption spectra in the parallel polariza-
tion,while fixedlateraldimensionofFCNTs reducesthe
S3
size sensitivity of the perpendicular absorption spectra.
This lateral confinement give rises to strong perpendic-
S2 ular LFEs and hence quenches the absorption peaks in
0 2 4 6 8 0 2 4 6 8 this polarization.
Energy (eV) For better understanding of the absorption spectra in
theparallelpolarization,thebehaviorofthefirsttwoab-
sorptionpeaks are highlighted by using two dashed lines
FIG. 4. Absorption cross section of the investigated FCNTs.
Thetubelengths areindicated in theright axis. Thesymbol in Fig. 4. The first dashedline (blue), which likely mim-
∞ stands for an infinite(periodic) (5,0) CNT. Theevolution icstrendoftheopticalgapofFCNTs,exhibitsanobvious
ofthetwofirstabsorptionpeaksoftheFCNTsarehighlighted redshift,morepronouncedintheshortertubes. Inother
by thedotted and dashed lines. words,theopticalgapofthesesystemsisrapidlydecreas-
ing from S2 to S7 and then converging to the obtained
opticalbandgapofperiodic(5,0)tube(∼1.2eV).Incon-
trasttoourresults,Monteroetal.11observedablueshift
minimizedenergieswhiletheFMstateismarginallymore intheopticalabsorptionof(5,0)FCNTs,computedusing
stable. We found that FM spin polarization has signifi- post Hartree-Fock method. Accurate inspection of their
cant effect on the electronic structure of S3, S4, and S5. published paper shows that they have likely constructed
As itwasexpected, spinpolarizationremovesthe degen- their finite tubes by repeating the relaxed units of the
eracy of the HOMO and LUMO levels in S3 and S4 and periodic tube and no further structural relaxation has
opens a gap in these systems. The enhanced energy gap been applied to the finite systems. Our calculations in-
of these systems in the FM state is presented in Fig. 3. dicate significant relaxation effects on edge atoms which
ItshouldbenotedthattheminimizedtotalenergyofS3, affects the optical spectra of finite systems, specially for
S4,andS5,presentedinFig.2aredeterminedintheFM shorter tubes. The bond pattern of S10, as a prototype,
state. is presented in Fig. 5.
7
parallel perpendicular
TABLE IV. Character of the first allowed optical transition
of S3, S5, S7, and S10 in the parallel and perpendicular po-
larizations. ThecapitallettersLandHstandforLUMOand 0.5
HOMO. 0 S5
FCNT parallel perpendicular -0.5
S3 H−2 → L+4 H−15 → L+5
0.5
S5 HH−1 →→ LL++14 H−6 → L+25 µ )B 0 S4
( -0.5
m
S7 H → L+6 H−9 → L+20
δ
H−10 → L+19 0.5
0 S3
S10 H−2 → L+5 H−39 → L+14
H−40 → L+13 -0.5
H−42 → L+16
H−43 → L+15
0 2 4 6 8 0 2 4 6 8
Energy (eV)
Comparedwiththeelectronicgap,presentedinFig.3,
optical gaps are significantly larger and display different FIG. 6. Calculated dynamical spin dipole of S3, S4, and,
trend. It may evidence absence of any HOMO-LUMO S5. δm(ω)=δn↑(ω)−δn↓(ω) where δn↑(↓)(ω) denotes the
optical transition in the (5,0) FCNTs. In order to ver- dynamical spin response of the system in the majority (mi-
ify this statement, we used the linear response Casida nority) channel.33 The energy range of the visible spectra is
approach to characterize the first allowed optical tran- highlighted on the energy axes.
sitions of four FCNTs. The results, given in table IV,
confirm absence of the HOMO-LUMO transition in the
systems. In the case of S3 and S10, we observe that the carbon atoms (see table III). As a result, the magnetic
first optical transition in the parallel polarizationoccurs response mainly comes from the edge atoms and hence
from the second level below HOMO to the fourth and is less sensitive to the polarization. The interplay be-
fifth levels above LUMO, respectively. In the case of S5, tween optical excitations and dynamical magnetizations
in the parallel polarization, and S7 and S10, in the per- of FCNTs is promising to develop new nano devices for
pendicular polarization, more than one joint levels was magneto-optical opportunities.
found to contribute to the first allowed optical transi-
tion. In the periodic CNTs, it is already discussed that
the first allowed transition, in the parallel polarization, VI. CONCLUSIONS
occurs between the valence and conduction bands which
have identical orbital character.41 Our results show that In summary, full-potential density functional compu-
lackoftranslationalsymmetrychangestheselectionrules tations was employed to investigate structural and elec-
and hence prevents the HOMO-LUMO transition in the tronic properties of (5,0) finite carbon nanotubes (FC-
finite CNTs. NTs), passivated with hydrogen atoms. The behavior
Inthesystemswithspinpolarizedelectronicstructure, of bond lengths and angles as a function of the tube
one may expect some magnetic responses to the photon length indicated that the ground state properties con-
(electromagnetic) irradiation, while non-magnetic sys- vergeto the periodic limit at a length of about 30˚A. We
temsexhibitonlyelectricresponsetoanexternalelectro- discussedthat the zone-foldedelectronicstructure ofthe
magneticfield.42 Therefore,wecalculatedthespindipole finiteCNTs,incontrasttotheperiodicsystem,showsno
response of S3, S4, and S5 FCNTs to an external mag- energy gap, while high curvature of (5,0) FCNTs opens
netic field, in the frameworkofTD-LSDA. The obtained agapinthe system. It wasarguedthat compensationof
spectra, presented in Fig. 6, illustrate stronger magnetic thecurvatureeffectsatalengthofabout6-8˚A,preserves
responseofS3andS4,likelyduetotheirhigherferromag- the sp2 character of carbon-carbonbonds and closes the
netic energy, comparedwith S5 (table III). It is interest- energy gapof FCNT in the nonmagnetic state, and then
ingtoseethatthehigherresponsesofS3andS4happens brings in a ferromagnetic state. Analysis of the second-
in the visible spectrum. Comparing magnetic responses orderdifferencesoftotalenergyshowedthatforthe(5,0)
of the systems in the parallel and perpendicular polar- FCNTsshorterthan40˚A,alengthofabout15˚Aexhibits
izations showsthat, in contrastto the electric responses, the highest relative stability. The behavior of the en-
the effects of magnetic LFEs are not evident. The rea- ergy gap and the lowest unoccupied molecular orbital
son is that the magnetic moment of the systems, as it indicated that, in contrast to the ground state proper-
wasmentionedearlier,isratherdistributedovertheedge ties, the excited state properties of the systems exhibits
8
long range edge effects. The real time propagation of nite CNTs. The spin dipole response of the systems,
Kohn-Sham orbitals was applied to determine the elec- which is found to be more pronounced in the visible or
tric dipole and spin dipole responses of FCNTs in the near visible region, is of order of Bohr magneton.
presence of electromagnetic pulses. The calculated opti-
cal absorption spectra show that the optical gap of the
systems is considerably larger than the Kohn-Sham en-
VII. ACKNOWLEDGMENTS
ergy gap. The obtained character of optical transitions
in the framework of Casida equation confirms absence
ofanyopticaltransitionbetweenthe highestunoccupied We acknowledge the helpful comments of Mojtaba
and the lowest unoccupied molecular orbitals in (5,0) fi- Alaei (Isfahan University of Technology).
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