DSpace at VNU: Nuclear magnetic resonance in one-dimensional spin chains

These effects should break down the symmetry of copper elec-tronic structure along spin chain and induce a co-existence of dif-ferent electronic spin states: a singlet state for segments

Nuclear magnetic resonance in one-dimensional spin chains

Faculty of Technical Physics and Nanotechnology, UET, Vietnam National University Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam

a r t i c l e i n f o

Article history:

Received 11 October 2009

Received in revised form 3 May 2010

Accepted 6 May 2010

Available online 3 June 2010

Keywords:

NMR

Spin chain

Ab initio

Quantum

Computer

a b s t r a c t

This paper shows how different lengths, terminations and electronic spin states of one-dimensional chains of Cu–O affected the shieldings in copper nuclear magnetic resonance signals The calculation was performed for both periodic structure and model clusters of different size and shape The obtained results showed that there was relatively large splitting of63Cu resonance for each copper position in the chain

Ó 2010 Elsevier B.V All rights reserved

1 Introduction

There are several solid-state compounds in which the

copper-to-oxygen (Cu–O) bonding exhibits only one-dimensional (1D) or

quasi 1D geometry In the compounds of general form A2CuO3

(A = Sr, Ca) there was observed a strong antiferromagnetic coupling

between Cu 3d9electrons in the 1D Cu–O chains Numerous

stud-ies have been presented for the magnetic propertstud-ies of these

com-pounds [1–3], including the experimental nuclear magnetic

resonance (NMR) studies[4–6] In modern science and technology,

the 1D spin systems are important in several aspects due to their

close connection to fundamental phenomena such as high Tc

superconductivity (e.g in 2D isostructural La2CuO4), Bose–Einstein

condensation (observed in 2D and 3D spin systems), spin-charge

separation (reported for Sr2CuO3 [7]), and quantum coherence

There were also a number of quantum computer models that have

been proposed on basis of 1D spin chain systems[8–10] The

mod-el quantum computers usually utilize nuclear spin but in some

in-trigued proposals the electronic spin and its hyperfine interaction

with spin of nucleus were also considered[8] The 1D spin chains

are also (almost) ideal candidates for quantum bus, as they

repre-sent the realistic 1D channel for quantum transportation[9,10]

However, for a possible application in quantum devices, it is

important for 1D spin chain system (e.g A2CuO3) to possess

differ-ent resonance frequencies for each copper site, so to provide a

tar-geting manipulation of each nuclear spin by separate RF pulse

Unfortunately, the existing studies showed only a broaden feature

which has even disappeared below 12 K (near TN 5 K)[4–6] For

A2CuO3, the development of nuclear spin–lattice relaxation rate 1/

T1and Gaussian spin-echo decay rate 1/T2Gfollowed the field the-ory prediction, i.e 1/T1= const and 1/T2G/ 1/pT (for a quantum critical region T  J, J = 1300 K for Ca2CuO3 and 2300 K for

Sr2CuO3) The peak broadening may be considered for Cu site due

to large electric field gradients for63,65Cu nuclei and anisotropy

of magnetic field at long data collection time but the position-dependent splitting at Cu sites is not expected in a highly symmet-ric cubic space group Immm (to which A2CuO3 belongs) In this group the copper atoms occupy two crystallographically equiva-lent positions (0, 0, 0) and (1/2, 1/2, 1/2), which are also electroni-cally equivalent, so the calculation performed for the periodic structure (discussed in the next section) really predicted a single resonance line Apart from the single crystals, the real nanocrystal-lites on the other hand exhibit features that are not usual for the large periodic structures, i.e they are exposed to effects of edge topology, chain termination and surface deviation of electronic structure In 2D spin 1/2 Heisenberg antiferromagnets, for exam-ples, the edge effects induced a smaller magnetic response of an edge in comparison with the bulk susceptibility due to singlet for-mation at the edge[12] For the 1D spin chains, there are mainly two reasons for systems to be decomposed into non-equivalent chain segments: the geometry defects and the spin fluctuation These effects should break down the symmetry of copper elec-tronic structure along spin chain and induce a co-existence of dif-ferent (electronic) spin states: a singlet state for segments with even number of Cu sites and a doublet state for segments with odd number of the same (Cu2+cation in 3d9 configuration pos-sesses one unpaired electron, 3d1x2 y 2, with spin ±1/2) The variation

0927-0256/$ - see front matter Ó 2010 Elsevier B.V All rights reserved.

* Corresponding author.

E-mail address: namnhat@gmail.com (H.N Nhat).

Contents lists available atScienceDirect

Computational Materials Science

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c o m m a t s c i

of density of electrons over each Cu position should introduce a

po-sition dependent shielding and consequently splitting of NMR

sig-nals at Cu sites Unfortunately, the experimental resolution as

obtained for the powder Ca2CuO3[5]inhibit detailed investigation

of fine structure of NMR shielding The purpose of this paper is to

demonstrate, on computational basis (with largest wave function

basis sets), the existence of such splitting in a series of

nanocrystal-lites and chain segment models

2 Calculation procedure and strategy

A2CuO3 is a typical system with strong electron correlation

[11,14] It is a common notion that to explore all aspects of

elec-tron correlation, the largest possible basis sets should be used

and all electrons, including the core ones, should be treated As this

requirement can be fulfilled only for small fragments, we are

fre-quently approached the restriction of computational cost and

capability of available softwares Therefore, it is critical for the real

system to be properly modeled Usually, to achieve the consistent

results, a systematic exploration of model chemistries should be

taken into account Basically, the shielding tensorrji(N) for a

nu-cleus N is given as[19]:

rjiðNÞ ¼ @

2E

where E is energy, Bia component of external magnetic field, mj(N) a

component of induced magnetic moment for nucleus N To

calcu-late (1), the field-dependent basis functions should be used The

Gauge-Independent Atomic Orbital (GIAO) method implemented

in GAUSSIAN 03[15]utilizes the exponential function to shift the

gauge origin The NMR implementation in CASTEP[13], on the other

hand, uses the so-called Gauge-Including Projector

Augmented-Wave (GIPAW) approach of Pickard and Mauri [18] to describe

the behavior of wavefunction in magnetic field There is also

an-other possibility to calculaterji(N) through the evaluation of

in-duced first-order electronic current density J(1)(r)[19]:

rN

2E

@Bi@mNj

¼ 1

Bc

Z

drN

rN Jð1Þi ðrÞ

r3 N

" #

j

where i, j are the components of external magnetic field and

in-duced moment In the Continuous Set of Gauge Transformations

(CSGT) method by Keith and Bader [22], which is available in

GAUSSIAN 03 [15], J(1)(r) is evaluated by performing the gauge

transformation (via a shift in gauge origin d(r)) for each point in

space (see Eq (25) in Ref [19]) There are many other methods

for computing magnetic shieldings but in the limit of large basis

sets they should all give comparable results

In this work, the calculation for periodic structure was

per-formed by using CASTEP code[13]on basis of optimized geometry

at the same level theory Two symmetry settings were involved,

the Immm cell with original dimension and the P1 cell with

dou-bled b-axis The LDA and GGA schemes were investigated

sepa-rately and the dependence of shielding on applied functional and

wave function (i.e plane wave ei(k+G)r) cut-offs (2|k + G|26Ecut)

were also examined A limit of CASTEP code in evaluating NMR

spectra is that it cannot afford the calculation of shielding with

polarized spin and LDA + U functional, therefore the (electronic)

antiferromagnetic spin coupling (between Cu 3d1x2 y 2 electrons)

could not be properly set for the P1 cell and the corresponding

insulating ground state[11]was not correctly reproduced (in fact,

CASTEP produced a metallic ground state instead)

For cluster models, the calculation was performed using

GAUSS-IAN 03 software package[15] We examined the single chain

mod-els n(O–Cu) up to n = 14 (length = 5.3 nm) with and without

oxygen termination (n(O–Cu)–O and n(O–Cu)) Systematically, by insertion of chains between sufficiently large CaO bilayers (carry-ing both zeroed charge and spin) of different size we investigated the influence of model topology on resulted shieldings We showed that the values of shielding in A2CuO3 system were solely con-trolled by the electronic structure of 1D spin chain The obtained shieldings varied less than 5 ppm for the models with and without CaO bilayers attached Recall that, except for the part of density of states (DOS) with E > 2 eV above Fermi level, the electronic struc-ture of A2CuO3 has been demonstrated that it depends only on Cu–O bonding in 1D chain[11,20] It is important to mention here that unlike the situation in La2CuO4[16]where every copper posi-tion is symmetric in a square-planar configuraposi-tion, the 1D bonding

of copper in A2CuO3rules out all considerations of symmetry ex-cept for a case of centrosymmetric n(O–Cu)–O chains Theoreti-cally, the equal electronic structure for all copper atoms in 1D Cu–O chain might only be expected for the chains infinitely long, but not for the chains of finite size and different termination

With-in the size of clusters With-investigated so far, our results showed that the chain termination and spin state sufficiently influenced the shielding values We have observed that the antiferromagnetic ex-change of Cu 3d1x2 y 2 electron spins was not perfect in the long chains: the zones with alternating spin densities ±1 were only seen for limited segments, usually shorter than 1–2 nm The longer chains were often split into parts of different spin states which in-duced different spin densities over copper positions[20] There-fore, a position-dependent splitting of NMR shielding for A2CuO3

is not an accidental but a fundamental aspect of A2CuO3

nanocrystallites

In agreement with the results previously reported in Ref.[16]

and the analysis there given, we adopted the CSGT scheme with diffuse function added basis sets Where not particularly stated, the open-shell model (unpaired electrons) with Beck–Lee–Yang– Parr hybrid functional (UB3LYP) and the largest basis set 6-311++G(3df,3pd) were selected

For reference purpose, the shieldings for solid CuCl were also computed The consistent results were obtained using PBE func-tional with DNP basis set and at k = 6  6  6 For an experimental cell (a = 5.406 Å), the isotropicr(CuCl) was 364 ppm (±1 ppm var-iation on change of cut-off from 350 to 450 eV) but for an opti-mized cell (a = 5.417 Å),r(CuCl) = 379 ppm This result is almost half of 700 ± 200 ppm as obtained for the CuCl4 cluster used in Ref.[16](for this cluster, our calculation reported 798 ppm) As the experimental values might be given for the referencing CuCl

in solution, we have also computed the shielding for [CuCl4]3

cluster in water based solution using largest basis set 6-311++g(3df,3pd) after geometry optimization at the same level (optimized Cu–Cl bond length 2.462 Å vs non-optimized 2.341 Å) The final shielding (absolute, unreferenced) was

1129 ppm (the similar calculation for shielding in methanol re-sulted at 793 ppm for an optimized cell 2.334 Å) It is worth to note that the shieldings over the copper and chlor were quite indepen-dent: while the shielding for chlor held within 1113 ± 4 ppm regardless of Cu–Cl bond length and kind of solvent used (CCl4,

H2O or methanol), the shielding for copper varied from 793 (meth-anol) to 1129 (H2O) and 1391 (CCl4) ppm

3 Results and discussion

The calculation for the periodic structure of Ca2CuO3by CASTEP code with PBE functional resulted atr(Cu) = 3550 ± 1 ppm for the cut-off of 500 eV and variation of k-space from 16 to 32 points in Monkhorst–Pack grid The LDA functional also produced a similar result 3558 ppm but the WC (Wu-Cohen) functional arrived at

r(Cu) = 3581 ppm These values showed a sufficiently larger

shielding for Cu in Ca2CuO3when compared to that of atomic Cu (2402 ppm) Scaling with respect to water-based [CuCl4]3solution would yield a relative shift of 4679 ± 1 ppm A calculation for a cell with doubled b-axis showed a negligible variation of shielding for four different positions of Cu (<1 ppm) Therefore, we conclude that there is no position-dependent splitting of shielding for Cu in the crystalline Ca2CuO3 This conclusion may be easily understood

as the periodic boundary condition applied to wave function al-ways disregards possible edge effects The doubling of b-axis in-duced some asymmetry to unit cell but it was evidently not enough to cause sufficient variation of copper shielding We should mention here that the existing experimental data showed broaden features, so they did not confirm a single resonance line as pre-dicted by the periodic structure model

For the cluster models, before going into details, let us briefly summarize the calculation of shielding for single Cu atom, Cu1+ and Cu2+ cation using Single Gauge Origin (SGO), CSGT and its slight variation Individual Gauges for Atoms in Molecules (IGAIM) methods The calculation provided considerable results only for Cu

Fig 1 Shielding for Cu in binary system CuO The convergence of values at various

functionals and basis sets (with largest extension of polarization) (a); the

convergence of values at UB3LYP level with various basis sets, with and without

diffuse functions (+, ++), at different polarization extensions (b); Cu and O

shieldings with respect to Cu–O bond distance (c).

Table 1

Results for Cu–O: a comparison of functionals at largest polarization extension.

Fig 2 A typical cluster with Cu–O spin chain inserted between two CaO bilayers as static potential layers (a); the largest cluster with 409 atoms, of which 13 Cu and 18

O (31 in total) were included in full ab initio treatment (b).

Fig 3 Shielding values at Cu sites as obtained for various model clusters: the centrosymmetric O-terminated clusters n(O–Cu)–O (type (a)); the clusters n(O–Cu) with no O-termination (type (b)); the singlet clusters in triplet excitation and the

Table 2

Copper shieldings for spin chain clusters n(O–Cu) with and without O-termination at ub3lyp/6-311++g(3df,3pd) model level.

n Cluster [charge]/spin/model chemistry

O–Cu–O [2]_2  5(CaO) ONIOM/doublet

O–Cu–O–Cu–O [2]_2  5(CaO) ONIOM/singlet

3(O–Cu)–O [2]_2  11(CaO) ONIOM/doublet

3(O–Cu)–O [2]_2  11(CaO) ONIOM/doublet

and Cu1+(whose values are similar to the ones of Cu) With +, ++

diffuse functions [17] added, the calculated isotropic magnetic

shieldings (unreferenced) with spin polarized UB3LYP functional,

regardless of adding polarization functions, were 2402 and

2404 ppm for 6-31 and 6-311 basis set particularly The value for

Cu1+was 2398 ppm and for Cu2+was very large We confirmed

here again the deficiency of GIAO scheme for obtaining the copper

shielding as reported in Ref.[16] The calculation with GIAO

usu-ally finalized in diverged, large values

Fig 1a shows the results as obtained for a single unit Cu–O by

different functionals upon using six basis sets with largest

exten-sion of polarization functions (3df,3pd) added The numeric values

are listed inTable 1 As seen, the agreement was seen only for the

basis sets containing diffuse functions InFig 1b the development

of shieldings according to implementation of polarization

func-tions as obtained with UB3LYP functional for two groups of basis

sets, with and without diffuse functions, is demonstrated The

con-vergence of results is clearly seen in both groups when larger

por-tion of polarizapor-tion is embedded However, the group with diffuse

functions added showed less pronounced shielding effect with

re-spect to atomic Cu in comparison with the group without diffuse

functions This means that the diffuse functions induced more

cir-culating electronic current over the oxygen than the copper The

inspection of17O shielding values confirmed this scenario.Fig 1c

shows the dependence of shieldingron distance R For the CuO

binary system, the geometry optimization yielded R(Cu–O) =

1.756(5) Å, which corresponds to the ground state E =

1715.66484375 a.u The experimental bond length is around

1.4–1.54 Å for gaseous CuO, and 1.96 Å for solid state CuO (which

showed the excellent agreement with an optimized value of 2.0 Å

by Dmol3using PBE/DNP setting; the Cu–O distance in A2CuO3is

1.9 Å) At optimized geometry, the shielding for copper and

oxy-gen is 509 and 4617 ppm, respectively (around 5000 and

5000 ppm in gaseous CuO, 1000 and 4350 ppm in solid CuO)

As one may observe, whiler(Cu) approaches a value 2402 ppm

of atomic copper when Cu–O distance increases, the shielding for

oxygen approximates 6321 ppm which is a value for atomic O

in triplet ground state (singlet O produced 7158, molecule O2

2903 and O2radical 408 ppm) The dependence ofr(R) for63Cu

resonances is almost linear within R = 1.8–2.0 Å (correlation

coeffi-cient > 0.99) and may be approximated by the equation:

Particularly, r(Cu) develops from 125 (1.8 Å) to 1027 ppm

(2.0 Å) This result demonstrates how sensitive is the shielding

for copper on Cu–O distance: the changeDr(Cu) per 0.1 Å is almost

573 ppm

We now discuss the shieldings as obtained for the linear chains

of form n(O–Cu)AO (O-terminated) (a) and n(O–Cu) (no

termina-tion) (b) The chains of type (a) is centrosymmetric and has

nega-tive charge (2) whereas the chain of type (b) is asymmetric

with respect to inversion and has zeroed charge The chains 5(O–

Cu)–O (5 units Cu–O) is shown inFig 2a for illustration This model

cluster has 141 atoms, of which 17 (7 Cu and 10 O) were treated

fully ab initio; the rest 124 atoms (two CaO bilayers) were

simu-lated as static potential layers The 5(O–Cu)–O cluster might also

be extended to cover a total of 409 atoms, 31 of which were

in-cluded in full ab initio treatment (Fig 2b) From analysis of

symme-try one may expect a half number of individual resonance lines in

the chains of type (a) in comparison with that of the chains of type

(b) By spin state, the chains in each type are divided into two

groups, one group has singlet spin state (even number of Cu atoms:

Cu2+has 3d9electronic configuration with one unpaired electron)

and one has doublet spin state (odd number of Cu atoms) For

sev-eral singlet chains, we have also calculated the resonances for

trip-let excitation, and for some doubtrip-let chains the resonances for

singlet state (by removing one electron) The results are summa-rized inFig 3andTable 2 From the beginning of calculation, all model chains were inserted between two CaO bilayers as static po-tential layers The subsequent investigation, however, revealed that the inclusion of CaO bilayers had a negligible effect on final shieldings as the obtained results with and without CaO bilayers varied less than 5 ppm in many cases The possibility of exclusion

of CaO bilayers from full ab initio treatment allowed us to simulate

in real-time the longer chains up to n = 14 (length = 5.3 nm) with largest basis set 6-311++G(3df,3pd) Overall, the calculation for type (b) clusters and doublet chains converged more slowly, or might even fail as for n = 9, 11 and 14, in comparison with type (a) clusters and singlet chains At first observation, the dispersion

of shielding values is very large, ranging from 14,000 to 12,000 ppm Fig 2 shows that shieldings for doublet chains are left-shifted (more shielding) in comparison with that of singlet chains The same is observed for the shieldings associated with triplet excitation in singlet chains (left-shifted) and with deduced singlet state of doublet chains (right-shifted) The presence of un-paired electron seemed to cause sufficiently large shifts to stronger shielding at copper sites This effect can also be observed in type (b) chains with no oxygen termination In general, the type (b) chains possess exact n different values of shielding, whereas the O-terminated type (a) chains exhibit only n/2 + 1 values (doublet chains) or less (singlet chains) Due to centrosymmetry of type (a) chains, the distribution of resonance lines also follows the sym-metry: two symmetrically equivalent positions always possess the same shielding value It is noteworthy that in the doublet chains the equivalent positions also possess the same Mulliken spin

den-Fig 4 The statistical distribution of copper resonances (a) and its decomposition

sity, i.e the distribution of Mulliken spin density is also

centrosym-metric However, the interplay between the spin density and

shielding is rather poor There is not any systematic development

of shielding upon specific positions such as central, edge and

mid-dle positions, although it seemed that in the doublet chains the

central positions tended to show excessively large shielding values

For type (b) chains, the first copper position after the beginning

oxygen often were largely deshielded (r(Cu) > 10,000 ppm) In

Fig 4a we show the statistic composition of shielding values at

copper sites to simulate the real63Cu NMR spectrum of A2CuO3

The decomposition of contribution from different parts is shown

inFig 4b As seen, there are two maxima for n(O–Cu) chains, one

at 3500 ± 500 ppm and another at 500 ± 500 ppm Smaller

maxi-mum may also be observed at around 4500 ppm The characteristic

peaks in the spectrum for type (a) clusters (n(O–Cu)O) are

suffi-ciently different These systems showed five main features: a large

broaden central peak at 1500 ± 500 ppm and two equidistant side

groups, about 4000 ppm apart from the central peak Each of these

groups again consists of two bands equally placed about 2000 ppm

apart each other The peak positions are: 4500, 2500, 1500,

6500, and 8500 ppm In the top part ofFig 4b we show the

dis-tribution of shielding for the chains with singlet and triplet states

As mentioned in the first part of this paper, in the spin chain

sys-tems there is always possible a spin fluctuation which

conse-quently leads to a fluctuation of chain’s spin states This is a

dynamic aspect of spin chain system and may have some impacts

on NMR shieldings Four main features may be recognized from a

top graph inFig 4b: 3500, 1500, 1500 and 3500 ppm The peaks

are again equally spaced (by 3000 ppm) but their intensities are

different, so they do not show the characteristics of the side bands

In general, we may conclude that the presence of peaks at 2500,

1500 and 3500 ppm is quite essential for the system under

investigation (Fig 4a) Recall that the value 1500 ± 10 ppm is

the experimentally determined diamagnetic shielding for Cu[21]

The value 2402 ± 2 ppm is the isotropic shielding for metallic

cop-per reported by many groups, e.g in Ref [16] The last value,

3500 ppm, agrees very well with our result of isotropic shielding

for Cu in crystalline Ca2CuO3,r(Cu) = 3550 ± 1 ppm as discussed

above This comparison demonstrates that despite a wide variation

of Cu electronic structure along the spin chain, the 1D Cu–O

struc-ture preferably created three electronic environments which are

similar to that of metallic copper and crystalline Ca2CuO3 A few

experimental data available for these systems partly confirm that

the values 2500, 1500 and 3500 ppm are not computationally arbitrary

At last, we show that the value near diamagnetic shielding

1500 ppm may be obtained on basis of cluster model Using a large cluster, which contains a copper atoms centered in the envi-ronment of 30 other atoms (Cu and O) and embedded by 378 point charges (Fig 5), we obtained 1430 ppm.Fig 5b shows that the (Mulliken) spin density over copper sites in the central spin chain follows in general the antiferromagnetic arrangement, although the magnitudes of spin densities are not exactly ±1 (may be caused

by the problem of wave function projection onto the atomic orbital basis) The antiferromagnetic setting was shown to represent the correct ground state of A2CuO3system[11] As noted before, there

is not clear a dependence between spin density and shielding

4 Conclusion

The experimentally observed featureless63Cu NMR spectrum of nanocrystallites of spin chain system A2CuO3[5]is indeed com-posed of rich structures of resonances The calculation showed that there is a large dispersion of shielding values for copper, ranging from low 10,000 to high 10,000 ppm The values are very sensi-tive to structural differences, geometry of model clusters and spin states of the chains under investigation However, the characteris-tic signatures are centered at several values 1500, 3500 and

2400 ppm These values can be used to distinguish the A2CuO3 sys-tem from other copper oxide based syssys-tems Although it is still far from the stage in which we are able to manipulate individual cop-per nuclear spins, the present results clearly showed that relatively large position-dependent splitting of copper resonances exits This might be of interest for the future development of copper oxide based quantum spintronics

Acknowledgments

This work is supported in part by the Grant-in-Aid for Scientific Research from Asian Research Center, Vietnam National University Hanoi ‘‘Materials containing nanoscale Cu–O spin chains”

2009-2011 and by the research project code No 103.02.19.09 from Na-tional Foundation for Science and Technology Development (NAFOSTED) The authors would like to express the sincere thanks

Fig 5 Cu and O shieldings for largest cluster (a) and Mulliken spin density distribution at each position (b).

to both referees for valuable comments we received during the

preparation of manuscript

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