J. Phys. Chem. C 2009, 113, 17313–17320
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Spontaneous Breaking of Rotation Symmetry in the Edge States of Zigzag Carbon Nanotubes† Weiliang Wang, Yu Xia, Ningsheng Xu, and Zhibing Li* State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics and Engineering, Sun Yat-sen UniVersity, Guangzhou 510275, China ReceiVed: May 14, 2009; ReVised Manuscript ReceiVed: July 16, 2009
Analytical solutions of the edge states were obtained for the (N, 0) type carbon nanotubes with distorted ending bonds. For N ) 5 and N g 7, it was found that the symmetric edge states are mixed via the distortion to form asymmetric edge eigenstates. The density of the π electrons in the last carbon layer exhibits axial rotation symmetry breaking explicitly. The distortion modes depend on the occupation of the edge states, implying that the symmetry of the electron density at the apex could be changed discontinuously by the external electronic field. 1. Introduction Generally, the performance of nanodevices depends strongly on boundary conditions. The nanotube-based field emission (FE) devices, such as the flat panel display, molecular sensor, and scanning tunneling microscope,1-6 are typical examples of which the ending structures play decisive roles. The process of FE from a carbon nanotube (CNT) is basically quantum tunneling of electrons through the potential barrier between the CNT apex and the vacuum. Therefore, it is very sensitive to both the macroscopic applied field and the electronic structure of the apex.7-12 The present paper is interested in the symmetry of the electronic structure at the apex. The zigzag single-walled carbon nanotube (ZSWCNT) is a graphene with zigzag edge rolled up into a cylinder. It had been known that the (N, 0) type ZSWCNTs are semimetallic if N has a factor of 3; otherwise, they are semiconducting. There would be edge states (ES) at the apex of ZSWCNTs, which is similar to the surface states (at the boundary of threedimensional crystals). Also, the ES can be generated by the broken bond induced local strain and quantum trapping and the associated local charge and energy densification and nonbonding (π electron in this case).13 The peculiar ESs of graphene ribbons with zigzag edges14,15 have attracted great interest as it had been predicted theoretically that the ES has surprising electronic and magnetic properties16-24 and as it had been possible to investigate the single-layered graphene25-28 and individual carbon nanotube29 experimentally. It is desirable to reinvestigate the ES of ZSWCNTs on this background since the FE is sensitive to the edge properties.30-35 Many efforts had been dedicated to the calculation of electronic structure of the SWCNT apex via, for instance, the tight binding (TB) method,9,10,36 semiclassical modeling,37 and ab initio simulation.33,38,39 However, in those studies, the coupling between the ES and the structure distortion has not been fully addressed. The present paper shows, however, that the symmetry breaking has a strong effect on the ES and would be observed through the FE image of the ZSWCNTs. The physical origin of the symmetry breaking resembles the Peierls instability or more generally the Jahn-Teller effect, where no external symmetry broken force is directly involved * Corresponding author. E-mail:
[email protected]. † PACS numbers; 73.22.-f; 11.30.Qc; 31.15.aq.
in this phenomenon. In the Peierls instability, the coupling of phonon modes and the electron states near the Fermi level leads to a lattice distortion that opens a gap between the conduction band and the valence band. In principle, this can happen, as had been discussed for the graphene ribbons with zigzag edges.14,40 However, it had been known that the increase of gap due to the Peierls instability in SWCNTs is greatly suppressed and thus hard to be observed.36 Instead, we found that the symmetry breaking of the ES of ZSWCNTs would be large because the distortion of the ending bonds is easier. The symmetry breaking can be manifested as an asymmetric FE image which would be detectable in the FE experiment. The spontaneous breaking of symmetry as one of the most profound physical concepts has shown its importance in the fields of condensed matters and elementary particles. It should be interesting to see an instance of spontaneous symmetry breaking in nanosystems.16,28,41 In section 2, the solution of the ES first given by Klein and Fujita et al. for the zigzag graphene14,15 will be recalled and applied to the ZSWCNTs, with presumed axial rotation symmetry. In section 3, the interaction potential originated from the distortion of the ending bonds is discussed. The spontaneous symmetry breaking of the ES is also presented. Section 4 presents the density of the asymmetric edge eigenstates. The last section is devoted to discussions and summary. 2. Edge States of ZSWCNT The atomic structure of the ZSWCNT is presented in Figure 1. The ending bonds are saturated by certain species of atoms. The system possesses the axial rotation symmetry natively. For concretion, the ending atoms are presumably hydrogen atoms. The ending bonds are thereby the carbon-hydrogen bonds (C-H bonds). The carbon sheet forms a bipartite lattice where the carbons can be divided into A and B sublattices, each of them forming a triangle lattice. Each row consists of N atoms. The carbons on the edge of the left-hand side in Figure 1 should be understood as the same as those on the edge of the righthand side. Ignoring the curvature effects, the ZSWCNT is treated as a planar zigzag graphene ribbon with periodic boundary condition, and the Cartesian coordinates are used as shown in Figure 1. The axial rotation corresponds to the transversal translation in the planar model. If N were infinite, this treatment
10.1021/jp9044868 CCC: $40.75 2009 American Chemical Society Published on Web 09/14/2009
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Wang et al. Neglecting the overlap integral, i.e., 〈Φm′l|Φml〉 ) δm′m, the least of E(0) l leads to a secular problem of an M × M matrix
(
0 h1 0 H(l) 0 ) 0 0 Figure 1. (7, 0) ZSWCNT with 8 rows of carbons. The nanotube is spread on a plane. The nodes are carbon atoms, with the top bonds saturated by hydrogen atoms. The dashed lines are the distorted C-H bonds. The sublattice spacing is indicated by “a”. The jth unit cell is indicated by the dotted square box in (a) and is regiven in (b) where carbons are labeled by the row number.
should be exact. For finite N, as far as the local properties are concerned, the ignorance of the curvature effects should be acceptable. The eigenstates of the Hu¨ckel model of the zigzag graphene ribbon had been solved, and the ES had been found.14,15 Their results can be applied to the ZSWCNT directly. However, the continuous transversal lattice wavenumber of the graphene is replaced by a discrete set of numbers in the ZSWCNT due to the periodic condition in the transversal direction. The main results will be reviewed with our notations in the following. Only the 2pz states (the z direction is actually the radial direction of the ZSWCNT) will be considered. For a ZSWCNT with M rows of carbons, there are M relevant atomic states, {|mj〉} in the jth cell. Their wave functions are
〈b|m r j〉 ) ψ(b r -b r mj)
∑
(2)
M
r ∑ CmlΦml(b)
(3)
m)1
The superposition coefficients {Cml} are determined by the ˆ 0 is the symmetric ˆ 0〉l ) 0, where H variational method, δ〈H Hamiltonian
ˆ 0〉l ) El(0) ) 〈H
ˆ 0 |Ψl〉 〈Ψl |H 〈Ψl |Ψl〉
0 0 0 ··· h2 0 · ··
)
(5)
ˆ 0|(m For our model, h1 ) 2t cos(φl) and h2 ) t, with t ) 〈mj|H + 1)j〉 and
φl ) kla/2
(6)
The eigenvalues of H0(l) are the energies of the molecular orbitals. Note that kN/2 ) π/a is the decoupling point (D point) where h1 ) 0. At this point,15 the electrons in the ES are exactly localized at both ends of the ZWCNT, with Cm,N/2 ) -iδm,1 or Cm,N/2 ) e-iWπ/ 2δm,M (the phase convention follows the solution of eq 9 for general kl) where W ) M/2 for even M. The ES at the D-point exists only for even N. For the ES with kl not at the D point, the eigenvalues are14
El(0)( ) ( √h21 + h22 + 2h1h2 cos θl
) -t√1 + 4 cos2 φl + 4 cos φl cos θl
sin((W + 1)θl) + cl sin(Wθl) ) 0
The periodic boundary condition in the x direction requires that kl ) 2πl/(Na), with l ) 0, 1, ..., N - 1. The sublattice spacing is known as a ) 2.494 Å. The molecular orbitals are linear combinations of the Bloch waves
Ψl(b) r )
0 0 h1 0 h2
(7)
where θl satisfies
N
1 eiklxmjψ(b r -b r mj) √N j)1
0 h2 0 h1 0 l
(1)
where b rmj is the position of the mth carbon in the jth unit cell. As shown in Figure 1(b), m is also the row label. The wave function ψ(r b) is presumably the 2pz orbital in an effective symmetric crystal potential, with the atomic energy ε0 (it is set to zero here). For a (N, 0) ZSWCNT, j ) 1, 2, ..., N. Due to the axial symmetry, the molecular orbitals can be expanded via Bloch waves. The Bloch waves are
Φml(b) r )
h1 0 h2 0 0
(4)
(8)
We have defined cl ) h2/h1 ) 1/(2cos φl) and assumed that M is even. (If M is odd, the atomic structures of two edges are different. However, for the edge states of long ZSWCNTs that are concerned in the present paper, the correlation of two edges is negligible, and therefore the parity of M has little physical effect.) There are W real θl being solutions of eq 8 for |cl| < (W + 1)/W, therefore there are M extended orbitals since each real θl associates with two levels as given by eq 7. It has been noted by Klein that, as |cl| increases, the largest solution of θl drifts toward π and will move off the real axis into the complex plane and become a complex solution θl ) π + iηl when |cl| > (W + 1)/W, and thereby an ES emerges. For infinite W, ηl is asymptotically given by ηl ) ln |cl|. The criterion for the ES turns out to be 2π/(3a) < kl < 4π/(3a), i.e., N/3 < l < 2N/3, with the boundary values being the so-called K points that correspond to two degenerated extended states. The energies of the ES approach zero asymptotically for large M. There are two ESs for each kl that satisfy the criterion, corresponding to two edges of the ZSWCNT, respectively. For the FE process, only the ES associated to the up-edge is relevant. The normalized amplitudes, for kl not at the D point, are known as
C2p-1,l ) cl-p√cl2 - 1(-eiφl)p, C2p,l ) 0,
p ) 1, 2, ...
p ) 1, 2, ...
(9) (10)
These states have the axial symmetry and are localized at the upedge; hereafter, they will be referred to as symmetric up-edge states
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b qj )
Figure 2. Distortion modes of the hydrogen ring. (a) The radial mode. (b) The twisting mode. The tube axis is perpendicular to the circles.
(SUES). At zero temperature and for zero chemical potential (the Fermi level is zero), each SUES is occupied by one electron (either spin up or spin down). The number of independent ESs is denoted as de, the dimension of the ES space, that is, the number of l satisfying the criterion of ES. In the range of 2 < N < 13, de ) 4 for N ) 11; de ) 3 for N ) 8, 10, and 12; de ) 2 for N ) 5, 7, and 9; de ) 1 for N ) 4 and 6; and de ) 0 for N ) 3.
N-1
-ik x ) ∑ (Qbleik x + Qb*e l l j
l j
(12)
l)0
As a consequence of this distortion, the SUES will mix with each other subject to the symmetry. Because V is a short-range interaction, it only affects atomic states that are near the H-ring (Figure 3). The extended states have very small amplitude at the edge sites, so they will not be affected by the distortion of the ending bonds. We first consider the x-component of the H-ring distortion. bNote that the wave function ψ(r b-b r1(j+1)) transforms to ψ(r b r1(j-1)) in the x-reflection with respect to x1j. Due to this fact, the uniform distortion in the x direction does not affect the SUES in the first-order approximation (i.e., the l ) 0 component of eq 12 has no contribution). The relevant part of the potential of x-component distortion that has nonzero matrix elements in the SUES representation is
3. Spontaneous Symmetry Breaking
N
Klein has argued that the Peierls gap does not exist for infinite wide graphene ribbon (here, it corresponds to an infinitely long ZSWCNT).14 However, since the SUES of different kl are almost degenerate, they will mix under the distortion of the ending bonds and lead to a gap via the Jahn-Teller mechanism. Now let us assume that the hydrogen atoms have small displacements with respect to their symmetric positions (the H-ring distortion), as indicated in Figure 1(a) by the dashed lines. Since the C-H bonds are polarized and charged,42,43 it will cause an extra potential proportional to the displacements of the hydrogen atoms (e.g., through the dipoles of the C-H bonds), which is important for the electrons in the edge states because they are localized near the ending bonds (see also argument below eq 12). The distortion causes the rise of elastic energy and the mixing of the edge states. The mixing may reduce the electronic energy. The total energy will be decreased if the reduction of the electronic energy is greater than the rise of elastic energy. This section aims to find the stable distortion by minimizing the total energy. 3.1. Distortion Potential. Let the displacement of the jth hydrogen be b qj. The H-ring as a three-dimensional object has three vibration modes. The longitudinal mode (in y direction) has a large elastic energy cost, so it can be ignored. On the plane of the H-ring, there is a radial mode and a twisting mode, as shown in Figure 2(a) and (b), respectively. The distortion potential acting on the π-orbitals can be generally written as
1 2
x Vrel )-
de-1
∑ ∑ (Qlxeik x + Qlx*e-ik x )(x - x0j)f(br - br 0j)
λ 2 j)1
l j
l j
l)1
(13) Before the H-ring distortion emerges, it is natural to suppose that x0j of the jth hydrogen atom is the same as the jth carbon in the first row, i.e., x0j ) x1j. Define
u ) 〈ψ(b r -b r 1j)|(x - x1j)f(b r -b r 0j)|ψ(b r -b r 1j+1)〉
(14)
Due to the x-reflection symmetry, one has
〈ψ(b r -b r 1j)|(x - x1j)f(b r -b r 0j)|ψ(b r -b r 1j-1)〉 ) -u (15) x between two SUES are The matrix elements of Vrel
x Q¯ll ≡ 〈Ψ¯l |Vrel |Ψl〉 ) de-1 * -i2λuC1l ¯C1l sin φl-l¯ cos φl+l¯
∑ Ql′x δ¯l,l+l′ + Ql′x*δ¯l,l-l′
l′)1
(16) where φl and C1l have been defined in eqs 6 and 9, respectively. The delta-functions originated from the axial rotation symmetry. For l > jl, using eq 9, one has
N
V(rF) ) -λ
∑ Fqj · (rF - rF0j)f(rF - rF0j)
(11)
j)1
where b r0j are coordinates of the jth hydrogen atom. λ is a constant. The function f(r b) ) f(x,y,z) is even in the reflection of x and z, and it decreases rapidly as |r b| increases. Since we have neglected the longitudinal mode, b qj has only the x- and z-components {qx,z j }. The potential V describes the coupling between the lattice variables {q bj} and the π-electron. Note that V is invariant under the CN axial rotation (including lattice and electronic variables, i.e, the combinational transformation: x f x,z x + a and qx,z j f qj-1). The displacement of the jth hydrogen atom can be expanded as
j
Ωjl l ) -i2λufjl leiφl-l Qx* l-lj sin φl-lj cos φl+lj
(17)
x 2 When l < jl, Qx* l-lj is replaced by Qjl-l. We have defined fjll ) [(cjl 2 1/2 j - 1)(cl - 1)] /(cjlcl) if l,l * N/2 and at the D point fjl,N/2 ) [(cj2l - 1)]1/2/cjl. The diagonal elements are zero, Ωll ) 0. Now, we consider the z-component of the distortion. It is expected that the C-H bonds are polarized and charged. Due to the repulsive electric interaction, the C-H bonds will bend outward so they have a uniform radial deformation, i.e., δ ) z0j - z1j > 0. The uniform radial deformation does not break the axial symmetry, and its effect has been accounted for in the symmetric atomic orbitals. Therefore, the z-component of symmetry broken distortion should be a further distortion on the background of a given uniform radial deformation. In other
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Wang et al. larger N is small since the stable distortion has a factor 1/N (section 3.4). 3.2. Electron Energy Gain in the Two-Dimensional ES Space. The 2-fold degenerate SUES are {Ψl0,Ψl0+1}. The distortion mixes them to form the energy eigenstates
Figure 3. Cj are the carbons nearest to the hydrogen Hj; Cj(1 are the second nearest to Hj; and C′j and C′j-1 are the third nearest to Hj.
words, the uniform radial deformation of the H-ring is irrelevant to the potential for the symmetry breaking. The relevant potential of the z-component distortion has the general form N
z Vrel
de-1
∑ ∑ (Qlzeik x + Qlz*e-ik x )(z - z0j)f(br - br 0j)
λ )2 j)1
l j
l j
l)1
(18) The matrix elements between two SUES states |Ψjl〉 and |Ψl〉, with l > jl, are z Σ¯ll ≡ 〈Ψ¯l |Vrel |Ψl〉 ) N
∑
λ * ¯ ¯ z* 1 - C1l V e-iklx1j+iklx1j-ikl-lx1j ¯C1lQl-l¯ 2 N j,j′,j′′)1 jjj
(19)
Ψ( )
1 iφ/2 (e Ψl0 ( e-iφ/2Ψl0+1) √2
(23)
Which have no axial rotation symmetry and will be referred to as asymmetric edge eigenstates (AEES). The corresponding eigenenergies are E( ) (|Vl0,l0+1|, with φ being the phase of Vl0,l0+1 and
1 z* Vl0,l0+1 ) -λeiφ1(2cos(φ1) - 1) i2uQx* 1 sin(φ1) - VδQ1 2 (24)
[
]
The occupation of the AEES depends on the chemical potential µ, that is, the energy for adding one electron to the apex. It can be adjusted by a macroscopic applied electric field. At zero temperature, the levels with energies smaller than (equal to) µ are double (single) occupied. Those with energy larger than µ are empty. Denote the occupation of Ψ- as n-. In the H-ring distortion, the electronic energy gain is
∆e ) -n-E- - n+E+
We have defined
Vj'jj'' ) 〈ψ(b r -b r 1j')|(z - z0j)f(b r -b r 0j)|ψ(b r -b r 1j'')〉 (20)
(25)
3.3. Electron Energy Gain in the Three-Dimensional ES Space. The 3-fold degenerate SUES are {ΨN/2-1,ΨN/2,ΨN/2+1}, where ΨN/2 is the D-point ES. The matrix elements are
As a consequence of the nonzero δ, the matrix element Vjjj is nonzero. Therefore, the off-diagonal elements with j′ * j or j′′ * j can be neglected. Since δ is small, one can expand Vjjj in δ and keep only the leading term. Due to the symmetry, the leading term is proportional to δ, i.e., Vjjj ) Vδ, with
VN/2-1,N/2 ) -VN/2,N/2+1 ) λeiφ1√(2 cos(φ2) - 1) 1 z* iuQx* 1 sin(φ2) - VδQ1 2
V)
VN/2-1,N/2+1 ) -λeiφ2(2 cos(φ2) - 1) 2iuQx* 2 sin(φ2) - 〈ψ(b r -b r 1j)|f(x - x1j, y - y0j, z - z1j)|ψ(b r -b r 1j)〉 + 〈ψ(b r -b r 1j)|(z - z1j) ∂f(x - x1j, y - y0j, z - z1j) |ψ(b r -b r 1j)〉 (21) ∂z1j
Then eq 19 becomes
1 iφl-lj Σjl l ) - λVδfjl lQz* l-lje 2
(22)
z When l < jl, Qz* l-lj is replaced by Qjl-l. The diagonal elements are zero. Following the standard perturbation theory for degenerate states, the physical realized ES can be found by diagonalizing x z + Vrel , which has elements Vjll ) Ωjll + the matrix Vrel ) Vrel j Σjll, with N/3 < l(l) < 2N/3 and |lj - l| < de. For the ES space of de ) 1, i.e., for N ) 4 and 6, there is only one ES, so no symmetry breaking can happen. We will focus on the ZSWCNTs whose ES space is two-dimensional or three-dimensional. The former includes N ) 5, 7, and 9. The latter includes N ) 8, 10, and 12. The distortion for
(
)
(26)
(
1 VδQz* 2 2
)
(27)
The distortion matrix can be parametrized as
(
0
AeiφA 0
Vrel ) Ae-iφA Be-iφB -Ae-iφA
BeiφB -AeiφA 0
)
(28)
Where VN/2-1,N/2 ≡ AeiφA and VN/2-1,N/2+1 ≡ BeiφB. Under the distortion potential of eq 28, the level of the degenerate SUES will split into three eigenenergies, E0 and E(. Correspondingly, three AEES are formed. In our notation, the levels are ordered as E- < E0 < E+. The explicit expressions of Eβ (β ) 0, () depend on the occupation. The electronic energy gain is ∆e ) -n-E- - n0E0 - n+E+, where nβ are electron numbers of energy levels Eβ. When all AEES are double occupied, ∆e ) 0, hence the symmetry breaking will not happen. To investigate other cases, it is convenient to study the phases of the distortion matrix elements first. Since the phases φA and φB only appear in the electron energy gain, a relation between them can be
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TABLE 1: Relative Phase of the Distortion Matrix Elements and the Corresponding Energy Levels of AEES in Various Occupations for N ) 8, 10, and 12a (n-, n0, n+) 2φA - φB energy levels of AEES ∆e a
I
II
III
(n, n, n+) π E0 ) -B E( ) (1/2)(B ( Λ1) [(n- - n+)(B + Λ1)]/2
(2, 1, 0) π/2 E0 ) 0 E( ) (Λ0 2Λ0
(2, 0, 0), (2, 1, 1), (1, 0, 0) 0 E0 ) B E( ) (1/2)(-B ( Λ1) [(n- - n+)(B + Λ1)]/2
The last row is the electronic energy gain.
found via maximizing ∆e. The relative phases, eigenenergies, and ∆e for various occupations are presented in Table 1 (referred to in the Appendix), where we have defined Λ0 ) (2A2 + B2)1/2 and Λ1 ) (8A2 + B2)1/2. Hereafter, A > B is assumed, which is the case when one adopts values of λu and λVδ estimated in section 4. For case I as shown in Table 1, the AEES are
Ψ(I) 0
(I) Ψ( )
1 ) (eiφAΨN/2-1 + e-iφAΨN/2+1) √2
(29)
1 [(Λ20 ( BΛ1)eiφAΨN/2-1 + A(3B ( Λ1)ΨN/2 Λ( (Λ20 ( BΛ1)e-iφAΨN/2+1] (30)
where Λ( ) [2Λ12(A2 + 2B2) ( 2B(7A2 + 2B2)Λ1]1/2 are normalization constants. For case II, the AEES are
Ψ(II) 0
1 ) (AeiφAΨN/2-1 + iBΨN/2 + Ae-iφAΨN/2+1) Λ0
(II) Ψ( )
(31)
(III) Ψ(
By minimizing the total energy, the stable distortion can be found. Because the elastic energy does not depend on the phases of the distortion and the electronic energy gain increases monotonically with |Vjll|, the phases of Qx|lj-l| and Qz|lj-l| (denoted by φx and φz, respectively) should maximize |Vjll|. Recalling eqs 17 and 22, the requisite of maximum |Vjll| leads to
φx - φz ) -π/2
1 ) - [-(Λ20 - BΛ1)eiφAΨN/2-1 + Λ A(3B - Λ1)ΨN/2 + (Λ20 - BΛ1)e-iφAΨN/2+1]
(32)
(33)
[
∆Et )
]
N K[|Qz1|2 + |Qx1|2] - (n- - n+)(2 cos φ1 - 1) 4 1 2|λu|sin φ1|Qx1|+ |λVδ||Qz1| (39) 2
(34) Qx1 )
(35)
]
The least total energy δ∆Et/δ|Qx,z l | ) 0 gives
(n- - n+)(2cos φ1 - 1) |λVδ|eiφz NK
(40)
4(n- - n+)(2 cos φ1 - 1)sin φ1 |λu|ei(φz-π/2) NK
(41) where φz ) φ1 - φ and φ is the arbitrary phase introduced in eq 23. Note that the symmetry is recovered when both levels are equally occupied. B. de ) 3. The elastic energy reads
N
∑
1 ) NK 4
(38)
1 ) ((2 cos φ1 - 1) 2|λu|sin φ1|Qx1|+ |λVδ||Qz1| 2
Qz1 )
3.4. Elastic Energy and Stable Distortion. The components (e.g., Ql)0 and Ql >de-2 components) of the distortion that can not reduce the electronic energy should be zero, as it raises the elastic energy and thus the total energy. Therefore, only the components (1 < l < de - 1) involved in ∆e should be considered. For small distortion, the elastic energy reads
1 ∆elast ) K q2 2 j)1 j
(37)
A. de ) 2. Since eq 37, the eigenenergies of the twodimensional ES space are written as
[
1 iφA (e ΨN/2-1 + e-iφAΨN/2+1) √2
(36)
The total energy reads
For case III, the AEES are
Ψ(III) ) 0
b ) ) ∆elast - ∆e ∆Et(Q
E( ) (|Vl0,l0+1|
1 [(iB ( Λ0)eiφAΨN/2-1 + 2AΨN/2 + 2Λ0 (iB - Λ0)e-iφAΨN/2+1]
where K is the spring constant of bending each C-H bond. The total energy for the distortion is the elastic energy subtracted by the electronic energy gain
∆elast )
NK z 2 [|Q1| + |Qx1|2 + |Qz2|2 + |Qx2|2] 4
(42)
de-1
∑ (|Qlx|2 + |Qlz|2) l)1
The electronic energy gain is a function of the amplitudes of distortion A and B, ∆e ) γ0Λ0(A,B) + γ1(B + Λ1(A,B)), where
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Wang et al.
constants γ0 and γ1 depend on the occupation and can be read out from Table 1. Since eq 37, one has
A ) |VN/2-1,N/2 | ) ax1 |Qx1 |+az1 |Qz1 |
(43)
B ) |VN/2-1,N/2+1 | ) ax2 |Qx2 |+az2 |Qz2 |
(44)
where axj ) j|λu|(2 cos φ2 - 1)j/2 sin φ2 and azj ) (1/2)|λVδ|(2 cos φ2 - 1)j/2, with j ) 1, 2. For given A and B, the requisite of the least total energy leads to
Qz1
)
Qz2 )
Qx1
)
Qx2 )
az1
Aeiφz
(45)
Figure 4. Densities of the asymmetric edge eigenstates. (a) and (b) are ψ( for N ) 7; (c), (d), and (e) are ψ-, ψ0, and ψ+ for N ) 8 and (n-, n0, n+) ) (2,2,0), respectively. The contour corresponds to 20% of the maximum density.
Beiφ′z
(46)
4. Density of Edge Eigenstates
(ax1)2 + (az1)2 az2 (ax2)2 + (az2)2 ax1
(ax1)2
+
Aei(φz-π/2)
(47)
Bei(φ′z-π/2)
(48)
(az1)2
ax2 (ax2)2 + (az2)2
From eq 26 and the definition of φA after eq 28, one can obtain φz ) φ1 + π - φA. From eq 27 and the definition of φB, one has φ′z ) φ2 - φB. The phases φA and φB are related to each other via the relation given in Table 1, hence there is only one arbitrary phase. (a) Case I and Case III. For these cases, γ0 ) 0 and γ1 ) (n- - n+)/2. The minimum of the total energy is corresponding to
A) 16√2γ1 [(ax1)2 + (az1)2]3/2√(4(ax1)2 + 4(az1)2 - (ax2)2 - (az2)2) NK 8(ax1)2 + 8(az1)2 - (ax2)2 - (az2)2 (49) B)
[
16γ1 8 1 - x2 NK (ax )2 + (az )2 (a ) + (az1)2 2 2 1
]
-1
(50)
Inserting them into eqs 45-48, one obtains the stable distortion up to an arbitrary phase. (b) Case II, (n-, n0, n+) ) (2, 1, 0). For this case, γ0 ) 2 and γ1 ) 0. By minimizing the total energy, one has
A)
4√2 x 2 [(a ) + (az1)2] NK 1
(51)
and B ) 0, i.e., Q2z ) Q2x ) 0. Therefore, there is only l ) 1 stable distortion in this case. Inserting eq 51 into eqs 45 and 47, the stable distortion components Q1z and Q1x for this case are obtained up to an arbitrary phase.
We have assumed that the ending bonds of the zigzag SWCNTs are saturated by hydrogen atoms, as a concrete example. Besides the known parameter a for the lattice spacing, the density of the edge eigenstates involves only two parameters, u and Vδ. The absence of the overlap integral parameter t is remarkable. To estimate the magnitude of the spring constant K, one may recall the inverse wavelength k/2π ∼ 1000 cm-1 for the vibration of the trans-polyacetylene. Then K ) mH(ck)2 ∼ 3.7 eV/Å2 for the C-H bond. Supposing the asymmetric distortion potential resembles the dipole potential of a charge χe shifted by a distance q, then the strengths of the asymmetric distortion potential λu and λVδ should be the same order as χe2/(4πε0a2) ∼ 2.3χ eV/Å, where χ < 1 takes into account the charge transfer between the hydrogen and carbon atoms and the overlap of adjacent atomic orbitals. Then the asymmetric distortion distance |q| is estimated by equating the elastic energy and the dipole potential energy, λu|q| ∼ Kq2, therefore, |q| ∼ λu/K ∼ 0.6χ Å. If the factor χ ∼ 0.1, roughly the same order of the second nearest neighbor transfer energy divided by the Hartree energy, one expects an asymmetric distortion distance of 0.06 Å. Although the amplitudes of the asymmetric distortion would be very small, it could have a profound impact on the local density of edge eigenstates, as the mixing of edge states exists even in the limitation of zero asymmetric distortion. The densities of the AEES of N ) 7 and 8, as examples, are given in Figure 4. The contour corresponds to 20% of the maximum density. The atomic orbital ψ(r b) has been assumed to a hydrogen-like 2pz orbital with wave function
ψ(b) r )
( ) Z 2a0
3/2
( )
Zr Zr Y1,0(θ) exp 2a 0 a0√3
(52)
The empirical parameter Z ) 3.25 for carbon44 and λu/K ) λVδ/K ) 0.06 Å have been used. Figure 4(a) and (b) are states ψ( for N ) 7 with wave functions given in eq 23. Figure 4(c), (d), and (e) are states ψ-, ψ0, and ψ+ for N ) 8 and (n-, n0, n+) ) (2, 2, 0), with wave functions given by eqs 29 and 30. 5. Discussions and Summary In summary, the theory suggests that the axial rotation symmetry of the open (N, 0) SWCNTs is broken spontaneously
Spontaneous Breaking of Rotation Symmetry
J. Phys. Chem. C, Vol. 113, No. 40, 2009 17319
if N ) 5 or N g 7, due to the coupling between the edge states and the distortion of the ending bonds. The eigenstates and eigenenergies have been given explicitly for N < 11 and N ) 12, in the frame of the tight-binding theory. It is straight forward to extend our results to N ) 11 and N g 13 numerically. However, it is expected that the effect of symmetry breaking is weaker for larger N. The stable geometric structure that minimizes the total energy generally has no axial rotation symmetry except that all edge eigenstates are equally occupied. The phases of various asymmetric distortion components and of different eigenstates are related to each other, leaving only one arbitrary phase. Remarkably, there is π/2 phase difference between the radial distortion and the twisting distortion. For N ) 5, 7, and 9, it is found that only distortion with wavenumber l ) 1 is possible. For N ) 8, 10, and 12, both l ) 1 and 2 are possible, but for the occupation (n-, n0, n+) ) (2, 1, 0), the l ) 2 mode is absent. Remarkably, the electron density of the asymmetric edge eigenstates generally depends on the occupation. Therefore it could be adjusted by the macroscopic external field and the temperature. As explicitly given for the example of N ) 8, the densities of the edge eigenstates for some occupations are obviously asymmetric. As the edge states are most relevant to the field electron emission experiment, the field-dependent symmetry of the edge states implies that the electron density distribution at the apex could change discontinuously and could be controlled by the external field. This would explain the change of FE pattern with the applied field that had been observed in the experiment.34 The relation between the observable FE images and the edge states would provide a technique to probe the quantum states of nanostructures. The appearance of the asymmetric edge states will affect the molecular adsorbate effect of the CNTs and be important for the application of CNTs as molecular sensors.45-47 Acknowledgment. The authors thank Z.G. Shuai and J. Iliopoulos for the valuable discussions. The project is supported by the National Natural Science Foundation of China (Grant Nos. 10674182, 90103028, and 90306016) and the National Basic Research Program of China (2007CB935500). Appendix Define
(
γ ) anti cos -
33/2A2B cos(2φA - φB) Λ30
)
(53)
where 0 e γ e π. The eigenvalues of eq 28 can be written as
E+ )
2 γ Λ0 cos 3 √3
(54)
E0 ) -
2 γ+π Λ0 cos 3 √3
(55)
E- ) -
2 γ-π Λ0 cos 3 √3
(56)
One can see that E+ > E0 > E-. If γ > π/2, E- < 0 and E0 > 0. While γ < π/2, E- < E0 < 0.
For the occupation (n-, n0, n+), nR ) 0, 1, 2, the electronic energy gain is
∆e ) )
∑ nRER
(57)
R
1 γ γ Λ0(n- + n0 - 2n+)cos + Λ0(n- - n0)sin 3 3 √3
If all edge states are equally occupied, i.e., n- ) n0 ) n+, then ∆e ) 0, and there will be no symmetry breaking. Otherwise, for the normal occupation with n- e n0 e n+, ∆e > 0, there will be symmetry breaking. Only the case of ∆e > 0 will be discussed in the following. Since the phases φA and φB are not involved in the elastic energy, they should be adjusted to maximize the electronic energy gain. According to eq 57, ∆e increases monotonically as γ decreases for n- ) n0. Then the maximum of ∆e corresponds to the minimum of γ, i.e., 2φA - φB ) π. For (n-, n0, n+) ) (2, 1, 0), ∆e is maximized at γ ) π/2. It is realized by 2φA - φB ) π/2. For the three other normal occupations, (n-, n0, n+) ) (2, 1, 1), (2, 0, 0), and (1, 0, 0), ∆e increases monotonically with γ; therefore, γ should take the maximum value that is given by 2φA - φB ) 0. References and Notes (1) Deheer, W. A.; Chatelain, A.; Ugarte, D. Science 1995, 270, 1179– 1180. (2) Choi, W. B.; Chung, D. S.; Kang, J. H.; Kim, H. Y.; Jin, Y. W.; Han, I. T.; Lee, Y. H.; Jung, J. E.; Lee, N. S.; Park, G. S.; Kim, J. M. Appl. Phys. Lett. 1999, 75, 3129–3131. (3) Collins, P. G.; Bradley, K.; Ishigami, M.; Zettl, A. Science 2000, 287, 1801–1804. (4) de Jonge, N.; Lamy, Y.; Schoots, K.; Oosterkamp, T. H. Nature 2002, 420, 393–395. (5) Hongjie, D.; Hafner, J. H.; Rinzler, A. G.; Colbert, D. T.; Smalley, R. E. Nature 1996, 384, 147–50. (6) Wong, S. S.; Harper, J. D.; Lansbury, P. T.; Lieber, C. M. J. Am. Chem. Soc. 1998, 120, 603. (7) Dean, K. A.; Chalamala, B. R. Appl. Phys. Lett. 2000, 76, 375– 377. (8) Smith, R. C.; Cox, D. C.; Silva, S. R. P. Appl. Phys. Lett. 2005, 87, 103112. (9) Zhou, G.; Kawazoe, Y. Chem. Phys. Lett. 2001, 350, 386–392. (10) Kim, C.; Choi, Y. S.; Lee, S. M.; Park, J. T.; Kim, B.; Lee, Y. H. J. Am. Chem. Soc. 2002, 124, 9906–9911. (11) Mayer, A.; Chung, M. S.; Kumar, N.; Weiss, B. L.; Miskovsky, N. M.; Cutler, P. H. J. Vac. Sci. Technol. B 2007, 25, 109–114. (12) Chen, G. H.; Li, Z. B.; Peng, J.; He, C. S.; Wang, W. L.; Deng, S. Z.; Xu, N. S.; Wang, C. Y.; Wang, S. Y.; Zheng, X.; Chen, G. H.; Yu, T. J. Phys. Chem. C 2007, 111, 4939–4945. (13) Sun, C. Q.; Fu, S. Y.; Nie, Y. G. J. Phys. Chem. C 2008, 112, 18927–18934. (14) Klein, D. J. Chem. Phys. Lett. 1994, 217, 261–265. (15) Fujita, M.; Wakabayashi, K.; Nakada, K.; Kusakabe, K. J. Phys. Soc. Jpn. 1996, 65, 1920–1923. (16) Yao, W.; Yang, S. A.; Niu, Q. Phys. ReV. Lett. 2009, 102, 096801. (17) Dutta, S.; Pati, S. K. J. Phys. Chem. B 2008, 112, 1333–1335. (18) Dutta, S.; Lakshmi, S.; Pati, S. K. Phys. ReV. B 2008, 77, 073412. (19) Gusynin, V. P.; Miransky, V. A.; Sharapov, S. G.; Shovkovy, I. A.; Wyenberg, C. M. Phys. ReV. B 2009, 79, 115431. (20) Hod, O.; Barone, V.; Peralta, J. E.; Scuseria, G. E. Nano Lett. 2007, 7, 2295–2299. (21) Munoz-Rojas, F.; Fernandez-Rossier, J.; Palacios, J. J. Phys. ReV. Lett. 2009, 102, 136810. (22) Yan, Q. M.; Huang, B.; Yu, J.; Zheng, F. W.; Zang, J.; Wu, J.; Gu, B. L.; Liu, F.; Duan, W. H. Nano Lett. 2007, 7, 1469–1473. (23) Dutta, S.; Manna, A. K.; Pati, S. K. Phys. ReV. Lett. 2009, 102, 096601. (24) Jiang, D. E.; Sumpter, B. G.; Dai, S. J. Chem. Phys. 2007, 126, 134701. (25) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Katsnelson, M. I.; Grigorieva, I. V.; Dubonos, S. V.; Firsov, A. A. Nature 2005, 438, 197–200.
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