PHYSICAL REVIEW B 88, 104421 (2013)
Crystal-field states of Pr
3+
in the candidate quantum spin ice Pr
2
Sn
2
O
7
A. J. Princep,
*
D. Prabhakaran, and A. T. Boothroyd
†
Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, United Kingdom
D. T. Adroja
ISIS Facility, Rutherford Appleton Laboratory, STFC, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom
(Received 4 June 2013; revised manuscript received 2 September 2013; published 23 September 2013)
Neutron time-of-flight spectroscopy has been employed to study the crystal-field splitting of Pr
3+
in the
pyrochlore stannate Pr
2
Sn
2
O
7
. The crystal field has been parameterized from a profile fit to the observed neutron
spectrum. The single-ion ground state is a well-isolated non-Kramers doublet of Ŵ
+
3
symmetry with a large
Ising-like anisotropy, χ
zz
/χ
⊥
≈ 60 at 10 K, but with a significant admixture of terms |M
J
=±J 〉, which can
give rise to quantum zero-point fluctuations. This magnetic state satisfies the requirements for quantum spin-ice
behavior.
DOI: 10.1103/PhysRevB.88.104421 PACS number(s): 71.70.Ch, 78.70.Nx, 75.10.Jm
I. INTRODUCTION
Magnetic moments in the pyrochlores A
2
B
2
O
7
(A =
rare earth, B = Ti, Zr, Ir, Sn, ...) are highly geometrically
frustrated and known to show complex magnetic ground states
ranging from spin liquids to spin glasses and spin ices.
1,2
This
diversity of phenomena arises out of the interplay between
crystal field, magnetic exchange, and dipolar interactions.
When the dipolar interactions dominate over exchange in-
teractions in systems with strong Ising-like anisotropy in
the 〈111〉 directions, spin-ice ground states are realized, for
example, in Ho
2
Ti
2
O
7
and Dy
2
Ti
2
O
7
. Spin ice is a very
special magnetically frustrated ground state in which the
spins at the corners of each tetrahedron on the pyrochlore
lattice freeze into a two-in–two-out configuration, analogous
to the proton correlations in water ice. Interest in spin-
ice materials has burgeoned thanks to the prediction and
subsequent experimental detection of magnetic monopole-like
quasiparticles, which are the fundamental excitations of the
spin-ice ground state.
3–6
In recent years there has been growing interest in the
possibility of a state known as dynamic, or quantum, spin
ice.
7
Whereas the dynamics of classical spin ice slow down
and eventually freeze as T → 0, a quantum spin ice exhibits
significant residual transverse spin fluctuations, even at the
lowest temperatures. If the nature and strength of the exchange
interactions between spins is favorable, the fluctuations can
become correlated, allowing quantum mechanical tunneling
within the ice rules manifold of states. It has been predicted
that this particular spin liquid state could realize a fully
dynamical, lattice analog of quantum electromagnetism with
linearly dispersing magnetic excitations exactly analogous to
photons,
8,9
in addition to other exotic excitations.
10,11
Most discussion of possible real-world candidates for
quantum spin ice has concerned the titanates Tb
2
Ti
2
O
7
and
Yb
2
Ti
2
O
7
, both of which exhibit spin liquid features according
to several different experimental probes.
12,13
At the same time,
arguably the most promising candidates for observation of
strong quantum effects are pyrochlores containing Pr
3+
, be-
cause the large ionic radius and small moment of Pr
3+
enhance
the exchange coupling and reduce the nearest-neighbor dipolar
interaction relative to the heavy rare earths.
14–16
Another
distinction of Pr systems is that quadrupolar interactions are
expected to be important, and these could lead to new types of
complex ground states, including states with nontrivial chiral
correlations (e.g., in Pr
2
Ir
2
O
7
, Ref. 17).
Evidence has been found for a dynamic spin-ice state at
low temperatures in Pr
2
Sn
2
O
7
, in which the nearest-neighbor
dipolar interaction strength D ≈ 0.13 K is considerably
weaker than the estimated exchange energy J ≈ 0.9 K.
18,19
The zero-point entropy of Pr
2
Sn
2
O
7
is about 25% higher than
that of the Ho/Dy-based spin ices, which indicates that the
spins are much more dynamic than in a classical dipolar
spin ice. The dynamic nature of the spins was confirmed
by observations of the quasielastic width in high-resolution
neutron spectra, which revealed that significant relaxation
persists down to temperatures as low as 0.2 K.
The feature that allows Pr
3+
pyrochlores to exhibit zero-
point fluctuations is the presence of terms with |M
J
=±J 〉
in the ground-state wave function imposed by the crystal-
field interaction.
14,16
Susceptibility measurements suggest that
the crystal-field ground state of Pr
3+
in Pr
2
Sn
2
O
7
is a non-
Kramers doublet with strong Ising-like single-ion anisotropy,
and the measured low-energy neutron spectra indicate that the
first excited state is about 18 meV above the ground state.
However, until now there has been no direct determination of
the crystal-field interaction in Pr
2
Sn
2
O
7
, or indeed in any other
Pr-containing pyrochlore.
In this work we used time-of-flight neutron inelastic
scattering to measure the spectrum of single-ion excitations
of Pr
3+
in Pr
2
Sn
2
O
7
up to 500 meV. We use a detailed model
of the Pr single-ion states, including intermediate coupling and
J mixing, to determine the single-ion Hamiltonian and hence
to calculate the magnetic properties. The analysis shows that,
as expected, the ground state is a non-Kramers doublet with a
strong Ising-like anisotropy, and that it contains a significant
admixture of terms with |M
J
=±J 〉. The results reinforce the
view that Pr
2
Sn
2
O
7
is a strong candidate for quantum spin ice.
II. EXPERIMENTAL DETAILS
Polycrystalline Pr
2
Sn
2
O
7
and Y
2
Sn
2
O
7
samples were
prepared by standard solid-state synthesis techniques as
104421-1 1098-0121/2013/88(10)/104421(5) ©2013 American Physical Society