# Magnetic structure of Yb2Pt2Pb: Ising moments on the Shastry-Sutherland
Lattice

W. Miiller1, L.S. Wu1, M. S. Kim1, T. Orvis1, J. W. Simonson1, M. Gamża2, D.
E. McNally1, C. S. Nelson3, G. Ehlers4, A. Podlesnyak4, J.S. Helton5, Y.
Zhao5,6, Y. Qiu5,6, J. R. D. Copley5, J. W. Lynn5, I. Zaliznyak2, and M. C.
Aronson1,2<EMAIL_ADDRESS>1 Department of Physics and Astronomy, Stony Brook
University, Stony Brook, NY 11794  2 Condensed Matter Physics and Materials
Science Department, Brookhaven National Laboratory, Upton, NY 11973  3
National Synchrotron Light Source, Brookhaven National Laboratory, Upton, NY
11973  4 Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak
Ridge, Tennessee 37831  5 NIST Center for Neutron Research, Gaithersburg, MD
20899  6 Department of Materials Science and Engineering, University of
Maryland, College Park, MD 20742

###### Abstract

Neutron diffraction measurements were carried out on single crystals and
powders of Yb2Pt2Pb, where Yb moments form planes of orthogonal dimers in the
frustrated Shastry-Sutherland Lattice (SSL). Yb2Pt2Pb orders
antiferromagnetically at TN=2.07 K, and the magnetic structure determined from
these measurements features the interleaving of two orthogonal sublattices
into a 5$\times$5$\times$1 magnetic supercell that is based on stripes with
moments perpendicular to the dimer bonds, which are along (110) and (-110).
Magnetic fields applied along (110) or (-110) suppress the antiferromagnetic
peaks from an individual sublattice, but leave the orthogonal sublattice
unaffected, evidence for the Ising character of the Yb moments in Yb2Pt2Pb.
Specific heat, magnetic susceptibility, and electrical resistivity
measurements concur with neutron elastic scattering results that the
longitudinal critical fluctuations are gapped with $\Delta$E$\simeq$0.07 meV.

###### pacs:

75.25.-j,75.40.-S

Much attention has focussed on coupled spin-dimer systems, where frustrated
magnetic order can lead to spin liquid states rice2002 ; balents2010 . The
Shastry-Sutherland Lattice (SSL), consisting of planes of orthogonal dimers
with intradimer exchange $J$ and interdimer exchange $J^{\prime}$, is of
particular interest as it has been solved shastry1981  to show that the ground
state is ordered antiferromagnetically (AF) for
$J/J^{\prime}\leq(J/J^{\prime})_{C}$ and otherwise has a singlet ground state
with gapped magnetic excitations. Insulating SrCu2(BO3)2 has such a SSL dimer
liquid ground state miyahara2003 ; gaulin2004 ; kakurai2005 , where frustrated
exchange interactions create complex ordered superstructures at high fields
momoi2000 ; kodama2002 ; takigawa2013 , resulting in quantized plateaux in the
magnetic field H dependence of the magnetization M(H)sebastian2008 ; jaime2012
; matsuda2013 .

Most SSL systems are magnetically ordered, where the closure of the singlet
dimer gap yields a nonzero M(H) that makes AF order possible even for H=0
michimura2006 ; siemensmeyer2008 ; yoshii2008 ; matas2010 ; kim2008 . In
metallic Yb2Pt2Pb, Yb moments form the SSL (Fig. 1a) and AF order appears
below TN=2.07 K kim2008 . Magnetic susceptibility $\chi$(T) measurements
kim2013  find that J$\simeq$5 K and $J/J^{\prime}\simeq$1, and a field of only
1.23 T, applied along the (110) or (-110) dimer bonds, suppresses the H=0 AF
order, accompanied by quantized M(H) steps shimura2012 ; iwakawa2012 ; kim2013
that culminate in the saturation of M(H) at $\simeq$3 T. The magnetic
structures of SSL systems reflect the near-frustration of short-ranged
magnetic interactions, and in Yb2Pt2Pb, where strong Ising anisotropy confines
large and classical Yb moments to the SSL planes, the H=0 AF structure is
analogous to but distinct from those found in SrCu2(BO3)2 at high fields,
where quantum Heisenberg spins S=1/2 are perpendicular to the SSL layers.

Neutron diffraction measurements presented here demonstrate that the Ising Yb
moments in Yb2Pt2Pb have a remarkably complex AF structure, with a
commensurate superlattice of ordered stripes consisting of moment-bearing and
magnetically-compensated Yb-pairs with moments perpendicular to the (110) and
(-110) dimer bonds. The magnetic cell contains 200 Yb atoms, arranged in two
orthogonal sublattices (SLs) that can be separately polarized by magnetic
fields. The persistence of AF order in the surviving SL shows that the two
orthogonal bonds of the H=0 SSL are decoupled by field, leading to a unique
high field ground state. The AF order parameter in Yb2Pt2Pb is mirrored in the
temperature dependencies of the specific heat C/T and the temperature
derivatives of $\chi$(T) and the electrical resistivity $\rho$(T), suggesting
that the stripe-ordered ground state is preceded by the dynamical freezing of
longitudinal critical fluctuations that are gapped.

Crystals of Yb2Pt2Pb were grown from Pb flux kim2008 . Neutron scattering
experiments were performed using an incident neutron wavelength of 2.359 $\AA$
on the BT-7 triple axis spectrometer at the NIST Center for Neutron Research
(NCNR) lynn2012  on 5 g of powder prepared by triturating single crystals, and
on a 60 mg single crystal. x-ray diffraction measurements were carried out on
this same crystal on beamline X21 at the National Synchrotron Light Source
(NSLS), with an incident x-ray energy of 10 keV. Neutron scattering
experiments were conducted on a 6 g collection of $\sim$ 300 oriented single
crystals of Yb2Pt2Pbwith an angular mosaic perpendicular to the c-axis od
$\simeq$ 2 degrees, using an incident energy Ei=3.27 meV at the Disk Chopper
Spectrometer (DCS) at NCNR copley2003 , while Ei=3.32 meV was used at the Cold
Neutron Chopper Spectrometer (CNCS) at the Spallation Neutron Source.
Measurements of $\chi$(T) were carried out using a Quantum Design Magnetic
Phenomena Measurement System, while $\rho$(T) and C were measured using a
Quantum Design Physical Phenomena Measurement System.

Neutron diffraction measurements show clear evidence for AF order below
TN=2.07 K kim2008 . Fig. 1a shows the elastic intensity in the (HHL)
scattering plane at 1.5 K, obtained at CNCS on the collection of aligned
Yb2Pt2Pb crystals. The (002) and (112) nuclear peaks are present at all
temperatures, however satellite peaks emerge around (000) and (111) for $T\leq
T_{N}$. The AF wave vector $\bf{q_{1}}=(\delta,\delta,0)$ with $\delta=0.2\pm
0.02$ in reciprocal lattice units (rlu) is deduced from Gaussian fits to the
(1-$\delta$,1-$\delta$,1) satellites(Fig. 1b), indicating that AF order
involves a $5\times 5\times 1$ superstructure. The temperature dependencies of
the integrated intensities of the ($\delta$,$\delta$,1) and
(1-$\delta$,1-$\delta$,1) AF peaks have the appearance of order parameters
that are substantially broadened near TN (Fig. 1b,inset).

Our scattering geometry accesses a strip of the (HK1) plane (Fig. 1c), where a
quartet of AF satellites forms around both (001) and (111), signalling the SSL
plane retains fourfold symmetry in AF Yb2Pt2Pb. Thus, there are two AF wave
vectors, $\bf{q_{1}}$=(0.2,0.2,0) and $\bf{q_{2}}$=(-0.2,0.2,0) that define
two orthogonal SLs, each containing half of the Yb moments. The fourth
(1+$\delta$,1+$\delta$,1) satellite around (111) is much weaker than can be
explained by either the Yb magnetic form factor or self-absorption
corrections, and we will show below that this is captured by our magnetic
structure model. A 3 T field H$\|$(-110) saturates all antiparallel moments
kim2008 ; ochiai2011 , and Fig. 1d shows that only the AF peaks with
$\bf{q_{1}}\perp$ H survive. Their Ising character restricts the Yb moments to
be along or perpendicular to the (110) and (-110) dimer bonds. Both
configurations are allowed by symmetry, but our magnetic structure produces no
AF peaks for moments oriented along the dimer bonds. We conclude that the Yb
moments are perpendicular to their own dimer bonds, so the
$\bf{q_{1}}$=(0.2,0.2,0) SL is formed by dimers oriented along (110) with
moments along (-110) and for the orthogonal $\bf{q_{2}}=$(-0.2,0.2,0) SL the
dimers are along (-110) with moments along (110). The Ising character of the
Yb moments is accommodated in Yb2Pt2Pb by the formation of two orthogonal SLs,
as the magnetization anisotropy suggests ochiai2011 .

Figure 1:  a) Elastic scattering in HHL plane at 1.5 K. Inset: orthogonal Yb
dimers along (110) and (-110) form the SSL. b) Intensity of the (0.8,0.8,1) AF
peak at different temperatures. Solid lines are Gaussian fits. Inset:
temperature dependencies of the intensities of the (0.2,0.2,1) (black points)
and (0.8,0.8,1) (red points) AF peaks. Solid lines are guides for the eye.
Elastic scattering at 1.5 K in the (HK1) planes for magnetic fields H=0 (c)
and at 1.6 K for H=3 T (d).

The magnetic structure of Yb2Pt2Pb is obtained from the refinement of neutron
diffraction data obtained on a 5g powder using BT-7. The 10 K neutron powder
pattern (Fig. 2a) is well refined using the reported U2Pt2Sn -type tetragonal
structure pottgen1999 . The 0.5 K neutron powder diffraction pattern has
additional peaks corresponding to AF order (Fig. 2a), also evident in the
(HH0) scan obtained at CNCS on the aligned crystal array (Fig. 2b). These
(1$\pm\delta$,1$\pm\delta$,0) AF satellites arise from Yb moments that are
perpendicular to (HH0) and to $\bf{q_{1}}$. No scans were performed along
directions parallel to (-HH0), however we have added to Fig. 2b the second set
of AF satellites from the moments perpendicular to $\bf{q_{2}}$. The 1.5 K
(HH0) neutron scan (Fig. 2c) carried out on BT-7 using a 60 mg single crystal
shows two satellites flanking each of the (110), (220), and (330) nuclear
Bragg peaks, while their absence in x-ray scans carried out on the same
crystal (Fig. 2c) indicates no structural distortion occurs at TN.

Figure 2:  a) Neutron powder diffraction patterns at 10 K (top) and 0.5 K
(bottom), with indexed AF and nuclear peaks. Red lines are Rietveld fits (see
text). b) HK0 plane for T$\leq$TN=2.07 K, indicating nuclear $\bf{\bullet}$
and main AF satellites $\bf{q_{1}}$($\bullet$) and $\bf{q_{2}}$ ($\circ$), as
well as AF harmonics 3$\bf{q_{1}}$ ($\blacksquare$) and 3$\bf{q_{2}}$
($\square$). 1.5 K CNCS data are overplotted. c) Neutron (BT-7, 1.5 K, top)
and x-ray (X21, 1.8 K, bottom) HH0 scans of 0.06 g single crystal, with sample
holder peaks(stars), nuclear peaks (Filled blue points), and first order
(first harmonic) AF peaks (full and open red points)from our magnetic model.
d) residual $\chi^{2}$ for 1024 stripe configurations. Red circle encloses the
10 solutions with lowest $\chi^{2}$=0.064.

Since the magnetic structure of Yb2Pt2Pb is quite complex, involving two
different AF wave vectors and the formation of a 5$\times$5$\times$1 supercell
with 200 Yb moments, it cannot be determined solely from conventional
representation analysis. However, the Ising character of the Yb moments
greatly constrains our magnetic model. The anisotropy of M(H) indicates that
the Yb moments lie in the SSL planes, and have no components perpendicular to
either (110) or (-110). No canting and no modulation of the Yb moment
amplitude is possible. The crystal structure consists of Yb dimers in the SSL
planes with long bonds (3.889 $\AA$) and short bonds(3.545 $\AA$), potentially
having different Yb moments M1 and M2. The unit cell contains $z=0$ and
$z=1/2$ planes with both types of dimers, but they are staggered so that along
the c-axis a long dimer with Yb moment $M1$ ($z=0$) is always below a short
dimer with Yb moment $M2$ ($z=1/2$).

Yb dimers are the building blocks of the magnetic structure. Since the Yb
moments of the crystal-field doublet ground state are large and classical, we
consider here the four configurations of two distinguishable moments (Fig. 3),
i.e. $\uparrow\downarrow$, $\downarrow\uparrow$, $\uparrow\uparrow$, and
$\downarrow\downarrow$, and not the singlet-triplet states of two
indistinguishable quantum spins as in SrCu2(BO3)2. The fourfold symmetry of
the SSL plane requires two interleaving and orthogonal magnetic SLs in each
SSL plane (Figs. 3a,b). Inspired by the stripelike structures of high field
SrCu2(BO3)2 kodama2002 ; momoi2000 ; miyahara2000 , we construct the
5$\times$5$\times$1 AF superlattice of Yb2Pt2Pb from stripes, each with single
configurations of Yb pairs having dimer bonds along (110) and moments parallel
to (-110) for the SL with $\bf{q_{1}}$ = (0.2,0.2.0) and with dimer bonds
along (-110) and moments parallel to (110) for the perpendicular SL with
$\bf{q_{2}}$=(-0.2,0.2.0) (Fig. 3). Each SSL plane consists of two
perpendicular arrangements of dimer stripes, and both orthogonal SLs are
present in the $z=0$ and $z=1/2$ planes (Figs. 3a,b). The AF wave vectors
$\bf{q_{1,2}}$=($\pm$ 0.2,0.2,0) indicate there must be a net AF alignment of
the neighboring $z=0$ and $z=1/2$ planes, where a moment bearing
$\uparrow\uparrow$ dimer stripe in the $z=0$ plane (Fig. 3a), requires an
antiparallel moment bearing $\downarrow\downarrow$ dimer stripe in the
neighboring $z=1/2$ plane (Fig. 3b). Similarly, stripes of
$\uparrow\downarrow$ dimers in the $z=0$ plane must be stacked beneath stripes
of $\downarrow\uparrow$ dimers in the $z=1/2$ plane. For both SLs and in both
$z=0$ and $z=1/2$ planes, stripes based on each of the four dimer
configurations can be arranged in a total of 1024 different ways within the
$5\times 5\times 1$ AF supercell of Yb2Pt2Pb.

Figure 3:  Key: the four configurations of classical Yb dimers, on the
$\bf{q_{1}}$ (filled symbols) and $\bf{q_{2}}$ (open symbols) sublattices.
Symbols are elongated in the direction of the dimer bond, larger (smaller)
symbol sizes indicate long (short) dimers. The z=0 (a) and z=1/2 (b) planes
are formed by interleaving orthogonal $\bf{q_{1}}$ (left) and $\bf{q_{2}}$
(right) sublattices, to form the 5$\times$5$\times$1 AF supercell (solid
lines, center panels). Each sublattice consists of ordered sequences of
stripes consisting of single dimer types (different colors). c) A fragment of
the complete magnetic structure, showing AF stacking of short and long dimers
along the c-axis.

These 1024 stripe configurations were refined against the form factor
corrected 0.5 K neutron powder data(Fig. 2a) using Fullprof rodriguez2001 .
The computed diffraction patterns were subtracted from the 0.5 K diffraction
pattern to generate a residual $\chi^{2}$ for each configuration. Fig. 2d
shows that there are 10 configurations that have markedly smaller values of
$\chi^{2}$=0.064, consisting of the five permutations of the stripe sequence
($\uparrow\downarrow$,$\uparrow\downarrow$,$\downarrow\uparrow$,$\downarrow\uparrow$,$\uparrow\uparrow$)
and its mirror reflection
($\downarrow\uparrow$,$\downarrow\uparrow$,$\uparrow\downarrow$,$\uparrow\downarrow$,$\downarrow\downarrow$)(Figs.
3a,b). The fit gives the same Yb moment on every site, $M1=M2$ = 3.8
$\pm$0.2$\mu_{B}$/Yb, in good agreement with the 4 $\mu_{B}$/Yb of the Yb3+
$|\pm$7/2$>$ Kramers doublet ground state aronson2010 , and with the measured
saturation moment M110 kim2008 ; ochiai2011 . The simulated powder pattern
that corresponds to the configuration with the lowest $\chi^{2}$ agrees very
well with the 0.5 K powder pattern (Fig. 2a). Computed nuclear and AF peak
intensities are also in excellent agreement with the peak heights found in the
0.5 K (HH0) single crystal scan (Fig. 2c).

The refined AF structure for Yb2Pt2Pb is remarkably intricate (Fig. 3). Both
SLs in each SSL layer are net-moment bearing, containing individual
$\uparrow\uparrow$ or $\downarrow\downarrow$ stripes, spatially separated by
two pairs of magnetically compensated $\downarrow\uparrow$ or
$\uparrow\downarrow$ stripes for a total of five stripes per magnetic unit
cell. The moment-bearing stripes in neighboring SSL layers are arranged AF
along the c-axis, and a fragment of the 5$\times$5$\times$1 magnetic unit cell
is depicted in Fig. 3c, emphasizing the three-dimensionality of the magnetic
structure of Yb2Pt2Pb. The Ising character of the Yb moments requires two
orthogonal SLs in each SSL layer, where half of the moments are parallel to
(110) and the other half to (-110). Consequently, only the $\bf{q_{1}}$ SL,
which has dimer bonds along (110) and moments perpendicular to (110), will be
saturated by fields in the (-110) direction. The AF peaks associated with the
$\bf{q_{2}}$ SL are unaffected by field (Fig. 1d), as is its phase line
TN(B)=2.07 K ochiai2011 , demonstrating that the two SLs are independent.

The directions but not the magnitudes of the Yb dimer moments are modulated by
a square wave with a periodicity of 5 chemical unit cells. We observed
3$\delta$ harmonics in the (HH1) scans of the 6 g array of aligned crystals at
1.5 K (CNCS) and at 0.1 K (DCS) (Fig. 4a). The ratio of the
($\delta$,$\delta$,1) AF peak to the (3$\delta$,3$\delta$,1) harmonic is 0.122
in our model, agreeing with the experimental data at 0.1 K. The model
correctly estimates the intensities of the nuclear and main AF peaks at 1.5 K,
although the 3$\delta$ harmonics are much weaker than predicted, implying the
presence of longitudinal critical fluctuations that are quenched between 1.5 K
and 0.1 K. C/T and the temperature derivatives of $\rho$(T) and $\chi$(T)
(Fig. 4b) all show peaks at TN=2.07 K, as well as an additional broad feature
at $\simeq$0.8 K. $\rho$(T) was measured with the current along the c-axis and
with a magnetic field oriented along (110), and it is plotted with its
temperature derivative $\partial\rho$/$\partial$T in Fig. 4c for H=0, 1T, and
5 T. The ordering anomaly remains fixed at TN=2.07 K, indicating that the
$\bf{q_{1}}$ SL continues to affect $\rho$(T), even when the $\bf{q_{2}}$ SL
has been fully polarized by the 5 T field. While the general appearance of the
broad maximum in $\partial\rho$/$\partial$T changes slightly in field, it
remains centered at 0.8 K in all fields.

The picture that emerges from the data in Fig. 4 is that individual Yb dimers
form above TN, consistent with the intradimer exchange J$\simeq$0.6 meV
deduced from fits to $\chi$(T) kim2013 . As T$\rightarrow$TN, these dimers
assemble into a striped magnetic structure, accompanied by the gradual
suppression of critical fluctuations in favor of static order. These
longitudinal fluctuations freeze between 1.5 K and 0.1 K and the much stronger
temperature dependencies in C/T and the temperature derivatives of $\chi$(T)
and $\rho$(T) above 0.8 K are together indications that these critical modes
are gapped by $\Delta$E/k${}_{B}\simeq 0.8$ K, where kB is Boltzmann’s
constant.

Figure 4:  a) (HH1) scans at 1.5 K (top, CNCS) and 0.1 K (bottom, DCS). Blue
circles are nuclear intensities, filled (open) red points are fundamental
(first harmonic) AF peaks from our magnetic model, $\star$ marks Al peaks. b)
Temperature dependencies of the specific heat C/T, and the temperature
derivatives of $\rho$ and $\chi$, the latter measured in a dc field of 0.1 T
along (110). All data are normalized at TN=2.07 K to emphasize similar
temperature dependencies. c) $\rho$(T) and $\partial\rho$/$\partial$T at
different fields H$\|$(110). Vertical dashed lines in b) and c) are at TN=2.07
K and 0.8 K.

The magnetic structure of Yb2Pt2Pb is based on stripes of moment-bearing and
non-moment bearing configurations of two classical Yb moments. It is
reminiscent of the striped order found by nuclear magnetic resonance
experiments in SrCu2(BO3)2 kodama2002 ; takigawa2013 , where increasing fields
drive the SSL dimers based on Cu-based S=1/2 quantum spins through a sequence
of discrete patterns of singlet and triplet states, reflecting the frustration
of short-ranged exchange interactions inherent to the SSL momoi2000 ;
miyahara2000 . We find a much larger separation between the moment-bearing
stripes in metallic Yb2Pt2Pb, suggesting that here the frustration may involve
long-ranged interactions, perhaps mediated by the conduction electrons. The
low fields required for the suppression of AF order, for the cascade of
magnetization steps that suggest intermediate structures shimura2012 ;
iwakawa2012 , and for SL saturation in Yb2Pt2Pb may permit future
investigations of the intermediate field states using neutron scattering,
currently impossible in SrCu2(BO3)2 where fields approaching 30 T are required
to drive analogous modulated phases sebastian2008 ; jaime2012 ; matsuda2013 .

The strong Ising anisotropy that restricts the Yb moments to the (110) and
(-110) directions has far-reaching implications for the AF order in Yb2Pt2Pb.
Unlike SrCu2(BO3)2 and also TmB4 siemensmeyer2008 , where the moments are
perpendicular to the SSL planes with a single AF wave vector, two
perpendicular SLs are required in Yb2Pt2Pb to accommodate the Ising moments
ochiai2011 , which can be independently polarized by field. Once one SL is
fully polarized, the SSL motif of orthogonal dimers is destroyed in favour of
a square lattice of the remaining Yb dimers with spacing that approaches 8
$\AA$, although the same TN=2.07 K that was found at H=0 when both SLs order
is maintained. This is a very different AF state than has been envisaged for
SSL systems with Heisenberg spins shastry1981 , and it is possible that SSL
physics does not act alone in determining the underlying magnetism of
Yb2Pt2Pb.

We acknowledge valuable discussions with T. Sakakibara. Work at Stony Brook
University was supported by NSF-DMR-1310008. Use of the NSLS was supported by
the U.S. Department of Energy, Office of Science, Office of Basic Energy
Sciences, under Contract No. DE-AC02-98CH10886. This work utilized facilities
at NCNR that were supported in part by the National Science Foundation under
Agreement No. DMR-0944772. Research conducted at SNS was sponsored by the
Scientific User Facilities Division, Office of Basic Energy Sciences, US
Department of Energy.

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