NANO LETTERS
Kohn Anomaly and Electron-Phonon Interaction at the K-Derived Point of the Brillouin Zone of Metallic Nanotubes
2009 Vol. 9, No. 9 3343-3348
Peter M. Rafailov,*,† Janina Maultzsch,‡ Christian Thomsen,‡ Urszula Dettlaff-Weglikowska,§,| and Siegmar Roth§,| Georgi NadjakoV Institute of Solid State Physics, Bulgarian Academy of Sciences, 72 Tzarigradsko Chaussee BouleVard, 1784 Sofia, Bulgaria, Institut fu¨r Festko¨rperphysik, Technische UniVersita¨t Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany, Max-Planck-Institut fu¨r Festko¨rperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany, and School of Electrical Engineering, Korea UniVersity, Anam-Dong, Sungbuk-Gu, 136-701 Seoul, Korea Received May 22, 2009; Revised Manuscript Received July 21, 2009
ABSTRACT We applied Raman spectroscopy to investigate the response to electrochemical doping of the second-order D* band in single-walled carbon nanotube (SWNT) bundles. Our study reveals a dramatic increase of the D* band sensitivity to doping upon moving the laser excitation to the red end of the visible spectrum and beyond. Using the double-resonance scattering model, we show that this phenomenon evidences a second Kohn anomaly in metallic SWNTs, located in the K-point-derived region of the Brillouin zone (BZ), which stems from the Kohn anomaly at the K-point of graphene. Our results will be compared to recent doping experiments on graphene with field-effect gating and can be used to investigate the wave-vector dependent electron-phonon coupling in the bulk of the BZ of metallic SWNTs.
The unusual electronic and elastic properties of single-walled carbon nanotubes (SWNTs) make them distinguished candidates for a variety of technological applications. Many of these applications are based on physical properties, such as electrical transport,1 electron tunneling,2 or optical transitions,3 that are significantly influenced by electron-phonon coupling (EPC) and doping. The correct description of all EPC mechanisms and their subtle interplay with doping is therefore of crucial importance for the success of both fundamental and technologic research on carbon nanotubes. One manifestation of the EPC is the Kohn anomaly (KA).4 A KA occurs at a certain point of the phonon dispersion when the lattice distortion created in this phonon state induces a dynamic band gap. This in turn lowers the energy needed for the lattice distortion, thus causing a softening of the pertinent phonon.5 The coupling strength of this phonon to the electronic system is usually governed by the position of the Fermi level EF. Especially, for the half-filled linear bands of graphene and metallic SWNTs the EPC strength is at maximum when the Fermi level is located at the band * To whom correspondence should be addressed. E-mail: rafailov@ issp.bas.bg. Telephone: +359-2-979-5718. Fax: +359-2-975-36-32. † Bulgarian Academy of Sciences. ‡ Technische Universita¨t Berlin. § Max-Planck-Institut fu¨r Festko¨rperforschung. | Korea University. 10.1021/nl901618q CCC: $40.75 Published on Web 08/20/2009
2009 American Chemical Society
crossing point (EF ) 0). Shifting the Fermi level away from this crossing point strongly decreases the energy saved by the lattice distortion, which leads to a hardening of the phonon. Raman spectroscopy combined with electrochemical doping or electric-field gating is a favorable technique to study KAs, because it simultaneously provides a monitoring of the EPC via the phonon behavior and a fine-tuning of the EPC strength. For both doping types, it is also possible to quantify the transferred charge and to relate it directly to the doping level.6-8 Strong KAs were established in the BZ center of both graphene and metallic SWNTs.8-12 Theory predicts a second KA, located in the TO phonon branch at the K-point in graphite and at the K-derived point (≈2π/3a) in metallic tubes,13 where a is the translational period of the SWNT. A detection of the KA at K in graphene by means of fieldeffect doping was reported recently.11 In this letter, we provide an experimental proof for the existence and wavevector dependence of a second KA at the K-derived point (referred to as K′ in what follows) in metallic SWNTs. We use the double-resonance scattering model to construct the phonon dispersion of metallic tubes in a vicinity of K′ from Raman spectra of the D* band (also known as 2D or D′ band) and show that the KA can be gradually removed by electrochemical doping.
Stripes of laser-ablation grown SWNT mats with a diameter distribution ranging from 1.2 to 1.5 nm were prepared as working electrodes in a three-electrode electrochemical cell equipped with quartz windows. The measurements were carried out using a Metrohm three-electrode potentiostat. The reference electrode was Ag/AgCl. The working electrode was partly dipped into the electrolyte solution and was electrically contacted with silver paint at its dry end. One molar aqueous solutions of KCl and NH4Cl were used instead of acids or bases to create mild doping conditions ensuring fine-tuning of the doping level. With the double-layer charging model it was established that increasing the applied potential from 0 to +1 V (-1 V) is equivalent to linearly increasing the doping level f from 0 to ∼0.005 holes (electrons)/C-atom, respectively,6,7 the latter being approximately equivalent to a surface carrier density of 1.5 × 1013 cm-2, which is a typical value in field-effect doping experiments on graphene. At the beginning of each measurement series the working electrode was cycled several times between 400 and -400 mV to ensure a maximum degree of wetting and to remove oxygen-containing functionalities possibly adsorbed on the SWNT sidewalls. The cell was purged with N2 gas prior to the measurements to remove dissolved oxygen. An Ar+/Kr+ laser, a dye laser and a Titansapphir laser were used for excitation. The Raman spectra of the D* band (overtone of the disorder-induced D band) were recorded at a Dilor XY800 spectrometer equipped with a CCD detector. It is of key importance to find a relation between the Fermi level shift ∆EF from the band crossing point and the doping level f. First, we assume that the transferred charge is uniformly distributed among all tubes in the sample. In view of the similarity of the work functions of metallic (∼4.5 eV) and semiconducting (∼5.5 eV) SWNTs,14 this approximation seems justified. Then, in a small vicinity of the band crossing point, where there is only the contribution from the linear bands of metallic tubes, their density of states (DOS) n(E) is a constant which can be approximated by the DOS of an armchair tube. Using eq 2 and Figure 2 of ref 15, we estimate n ≈ 0.015 states/eV·C-atom. As f ) n(E)·∆EF, we conclude that ∆EF varies linearly from 0 to 0.3 eV for applied potentials from 0 to 1 V. This relation is sufficiently precise for a correct interpretation of the results but should actually be regarded as a lower bound for ∆EF since the DOS of semiconducting tubes is zero near the band crossing point and hence metallic tubes should adopt more charge during the initial doping stages than semiconducting ones.16 However, even if the charge is initially concentrated only in metallic tubes, which is very unlikely, EF would not move beyond the first van Hove singularities of semiconducting tubes at ∼0.4 eV. Figure 1 shows the response of the D* band excited at 699 nm to a gradual increase of the electrochemical doping level from 0 to ∼0.005 holes (electrons)/C-atom. Obviously, doping causes a giant shift and substantially decreases the full width at half-maximum (fwhm) of the D* band. These effects are, however, more strongly pronounced for hole injection (applying positive potentials) than for electron 3344
Figure 1. (a) Raman spectra of the D* band of a SWNT-bundle sample at several potentials in a NH4Cl solution for λexc ) 699 nm. (b) fwhm of the D* band vs applied potential. (c) Same as panel b for the peak frequency of the D* band.
Figure 2. (Left axis) Doping-induced D* band shift at the same conditions (hole doping; potential of 1 V in a 1 M KCl solution) vs λexc (red squares). (Right axis) fwhm ratio of the D* band for potentials of 1 and 0 V vs λexc (blue diamonds, the solid line is a guide to the eye).
injection (negative potentials). We will discuss this asymmetry in the doping behavior below. For modeling of the observed phenomena and for quantitative estimates, we will use our hole doping results. The evolution of the D* sensitivity (doping-induced change in frequency and fwhm) is shown in Figure 2 for a fixed applied potential of 1 V using a variety of excitation wavelengths (λexc) all over the visible spectrum. Figure 2 presents our main experimental observation, a dramatic increase of the D* band sensitivity to doping upon moving the laser excitation to longer wavelengths up to the near IR. On the other hand, far from the red region, there is almost no doping-induced fwhm change and only a minor frequency shift that merely reflects the conventional lattice expansion/contraction upon doping. In the following discussion, we will show by means of the double resonance (DR) Raman scattering model that the observed effects are due to the predicted KA in the TO phonon branch at K′ in metallic SWNTs. While the disorder-induced D band can be excited by DR only in metallic SWNTs,17 the D* overtone has less restrictive selection rules and is allowed also for semiconducting tubes.18,19 We thus expect a contribution to the D* Nano Lett., Vol. 9, No. 9, 2009
Figure 3. Double-resonant Raman processes that reveal the D* band in the (10,10) tube upon the commonly used excitation at λexc ) 514.5 nm (2.41 eV). This plot is qualitatively representative for all metallic R ) 3 nanotubes.
band from semiconducting SWNTs at λexc from 477 to 568 nm. However, the red part of the visible spectrum coincides with the “metallic” resonance window, where the first optical transition energies of metallic tubes are located and where the electronic joint DOS of metallic tubes have their maxima, while those of semiconducting tubes reach their minima. The “metallic” resonance window relevant for the nanotube diameter distribution in our sample is 1.5-2.1 eV.20 Compared to the experimental data obtained from nanotubes in solution, it is expected that the transition energies downshift by approximately 50-100 meV due to bundling, see, for example, refs 21 and 22. This redshift does not affect our assignment of the metallic resonance window. Similarly to the single-resonance case, the DR Raman features of metallic tubes excited in this region, are expected to be further strongly enhanced18,23 and to dominate over the signal from semiconducting tubes. Therefore, we regard the D* band excited with red light as predominantly coming from metallic tubes. Without loss of generality we will model the observed effects using the “R ) 3” metallic tubes (n,m), for which (n - m)/3dc is an integer (dc being the greatest common divisor of n and m). The van Hove singularities in the electronic DOS of these tubes occur near K′.17 From this category the three armchair tubes (9,9), (10,10), and (11,11) and a couple of chiral tubes are present in our sample and we assume that their DR contributions add up to yield the observed D* band. As the diameter of (10,10) nearly equals the mean SWNT diameter in our sample, for modeling of our experimental results we use its electronic band structure,24 depicted in Figure 3, and appropriately modified to account for the bundling effects.21,22 In Figure 3, the two DR processes yielding the D* band are shown. In both of them an electron with wave vector ki is excited into the conduction band to a state (ki, E(ki)) with energy E(ki) and is then resonantly scattered by a phonon across the Γ-point to another state (ks, E(ks)). Then it is scattered back by another phonon with opposite wave vector and recombines in the valence band. For a given excitation energy, this four-step process can be accomplished only by a unique combination17 of phonon energy pωD*/2 and wave Nano Lett., Vol. 9, No. 9, 2009
Figure 4. DR phonon states of the (10,10) tube, obtained with the “q ) ki - ks” rule (see text), in pristine (squares) and in doped (∼0.005 holes/C-atom) state (triangles). Blue (red) squares correspond to the blue-line (red-line) process shown in Figure 3. The relevant wave vectors q1 and q2 undergo an Umklapp at the BZ boundary. The blue (red) thick vertical arrows denote the phonon states excited in the DR process shown in Figure 3 (λexc ) 514.5 nm). Theoretical TO branches of graphite (DFT, solid line and GW, dashed line) from ref 27 and of the (10,10) tube from ref 13 (DFT - dash-dotted line) are shown for comparison.
vector q from the TO phonon branch. This DR phonon wave vector can be approximated by q ≈ ki - ks, while simultaneously the condition E(ki) – E(ks) ) pωD*/2 must be satisfied.17 The two DR processes shown in Figure 3 yield different DR wave vectors:17 q1 < 4π/3a ) 2kF and q2 > 4π/3a ) 2kF (kF ) K′ ) 2π/3a being the Fermi wave vector). As both q1 and q2 exceed the nanotube BZ, we apply the Umklapp rules, which are equivalent to changing the phonon band quantum number from m ) 0 to m ) n.18 The values ωD*/2 (ωD* being the measured D* band frequency) can then be plotted at q1 and q2 to obtain a mapping of the TO branch of a metallic SWNT on both sides of K′. Consequently, for each λexc there should be two components in the D* band, slightly separated in frequency due to trigonal warping in the electronic bands around K′.19 The possible presence of charged defects in the SWNTs may also be a source of additional splitting in the D* band.25 Indeed, some of our D* spectra exhibited slight deviations from a fully symmetric shape, however, these deviations were not systematic and the D* lines could not be separated into two components in an consistent way. We only obtained reasonable results when fitting the D* band with a single Voigt profile. Therefore, for each λexc we assign the same ωD*/2 value to the wave vectors q1 and q2, corresponding to both possible DR processes for this λexc. This is justified, considering that (i) the D* signal comes from an ensemble of metallic tubes, so there is anyway some spread in the contributions to the D* band and (ii) we are interested in a narrow vicinity of K′, accessible by extending the laser excitation to the near IR, where the TO branch is expected to be less anisotropic.13 3345
Figure 4 presents the TO branch mapping results as obtained for each λexc in the pristine state26 and in a doped state at applied potential of 1 V (f ≈ 0.004-0.005 holes/Catom). As already indicated in Figure 2, the TO branch undergoes a dramatic doping-induced hardening in the vicinity of K′, while far from that point there are hardly any changes in the TO phonon frequency. The maximum of the hardening amounts to ∼20 cm-1 and seems to coincide with K′. In perfect correlation with the hardening, there is also a wave-vector dependent doping-induced narrowing of the D* band, as can be seen in Figures 1 and 2. Near to K′ (λexc ) 699 nm) doping causes the D* bandwidth to decrease by half, while far from that point (e.g., λexc ) 514.5 nm) the D* bandwidth is almost not doping-sensitive. Therefore, we regard the present results as the first direct proof for the existence of a KA at K′ and for a considerable impact of the KA on phonons around K′ in metallic SWNTs. Two theoretical dispersion curves of the TO phonon branch in graphite, taken from Figure 2 of ref 27 are included in Figure 4 for comparison. The steeper one was computed using GW corrections to renormalize the EPC, while the other one was obtained with conventional density functional theory (DFT).27 Also plotted in Figure 4 is a theoretical TO dispersion branch of the (10,10) tube derived from an interpolation using the corresponding DFT dispersion branches of the tubes (6,6), (11,11), and (18,18) of ref 13. The interpolation is justified since the dispersion curves of these three tubes almost completely coincide and differ only in the depth of the KA-related frequency drop, which, according to ref 13, depends solely on the SWNT diameter. As can be appreciated from Figure 4, our TO mapping points for zero doping are in qualitative agreement with the theoretical dispersion curves of graphite and the (10,10) tube in reproducing the KA-related minimum in phonon energy. The lack of accurate quantitative agreement is not surprising considering that (i) SWNTs with different chiral angles and diameters contributing to our measured D* band may have differently steep phonon and electron slopes;13 and (ii) in SWNTs with different diameters DR phonon states at different distances to K′ are excited with a considerable spread in frequency. On the other hand, our results for different λexc provide an estimate for the size of the K′ point vicinity, where the KA is still operative, as well as for its strength at different distances from the K′ point, which had not so far been done experimentally for SWNTs. The comparison curve with the theoretical phonon dispersion of the (10,10) tube in Figure 4 can be utilized to assess the error in the determination of the DR wave vectors q1 and q2. For each (q1, q2) pair q1 is situated closer to K′ than q2 because the conduction band is asymmetric with respect to its minimum and this minimum is shifted from K′ towards the BZ edge (see Figure 3). The relative difference in the distances of q1 and q2 to K′ increases on approaching K′, causing the resulting experimental TO phonon dispersion to look artificially left-shifted with respect to K′. Hence the energy values at q1 and q2 in the comparison curve for (10,10) would increasingly differ from each other on approaching K′. Another error source relevant for points far 3346
from K′ is the asymmetry in the TO phonon dispersion itself with respect to K′. An assessment of the error in the q1,2 values may be obtained considering a hypothetical shift q1 and q2 should undergo in order to correspond to the same energies in the theoretical dispersion curve. Thus obtained values of the uncertainty in q1,2 are added in Figure 4 as error bars to the points representing the DR phonons in the undoped state. The maximum uncertainty amounts to ∆q ≈ 0.04(π/a). The corresponding uncertainty in frequency ∆ω, which is the difference in the frequency values in the theoretical dispersion curve at the originally obtained q1 and q2, has its maximum at ∼50 cm-1. The uncertainty in q-vector and frequency affects most strongly the points closest to K′ in accordance with the broad measured D* band shapes corresponding to those points in the undoped state. The Kohn anomaly at Γ has been studied in both graphene and SWNTs at different Raman excitations, chosen to resonantly select the signal of metallic SWNTs;8-12 a 10-20 cm-1 upshift and significant narrowing of the Raman G-mode upon a EF shift of 0.3 eV was reported. In terms of both frequency shift and doping level these results are fully comparable with the observed behavior of the TO phonons near K′ in this work. In two cases, the KA at K in graphene was examined with field-effect gating at λexc ) 488 nm11 and with top gating by a polymer electrolyte at 514.5 nm.28 For ∆EF ≈ 0.25-0.3 eV a D* band shift ∼2 cm-1 was obtained in ref 11, while the corresponding shift for λexc ) 488 nm in our study under comparable conditions does not exceed 1 cm-1. Similarly, the D* band shift of ∼10 cm-1 obtained upon ∆EF ≈ 0.35 eV in ref 28 is well above our measured corresponding shift of ∼2 cm-1 at 514.5 nm. This discrepancy may be due to the lack of resonance in our examined metallic tubes at λexc ) 488 and 514.5 nm leading to a D* band predominantly consisting of contributions from semiconducting SWNTs with small doping-induced phonon frequency shifts. On the other hand, a recent theoretical study of the doping behavior of the TO phonons at K in graphene29 estimated a hardening of ∼15-20 cm-1 upon ∆EF ≈ 0.35 eV (hole density of 1013 cm-2) in good agreement with our results. The estimated magnitude of the maximal Fermi level shift in our experiments (0.3-0.4 eV at 1 V) confirms the detection of a KA and rules out any other explanation for the observed effects. The small ∆EF ) 0.3 eV at 1 V applied potential warrants that no van Hove singularity of any tube in our sample has been crossed by the Fermi level. This ensures that no band-structure renormalization effects occur in our experiment. On the other hand, ∆EF (1 V) is large enough to close a possible dynamic band gap created by the KA-affected phonon. In ref 5, this dynamic band gap was estimated to 0.5 eV, so even a ∆EF of 0.3 eV suffices to remove this Kohn anomaly. ∆EF (1 V) is also substantially bigger than the phonon energy pωD*/2. For ∆EF ) (1/2) pωD*/2 time-dependent corrections to the static description of the EPC predict a logarithmic singularity in the phonon energy, because the latter is exactly matched by the energy of the low-lying electronic transitions and the EPC strength is at maximum.15 For ∆EF > (1/2) pωD*/2 these transitions Nano Lett., Vol. 9, No. 9, 2009
are effectively blocked and the line width of the TO phonon is expected to decrease because it is prevented from decaying into an electron-hole pair. In our particular case, (1/2) pωD*/2 ≈ 0.08 eV and it is nicely seen in Figure 1 that the fwhm decrease of the D* band starts indeed at ∆EF ≈ 0.1 eV. However, we cannot attribute the observed sharp D* band narrowing by more than 50% only to the longer phonon lifetime. The D* band may have a large fwhm in the undoped state due to the predicted diameter dependence of the KAinduced softening13 with thinner tubes undergoing much stronger frequency drops than the thicker ones. Upon doping, the D* band phonons of all tubes would shift to the frequencies they would have in absence of a KA, which are not expected to be largely spread. The TO phonon states near K′ of distinct tubes may thus shift closer together in frequency which would also contribute to the D* band narrowing. As mentioned above, Figure 1 shows a clear electron-hole (e-h) asymmetry in the response to doping. Hardening and narrowing of the TO phonon peaks close to K′ (λexc ) 699, 716, and 733 nm) are definitely weaker for negative potentials (e doping) than for positive ones (h doping). At λexc ) 699 nm (see Figure 1) the e-doping induced shift (∼8 cm-1) is about one-third of the h-doping induced one (∼24 cm-1). Actually, e doping leads to an effective D* band softening and broadening at λexc from 477 to 637 nm30 (see also Figure S3 in the Supporting Information). The different response to e/h doping of the lattice constant6,7 definitely introduces a certain asymmetry into the TO phonon behavior, in close analogy to doped graphene.29 However, a D* band shift directly coming from a lattice constant change is expected to be countered by the change in DR phonon selection upon a possible slight electronic-band shift caused by the same lattice constant change. The resulting latticedriven effect on the D* band does not exceed 4 cm-130 and cannot alone account for the peculiar response difference between e and h doping. Another possible cause for this may be a difference in the reactions to e/h doping of the quasiparticle interactions whose existence in pristine metallic SWNTs was recently established.31 It appears, that e doping (cathodic charging) involves much more complex effects, in close analogy to the complexity of the G-band behavior upon e doping,32 and the KA removal is able to overcome these effects only in a narrow vicinity of K′. The strongly varying response of the D* band to negative potentials implies that e doping is more susceptible to individual behaviors of the distinct SWNTs. A variety of such behaviors was detected recently from individual tubes.33 Upon h doping, the phonon frequencies shift in a more unified manner, and for quantitative estimates we use our h-doping results. The comparison of the TO phonon frequency shift of about 20 cm-1 at K′ with our previously obtained shift of ≈8 cm-1 of the zone-center high-energy mode of metallic SWNTs for the same doping level,8 provides a confirmation for the SWNT case of the DFT result of ref 4: ≈ 2 connecting the EPC magnitudes at K and Γ in graphite. Having the Γ-point value for metallic SWNTs ) 37 (eV/Å)2 from ref 34, we estimate the average EPC strength Nano Lett., Vol. 9, No. 9, 2009
at K′ for our ensemble of metallic SWNTs as ≈ 90 (eV/Å)2. In summary, we use the double-resonance Raman scattering model to construct the TO phonon dispersion of doped metallic SWNTs in a vicinity of K′ ) 2π/3a that is derived from the K-point of graphite. We establish an increasing doping sensitivity of this TO branch with maximum sensitivity at K′. We show that this result is an experimental evidence for a second Kohn anomaly in metallic SWNTs, located at K′, that is related to the Kohn anomaly at K in graphite. We confirm also for carbon nanotubes the DFT result for graphite,4 that the EPC at K is about twice stronger than the EPC at Γ. Our results provide an estimate of ≈ 90 (eV/Å)2 for the EPC magnitude at K′ and can be used to investigate the wave-vector dependent EPC in the bulk of the BZ of metallic SWNTs. Acknowledgment. P.M.R. acknowledges support from the NATO Reintegration Grant CBP.EAP.RIG.982322 and thanks E. Penev for useful discussions. J.M. and C.T. acknowledge partial support from the center of Excellence UniCat funded by the DFG. U.D. and S.R. acknowledge support from the World Class University Project of the Korean Ministry of Education, Science and Technology (WCU, R32-2008-00010082-0). Supporting Information Available: (Figure S1) Kataura plot of the SWNT electronic transition energies versus nanotube diameter. The data points are uniformly upshifted by 300 meV to reproduce the experimental results from Raman and photoluminescence measurements on nanotubes in solution. The plot is taken from the work of V. Popov et al. Phys. ReV. B 2005, 72, 035436. EiiS (i ) 1, 2, 3) denotes the regions of the first, second, and third transition of semiconducting tubes, respectively, and E11M denotes the region of the first transition of metallic tubes. The straight solid lines delineate the “metallic” resonance window relevant for our sample. (Figure S2) Dispersion of the D* band (frequency vs laser energy). The black squares are our experimental values for the pristine state with approximate dispersion slope of 96 cm-1/eV. The red solid line is obtained with GW calculations in ref 27 and is consistent with experimental results on graphene. For comparison, we included experimental results of ref 17 from measurements on the first-order D band of SWNTs with mean diameter of 1.3 nm (red triangles, frequency values are multiplied by a factor of 2). (Figure 3S) Raman spectra of the D* band of our SWNT-bundle sample at several potentials in a NH4Cl solution for λexc ) 637 nm. Peak frequency (cm-1) and full width at half-maximum (fwhm) of the D* band are quoted for each spectrum. This material is available free of charge via the Internet at http://pubs.acs.org. References ¨ stu¨nel, H.; Braig, (1) Park, J.-Y.; Rosenblatt, S.; Yaish, Y.; Sazonova, V.; U S.; Arias, T. A.; Brouwer, P. W.; McEuen, P. L. Nano Lett. 2004, 4, 517–520. (2) Sapmaz, S.; Jarillo-Herrero, P.; Blanter, Y. M.; Dekker, C.; van der Zant, H. S. J. Phys. ReV. Lett. 2006, 96, 026801-1–026801-4. 3347
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