INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 14 (2002) L319–L325

PII: S0953-8984(02)34996-8

LETTER TO THE EDITOR

Cu nuclear spin–spin coupling in the dimer singlet
state in SrCu2 (BO3 )2
K Kodama, J Yamazaki, M Takigawa1 , H Kageyama, K Onizuka andY UedaInstitute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa-shi,
Chiba 277-8581, Japan
E-mail: masashi@issp.u-tokyo.ac.jp (M Takigawa)

Received 25 March 2002
Published 18 April 2002
Online at stacks.iop.org/JPhysCM/14/L319
Abstract
We report results of nuclear magnetic resonance experiments in SrCu2 (BO3 )2 , aquasi-two-dimensional spin system with a singlet ground state. When magnetic
field is applied along the c-axis, each of the quadrupole-split Cu resonancelines splits further into four lines. The spin-echo intensity for some of the
split lines oscillates against the separation time between π/2 and π rf pulses.
These phenomena are due to strong nuclear spin–spin coupling mediated by
the electronic spin system, which exists only within a pair of nuclei. Thus
the results provide direct evidence for the dimer singlet ground state in thismaterial.

The discovery of an excitation gap and quantized magnetization plateaus in the quasi-twodimensional spin system SrCu2 (BO3 )2 by Kageyama et al [1] has stimulated a vast amountof experimental and theoretical work. As shown in figure 1, the Cu2+ spins (s = 1/2) in thiscompound form a planar network of dimers, which is identical to the Shastry–Sutherland (SS)model [2] when only the nearest-neighbour intra-dimer exchange (J ) and the second-nearest
inter-dimer exchange (J  ) interactions are retained.
Extensive studies on the SS model have revealed interesting novel aspects [3–11]. First,
the simple direct product of the dimer singlet is an exact eigenstate for any value of J  /J andis the ground state when J  /J is smaller than a certain critical value [2, 4]. In the opposite
limits, J  /J  1, the model reduces to the nearest-neighbour Heisenberg model on a squarelattice with the obvious Néel order. It was proposed that the two phases are separated by afirst-order transition at T = 0 near J  /J = 0.7 [4]. For SrCu2 (BO3 )2 , the exponential decreaseof susceptibility and Cu nuclear spin–lattice relaxation rate at low temperatures [1] indicatethat the ground state is a singlet with a finite energy gap for excited states. The magnitude of1

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L319

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Letter to the Editor

J'
J
Cu
B
O
Figure 1. Structure of the magnetic layer in SrCu2 (BO3 )2 (left-hand panel) and the SS model(right-hand panel).

the excitation gap  is determined to be 35 K from the inelastic neutron scattering [12] andthe electron spin resonance [13] experiments.
Secondly, excitations from the dimer singlet ground state in the SS model have extremely
localized character. Using perturbation expansion in J  /J , Miyahara and Ueda [4] found that
hopping of an excited triplet from one dimer to another is allowed only from sixth order. Avery small dispersion width of 0.2 meV for the magnetic excitations was indeed observed by
the neutron inelastic scattering experiments in SrCu2 (BO3 )2 [12]. The most striking feature of
SrCu2 (BO3 )2 is the plateaus in the magnetization curve under high magnetic field at fractional
values (1/8, 1/4 and 1/3) of the fully saturated magnetization [1, 14]. It is proposed that thesmall kinetic energy of the triplets and weak repulsive interaction between them are responsible
for the formation of superstructure of triplet dimers at these plateaus [6, 7], although no direct
experimental evidence has been obtained yet.
The nature of quantum phase transition in the SS model, however, is not well understoodyet. Possibilities have been discussed for intermediate phases between the dimer singlet andthe Néel state for a certain range of J  /J such as a plaquette-type singlet state [10] or ahelical spin order [3]. The exchange parameters appropriate for SrCu2 (BO3 )2 were estimatedas J = 85 K and J  = 54 K (J  /J = 0.64) by fitting the magnetic susceptibility data tonumerical simulation of the SS model [6]. Analysis of the energy of various excitation modes
has lead to slightly different values, J = 72 K and J  = 43 K (J  /J = 0.60) [11]. For suchparameter values, theories appear to agree that the ground state is the dimer singlet. Various
properties of SrCu2 (BO3 )2 known so far are indeed compatible with the dimer singlet groundstate, although it may be in close proximity to a quantum critical transition.
In this letter, we report the results of nuclear magnetic resonance (NMR) experiments
at 63 Cu and 11 B nuclei (both with nuclear spin 3/2) in SrCu2 (BO3 )2 . We observed fourfold
splitting of Cu NMR lines and sinusoidal oscillation of Cu spin echo intensity against the
separation time between the π/2 and π rf pulse. These phenomena are naturally explainedby strong nuclear spin–spin coupling acting only within a pair, and therefore provide direct
evidence for the dimer singlet ground state.
The NMR measurements were performed on a single crystal prepared by the travellingsolvent-floating-zone method and cut into an approximately cubic shape (2.1×2.3×1.9 mm3 ).
The applied magnetic field H did not exceed 8 T; therefore, the Zeeman energy was muchsmaller than the excitation gap. The NMR spectra were obtained from the Fourier transformof the spin-echo signal. SrCu2 (BO3 )2 has tetragonal structure with I 42m space group, in
which Cu(BO3 ) and Sr layers stack alternately [15]. The atoms in the magnetic Cu(BO3 ) layer
depicted in figure 1 are not strictly coplanar, and the only symmetry operation at either Cu

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H=8T
T=3K

B

Intensity

11

107.96 108.00 109.20 109.24 110.45 110.49

Frequency (MHz)Figure 2. The 11 B spectrum at T =3 K. The magnetic field is 7.9967 T parallel to the c-axis.

or B sites is the mirror reflection normal to 110. Then the direction normal to the mirror
plane coincides with one of the principal axes of the electric field gradient (EFG) and themagnetic hyperfine shift (K-) tensors; however, neither possesses axial symmetry. When themagnetic field H is along the c-axis, the quadrupolar and the magnetic shifts are identical for
all Cu (B) sites but the c-axis is not a principal axis of the EFG or K-tensors. For other field
directions, there are more than two inequivalent sites. For example, there are two Cu (B) sitesfor H  [110]. At one of these sites, H is normal to the mirror plane and thus along one of theprincipal axes of EFG and K-tensors.
Figure 2 shows the 11 B NMR spectrum at T = 3 K for H = 7.9967 T along the c-axis. The
spectrum consists of three lines split by electric quadrupole interaction, yielding the quadrupole
splitting 11 νcc = 11 Vcc e11 Q/2h = 1.25 MHz, where 11 Vcc is the cc-component of the EFG
tensor and 11 Q is the nuclear quadrupole moment. Other components of quadrupole splitting
are obtained from the spectrum for H  [110] as 11 ναα = 0.694 MHz and 11 νββ = 0.555 MHz,
where α is a principal axis and β is perpendicular to it.
The 11 K magnetic hyperfine shifts at B sites along three directions c, α and β are
determined from the position of the central line for the transition Iz = 1/2 ↔ −1/2 at H = 8 T.
There are good linear relations between the shifts and the magnetic susceptibility χ , as shownin figure 3. From the slope of the K versus χ plot, three components of the hyperfine coupling
tensor defined as 11 Aii = 11 Kii /(µB NA χii ) are determined as 11 Acc = −0.259, 11 Aαα =−0.202 and 11 Aββ = 0.115 in units of T /µB . The coupling constant 11 Aii represents the
i-component of the hyperfine field at B nuclei provided that each Cu has a uniform moment of
1 µB along the i-direction. We found that the dipolar field from Cu spins alone does not accountfor the anisotropy of 11 Aii , indicating sizable anisotropy of the transferred hyperfine interaction.
The temperature dependence of χcc is shown in the inset of figure 3. Since the susceptibility wasmeasured at 1 T and the magnetization is not linear in field when the temperature is much lower
than the gap, only the data above the peak temperature of χ are used in the K versus χ plot.
We now turn to the results at Cu sites. Figure 4 shows the 63 Cu NMR spectrum for thecentral transition at H = 8.001 T along the c-axis at T = 3.0 K. The spectrum consists offour equally spaced lines in spite of the fact that all Cu sites are equivalent for H  c. Similar
four-peak structure was observed for the quadrupole-split satellite spectra. The quadrupolar
splitting for 63 Cu nuclei was obtained as 63 νcc = 22.13 MHz. As will be explained shortly,

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Letter to the Editor

K (%)
11

Kβ

0
11

-0.05

-0.15

Kα

11

-0.1

0.001

4

χ (10-3emu/Cu mol)

11

0.05

3

H//c

2
1
0
0

100

200

300

T (K)

Kc
0.002

χ (emu/Cu mol)

0.003

Figure 3. The shifts at B sites are plotted against the susceptibility. The solid lines show fitting toa linear relation.

Cu-center

Intensity

63

A

91.4

T=3K
H=8T

B

91.6

C

D

91.8

92

frequency (MHz)Figure 4. The 63 Cu NMR spectrum for the central line at T = 3 K. The magnetic field is 8.001 T
parallel to the c-axis.

the fourfold splitting is due to the coupling of two Cu nuclear spins on the same dimer. The
magnetic shift is then determined from the average frequency of the four peaks. The observationof Cu NMR signal is limitted to T  4.2 K because of the short spin–spin relaxation time athigher temperatures. The hyperfine coupling constant at Cu nuclei 63 Acc was determined byplotting the shift for 63 Cu against the shift for 11 B, both taken at the same field and several
temperatures below 4.2 K as shown in figure 5. From the slope of this plot and the value of11
Acc determined above, we obtain 63 Acc = −23.76 ± 0.15 T/µB . Complete determinationof the EFG and K-tensors at B and Cu sites will be presented in a separate paper.
We found remarkable oscillation of the spin-echo intensity of the 63 Cu resonance as shownin figure 6. Here the spin-echo intensity of peak B of the central line is plotted against twice
the separation time τ between π/2 and π rf pulses. Such oscillation is observed only for peaks
B and C. Similar oscillation was observed for the satellite spectra, although at different peaks.
For the high-frequency satellite only peaks C and D show the echo oscillation, while for the
low-frequency satellite oscillation was observed for peaks A and B.

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63

Kc (%)

1.5

1

0.5-0.008-0.004110

Kc (%)

0.004

0.008

Figure 5. The magnetic hyperfine shift at Cu sites (63 Kc ) is plotted against the shift at B sites(11 Kc ). Both were taken at the same field, H = 8 T, along the c-axis in the temperature range
below 4.2 K.

163

Cu-center

T=3K
H=3.9T

Intensity

0.8
0.6
0.4
0.2
0

50

100

150

200

2τ (µs)Figure 6. The spin-echo intensity of peak B in figure 4 for the centre line of the 63 Cu NMR
spectrum is plotted as a function of 2τ , where τ is the separation time between π/2 and π pulses.
The data are taken at H = 3.9 T along the c-axis and T = 3 K.

The line splitting and the echo oscillation are both explained easily by Rudermann–Kittel–
Kasuya–Yosida-type indirect interaction between Cu nuclear spins mediated by the electron
spin system. The indirect coupling between two nuclear spins I1 and I2 is generally expressedas [16]
Hind = −az I1z I2z ,(1)

az = h̄ γN 1 γN 2 (gz Az ) χ12
2

2

(2)

where γN i is the nuclear gyromagnetic ratio for Ii and only the on-site hyperfine coupling
constant along the field direction Az is considered. The non-local electron spin susceptibilityχ12 describes the spin polarization at site 1 when a fictitious magnetic field is applied only tothe spin at site 2. At T = 0 this is expressed as 0|S2z |nn|S1z |0χ12 =+ c.c.,
(3)
En
n

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Letter to the Editor

where |0 is the ground state and |n is an excited state with energy En . In the case of an
isolated dimer with exchange J ,1
χ12 = − .
(4)2J
Equation (1) implies that local magnetic field acting on I1 is produced by I2z , which can takefour different values ±1/2 or ±3/2. This results in the fourfold splitting of the NMR line forI1 . Since I2 can be either of the two isotopes 63 Cu or 65 Cu, there should strictly speaking beeightfold splitting. However, both isotopes have spin 3/2 and the difference in γ is only 7%.
Thus they are not experimentally resolved. Assuming that the spectrum in figure 4 represents
the isotopic average for I2 , the value of az for the case when I2 is a 63 Cu nucleus is deduced
as az / h = 119 kHz.
It is well known that such a nuclear spin coupling gives rise to spin-echo oscillation [17]I (2τ ) = cos(az τ/h̄)(5)when the two spins are identical (like spins). Because of the quadrupole splitting, only those
63
Cu nuclei with Iz = ±1/2 behave as like spins when the central line for the transition
Iz = 1/2 ↔ −1/2 is being observed. This is why only peaks B and C show oscillation.
The selection rules for the satellite lines is explained in a similar manner. The data infigure 6 are fitted to a sum of oscillatory and non-oscillatory parts, both decaying exponentiallywith different time constants. The oscillation frequency is obtained as az / h = 119 kHz, inagreement with the value deduced from the splitting of the spectrum.
The fourfold splitting of the spectrum and the coherent spin-echo oscillation indicate that
nuclear spins are strongly coupled in pairs and the couplings between pairs are very weak.
The results thus provide direct evidence for the dimer singlet ground state in SrCu2 (BO3 )2 .
If one nucleus were coupled to many neighbours, for example as one would expect for theplaquette-type singlet state, the spectrum should consist of many lines, which with random
isotopic configuration would result in a single broad peak. Likewise, random superpositionof echo oscillation containing many frequency components would show rapid damping. The
long-lived coherent echo oscillation implies that the inter-dimer coupling if any is orders ofmagnitude smaller than the intra-dimer coupling. Similar phenomena have been observed also
for the chain Cu sites in the ladder–chain composite material Sr14 Cu24 O41 , where the dimer
singlet state is formed due to charge order in the one-dimensional chain subsystem [18].
For the case of an isolated dimer, the coupling constant az / h = 119 kHz corresponds tothe exchange constant J = 75 K. Here we have used equations (1), (2) and (4) with the values
Ac = −23.76 T/µB and gc = 2.28, the latter being obtained from the ESR measurements [13].
This value of J is remarkably close to the intra-dimer exchange estimated for SrCu2 (BO3 )2 .
Although J  /J is not a small number for SrCu2 (BO3 )2 , the spin correlation within a dimer
must remain almost unchanged from an isolated dimer.
In conclusion, 63 Cu and B NMR measurement has been performed on a single crystal of
SrCu2 (BO3 )2 . The values of the hyperfine coupling constant and the quadrupole splitting were
determined. The spectrum of Cu splits into four lines and the spin-echo intensity oscillates
against 2τ with the oscillation frequency equal to the line splitting. These are caused by the
nuclear spin–spin coupling mediated by the strong intradimer coupling of the electron spins.
These results provide firm evidence that the ground state is indeed the dimer singlet state.
This work was supported by Grant-in Aid for Scientific Research 10304027 from the JapaneseSociety of the Promotion of Science and Priority Area (A) on ‘Novel quantum phenomena intransition metal oxides’ and Priority Area (B) on ‘Field-induced new quantum phenomena inmagnetic systems’ from the Ministry of Education, Culture, Sports, Science and Technologyof Japan.

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[14] Onizuka K, Kageyama H, Ueda Y, Goto T, Narumi Y and Kindo K 2000 J. Phys. Soc. Japan 69 1016
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[17] Abragham A 1961 The Principles of Nuclear Magnetism (Oxford: Oxford University Press) p 497[18] Takigawa M, Motoyama N, Eisaki H and Uchida S 1998 Phys. Rev. B 57 1124