- Email: [email protected]
Optical-absorption spectra in armchair tubes
7 July 1997
PHYSICS
LETTERS
A
Physics Letters A 231 (1997) 259-264
ELSJYIER
Optical-absorption spectra in armchair tubes Jie Jiang a, Rui-hua Xie b**,Jun Maa, Feng Yana a Department of Physics, Nanjing University. Nanjing 210093, China b CCAST (WorM Laboratory), PO. Box 8730, Beging 100080, China
Received 4 February 1997; revised manuscript received 12 April 1997; accepted for publication 16 April 1997 Communicated by L.J. Sham
Abstract The optical-absorption spectra of armchair tubes have been studied by using the extended Su-Schrieffer-Heeger (SSH) model with the Coulomb interaction included. It is found that the line shape of the absorption spectra of armchair tubes
is anisotropic, not depending on whether the Coulomb interactions are present or not; with the increase of the carbon atom number, the position of the first peak shifts to lower energies and the oscillator strengths of the lower-energy peaks are relatively larger than those of the higher-energy ones; because of the Coulomb interactions, overall peak positions are shifted to higher energies and the oscillator strengths of higher-energy peaks have relatively larger weight than those in the free-electron case. @ 1997 Published by Elsevier Science B.V. PACS: 6150.-f
Very recently, multi- and single-layer graphitic nanotubes have been discovered using high-resolution transmission electron microscopy [ 1,2]. Such nanotubes have cylindrical shapes and their cylinder surfaces have honeycomb-lattice patterns just as in a two-dimensional graph& plane. It is known that a microtube can show metallic or semiconductor properties depending on its diameter and geometric symmetry; the diameters of single-layer carbon nanotubes are in the range 0.7-1.6 nm and most of them are approximately 1 nm or 1.2 nm [ 1,2]. Because the energy gap of a tube is approximately in inverse proportion to its diameter, the interest of researchers has been attracted to the tubes with the smallest available diameters, where the geometric effects are predicted to be largest, for example, the Cm-
derived nanotube [ 3, 41. By adding one row of five armchair hexagons to Cm along the equator normal to a fivefold axis, one can get C7c. More generally, by adding i rows of armchair hexagons, one can obtain a Cm+iai molecule, which is in the form of a monolayer graphene tube (armchair tube). The study of such tubes could provide interesting theoretical limits for the behavior of carbon fibers, especially for vaporgrown carbon fibers which have a similar structural arrangement. The lattice and electronic structures of C~+ici will greatly change with increasing carbon numbers. On the other hand, the lattice and electronic structures have great effects on their optical properties. Therefore, it is interesting to investigate the optical properties of C~+iei~ In this paper, we apply the extended Su-Schrieffer-Heeger (SSH) model including the Coulomb interaction to calculate the electronic struc-
* Corresponding author. E-mail: [email protected]. 0375-9601/97/$17.00 @ 1997 Published by Elsevier Science B.V. All rights reserved. PUSO375-9601(97)00311-3
J. Jiang et al/Physics
260
tures of armchair tubes, and then obtain the optical absorption spectra of C6a+iOiby using the electronic structures obtained and the sum-over-states (SOS) approach. The model for C60+iai can be written as
Letters A 231 (1997) 259-244
HSSH+ Hee within the single-excitation subspace and diagonal& the obtained matrix. Since photoexcitation does not include spin flip, the excitation bases can be written as follows
Ilm) = (1/JZ)(+rz, H= HSSH + f&e,
where the first term is the extended SSH model given by [51 H SSH =
+ h*c.) c (-to - SYij)(c/,sCj,s (ii)3
+ aKC_Yzj*
(2)
(9)
Here to is the hopping integral of the undimerized system; 5 is the electron-phonon couplin contant; K is 9 annihilates the spring constant; the operator ci,s ( ci,$) (creates) a v electron at the ith carbon atom with spin s; yij is the change of the bond length between the ith and jth atoms; and the sum (ij) is taken over the nearest-neighbor pairs. It can be seen clearly from Ref. [6] that the parameters in the extended SSH model do not sensitively depend on the shape and size of the molecule. So, in the free-electron cases for all Ca+iai, we take the same parameters as those in C60 and C70, i.e., to = 2.5 eV, 5 = 6.31 eV/& and K = 49.1 eV/A2. The interaction among T electrons is described by the following model, Hee = fJ C
PiTPi + C
i
i
C
C
(3) KjPi,sPj,s’ r
j( +i) s,s’
where pi,s = C/,,Zi,,- 1; U is the usual on-site Hubbard repulsion for the carbon atom; Nj is the Coulomb interaction between the ith and jth carbon atoms; and the summation over (i, j) is carried out only for the nearest and next-nearest neighbors due to screening of the T electrons. In the actual calculation, the off-site interaction is taken as l$j
=
lflrijl.
+&l,)Ig),
(5)
(1)
where + is for the spin singlet; - for one of the triplets; Zfs is the creation operator of a rr electron at an unoccupied level with spin s; E,,, is an annihilation operator of a T electron at an occupied level with spin s; and ]g) is the ground state. Furthermore, by using Wicker’s theorom and (pi) = 0, we have obtained the matrix elements in the following form, (m’l’jA - Ei$m) = 2&E, - E, + &,&,,~,,,,[6 - 61,
(6)
where EO = (A), S, = 1 for the singlet, 6, = 0 for the triplet,
= &[+ C
xj(?[,,tl)
(Zjt/ ) (ZiC?~,,)
9
(7)
i,j( Zi) Cn = -(6,& = Em -
C
Eo)$,,) xj
(C,,$J)
(tjict
) (t$~,,),
(8)
i,j( #i)
and the quantities ~1 and cm are the unoccupied and occupied single-electron state energies, respectively. In addition, the terms Ex and EC ace given by
+
C
~j(a,S~et)(~je:,)(e:,,~i)(~~~~,s),
(9)
i.j( +i)
(4) +
We use the standard exciton theory [7,8] to treat the model Hamiltonian. First, we obtain the singleelectron states of the extended SSH Hamiltonian HSSH and construct the ground state and the single electronhole pair excited states of the system. Then, we cdculate the matrix elements of the total Hamiltonian H =
C
~j(er~,,~~~)(~jE:,)(E~,,,,ei)(C!e,,,).
i,j( Zi) (10)
Then, by diagonalizing the matrix given above, we get a set of excited state energies E,, and corresponding wave functions fn (i).
261
J. Jiang et al./Physics Letters A 231 (1997) 259-264
Finally, by using the SOS approach, we arrive at the optical absorption
4w) =
c
4=gni%g
n
X
(
1
E,, -w
-iq
+
1
E,, + w +iq > ’
(11)
where p,,s is the dipole transition matrix element between the excited state In) and the ground state ]g), v is a lifetime broadening factor, which suppresses the height of the resonance peaks and is taken to be 1.6 x lo-* eV in the actual calculation. If we assume In) = xi fn (i)& (note: here &, implies that the ith exciton state refers to the Slater determinant where one electron has been promoted from the occupied level a to the unoccupied level b), we then have the following transition matrix,
b44d= c f!!(i)WM.
kg =
E//Y
(b)
( 12)
i
When U = V = 0, Q. ( 11) above turns out to be [9] a(o)
=2
PmlPIm
c Kunocc.; mEocc.
X (
1 1 + ~1, - w - i77 qrn + w +iv > ’
(13)
where cl,,, = cl - a,,,, and ,u,,,l = (m]r]l). Varying the parameters of the Coulomb interaction within 0 < V < U < 30, in this paper we report the results of the free-electron case (U = V = 0) and the Coulomb interaction case (U = 2V = 4ta) as studied in Refs. [ 10, 1 1] . We expect that the absorption becomes anisotropic with respect to the direction of the electric field of light and the optical properties of the armchair tubes change with the increase of the atom number. In the actual calculation, the z-axis is taken along the direction from the bottom pentagon to the top pentagon and is located as the long axis of the molecule. For nanotubes, if the length of every bond is given, the coordinates of every atom will be obtained. However, we find that the cy values are not sensitive to small changes of the atomic coordinates. So, for simplicity, we take the same bond length 1.4225 A for all molecules, which is the average bond length of CTO given by the experiment [ 121. The free-electron
case: U = V = 0. Figs.
la-lc
0
2
4
6
ad 6
w Fig. 1. Optical-absorptionspectra a for & in the free-electron case: .!I = V = 0. Electric field parallel to the (a) x, (b) y and (c) z axes. Q is in units of A3 and o is in units of eV.
show the optical absorption spectra of Cw in the freeelectron case when the electric field is in the x, y and z, directions, respectively. Figs. la,lb have almost the same spectral shape. But the spectral shape shown in Fig. lc is very different when the electric field is parallel to the z axis: the optical gap is about 2 eV in Figs. la,lb while it is 1.8 eV in Fig. lc, and the oscillator strengths in Fig. lc are relatively larger than those in Figs. la,lb. This shows that the absorption spectra become anisotropic with respect to the direction of the
262
J. Jiang et al./Physics
Letters A 231 (1997) 259-264
1 .o
0.5
0.0 1.0
a 0.5
0.0 6
4
2
0 0
2
4
6
8
cd Fig. 2. Optical-absorptionspectra (I for C~X, in the free-electron case: 21 = V = 0. Eiectric field paraiiel to the (a) x, (b) y and (c) E axes. CIis in units of A3 and w is in units of eV.
electric field of light. Similar results are obtained for Clso. See Fig. 2 for details. Furthermore, by comparing Fig. 1 with Fig. 2, one may notice that when the atom number Nb = 60 + lOi increases, the position of the first peak shifts toward lower energies and the oscillator strengths of the lower-energy peaks have relatively larger weight than those of the higher-energy peaks. The shift can be understood as follows. Because the first peak is produced by a transition between two single-electron en-
ergy levels around the Fermi energy with one in the conduction band and the other in the valence band, the position of the first peak is clearly related to the energy gap Eg. But, Eg decreases with the increase of Nb. Therefore, the position of the first peak shifts to the lower frequency region with the increase of the atom number. According to these analysis, naturally, it is not strange that a resonance peak at w N 0.01 eV can be observed in (Yspectra when Nb is very large. The Coulomb interaction case: U = 2V = 4te. Next we turn to the study of the Coulomb interaction effects on the optical-absorption spectra. When the Coulomb interactions are strong, the effective hopping integral should be chosen smaller than in the free-electron systems (see Ref. [ lo] ). So, we choose to = 1.8 eV in this case with the Coulomb interaction included. In Fig. 3 we present the absorption spectra for Ctso in the case U = 2V = 4to. Also, we notice that when the electric field is in the x and y directions, as shown in Figs. 3a,3b, the spectral shape is almost the same. But the spectral shape displayed in Fig. 3c is very different when the electric field is parallel to the z axis: the optical gap is about 2.2 eV in Figs. 3a,3b while it is 1.7 eV in Fig. 3c, and the oscillator strengths in Fig. 3c are relatively larger than those in Figs. 3a,3b. Obviously, the property of the absorption spectra to become anisotropic with respect to the direction of the electric field of light, does not depend on whether the Coulomb interactions are present or not. Similar results are obtained for Cm. See Fig. 4 for details. Moreover, by comparing Fig. 4 with Fig. 3, one may obtain the same relation obtained by comparing Fig. 1 with Fig. 2. Furthermore, by comparing Fig. 3 (or Fig. 4) with Fig. 2 (or Fig. 1), we conclude that that a strong anisotropy also seems to exist when the Coulomb interactions are present. The effects of the Coulomb interactions are similar as in the C60 and C70 cases: the Coulomb interactions tend to shift peaks toward higher energies and oscillator strengths at higher energies have relatively larger weight than those in the free-electron case. In addition, the static cyvalues have decreased due to the Coulomb interactions. When the atom number increases, the length of the tube increases. Then, the dipole matrix elements of (mlz II) increase, and the intervals between neighboring energy levels decrease. Thus, we get the following result: the bigger the atom number, the larger the (Y
J. Jiang et al./Physics
263
Letters A 231 (1997) 259-264
r 1.0
0.5
0.0
1.0
a 0.5
0.0 2
1
0 0
w
2
4
6
(I
cd
Fig. 3. Optical-absorption spectra a for CIX, in the case U = 2V = 4ro. Electric field parallel to the (a) X, (b) p and (c) z axes. a is in units of A3 and o is in units of eV.
Fig. 4. Optical-absorption spectra a for Cw in the case U = 2V = 410. Electric field parallel to the (a) n, (b) y and (c) z axes. LYis in units of A3 and o is in units of eV.
value, which can be seen by comparing Fig. 4 (or Fig. 1) with Fig. 3 (or Fig. 2). In addition, it is seen that the static absorption values LYof Cga and Cisa in both cases, as shown in Figs. 1-4, do not equal zero, i.e., the value of a(w = 0) is finite. Such a result is actually caused by the finite broadening 71assumed in this paper (see Eqs. ( 11) and (13)). In summary, we have reported the details of the optical-absorption spectra of armchair tubes by using
the extended Su-Schrieffer-Heeger (SSH) model with the Coulomb interaction included. It is found that ( 1) the line shape of the absorption spectrum for the GO+ Ioi molecule is anisotropic, independently of whether or not the Coulomb interactions are present; (2) the number of carbon atoms can greatly change the optical properties of the armchair tubes: with the increase of the atom number, the position of the first peak shifts to lower energies, and the oscillator strengths of the lower-energy peaks have relatively
264
J. Jiang et al./Physics Letters A 231 (1997) 259-264
larger weight than those of the higher-energy peaks; (3) the Coulomb interaction tends to shift the overall peak positions to higher energies, and the oscillator strengths of higher-energy peaks, due to the Coulomb interactions, have relatively larger weight than those in the free-electron case. Acknowledgement
The authors are grateful to the referee for his constructive comments and useful suggestions in improving this paper. References [ I] S. Iijima, Nature 354 ( 1991) 56; S. Iijima and T. Ichihashi, Nature 363 (1993) 603; T.W. Ebbesen and P.M. Ajayan, Nature 358 (1992) 220.
[2] D.S. Bethune, C.H. Kiang, MS. devries, G. Gorman, R.
Saroy, J. Vazquez and R. Beyers, Nature 363 (1993) 605; R. Saito, M. Fujita, G. Dresselhaus and M.S. Dresselhaus, Appl. Phys. Lett 60 ( 1992) 2204. [3] R. Saito, M. Fuji& G. Dresselhaus and M.S. Dtesselhaus, Phys. Rev. B 46 ( 1992) 1804; K. Harigaya, Phys. Rev. B 45 ( 1992) 12071. [4] M.S. Dresselhaus, G. Dresselhaus and R. Saito, Phys. Rev. B 45 (1992) 6234. [5] W.l? Su, J.R. Schrieffer and A.J. Heeger, Phys. Rev. Lett. 42 (1979) 1698; Phys. Rev. B 22 (1980) 2099. [6] K. Harigaya, Phys. Rev. B 45 ( 1992) 13676. [7] S. Abe, J. Yu and W.P Su, Phys. Rev. B 45 ( 1992) 8264. [8] S. Abe, M. Schreiber, W. P.Su and J. Yu, Phys. Rev. B 45 ( 1992) 9432. [9] Q. Xu, J. Jiang, J. Dong and D.Y. Xing, Phys. Status Solidi b 193 (1996) 205. [lo] K. Harigaya and S. Abe, Phys. Rev. B 49 (1994) 16746. [ 111 K. Harigaya and S. Abe, Mol. Liq. Cryst. 256 (1994) 825. [ 121 D.R. Mckenzie, C.A. Davis, D.J.H. Cockayne, D.A. Muller and A.W. Vassallo, Nature 355 (1992) 622.
PHYSICS
LETTERS
A
Physics Letters A 231 (1997) 259-264
ELSJYIER
Optical-absorption spectra in armchair tubes Jie Jiang a, Rui-hua Xie b**,Jun Maa, Feng Yana a Department of Physics, Nanjing University. Nanjing 210093, China b CCAST (WorM Laboratory), PO. Box 8730, Beging 100080, China
Received 4 February 1997; revised manuscript received 12 April 1997; accepted for publication 16 April 1997 Communicated by L.J. Sham
Abstract The optical-absorption spectra of armchair tubes have been studied by using the extended Su-Schrieffer-Heeger (SSH) model with the Coulomb interaction included. It is found that the line shape of the absorption spectra of armchair tubes
is anisotropic, not depending on whether the Coulomb interactions are present or not; with the increase of the carbon atom number, the position of the first peak shifts to lower energies and the oscillator strengths of the lower-energy peaks are relatively larger than those of the higher-energy ones; because of the Coulomb interactions, overall peak positions are shifted to higher energies and the oscillator strengths of higher-energy peaks have relatively larger weight than those in the free-electron case. @ 1997 Published by Elsevier Science B.V. PACS: 6150.-f
Very recently, multi- and single-layer graphitic nanotubes have been discovered using high-resolution transmission electron microscopy [ 1,2]. Such nanotubes have cylindrical shapes and their cylinder surfaces have honeycomb-lattice patterns just as in a two-dimensional graph& plane. It is known that a microtube can show metallic or semiconductor properties depending on its diameter and geometric symmetry; the diameters of single-layer carbon nanotubes are in the range 0.7-1.6 nm and most of them are approximately 1 nm or 1.2 nm [ 1,2]. Because the energy gap of a tube is approximately in inverse proportion to its diameter, the interest of researchers has been attracted to the tubes with the smallest available diameters, where the geometric effects are predicted to be largest, for example, the Cm-
derived nanotube [ 3, 41. By adding one row of five armchair hexagons to Cm along the equator normal to a fivefold axis, one can get C7c. More generally, by adding i rows of armchair hexagons, one can obtain a Cm+iai molecule, which is in the form of a monolayer graphene tube (armchair tube). The study of such tubes could provide interesting theoretical limits for the behavior of carbon fibers, especially for vaporgrown carbon fibers which have a similar structural arrangement. The lattice and electronic structures of C~+ici will greatly change with increasing carbon numbers. On the other hand, the lattice and electronic structures have great effects on their optical properties. Therefore, it is interesting to investigate the optical properties of C~+iei~ In this paper, we apply the extended Su-Schrieffer-Heeger (SSH) model including the Coulomb interaction to calculate the electronic struc-
* Corresponding author. E-mail: [email protected]. 0375-9601/97/$17.00 @ 1997 Published by Elsevier Science B.V. All rights reserved. PUSO375-9601(97)00311-3
J. Jiang et al/Physics
260
tures of armchair tubes, and then obtain the optical absorption spectra of C6a+iOiby using the electronic structures obtained and the sum-over-states (SOS) approach. The model for C60+iai can be written as
Letters A 231 (1997) 259-244
HSSH+ Hee within the single-excitation subspace and diagonal& the obtained matrix. Since photoexcitation does not include spin flip, the excitation bases can be written as follows
Ilm) = (1/JZ)(+rz, H= HSSH + f&e,
where the first term is the extended SSH model given by [51 H SSH =
+ h*c.) c (-to - SYij)(c/,sCj,s (ii)3
+ aKC_Yzj*
(2)
(9)
Here to is the hopping integral of the undimerized system; 5 is the electron-phonon couplin contant; K is 9 annihilates the spring constant; the operator ci,s ( ci,$) (creates) a v electron at the ith carbon atom with spin s; yij is the change of the bond length between the ith and jth atoms; and the sum (ij) is taken over the nearest-neighbor pairs. It can be seen clearly from Ref. [6] that the parameters in the extended SSH model do not sensitively depend on the shape and size of the molecule. So, in the free-electron cases for all Ca+iai, we take the same parameters as those in C60 and C70, i.e., to = 2.5 eV, 5 = 6.31 eV/& and K = 49.1 eV/A2. The interaction among T electrons is described by the following model, Hee = fJ C
PiTPi + C
i
i
C
C
(3) KjPi,sPj,s’ r
j( +i) s,s’
where pi,s = C/,,Zi,,- 1; U is the usual on-site Hubbard repulsion for the carbon atom; Nj is the Coulomb interaction between the ith and jth carbon atoms; and the summation over (i, j) is carried out only for the nearest and next-nearest neighbors due to screening of the T electrons. In the actual calculation, the off-site interaction is taken as l$j
=
lflrijl.
+&l,)Ig),
(5)
(1)
where + is for the spin singlet; - for one of the triplets; Zfs is the creation operator of a rr electron at an unoccupied level with spin s; E,,, is an annihilation operator of a T electron at an occupied level with spin s; and ]g) is the ground state. Furthermore, by using Wicker’s theorom and (pi) = 0, we have obtained the matrix elements in the following form, (m’l’jA - Ei$m) = 2&E, - E, + &,&,,~,,,,[6 - 61,
(6)
where EO = (A), S, = 1 for the singlet, 6, = 0 for the triplet,
= &[+ C
xj(?[,,tl)
(Zjt/ ) (ZiC?~,,)
9
(7)
i,j( Zi) Cn = -(6,& = Em -
C
Eo)$,,) xj
(C,,$J)
(tjict
) (t$~,,),
(8)
i,j( #i)
and the quantities ~1 and cm are the unoccupied and occupied single-electron state energies, respectively. In addition, the terms Ex and EC ace given by
+
C
~j(a,S~et)(~je:,)(e:,,~i)(~~~~,s),
(9)
i.j( +i)
(4) +
We use the standard exciton theory [7,8] to treat the model Hamiltonian. First, we obtain the singleelectron states of the extended SSH Hamiltonian HSSH and construct the ground state and the single electronhole pair excited states of the system. Then, we cdculate the matrix elements of the total Hamiltonian H =
C
~j(er~,,~~~)(~jE:,)(E~,,,,ei)(C!e,,,).
i,j( Zi) (10)
Then, by diagonalizing the matrix given above, we get a set of excited state energies E,, and corresponding wave functions fn (i).
261
J. Jiang et al./Physics Letters A 231 (1997) 259-264
Finally, by using the SOS approach, we arrive at the optical absorption
4w) =
c
4=gni%g
n
X
(
1
E,, -w
-iq
+
1
E,, + w +iq > ’
(11)
where p,,s is the dipole transition matrix element between the excited state In) and the ground state ]g), v is a lifetime broadening factor, which suppresses the height of the resonance peaks and is taken to be 1.6 x lo-* eV in the actual calculation. If we assume In) = xi fn (i)& (note: here &, implies that the ith exciton state refers to the Slater determinant where one electron has been promoted from the occupied level a to the unoccupied level b), we then have the following transition matrix,
b44d= c f!!(i)WM.
kg =
E//Y
(b)
( 12)
i
When U = V = 0, Q. ( 11) above turns out to be [9] a(o)
=2
PmlPIm
c Kunocc.; mEocc.
X (
1 1 + ~1, - w - i77 qrn + w +iv > ’
(13)
where cl,,, = cl - a,,,, and ,u,,,l = (m]r]l). Varying the parameters of the Coulomb interaction within 0 < V < U < 30, in this paper we report the results of the free-electron case (U = V = 0) and the Coulomb interaction case (U = 2V = 4ta) as studied in Refs. [ 10, 1 1] . We expect that the absorption becomes anisotropic with respect to the direction of the electric field of light and the optical properties of the armchair tubes change with the increase of the atom number. In the actual calculation, the z-axis is taken along the direction from the bottom pentagon to the top pentagon and is located as the long axis of the molecule. For nanotubes, if the length of every bond is given, the coordinates of every atom will be obtained. However, we find that the cy values are not sensitive to small changes of the atomic coordinates. So, for simplicity, we take the same bond length 1.4225 A for all molecules, which is the average bond length of CTO given by the experiment [ 121. The free-electron
case: U = V = 0. Figs.
la-lc
0
2
4
6
ad 6
w Fig. 1. Optical-absorptionspectra a for & in the free-electron case: .!I = V = 0. Electric field parallel to the (a) x, (b) y and (c) z axes. Q is in units of A3 and o is in units of eV.
show the optical absorption spectra of Cw in the freeelectron case when the electric field is in the x, y and z, directions, respectively. Figs. la,lb have almost the same spectral shape. But the spectral shape shown in Fig. lc is very different when the electric field is parallel to the z axis: the optical gap is about 2 eV in Figs. la,lb while it is 1.8 eV in Fig. lc, and the oscillator strengths in Fig. lc are relatively larger than those in Figs. la,lb. This shows that the absorption spectra become anisotropic with respect to the direction of the
262
J. Jiang et al./Physics
Letters A 231 (1997) 259-264
1 .o
0.5
0.0 1.0
a 0.5
0.0 6
4
2
0 0
2
4
6
8
cd Fig. 2. Optical-absorptionspectra (I for C~X, in the free-electron case: 21 = V = 0. Eiectric field paraiiel to the (a) x, (b) y and (c) E axes. CIis in units of A3 and w is in units of eV.
electric field of light. Similar results are obtained for Clso. See Fig. 2 for details. Furthermore, by comparing Fig. 1 with Fig. 2, one may notice that when the atom number Nb = 60 + lOi increases, the position of the first peak shifts toward lower energies and the oscillator strengths of the lower-energy peaks have relatively larger weight than those of the higher-energy peaks. The shift can be understood as follows. Because the first peak is produced by a transition between two single-electron en-
ergy levels around the Fermi energy with one in the conduction band and the other in the valence band, the position of the first peak is clearly related to the energy gap Eg. But, Eg decreases with the increase of Nb. Therefore, the position of the first peak shifts to the lower frequency region with the increase of the atom number. According to these analysis, naturally, it is not strange that a resonance peak at w N 0.01 eV can be observed in (Yspectra when Nb is very large. The Coulomb interaction case: U = 2V = 4te. Next we turn to the study of the Coulomb interaction effects on the optical-absorption spectra. When the Coulomb interactions are strong, the effective hopping integral should be chosen smaller than in the free-electron systems (see Ref. [ lo] ). So, we choose to = 1.8 eV in this case with the Coulomb interaction included. In Fig. 3 we present the absorption spectra for Ctso in the case U = 2V = 4to. Also, we notice that when the electric field is in the x and y directions, as shown in Figs. 3a,3b, the spectral shape is almost the same. But the spectral shape displayed in Fig. 3c is very different when the electric field is parallel to the z axis: the optical gap is about 2.2 eV in Figs. 3a,3b while it is 1.7 eV in Fig. 3c, and the oscillator strengths in Fig. 3c are relatively larger than those in Figs. 3a,3b. Obviously, the property of the absorption spectra to become anisotropic with respect to the direction of the electric field of light, does not depend on whether the Coulomb interactions are present or not. Similar results are obtained for Cm. See Fig. 4 for details. Moreover, by comparing Fig. 4 with Fig. 3, one may obtain the same relation obtained by comparing Fig. 1 with Fig. 2. Furthermore, by comparing Fig. 3 (or Fig. 4) with Fig. 2 (or Fig. 1), we conclude that that a strong anisotropy also seems to exist when the Coulomb interactions are present. The effects of the Coulomb interactions are similar as in the C60 and C70 cases: the Coulomb interactions tend to shift peaks toward higher energies and oscillator strengths at higher energies have relatively larger weight than those in the free-electron case. In addition, the static cyvalues have decreased due to the Coulomb interactions. When the atom number increases, the length of the tube increases. Then, the dipole matrix elements of (mlz II) increase, and the intervals between neighboring energy levels decrease. Thus, we get the following result: the bigger the atom number, the larger the (Y
J. Jiang et al./Physics
263
Letters A 231 (1997) 259-264
r 1.0
0.5
0.0
1.0
a 0.5
0.0 2
1
0 0
w
2
4
6
(I
cd
Fig. 3. Optical-absorption spectra a for CIX, in the case U = 2V = 4ro. Electric field parallel to the (a) X, (b) p and (c) z axes. a is in units of A3 and o is in units of eV.
Fig. 4. Optical-absorption spectra a for Cw in the case U = 2V = 410. Electric field parallel to the (a) n, (b) y and (c) z axes. LYis in units of A3 and o is in units of eV.
value, which can be seen by comparing Fig. 4 (or Fig. 1) with Fig. 3 (or Fig. 2). In addition, it is seen that the static absorption values LYof Cga and Cisa in both cases, as shown in Figs. 1-4, do not equal zero, i.e., the value of a(w = 0) is finite. Such a result is actually caused by the finite broadening 71assumed in this paper (see Eqs. ( 11) and (13)). In summary, we have reported the details of the optical-absorption spectra of armchair tubes by using
the extended Su-Schrieffer-Heeger (SSH) model with the Coulomb interaction included. It is found that ( 1) the line shape of the absorption spectrum for the GO+ Ioi molecule is anisotropic, independently of whether or not the Coulomb interactions are present; (2) the number of carbon atoms can greatly change the optical properties of the armchair tubes: with the increase of the atom number, the position of the first peak shifts to lower energies, and the oscillator strengths of the lower-energy peaks have relatively
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larger weight than those of the higher-energy peaks; (3) the Coulomb interaction tends to shift the overall peak positions to higher energies, and the oscillator strengths of higher-energy peaks, due to the Coulomb interactions, have relatively larger weight than those in the free-electron case. Acknowledgement
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