Electron spin resonance signal of Luttinger liquids and single-wall carbon nanotubes
Abstract
A comprehensive theory of electron spin resonance (ESR) for a Luttinger liquid (LL) state of correlated metals is presented. The ESR measurables such as the signal intensity and the line-width are calculated in the framework of Luttinger liquid theory with broken spin rotational symmetry as a function of magnetic field and temperature. We obtain a significant temperature dependent homogeneous line-broadening which is related to the spin symmetry breaking and the electron-electron interaction. The result crosses over smoothly to the ESR of itinerant electrons in the non-interacting limit. These findings explain the absence of the long-sought ESR signal of itinerant electrons in single-wall carbon nanotubes when considering realistic experimental conditions.
pacs:
71.10.-w,73.63.Fg,76.30.-vThe experimental and theoretical studies of strong correlation effects are in the forefront of condensed matter research. Low-dimensional carbonaceous systems, fullerenes, carbon nanotubes (CNTs), and graphene exhibit a rich variety of such phenomena including superconductivity in alkali doped fullerenes GunnRMP , quantized transport in SWCNTs, and massless Dirac quasi-particles showing a half integer quantum Hall-effect in graphene even at room temperature NovoselovNature . A compelling correlated state of one-dimensional systems is the Luttinger liquid (LL) state. There is now abundant evidence from both theoretical eggergog ; kane ; balents ; yoshioka ; krotov and experimental bockrath ; BachtoldPRL2004 ; ishii ; rauf side that the low energy properties of CNTs with a single shell, the single-wall carbon nanotubes (SWCNTs) can be described with the LL state. As a result, SWCNTs are regarded as a model system of the LL state, which could be exploited to further test the theories and novel experimental methods.
Electron spin resonance (ESR) is a well established and powerful method to characterize correlated states of itinerant electrons. It helped to resolve e.g. the singlet nature of superconductivity in elemental metals VierSchultz , the magnetically ordered spin-density state in low-dimensional organic metals TorrancePRL1982 and in alkali doped fullerides JanossyPRL1997 . In three-dimensional metals, the ESR signal intensity is proportional to the Pauli spin-susceptibility, the ESR line-width and -factor are determined by the mixing of spin up and down states due to spin-orbit (SO) coupling in the conduction band. These ESR measurables are affected when correlations are present and thus their study holds information about the nature of the correlated state.
This motivated a decade long quest to find the ESR signal of itinerant electrons in SWCNTs and to characterize its properties in the framework of the expected correlations PetitPRB1997 ; NemesPRB2000 ; SalvetatPRB2005 . Detection of ESR in SWCNTs is also vital for applications as it enables to determine the spin-lattice relaxation time, , which determines the usability for spintronics FabianRMP . However, to our knowledge no conclusive evidence for this observation has been reported. An often cited argument for this anomalous absence of the ESR signal is the large heterogeneity of the system, the lack of crystallinity, and the presence of magnetic catalyst particles PetitPRB1997 ; NemesPRB2000 ; SalvetatPRB2005 . However, ESR signal of conduction electrons has been indeed observed for electron doped SWCNTs NemesPRB2000 , which are known to be Fermi liquids rather than the pristine SWCNTs rauf . Thus the above properties of SWCNTs should hinder the observation of the ESR signal also for the Fermi liquid state, which is clearly not the case. As a result, we suggest that the LL state inherently prohibits the observation of ESR of the itinerant electrons, calling for a realistic description of such experiments. A recent experiment by Kuemmeth et al. kuemmeth shed new light on the spin degree of freedom of SWCNTs. It was shown that SO coupling and correspondingly the lifting of the spin rotational invariance is unexpectedly large. As we show below, this results in a uniquely large homogeneous broadening of the ESR line which explains the absence of an intrinsic ESR signal of SWCNTs. So far, the theory of ESR in the SWCNTs was limited to SU(2) symmetric models martino ; martino1 .
Here, we study the ESR signal in a Luttinger liquid with broken spin rotational symmetry. While at low temperatures the characteristic non-integer power laws characterize the response, the high temperature behavior crosses over to the standard Lorentzians, whose width, in contrast to the Fermi liquid picture Slichterbook , is determined by the Luttinger liquid parameters. We show that this explains the absence of itinerant electron spin resonance in this system by combining DFT calculations of the spin-susceptibility on metallic SWCNTs with a critical evaluation of the experimental conditions.
To describe a metallic SWCNT, we apply an effective low-energy theory. We neglect the ”flavour” index coming from the two points martino1 since we are interested in the spin properties only. The standard Luttinger liquid Hamiltonian is expressed as a sum of independent spin and charge excitations as
(1) |
where ’s are the Luttinger liquid parameters, denotes the charge and spin sector, respectively, and are canonically conjugate fields with velocity . The Luttinger liquid parameter in the spin sector, for SU(2) symmetric models as these preserve the spin rotational symmetry. However, the presence of spin-orbit and magnetic dipole-dipole interaction between the conduction electrons spinanis ; nersesyan produces spin dependent interactions and breaks the spin rotational symmetry, leading to . In addition, these processes are also responsible for the -factor anisotropy.
The original fermionic field operators are expressed in terms of the bosons as
(2) |
which are needed to express the spin operators in the bosonic language, is the Klein factor, , =R/L= denotes the chirality of the electrons, and is the electron spin.
The ESR experiments are performed in a longitudinal static magnetic field, , applying a transversal perturbing microwave radiation with a magnetic component, . For the ESR description, the above Hamiltonian is completed with the Zeeman term:
(3) |
The ESR signal intensity is given by the absorbed microwave power that is Slichterbook :
(4) |
where is the permeability of the vacuum, is the imaginary part of the retarded spin-susceptibility for the transversal direction, and is the sample volume. The spin operators required to calculate are . Since ESR measures the response, only the terms contribute, the others contain fast oscillating terms and average to zero.
The Zeeman term in Abelian bosonization is the simplest when the longitudinal magnetic field points in the spin quantization axis (the -axis). For a different field orientation, the Zeeman term becomes more complicated but it can be rotated along the direction, at the expense of changing the Luttinger liquid parameters spinanis . The external magnetic field further lowers the SU(2) symmetry in addition to the spin-orbit and dipole-dipole interactions, resulting in a further renormalization of .
From now on, we set and they will be reinserted whenever necessary. The retarded spin-susceptibility is built up from correlators of the type schulz
(5) |
where , is determined by the short distance behavior and cannot be obtained by the methods used here. Here we introduced the parameter, where , and encodes information about spin symmetry breaking processes. Upon Fourier transformation, we obtain the retarded spin-susceptibility. From a simple scaling analysis, we can conjecture the behaviour of the retarded spin-susceptibility as which is confirmed later by a careful investigation in Eqs. (9) and (10).
Putting all this together, we find for the ESR intensity:
(6) |
where
(7) | |||
(8) |
where is Euler’s beta function, is Euler’s gamma function, is a constant, whose value is determined further below. In the limit, SU(2) spin symmetry is conserved by the Hamiltonian and the ESR resonance becomes completely sharp, located at as .
The influence of interactions is most clearly seen at , when the ESR signal is completely asymmetric around , and cannot be approximated by Lorentzians:
(9) |
The ESR intensity vanishes below a threshold set by the magnetic field, and falls off in a power law fashion, depending on the explicit value of .
However, the sharp threshold disappears with increasing temperature and the spectrum broadens. In the limit of and , which is relevant for realistic experiments, the intensity can be approximated by (upon reinserting original units)
(10) |
where . This expression works well outside of its range of validity and it consists of two Lorentzians, centered around , characterized by a width of . Hence, the interaction () together with the temperature determines the width of the resonance and shifts the resonance center as well, as is seen in Fig. 1.
This expression allows us to make contact with the conventional Fermi liquid case. In that case, (together with ), and the ESR intensity reduces to
(11) |
Thus the integrated ESR intensity reads as . In a Fermi liquid, this is expressed in terms of the static spin-susceptibility Slichterbook , , as . This fixes the so far unknown numerical prefactor as . In summary, the ESR signal of a Luttinger liquid with broken spin rotational symmetry i) is significantly broadened due to the interaction and spin symmetry breaking and ii) has a signal intensity which matches that of the non-interacting state.
These results are similar to those found for the 1D antiferromagnetic Heisenberg model oshikawa , whose low energy theory is identical to the spin sector of a Luttinger liquid, Eq. (1). The ESR line-width also scales with at low temperatures. However, the spin in the Heisenberg model, when represented in terms of fermionic variables via the Jordan-Wigner transformation, usually contains non-local string operators giamarchi and acquires a different scaling dimension than the spin of itinerant electrons. The exchange anisotropy, causing the broadening of the ESR signal, shares common origin with the -factor anisotropy in terms of spin-orbit coupling, scaling with .
The spin orbit coupling in SWCNTs was found to be unexpectedly large, around 1 meV for a nanotube with diameter of 1 nm, resulting in a -factor enhancement in a few electron carbon nanotube quantum dot kuemmeth . In the presence of many electrons, the interplay of interactions, low dimensionality and spin-orbit coupling determines the strongly correlated ground state and it can further enhance spin symmetry breaking. for quantum wires gritsev like InAs, which is another possible realization of Luttinger liquids. These materials possess a spin orbit coupling of the same order of magnitude than SWCNTs but they have a smaller Fermi velocity. Following a similar line of reasoning for SWCNTs, we take a conservative estimate of to be around 1.1.
In the following, we discuss the relevance of the results on the absence of an ESR signal in SWCNTs. As we showed above, the ESR signal intensity of a Luttinger liquid crosses over smoothly for the non-correlated case to the static susceptibility that is the Pauli susceptibility of metallic SWCNTs ashcroft : , where is the density of states (DOS) at the Fermi energy. To have an accurate value for the DOS in a realistic sample, we performed density functional theory calculations with the Vienna ab initio Simulation Package KresseG_1996_2 within the local density approximation for metallic nanotubes. The projector augmented-wave method was used with a plane-wave cutoff energy of 400 eV. The DOS was obtained with a Green’s function approach from the band structure that was calculated with a large -point sampling. We considered the (9,9), (15,6), (10,10), (18,0), and (11,11) SWCNTs in order of increasing diameter. Here denotes the lattice vectors on the graphene basis along which a cut-out stripe is folded up to represent a SWCNT DresselhausTubes . These tubes are within the Gaussian diameter distribution of a usual SWCNT sample with a mean diameter of 1.4 nm and a variance of 0.1 nm. Calculating the DOS for chiral SWCNTs with a large number of atoms in the elementary cell is prohibitively long. Therefore, we confirmed by nearest-neighbor tight binding calculations on all the metallic SWCNTs in the above diameter distribution that the DOS depends very weakly on the chirality, thus the above SWCNTs chiralities are indeed representative for the ensemble of the metallic tubes.
We obtain that such a tube ensemble has an effective DOS of states/eV/atom by averaging the DOS for the above SWCNTs and taking into account that only one third of the tubes are metallic for this diameter range DresselhausTubes . This is a very low DOS which results from the one-dimensionality of the tubes and from the fact that the majority of the tubes are non-metallic. It is 50 times smaller than in KC ( states/eV/atom GunnRMP ) and is comparable to the well known low DOS of pristine graphite ( states/eV/atom Dresselhaus_AP_Review ). With the above DOS, we obtain that a typical 2 mg SWCNT sample gives a practically detectable signal-to-noise ratio of for a spectrum measured for 1000 seconds provided the ESR line is not broader than 110 mT. To obtain this value, we considered that the state-of-the-art ESR spectrometers give a for S=1/2 spins at 300 K provided the ESR line-width is 0.1 mT and each spectra points (typically 1000) are measured for 1 sec. We also took into account that the drops with the square of the line-width for broadening beyond 1 mT.
The above calculated homogeneous broadening of the ESR line of a Luttinger liquid is in units of the magnetic field. Thus at 4 K, which is the lowest available temperature for most ESR spectrometers, one has a broadening of Tesla. This, together with the above detectability criterion gives an upper limit of for the detection of the ESR signal ^{1}^{1}1This is an overestimate of the limit as many other factors such as more limited microwave penetration into the sample makes the ESR experiment less sensitive and thus reduce this limit, which further justifies our argument.. Clearly, the above conservative estimate of based on the the value is close to this limit, which explains why careful studies have not yet yielded a conclusive ESR signal of itinerant electrons is SWCNTs. This argument can be also turned around: the fact that no ESR signal of the itinerant electron has been observed in the SWCNTs means that the line is broadened beyond observability, which means that the real is larger than putting also .
We finally comment on the future viability of this observation. Clearly, ESR spectrometers operating to sub Kelvin temperatures are required. Observation of linearly temperature dependent ESR line-width would be an unambiguous evidence for the observation of the ESR signal of itinerant electrons in the Luttinger liquid state. We note that such a temperature dependence is fairly unusual as ESR line-width in metals normally tends to a residual value similar to the resistivity.
In summary, we extended the theory of electron spin resonance in a Luttinger liquid for the case of broken spin-symmetry. We obtain a significant homogeneous broadening of the ESR line-width with increasing temperature, which explains the unobservability of ESR in single-wall carbon nanotubes and puts severe constraints on the usability of SWCNTs for spintronics.
Acknowledgements.
The authors acknowledge useful discussions with L. Forró and an illuminating exchange of e-mails with A. De Martino. Supported by the Hungarian State Grants (OTKA) F61733, K72613, NK60984, F68852, and K60576. VZ acknowledges the Bolyai programme of the HAS for support.References
- (1) O. Gunnarsson, Rev. Mod. Phys. 69, 575 (1997).
- (2) K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005).
- (3) R. Egger and A. O. Gogolin, Eur. Phys. J. B 3, 281 (1998).
- (4) C. Kane, L. Balents, and M. P. A. Fisher, Phys. Rev. Lett. 79, 5086 (1997).
- (5) L. Balents and M. P. A. Fisher, Phys. Rev. B 55, R11973 (1997).
- (6) H. Yoshioka and A. A. Odintsov, Phys. Rev. Lett. 82, 374 (1999).
- (7) Y. A. Krotov, D.-H. Lee, and S. G. Louie, Phys. Rev. Lett. 78, 4245 (1997).
- (8) M. Bockrath, D. H. Cobden, J. Lu, A. G. Rinzler, R. E. Smalley, L. Balents, and P. L. McEuen, Nature 397, 598 (1999).
- (9) B. Gao, A. Komnik, R. Egger, D. C. Glattli, and A. Bachtold, Phys. Rev. Lett. 92, 216804 (2004).
- (10) H. Ishii, H. Kataura, H. Shiozawa, H. Yoshioka, H. Otsubo, Y. Takayama, T. Miyahara, S. Suzuki, Y. Achiba, M. Nakatake, T. Narimura, M. Higashiguchi, et al., Nature 426, 540 (2003).
- (11) H. Rauf, T. Pichler, M. Knupfer, J. Fink, and H. Kataura, Phys. Rev. Lett. 93, 096805 (2004).
- (12) D. C. Vier and S. Schultz, Phys. Lett. A 98A, 283 (1983).
- (13) J. B. Torrance, H. Pedersen, and K. Bechgaard, Phys. Rev. Lett. 49, 881 (1982).
- (14) A. Jánossy, N. Nemes, T. Fehér, G. Oszlányi, G. Baumgartner, and L. Forró, Phys. Rev. Lett. 79, 2718 (1997).
- (15) P. Petit, E. Jouguelet, J. E. Fischer, A. G. Rinzler, and R. E. Smalley, Phys. Rev. B 56, 9275 (1997).
- (16) A. S. Claye, N. M. Nemes, A. Jánossy, and J. E. Fischer, Phys. Rev. B 62, 4845 (2000).
- (17) J.-P. Salvetat, T. Fehér, C. L’Huillier, F. Beuneu, and L. Forró, Phys. Rev. B 72, 075440 (2005).
- (18) I. Žutić, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004).
- (19) F. Kuemmeth, S. Ilani, D. C. Ralph, and L. M. McEuen, Nature 452, 448 (2008).
- (20) A. De Martino, R. Egger, K. Hallberg, and C. A. Balseiro, Phys. Rev. Lett. 88, 206402 (2002).
- (21) A. De Martino, R. Egger, F. Murphy-Armando, and K. Hallberg, J. Phys. Cond. Matter 16, S1427 (2004).
- (22) C. P. Slichter, Principles of Magnetic Resonance (Spinger-Verlag, New York, 1989), 3rd ed.
- (23) T. Giamarchi and H. J. Schulz, J. Phys. France 49, 819 (1988).
- (24) A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik, Bosonization and Strongly Correlated Systems (Cambridge University Press, Cambridge, 1998).
- (25) H. J. Schulz, Phys. Rev. B 34, 6372 (1986).
- (26) M. Oshikawa and I. Affleck, Phys. Rev. B 65, 134410 (2002).
- (27) T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press, Oxford, 2004).
- (28) V. Gritsev, G. Japaridze, M. Pletyukhov, and D. Baeriswyl, Phys. Rev. Lett. 94, 137207 (2005).
- (29) N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt Rinehart and Winston, 1976).
- (30) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
- (31) R. Saito, G. Dresselhaus, and M. Dresselhaus, Physical Properties of Carbon Nanotubes (Imperial College Press, 1998).
- (32) M. Dresselhaus and G. Dresselhaus, Adv. Phys. 30, 139 (1981).