Chiral symmetry analysis and rigid rotational invariance for the lattice dynamics of single-wall carbon nanotubes

0.57 MB·English

by  Jin-Wu Jiang

Chiral symmetry analysis and rigid rotational invariance for the lattice dynamics of
single-wall carbon nanotubes
Jin-Wu Jiang,1 Hui Tang,1 Bing-Shen Wang,2 and Zhao-Bin Su1,3
1Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China
2National Laboratory of Semiconductor Superlattice and Microstructure
6 and Institute of Semiconductor, Chinese Academy of Sciences, Beijing 100083, China
0 3Center for Advanced Study, Tsinghua University, Beijing 100084, China
0
2 Inthispaper,weprovideadetailedexpressionofthevibrationalpotentialforthelatticedynamics
of thesingle-wall carbon nanotubes (SWCNT) satisfying the requirements of the exact rigid trans-
n
lational as well as rotational symmetries, which is a nontrivial generalization of the valence force
a
modelfor theplanargraphenesheet. Withthemodel, thelowfrequencybehaviorofthedispersion
J
of the acoustic modes as well as the flexure mode can be precisely calculated. Based upon a com-
8
prehensivechiralsymmetryanalysis,thecalculated modefrequencies(includingalltheRamanand
1
infrared active modes), velocities of acoustic modes and the polarization vectors are systematically

fittedintermsofthechiralangleandradius,wheretherestrictionsofvarioussymmetryoperations
]
i of theSWCNT are fulfilled.
c
s
PACSnumbers: 81.07.De,63.22.+m
-
l
r
t
m I. INTRODUCTION SWCNT with various vibrational energy terms such as
. springenergy,in-plane orout-of-planebond bending po-
t
a Sincethediscovery1ofcarbonnanotubes,itstimulated tential energies successively, and calculated the phonon
m spectrum for the achiral as well as chiral22 tubes. They
extensive studies on the lattice dynamics of the single-
- wall carbon nanotubes (SWCNT).2,3,4,5,6 The SWCNT realized that, to keep the twisting mode to be a zero
d mode in the long wave length limit, all the pieces of vi-
can be viewed as rolling up a two-dimensional graphitic
n
lattice into a seamless cylinder.7 Among various stud- bration energies under their consideration violating the
o
global rotational invariance around axial axis should be
c ies, the zone-folding method was firstly applied to study
[ the lattice dynamics of the SWCNT, which is migrated ignored. The authors also interpreted the existence con-
from that for the planar graphite sheet. It successfully ditionoftheflexuremode,whichisaninherentcharacter
1
for rods and plates. This mode also shows up in Popov
v accounts the general features of the phonon spectrum
et. al.’s calculation but is absent in Damnjanovic et al.’s
7 in SWCNT. However, unlike the electronic problems, on
results.
9 which this method is workable except for very narrow
3 SWCNT,theresultsforthelatticevibrationbythezone- However,aspointedoutinRef.21,thecommonlyused
1 folding were not very satisfactory,2 particularly, it can expression for bond bending potential, even in addition
0 notprovidecorrectnumbers ofRamanandinfrared(IR) withthekinematicconstraintsasinRefs17and18,does
6
modes of SWCNT.8 notsatisfytherotationalinvariancecondition,wherethe
0
/ In recent a few years, the theoretical investigation on missing ofthe flexure mode is a kindof evidence. At the
at lattice dynamics of SWCNT has achieved many success- same time, the discussion on the rotational symmetry
m ful and interesting results. Using the rotational and he- requirements to the potential terms in connection with
lical symmetries, Popov et al.9,10 decomposed for the the flexure mode is restricted only to those around the
-
d first time the lattice dynamics equation of the SWCNT tube axis. The force model10 taken from that for the
n intoasix-dimensionaleigenvalueproblem. Analyticalex- graphite23 obeys the translational and rotational sum
o pression for the velocities of the acoustic modes are de- rules of the planar sheet but does not meet precisely the
c
rived using Born’s perturbation technique.11 The trans- symmetry requirements of cylindrical warped hexagonal
:
v lational and rotational sum rules expressed in terms of lattice sheet.
Xi the force constant matrix11 of the helical tube are also Actually, the rotational invariance symmetry, i.e., the
addressedinRef.10. Damnjanovicet al. appliedtheline potential energy must keep unchanged when a system is
r
a group analysis to the lattice dynamics of SWCNT,12,13 rigidly rotated, should play a nontrivial role for rod-like
and also obtained the generalized Bloch form by modi- cylindricallatticesheet. Ontheotherhand,the intrinsic
fied group projectors technique.14,15,16 They also calcu- symmetry of the tube lattice structure would also lead
lated the phonon spectra in SWCNT with a modified to specific restriction on dynamic matrix. For a com-
force constant model17,18,19 in which certain kinematic plete account of the symmetries of the lattice dynamics
constraints were imposed to achieve the twisting mode. for the SWCNT, both aspects should be considered in a
The authors tried also to fit their calculated frequencies consistent way. This constitutes the basic scope of this
of the Raman and IR modes as functions of tube radius paper.
and chiral angel. In a series of papers Mahan and his As a continuation of the previously mentioned series
collaborators20,21 worked on the lattice dynamics of the of works, in this paper, we show the vibration potential,
2
i.e. force model, up in a detailed expression with the
rigid translational and rotational invariance being pre-
cisely guaranteed, in which the curvature effect on bond
lengthsandbondanglesiscarefullyconsidered. Asitsdi-
rectconsequences,the flexuremode aswellasthe acous-
tic torsion mode can be correctly treated. Moreover we
calculate the phonon spectrum with the proposed force
model and carry out a full chiralsymmetry analysis. In-
stead of previous case-to-case empirical forms,18 the fre-
quenciesofallRamanandIRactivemodes,thepolariza-
tioncharacteristicsofthemodes,aswellasthevelocities
of all four acoustic modes can be systematically fitted in
terms oftube radiusandchiralanglewith the symmetry
considerations.
As expected, our calculated results show that the cur-
vature effect on bond lengths and bond angles would FIG. 1: Thesketch of carbon atoms on a cylindric surface.
be no more negligible only for nanotubes of smaller ra-
dius. Meanwhile the nontrivial contribution for the θ
They have the relations as
dependence of varies physical quantities is also effec-
tive only for tubes with enough small radius. For thin
~a √3n
tubes with the diameter range such as 4 10 ˚A, the r = | 1| n2+n n +n2, θ =atan 2 . (1)
warped graphene sheet model, on which ou∼r discussions 2π q 1 1 2 2 2n1+n2
are based, should be fairly applicable to the SWCNT.
Fortubeswithdiametersmallerthan4˚Aapproximately, We use the helical and rotational symmetry descrip-
the warped graphene sheet model might have to be tions for the SWCNT throughout this paper. Introduce
improved by further optimizing its equilibrium lattice areallatticevectorH~ =p ~a +p ~a describingthescrew
1 1 2 2
configuration,24,25,26,27 and the part of chiral symme- operation7 withn p n p =N (N isthegreatestcom-
1 2 2 1
−
try analysis in this paper were not be valid any more. mon divisor of n and n ). The rotational angle of the
1 2
Such estimation is entirely consistent with the existing screw operation is α = (H~ R~)/R~ 2 (in the unit of 2π).
literatures.18 A unit cell in the (n ,n ) t·ubule| c|an be notated by two
1 2
Thepresentpaperisorganizedasfollowing. InSec.II, equivalent ways, either based on (H~, R~), or on (~a ,~a )
N 1 2
after a brief description on notations, we present the de- as
tailed expression of the vibrational potential. The rigid
rotational invariance and the comparisons with previ- ~r =mH~ +lR~ , or ~r =q ~a +q ~a (2)
ous works are discussed. Subsection III A is devoted to m,l N q1,q2 1 1 2 2
a generic chiral symmetry analysis for various physical
with the relations
quantities, while the main results and relevant discus-
sions on phonon spectrum calculations are presented in
m=(n q n q )/N, l=q p q p . (3)
1 2 2 1 1 2 2 1
Subsec. III B. The paper ends with a brief summary in − −
Sec. IV. In the following,we use (H~, R~) as the basic vectors,and
N
~b and~b aretheircorrespondingreciprocalunitvectors.
H R
Any wave vector in the reciprocal space can be written
as
II. MODEL
κ n
K~ = ~b + ~b . (4)
H R
2π N
A. Notations
InthefirstBrillouinzone,κ ( 1,1]andnisaninteger
∈ −2 2
in ( N,N].
The z axis in this paper is set along the tube axis and −2 2
thexaxiscomesacrossthemiddlepointofaC-Cbondin
Fig. 1. It is understoodin this subsectionthat notations B. Potentials
for cylindrically warped tubes are often introduced with
the aid of those defined on the planar sheet.
It had been proposed23 that there are five distinctive
As is well known, SWCNT can be notated by a chiral terms for the vibrationalpotential of the graphene sheet
vector R~ = n ~a + n ~a or equivalently, the radius r satisfying the rigid translational and rotational invari-
1 1 2 2
and chiral angle θ. Where~a and~a being the primitive anceoftheplanarsheet. Wehereinpresentadetailedex-
1 2
lattice vectors in graphite lattices,2 ~a = ~a =2.46 ˚A. pressionofthevibrationalpotentialfortheSWCNTwith
1 2
| | | |
3
the curvature effect being carefully in-cooperated,which to an arbitrary axis can be kept only when the differ-
is essentially the generalization of that for the graphene ences among bond lengths and bond angles be carefully
sheet. It satisfies precisely the requirements of the rigid accounted.
translational and rotational invariance and realizes the (4) The potential of the out-of-surface bond bending
corresponding general symmetry sum rules in Ref. 10, V is the energy between atom i and the three nearest-
rc
i.e., the potential energy must keep vanish term by term neighbor atoms j with four atoms simultaneously con-
i
whenthetubeisrigidlytranslatedorrotatedaroundany tained,
axis. Introduce ~r as the equilibrium position of atom i
i
avnecdto~urifraosmitastdomispilatcoemj ienntthveecntaonro.tu~rbije=wh~rilje−th~reimisodthue- Vrc = k2rc[(3~ui− ~uji)·~eric]2, (8)
lus r representsthe length of C-C bond between atoms Xji
ij
i and j. ~r is determined following the geometry of a
i
warped graphene sheet so that the three tridental bonds
~r
nearest-neighboredwith the atom i as well as the angles ~erc = ji ji , (9)
betweenanyofthetwobondsarenotequaltoeachother, i −|Pji~rji|
especially for thin tubes. P
ThefivepotentialtermsfortheSWCNTareexpressed Whereitakes1or2withji runningoverthe threenear-
in the following. est neighbors of atom i. This potential has the physical
(1) V is the potential of the spring force between the intuitionasthatresponsiblefortheradialopticalmodein
l
nearest-neighbor atom pair, tubesandistryingtokeepthefouratomsonthecylinder
surface. Hereinweintroduceanunitvector~ercdefinedin
i
k 4 Eq. (9), which is rather close but not equal to the radial
Vl = 2l [(~ui−~u1)·~el1i]2 , (5) unit vector of site 1 shown in Fig. 2. When the radius
Xi=2 is large enough, the~erc would be close enough to the~er.
i 1
However, we stress that the potential term V with ~erc
where kl is the first-order force constant and ~el1i = |~~rr11ii|. keeps the rigid rotational invariance while tharct with ~eiric
We’d like to point out that the components of the dis- substitutedby~er wouldbreakthe rotationalsymmetry.
placementvectorsperpendicularto~el whichviolatesthe i
1i
rigid rotational invariance are forbidden.
(2) V is also the potential of the spring force but
sl
between the next nearest-neighbored atoms illustrated
as (1, 5...10) in Fig. 1,
10
k
V = sl [(~u ~u ) ~el ]2 (6)
sl 2 i− 1 · 1i
Xi=5
with k the second-order force constant.
sl
(3)The potentialenergyforthe in-surfacebondbend-
ing V is actually a term associated with bond angle
BB
variations. Three atoms have to be considered together,
and
k ~u ~u
VBB = B4B Xji (jXi′j6=i′ji)(cid:20) jiri−ji i ·(~eliji′ −cosθjiiji′~eliji) eFnvI∈eGn.[1i2n,:1t5hT].ehIsetmspahrloolwjreascdttiihouanst(o~e2fr1,c~e1r1o)cntluoynbde~e.er1viafotrestaubboeust(22%n,nfr)omwit~ehr1
+~uji′ri−j′~ui ·(~eliji −cosθji′iji~eliji′)(cid:21)2 (5) The twist potential energy for bond~r1k is
= kB4B Xji (jXi′j6=i′ji)(cosθj′iiji′ −cosθjiiji′)2 . (7) Vtw = k2tw <Xi,j>[(~ui−~uj −(~ui′ −~uj′))·~er1k]2 , (10)
where ~er is the unit vector along the radial direction
If i is 1 or 2, j ,j′ take the sites 2,3,4 or 1,5,6 respec- 1k
i i of the middle point of ~r , < i,j > represents a pair of
1k
tively. In Eq. (7), θjiiji′ stands for the equilibrium angle atoms nearest-neighbored with atom 1 while k the third
betweenthebonds~riji and~riji′,whileθj′iiji′ forthecorre- of its nearest neighbors. Pair < i′,j′ > is the image of
sponding angle in vibration. The expression in terms of < i,j > referring to a C rotation around the axis in
2
~uji −~ui and ~uji′ −~ui is an honest representation to that ~er1k. It has the intuition that responsible for modes with
of cosθ′, where the rigid rotational invariance referred twisted vibrations.
4
C. Some details about the model
Obviously, all the five potential energy terms respect
the translational invariance. When ~u =~u , ~u ~u = 0
i j i j
−
leads to
V =V =V =V =V =0.
l sl BB rc tw
For the rotational invariance, we have to consider the
FIG. 3: The effect of bond lengths on TW mode in tube (5,
five potentials term by term separately.11,28 When the
2). (a)Allbondsareassumedtobethesame. Thefrequency
tube rotates rigidly around an arbitrary axis for a small of TW mode is nonzero. (b) The differences between bonds
angle δ~ω with its direction along the axis δω~ and its are considered. Thefrequency of TW mode is presicely zero.
|δω~|
magnitude δ~ω beingthe totationangle,eachlatticesite
| |
acquires a displacement ~u =δ~ω ~r ,
i i
× toverifythatV =0whenthetuberotatesaroundthez
τi
~ui ~uj =δ~ω (~ri ~rj)=δ~ω ~rji . (11) axis. ButthereisnoflexuremodewhenVτi istakeninto
− × − × account.18ThisisbecausethatV wouldbenomorezero
τi
Substituting Eq. (11) into the first two potential terms when the tube rotates around any axis perpendicular to
(5) and (6), it is straightforwardto have (~u ~u ) ~el = the z axis.
j− i · ij
r (δ~ω ~el ) ~el =0. Then
ij × ij · ij
V =V =0. III. RESULTS AND DISCUSSIONS
l sl
Substituting Eq.(11)intothe thirdpotentialterm(7), a
A. Further symmetry considerations
typical representative term in summation becomes
k We may establish a mapping from the space of chi-
VBB ∼ B4B[δ~ω·(~el12×~el13+~el13×~el12)]2 =0. ral vectors on the planar sheet to that of the nanotube
structure in a fixed way of wrapping29
Inwhicha facthasbeen usedthatr inthe denominate
ij
is cancelled by that in the numerator when Eq. (11) is f : chiral vector set nanotube set .
{ }7→{ }
applied. Moreover, for each typical representative term
Howeverthisisnotaone-to-onemapping. Asmentioned
in potentials (8) and (10), we have similarly,
inRef.29,agivenSWCNTcanbeequivalentlycomposed
Vrc ∼ k2rc[δ~ω×(~r12+~r13+~r14)·~er1c]2 =0, absyθt,hrθe+e c2hπi/ra3lavnedctθor+s R4~π0,/3R~r2easpnedctR~iv4ewlyitwhitchhiinratlhaencgolers-
Vtw ∼ kt2w[δ~ω×(~r43−~r56)·~er12]2 =0. responding graphite sheet. Due to the Cˆ symmetry of
6v
As an example for further clarification, one might cal- thehexagonalgraphenelatticesheet,thereisanotherset
culatethephononspectrumwithallthebondlengthsand of vectors R~ , R~ and R~ with chiral angles as θ+π/3,
1 3 5
bondanglesassumedto be equalto thatofthe graphite. θ +π and θ +5π/3 respectively on the graphite sheet.
It results that the twisting mode (TW) at (κ,n)=(0,0) These three chiral vectors correspond again to the same
isnolongerazeromodeandthereisafinitegap. Forthe SWCNT which is actually the nanotube by rotating the
SWCNT (5,2), the error introduced by the equal-bond- tube formed by R~ upside down, i.e. a Cˆ operation.
0 2x
length creates a gap of the order 0.5 cm−1 as shown in The net result of the operation is an exchange of A and
Fig.3,wherethe radiusr=2.45˚A,thethreeC-Cbonds B carbon atoms in unit cells with the sign of its chiral
are about 1.3%, 0.3% and 0.0% shorter than that of indexν =mod n n ,3 alsochanged. Thereisonean-
graphite 1−.42 ˚A re−spectively. Although it is a minute other operation{σˆ1−co2nn}ecting a pair of SWCNT which
xz
number and entirely negligible in practice, it is of quali- isthemirrorreflectionontoeachotherwithrespecttoxz
tative significance. plane. Correspondingly R~ in the sheet is changed into
0
As a distinguished feature of the nanotubes that the its countpart R~′ with θ θ but ν kept unchanged.
two degenerate transversal acoustic (TA) modes shown As one of the0direct c→ons−equences of the above obser-
up at (κ,n) = (α,1) are flexure modes. Instead of the vations,any physicalquantityQν(r,θ) of the SWCNT is
±
conventional linear behaviors, the low frequency limits aperiodicalfunctionofthechiralangleθ withperiod 2π,
of their dispersions are parabolic as ω2 = β2(κ α)4. i.e., Qν(r,θ) can be expanded as29 3
∓
We address that the rigidrotationalinvariance aroundz
axis itself is not a sufficient condition for the existence ∞
Qν(r,θ)= aν cos(3nθ)+bνsin(3nθ). (12)
of the flexure modes cf. Ref. 21. As a counter exam- n n
ple, we introduce a term Vτi = k2τi[(~u2−~u1)·~eτ12i]2 with nX=0
~eτi = ~er ~el , which is the elastic energy due to the Furthermore, under the operation θ θ +π/3, any
12 12 × 12 →
tangentialspringforce betweenatoms1 and2. Itis easy scalarsS wouldkeepunchanged,Fornormalvectors,the
5
radial components have the same behavior, but the az-
imuthal and axial components change signs29. That is
Sν(θ+ π)=S−ν(θ),
3 (13)
(cid:26)~vν(θ+ π3)=Cˆ2x~v−ν(θ)
with Cˆ ~e = ~e , Cˆ ~e = ~e , Cˆ ~e = ~e . Here ~e ,
2x r r 2x φ φ 2x z z r
− −
~e and ~e are unit vectors oriented towards radial, az-
φ z
imuthal andaxialdirections upon the cylindricalsurface
respectively.
In addition, operation σˆ gives that any scalars or
xz
radialandaxialcomponentsofvectorsareevenfunctions
of θ, while azimuthal components are odd functions of
θ,29
Sν( θ)=Sν(θ),
(cid:26)~vν(−θ)=σˆxz~vν(θ) (14)
−
with σˆ ~e =~e , σˆ ~e = ~e , σˆ ~e =~e . The detailed
xz r r xz φ φ xz z z
−
expressions reduced from Eq. (12) due to the symmetry
restrictionsEqs(13)and(14)areshownintheappendix.
B. Frequencies, acoustic velocities, and
eigenvectors
We calculated the phonon spectrum in the (κ,n) rep-
resentation with the vibrational potential Eqs (5)-(10).
By tuning the calculated results to the experimental
data30 for the Raman modes of (10, 10) tube (see Ta-
ble I), we fit the corresponding force constants as k =
l
364.0 Nm−1, k = 62.0 Nm−1, k =1.07 10−11 erg,
sl BB
k =14.8Nm−1,andk =6.24Nm−1. We×successfully
rc tw
identified 4 zero modes, 14 Raman active modes and 6
IR active modes for the chiral SWCNT.8 Two of the six
modes at (κ,n) = (0,0) are zero modes. They are lon-
gitudinal acoustic (LA) and TW modes both belonging
to A− representation (Reps). Another mode belonging
0 0
also to 0A−0 Reps is the IR active ~er optical (OP) mode FIG.4: Asfunctionsofr andθ,weshowin(a)thefrequency
withAandBatomsoscillatingoutofsurfaceinradialdi- of ~ez OP mode at (κ,n) = (0,0); (b) CTW for small radius
rection. Theotherthreenonzeromodesat(κ,n)=(0,0) tubes;(c)thez componentoftheeigenvectorof~er ACmode.
belonging to A+ Reps are Raman active, they are ~e
0 0 r
acoustic (AC), ~e OP and ~e OP. There are one flexure
θ z
and five nonzero modes at(κ,n)=(α,1). The latter are
both Raman and IR active assigned to E Reps. At parabolic,andthe polarizationvectorsofnonzeromodes
α 1
(κ,n)=(2α,2), six nonzero modes are all Raman active at (κ,n) = (0,0) are listed in Tables II, III, and IV re-
belonging to the same E Reps. spectively. In this paper particular emphasis is drown
2α 2
We alsocalculatedthe soundvelocitiesofLAandTW to tubes with smaller radius. So we take the parameter
modes,theflexureparaboliccoefficientfortheTAmode, regime for frequencies and velocities in r [4.0,10.0] ˚A
and polarizations of nonzero modes at (κ,n)=(0,0) for and θ ∈ [−π6,π6], while that for polarizat∈ion vectors in
varietiesoftubes asthe functions ofr andθ. The modes r [2.0,10.0] ˚A and θ [ π,π]. We keep the relative
∈ ∈ −3 3
at (κ,n) = (α,1) as well as (2α,2) are doubly degen- errorsinfitting lessthan5 10−4. The aboveparameter
×
erated with their one-to-one degenerate counterparts at regimes are such chosen only for tubes of smaller radius,
( α, 1) and ( 2α, 2) respectively. In Fig. 4, we pro- in which the contributions of θ dependence are notable
− − − −
vide the three-dimensional plots for some of these quan- comparing to those of r dependence. The velocity of
tities as functions of r and θ. the TW mode (in Table III) as well as the polarization
Our calculated results are further fitted following the vectors of modes with A+ Reps (in Table IV) are typi-
0 0
parameterization shown in Eq. (12). The correspond- calexamplesshownupanevidentθ dependence. Infact,
ing expressions for the frequencies of above 15 Ra- thecorrespondingcalculationshavealsobeencarriedout
man and IR active modes, the velocities and flexure in the parameter regime r [3.13,58.7] ˚A. The results
∈
6
TABLE I: Comparison between the calculated results (following our model) and the experimental values for several mode
frequencies (in theunit of cm−1) of SWCNT (10,10).
Reps 0A+0 0A+0 αE1− αE1− 2αE2+ 2αE2+ 2αE2+ 2αE2+
theory 167 1588 105 1588 21 367 873 1584
experiment30 186 1593 118 1567 / 377 / 1606
TABLE II:Frequencies (in theunit of cm−1) of 15 Raman and IR active modes as functions of r (in ˚A) and θ.
Reps Mode/Order ω(θ) fi(r)
0A+0 R ~er AC, 1 f0(r) f0(r)= 113r3.86 − 13r93.65
~eθ OP, 2 f0(r)+f1(r)cos6θ f0(r)=1594.00− 26r62.98, f1(r)= 8r.625
~ez OP, 3 f0(r)+f1(r)cos6θ+f2(r)cos12θ f0(r)=1594.00− 91r.281, f1(r)=−15r.268, f2(r)=−0r.628
0A−0 Ir ~er OP, 4 f0(r)+f1(r)cos6θ f0(r)=864.81+ 99r02.22 − 111r74.30, f1(r)= 9r.126
αE1 R Ir 1 f0(r)+f1(r)cos6θ f0(r)= 710r.16 + 45r.307, f1(r)= 1r.939 − 31r.455
2 f0(r)+f1(r)cos6θ f0(r)= 160r3.51 − 74r63.51, f1(r)=−11r53.54
3 f0(r)+f1(r)cos6θ f0(r)=864.84+ 86r02.00 − 175r84.70, f1(r)= 11r.263 − 20r64.52
ν =±1, 4 f0(r)±f1(r)cos9θ f0(r)=1594.13− 31r62.67, f1(r)= 31r.392
ν =0, 4 f0(r)+f1(r)cos6θ+f2(r)cos12θ f0(r)=1594.14− 31r82.48, f1(r)= 7r.823 − 19r.403
f2(r)= 2r.720 + 0r.640
5 f0(r)+f1(r)cos6θ f0(r)=1593.97− 27r72.49, f1(r)=−12r.245
2αE2 R 1 f0(r)+f1(r)cos6θ+f2(r)cos12θ f0(r)= 95r92.33 − 73r64.60 + 77r95.59
f1(r)= 6r.139 + 73r.437, f2(r)=−0r.036 + 9r.344
2 f0(r)+f1(r)cos6θ f0(r)= 142r0.21 + 54r.352 − 124r65.29, f1(r)= 20r43.34
3 f0(r)+f1(r)cos6θ f0(r)= 253r5.48 − 242r63.65, f1(r)=−41r23.23
4 f0(r)+f1(r)cos6θ f0(r)=864.80+ 48r62.71 − 471r14.81 + 124r265.61
f1(r)= 9r.829 − 52r44.74
5 f0(r)+f1(r)cos6θ f0(r)=1594.00− 86r92.19 + 97r84.77
f1(r)=−16r.215 + 36r34.41
6 f0(r)+f1(r)cos6θ f0(r)=1594.01− 39r22.92 − 216r04.15 + 541r66.26
f1(r)= 7r.828 − 29r74.88
canbe fitted very wellby the same functions with minor formation properties with respect to θ θ+ π as well
→ 3
change of coefficients, i.e., our fitting formulae could be as θ θ need to be generalized to
→−
applied to a much wider parameter regime.
vν(m)(θ+ π)=λ(m)(Cˆ )Cˆ v−ν(m)(θ),
It is interesting to see that all the numerically fitting  P 3 2x 2x P (15)
expressionsmeetthesymmetryrequirementsofEqs(13)  vPν(m)(−θ)=λ(m)(σˆxz)σˆxzvPν(m)(θ),
and (14), where the velocities and frequencies are of

scalarswhilepolarizationsareofvectors. We noticethat respectively, in which λ(m)(oˆ) is the phase factor taking
inTable II, oneoftheαE1 modes manifests differentpa- value1or−1correspondingtomodemsuchas~er AC,~eφ
rameter dependence for different chiral index ν, i.e. it OP and~ez OP and referring to the symmetry operation
has different expressions for ν = 1 and ν = 0 respec- oˆsuch as Cˆ2x and σˆxz. In particular, we have
±
tively. However, it is still consistent with the general
expressions Eqs (13) and (14). We further notice that, λ(~erAC)(Cˆ2x)= λ(~erAC)(σˆxz)= 1,
differentfromthosephysicalquantitiesofnormalvectors, λ(~eφOP)(Cˆ2x)= λ(~eφOP)(σˆxz)= 1,
the polarization vector can be measured up to a global −
phase factor as 1. Therefore, the corresponding trans- λ(~ezOP)(Cˆ2x)= λ(~ezOP)(σˆxz)= 1.
± − −
7
TABLE III: Soundvelocities (in kms−1) of the TW and LA modes, and β (in 10−6m2s−1) of theflexuremode as functions of
r (in ˚A) and θ.
Velocity Velocity(θ) fi(r)
CTW f0(r)+f1(r)cos6θ f0 =13.5− 1r.623, f1 = 2r.328
CLA f0(r)+f1(r)cos6θ f0 =21.0706+ 0.0r055 − 0.6r8260,f1 = 0.00r091 − 0.0r12679
β f0(r)+f1(r)cos6θ f0 =1.3767r−0.00142r2−5.8×10−5r3, f1 =−0.1r43 + 0.0r43994
TABLEIV:Polarizationvectors~u≡(~u(A),~u(B))asfunctionsofr(in˚A)andθ. Where~u(A)and~u(B)indicatethedisplacement
vectorsofatomsAandBinthe(0,0)unitcellrespectively. Forthethreemodesinthistable,ur(B)=ur(A),uφ(B)=−uφ(A),
uz(B)=−uz(A).
(κ,n)=(0,0) Vector(θ) fi(r)
R1 (~er AC) ur(A) f0(r) f0(r)=0.7071− 0.0r0228
uφ(A) f1(r)sin3θ f1(r)= 0.0r518 + 0.0r4268
uz(A) f1(r)cos3θ f1(r)= 0.0r517 + 0.0r7249
R2 (~eφ OP) ur(A) f1(r)sin3θ f1(r)=−0.0r542 − 0.0r4255
uφ(A) f0(r)+f1(r)cos12θ f0(r)=0.7056+ 0.0r0219, f1(r)=0.0015− 0.r0203
uz(A) f1(r)sin6θ+f2(r)sin12θ f1(r)=0.0656− 0.0r8201, f2(r)=0.0048− 0.0r1212
R3 (~ez OP) ur(A) f1(r)cos3θ f1(r)=−0.0r447 − 0.0r4217
uφ(A) f1(r)sin6θ+f2(r)sin12θ f1(r)=−0.0656+ 0.0r7273, f2(r)=−0.0048+ 0.0r1211
uz(A) f0(r)+f1(r)cos12θ f0(r)=0.7056+ 0.0r0219, f1(r)=0.0015− 0.0r0233
It can be easily verified that the parameterized expres- flexure modes with the cylindrical symmetry while the
sions for the modes ~e AC, ~e OP and ~e OP listed in plate-like graphene sheet breaks the symmetry so as to
r φ z
Table IV satisfy exactly Eq. (15). Interestingly, the val- have only one flexure mode. There is no way to cross
ues taken by λ(m)(oˆ) coincide precisely with the signa- continuously from the former to the latter.
turesofthemajorcomponentofthecorrespondingmodes The sound velocities, slopes of the dispersions, of
withrespecttotheoperationsCˆ andσˆ . Thedetailed zero modes can be determined correctly only if we have
2x xz
expressions for polarization vectors are given in the ap- enoughdataofthe dispersionswith highprecisioninthe
pendix. low frequency limit around (κ,n) = (0,0). As one of
It is known that the planar graphene is the r the immediate consequences of our calculations, besides
→ ∞
limit of the SWCNT. We notice that six modes in Ta- that for C , our calculations provide a credible value
LA
bleIIevolvetothetwodegeneratein-planeopticalmodes for C by taking the advantage that our model keeps
TW
of the graphene with approximately the same frequency exactlythetranslationalandrotationalinvariance. With
limit 1594.0. While three modes with f (r) 864.8 the velocities C and C we can extract the chirality-
approach the out-of-plane optical mode of0the g→raphene dependent exprLeAssion forTWthe Poissonratio µ.10 Ignoring
sheet. Moreovershown in Table III, the sound velocities 1 and higher order terms, we get
r4
ofthe twozeromodes,i.e. LAandTWmodesbelonging
0.206 0.43
to0A−0,havenonzerolimitswithdifferentvalues. There- µ=0.21+ r2 − r2 cos6θ . (16)
fore,itisexpectedthatthosetwomodeswouldapproach
two acoustic modes of the graphene sheet. We further We may take the values of θ as either π (armchair) or
6
notice that the frequency of the ~er AC mode belonging 0 (zigzag) in the above Eq. (16) to extract the Poisson
to A+ (Table II) vanishes with its polarization vector ratios for the achiral cases:
0 0
perpendicular to the limiting sheet. We might interpret
thatthis mode is a kind ofprecursorofthe flexuremode µ=0.21+ 0.r6236, for θ = π6 ;
of the graphitic sheet. However, at (κ,n)= (α,1), the µ=0.21 0.224, for θ =0.
dispersion of the TA branch is quadratic in κ±. We stress − r2
that the parameterizationfor the coefficient β cannotbe The correspondingresultsreportedinRef.10derivedby
extrapolated to r . This is prohibited by a kind theperturbationapproachcanbefittedbyourexpression
→ ∞
of symmetry argument that the rod-like tube has two in good agreement.
8
IV. CONCLUSIONS = νa +a cos3θ+νa cos6θ+... . (A2)
0 1 2
In conclusion, we presented a detailed expression for Moreover, for polarization vectors, when λm(σxz) =
the vibrational potential of the SWCNT. On the one 1, and λm(C2x)= 1, we have
− −
hand, they obey precisely the rigid rotationaland trans-
∞
laastiwonelallassymthmoseetroiefs,bionnwdhainchgltehseabreonpdrolepnegrltyhcdoiffnesrideenrceeds vrν(θ) = ν[1+(−21)n]bnsin3nθ
nX=0
following the geometry of cylindrically warped graphene
= b sin3θ+νb sin6θ+... ,
sheet. Ontheotherhand,theyexhibitthechiralsymme- 1 2
∞
tmryodinelhwereitceadlcfurolamtetdhethpelapnhaornhoenxsapgeocntarullmatotifcteh.eWSWithCtNhTis vφν(θ) = ν[1−(−21)n]ancos3nθ
and parameterized the physical quantities in terms of nX=0
= a +νa cos3θ+a cos6θ+... ,
0 1 2
tube diameter and chiral angle to make the chiral sym-
∞
metry to be explicit. Our calculated results are mainly vzν(θ) = ν[1−(−21)n]bnsin3nθ
illustrated and summarized in Fig. 4 and Tables II, III, nX=0
andIV. Thephysicsofrelevantquantitiessuchasflexure
= νb sin3θ+b sin6θ+... . (A3)
1 2
modes, some ofthe immediate consequenceslike Poisson
ratio, and the symmetry properties of, for example, the When λm(σ )=1, and λm(C )= 1, we have
xz 2x
polarizationvectors are discussed. Our paper provides a −
comprehensiveandsystematicunderstandingforthe lat- ∞
ticedynamicsofthe SWCNT,andformsakindofsound vrν(θ) = ν[1+(−21)n]ancos3nθ
basis for further improvements and developments. nX=0
= νa +a cos3θ+νa cos6θ+... ,
0 1 2
∞
APPENDIX A: THE EXPRESSION FOR vφν(θ) = ν[1−(−21)n]bnsin3nθ
PHYSICAL QUANTITY
nX=0
= νb sin3θ+b sin6θ+... ,
1 2
With the symmetry restriction as Eqs (13) and (14), ∞
i.e. λ(m)(C2x) = λ(m)(σxz) = 1 in Eq. (15), the general vzν(θ) = ν[1−(−21)n]ancos3nθ
expression Eq. (12) would reduce to the following. nX=0
Since the frequency ω is a scalar, we have = a +νa cos3θ+a cos6θ+... . (A4)
0 1 2
ω±(θ)=a a cos(3θ)+a cos(6θ) a cos(9θ)+...; When λm(σ )= 1, and λm(C )=1, we have
0± 1 2 ± 3 (A1) xz − 2x
ω0(θ)=a +a cos(6θ)+... .
0 2 ∞
vrν(θ) = ν[1−(−21)n]bnsin3nθ
The three components of a vector can be expanded as
nX=0
= νb sin3θ+b sin6θ+... ,
∞ 1 2
vrν(θ) = nX=0ν[1−(−21)n]ancos3nθ vφν(θ) = ∞ ν[1+(−21)n]ancos3nθ
= a +νa cos3θ+a cos6θ+... , nX=0
0 1 2
= νa +a cos3θ+νa cos6θ+... ,
∞ 0 1 2
vφν(θ) = nX=0ν[1+(−21)n]bnsin3nθ vzν(θ) = ∞ ν[1+(−21)n]bnsin3nθ
= b sin3θ+νb sin6θ+... , nX=0
1 2
= b sin3θ+νb sin6θ+... . (A5)
∞ 1 2
vzν(θ) = ν[1+(−21)n]ancos3nθ
nX=0
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