From analysis of density of states, this mode may be associated with the excitation across the lowest LUMO bands with changing in spin state.. Table 1 Optical modes, frequencies and Rama
Trang 1Optical modes in nanoscale one-dimensional spin chains
a
Center for Materials Science, College of Science, Vietnam National University, 334 Nguyen Trai, Hanoi, Vietnam
b
Department of Physics, Chungbuk National University, Cheongju 361-763, Republic of Korea
a r t i c l e i n f o
Article history:
Received 4 January 2009
Received in revised form
1 August 2009
Available online 19 August 2009
Keywords:
Optical phonon
Spin chain
Ab initio
Nanoscale
a b s t r a c t
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 configuration interaction singles for excited states, density functional theory and second-order M ¨oller-Plesset perturbation In general, the number of modes increases with chain length due to growing asymmetry of atomic positions when chain exceeds 5 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 cm1) 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
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1 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 A2CuO3have 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 cm1[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
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 Hartree-Fock (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 M ¨oller-Plesset perturbation theory (MP2), the density functional theory (DFT) with Beck’s style hybrid func-tional 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 0.8EX
LDA+0.2EX
HFand Beck’s gradient-corrected exchange 0.72DEXB88 and a correlation part consisted of the local (Vosko–Wilk–Nusair) and gradient-corrected (Lee–Yang–Parr) correlation, 0.19EC
VWN3+0.81EC
LYP 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
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Journal of Magnetism and Magnetic Materials
0304-8853/$ - see front matter & 2009 Elsevier B.V All rights reserved.
Corresponding author Tel.: +84 98 300 6668; fax: +84 4 768 2007.
E-mail address: namnhat@gmail.com (Hoang Nam Nhat).
Trang 2HF 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
2 Definition of model cluster
The model chain (Cu–O)nis defined as a linear chain composed
of the connected Cu–O units The chains are then put into the
A2CuO3unit 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/6-31G(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 6 5 2 are given inFig 1 The resemblance
between the calculated total DOS and photoemission spectrum of
Ca2CuO3is obvious So the removing of the (CaO)2bilayer 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 1.72 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 3d1x2 y 2 to 4s0orbital, 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 0.2e) and has been estimated to be 0.11e 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 2.75 eV and another at
0.57 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 15 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
3 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 6-311++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 inFig 2 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 cm1
Fig 1 The total DOS (a) and p- and d- partial DOS (b) for Cu–O strings (in
structural packing of Ca 2 CuO 3 ) as obtained from the PBE/6-31G(d) level theory.
The inset in (a) shows the photoemission spectrum of Ca 2 CuO 3 as excited by a
325 nm laser and the inset in (b) shows the DOS for the single Cu–O unit after Fig 2 Theoretical frequencies for a sole optical mode in single Cu–O unit (1.889 ˚A)
Trang 3The CIS also provided good approximation for two IR-active
modes, 450 and 530 cm1, 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
(DGDZVP-DGauss double zeta valence polarization, LANL2DZ-Los Alamos
double zeta and 6-31++G(3df, 2dp), the results were concentrated within a limited range from 380 to 480 cm1 Two important first-order modes in Ca2CuO3(435 and 470 cm1) 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 3 Optical modes in the double unit (Cu–O) 2 The measured data are taken from Ca 2 CuO 3 [5]
Table 1
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
MPW1PW91/
LANL2DZ
6 2.1 6800 189 200–270 412 507 640 B3LYP/LANL2DZ
6 2.1 13600 168 199 361 240/454/484 504 B3LYP/6-31G
4 1.3 135 94 200 422 545 598/638 B3LYP/LANL2DZ
4 1.3 290 85 218 377 240/488 535/572 B3LYP/6-311G(d)
4 1.3 1160 87 220/285 368 225/471/483 505 B3LYP/6-31G
Trang 44 The double Cu–O unit
The optical modes in the chain (Cu–O)n ¼ 2 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 cm1), (ii) a symmetric
movement of oxygen in static host lattice of copper atoms in
the perpendicular direction to the chain (240–265 cm1), (iii) a
movement of a boundary oxygen along the chain (364 cm1) and
(iv) a movement of a middle oxygen (540 cm1) 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 cm1) 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 cm1 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 cm1 modes The 235 and 280 cm1 modes are associated with the perpendicular movement of oxygen to the chain direction The calculation also did not reveal the 435 cm1 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 cm1 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 cm1) grows exponentially with the chain length and reaches maximum value at n ¼ 12 (i.e., at 4.3 nm)
5 Optical modes in nanoscale chains For the singlet (even n) spin chain with length varying from 1.3 (n ¼ 4) to 6 nm (n ¼ 16), the optical excitation was studied by B3LYP/LANL2DZ and B3LYP/DGDZVP level theories The results are summarized inTable 1and the graphs are shown inFig 4 The
Fig 4 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.
Fig 5 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).
Trang 5modes may be addressed as follows There are two basic modes, a
symmetric mode (440, 470 cm1) and an asymmetric one (505,
545, 640 cm1) 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 cm1) The intensity arising from those
modes are too low in comparison with the allowed Ag-mode
phonons seen at 307 and 530 cm1 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 inFig 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
clusters as shown inFig 5 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
6 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
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