- Email: [email protected]
Exciton effects in carbon nanotubes
Carbon 42 (2004) 1007–1010 www.elsevier.com/locate/carbon
Exciton effects in carbon nanotubes Thomas G. Pedersen
*
Institute of Physics, Aalborg University, Pontoppidanstræde 103, DK-9220 Aalborg Øst, Denmark Available online 25 January 2004
Abstract Exciton effects in quasi-one-dimensional materials such as carbon nanotubes are expected to be important due to confinement of electrons and holes. We present a simple picture of these effects based on an effective mass description of electron–hole pairs moving on the surface of a cylinder. A compilation of results for different geometries shows that the exciton binding energy follows the same dependence on diameter and chiral angle as the measured excitation energy. The exciton effects are found be substantial and produce a complete rearrangement of the absorption spectrum. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: A. Carbon nanotubes; D. Optical properties
1. Introduction Recent advances in fluorescence spectroscopy [1,2] have finally achieved the goal of connecting experimental optical gaps of carbon nanotubes (CNs) to their geometric structure. A highly regular pattern has been established in which the fundamental excitation energy is simply related to nanotube diameter and chiral angle [2]. It is, hence, a challenge to theoretical physics to explain the observed regularity as well as the overall magnitude of the gaps. Much insight has been gained from the zone-folding tight-binding model of the band structure of CNs [3–5]. This simple method predicts an excitation energy varying approximately as the inverse of the CN diameter and having only a weak dependence on chiral angle [3,4]. More elaborate tight-binding calculation including orbital overlap and interactions beyond nearest neighbours [2,6] partly account for the observed dependence on chirality. However, both experimental [7,8] and theoretical [9,10] evidence point to the fact that a simple band-structure calculation is insufficient for an accurate description of optical excitations in CNs. This is due to the omission of two highly important effects in low-dimensional semiconductors: (i) quasi-particle corrections to the simple band-structure and (ii) exciton effects. Both quasi-particle and exciton
*
Tel.: +45-9635-9228; fax: +45-9815-6502. E-mail address: [email protected] (T.G. Pedersen).
0008-6223/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2003.12.028
shifts are due to screened Coulomb interactions. The former describes the repulsive energy needed to add an additional electron to the system and, hence, raises the energy of the conduction band or, equivalently, increases the band gap. In contrast, the exciton shift describes the attractive Coulomb interaction between electrons and holes, which lowers the excitation energy. The experimental results in Refs. [7,8] demonstrate that, in fact, the overall effect is a blue-shift so that the positive quasi-particle correction actually dominates over the negative exciton binding energy. It is the aim of the present paper to provide a simple model for the exciton shift and to evaluate the influence on the optical properties of CNs. Excitonic effects in low-dimensional semiconductors such as CNs are greatly enhanced over the bulk counterparts because confinement leads to a larger overlap between electrons and holes and, thereby, to an enhanced Coulomb binding energy. For ideal quantum wells, the binding energy is enhanced by a factor of four [11]. However, for quasi-one-dimensional structures such as quantum wires and nanotubes, the binding effect diverges as the diameter decreases [12]. Hence, binding energies readily reach )100 meV or more, several orders of magnitude larger that typical bulk values. In our previous work [10], we applied a variational approach to provide an estimate of the exciton effect in zigzag CNs. In the present paper, we extend the analysis to arbitrary chiral vectors covering all semiconducting CNs, i.e. all (n; m) in the usual notation [3] for which n m is not
T.G. Pedersen / Carbon 42 (2004) 1007–1010
ð1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 þ m2 þ nm=p is the diameter and where d ¼p2:46 A ffiffiffi a ¼ tan1 ½ 3m=ð2n þ mÞ the chiral angle. This characteristic dependence is seen to resemble that of the experimental excitation energy Eexc quite closely. As shown by Bachilo et al. [2], Eexc follows the empirical relation (converted to eV and A):
ðn mÞ mod 3 ¼ 1
:
ðn mÞ mod 3 ¼ 2
ð2Þ
Most notably, the reduced mass and the excitation energy depend in similar manners on both the diameter and the chiral angle. The similarities are clearly recognized from Fig. 1, in which l and Eexc are illustrated Also, the together for CNs in the diameter range 6–16 A. lines highlight the perceived patterns in both quantities. The marked similarity between l and Eexc leads to equally striking results for the exciton binding energy Ebind illustrated in Fig. 2. This is a consequence of the conversion from natural excitonic to absolute units using the relation Ry ¼ 13:6 eV l=e2 . Hence, the conversion factor has precisely the same d and a dependence as the reduced mass l. This similarity indicates that the geometry dependence of the reduced mass plays an important role for the excitation energy. Also, it may readily be appreciated from Fig. 2 that the binding energy amounts to a sizeable fraction of the excitation energy. This is seen more clearly in Fig. 3, in
0.10
1.4
0.08
1.2 1.0
0.06
0.8 0.04
0.6
0.02 0.00 6
0.4 8
10
12
14
0.2 16
Excitation energy E exc [eV]
In our effective mass description of CNs [10], the exciton is described as an electron–hole pair moving on the surface of a cylinder. In addition to the effective mass approximation for electrons and holes, screening is introduced via a structure-independent dielectric constant. The resulting two-dimensional eigenproblem for the fundamental exciton is solved using a simple variational approach. Taking x and y as relative coordinates running along the circumference and long axis of the cylinder, respectively, we approximate the exciton wave 1=2 function by the trial expression exp½ðx2 =q2 þ y 2 =k 2 Þ with q and k as variational parameters. This procedure shows that the exciton binding energy diverges as a power law for small-diameter CNs. For large tubes, the results approach those of a flat ‘‘quantum well’’ and so the binding energy becomes –4 in units of effective Rydbergs Ry . The variational results expressed in natural exciton units (Ry and aB ) must subsequently be via the relaconverted into ordinary units (eV and A) 2 e=l, tions Ry ¼ 13:6 eV l=e and aB ¼ 0:529 A where l is the reduced effective mass in units of the freeelectron mass m0 and e 3:5 is the static dielectric constant. Here, an important point is that the conversion factors differ between different CNs since the reduced effective mass is a function of both diameter and chiral angle. We have used the zone-folding method to calculate this dependence. Valence band maxima and conduction band minima are carefully located from the numerical band structure curves and fitted to parabolas. It turns out that the observed dependence is extremely accurately described by the expression 0:3871 A l¼ d ( 2 cosð3aÞ=d 2 ; ðn mÞ mod 3 ¼ 1 0:5083 A þ ; 2 cosð3aÞ=d 2 ; 0:8533 A ðn mÞ mod 3 ¼ 2
eV 11:62 A d þ 1:476 A ( 2 eV cosð3aÞ=d 2 ; 8:803 A þ 2 eV cosð3aÞ=d 2 ; 4:575 A
Nanotube diameter [Å] Fig. 1. Reduced effective mass (left scale) and experimental excitation The lines energy (right scale) for CNs in the diameter range 6–16 A. highlight the similarity of the patterns perceived in both quantities.
0.7
1.4
0.6
1.2
0.5
1.0
0.4
0.8
0.3
0.6
0.2
0.4
0.1
6
8
10
12
14
0.2 16
Excitation energy Eexc [eV]
2. Theoretical model
Eexc ¼
Reduced mass µ [m0]
divisible by three. We find two important results: (i) the calculated binding energy displays the same regular pattern as the experimental excitation energy and (ii) for all CNs the binding energy exceeds 30% of the experimental excitation energy. We then utilize the experimentally determined excitation energy to fix the single-particle band gap including quasi-particle shift and subsequently, as an example, calculate the absorption spectrum for a (7,6)-nanotube.
Binding energy | E bind | [eV]
1008
Nanotube diameter [Å] Fig. 2. Illustration of the similarities between excitation energy and absolute exciton binding energy for different CNs.
T.G. Pedersen / Carbon 42 (2004) 1007–1010 0.40
Absorpsion [arb. units]
1.00
| Ebind | / Eexc
0.38 0.36 0.34 0.32 0.30
8
10
12
14
which the ratio between the two quantities is plotted for different CNs. In all cases, the ratio exceeds 30% with a slight decreasing tendency for larger tubes. The smaller exciton effect for larger tubes is a consequence of the decreasing degree of confinement. The large ratio found for all CNs is a clear demonstration of the importance of exciton effects in these systems.
3. Absorption spectrum By combining the variational binding energy Ebind and the observed excitation energy Eexc we are now in a position to determine the single-particle energy gap Eg via the relation Eexc ¼ Eg þ Ebind . It should be noticed that Eg is the true gap including quasi-particle corrections. If, furthermore, identical quasi-particle corrections are assumed for higher as well as the fundamental exciton, we can calculate the absorption spectrum from the approximate expression [13]
n
ðEn þ Eg Þ½ðEn þ Eg hxÞ þ ð hCÞ
2
2
:
0.8
1.0
1.2
1.4
1.6
Photon energy hω [eV]
Fig. 3. Plot of the ratio between absolute exciton binding energy and measured excitation energy.
jwn ð0; 0Þj2
Exciton
0.25
16
X
Free carrier
0.50
Nanotube diameter [Å]
aðxÞ /
(7,6) Nanotube
0.75
0.00 6
1009
ð3Þ
Here, wn ðx; yÞ is the nth exciton state with energy En , hx is the photon energy and hC ¼ 0:05 eV is a phenomenological line width. In order to obtain the required exciton states, we expand the relative-coordinate Schr€ odinger equation in an orthogonal basis. The variational approach described above cannot be applied here since it only provides the lowest exciton state. Our choice of basis is of the form upq ðx; yÞ ¼ fp ðxÞgq ðyÞ. Actually, the xdependence is ‘‘frozen’’ to a large extent, as this is the confined dimension for which excitations are energetically costly. The y-axis, on the other hand, is along the nanotube and excitations that mainly involve the ymotion are not restricted by any confinement. This means that a relatively large number of basis states gq ðyÞ are needed to accurately capture the y-motion, whereas the xmotion is adequately described using just a few terms fp ðxÞ. In addition, we are only interested in the optically
Fig. 4. Theoretical absorption spectra for a (7,6)-nanotube. The freecarrier absorption peak is near the band gap of 1.49 eV whereas the exciton spectrum is red-shifted by a binding energy of approximately )0.38 eV.
active excitons, i.e. those for which wn ð0; 0Þ is non-zero. Hence, only basis functions that are even in x and y are needed. For the y-motion, a Laguerre type basis has been used: gq ðyÞ ¼ ð2kÞ1=2 Lq ðjyj=kÞ expðjyj=2kÞ, where Lq is a Laguerre polynomial and k ¼ 1:35 (in natural units) has been selected since this value has been found to minimize the lowest exciton eigenvalue. For the x-motion, only two terms have been used in the expansion: a constant function and a function linear in jxj. The inclusion of higher order terms has a marginal effect on the states in the relevant energy range. Using 51 members in the Laguerre basis, we have found excellent agreement with the variational results for the fundamental exciton. As an example of the applicability of the present approach, we have considered a (7,6)-nanotube. Experimentally [2], the excitation energy is found to be Eexc ¼ 1:105 eV. The diameter and chiral angle forpffiffiffiffiffiffiffiffi 127=p 8:82 A and a ¼ mulas pyield d ¼ 2:46 A ffiffiffi 1 tan ½3 3=10 27:5° and, consequently, l ¼ 0:043 m0 , The calculated exciton Ry ¼ 0:048 eV and aB ¼ 43 A. binding energy converted using these values is then Ebind 0:38 eV or, roughly, )34% of the excitation energy. In turn, the true band gap must be Eg 1:49 eV. Using this value in Eq. (3) leads to the absorption spectrum depicted in Fig. 4. In the figure, the free-carrier spectrum given by a broadened square-root singularity at the band gap has been included for comparison. The most striking difference between the two spectra is obviously the red-shift of the exciton curve. Also, the free-carrier curve is highly asymmetric whereas the exciton spectrum consists of a symmetric peak at the location of the fundamental resonance and a flat continuum with an onset located near the band gap. 4. Summary In the present paper, exciton effects in CNs have been considered within an effective mass description of
1010
T.G. Pedersen / Carbon 42 (2004) 1007–1010
carriers confined to the surface of a cylinder representing the nanotube. We have shown that the exciton binding energy follows a diameter- and chiral angle-dependence, which closely resembles the behavior of measured excitation energies. Moreover, the magnitude of the binding energy amounts to more that 30% of the excitation energy in all cases. As a consequence, several important differences are predicted for exciton and free-carrier absorption spectra. References [1] O’Connell MJ et al. Band gap fluorescence from individual singlewalled carbon nanotubes. Science 2002;297:593–6. [2] Bachilo SM et al. Structure-assigned optical spectra of singlewalled carbon nanotubes. Science 2002;298:2361–6. [3] Saito R, Fujita M, Dresselhaus G, Dresselhaus MS. Electronic structure of chiral graphene tubules. Appl Phys Lett 1992;60:2204–6. [4] Saito R, Dresselhaus G, Dresselhaus MS. Physical Properties of Carbon Nanotubes. London: Imperial College Press; 1998.
[5] Saito R, Dresselhaus G, Dresselhaus MS. Trigonal warping effect of carbon nanotubes. Phys Rev B 1999;61:2981–90. [6] Reich S, Maultzsch J, Thomsen C, Ordej on P. Tight-binding description of graphene. Phys Rev B 2002;66:035412. [7] Ichida M, Mizuno S, Saito Y, Kataura H, Achiba Y, Nakamura A. Coulomb effect on the fundamental optical transition in semiconducting single-walled carbon nanotubes: divergent behavior in the small-diameter limit. Phys Rev B 2002;65:241407. [8] Ichida M, Mizuno S, Tani Y, Saito Y, Nakamura A. Exciton effects of optical transitions in single-wall carbon nanotubes. J Phys Soc Jpn 1999;68:3131–3. [9] Ando T. Excitons in carbon nanotubes. J Phys Soc Jpn 1997;66:1066–73. [10] Pedersen TG. Variational approach to excitons in carbon nanotubes. Phys Rev B 2003;67:073401. [11] Shinada M, Sugano S. Interband optical transitions in extremely anisotropic semiconductors I. Bound and unbound exciton absorption. J Phys Soc Jpn 1966;21:1936–46. [12] Loudon R. One-dimensional hydrogen atom. Am J Phys 1959;27:649–55. [13] Haug H, Koch SW. Quantum Theory of the Optical and Electronic Properties of Semiconductors. Singapore: World Scientific; 1993.
Exciton effects in carbon nanotubes Thomas G. Pedersen
*
Institute of Physics, Aalborg University, Pontoppidanstræde 103, DK-9220 Aalborg Øst, Denmark Available online 25 January 2004
Abstract Exciton effects in quasi-one-dimensional materials such as carbon nanotubes are expected to be important due to confinement of electrons and holes. We present a simple picture of these effects based on an effective mass description of electron–hole pairs moving on the surface of a cylinder. A compilation of results for different geometries shows that the exciton binding energy follows the same dependence on diameter and chiral angle as the measured excitation energy. The exciton effects are found be substantial and produce a complete rearrangement of the absorption spectrum. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: A. Carbon nanotubes; D. Optical properties
1. Introduction Recent advances in fluorescence spectroscopy [1,2] have finally achieved the goal of connecting experimental optical gaps of carbon nanotubes (CNs) to their geometric structure. A highly regular pattern has been established in which the fundamental excitation energy is simply related to nanotube diameter and chiral angle [2]. It is, hence, a challenge to theoretical physics to explain the observed regularity as well as the overall magnitude of the gaps. Much insight has been gained from the zone-folding tight-binding model of the band structure of CNs [3–5]. This simple method predicts an excitation energy varying approximately as the inverse of the CN diameter and having only a weak dependence on chiral angle [3,4]. More elaborate tight-binding calculation including orbital overlap and interactions beyond nearest neighbours [2,6] partly account for the observed dependence on chirality. However, both experimental [7,8] and theoretical [9,10] evidence point to the fact that a simple band-structure calculation is insufficient for an accurate description of optical excitations in CNs. This is due to the omission of two highly important effects in low-dimensional semiconductors: (i) quasi-particle corrections to the simple band-structure and (ii) exciton effects. Both quasi-particle and exciton
*
Tel.: +45-9635-9228; fax: +45-9815-6502. E-mail address: [email protected] (T.G. Pedersen).
0008-6223/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2003.12.028
shifts are due to screened Coulomb interactions. The former describes the repulsive energy needed to add an additional electron to the system and, hence, raises the energy of the conduction band or, equivalently, increases the band gap. In contrast, the exciton shift describes the attractive Coulomb interaction between electrons and holes, which lowers the excitation energy. The experimental results in Refs. [7,8] demonstrate that, in fact, the overall effect is a blue-shift so that the positive quasi-particle correction actually dominates over the negative exciton binding energy. It is the aim of the present paper to provide a simple model for the exciton shift and to evaluate the influence on the optical properties of CNs. Excitonic effects in low-dimensional semiconductors such as CNs are greatly enhanced over the bulk counterparts because confinement leads to a larger overlap between electrons and holes and, thereby, to an enhanced Coulomb binding energy. For ideal quantum wells, the binding energy is enhanced by a factor of four [11]. However, for quasi-one-dimensional structures such as quantum wires and nanotubes, the binding effect diverges as the diameter decreases [12]. Hence, binding energies readily reach )100 meV or more, several orders of magnitude larger that typical bulk values. In our previous work [10], we applied a variational approach to provide an estimate of the exciton effect in zigzag CNs. In the present paper, we extend the analysis to arbitrary chiral vectors covering all semiconducting CNs, i.e. all (n; m) in the usual notation [3] for which n m is not
T.G. Pedersen / Carbon 42 (2004) 1007–1010
ð1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 þ m2 þ nm=p is the diameter and where d ¼p2:46 A ffiffiffi a ¼ tan1 ½ 3m=ð2n þ mÞ the chiral angle. This characteristic dependence is seen to resemble that of the experimental excitation energy Eexc quite closely. As shown by Bachilo et al. [2], Eexc follows the empirical relation (converted to eV and A):
ðn mÞ mod 3 ¼ 1
:
ðn mÞ mod 3 ¼ 2
ð2Þ
Most notably, the reduced mass and the excitation energy depend in similar manners on both the diameter and the chiral angle. The similarities are clearly recognized from Fig. 1, in which l and Eexc are illustrated Also, the together for CNs in the diameter range 6–16 A. lines highlight the perceived patterns in both quantities. The marked similarity between l and Eexc leads to equally striking results for the exciton binding energy Ebind illustrated in Fig. 2. This is a consequence of the conversion from natural excitonic to absolute units using the relation Ry ¼ 13:6 eV l=e2 . Hence, the conversion factor has precisely the same d and a dependence as the reduced mass l. This similarity indicates that the geometry dependence of the reduced mass plays an important role for the excitation energy. Also, it may readily be appreciated from Fig. 2 that the binding energy amounts to a sizeable fraction of the excitation energy. This is seen more clearly in Fig. 3, in
0.10
1.4
0.08
1.2 1.0
0.06
0.8 0.04
0.6
0.02 0.00 6
0.4 8
10
12
14
0.2 16
Excitation energy E exc [eV]
In our effective mass description of CNs [10], the exciton is described as an electron–hole pair moving on the surface of a cylinder. In addition to the effective mass approximation for electrons and holes, screening is introduced via a structure-independent dielectric constant. The resulting two-dimensional eigenproblem for the fundamental exciton is solved using a simple variational approach. Taking x and y as relative coordinates running along the circumference and long axis of the cylinder, respectively, we approximate the exciton wave 1=2 function by the trial expression exp½ðx2 =q2 þ y 2 =k 2 Þ with q and k as variational parameters. This procedure shows that the exciton binding energy diverges as a power law for small-diameter CNs. For large tubes, the results approach those of a flat ‘‘quantum well’’ and so the binding energy becomes –4 in units of effective Rydbergs Ry . The variational results expressed in natural exciton units (Ry and aB ) must subsequently be via the relaconverted into ordinary units (eV and A) 2 e=l, tions Ry ¼ 13:6 eV l=e and aB ¼ 0:529 A where l is the reduced effective mass in units of the freeelectron mass m0 and e 3:5 is the static dielectric constant. Here, an important point is that the conversion factors differ between different CNs since the reduced effective mass is a function of both diameter and chiral angle. We have used the zone-folding method to calculate this dependence. Valence band maxima and conduction band minima are carefully located from the numerical band structure curves and fitted to parabolas. It turns out that the observed dependence is extremely accurately described by the expression 0:3871 A l¼ d ( 2 cosð3aÞ=d 2 ; ðn mÞ mod 3 ¼ 1 0:5083 A þ ; 2 cosð3aÞ=d 2 ; 0:8533 A ðn mÞ mod 3 ¼ 2
eV 11:62 A d þ 1:476 A ( 2 eV cosð3aÞ=d 2 ; 8:803 A þ 2 eV cosð3aÞ=d 2 ; 4:575 A
Nanotube diameter [Å] Fig. 1. Reduced effective mass (left scale) and experimental excitation The lines energy (right scale) for CNs in the diameter range 6–16 A. highlight the similarity of the patterns perceived in both quantities.
0.7
1.4
0.6
1.2
0.5
1.0
0.4
0.8
0.3
0.6
0.2
0.4
0.1
6
8
10
12
14
0.2 16
Excitation energy Eexc [eV]
2. Theoretical model
Eexc ¼
Reduced mass µ [m0]
divisible by three. We find two important results: (i) the calculated binding energy displays the same regular pattern as the experimental excitation energy and (ii) for all CNs the binding energy exceeds 30% of the experimental excitation energy. We then utilize the experimentally determined excitation energy to fix the single-particle band gap including quasi-particle shift and subsequently, as an example, calculate the absorption spectrum for a (7,6)-nanotube.
Binding energy | E bind | [eV]
1008
Nanotube diameter [Å] Fig. 2. Illustration of the similarities between excitation energy and absolute exciton binding energy for different CNs.
T.G. Pedersen / Carbon 42 (2004) 1007–1010 0.40
Absorpsion [arb. units]
1.00
| Ebind | / Eexc
0.38 0.36 0.34 0.32 0.30
8
10
12
14
which the ratio between the two quantities is plotted for different CNs. In all cases, the ratio exceeds 30% with a slight decreasing tendency for larger tubes. The smaller exciton effect for larger tubes is a consequence of the decreasing degree of confinement. The large ratio found for all CNs is a clear demonstration of the importance of exciton effects in these systems.
3. Absorption spectrum By combining the variational binding energy Ebind and the observed excitation energy Eexc we are now in a position to determine the single-particle energy gap Eg via the relation Eexc ¼ Eg þ Ebind . It should be noticed that Eg is the true gap including quasi-particle corrections. If, furthermore, identical quasi-particle corrections are assumed for higher as well as the fundamental exciton, we can calculate the absorption spectrum from the approximate expression [13]
n
ðEn þ Eg Þ½ðEn þ Eg hxÞ þ ð hCÞ
2
2
:
0.8
1.0
1.2
1.4
1.6
Photon energy hω [eV]
Fig. 3. Plot of the ratio between absolute exciton binding energy and measured excitation energy.
jwn ð0; 0Þj2
Exciton
0.25
16
X
Free carrier
0.50
Nanotube diameter [Å]
aðxÞ /
(7,6) Nanotube
0.75
0.00 6
1009
ð3Þ
Here, wn ðx; yÞ is the nth exciton state with energy En , hx is the photon energy and hC ¼ 0:05 eV is a phenomenological line width. In order to obtain the required exciton states, we expand the relative-coordinate Schr€ odinger equation in an orthogonal basis. The variational approach described above cannot be applied here since it only provides the lowest exciton state. Our choice of basis is of the form upq ðx; yÞ ¼ fp ðxÞgq ðyÞ. Actually, the xdependence is ‘‘frozen’’ to a large extent, as this is the confined dimension for which excitations are energetically costly. The y-axis, on the other hand, is along the nanotube and excitations that mainly involve the ymotion are not restricted by any confinement. This means that a relatively large number of basis states gq ðyÞ are needed to accurately capture the y-motion, whereas the xmotion is adequately described using just a few terms fp ðxÞ. In addition, we are only interested in the optically
Fig. 4. Theoretical absorption spectra for a (7,6)-nanotube. The freecarrier absorption peak is near the band gap of 1.49 eV whereas the exciton spectrum is red-shifted by a binding energy of approximately )0.38 eV.
active excitons, i.e. those for which wn ð0; 0Þ is non-zero. Hence, only basis functions that are even in x and y are needed. For the y-motion, a Laguerre type basis has been used: gq ðyÞ ¼ ð2kÞ1=2 Lq ðjyj=kÞ expðjyj=2kÞ, where Lq is a Laguerre polynomial and k ¼ 1:35 (in natural units) has been selected since this value has been found to minimize the lowest exciton eigenvalue. For the x-motion, only two terms have been used in the expansion: a constant function and a function linear in jxj. The inclusion of higher order terms has a marginal effect on the states in the relevant energy range. Using 51 members in the Laguerre basis, we have found excellent agreement with the variational results for the fundamental exciton. As an example of the applicability of the present approach, we have considered a (7,6)-nanotube. Experimentally [2], the excitation energy is found to be Eexc ¼ 1:105 eV. The diameter and chiral angle forpffiffiffiffiffiffiffiffi 127=p 8:82 A and a ¼ mulas pyield d ¼ 2:46 A ffiffiffi 1 tan ½3 3=10 27:5° and, consequently, l ¼ 0:043 m0 , The calculated exciton Ry ¼ 0:048 eV and aB ¼ 43 A. binding energy converted using these values is then Ebind 0:38 eV or, roughly, )34% of the excitation energy. In turn, the true band gap must be Eg 1:49 eV. Using this value in Eq. (3) leads to the absorption spectrum depicted in Fig. 4. In the figure, the free-carrier spectrum given by a broadened square-root singularity at the band gap has been included for comparison. The most striking difference between the two spectra is obviously the red-shift of the exciton curve. Also, the free-carrier curve is highly asymmetric whereas the exciton spectrum consists of a symmetric peak at the location of the fundamental resonance and a flat continuum with an onset located near the band gap. 4. Summary In the present paper, exciton effects in CNs have been considered within an effective mass description of
1010
T.G. Pedersen / Carbon 42 (2004) 1007–1010
carriers confined to the surface of a cylinder representing the nanotube. We have shown that the exciton binding energy follows a diameter- and chiral angle-dependence, which closely resembles the behavior of measured excitation energies. Moreover, the magnitude of the binding energy amounts to more that 30% of the excitation energy in all cases. As a consequence, several important differences are predicted for exciton and free-carrier absorption spectra. References [1] O’Connell MJ et al. Band gap fluorescence from individual singlewalled carbon nanotubes. Science 2002;297:593–6. [2] Bachilo SM et al. Structure-assigned optical spectra of singlewalled carbon nanotubes. Science 2002;298:2361–6. [3] Saito R, Fujita M, Dresselhaus G, Dresselhaus MS. Electronic structure of chiral graphene tubules. Appl Phys Lett 1992;60:2204–6. [4] Saito R, Dresselhaus G, Dresselhaus MS. Physical Properties of Carbon Nanotubes. London: Imperial College Press; 1998.
[5] Saito R, Dresselhaus G, Dresselhaus MS. Trigonal warping effect of carbon nanotubes. Phys Rev B 1999;61:2981–90. [6] Reich S, Maultzsch J, Thomsen C, Ordej on P. Tight-binding description of graphene. Phys Rev B 2002;66:035412. [7] Ichida M, Mizuno S, Saito Y, Kataura H, Achiba Y, Nakamura A. Coulomb effect on the fundamental optical transition in semiconducting single-walled carbon nanotubes: divergent behavior in the small-diameter limit. Phys Rev B 2002;65:241407. [8] Ichida M, Mizuno S, Tani Y, Saito Y, Nakamura A. Exciton effects of optical transitions in single-wall carbon nanotubes. J Phys Soc Jpn 1999;68:3131–3. [9] Ando T. Excitons in carbon nanotubes. J Phys Soc Jpn 1997;66:1066–73. [10] Pedersen TG. Variational approach to excitons in carbon nanotubes. Phys Rev B 2003;67:073401. [11] Shinada M, Sugano S. Interband optical transitions in extremely anisotropic semiconductors I. Bound and unbound exciton absorption. J Phys Soc Jpn 1966;21:1936–46. [12] Loudon R. One-dimensional hydrogen atom. Am J Phys 1959;27:649–55. [13] Haug H, Koch SW. Quantum Theory of the Optical and Electronic Properties of Semiconductors. Singapore: World Scientific; 1993.