In recent years, frustrated magnetic systems have attracted significant interest as potential hosts of exotic magnetic excitations. These materials impose constraints on their magnetic moments that prevent ordering down to temperatures well below their Curie-Weiss temperatures. The magnetic pyrochlores, such as Dy2Ti2O7(DTO227), Ho2Ti2O7, Pr2Sb2O7, are the most famous examples of frustrated magnetic crystals. Dy2Ti2O7 and Ho2Ti2O7 contain highly-magnetic ions that lie on a lattice of corner-sharing tetrahedra (Figure 1A). In the ground state configurations of these moments, each tetrahedron has two moments pointing toward its center and two moments pointing away from its center (Figure 1B). This is analogous to the rules that govern hydrogen bonding in water ice (H2O), so Dy2Ti2O7 and Ho2Ti2O7 are known as “spin ices”. These materials have also been theorized to be potential hosts of mobile magnetic quasiparticle “monopole” excitations that interact via magnetic Coulomb interactions. However, until recently there existed no good quantitative model that accurately described the low-temperature magnetic relaxation dynamics of Dy2Ti2O7 and Ho2Ti2O7. Furthermore, the characteristic relaxation times of these materials show a divergence with decreasing temperature that is significantly more severe than what is expected from the most prominent microscopic models.
[Figure 1] A. The Dy3+ moments (black circles) in DTO227 are located on a lattice comprised of equilateral corner-sharing tetrahedra. The centers of the tetrahedra themselves form a diamond lattice. B. A configuration of Dy moments (blue and yellow arrows) that satisfies the “2-in, 2-out” rules for the magnetic ground states of DTO227.Instrumentation
We approached this problem by developing unique boundary-free magnetization transport techniques based on toroidal sample and measurement geometries (Figure 2). After making holes in single-crystal disks to form sample tori, we wind fine superconducting wire around each sample to create a superconducting toroidal solenoid (STS). We apply currents to the STS to generate toroidal magnetic fields, and the resulting time-varying sample magnetization generates measurable EMFs. This experimental topology is a physical realization of periodic boundary conditions. The predictions of exotic physical models are very often made using simulations in the same boundary conditions, so we can use our novel measurement technique to directly test various models of DTO227 dynamics. These measurements have allowed us to achieve a new understanding of the time- and frequency-domain magnetization dynamics in DTO227. Our measurements are performed on a dilution refrigerator at temperatures ranging from ~20mK up to several Kelvin (Figure 3, see below).
[Figure 2] The circuit and sample geometry used to probe DTO227 dynamics in our recent work. We wrap superconducting wire around each toroidal sample, creating a superconducting toroidal solenoid (STS) that can both generate fields and measure the EMF resulting from sample dynamics. An applied toroidal B-field (blue arrows) will move positive (green dots) and negative (red dots) monopoles in opposite directions, generating net magnetization currents that can be measured via the resulting time-varying EMFs.
[Figure 3] A. The dilution refrigerator on which we mount our experiments. Using this system we can reliably perform measurements from ~10mK up to several Kelvin. B. Experimental insert on the bottom of our dilution refrigerator. Our setup has ample space and flexibility, giving us a substantial amount of creative freedom when designing and building our experiments.
Recent Research Achievements
[Figure 4] Time-varying EMFs resulting from the decays of DTO227 magnetization (symbols) after turning off DC fields. We can describe our time-domain relaxation data very well with fits to a KWW function (lines).
We recently showed that the time-domain dynamics of toroidal DTO227 samples are described very well by a general Kohlrausch-Williams-Watts (KWW) function given by Χ(t)=C·exp[-(t/τ)β] (Figure 4), while the frequency-domain dynamics are described by a general Havriliak-Negami (HN) function given by Χ(ω)=Χ∞+Χ0/[1+(iωτ)α]β (Figure 5). From these relaxation measurements we also found that we could accurately describe the divergence of characteristic relaxation times using the Vogel-Tamman-Fulcher (VTF) function τ(T)=τ0exp[DT0/(T-T0)].
[Figure 5] A. Real part of the complex frequency-domain magnetic susceptibility (symbols) calculated from DTO227 responses to sinusoidal magnetic fields. Fits to a magnetic HN relaxation function (lines) show that this quantitative model is an excellent description of our data over a wide ranges of temperature and frequency. B. Imaginary part of the complex frequency-domain magnetic susceptibility (symbols) calculated from DTO227 responses to sinusoidal magnetic fields. As in A, HN fits to the data are excellent (lines). We have therefore achieved a self-consistent quantitative description of our observations over a wide range of parameters.
KWW time-domain relaxation, HN frequency-domain relaxation, and VTF relaxation time divergence have been known for decades to describe the dynamics of dielectric supercooled liquids. Materials in this phase remain liquid even when cooled below their typical ordering temperature. As supercooled liquids are cooled, their relaxation times eventually increase so rapidly that they fall out of equilibrium and become glasses. Our measurements of DTO227 relaxation, along with heat capacity measurements performed previously by other groups, show that at temperatures of a few Kelvin or less DTO227 dynamics are strongly analogous to the dynamics of a supercooled liquid approaching a glass transition. DTO227 therefore behaves as a supercooled magnetic liquid at low temperatures; our measurements indicate that it approaches a glass transition at temperatures ~250mK.
The general causes of glass transitions are still under intense debate; theories usually focus on disorder as a generator of the persistent heterogeneity present in glasses and their antecedent supercooled liquids. However, the dynamics in DTO227 take place in crystals with little structural disorder; this differentiates DTO227 phenomena from the behavior of spin glasses, which exhibit glassy behavior due to disorder in the positions of their microscopic magnetic moments. The possibility of glass formation in ordered environments has attracted a substantial amount of recent theoretical interest, since understanding the properties of such systems could lead to an improved understanding of the glass phase in general. DTO227 could therefore offer a new approach to what is considered by many scientists to be one of the great unsolved problems in physics.