DSpace at VNU: Optical modes in nanoscale one-dimensional spin chains

ARTICLE IN PRESS Journal of Magnetism and Magnetic Materials 321 (2009) 4123–4127 Contents lists available at ScienceDirect Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm Optical modes in nanoscale one-dimensional spin chains Hoang Nam Nhat a,Ã, Phung Quoc Thanh a,b, Le Thi Anh Thu a a b Center for Materials Science, College of Science, Vietnam National University, 334 Nguyen Trai, Hanoi, Vietnam Department of Physics, Chungbuk National University, Cheongju 361-763, Republic of Korea a r t i c l e in fo abstract Article history: Received January 2009 Received in revised form August 2009 Available online 19 August 2009 The spin chain systems with one-dimensional magnetic ordering are promising candidates for quantum optical devices This paper shows how the optical excitation can induce various phonon modes in an ideal Cu–O chain at various lengths The calculation was carried out at different level theories including ă conguration interaction singles for excited states, density functional theory and second-order MollerPlesset perturbation In general, the number of modes increases with chain length due to growing asymmetry of atomic positions when chain exceeds nm There were, however, only two basic modes: one is associated with the symmetric oscillation of oxygen and another with the asymmetric motion of the same along the chain At the length below 4.3 nm, the Raman activity of the symmetric mode (440 cmÀ1) dominates From analysis of density of states, this mode may be associated with the excitation across the lowest LUMO bands with changing in spin state & 2009 Elsevier B.V All rights reserved Keywords: Optical phonon Spin chain Ab initio Nanoscale Introduction For modern quantum devices, the one-dimensional spin chains with strong antiferromagnetic interaction offer a promising opportunity for setting up the devices which are based on spin switching Such spin chains occur, for example, in A2CuO3 (A ¼ Sr, Ca), where the spin and charge exchange along the Cu–O chains was coupled with many strange optical modes that could not be explained by the group theory Therefore, the insight into the optical excitation in this system plays a key role to understand the spin and charge transport mechanism which may lead to a successful spin manipulation in the near future The extensive efforts to shed light onto a complicated system of the forbidden optical modes in Raman scattering spectra of the pure and doped A2CuO3 have been found in the past [1–5] Among seventeen observed modes, fifteen were addressed as forbidden One half of this set was ascribed as the multi-phonon bands, originating possibly from the three intrinsic modes of the spin chain: 235, 440 and 670 cmÀ1 [5] This paper concentrates on the proper vibrational modes of the Cu–O spin chains at different lengths from 0.19 to 7.0 nm Because of the computational cost, we were restricted to the singlet spin state, i.e to the chains with even number of Cu atoms The chains with doublet (and higher) spin state are the subjects for another work For the spin chains, the spin–spin interaction between the electrons and nuclei, which leads to the hyperfine splitting of spin à Corresponding author Tel.: +84 98 300 6668; fax: +84 768 2007 E-mail address: namnhat@gmail.com (Hoang Nam Nhat) 0304-8853/$ - see front matter & 2009 Elsevier B.V All rights reserved doi:10.1016/j.jmmm.2009.08.015 states, is expected This problem complicates the estimation of ground state and is known to be a difficult problem for quantum computation Therefore, we utilized various level theories for the purpose of correct identification of the most appropriate model chemistry for the case under investigation Although the HartreeFock (HF) level, which contains only a minimal amount of electron correlation, has accurately predicted some resonance frequencies for Ca2CuO3 [2], we preferred here the higher level theories which treat the electron correlation more extensively Particularly, the ¨ second-order Moller-Plesset perturbation theory (MP2), the density functional theory (DFT) with Beck’s style hybrid functional B3LYP and the configuration interaction singles (CIS) for excited state have been chosen These theories represent different approaches to the correction of HF energy For the MP2 level, the total electronic energy is the sum of the HF energy and the second-order correction, which is negative So the MP2 energy is always lower than the upper bound generated by HF estimation It turned out that the MP2 level correctly predicted the resonances for shorter spin chains but failed to converge for the longer ones In the DFT approach, the electron correlation is partitioned into the exchange and correlation parts Both parts depend on electron density and its gradient The hybrid functional B3LYP used in this work is the sum of LDA and HF local exchange 0.8EXLDA+0.2EXHF and Beck’s gradient-corrected exchange 0.72DEXB88 and a correlation part consisted of the local (Vosko–Wilk–Nusair) and gradientcorrected (Lee–Yang–Parr) correlation, 0.19ECVWN3+0.81ECLYP The DFT results were found to be correct for longer chains at relatively good convergence and moderate computational cost The CIS method accounts for the electron correlation in terms of the additional determinant: it constructs a new determinant from the ARTICLE IN PRESS 4124 H.N Nhat / Journal of Magnetism and Magnetic Materials 321 (2009) 4123–4127 HF one by replacing the occupied state by a virtual orbital This is equivalent to exciting one electron For shorter chains, the CIS proved to be sufficiently accurate but it failed for longer chains where the excitation, e.g singlet–triplet, involves many electrons Definition of model cluster The model chain (Cu–O)n is defined as a linear chain composed of the connected Cu–O units The chains are then put into the A2CuO3 unit cell to complete the 3D structure Thus the model cluster may be considered as the A2CuO3 structure with the bilayer (AO)2 removed The previous study showed that the phonon structure become stable within the range n ¼ 8C12 [5] For this limited segment, the electronic structure may be accurately obtained by the higher level theory such as PBE/631G(d) The calculation was carried out using the full ab initio software package Gaussian 03 [6] The structural parameters of Ca2CuO3 [4] were used The resulting density of state (DOS) and spin density for k ¼   are given in Fig The resemblance between the calculated total DOS and photoemission spectrum of Ca2CuO3 is obvious So the removing of the (CaO)2 bilayer did not seem to bring much change to the valence behaviour of Ca2CuO3 From the figures given we can observe that the highest peak, which occurs at À1.72 eV, has mainly d-character This depicts the density of the unpaired electron over the copper atoms (Cu2+ has (3d94s0) configuration) Since the total spin density shows the polarization, there is a portion of the unpaired d-electrons which expresses splitting of energy for the spin up and spin down state This behaviour may be associated with the creation of d-hole by transferring some electrons from 3dx2 Ày2 to 4s0 orbital, i.e with direct 3d–4s coupling between the two copper positions in two neighbour parallel chains Such 3d–4s interaction has been experimentally observed in Cu2O [7] (d-hole about 0.2eÀ) and has been estimated to be 0.11eÀ in Ca2CuO3 [8] Not all the unpaired d-electrons are therefore involved in the antiferromagnetic interaction along the chain The two shoulders of the main peak, one at À2.75 eV and another at À0.57 eV, have both p- and d-character and correspond to the bonding electrons (pd-hybridization) These electrons are located primarily on oxygen and have probably the parallel spin Near HOMO level (0 eV) the DOS is still dominated by the pd-electrons but some HOMO-LUMO excitations express the spin switching The area below À15 eV corresponds to the s-electron density and clearly shows the spin polarization The area above 4.5 eV has mainly p-character, but some portion of pd-hybridization also appears above 12.0 eV The single Cu–O unit There is theoretically only one optical mode in the single Cu–O unit To predict this value, the largest possible basis set 6311++G(3df, 3dp) (augmented split valence set with adding polarization and diffuse function) was used for MP2, DFT and CIS calculation The numeric value for this sole phonon mode is interesting just because it reflects the accuracy in the estimation of force constant for the Cu–O string The obtained results may be confronted with the photoemission data for the gas phase CuO but unfortunately the IR and far IR emission data exist only for a narrow band of photon energy from 1.5 to 2.0 eV [9] The calculated DOS for the Cu–O single unit (after optimization of geometry) is shown in Fig 1(b), the inset, and appears more complex than just to have a few peaks at 1.5–2.0 eV The obtained frequencies are given in Fig The data show that despite the structural simplicity, the calculated frequencies differ from each other for all level theories While the HF and MM approximation failed to predict any of the observed frequencies, the MP2 level offered a good estimation for one first-order mode at 470 cmÀ1 Fig The total DOS (a) and p- and d- partial DOS (b) for Cu–O strings (in structural packing of Ca2CuO3) as obtained from the PBE/6-31G(d) level theory The inset in (a) shows the photoemission spectrum of Ca2CuO3 as excited by a 325 nm laser and the inset in (b) shows the DOS for the single Cu–O unit after geometry optimization ˚ Fig Theoretical frequencies for a sole optical mode in single Cu–O unit (1.889 A) as obtained from various level theories ARTICLE IN PRESS H.N Nhat / Journal of Magnetism and Magnetic Materials 321 (2009) 4123–4127 The CIS also provided good approximation for two IR-active modes, 450 and 530 cmÀ1, which were seen in Ca2CuO3 [4,5] The most consistency upon the substitution of various wave functions seemed to be achieved by the DFT calculation using B3LYP hybrid functionals For the larger basis sets (DGDZVPDGauss double zeta valence polarization, LANL2DZ-Los Alamos 4125 double zeta and 6-31++G(3df, 2dp), the results were concentrated within a limited range from 380 to 480 cmÀ1 Two important firstorder modes in Ca2CuO3 (435 and 470 cmÀ1) fall within this range Therefore, we preferred the DFT/B3LYP level theory with 6–31 G and the double zeta basis sets LANL2DZ, DGDZVP and SDD, for the rest cases with n41 Fig Optical modes in the double unit (Cu–O)2 The measured data are taken from Ca2CuO3 [5] Table Optical modes, frequencies and Raman activities for the singlet spin chains with varying lengths n Length (nm) Highest activity Stretching Cu–O Perpendicular movement Boundary oxygen Symmetric movement Asymmetric movement Level theory 16 5.9 35000 409 457 641 14 5.1 43000 395 590 616/640 700000 333000 120000 110700 28000 30000 6800 5000 13600 135 640 290 1160 687 80 100 64 156 386 151 400 360 410 360 411 355 412 356 361 422 357 377 368 385 423 368 364 352 350 383 442 249/440 480 247/440 490 419 507 431/473 240/454/484 545 458/516 240/488 225/471/483 486/536 423 641 507 641 506 639 547 640 546 504 598/638 542 535/572 505 570 613 528 12 12 10 10 8 6 4 4 2 2 2 4.3 4.3 3.6 3.6 2.8 2.8 2.1 2.1 2.1 1.3 1.3 1.3 1.3 1.3 0.6 0.6 0.6 0.6 0.6 0.6 189 168 94 85 87 190 168 168 165 200–270 211 199 200 218 220/285 209 234 220 195/242/281 478 474 474 552 B3LYP/LANL2DZ B3LYP/LANL2MB MPW1PW91/ LANL2DZ B3LYP/6-31G B3LYP/LANL2DZ B3LYP/6-31G B3LYP/LANL2DZ B3LYP/DGDZVP B3LYP/LANL2DZ B3LYP/DGDZVP B3LYP/6-31G B3LYP/LANL2DZ B3LYP/DGDZVP B3LYP/6-311G(d) B3LYP/6-31G B3PW91/SDD B3LYP/LANL2DZ B3LYP/DGDZVP B3LYP/6-31G PBE/6-31G MPW1PW91/6-31G B3LYP/6-31++G ARTICLE IN PRESS 4126 H.N Nhat / Journal of Magnetism and Magnetic Materials 321 (2009) 4123–4127 Fig Effect of spin coupling in two doublet chains (a) and the redistribution of spin density in the triplet excitation of a longer singlet chain in the presence of a shorter doublet chain (b) Fig The development of intrinsic modes according to the chain length The graphs are re-scaled for clarity since in comparison with the activity arising from a chain having 12 Cu–O units the activities from the rest are very small The Raman activity of a symmetric mode developed exponentially according to chain length until n ¼ 12 The double Cu–O unit The optical modes in the chain (Cu–O)n ¼ with two Cu–O units contain four IR and two Raman-active modes (Fig 3) At UB3LYP/6-311++G(3df, 3dp) level theory (with largest available wave function basis set and separate treatment for spin up and down) the obtained IR-active modes include: (i) an asymmetric movement of both oxygen and copper in the perpendicular direction to the chain (150–160 cmÀ1), (ii) a symmetric movement of oxygen in static host lattice of copper atoms in the perpendicular direction to the chain (240–265 cmÀ1), (iii) a movement of a boundary oxygen along the chain (364 cmÀ1) and (iv) a movement of a middle oxygen (540 cmÀ1) along the chain Except the first mode, all other ones were observed in the Ca2CuO3 system [1,4,5] The modes obtained by the B3LYP/ DGDZVP setting are quite similar but a systematic shift to lower frequencies (about 15 cmÀ1) was seen This was probably caused by an additional gain in bonding energy when modeling with the 6-311++G(3df, 3dp) basis set which contains a more complete diffuse functions The inclusion of diffuse function, however, failed to bring the accurate results for the Raman-active modes as the B3LYP/6-31G(d) and UB3LYP/6-311++G(3df, 3dp) settings did not reveal the highly active mode at 470 cmÀ1 This mode corresponds to a symmetric movement of the middle oxygen and its activity grows very fast at high n On the other hand, with no diffuse function added, all functionals, PBE1, MPW1PW91 and B3LYP, showed the excellent matches with experimental data for 235, 280 and 470 cmÀ1 modes The 235 and 280 cmÀ1 modes are associated with the perpendicular movement of oxygen to the chain direction The calculation also did not reveal the 435 cmÀ1 mode (except MP2/DGDZVP), which is also composed of the symmetric movement of the middle oxygen along the chain Since in (Cu–O)2 the symmetric and asymmetric modes cannot be distinguished from each other, the activities at 635 and 670 cmÀ1 are absent It is important to note that by changing the chain length from one to two Cu–O units (i.e., from 0.19 to 0.57 nm), the number of modes increased seven times and the Raman activity has grown more than ten times As we show below, the Raman activity of the symmetric mode (435, 470 cmÀ1) grows exponentially with the chain length and reaches maximum value at n ¼ 12 (i.e., at 4.3 nm) Optical modes in nanoscale chains For the singlet (even n) spin chain with length varying from 1.3 (n ¼ 4) to nm (n ¼ 16), the optical excitation was studied by B3LYP/LANL2DZ and B3LYP/DGDZVP level theories The results are summarized in Table and the graphs are shown in Fig The ARTICLE IN PRESS H.N Nhat / Journal of Magnetism and Magnetic Materials 321 (2009) 4123–4127 modes may be addressed as follows There are two basic modes, a symmetric mode (440, 470 cmÀ1) and an asymmetric one (505, 545, 640 cmÀ1) Both modes are associated with vibration of the middle oxygen along chain direction in static host lattice of copper atoms The activity of a symmetric mode grows very fast when the chain length increases and this mode becomes dominated at 4.3 nm (n ¼ 12), then its activity decreases This development has several consequences First, it explains why at 20 nm length, the spin chains in Ca2CuO3 [1,4,5] showed only weak first-order modes (235, 280, 435, 470, 630 and 670 cmÀ1) The intensity arising from those modes are too low in comparison with the allowed Ag-mode phonons seen at 307 and 530 cmÀ1 Recall that, these Ag-mode bands are not originated in the spin chain but from the Ca movement along c-axis Second, one may expect the growing activity of the spin chain intrinsic modes when the chain length reduces towards 4.3 nm At this length, the activity from the symmetric mode would dominate over all other modes and the Raman scattering spectra would contain only two mentioned basic modes plus a mode arising from the vibration of boundary oxygen It is interesting to observe that, although the number of modes increases with chain length, the Raman scattering output contains only a few peaks at 4.3 nm (i.e., graph for n ¼ 12 in Fig 4) The reason for this disappearance of many intrinsic features can also be found in the dominating activity of a symmetric mode: the intensity of this mode is too high so that no other activity can be observed Looking at the energy level diagram and DOS (Fig 1), the symmetric mode corresponds to 0.05–0.06 eV excitation which occurs in the middle segment of the lowest LUMO bands The existence of a maximum intensity of symmetric mode at 4.3 nm and the growing number of modes at longer chains seem to have the same origin in the antiferromagnetic interaction along the spin 1/2 chain It is well-known that the quantum fluctuation of spin prevents the long-range antiferromagnetic ordering in an ideal (infinite) chain A random spin flip may propagate along the chain to opposite directions like two spinons whose attraction at characteristic length is mediated by the induced staggered field The spinon excitation has been experimentally observed in many copper-based quasi 1D Heisenberg spin 1/2 chain systems (see Ref [10]) The occurrence of spinon (and two-spinon or alias psinon) segments the spin chain into the 1D magnetic domains Thus the chains longer than 4.3 nm may behave like the assemblages of asymmetric divisions whose consequences are the reduction of activity of the symmetric mode and the appearance of additional vibrational modes Further aspect to this asymmetric segmentation of spin chain may be considered as follows In the real structure with defects, some chains with different lengths and spin states may occur close to each other so the spin coupling between them become an essential factor for redistribution of spin density over the neighbouring chains This scenario may be verified on the model 4127 clusters as shown in Fig The part (a) of this figure shows the spin density for the two chains having relatively small size (1.7 nm) with doublet ground state When the chains are enough separated, the two spin densities appear independent to each other but when the separation reduces towards 0.6 nm, the coupling effect becomes visible The part (b) illustrates another example: it compares the spin density in the 7.0 nm singlet chain in triplet excitation in the absence and presence of a neighbour 1.7 nm doublet chain As seen, the redistribution of spin density in the chain is well observed Within the frame of 1D Heisenberg model with known Bethe ansatz solutions, the spinons behave like the magnetic defects (observed ridges in the neutron scattering data) when the chain is subjected to the (uniform) outer magnetic field Although the local magnetic field created by one doublet chain is neither uniform nor strong enough to induce the spinon defects on the near-by singlet chains, the total field created by all doublet chains in the bulk crystal may be enough to cause such effect Conclusion The most exciting result from this investigation was that despite a variety of excitation states, there were only two basic modes with superior activity which could finally lead to the observation The reason for this may lie in the nature of the antiferromagnetic interaction along the chain which forces the segmentation of chain into local asymmetric units Each of these units is hosting its own vibrational modes Acknowledgment This work was funded by the National Foundation for Science and Technology Development, Vietnam (NAFOSTED), research project NCCB 2009 The authors are grateful to their support References [1] M Yoshida, S Tajima, N Koshizuka, S Tanaka, S Uchida, S Ishibashi, Phys Rev B 44 (1991) 11997 [2] N.N Hoang, T.H Nguyen, C Nguyen, J Appl Phys 103 (2008) 093524 [3] S.-L Drechsler, J Malek, M Yu Lavrentiev, H Koppel, Phys Rev B 49 (1994) 233 [4] N.N Hoang, D.C Huynh, T.T Nguyen, D.T Nguyen, D.T Ngo, M Finnie, C Nguyen, Appl Phys A 92 (2008) 715–725 [5] N.N Hoang, T.T Nguyen, H.V Bui, D.T Nguyen, J Raman Spectr 40 (2) (2008) 170–175 [6] M.J Frisch, G.W Trucks, H.B Schlegel, et al., GAUSSIAN 03, Revision B 03, Gaussian Inc., Pittsburgh, PA, 2003 [7] J.M Zuo, M Kim, M O’Keeffe, J.C.H Spence, Nature 401 (1999) 49 [8] D.-C Huynh, D.T Ngo, N.N Hoang, J Phys.: Condens Matter 19 (2007) 106215 [9] O Appelblad, A Lagerqvist, I Renhorn, Phys Scr 22 (1981) 603–608 [10] M.B Stone, D.H Reich, C Broholm, K Lefmann, C Rischel, C.P Landee, M.M Turnbull, Phys Rev Lett 91 (2003) 037205  ... maximum value at n ¼ 12 (i.e., at 4.3 nm) Optical modes in nanoscale chains For the singlet (even n) spin chain with length varying from 1.3 (n ¼ 4) to nm (n ¼ 16), the optical excitation was studied... at 307 and 530 cmÀ1 Recall that, these Ag-mode bands are not originated in the spin chain but from the Ca movement along c-axis Second, one may expect the growing activity of the spin chain intrinsic... maximum intensity of symmetric mode at 4.3 nm and the growing number of modes at longer chains seem to have the same origin in the antiferromagnetic interaction along the spin 1/2 chain It is