Quantum Magnetism in the New Pyrochlore Compound Pr2Zr2O7
Alison Pawlicki, Dr. Christopher Wiebe
Florida State University, NHMFL
Abstract: The pyrochlores, of chemical formula A2B2O7, have been the target of intense investigation for the last few decades due to a phenomenon called geometric frustration. The electronic spins on the
A-sites are located on the vertices of a corner-sharing tetrahedral pattern and cannot align in a conventional antiferromagnetic ground state in such a way as to simultaneously minimize the energy associated
with the competing interactions. The result is that many of these systems do not order at low temperatures and appear to violate the third law of thermodynamics with an enormous amount of residual
entropy. The new compound Pr2Zr2O7 has been synthesized at the National High Magnetic Field Laboratory to search for such exotic states. Single crystals have been grown using the floating zone image
furnace and x-ray diffraction. Low temperature susceptibility measurements and heat capacity experiments have been completed to learn about how the entropy is released in the limit of zero Kelvin.
Introduction
The main focus of solid state physics is on macroscopic properties of solid materials that
result from interactions between atoms. One emergent property is disorder that arises from
geometric frustration. Frustration is the inability of a material to find a single ground state
and instead has a highly degenerate ground state. The reason for this is because the geometry
of the crystal is such that any arrangement of spins on the lattice cannot simultaneously
satisfy all interactions. For instance, there is no possible configuration of four Ising spins on
the vertices of a tetrahedron where they will all align antiferromagnetically (see Fig. 1). The
size of the frustration parameter f, defined by the ratio of the Curie-Weiss temperature and the
expected mean field ordering temperature (Néel temperature) or θCW/TN, is used to determine
the extent of the frustration in a material.
Figure 1. Antiferromagnetic
interacting spins arranged on the
vertices of a tetrahedral. Because of
the geometry, it is not possible to
satisfy the interactions between all of
the spins and thus it is said to be
geometrically frustrated.
Figure 2. The cubic
pyrochlore crystal
structure is composed
cored sharing
tetrahedra and its
geometry leads to
frustration in materials.
The 3D structure is
pictured in green and
the 2D structure is
shown as a shadow in
black.
In this study, we focus on a pyrochlore compound, Pr2Zr2O7. Its crystal structure is
composed of corner sharing tetrahedra and by extending the instance explained above, it is
clear that this structure exhibits geometric frustration (see Fig. 5). An interesting feature of
the heat capacity, which will be discussed further later on, is that there may be a quantum
phase transition in applied magnetic fields. Due to the uncertainty principle, quantum
fluctuations exist in materials, and these effects can even induce a phase transition at zero
Kelvin. This is analogous to thermal fluctuations causing a phase transition, such as from
water to ice, but the power laws governing the order parameters at quantum phase transitions
is still a matter of debate.
The heat capacity was measured at constant pressure using a PPMS (Physical Property
Measurement System, Quantum Design). To do this, PPMS uses an adiabatic relaxation
process where a known heat pulse is added to the system, and the corresponding temperature
change is noted. The magnetic susceptibility was measured using a SQUID
(Superconducting Quantum Interference Device) where small changes in the magnetic field
gives rise to resistance that is detected.
Results and Analysis
X-Ray Diffraction: The x-ray diffraction pattern of crystallized Pr2Zr2O7, minus the
background, is shown in Fig. 4. The background data was subtracted and the raw data was fit
by varying parameters, like scale factors and peak width, in a program called FullProf. This
same program was able to locate the Bragg peaks, marked with black lines in Figure 4. These
peaks are the points at which an atom is located and using a transformation, or in this case, a fit
to a previously known diffraction pattern, the distance of the atoms can be found. After the
data fit, the lattice parameters a, b, and c are determine to be 10.7146 ± 0.00029 Å and
confirmed to have a cubic pyrochlore crystal structure of single phase (Fig. 5).
Figure 4. X-ray diffraction
pattern for crystallized Pr2Zr2O7.
The background (purple) was
subtracted from the raw data and
was fit to a previously known
diffraction pattern (black) to
determine the lattice parameters
a, b, and c. From this pattern, it
is confined that Pr2Zr2O7 has a
cubic pryochlore crystal structure
of single phase.
The rods were placed in the center of an image furnace where one rod was positions just
above and the other rod was positioned just below an area of focused light (see Fig. 3). The
light was produced by a special bulb and focused by two concave gold mirrors at either end
of the furnace. The rods were slowly rotated in oppositions and brought close together so that
the top rod melted into the lower. The material was gradually cooled to form a crystal.
Powder x-ray diffraction was used to determine the structure of the crystal.
Figure 3. Left: Side view of the Image Furnace. The clear center section is where the light is focused and
where the rods were placed. Right: End view of the Image Furnace. This is where one of the concave gold
mirrors, used to focus the light, is located.
Figure 7. The entropy per mole of Pr2Zr2O7
does not reach zero in the limit of zero
Kelvin. Instead, there is a large residual
entropy. This seems to be contrary to the
Third Law of Thermodynamics.
Pr
Zr
Experimental Procedure
Two rods (10g total) of Pr2Zr2O7 powder were made by a solid state reaction between Pr2O3
and ZrO3.
Pr2O3 + 2ZrO2 → Pr2Zr2O7
Figure 6. The heat capacity of Pr2Zr2O7 fits Einstein’s model above 30 K and deviates below 30 K.
There appears to be a second order phase transition at about 4 K . It is clear from the graph on the
right that the heat capacity does not reach zero in the limit of zero Kelvin, which is seems to
contradict the Third law of Thermodynamics.
O
Figure 5. Unit cell of crystallized Pr2Zr2O7. The lattice parameters a, b, and c are the dimensions of the
cube. The pink bonds between the atoms forms a corner-sharing tetrahedral pattern which is
characteristic of a cubic pryochlore crystal structure. Left: All bonds are displayed where the yellow
bonds are between the zirconium oxide molecules and the pink bonds are between the praseodymium
atoms . Right: Only the bonds between the praseodymium atoms are displayed to bring attention to the
similarities in figure 2.
Heat Capacity: At high temperature (above 30 K), the heat capacity of Pr2Zr2O7 was fit to
Einstein's heat capacity model to see how it deviates at low temperature. These deviations are
magnetic in origin, and either due to crystal-field transitions in Pr3+ ions, or due to magnetic
short-ranged ordering. The crystal-field hypothesis was tested by fitting our data to a Schottky
anomaly feature, but with no success. Just above 25 K, the heat capacity departs from the
Einstein model and there seems to be a feature at 4 K. It is clear from Fig. 6 and Fig. 7 that there
is a magnetic component to the heat capacity at low temperatures. We believe that this could be
due to some sort of magnetic short-ranged ordering.
Figure 8. The inverse of the magnetic
susceptibility is linear which is characteristic of
paramagnetism. The data was fit to the CurieWeiss Law to determine the Curie constant and
Curie-Weiss temperature.
Magnetic Susceptibility: The magnetic susceptibility of Pr2Zr2O7 is what was expected
and this material is confirmed to be paramagnetic. At high temperature (300 K to 100 K),
the susceptibility was fit to the Curie-Weiss Law with an adjusted R-squared value of
0.99913. From this, the Curie constant is found to be 2.678 ± 0.002 and the Curie-Weiss
temperature is found to be -4.2 ± 0.4 K. The Curie constant is consistent with a moment
of 3.27 Bohr magnetons on the Pr3+ site (which is close to the literature value of 3.5).
Using the Curie-Weiss temperature and 1.8 K as an upper bound on the Néel ordering
temperature, the lower bound on the amount of frustration f is calculated to be 2.33.
Conclusions
The new pryochlore compound Pr2Zr2O7 was successfully made into a crystal of single
phase. Because of the geometry of the crystal structure, Pr2Zr2O7 is frustrated and exhibits
novel properties. From the heat capacity measurements and fit to Einstein's model, it is
clear that there is interesting magnetic behavior at low temperatures. Since the data cannot
be fit to a Schottky anomaly feature, it is concluded that the magnetic behavior is not due
to crystal-field transitions. It seems that there is a phase transition at about 4 K where the
material attempts to order. Another noticeable feature of the heat capacity is that that there
is a large residual entropy in the limit of low temperatures where theses highly degenerate
ground states are targets of study by many research groups. From the magnetic
susceptibility measurements and calculations the dominate interaction is found to be
antiferromagnetic and the amount of frustration is found to have a lower bound of 2.33,
which indicates that Pr2Zr2O7 is potentially highly to moderately frustrated, . To further
investigate this novel behavior in Pr2Zr2O7, heat capacity measurements below 1.8 K and
a neutron scattering experiment would provide better insight into the magnetic ordering of
this material.
Acknowledgements
Special thanks to Chris Wiebe, who oversaw this project, for providing insight and
guidance. Thanks to Haidong Zhou for helping to acquire the data . Also, thanks to the
National High Magnetic Field Laboratory where the research was carried out. This work
was funded by the National Science Foundation and the EIEG program.