# Dynamics of the two-dimensional dimer system

###### Abstract

Inelastic neutron scattering was used to study a quantum
*S*=1/2 antiferromagnetic Heisenberg
system—Bis(2-amino-5-fluoropyridinium) Tetrachlorocuprate(II). The
magnetic excitation spectrum was shown to be dominated by long-lived
excitations with an energy gap of =1.07(3) meV. The measured
dispersion relation is consistent with a simple two-dimensional
square lattice of weakly-coupled spin dimers. Comparing the data to
a random phase approximation treatment of this model gives the
intra-dimer and inter-dimer exchange constants =1.45(2) meV and
=0.31(3) meV, respectively.

###### pacs:

75.10.Jm, 75.50.EeQuantum spin liquids, where strong zero point fluctuation destroy magnetic long-range order even at =0, have been a subject of intense studies for over two decades. Of particular interest are two-dimensional systems of this type. In addition to their relevance to layered superconducting cuprates,Anderson87:235 ; Lee06:78 they are of interest in the context of field-induced quantum phase transitions,Sach99 ; Thier08:4 where, to date, most experimental work was performed on either spin chainsStone03:91 ; Hong09:80 and ladders,Masuda06:96 ; Hong10:105 or 3D systems.Stone02:65 ; Ruegg03:423 Several 2D spin liquids including ,Sebastian06:441 ,Gaulin04:93 and PHCCStone01:64 have been studied in great detail, but show rather complex behavior due to geometrically frustrated interactions.

Recently, a new quantum antiferromagnet Bis(2-amino-5-fluoropyridinium) Tetrachlorocuprate(II) (5FAP-CuCl, for short) has been discovered and characterized by bulk measurements.Li07:46 The ground state is a spin liquid. The
energy gap to the lowest excited (triplet) state was determined to be 0.97(4) meV from the high-field magnetization data. 5FAP-CuCl () crystallizes in a monoclinic space group *P*2/c with lattice constants =6.926(7) , =21.73(2) , =10.911(10) , and =100.19(2). There are four inequivalent Cu atoms per unit cell as shown in Fig. 1(a). In view of crystal structure, 5FAP-CuCl consists of anion layers separated by organic cations. Those anions form zigzag
layers which are stacked along the the crystallographic *a* axis as shown in Fig. 1(b).
Fig. 1(c) shows the arrangement of anions and organic cations viewing from the *a* axis. The yellow and purple lines indicate the most plausible super-exchange
interactions that could be carried across Cu-Cl-Cl-Cu bonds. The former involves the shortest Cl-Cl contact with a separation of 3.66 , while for the latter the relevant Cl-Cl distance is 4.07 . One can expect inter-layer coupling along the *a* axis to be exponentially weak due to a much larger Cl-Cl separation 5.07 . Even with this in mind, it is not obvious which in-lane interactions are the most relevant. To better understand the spin Hamiltonian for 5FAP-CuCl, in this paper, we report inelastic neutron scattering (INS) measurements of magnetic
excitations and their dispersion relation. We show that 5FAP-CuCl can be
adequately described by a minimal model that is equivalent to a
square lattice of weakly interacting antiferromagnetic
dimers with no geometric frustration.

Small size fully deuterated single crystalline samples of 5FAP-CuCl were
prepared by the procedure described in Ref. Li07:46, .
About twenty single crystals (0.3 g total) were co-aligned
with a mosaic about 4 and used for INS measurements on the
cold-neutron triple-axis spectrometer IN14 at Institut
Laue-Langevin. Another assembly consisting of 20 single crystals
(0.5 g) with a total 5 mosaic was used for the
measurements on TASPSemadeni01:297 spectrometer at SINQ, Paul
Scherrer Institute. Horizontal beam divergences were given by
Ni guide-60’-open-open at IN14 and by -open-open-open at TASP.
Both experiments were performed with the
scattered wave vector fixed at 1.5 and Be
filter placed after the sample was used to remove higher order beam
contamination. The sample was oriented horizontally in a
(0,*k*,*l*) reciprocal lattice plane and cooled in a
standard He-4 cryostat.

The data were collected in a series of constant-*q*
scans at K along the high symmetry directions in the
(0,*k*,*l*) reciprocal plane. The inset of
Fig. 2(a) shows the raw data of a typical scan performed at
*q*=(0,0,-1.5), where a sharp resolution-limited peak
due to a magnetic excitation is clearly visible. The background
(shown as a solid line) was determined by fitting the data over the
range where no magnetic excitations are apparent, to a term linear
in energy, plus a resolution-limited Gaussian peak at zero energy
transfer. The latter accounted for incoherent elastic scattering.
Background-subtracted data at several wave vectors are shown in
Fig. 3. At certain wave vectors the observed signal has a
distinct double-peak structure as shown in Figs. 3(c) and
(d). Solid lines in Fig. 3 are fits to Gaussian profiles of
fixed widths, determined by instrumental resolution, as calculated
using the Cooper-Nathans approximation.Cooper67:23 Excitation energies (dispersion relation)
obtained from such fits are summarized in Fig. 2. The
spectrum appears to consist of two equivalent branches, an “optic”
and an “acoustic” one, although the former is not visible at the
majority of the wave vectors studied.

The most comprehensive method of determining the relevant exchange parameters in a gapped system with weakly dispersive excitations is based on the application of the first-moment sum rule for the magnetic dynamic structure factor.Hohenberg74:10 This procedure involves quantitative measurements of inelastic intensities as a function of wave vector and was successfully applied to the rather complex interaction geometry in CuHpCl.Stone02:65 Due to technical complications, associated with the sample fracturing upon cooling, and the data being collected in several disjoint experiments, in the present study we adopted a simpler approach. Only the measured excitation energies were analyzed. A parameterized model was refined to reproduce the experimental dispersion relations. The proposed minimal model is visualized in Fig. 4, which shows the exchange constants a single layer of magnetic Cu ions. The dimers are formed by the stronger antiferromagnetic bonds , which ensures a spin-singlet ground state and an energy gap. The dispersion of singlet-triplet dimer excitations is due to the weaker inter-dimer exchange constant . Topologically, this spin network can be viewed as a square lattice of dimers. Nevertheless, its actual realization in the crystal has two dimers per crystallographic unit cell. This leads to a folding of the Brillouin zone and the appearance of two excitation branches. As shown below, this minimal model seems to be fully consistent with experimental observations.

We can treat the weak inter-dimer coupling at the Random Phase Approximation (RPA) level.Haley72:5 This method has been applied to numerous spin systems, including ,Leue84:30 ,Sasago97:55 ,Zhel00:62 ,Stone08:77 and .Kofu09:102-1 In application to 5FAP-CuCl , we can treat two decoupled dimers in a unit cell as the basic spin cluster (tetramer). These tetramers will then form a Bravais lattice. With four sites in each tetramer, their bare susceptibilities are written as a matrix, the indexes enumerating the four spins in each unit:

(1) |

Here is the difference in population of the excited and ground states of the dimers. The weak interaction couples adjacent tetramers, but also dimers within each tetramer. Its effect is taken into account within the RPA. The corresponding RPA equation is written as:

(2) |

Here is the Fourier transform of exchange interactions due to . It is also written as a matrix:

(3) |

The dispersion relation is then readily obtained from the location of poles in :

(4) |

The solid and dashed lines in Fig. 2 are fits of this dispersion relation to the experimental data. A good agreement is obtained with meV and meV.

In summary, 5FAP-CuCl can be described as a simple 2D network of weakly interacting spin dimers, which is topologically equivalent to a dimer square lattice. The magnon dispersion bandwidth is 0.70(4) meV, smaller than gap energy meV, but still substantial. Future work will focus on the high-field study of quantum phase transition and critical phenomena in 5FAP-CuCl.

###### Acknowledgements.

One of the authors (AZ) would like to thank B. Normand for discussions. The work at ORNL was partially funded by the Division of Scientific User Facilities, Office of Basic Energy Sciences - Materials Science, Department of Energy.## References

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