ARTICLE pubs.acs.org/JPCC
Quantum Optics Effects in Quasi-One-Dimensional and Two-Dimensional Carbon Materials Dmitri Yerchuck*,† and Alla Dovlatova‡ † ‡
Heat-Mass Transfer Institute of National Academy of Sciences of Belarus, Brovka Street, 15, Minsk, 220072, Republic of Belarus M. V. Lomonosov Moscow State University, Moscow, 119899, Russian Federation ABSTRACT: It is argued that the two-dimensional (2D)onedimensional (1D) transition (quantum size effect) in all physical properties of carbon nanotubes takes place with a decrease in their diameter. It has been established that the π-electronic subsystem is inactive in optical spectra of quasi-1D carbon zigzag-shaped nanotubes (CZSNTs), produced by means of high energy ion implantation, that leads to vanishing in Raman spectra of longitudinal and transverse optical phonon G+ and Gmodes and the outof-plane radial breathing mode, observed in 2D single-walled nanotubes. The SuSchrifferHeeger (SSH) model of organic conductors was developed and used to establish the nature of optically active centers in quasi-1D CZSNTs. They are SSH σ-polarons. Raman spectra in quasi-1D CZSNTs, which were produced by high energy ion implantation of diamond single crystals, are characterized by the only localized vibronic mode of the antiferroelectrically ordered lattice, formed by SSH σ-polarons. It has been found that Raman spectra are strongly dependent on the laser excitation beam direction, consisting in appearance additionally of antiferroelectric spin wave resonance modes and the mode, corresponding to the Fr€ohlich σ-polaron lattice sliding itself by the excitation beam direction, being opposite to the ion beam direction. A new quantum optics phenomenon—Rabi wave packet formation and propagation in space—has experimentally been identified for the first time in CZSNTs, in carbynoid films, and in graphene. It is a consequence of strong electronphoton coupling, and it leads to the appearance of additional lines, corresponding to Fourier transform of the revival part of the time dependence of integral inversion of coupled qubits.
matter and the EM field confined within a resonator structure, and is providing a useful platform for developing concepts in quantum information processing and quantum state engineering; see, for instance, refs 6 and 7. Thus, QED, is in fact, a working instrument in transient spectroscopy studies and industrial transient spectroscopy control. At the same time, in all theoretical and experimental studies of the matter systems by means of stationary optical and radio spectroscopy methods, the EM field is considered only classically. For example, in ref 8 the review of development in the theory of resonant Raman scattering (RS) in 1D electron systems is given by using several different theoretical models: the Fermi liquid model, the Luttinger liquid model, and the Hubbard model. The authors describe the interaction between the electrons in the matter and the radiation field by the following Hamiltonian: " # 2 N 2 e e ^ ¼H ^e þ p^ B ! Ai þ B Ai ð1Þ H mi c i 2mi c2 i¼1
I. INTRODUCTION Quantum electrodynamics (QED) takes on more and more significance for its practical application by using most of the analytical solutions of the tasks of the interaction of a quantized electromagnetic (EM) field with also quantized matter systems. There exists at present in QED theory the analytical solution of the task of the interaction of a quantized single-mode EM field with a two-level matter subsystem. It is a well known one qubit JaynesCummings model (JCM).1 There exists also the analytical solution of a multiqubit system without interaction between qubits, that is, the TavisCummings model.2 There is found in ref 3 the analytical solution of the multiqubit model, which is the generalization of the TavisCummings model by taking into account the one-dimensional (1D) coupling between qubits. An analytically solvable QED model for multichain coupled qubit quasi-1D system is proposed in ref 4. It is a generalization of the model described in ref 3 by taking into account both intrachain coupling and interchain coupling between qubits. The quantization of an EM field by the description of experimental results is taken into account at present in transient optical and radio spectroscopies, mainly in transient atomic spectroscopy. The first work where the predictions of JCM were experimentally confirmed is ref 5. The authors have observed quantum Rabi oscillations having time discrete character predicted by JCM by the study of the dynamics of interaction of a single Rydberg (rubidium) atom with a single-mode EM field. In transient spectroscopy strong development has obtained cavity quantum electrodynamics, which describes the coherent interaction between r 2011 American Chemical Society
∑
B(xi,t) is the vector potential of photon at the ith where A Bi = A p^ i is the electron position, i = 1, N, N is number of electrons, ! momentum operator of the ith electron, c is the light velocity, ^ e is the Hamiltonian of and mi is the effective electron mass. H electrons interacting with the Coulomb potential without the Received: June 14, 2011 Revised: October 31, 2011 Published: November 02, 2011 63
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^p A radiation field. One can simply neglect the (e/mic)! i Bi term in eq 1 if only the nonresonant Raman scattering spectroscopy is considered, where the photon frequency is far from the band gap energy. The resulting Raman scattering intensity will be then equivalent to the imaginary part of the time-ordered density correlation function in the linear response theory. It is evident from the structure of the Hamiltonian (eq 1) that by means of the B(xi, t) the classical set of continuous vector functions A Bi = A description of an EM field is given. At the same time the QED model for a multichain coupled qubit system proposed in ref 4 predicts that by strong electron photon interaction the quantum nature of an EM field can become apparent in any stationary optical experiment. In particular, a new quantum optics phenomenon—Rabi waves and Rabi wave packet formation—which was theoretically predicted for the first time in ref 3 can give an essential contribution in the stationary spectral distribution of Raman scattering intensity and spectral distributions of infrared (IR), visible, or ultraviolet absorption, reflection, and transmission intensities. The calculation is illustrated by the example of perfect quasi-one-dimensional carbon zigzag-shaped nanotubes (CZSNTs). In fact, a two-dimensional-one-dimensional (2D1D) transition appearance (new quantum size effect) in physical properties of carbon nanotubes, which is realized with a decrease in diameter, is theoretically argued. It is predicted that the 2D1D transition leads of necessity to a qualitatively different electronic model for quasi-1D CZSNTs with a cardinal change of their physical properties. It is the following. The single quasi-1D CZSNT can be modeled by an autonomous dynamic system with discrete circular symmetry, consisting of a finite number n ∈ N of carbon backbones of trans-polyacetylene (t-PA) chains, which are placed periodically along the transverse angle coordinate. Longitudinal axes {xi}, i = (1, n), of individual chains can be directed both along the element of a cylinder and along the generatrix of any other smooth figure with axial symmetry. It is similar to the Su, Schrieffer, and Heeger (SSH)9,10 model of 1D organic conductors in the part concerning the choice of active degrees of freedom, which allows consideration of all n carbon chains in the model of quasi-1D CZSNTs to be equivalent to each other, while in real quasi-1D CZSNTs the adjacent chains are a mirror of each other's relatively corresponding planes, passing through the NT axis. A given n-chain set can be considered to be a single whole, which holds the quasi-one-dimensionality of a single chain. It seems to be correct for perfect CZSNTs, if their diameter is ðn nvks Þ > 0 : k ; Ek Ek ks for the SSH solution and 8 9 < = εk Δk 2 c εk 1 þ < ðnks nvks Þ < 0 : ; Ek Ek 8 9 < = 2 εk Δk c εk 1 þ > ðnks nvks Þ > 0 : ; Ek Ek
½u
E0 ðuÞ ¼
!2 Ek 2 þ
3 2 Δk > 0 4
ð14Þ
Then, calculating the integral, we obtain ( 4Nt0 π 1 þ z2 π ½u 2 , 1 z F , 1 z2 þ E0 ðuÞ ¼ E 2 2 π 1 z2 π F , 1 z2 ð15Þ þ 2NKu2 2 where F(π/2, 1 z2) is the complete elliptic integral of the first kind, E(π/2, 1 z2) is the complete elliptic integral of the second kind, and z2 = 2αu/t0. Approximation of eq 15 at z , 1 gives ½u E0 ðuÞ
( ) 4t0 6 2t0 4α2 u2 28α2 u2 ¼N þ þ ::: þ 2NKu2 ln π π αu t0 πt0
ð16Þ It is seen from eq 16 that the energy of quasiparticles, described by the solution, which corresponds to the upper signs in eq 3, has the form of the ColemanWeinberg potential with two minima at the values of the dimerization coordinate u0 and u0 like the energy of quasiparticles, described by the SSH solution.10 Therefore, all qualitative conclusions of the model proposed in ref 10 are holding, however, for the quasiparticles, corresponding to the second-branch solution. III.B.2. Slater Principle and SSH Model. In ref 4 it was suggested that the success of the SSH model is the consequence of some general principle and it was shown that the given general principle really exists. The main idea was proposed by Slater at the earliest stage of the quantum physics era in 1924, that is, before the creation of quantum mechanics and quantum electrodynamics. It is, “Any atom may in fact be supposed to communicate with other atoms all the time it is in stationary state, by means of virtual field of radiation, originating from oscillators having the frequencies of possible quantum transitions ...”.11 The development of the given idea is based on the results of the work.12 It has been found in ref 12 that a Coulomb field in 1D systems or 2D systems can be quantized; that is, it has the character of a radiation field and it can exist without the sources that have created the given field. Consequently, the Coulomb field can be considered to be a “virtual” field in Slater principle and it can be applied to both t-PA and to quasi-1D NTs. It produces in t-PA the preferential direction in atom communication only along the chain axis (to be consequence of quasi-one-dimensionality). It is reasonable to suggest that the given direction remains to be also preferential by interaction with an external EM field; then the explanation of the success of SSH model, taking into consideration
ð10Þ
for the additional solution. It is seen that the first condition is realizable for the quasiparticles of both kinds, at that in equilibrium (ncks nvks < 0) and in nonequilibrium (ncks nvks > 0) conditions. b. The Second Condition. The second condition is the same for both the solutions, and it is εk 2 Δk 2 2 Ek Ek
2Na Z π=2a Δk 2 εk 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dk þ 2NKu2 π 0 Δk 2 þ εk 2
ð11Þ
It is realizable, which means that the quasiparticles of both kinds are satisfying to the second condition. c. The Third Condition. For the SSH solution we have ! Δk 2 εk 2 3 þ 4 ð12Þ ðncks nvks Þ > 0 Ek Ek This means that the SSH solution is unapplicable for the description of standard processes, passing the near equilibrium state by any parameters. The quasiparticles, described by the SSH solution, can be created only in the strongly nonequilibrium state with inverse population of the levels in C- and V-bands. At the 68
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the existence of solitons, polarons, and breathers, formed in a πelectronic subsystem (π-solitons, π-polarons, π-breathers), contains in implicit form also the basis for the existence of similar quasiparticles in a σ-electronic subsystem; that is, the SSH model can be developed. It was done in brief form in ref 36 and in ref 4. In section III.B.3 a more detailed description of the role of the σelectronic subsystem is given. III.B.3. σ-Quasiparticles in the SSH Model. The origin of quasiparticle formation in a σ-electronic subsystem is the same 2-fold degeneration of the ground state of the whole electronic system, the energy of which in the ground state has in correspondence with results in section III.B.1 the form of the ColemanWeinberg potential with two minima at the values of dimerization coordinate u0 and u0 for both kinds of quasiparticles in the SSH model. Really the appearance of u0 6¼ 0 and u0 6¼ 0 denotes the alternation in interatomic distance. This means that, simultaneously with the π-subsystem, the σ-subsystem will also be dimerized. The shapes, for instance, of π-solitons and σ-solitons can be given by the expression with the same mathematical form: " # 1 nπ 2 2 ðn n0 Þa ð17Þ sech vπðσÞ t cos2 jϕðnÞj ¼ ξπðσÞ ξπðσÞ 2
only one degree of freedom of the nth CH group instead of six degrees of freeedom becomes natural. It demonstrates the deep insight of Su, Schrieffer, and Heeger in the field. The Slater principle determines the applicability of the SSH model [in the part concerning active degrees of freedom] to quasi-one-dimensional CZSNTs by their interaction with an external EM field, though the interaction between the chains, which produce CZSNTs, is strong (see further) in the sense that it is described by the same hopping integral. At the same time transversal to CZSNT axis communication by means of the radiation field is absent. In other words, in the quasi-1D case the CZSNT can be considered to be a system of N independent to a sufficient extent subsystems, consisting of N chains, in which the interaction between the chains can be renormalized into the only intrachain interaction. The Slater principle can be applied also to 2D SWNTs. However, the physical consequences of the Slater principle application are quite different for quasi-1D and 2D systems. Unlike quasi-1D systems, that is, t-PA and quasi-1D SWNTs, where the Coulomb field can be considered to be a “virtual” field with the propagation direction only along the t-PA chain and NT-axis correspondingly, there exist in 2D systems including 2D SWNTs two preferential directions of Coulomb “virtual” field propagation. This means that for 2D SWNTs all degrees of freedom, which are relevant to the bonds in the rolled graphene sheet, will be essential by interaction with the external EM field and the simplification, which is fruitfully used in the 1D SSH model, becomes incorrect for 2D SWNTs in full accordance with existing 2D SWNT theory and Raman scattering experiments; see, for instance, refs 26, 28, 29, and 30. Experimental confirmation for applicability of the Slater principle to quasi-1D CZSNTs and for the model of quasi-1D CZSNTs proposed above follows from ESR studies in rather perfect quasi-1D CZSNTs, produced by HEIBM of diamond single crystals, by which the appearance of the Peierls transition and neutral paramagnetic (with spin S = 1/2) SSH π-soliton formation were established.16,17 By the way, this means that perfect quasi-1D CZSNTs, characterized by (m, 0) indices, will have band-gap like to classical semiconductors at any m, including the case m = 3q, q ∈ N, for which the 2D theory existing at present predicts the metallic properties. It follows also from the Slater principle the following. Longitudinal and transverse optical phonon graphite-like G modes,28 undergoing in 2D SWNTs splitting into G+ and G modes, respectively,30 because of the curvature effect and the out-of-plane RBM,29 that is, all the modes which are observed in Raman spectra of 2D SWNTs, have to disappear in perfect quasi-1D CZSNTs. Really, the fact that neutral SSH π-solitons are responsible for ESR spectra in quasi1D CZSNTs incorporated in diamond matrix is direct confirmation of the given conclusion. Neutral (zero charged) SSH πsolitons are optically inactive.38 Consequently, all π-subsystems will be inactive in optical spectra of quasi-1D CZSNTs. It seems to be one of the most substantial characteristics of the 2D1D transition in physical properties of CZSNTs. It seems to be interesting to predict what kind of lines have to appear in Raman spectra instead of G+, G, and RB modes by the 2D1D transition. The model of quasi-1D CZSNTs proposed in ref 4 and experimental results on ESR studies of quasi-1D CZSNTs and on optical studies of related carbon chain material— carbynes—allow obtaining a given prediction a priori without detailed analytical calculation. It is sufficient to take into consideration that the SSH model along with the physical basis of
where n and n0 are variable and fixed numbers of CH units in the CH chain, a is the CC interatomic spacing projection on the chain direction, vπ(σ) is the π (σ)-soliton velocity, t is time, and ξπ(σ) is the π(σ) coherence length. It is seen that π-solitons and σ-solitons differ in fact only by the numerical value of the coherence length. The given difference can be evaluated even without numerical calculation of the relation, which determines the shift of the ground state energy of the extended system by the presence of localized perturbation. Actually, it is sufficient to take into account the known value of ξπ and relationships46 ξ0π ¼
pvF , Δ0π
ξ0σ ¼
pvF Δ0σ
ð18Þ
where Δ0σ and Δ0π are σ- and π-band-gap values at T = 0 K; vF is the Fermi velocity. The theoretical value of ξπ in t-PA is 7a, and it is the low boundary in the range 7a11a, obtained for ξπ from experiments.38 Taking into account the relationships (18), using the value Δσ/Δπ ≈ 8.8, which was evaluated from the t-PA band structure calculation in ref 44, and the mean experimental value of the coherence length ξπ = 9a, we obtain the value ξσ ≈ 0.125 nm. This means that the half-width of the space region occupied by σ-soliton in t-PA is ≈0.25 nm; that is, SSH σ-solitons are much more localized in comparison with SSH π-solitons. A similar conclusion takes place for SSH σ-polarons representing itself the solitonantisoliton pair. SSH σ-polarons have been experimentally detected in carbynes,36 where the formation of a polaron lattice (PL) was established. It was found that two components of each PL elementary unit, that is, of each polaron, possess by two equal in absolute values electrical dipole moments, but with opposite directions. The value of the own electrical dipole moment, which was called the electrical spin moment (ESM), is proportional to the spin value. It was shown that experimental results agree well with PL formation, which means in fact the formation of an antiferroelectrically ordered lattice of quasiparticles. The given lattice consists of two sublattices, corresponding to soliton and antisoliton components of a polaron. The corresponding chain state 69
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is optically active, and it is characterized by the set of lines in IR spectra, which were assigned with a new optical phenomenon— antiferroelectric spin wave resonance (AFESWR). Let us remember that carbynes are organic quasi-1D conductors with the simplest chain structure, consisting only of the carbon atoms. At the same time the presence of two electronic πx and πy subsystems, which are “hung” on a single σ subsystem, means that the ground electronic state is similar to a two-dimensional ColemanWeinberg potential with four minima at the values of the dimerization coordinate u0 and u0. In other words, the ground electronic state in carbynes is 4-fold degenerate, which leads to a substantially more rich spectrum of possible quasiparticles, discussed in ref 48 and in ref 37. Taking into account that σ-subsystems are very similar in carbynes and in t-PA and consequently in an arbitrary chain of Nchained quasi-1D CZSNTs, we can evaluate the numerical characteristics of AFESWR in t-PA and in quasi-1D CZSNTs. Really, the central AFESWR mode, that is, the antiferroelectric resonance (AFR) mode, has in carbynoid sample by IR spectroscopy detection the frequency value νσp (C) equal to 477 cm1; the splitting parameter in IR detected AFESWR spectra was equal to 150 cm1. The given values of νσp(C) and AFESWR splitting parameter in carbynoids allow estimate the range for expected values of analogous parameters in t-PA and in quasi-1D CZSNTs in the following way. The AFESWR splitting parameter is determined by exchange integrals in the σ-electronic subsystem,36 which seems to be practically the same in carbynoids and in t-PA and in quasi-1D CZSNTs, since the role of quite different π-subsystems in carbynoids and in t-PA and quasi-1D CZSNTs can be neglected to a first approximation. Consequently, the value of the AFESWR splitting parameter in quasi-1D CZSNTs and in t-PA must be close to 150 and 300 cm1 by IR and RS-AFESWR detection correspondingly.39 The frequencies νσp(C), νσp(t-PA), and νσp (NT) of the main AFESWR mode in SSH σ-polaron lattice in carbynoids, in t-PA, and in quasi-1D CZSNTs depend on the intracrystalline field,36 which means that their values will be different. However, the values of νσp (t-PA) and νσp(NT) can be evaluated, if the known relation for the vibration frequencies of similar centers and the fact of σ-polaron and π-soliton lattice formation in carbynoids are taken into account, leading to change in effective masses.37 Naturally, the presence of two π-subsystems in carbynoids, the difference of coherence lengths in accordance with eq 18 of σ-solitons (ξσ) and π-solitons (ξπ), and band structure data for t-PA44 and carbynoids45 have to be also taken into consideration. We have obtained the following frequency ranges for IR SSH σ-polaron lines in t-PA and in quasi-1D CZSNTs: νσp (t-PA) ∈ (386.7, 603) cm1 and νσp (NT) ∈ (402.5, 627.6) cm1. The known IR mode with the frequency near 540 cm1, in t-PA,38 belongs to the interval (386.7, 603) cm1 and can represent itself the AFR mode of the σ-polaron lattice; that is, there is an alternative interpretation of the given IR mode, ascribed earlier to the Goldstone SSH π-soliton vibration mode.38 It follows from the results of ref 47 that the same spectral interval with a slightly different right-hand value, equal to 673.7 cm1, is the evaluation for the frequency of σ-polaron main AFESWR modes in t-PA and in quasi-1D CZSNTs, which are RS-active. The calculation in ref 47 does not take into consideration the soliton and polaron formation. However, σ-polaron formation does not violate the symmetry of task, which allows the conclusion that the same value of asymmetry extent will be retained in IR and RS spectral distributions. Therefore, the qualitative semiclassical consideration predicts, a priori, that the Raman spectrum of quasi-1D CZSNTs will 70
)
)
consist of only one line, representing itself the AFR mode of the σ-polaron lattice with peak frequency position in the range 386.7673.7 cm1. It can be split into a series of AFESWR modes with average splitting parameter near 300 cm1. The possibility of AFESWR splitting depends on experimental geometry conditions (that is, whether an antiferroelectric spin wave mode’s excitation is allowed by the experimental geometry). It will be shown further that the prediction must be completed, if to take into account the quantum nature of EM field. Experimental confirmation for conclusion on the 2D1D transition and its main physical properties predicted follows from ESR studies, performed earlier on rather perfect CZSNTs, produced by HEIBM of diamond single crystals,1517 and from Raman scattering data, presented in section II, obtained on the same samples. Let us reproduce the formulation of the model of quasi-1D CZSNTs, given in ref 4, once again. Quasi-1D CZSNT can be modeled by an autonomous dynamic system with discrete circular symmetry consisting of a finite number n ∈ N of carbon backbones of t-PA chains, which are placed periodically along the transverse angle coordinate. Longitudinal axes {xi}, i = 1, n, of individual chains can be directed both along the element of the cylinder and along the generatrix of any other smooth figure with axial symmetry. It is taken into account that the Slater principle like the SSH model for t-PA allows considering to be active the only degree of freedom along axes {xi}, i = 1, n, of individual chains or, in other words, along a single hypercomplex axis in a hypercomplex number language of the task formulation. This is the reason that the adjacent chains, which are a mirror of each other in real structure, become equivalent in the model; that is, they will be indistinguishable. Actually, only one degree of freedom—the dimerization coordinate um along the hypercomplex axis x in a real sub-ring of the n-dimensional hypercomplex ring Zn, which is the direct sum of n complex spaces C (see for details section III.C), of n mth C-atoms, m = 1, N, placed on the mth site along the hypercomplex axis x of CZSNT (that is along n-chain molecular symmetry axes {xi}, i = 1, n)—is substantial for determination of the main physical properties in the frames of the model proposed. III.C. Discussion. III.C.1. Summarization of Results on HEIBM Method of Nanotube Production. Let us give a short review concerning the HEIBM method of production of incorporated in diamond matrix carbon NTs and to represent the exact experimental proof of their real formation. The first report on the discovery of a new carbon phase— carbon nanotubes, incorporated in diamond matrix—is related to 1990, and it was made during the 1990 IBMM Conference, Knoxville, TN, USA. A similar report was also presented at the E-MRS 1990 Fall Meeting, Strasbourg, France;14 that is, the conclusion to a new carbon phase formation became known substantially prior to the now well-known Japanese discovery of freestanding NTs, reported in ref 49, which is related to 1991. The basis experimental method was ESR. It was found that in ESR spectra of diamond single crystals, modified by high energy ion implantation, a single intensive line with very unusual radio spectroscopic properties takes a central place. It was anisotropic; however, the anisotropy was weak in comparison with the anisotropy of point centers in diamond. Especially interesting, along with the g value the line width ΔHpp was also found to be a tensor quantitity, and the g-tensor and ΔHpp-tensor were characterized by the same axial symmetry group.15 Moreover, only one equivalent configuration in the diamond lattice was presented with the g and ΔHpp principal directions of the axial dx.doi.org/10.1021/jp205549b |J. Phys. Chem. C 2012, 116, 63–80
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in phase with high-frequency modulation and in quadrature with high-frequency modulation.15,16 3. It was found unusual for nonsaturating resonance systems the dependence of resonance signal amplitude on the value of modulation frequency, which was characterized by a substantial increase (instead of decrease) of resonance signal amplitude with increasing modulation frequency.15,16 4. The effect of the appearance of “phase angle” characterizing the absorption process was established. It consists in that, although absorption kinetics is nonsaturating, the maximum of absorption is achieved not in phase with the modulation field, but at the modulation phase value which is nonequal to zero. “Phase angle” is anisotropic in the general case, and its maximum value was found to be equal to 20 degrees.16 The analysis of given peculiarities16,17 has led to the conclusion that PC, which are responsible for strong non-Blochian absorption, are mobile and are characterized by two times of spin lattice relaxation: by T1, which is comparable with the relaxation time of usual point PC in diamond single crystals (103105 s), and by a very short time τ (up to 1013 s) of conversion of the energy of the spin system into mechanical kinetic energy of PC motion, which can be considered to be a consequence of a fundamental in the magnetic resonance phenomenon of gyromagnetic coupling between magnetic and mechanical moments. Therefore, the τ process is a very fast process in comparison with any process realized by means of pure magnetic interaction, the relaxation times of which are not shorter than 1010 s.13 The possibility of a given conversion can be realized naturally if the energy of the motion activation of absorbing PC is very small, that is typical for mobile topological solitons like those identified in t-PA. Moreover, nonsaturating absorption kinetics and superLorentzian shape of resonance lines have been observed by ESR study on SSH electrically neutral spin S = 1/2 topological πsolitons in t-PA in ref 18. Theoretical analysis of the given line shape in ref 18 allowed conclusions on the 1D movement of given PC and their responsibility for the so-called spin-diffusion process with a very short characteristic time τ ≈ 1011 s. Let us remark that the most direct experimental proof of SSH neutral π-soliton movement was obtained for the first time in ref 19 by the study of the Overhauser effect in t-PA samples, at that characteristic time of the spin-diffusion process was evaluated to be more short than τ ≈ 1012 s. We see that the very high rate of the τ process, the upper boundary of which was evaluated in refs 16 and 17 to be determined by times up to 1013 s, is in qualitative agreement with the very high rate of the spin-diffusion process which was found in refs 18 and 19. In fact, it seems to be physically the same process, and in refs 16 and 17 the origin of the given very fast spin diffusion is established more exactly. This allows the proposal of a qualitative explanation for difference in characteristic times in the same material,18,19 by a slightly different activation energy of soliton motion, which seems to be dependent on t-PA technology. It was found also that there is a numerical coincidence of the characteristics of SSH topological πsolitons in t-PA and in Æ111æ incorporated NTs. So, the values of g-tensor components in Cu-implanted sample are g1 = 2.00255 (the minimal g value and g principal direction), g2 = g3 = g^ = 2.00273 and the accuracy of relative g-value measurements is (0.00002.15 We see, that the g value of paramagnetic π-solitons in trans-polyacetylene, equaled 2.00263,20 is in the middle of a given rather narrow interval of g-value variation of PC in ion )
)
)
g- and ΔHpp-tensors, which precisely coincided with the ion pp beam direction; at that g is the minimal g value and ΔH is the maximal ΔHpp value in g and ΔHpp angular dependences. The other lattice equivalent configurations were absent. The kind of axial symmetry group corresponding to g-tensor and ΔHpptensor symmetry was strongly dependent on the choice of ion beam direction relative to the crystallographic axes of the diamond lattice. The symmetry axis of the structure produced by a single high energy ion along its track at the Æ111æ implantation direction was C∞; however, at the Æ100æ implantation direction it was C4. Further, it has been shown for the first time in radio spectroscopy that g-tensor and ΔHpp-tensor symmetry corresponding to the most intensive line in the spectrum observed are the mapping of the symmetry of lengthy objects with macrosizes along the implantation direction.15 Therefore, detailed studies of ESR spectral angular dependences in appropriate crystallographic planes allowed establishing that the structures formed by Æ111æ HEIBM in diamond have a tracklike cylindrical shape; more strictly, they represent a near cylindrical smooth figure with C∞ symmetry axis, which seems to have an expanding onionlike shape at the end of the ion run. The given conclusion agrees with different values of the g-tensor and ΔHpp-tensor along the symmetry axis and in the transversal direction. At the same time they are situated precisely in the ion beam direction, coinciding with accuracy 0.7 degree with Æ111æ direction and have size varying from several micrometers to several tens of micrometers, depending on the ion energy used. The deviation of diameter size along the symmetry axis from a constant value is relatively small. In other words, the first studies in 1990 allowed establishment of the formation of carbon nanotubes of cylindical symmetry shape incorporated in diamond matrix. Simultaneously the formation of quite different nanotubes was found. They represent themselves crimped cylinders with crimping corresponding to a four-petal structure in the cross section. They are produced by the implantation direction, coinciding with the Æ100æ axis of diamond lattice.15 The next step was the additional confirmation of the structure of cylindical symmetry shape to be really Æ111æ-incorporated nanotubes with a rolled-up graphene sheet in a zigzag-shaped configuration (but not, for example, a cylindical rod). It was done by the study of the radio spectroscopic properties of spin carriers in given nanotubes. It was found that the system of paramagnetic centers (PC), which are responsible for appearance of strong single line absorption described above, is a non-Blochian system. A number of distinctive peculiarities have been observed for the first time in radio spectroscopy. The main peculiarities are the following. 1. It was found that the shape of resonance lines cannot be approximated by Lorentzian, by Gaussian, or by their convolution. The shape is characterized by an essentially slower decrease of the absorption intensity in the wings in comparison with known shape functions. It was called “super-Lorentzian”. It was proved that super-Lorentzian is a genuine line shape of resonance absorption (that is, it does not represent the superposition of Lorentzians with different line widths).16 2. It was established that the PC system is nonsaturating with superlinear absorption kinetics (that is, the dependence of the resonance signal amplitude on the amplitude of the magnetic component of the microwave field is superlinear). Especially interesting, superlinear absorption kinetics has been observed by the registration of resonance signal both 71
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The Journal of Physical Chemistry C
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produced NTs. Although anisotropy of paramagnetic π-solitons in t-PA, mapping the distribution of π-electron density along the whole individual t-PA chain, is not resolved by ESR measurements directly (which in fact is the indication that chemically produced tPA is less perfect in comparison with NTs in diamond matrix), there is indirect evidence of the axial symmetry of ESR absorption spectra in t-PA, too.21,22 Consequently, the value 2.00263 is the mean value and it coincides with an accuracy of 0.00002 with the mean value of the aforecited principal g-tensor values of PC in NTs. Therefore, the conclusion that the strong non-Blochian ESR absorption by HEIBM of diamond is determined by mobile topological SSH π-solitons is obtained by a natural way. Thereby immediately from ESR studies was found that the structural element of the new carbon phase with near-cylindrical axial symmetry produced in diamond by Æ111æ high energy implantation is the t-PA chain backbone. At the same time the t-PA chain backbone is a basic element for graphene, graphite, and NT formation and also for carbynoid formation. Carbynoids were studied by ESR, and they were characterized by quite different ESR spectra.35 The possibility of formation of cylindrical graphite rod is also not corresponding, since graphite is characterized by other ESR spectra. Thus it was concluded unambiguously that the spectra observed by Æ111æ high energy implantation correspond only to the graphene sheet, which is rolled up in that way, so that t-PA chain backbones were directed along the Æ111æ axis of diamond lattice, coinciding with ion beam direction, which results in CZSNTs production. All subsequent studies have confirmed the given conclusion. In particular, the theoretical model of quasi-1D CZSNTs4 is very similar to the SSH model of t-PA that gives theoretical substantiation of the similarity of their paramagnetic characteristics, including numerical coincidence of mean values of g-tensor components. The main differences of both models consist in the following. The carbon chain backbone of the t-PA single chain can be considered to be a 1D object in the space 3D X Z1; the carbon chain backbone of a single CZSNT can be considered to be a 1D object in 3D X Zn space. Here 3D is real Euclidian space, Zn is the aforesaid hypercomplex commutative ring, that is, the direct sum of n complex spaces C: Zn ¼ C x C x ::: x C
diamond single crystals, by means of which exactly identical implantation conditions were ensured, does not lead to silicon nanotube production; that follows also from ESR studies.23,24 The given difference becomes understandable if we take into account that carbon in the allotropic form of diamond single crystals is a metastable system in contrast to silicon. By the way, the given result seems to be experimental confirmation that carbon in the allotropic form of nanotubes is a more energetically favorable carbon system in comparison with diamond. It is substantial that the CZSNTs produced by HEIBM are not random in orientation. They are produced precisely in the ion beam direction and they have the usual length by ∼1 MeV per nucleon ion energy; it is proved in detail in the works cited above. Naturally, random processes are also presented by HEIBM. The knocked diamond atoms produce point defects in lateral to ion track directions; their concentration distribution in the sample is formed by overlapping the overlap of random distributions along directions which are perpendicular to each single ion track axis. Given processes were also detected and studied by ESR; see, for instance, ref 25. The main difference of our results on NTs from well-known ones is based on another dimensionality of CZSNTs studied. We are dealing with quasi-one-dimensional CZSNTs, which have to possess and really possess quite other physical properties including optical properties in comparison with two-dimensional CZSNTs, which have naturally graphite-like optical spectra. In addition, we see also that the spectra observed by Raman spectroscopy studies in quasi-1D CZSNTs can be predicted qualitatively, if we take into consideration only ESR data (optical data in carbynes were used additionally for quantitative evaluation of optical characteristics, that is, AFESWR characteristics). It was indicated above that freestanding nanotubes were considered theoretically only to be 2D strutures. At the same time experimental ESR and Raman scattering results, which were obtained on narrow freestanding tubes with diameter
