Spin-Charge Separation in Finite Length Metallic Carbon Nanotubes

Letter Cite This: Nano Lett. XXXX, XXX, XXX-XXX

pubs.acs.org/NanoLett

Spin-Charge Separation in Finite Length Metallic Carbon Nanotubes Yongyou Zhang,†,‡ Qingyun Zhang,† and Udo Schwingenschlögl*,† †

Physical Science and Engineering Division (PSE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia ‡ Beijing Key Lab of Nanophotonics & Ultrafine Optoelectronic Systems and School of Physics, Beijing Institute of Technology, Beijing 100081, China S Supporting Information *

ABSTRACT: Using time-dependent density functional theory, we study the optical excitations in finite length carbon nanotubes. Evidence of spin-charge separation is given in the spacetime domain. We demonstrate that the charge density wave is due to collective excitations of electron singlets, while the accompanying spin density wave is due to those of electron triplets. The Tomonaga−Luttinger liquid parameter and density−density interaction are extrapolated from the first-principles excitation energies. We show that the density−density interaction increases with the length of the nanotube. The singlet and triplet excitation energies, on the other hand, decrease for increasing length of the nanotube. Their ratio is used to establish a first-principles approach for deriving the Tomonaga−Luttinger parameter (in excellent agreement with experimental data). Time evolution analysis of the charge and spin line densities evidences that the charge and spin density waves are elementary excitations of metallic carbon nanotubes. Their dynamics show no dependence on each other. KEYWORDS: Spin-charge separation, carbon nanotube, singlet excitation, triplet excitation

T

Because of strong correlation effects, 1D electronic systems typically cannot be described by Fermi liquid theory, but behave as Tomonaga−Luttinger liquids.33 For a linear Fermionic dispersion the Tomonaga−Luttinger liquid model is exactly solvable to give the power law of the conductance variation with temperature.34,35 Since the electronic bands of metallic carbon nanotubes, for example the (N, N) armchair nanotubes, are linear near the two Dirac points, they can be described by Tomonaga−Luttinger liquid theory,36−38 as has been proven by measuring the conductance power law.21,39,40 Recently, direct observation of the Tomonaga−Luttinger liquid plasmon (charge density wave, corresponding to a collective excitation of the 1D Dirac electrons) has become possible in long carbon nanotubes by determining the near-field infrared scattering intensity.41 On the other hand, spin density oscillations so far have not been observed, though there have been several experimental attempts.42−45 Besides the Tomonaga−Luttinger model, density functional theory can be used to study the properties of 1D electron systems, including the 1D Hubbard model46−48 and real materials.49,50 Density matrix renormalization group theory has been employed to improve the exchange-correlation potential.51−53 In this context, the present work analyzes the spin-charge separation in carbon

here is a huge variety of applications of carbon-based materials in optoelectronic devices, as there are plenty of carbon allotropes, such as graphene,1−3 fullerenes,4 and nanotubes.5,6 The high breaking strength and electrical/thermal conductivity of graphene make the material interesting for electronic high-frequency devices, touch screens, light panels, and solar cells,7−9 for example. Fullerenes have potential in biomedical applications.10 Both metallic and semiconducting carbon nanotubes are used for energy storage, electronics, and electromagnetic shielding,11 for example. Carbon-based materials also provide chances for studying fundamental physics. Graphene may be useful to obtain insights into twodimensional spintronics, topological states, and proximity phenomena,12−14 whereas carbon nanotubes play an important role in theory15−17 and experiments18−25 on one-dimensional (1D) electronics and spintronics. Since the reduced dimensionality provides access to interaction effects in many body systems, we study in the present work the collective excitations, i.e., charge and spin density waves, in finite length carbon nanotubes. Alternative systems include semiconductor wires,26,27 edge modes of quantum Hall liquids,28,29 antiferromagnetic spin chains,30 and cold atoms in 1D optical traps.31 In particular, spin-charge separation in a chiral Tomonaga−Luttinger liquid with quantum Hall edge channels has been demonstrated experimentally in ref 32 by timeresolved measurements of the spin and charge density waves. This technique can provide full information on the time evolution of 1D electronic systems. © XXXX American Chemical Society

Received: July 7, 2017 Revised: September 25, 2017

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DOI: 10.1021/acs.nanolett.7b02880 Nano Lett. XXXX, XXX, XXX−XXX

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Nano Letters nanotubes by time-dependent density functional theory. We demonstrate that the charge/spin density wave is due to collective excitations of electron singlets/triplets and clarify the dependence of the singlet and triplet excitation energies on the length of the nanotube. The ratio of the excitation energies is used to establish a first-principles approach for deriving the Tomonaga−Luttinger parameter. As an example, we consider (5, 5) armchair carbon nanotubes (at 0 K). Calculations for finite length nanotubes are performed by the real-space electronic structure code OCTOPUS54 (Troullier−Martins pseudopotentials) and calculations for infinite nanotubes by the reciprocal-space Vienna Ab-initio Simulation Package55 (pseudopotential generation PAW-08Apr2002). Simulation sphere radii of 5 Å are used for all atoms, and the spacing of the real-space grid is set to 0.1 Å. We have checked that the total energy does not diverge in time steps of 0.00075 fs (20 fs total simulation time). The generalized gradient approximation (Perdew−Burke−Ernzerhof) for the exchange correlation functional is adopted. Figure 1 shows the structure of the infinite (5, 5) armchair carbon

Figure 2. Black lines represent densities of states obtained for the (a) C120, (b) C240, (c) C480, and (d) infinite (5, 5) armchair carbon nanotubes (normalized with respect to the number of carbon atoms). Smoothed curves (red lines) are given to aid the comparison between the different cases. In panel d, the contributions of the 2s and 2p orbitals are shown by magenta and green lines, respectively.

shown in Figures 3a−d by solid and dashed lines, respectively, with down and up arrows marking the charge and spin density waves. The same absorption spectra obtained for charged nanotubes are given in the Supporting Information. The fact that the peaks assigned to the charge and spin density waves hardly depend on the number of electrons shows that these are collective excitations. The low energy excitations of infinite carbon nanotubes are described by Tomonaga−Luttinger liquid theory, where the parameter

Figure 1. (a) Structure and (b) band structure of the infinite (5, 5) armchair carbon nanotube with lattice constant a = 2.47 Å and diameter 2R = 6.84 Å. The two Dirac cones are denoted by α = ± and the two branches of each Dirac cone by η = ±. The energy is given with respect to the Fermi level.

g −1 =

nanotube (lattice constant a = 2.47 Å and diameter 2R = 6.84 Å; the unit cell contains 20 carbon atoms) and the corresponding band structure. Two Dirac cones at the Fermi level give rise to a 1D metal with Fermi velocity ℏvF = 4.9 eV Å, in agreement with ref 56. We show the ground state densities of states of finite length carbon nanotubes in Figures 2a−c and compare them to the corresponding infinite nanotube in Figure 2d. The notation Cn means that the total number of carbon atoms is n. For the short C120 nanotube (length/diameter = 2.1), the density of states is composed of isolated peaks, resembling a quantum dot, while for increasing length it approaches that of the infinite nanotube (1D system). The C480 nanotube (length/diameter = 8.6) is already close to this limit and therefore is the largest system considered here. Absorption spectra of the singlet (charge density polarization) and triplet (spin density polarization) excitations are

1+

8G π ℏvF

(1) 39

is determined by the density−density interaction G. A free electron gas is described by g−1 = 1, while attractive and repulsive interaction corresponds to g−1 < 1 and g−1 > 1, respectively. We can estimate g−1 and G by extrapolating our results for finite length carbon nanotubes (if the charge and spin density waves have the same wavelength, which we demonstrate later). It is possible to derive results for long nanotubes from those of short nanotubes, because the 1D character of a carbon nanotube increases with its length. According to Figure 3e, the energies of the charge and spin density waves, εc and εs, redshift for increasing length of the nanotube. The value of εc/εs, see Figure 3f, turns out to be larger than 1 and increases with the length of the nanotube (increasing 1D character). As εc/εs will approach g−1 for long B

DOI: 10.1021/acs.nanolett.7b02880 Nano Lett. XXXX, XXX, XXX−XXX

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Figure 3. (a−d) Absorption spectra of the singlet (solid lines) and triplet (dashed lines) excitations. Down and up arrows denote the charge and spin density waves, respectively. Their energies are given in (e) as functions of the number of atoms and the ratio εc/εs is shown in (f).

−1 nanotubes, we use the expression g −1 = g∞ n

g−1 ∞

0

n +n

with the

parameters and n0 to fit the data in Figure 3f. We obtain g−1 ∞ G = 3.1 and, using eq 1, π ∼ 1.1ℏvF = 5.3 eV Å. Agreement of g−1 ∞ with the experimental value of 3.2 obtained in ref 41 for individual nanotubes indicates that the charge and spin density waves are collective singlet and triplet excitations, respectively.57 Our results consequently demonstrate that it is possible to obtain the Tomonaga−Luttinger parameter and density−density interaction in a 1D system by calculating the singlet and triplet absorption spectra within time-dependent density functional theory. Although the spin density wave results from the spin up and down charge densities, its transport in the 1D system is independent of the charge density wave, as shown in Figure 4a, where the x components of the charge and spin density polarizations Pxc/s =

∫ xρc/s (x) dx

Figure 4. C480 nanotube: Time dependences of the (a) charge and spin density polarizations, (b) charge line density, and (c) spin line density. (d) Contour plots of the charge (left) and spin (right) densities at different times. (e) Distributions of the charge and spin line densities at time 20.7 ℏ/eV (thick lines representing the results after Fourier transform filtering).

ρc/s (x) =

∫ [ρ↑(x , y , z) ± ρ↓(x , y , z)]dy dz

(3)

are given by the spin up and down electron densities (ρ↑(x, y, z), ρ↓(x, y, z)). The ground state of the C480 nanotube is perturbed by an external electric field prior to time zero, so that the charge and spin density waves can appear simultaneously. After time zero the system evolves freely (no external electric

(2)

are plotted as functions of time. Their oscillation frequencies correspond to the energies of the charge and spin density waves. The charge and spin line densities C

DOI: 10.1021/acs.nanolett.7b02880 Nano Lett. XXXX, XXX, XXX−XXX

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field). Pcx and Psx show perfect sine-type oscillations with periods of 8.4 ℏ/eV and 22.8 ℏ/eV, respectively, which are consistent with the singlet and triplet excitation energies, namely, 0.78 and 0.33 eV, of the C480 nanotube; see Figure 3e. To illustrate the oscillations of the charge and spin density waves, we plot the time evolution of ρc/s(x) in Figure 4b/c. At time zero ρc(x) and ρs(x) coincide. Since the period of the charge density wave is shorter than that of the spin density wave, the coincidence is lost after time zero, that is, we have spin-charge separation. Importantly, the two time evolutions show no sign of mutual influence, at least in the calculated time range up to 45 ℏ/eV, which indicates that the spin and charge density waves are elementary excitations of metallic carbon nanotubes. The spatial distributions of the charge and spin densities in Figure 4d confirm spin-charge separation. They overlap qualitatively at times 3.9 ℏ/eV and 16.8 ℏ/eV, but not at the other times shown. Figure 4b−d also demonstrates that the charge and spin density waves have the same wavelength (about two times the length of the nanotube, as these are the lowest energy excitations). For clarity, charge and spin line densities at time 20.7 ℏ/eV after removal of the high frequency spatial components due to the atomic structure (Fourier transform filter method) are shown in Figure 4e. These curves reveal clear oscillations along the x direction. Because maxima appear close to x = ±25 Å, which is about half of the length of the C480 nanotube (58.6 Å), we can estimate that the velocity of the charge density wave is ℏvc = εc/q ∼ 0.78 × 50/π eV Å = 12.4 eV Å. This value agrees with that of the infinite nanotube, −1 ℏvFg−1 ∞ = 15.2 eV Å (as obtained for g∞ = 3.1), given that the density−density interaction increases with the length of the nanotube. In conclusion, we have demonstrated spin-charge separation in metallic carbon nanotubes by time-dependent density functional theory. It turns out that the charge and spin density waves correspond to collective excitations of electron singlets and triplets, respectively. Using the singlet and triplet absorption spectra, the Tomonaga−Luttinger liquid parameter and density−density interaction have been calculated by extrapolation of first-principles excitation energies. The density−density interaction is found to increase with the length of the nanotube. All of the obtained data are consistent with experiment, demonstrating that long carbon nanotubes resemble the low energy excitations of infinite carbon nanotubes. The time evolutions of the charge and spin line densities have been used to show that the charge and spin density waves are elementary excitations of metallic carbon nanotubes, without mutual influence on their dynamics.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +966(0) 544700080. ORCID

Udo Schwingenschlögl: 0000-0003-4179-7231 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

The research reported in this publication was supported by funding from King Abdullah University of Science and Technology (KAUST). It was also supported by the National Natural Science Foundation of China (NSFC grant no. 11304015).

(1) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. Science 2004, 306, 666−669. (2) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Katsnelson, M. I.; Grigorieva, I. V.; Dubonos, S. V.; Firsov, A. A. Nature 2005, 438, 197−200. (3) Yang, X.; Dou, X.; Rouhanipour, A.; Zhi, L.; Räder, H. J.; Müllen, K. J. Am. Chem. Soc. 2008, 130, 4216−4217. (4) Kroto, H. W.; Heath, J. R.; O’Brien, S. C.; Curl, R. F.; Smalley, R. E. Nature 1985, 318, 162−163. (5) Iijima, S.; Ichihashi, T. Nature 1993, 363, 603−605. (6) Bethune, D. S.; Kiang, C. H.; de Vries, M. S.; Gorman, G.; Savoy, R.; Vazquez, J.; Beyers, R. Nature 1993, 363, 605−607. (7) Das Sarma, S.; Adam, S.; Hwang, E. H.; Rossi, E. Rev. Mod. Phys. 2011, 83, 407−470. (8) Novoselov, K. S.; Fal’ko, V. I.; Colombo, L.; Gellert, P. R.; Schwab, M. G.; Kim, K. Nature 2012, 490, 192−200. (9) Han, W.; Kawakami, R. K.; Gmitra, M.; Fabian, J. Nat. Nanotechnol. 2014, 9, 794−807. (10) Lalwani, G.; Sitharaman, B. Nano LIFE 2013, 3, 1342003. (11) De Volder, M. F. L.; Tawfick, S. H.; Baughman, R. H.; Hart, A. J. Science 2013, 339, 535−539. (12) Zhu, L.; Ma, R.; Sheng, L.; Liu, M.; Sheng, D. N. Phys. Rev. Lett. 2010, 104, 076804. (13) Horng, J.; Chen, C. F.; Geng, B.; Girit, C.; Zhang, Y.; Hao, Z.; Bechtel, H. A.; Martin, M.; Zettl, A.; Crommie, M. F.; Shen, Y. R.; Wang, F. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 83, 165113. (14) Abanin, D. A.; Morozov, S. V.; Ponomarenko, L. A.; Gorbachev, R. V.; Mayorov, A. S.; Katsnelson, M. I.; Watanabe, K.; Taniguchi, T.; Novoselov, K. S.; Levitov, L. S.; Geim, A. K. Science 2011, 332, 328− 330. (15) Egger, R. Phys. Rev. Lett. 1999, 83, 5547−5550. (16) Recher, P.; Kim, N.; Yamamoto, Y. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 74, 235438. (17) Dóra, B.; Gulácsi, M.; Koltai, J.; Zólyomi, V.; Kürti, J.; Simon, F. Phys. Rev. Lett. 2008, 101, 106408. (18) Bockrath, M.; Cobden, D. H.; Lu, J.; Rinzler, A. G.; Smalley, R. E.; Balents, L.; McEuen, P. L. Nature 1999, 397, 598−601. (19) Tarkiainen, R.; Ahlskog, M.; Penttilä, J.; Roschier, L.; Hakonen, P.; Paalanen, M.; Sonin, E. Phys. Rev. B: Condens. Matter Mater. Phys. 2001, 64, 195412. (20) Ishii, H.; Kataura, H.; Shiozawa, H.; Yoshioka, H.; Otsubo, H.; Takayama, Y.; Miyahara, T.; Suzuki, S.; Achiba, Y.; Nakatake, M.; Narimura, T.; Higashiguchi, M.; Shimada, K.; Namatame, H.; Taniguchi, M. Nature 2003, 426, 540−544. (21) Monteverde, M.; Núñez-Regueiro, M.; Garbarino, G.; Acha, C.; Jing, X.; Lu, L.; Pan, Z. W.; Xie, S. S.; Souletie, J.; Egger, R. Phys. Rev. Lett. 2006, 97, 176401.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.7b02880. Absorption spectra of the singlet and triplet excitations for neutral and charged nanotubes. The data for the neutral nanotubes are taken from Figure 3a−d. No qualitative modifications are observed for the charged nanotubes. (PDF) D

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Nano Letters (22) Hueso, L. E.; Pruneda, J. M.; Ferrari, V.; Burnell, G.; ValdésHerrera, J. P.; Simons, B. D.; Littlewood, P. B.; Artacho, E.; Fert, A.; Mathur, N. D. Nature 2007, 445, 410−413. (23) Kuemmeth, F.; Ilani, S.; Ralph, D. C.; McEuen, P. L. Nature 2008, 452, 448−452. (24) Ihara, Y.; Wzietek, P.; Alloul, H.; Rümmeli, M. H.; Pichler, T.; Simon, F. EPL 2010, 90, 17004. (25) Morgan, C.; Misiorny, M.; Metten, D.; Heedt, S.; SchäPers, T.; Schneider, C. M.; Meyer, C. Phys. Rev. Appl. 2016, 5, 054010. (26) Auslaender, O. M.; Steinberg, H.; Yacoby, A.; Tserkovnyak, Y.; Halperin, B. I.; De Picciotto, R.; Baldwin, K. W.; Pfeiffer, L. N.; West, K. W. Solid State Commun. 2004, 131, 657−663. (27) Laroche, D.; Gervais, G.; Lilly, M. P.; Reno, J. L. Science 2014, 343, 631−634. (28) Chang, A. M. Rev. Mod. Phys. 2003, 75, 1449−1505. (29) Fertig, H. A.; Brey, L. Phys. Rev. Lett. 2006, 97, 116805. (30) Lake, B.; Tennant, D. A.; Frost, C. D.; Nagler, S. E. Nat. Mater. 2005, 4, 329−334. (31) Görlitz, A.; Vogels, J. M.; Leanhardt, A. E.; Raman, C.; Gustavson, T. L.; Abo-Shaeer, J. R.; Chikkatur, A. P.; Gupta, S.; Inouye, S.; Rosenband, T.; Ketterle, W. Phys. Rev. Lett. 2001, 87, 130402. (32) Hashisaka, M.; Hiyama, N.; Akiho, T.; Muraki, K.; Fujisawa, T. Nat. Phys. 2017, 13, 559−562. (33) Voit, J. Rep. Prog. Phys. 1995, 58, 977−1116. (34) Giamarchi, T.; Millis, A. Phys. Rev. B: Condens. Matter Mater. Phys. 1992, 46, 9325−9331. (35) Kane, C. L.; Fisher, M. P. A. Phys. Rev. Lett. 1996, 76, 3192− 3195. (36) Bockrath, M. W.; Cobden, D. H.; Lu, J.; Rinzler, A. G.; Smalley, R. E.; Balents, L.; McEuen, P. L. Nature 1999, 397, 598−601. (37) Auslaender, O. M.; Yacoby, A.; de Picciotto, R.; Baldwin, K. W.; Pfeiffer, L. N.; West, K. W. Phys. Rev. Lett. 2000, 84, 1764−1767. (38) Que, W. Phys. Rev. B: Condens. Matter Mater. Phys. 2002, 66, 193405. (39) Kane, C.; Balents, L.; Fisher, M. P. A. Phys. Rev. Lett. 1997, 79, 5086−5089. (40) Sweeney, M. C.; Eaves, J. D. Phys. Rev. Lett. 2014, 112, 107402. (41) Shi, Z.; Hong, X.; Bechtel, H. A.; Zeng, B.; Martin, M. C.; Watanabe, K.; Taniguchi, T.; Shen, Y. R.; Wang, F. Nat. Photonics 2015, 9, 515−519. (42) Kim, C.; Matsuura, A. Y.; Shen, Z. X.; Motoyama, N.; Eisaki, H.; Uchida, S.; Tohyama, T.; Maekawa, S. Phys. Rev. Lett. 1996, 77, 4054− 4057. (43) Auslaender, O. M.; Steinberg, H.; Yacoby, A.; Tserkovnyak, Y.; Halperin, B. I.; Baldwin, K. W.; Pfeiffer, L. N.; West, K. W. Science 2005, 308, 88−92. (44) Kim, B. J.; Koh, H.; Rotenberg, E.; Oh, S. J.; Eisaki, H.; Motoyama, N.; Uchida, S.; Tohyama, T.; Maekawa, S.; Shen, Z. X.; Kim, C. Nat. Phys. 2006, 2, 397−401. (45) Jompol, Y.; Ford, C. J. B.; Griffiths, J. P.; Farrer, I.; Jones, G. A. C.; Anderson, D.; Ritchie, D. A.; Silk, T. W.; Schofield, A. J. Science 2009, 325, 597−601. (46) Lima, N. A.; Silva, M. F.; Oliveira, L. N.; Capelle, K. Phys. Rev. Lett. 2003, 90, 146402. (47) Gao, X. Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 81, 104306. (48) Akande, A.; Sanvito, S. Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 82, 245114. (49) Popović, Z. S.; Satpathy, S. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 74, 045117. (50) López-Lozano, X.; Barron, H.; Mottet, C.; Weissker, H. C. Phys. Chem. Chem. Phys. 2014, 16, 1820−1823. (51) Stoudenmire, E. M.; Wagner, L. O.; White, S. R.; Burke, K. Phys. Rev. Lett. 2012, 109, 056402. (52) Wagner, L. O.; Stoudenmire, E. M.; Burke, K.; White, S. R. Phys. Rev. Lett. 2013, 111, 093003.

(53) Wagner, L. O.; Baker, T. E.; Stoudenmire, E. M.; Burke, K.; White, S. R. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90, 045109. (54) Andrade, X.; Strubbe, D.; De Giovannini, U.; Larsen, A. H.; Oliveira, M. J. T.; Alberdi-Rodriguez, J.; Varas, A.; Theophilou, I.; Helbig, N.; Verstraete, M. J.; Stella, L.; Nogueira, F.; Aspuru-Guzik, A.; Castro, A.; Marques, M. A. L.; Rubio, A. Phys. Chem. Chem. Phys. 2015, 17, 31371−31396. (55) Kresse, G.; Joubert, D. Phys. Rev. B: Condens. Matter Mater. Phys. 1999, 59, 1758−1775. (56) Laird, E. A.; Kuemmeth, F.; Steele, G. A.; Grove-Rasmussen, K.; Nygård, J.; Flensberg, K.; Kouwenhoven, L. P. Rev. Mod. Phys. 2015, 87, 703−764. (57) Nakamura, M.; Kitazawa, A.; Nomura, K. Phys. Rev. B: Condens. Matter Mater. Phys. 1999, 60, 7850−7862.

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