Large Anisotropy in the Magnetic Susceptibility of ... - Semantic Scholar
PRL 105, 017403 (2010)
PHYSICAL REVIEW LETTERS
week ending 2 JULY 2010
Large Anisotropy in the Magnetic Susceptibility of Metallic Carbon Nanotubes T. A. Searles,1 Y. Imanaka,2 T. Takamasu,2 H. Ajiki,3 J. A. Fagan,4 E. K. Hobbie,4 and J. Kono1,* 1
Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005, USA 2 National Institute for Materials Science, 3-13 Sakura, Tsukuba, Ibaraki 305-0003, Japan 3 Photon Pioneers Center, Osaka University, Suita 565-0871, Japan 4 National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA (Received 4 January 2010; published 2 July 2010)
Through magnetic linear dichroism spectroscopy, the magnetic susceptibility anisotropy of metallic single-walled carbon nanotubes has been extracted and found to be 2–4 times greater than values for semiconducting nanotubes. This large anisotropy can be understood in terms of large orbital paramagnetism of metallic nanotubes arising from the Aharonov-Bohm-phase-induced gap opening in a parallel field, and our calculations quantitatively reproduce these results. We also compare our values with previous work for semiconducting nanotubes, which confirm that the magnetic susceptibility anisotropy does not increase linearly with the diameter for small-diameter nanotubes. DOI: 10.1103/PhysRevLett.105.017403
PACS numbers: 78.67.Ch, 73.22.f, 78.20.Ls
Unusual magnetic properties have been predicted for single-walled carbon nanotubes (SWNTs) due to the combined effects of the Aharonov-Bohm effect and the large diamagnetic susceptibility of the graphene lattice [1–9]. Their orbital magnetic susceptibility () is expected to be large, 2 orders of magnitude larger than their spin magnetic susceptibility, and the value of is predicted to be strongly dependent on the strength of the applied magnetic field, B, and the Fermi energy. Furthermore, the sign of can be either positive (paramagnetic) or negative (diamagnetic), depending on the nanotube chirality as well as the field orientation. We have calculated the orientation-dependent of SWNTs within the k p effective-mass approximation [5], as shown in Fig. 1 [10]. Starting from a massless Dirac Hamiltonian, we took into account the effect of B through the Landau-Peierls substitution k^ ¼ ir þ eA=c@, where k^ is the wave vector, A is the vector potential (B ¼ r A), c is the speed of light, and @ is the reduced Planck constant. When a parallel magnetic flux threads the nanotube, the Aharonov-Bohm effect modifies the circumferential boundary condition through a phase factor expð2i=0 Þ, where 0 ¼ ch=e is the magnetic flux quantum, and, consequently, the band gap oscillates with period 0 [1,5,10]. As increases from zero, a band gap opens up in a metallic tube (the band gap shrinks in a semiconducting tube), which leads to orbital paramagnetism (diamagnetism). On the other hand, both metallic and semiconducting SWNTs show diamagnetism in a perpendicular field. Thus, all SWNTs are expected to have positive magnetic susceptibility anisotropy ¼ k ? > 0, where k (? ) is the parallel (perpendicular) susceptibility. In addition, metallic SWNTs are expected to have greater values of than semiconducting SWNTs. These trends can be confirmed in the calculated for ð6; 6Þ (metallic) and ð6; 5Þ (semiconducting) nanotubes shown in Fig. 1 [10]. 0031-9007=10=105(1)=017403(4)
A finite > 0 results in the alignment of SWNTs in the B direction, which, combined with the anisotropic optical properties of SWNTs, allows for an estimation of values for SWNTs through magneto-optical spectroscopy. Previous magneto-optical studies found that 1:5 105 emu=mol for semiconducting SWNTs with 1 nm diameters [9,11–15], while no information on is currently available for metallic SWNTs. Below, we present the first experimental estimation of the of metallic SWNTs through high-field magnetic linear dichroism spectroscopy of cobalt molybdenum catalyst-synthesized nanotubes (mean diameter 0:8 nm) individually suspended in aqueous solution at room temperature. The sample had a much smaller diameter distribution than the high-pressure carbon monoxide-produced samples used in the previous studies [11,12], which allowed us to clearly identify and closely examine the mag-
FIG. 1 (color online). Calculated magnetic susceptibility anisotropy of single-walled carbon nanotubes. The susceptibility is expressed in units of ¼ ð2=aÞða2 =0 Þ2 a2 ¼ 1:46 a ¼ 2:5 A, and 0 ¼ 104 emu=mol, where ¼ 6:46 eV A, ch=e is the magnetic flux quantum. See supplementary material [10] for details on the calculation methods used.
017403-1
Ó 2010 The American Physical Society
Ak þ 2A? ; 3
(1)
where Ak (A? ) is the absorption for light polarized parallel (perpendicular) to the orientation axis, i.e., the magnetic field direction. We calculated A0 at 35 T by using Eq. (1) and plotted it in Fig. 2(a) as a red dashed line. The agreement between the calculated A0 and the zero-field absorption spectrum confirms that A0 is independent of alignment (or B), because at 0 T Ak ¼ A? ¼ A0 . We further confirmed that at any fields between 0 and 35 T the increase in Ak and decrease in A? are such that A0 is preserved through Eq. (1). Linear dichroism, LD ¼ Ak A? , is a measure of the degree of alignment. However, since it directly depends on the absorbance, LD alone cannot be used for comparing different spectral features. To adjust for differences in
Absorbance
0.8
(6,5)
(a)
Data A0 = (A// + 2A⊥)/3
(6,4)
0.6
(6,6) (7,4)
(7,5) (8,3)
(6,5) phonon
A// @ 35 T (5,5)
0 T & A0
0.4
A @ 35 T
(6,6) (7,4)
35 T
0.6
(b)
r
netic field dependence of absorption peaks for metallic nanotubes. We made a detailed comparison of the of metallic and semiconducting nanotubes with similar diameters and lengths within the same sample and found that the values of are 2–4 times greater in metallic tubes. We also compared our values with previous work for semiconducting nanotubes [14], which confirm a break from the prediction that should increase linearly with the tube diameter. Polarized magneto-optical absorption measurements on length-sorted, ð6; 5Þ-enriched cobalt molybdenum catalyst SWNTs were performed by using the 35 T hybrid magnet at the National Institute for Materials Science in Tsukuba, Japan. The SWNTs were suspended in 1% sodium deoxycholate and length sorted by dense liquid ultracentrifugation [16] to have an average length of 500 nm. A Xe lamp, fiber coupled to a custom optical probe, allowed for broadband white-light excitation of the E11 metallic, E22 semiconductor, and E33 semiconductor interband transitions of SWNTs. The sample was held in a cuvette with a film polarizer directly on the front face to ensure that the incident light was linearly polarized. The transmitted light was collimated by a lens, collected with another fiber, and dispersed through a monochromator equipped with a Si CCD. All measurements were done at 300 K. Figure 2(a) shows polarization-dependent optical absorption spectra at 0 and 35 T. None of the spectra are intentionally offset, indicating an increase (decrease) in absorbance for light polarized parallel (perpendicular) to the field. This is a direct result of magnetic alignment, together with the fact that only the light-field component parallel to the tube axis is strongly absorbed in SWNTs. Namely, at 35 T there is a finite degree of alignment in the magnetic field direction within the nanotube ensemble, resulting in stronger (weaker) absorption for light polarized parallel (perpendicular) to the field. From the theory of linear dichroism for an ensemble of anisotropic molecules [17], the following quantity can be shown to be constant, independent of the degree of alignment: A0
week ending 2 JULY 2010
PHYSICAL REVIEW LETTERS
LD
PRL 105, 017403 (2010)
0.5
(5,5)
(6,5) (6,4)
0.4 1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
Energy (eV) FIG. 2 (color online). (a) Absorption spectra (solid black line) for 0 and 35 T with all peaks assigned to specific chiralities. No trace is intentionally offset, indicating greater (smaller) absorption for parallel (perpendicular) polarization. The unpolarized isotropic absorbance (dashed red line), calculated from the 35 T spectra via Eq. (1), agrees well with the 0 T data. (b) Reduced linear dichroism versus energy from data taken at 35 T. The largest peak is from metallic nanotubes ð6; 6Þ and ð7; 4Þ.
relative absorbances due to the fact that our sample is enriched for ð6; 5Þ, the LD was divided by A0 , yielding the reduced linear dichroism, LDr ¼ LD=A0 [17,18], which is plotted in Fig. 2(b) for 35 T. It is immediately evident from this plot that the spectral region where metallic peaks [ð6; 6Þ and ð7; 4Þ] exist has much larger LDr than the region where the most prominent semiconducting peaks [ð6; 5Þ and ð6; 4Þ] exist. This is evidence that the ð6; 6Þ and ð7; 4Þ tubes are aligning more strongly than the ð6; 5Þ and ð6; 4Þ tubes. In order to quantitatively determine the magnetic alignment properties of each chirality present in our sample, we performed detailed spectral analysis, as described below. Each absorption spectrum was fit by a superposition of multiple Lorentzian peaks plus a polynomial offset. The zero-field spectrum was fit first, as shown in Fig. 3(a), with each Lorentzian assigned to a specific chirality. The offset was included to account for the contributions from the plasmon peaks, bundles, and light scattering [19–21]. For each magnetic field, the parallel and perpendicular spectra were fit simultaneously with the constraint that each Lorentzian independently satisfy Eq. (1). Figure 3(b) shows representative results for the ð7; 5Þ, ð6; 5Þ, ð6; 6Þ, and ð5; 5Þ nanotubes at 0 and 35 T. Note that we required the width and position of each Lorentzian to be the same as the values obtained at 0 T, which ensured that the calculated LDr for each chirality is constant as a function of
017403-2
PRL 105, 017403 (2010) 0.6
0 T, data 0 T, fit (total) individual fits
Absorbance
0.5 0.4
and ð5; 5Þ] show much stronger alignment than the semiconducting nanotubes [ð6; 5Þ and ð6; 4Þ]. To model the observed magnetic field dependence of S, we approximate the angular distribution Pu ðÞ of nanotubes at temperature T in a magnetic field B by the Maxwell-Boltzmann distribution function [11–14,22]
(a)
(6,5)
0.3 0.2 0.1
(7,5) (8,3)
(6,5) phonon
(6,4)
(6,6) (7,4)
(5,5)
expðu2 sin2 Þ sin Pu ðÞ ¼ R=2 ; (2) expðu2 sin2 Þ sind 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where u ¼ B2 N=2kB T is a dimensionless measure of the relative importance of the magnetic alignment energy (B2 N=2) [22] and the thermal energy (kB T), N is the number of carbon atoms in each nanotube with a length of 500 nm (in moles), and kB represents Boltzmann’s constant. Using this distribution function and the definition of S, one can derive Z =2 3cos2 1 d SðuÞ ¼ Pu ðÞ 2 0 1 3 u pffiffiffi ¼ þ 1 ; (3) u2 2 4u2 erfiðuÞ 2 e
(7,5) E33
0.0 (6,5)
(b)
Absorbance
0.4 35 T, parallel 0T 35 T, perpendicular
0.3 0.2 (7,5)
(6,6) (5,5)
0.1 0.0 Metal
(6,6) (5,5) (7,4)
1.0 (8,3)
r
LD @ 35 T
(c)
Semiconductor
1.5
week ending 2 JULY 2010
PHYSICAL REVIEW LETTERS
where erfi is the imaginary error function [erfiðuÞ ¼ pffiffiffiffi R t2 ð2=i Þ iu dt]. Equation (3) is plotted in Fig. 4(b). 0 e We were able to fit the S versus B curves with Eq. (3), as shown for ð6; 6Þ and ð6; 5Þ nanotubes in Fig. 4(c), to extract
3.0
FIG. 3 (color online). (a) Fitting results for 0 T. (b) Lorentzians for different chirality nanotubes at 0 and 35 T. (c) LDr versus energy (eV) derived from fitting for each individual chirality nanotube in our sample. The metallic tubes (red line) are higher than the semiconducting nanotubes (blue dotted line).
photon energy, as shown in Fig. 3(c) for 35 T. Here, each LDr line is a direct result of calculation from each Lorentzian of different chirality, while LDr in Fig. 2(b) was calculated from entire spectra. As a result, one can extract chirality-specific information on the alignment degree from Fig. 3(c). It is evident that the values are much larger for the metallic (red lines) than the semiconducting (blue lines) nanotubes. For anisotropic molecules with the dipole moment fully oriented along the long axis, the reduced linear dichroism can be expressed as LDr ¼ 3S [17,18]. Here, the nematic order parameter S ¼ ð3hcos2 i 1Þ=2 is a dimensionless measure of the degree of alignment, being equal to zero for completely randomly oriented nanotubes and one for completely aligned nanotubes, and is the angle between the nanotube axis and the orientation axis (i.e., the magnetic field direction in the present case). Values for S were calculated for representative nanotubes from the LDr values in Fig. 3(c) and plotted as a function of magnetic field in Fig. 4(a). Here, we can clearly see that the ðn; nÞ-chirality, or armchair, metallic nanotubes [ð6; 6Þ
1.0
(a)
0.5
(b) 0.8
0.4
(6,6) (5,5) (6,5) (6,4)
0.3
0.6
S
2.5 Energy (eV)
0.2
0.4
0.1
0.2
0.0
0
10 20 30 Magnetic Field (T)
40
0.0 0.1
2
4
2
1
u
4
2
4
10
3.0 0.6
0.4
(d)
(c)
(6,6)
2.5
(6,6) fit for (6,6) (6,5) fit for (6,5)
χ// /χ∗
2.0
S
0.5
(6,5) (6,4)
S
(7,5)
30 K
2.0
0.2 300 K
1.5 0.0 5
6 7 8 9
2
10 Magnetic Field (T)
3
4
0
10 20 30 Magnetic Field (T)
40
FIG. 4 (color online). (a) Nematic order parameter S versus magnetic field for armchair nanotubes ð5; 5Þ and ð6; 6Þ and semiconducting nanotubes ð6; 5Þ and ð6; 4Þ, calculated from the measured reduced linear dichroism values. (b) S versus u [Eq. (3)]. (c) Comparison of S versus magnetic field for ð6; 5Þ and ð6; 6Þ nanotubes. (d) Calculated magnetic field dependence of the parallel magnetic susceptibility of ð6; 6Þ nanotube for T ¼ 30 and 300 K in units of ¼ 1:46 104 emu=mol [10].
017403-3
PRL 105, 017403 (2010)
PHYSICAL REVIEW LETTERS
TABLE I. Comparison of theoretical and experimental values of magnetic susceptibility anisotropy for seven types of SWNTs. For each chirality ðn; mÞ, the diameter d, the chiral index ¼ ðn mÞ mod 3, and the chiral angle are given, followed by theoretical (30 and 300 K) and experimental (300 K) values of . All values for are 105 emu=mol. ðn; mÞ
d (nm)
( )
K 30 th
K 300 th
K 300 exp
ð6; 6Þ ð5; 5Þ ð7; 4Þ ð8; 3Þ ð7; 5Þ ð6; 5Þ ð6; 4Þ
0.83 0.69 0.77 0.78 0.83 0.76 0.69
0 0 0 1 1 1 1
30 30 21 15 24 27 23
5.78 4.95 5.42 1.49 1.58 1.45 1.33
3.92 3.39 3.70 1.46 1.55 1.42 1.29
3.63 3.35 2.44 2.13 1.66 1.01 1.24
for each nanotube chirality. However, care must be taken in using this procedure, since the values of k are expected to depend on B and T, especially for metallic nanotubes. Thus, we calculated as a function of B and T for all seven SWNTs studied in this work [10]. Figure 4(d) shows a theoretical calculation of the B dependence of k for the ð6; 6Þ nanotube for 30 and 300 K. Although k decreases by 27% at 30 K as the field increases from 0 to 40 T, it stays constant within 0:7% at 300 K in the same field range, ensuring that the magnetic field dependence can be neglected for our room temperature measurements. Experimentally estimated values are summarized in Table I, together with the theoretical values we calculated through the k p method [10]. The experimental values agree well with the theoretical values at 300 K. The values for the three metallic nanotubes, ð7; 4Þ, ð5; 5Þ, and ð6; 6Þ, are all higher than those for the semiconducting nanotubes. In particular, the of armchair, or ðn; nÞ, carbon nanotubes are 2–4 times larger than those in semiconducting nanotubes, depending on the diameter. This large difference in magnetic susceptibility anisotropy is consistent with theoretical predictions, a direct consequence of the AharonovBohm physics. Finally, the experimental values for metallic nanotubes in Table I do not follow a strict diameter dependence, something that is predicted [8] and shown experimentally [14] for some semiconducting tubes. For smaller diameter nanotubes, like the ones seen Table I, the chiral angle plays an important role in determining the values of . Furthermore, all the estimated values for all SWNTs (both metallic and semiconducting) are larger than those found in earlier studies for multiwalled nanotubes [23,24]. It is also important to note that the ð7; 4Þ nanotube has a larger value of than the semiconducting tubes, but it is not as large as those of the armchair tubes. Unfortunately, it is the only nonarmchair mod-3 nanotube in our sample, but a detailed study on a metallic-enriched sample [25] should yield many more metallic nanotubes to investigate in the future.
week ending 2 JULY 2010
In conclusion, we have successfully measured for metallic single-walled carbon nanotubes for the first time and confirmed that they are much larger than those of semiconducting nanotubes. We also calculated the magnetic field and temperature dependence of magnetic susceptibilities of the nanotubes studied experimentally, which support the experimental findings. Last, we were able to confirm previous experimental results for the chirality dependence of the magnetic susceptibility anisotropy in semiconducting nanotubes and found that this is also true for metallic nanotubes. This work was supported by the DOE-BES (Grant No. DE-FG02-06ER46308), the NSF (Grant No. OISE0530220), the Robert A. Welch Foundation (Grant No. C-1509), and Grant-in-Aid for Scientific Research under Grant No. 19054013 from the Ministry of Education, Culture, Sports, Science and Technology of Japan. We thank Noe Alvarez and Ajit Srivastava for their help with atomic force microscope length measurements and data analysis, respectively. This work is an official contribution of the National Institute of Standards and Technology and not subject to copyright in the United States.
*Corresponding author. [email protected] H. Ajiki and T. Ando, J. Phys. Soc. Jpn. 62, 1255 (1993). H. Ajiki and T. Ando, J. Phys. Soc. Jpn. 62, 2470 (1993). W. Tian and S. Datta, Phys. Rev. B 49, 5097 (1994). J. P. Lu, Phys. Rev. Lett. 74, 1123 (1995). H. Ajiki and T. Ando, J. Phys. Soc. Jpn. 64, 4382 (1995). S. Roche et al., Phys. Rev. B 62, 16 092 (2000). N. Nemec and G. Cuniberti, Phys. Rev. B 74, 165411 (2006). [8] M. A. L. Marques et al., Phys. Rev. B 73, 125433 (2006). [9] J. Kono and S. Roche, in Carbon Nanotubes: Properties and Applications, edited by M. J. O’Connell (CRC Press, Boca Raton, 2006), Chap. 5, pp. 119–151. [10] See supplementary material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.105.017403 for details of our theoretical calculations. [11] S. Zaric et al., Science 304, 1129 (2004). [12] S. Zaric et al., Nano Lett. 4, 2219 (2004). [13] M. F. Islam et al., Phys. Rev. B 71, 201401(R) (2005). [14] O. N. Torrens et al., J. Am. Chem. Soc. 129, 252 (2007). [15] J. Kono, R. J. Nicholas, and S. Roche, in Carbon Nanotubes, edited by A. Jorio et al. (Springer, Berlin, 2008), pp. 393–421. [16] J. A. Fagan et al., Adv. Mater. 20, 1609 (2008). [17] A. Rodger and B. Norden, Circular Dichroism and Linear Dichroism (Oxford University Press, Oxford, 1997). [18] J. Shaver et al., ACS Nano 3, 131 (2009). [19] Y. Murakami et al., Phys. Rev. Lett. 94, 087402 (2005). [20] N. Nair et al., Anal. Chem. 78, 7689 (2006). [21] J. A. Fagan et al., Appl. Phys. Lett. 91, 213105 (2007). [22] M. Fujiwara et al., J. Phys. Chem. B 103, 2627 (1999). [23] X. K. Wang et al., Appl. Phys. Lett. 62, 1881 (1993). [24] O. Chauvet et al., Phys. Rev. B 52, R6963 (1995). [25] E. H. Ha´roz et al., ACS Nano 4, 1955 (2010). [1] [2] [3] [4] [5] [6] [7]
017403-4
PHYSICAL REVIEW LETTERS
week ending 2 JULY 2010
Large Anisotropy in the Magnetic Susceptibility of Metallic Carbon Nanotubes T. A. Searles,1 Y. Imanaka,2 T. Takamasu,2 H. Ajiki,3 J. A. Fagan,4 E. K. Hobbie,4 and J. Kono1,* 1
Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005, USA 2 National Institute for Materials Science, 3-13 Sakura, Tsukuba, Ibaraki 305-0003, Japan 3 Photon Pioneers Center, Osaka University, Suita 565-0871, Japan 4 National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA (Received 4 January 2010; published 2 July 2010)
Through magnetic linear dichroism spectroscopy, the magnetic susceptibility anisotropy of metallic single-walled carbon nanotubes has been extracted and found to be 2–4 times greater than values for semiconducting nanotubes. This large anisotropy can be understood in terms of large orbital paramagnetism of metallic nanotubes arising from the Aharonov-Bohm-phase-induced gap opening in a parallel field, and our calculations quantitatively reproduce these results. We also compare our values with previous work for semiconducting nanotubes, which confirm that the magnetic susceptibility anisotropy does not increase linearly with the diameter for small-diameter nanotubes. DOI: 10.1103/PhysRevLett.105.017403
PACS numbers: 78.67.Ch, 73.22.f, 78.20.Ls
Unusual magnetic properties have been predicted for single-walled carbon nanotubes (SWNTs) due to the combined effects of the Aharonov-Bohm effect and the large diamagnetic susceptibility of the graphene lattice [1–9]. Their orbital magnetic susceptibility () is expected to be large, 2 orders of magnitude larger than their spin magnetic susceptibility, and the value of is predicted to be strongly dependent on the strength of the applied magnetic field, B, and the Fermi energy. Furthermore, the sign of can be either positive (paramagnetic) or negative (diamagnetic), depending on the nanotube chirality as well as the field orientation. We have calculated the orientation-dependent of SWNTs within the k p effective-mass approximation [5], as shown in Fig. 1 [10]. Starting from a massless Dirac Hamiltonian, we took into account the effect of B through the Landau-Peierls substitution k^ ¼ ir þ eA=c@, where k^ is the wave vector, A is the vector potential (B ¼ r A), c is the speed of light, and @ is the reduced Planck constant. When a parallel magnetic flux threads the nanotube, the Aharonov-Bohm effect modifies the circumferential boundary condition through a phase factor expð2i=0 Þ, where 0 ¼ ch=e is the magnetic flux quantum, and, consequently, the band gap oscillates with period 0 [1,5,10]. As increases from zero, a band gap opens up in a metallic tube (the band gap shrinks in a semiconducting tube), which leads to orbital paramagnetism (diamagnetism). On the other hand, both metallic and semiconducting SWNTs show diamagnetism in a perpendicular field. Thus, all SWNTs are expected to have positive magnetic susceptibility anisotropy ¼ k ? > 0, where k (? ) is the parallel (perpendicular) susceptibility. In addition, metallic SWNTs are expected to have greater values of than semiconducting SWNTs. These trends can be confirmed in the calculated for ð6; 6Þ (metallic) and ð6; 5Þ (semiconducting) nanotubes shown in Fig. 1 [10]. 0031-9007=10=105(1)=017403(4)
A finite > 0 results in the alignment of SWNTs in the B direction, which, combined with the anisotropic optical properties of SWNTs, allows for an estimation of values for SWNTs through magneto-optical spectroscopy. Previous magneto-optical studies found that 1:5 105 emu=mol for semiconducting SWNTs with 1 nm diameters [9,11–15], while no information on is currently available for metallic SWNTs. Below, we present the first experimental estimation of the of metallic SWNTs through high-field magnetic linear dichroism spectroscopy of cobalt molybdenum catalyst-synthesized nanotubes (mean diameter 0:8 nm) individually suspended in aqueous solution at room temperature. The sample had a much smaller diameter distribution than the high-pressure carbon monoxide-produced samples used in the previous studies [11,12], which allowed us to clearly identify and closely examine the mag-
FIG. 1 (color online). Calculated magnetic susceptibility anisotropy of single-walled carbon nanotubes. The susceptibility is expressed in units of ¼ ð2=aÞða2 =0 Þ2 a2 ¼ 1:46 a ¼ 2:5 A, and 0 ¼ 104 emu=mol, where ¼ 6:46 eV A, ch=e is the magnetic flux quantum. See supplementary material [10] for details on the calculation methods used.
017403-1
Ó 2010 The American Physical Society
Ak þ 2A? ; 3
(1)
where Ak (A? ) is the absorption for light polarized parallel (perpendicular) to the orientation axis, i.e., the magnetic field direction. We calculated A0 at 35 T by using Eq. (1) and plotted it in Fig. 2(a) as a red dashed line. The agreement between the calculated A0 and the zero-field absorption spectrum confirms that A0 is independent of alignment (or B), because at 0 T Ak ¼ A? ¼ A0 . We further confirmed that at any fields between 0 and 35 T the increase in Ak and decrease in A? are such that A0 is preserved through Eq. (1). Linear dichroism, LD ¼ Ak A? , is a measure of the degree of alignment. However, since it directly depends on the absorbance, LD alone cannot be used for comparing different spectral features. To adjust for differences in
Absorbance
0.8
(6,5)
(a)
Data A0 = (A// + 2A⊥)/3
(6,4)
0.6
(6,6) (7,4)
(7,5) (8,3)
(6,5) phonon
A// @ 35 T (5,5)
0 T & A0
0.4
A @ 35 T
(6,6) (7,4)
35 T
0.6
(b)
r
netic field dependence of absorption peaks for metallic nanotubes. We made a detailed comparison of the of metallic and semiconducting nanotubes with similar diameters and lengths within the same sample and found that the values of are 2–4 times greater in metallic tubes. We also compared our values with previous work for semiconducting nanotubes [14], which confirm a break from the prediction that should increase linearly with the tube diameter. Polarized magneto-optical absorption measurements on length-sorted, ð6; 5Þ-enriched cobalt molybdenum catalyst SWNTs were performed by using the 35 T hybrid magnet at the National Institute for Materials Science in Tsukuba, Japan. The SWNTs were suspended in 1% sodium deoxycholate and length sorted by dense liquid ultracentrifugation [16] to have an average length of 500 nm. A Xe lamp, fiber coupled to a custom optical probe, allowed for broadband white-light excitation of the E11 metallic, E22 semiconductor, and E33 semiconductor interband transitions of SWNTs. The sample was held in a cuvette with a film polarizer directly on the front face to ensure that the incident light was linearly polarized. The transmitted light was collimated by a lens, collected with another fiber, and dispersed through a monochromator equipped with a Si CCD. All measurements were done at 300 K. Figure 2(a) shows polarization-dependent optical absorption spectra at 0 and 35 T. None of the spectra are intentionally offset, indicating an increase (decrease) in absorbance for light polarized parallel (perpendicular) to the field. This is a direct result of magnetic alignment, together with the fact that only the light-field component parallel to the tube axis is strongly absorbed in SWNTs. Namely, at 35 T there is a finite degree of alignment in the magnetic field direction within the nanotube ensemble, resulting in stronger (weaker) absorption for light polarized parallel (perpendicular) to the field. From the theory of linear dichroism for an ensemble of anisotropic molecules [17], the following quantity can be shown to be constant, independent of the degree of alignment: A0
week ending 2 JULY 2010
PHYSICAL REVIEW LETTERS
LD
PRL 105, 017403 (2010)
0.5
(5,5)
(6,5) (6,4)
0.4 1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
Energy (eV) FIG. 2 (color online). (a) Absorption spectra (solid black line) for 0 and 35 T with all peaks assigned to specific chiralities. No trace is intentionally offset, indicating greater (smaller) absorption for parallel (perpendicular) polarization. The unpolarized isotropic absorbance (dashed red line), calculated from the 35 T spectra via Eq. (1), agrees well with the 0 T data. (b) Reduced linear dichroism versus energy from data taken at 35 T. The largest peak is from metallic nanotubes ð6; 6Þ and ð7; 4Þ.
relative absorbances due to the fact that our sample is enriched for ð6; 5Þ, the LD was divided by A0 , yielding the reduced linear dichroism, LDr ¼ LD=A0 [17,18], which is plotted in Fig. 2(b) for 35 T. It is immediately evident from this plot that the spectral region where metallic peaks [ð6; 6Þ and ð7; 4Þ] exist has much larger LDr than the region where the most prominent semiconducting peaks [ð6; 5Þ and ð6; 4Þ] exist. This is evidence that the ð6; 6Þ and ð7; 4Þ tubes are aligning more strongly than the ð6; 5Þ and ð6; 4Þ tubes. In order to quantitatively determine the magnetic alignment properties of each chirality present in our sample, we performed detailed spectral analysis, as described below. Each absorption spectrum was fit by a superposition of multiple Lorentzian peaks plus a polynomial offset. The zero-field spectrum was fit first, as shown in Fig. 3(a), with each Lorentzian assigned to a specific chirality. The offset was included to account for the contributions from the plasmon peaks, bundles, and light scattering [19–21]. For each magnetic field, the parallel and perpendicular spectra were fit simultaneously with the constraint that each Lorentzian independently satisfy Eq. (1). Figure 3(b) shows representative results for the ð7; 5Þ, ð6; 5Þ, ð6; 6Þ, and ð5; 5Þ nanotubes at 0 and 35 T. Note that we required the width and position of each Lorentzian to be the same as the values obtained at 0 T, which ensured that the calculated LDr for each chirality is constant as a function of
017403-2
PRL 105, 017403 (2010) 0.6
0 T, data 0 T, fit (total) individual fits
Absorbance
0.5 0.4
and ð5; 5Þ] show much stronger alignment than the semiconducting nanotubes [ð6; 5Þ and ð6; 4Þ]. To model the observed magnetic field dependence of S, we approximate the angular distribution Pu ðÞ of nanotubes at temperature T in a magnetic field B by the Maxwell-Boltzmann distribution function [11–14,22]
(a)
(6,5)
0.3 0.2 0.1
(7,5) (8,3)
(6,5) phonon
(6,4)
(6,6) (7,4)
(5,5)
expðu2 sin2 Þ sin Pu ðÞ ¼ R=2 ; (2) expðu2 sin2 Þ sind 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where u ¼ B2 N=2kB T is a dimensionless measure of the relative importance of the magnetic alignment energy (B2 N=2) [22] and the thermal energy (kB T), N is the number of carbon atoms in each nanotube with a length of 500 nm (in moles), and kB represents Boltzmann’s constant. Using this distribution function and the definition of S, one can derive Z =2 3cos2 1 d SðuÞ ¼ Pu ðÞ 2 0 1 3 u pffiffiffi ¼ þ 1 ; (3) u2 2 4u2 erfiðuÞ 2 e
(7,5) E33
0.0 (6,5)
(b)
Absorbance
0.4 35 T, parallel 0T 35 T, perpendicular
0.3 0.2 (7,5)
(6,6) (5,5)
0.1 0.0 Metal
(6,6) (5,5) (7,4)
1.0 (8,3)
r
LD @ 35 T
(c)
Semiconductor
1.5
week ending 2 JULY 2010
PHYSICAL REVIEW LETTERS
where erfi is the imaginary error function [erfiðuÞ ¼ pffiffiffiffi R t2 ð2=i Þ iu dt]. Equation (3) is plotted in Fig. 4(b). 0 e We were able to fit the S versus B curves with Eq. (3), as shown for ð6; 6Þ and ð6; 5Þ nanotubes in Fig. 4(c), to extract
3.0
FIG. 3 (color online). (a) Fitting results for 0 T. (b) Lorentzians for different chirality nanotubes at 0 and 35 T. (c) LDr versus energy (eV) derived from fitting for each individual chirality nanotube in our sample. The metallic tubes (red line) are higher than the semiconducting nanotubes (blue dotted line).
photon energy, as shown in Fig. 3(c) for 35 T. Here, each LDr line is a direct result of calculation from each Lorentzian of different chirality, while LDr in Fig. 2(b) was calculated from entire spectra. As a result, one can extract chirality-specific information on the alignment degree from Fig. 3(c). It is evident that the values are much larger for the metallic (red lines) than the semiconducting (blue lines) nanotubes. For anisotropic molecules with the dipole moment fully oriented along the long axis, the reduced linear dichroism can be expressed as LDr ¼ 3S [17,18]. Here, the nematic order parameter S ¼ ð3hcos2 i 1Þ=2 is a dimensionless measure of the degree of alignment, being equal to zero for completely randomly oriented nanotubes and one for completely aligned nanotubes, and is the angle between the nanotube axis and the orientation axis (i.e., the magnetic field direction in the present case). Values for S were calculated for representative nanotubes from the LDr values in Fig. 3(c) and plotted as a function of magnetic field in Fig. 4(a). Here, we can clearly see that the ðn; nÞ-chirality, or armchair, metallic nanotubes [ð6; 6Þ
1.0
(a)
0.5
(b) 0.8
0.4
(6,6) (5,5) (6,5) (6,4)
0.3
0.6
S
2.5 Energy (eV)
0.2
0.4
0.1
0.2
0.0
0
10 20 30 Magnetic Field (T)
40
0.0 0.1
2
4
2
1
u
4
2
4
10
3.0 0.6
0.4
(d)
(c)
(6,6)
2.5
(6,6) fit for (6,6) (6,5) fit for (6,5)
χ// /χ∗
2.0
S
0.5
(6,5) (6,4)
S
(7,5)
30 K
2.0
0.2 300 K
1.5 0.0 5
6 7 8 9
2
10 Magnetic Field (T)
3
4
0
10 20 30 Magnetic Field (T)
40
FIG. 4 (color online). (a) Nematic order parameter S versus magnetic field for armchair nanotubes ð5; 5Þ and ð6; 6Þ and semiconducting nanotubes ð6; 5Þ and ð6; 4Þ, calculated from the measured reduced linear dichroism values. (b) S versus u [Eq. (3)]. (c) Comparison of S versus magnetic field for ð6; 5Þ and ð6; 6Þ nanotubes. (d) Calculated magnetic field dependence of the parallel magnetic susceptibility of ð6; 6Þ nanotube for T ¼ 30 and 300 K in units of ¼ 1:46 104 emu=mol [10].
017403-3
PRL 105, 017403 (2010)
PHYSICAL REVIEW LETTERS
TABLE I. Comparison of theoretical and experimental values of magnetic susceptibility anisotropy for seven types of SWNTs. For each chirality ðn; mÞ, the diameter d, the chiral index ¼ ðn mÞ mod 3, and the chiral angle are given, followed by theoretical (30 and 300 K) and experimental (300 K) values of . All values for are 105 emu=mol. ðn; mÞ
d (nm)
( )
K 30 th
K 300 th
K 300 exp
ð6; 6Þ ð5; 5Þ ð7; 4Þ ð8; 3Þ ð7; 5Þ ð6; 5Þ ð6; 4Þ
0.83 0.69 0.77 0.78 0.83 0.76 0.69
0 0 0 1 1 1 1
30 30 21 15 24 27 23
5.78 4.95 5.42 1.49 1.58 1.45 1.33
3.92 3.39 3.70 1.46 1.55 1.42 1.29
3.63 3.35 2.44 2.13 1.66 1.01 1.24
for each nanotube chirality. However, care must be taken in using this procedure, since the values of k are expected to depend on B and T, especially for metallic nanotubes. Thus, we calculated as a function of B and T for all seven SWNTs studied in this work [10]. Figure 4(d) shows a theoretical calculation of the B dependence of k for the ð6; 6Þ nanotube for 30 and 300 K. Although k decreases by 27% at 30 K as the field increases from 0 to 40 T, it stays constant within 0:7% at 300 K in the same field range, ensuring that the magnetic field dependence can be neglected for our room temperature measurements. Experimentally estimated values are summarized in Table I, together with the theoretical values we calculated through the k p method [10]. The experimental values agree well with the theoretical values at 300 K. The values for the three metallic nanotubes, ð7; 4Þ, ð5; 5Þ, and ð6; 6Þ, are all higher than those for the semiconducting nanotubes. In particular, the of armchair, or ðn; nÞ, carbon nanotubes are 2–4 times larger than those in semiconducting nanotubes, depending on the diameter. This large difference in magnetic susceptibility anisotropy is consistent with theoretical predictions, a direct consequence of the AharonovBohm physics. Finally, the experimental values for metallic nanotubes in Table I do not follow a strict diameter dependence, something that is predicted [8] and shown experimentally [14] for some semiconducting tubes. For smaller diameter nanotubes, like the ones seen Table I, the chiral angle plays an important role in determining the values of . Furthermore, all the estimated values for all SWNTs (both metallic and semiconducting) are larger than those found in earlier studies for multiwalled nanotubes [23,24]. It is also important to note that the ð7; 4Þ nanotube has a larger value of than the semiconducting tubes, but it is not as large as those of the armchair tubes. Unfortunately, it is the only nonarmchair mod-3 nanotube in our sample, but a detailed study on a metallic-enriched sample [25] should yield many more metallic nanotubes to investigate in the future.
week ending 2 JULY 2010
In conclusion, we have successfully measured for metallic single-walled carbon nanotubes for the first time and confirmed that they are much larger than those of semiconducting nanotubes. We also calculated the magnetic field and temperature dependence of magnetic susceptibilities of the nanotubes studied experimentally, which support the experimental findings. Last, we were able to confirm previous experimental results for the chirality dependence of the magnetic susceptibility anisotropy in semiconducting nanotubes and found that this is also true for metallic nanotubes. This work was supported by the DOE-BES (Grant No. DE-FG02-06ER46308), the NSF (Grant No. OISE0530220), the Robert A. Welch Foundation (Grant No. C-1509), and Grant-in-Aid for Scientific Research under Grant No. 19054013 from the Ministry of Education, Culture, Sports, Science and Technology of Japan. We thank Noe Alvarez and Ajit Srivastava for their help with atomic force microscope length measurements and data analysis, respectively. This work is an official contribution of the National Institute of Standards and Technology and not subject to copyright in the United States.
*Corresponding author. [email protected] H. Ajiki and T. Ando, J. Phys. Soc. Jpn. 62, 1255 (1993). H. Ajiki and T. Ando, J. Phys. Soc. Jpn. 62, 2470 (1993). W. Tian and S. Datta, Phys. Rev. B 49, 5097 (1994). J. P. Lu, Phys. Rev. Lett. 74, 1123 (1995). H. Ajiki and T. Ando, J. Phys. Soc. Jpn. 64, 4382 (1995). S. Roche et al., Phys. Rev. B 62, 16 092 (2000). N. Nemec and G. Cuniberti, Phys. Rev. B 74, 165411 (2006). [8] M. A. L. Marques et al., Phys. Rev. B 73, 125433 (2006). [9] J. Kono and S. Roche, in Carbon Nanotubes: Properties and Applications, edited by M. J. O’Connell (CRC Press, Boca Raton, 2006), Chap. 5, pp. 119–151. [10] See supplementary material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.105.017403 for details of our theoretical calculations. [11] S. Zaric et al., Science 304, 1129 (2004). [12] S. Zaric et al., Nano Lett. 4, 2219 (2004). [13] M. F. Islam et al., Phys. Rev. B 71, 201401(R) (2005). [14] O. N. Torrens et al., J. Am. Chem. Soc. 129, 252 (2007). [15] J. Kono, R. J. Nicholas, and S. Roche, in Carbon Nanotubes, edited by A. Jorio et al. (Springer, Berlin, 2008), pp. 393–421. [16] J. A. Fagan et al., Adv. Mater. 20, 1609 (2008). [17] A. Rodger and B. Norden, Circular Dichroism and Linear Dichroism (Oxford University Press, Oxford, 1997). [18] J. Shaver et al., ACS Nano 3, 131 (2009). [19] Y. Murakami et al., Phys. Rev. Lett. 94, 087402 (2005). [20] N. Nair et al., Anal. Chem. 78, 7689 (2006). [21] J. A. Fagan et al., Appl. Phys. Lett. 91, 213105 (2007). [22] M. Fujiwara et al., J. Phys. Chem. B 103, 2627 (1999). [23] X. K. Wang et al., Appl. Phys. Lett. 62, 1881 (1993). [24] O. Chauvet et al., Phys. Rev. B 52, R6963 (1995). [25] E. H. Ha´roz et al., ACS Nano 4, 1955 (2010). [1] [2] [3] [4] [5] [6] [7]
017403-4
