Microwave spin resonance in epitaxial thin films of spin liquid candidate TbInO3

Abstract

Minimizing the energy of a many body system tends to favor order, but classical frustration and quantum fluctuations destabilize that order. The tension between these effects can produce exotic quantum states of matter. Quantum spin liquid (QSL) states emerge in models of localized magnetic moments where the crystal lattice connectivity frustrates ordering, and the exchange interaction of neighboring spins strengthens quantum fluctuations. Experimentally identifying a QSL in a real material is challenging from the lack of an order parameter. Piecing together evidence from varied techniques is necessary for diagnosing the nature of the ground state -- QSL or otherwise -- of a frustrated spin system. In this work, we use coplanar superconducting resonators to probe magnetic excitations in epitaxially grown thin films of a spin liquid candidate TbInO3. Adapting microwave techniques from the field of circuit quantum electrodynamics, we measure responses of these thin films whose volume is too low for applying conventional bulk techniques. In-plane susceptibility extracted from the spin resonance signal indicates extreme frustration of magnetic order down to 20 mK, over two orders of magnitude lower than the Curie-Weiss energy scale. Through a crystal field analysis, we identify the doublet eigenstates comprising the ground state. As a consequence of improper ferroelectricity, Tb moments split into two flavors with distinct g-factors reflecting the local crystal field environment of each site. Spin-orbit coupling, crystal fields, magnetic frustration and improper ferroelectricity distinctively combine to shape the magnetic ground state of TbInO3. This work establishes a measurement technique using superconducting resonators to probe thin films of frustrated magnets, and applies this technique towards building a coherent understanding of the magnetic properties of TbInO3.

Figures (10)

• Figure 1: Device schematic and basic principles of spin resonance detection: (a) Tb$^{3+}$ ions with a partially filled $4f^8$ subshell form a frustrated triangular lattice. At low temperatures ($T \lesssim 10\,K$), each Tb$^{3+}$ is in a doublet state. (b) Schematic of coplanar microwave resonator fabricated by etching a NbTiN superconductor (blue) deposited on a TbInO$_3$ film (golden). The microwave magnetic field $B_{mw}$ points along paths concentric with the center pin of the waveguide. An external static magnetic field $B_0$ is applied in the plane of the film such that the static field is perpendicular to the microwave field. (c) A spin embedded in a coplanar microwave resonator interacts with the resonator's electromagnetic field. When the frequency of the resonator is equal to the Zeeman splitting of a spin or the energy of a collective excitation, a resonant exchange occurs between photons in the resonator and the spins. This process causes an increase in the decay rate of the resonator. (d) $S_{21}$ (dB) of the microwave feedline coupled to the resonator as a function of $B_0$ and frequency. The white dots are a guide to the eye and track the resonator width. (e) The decay rate of the resonator as a function of an extracted $g$-factor, where $g = \frac{h f}{\mu_B B_0}$ where $f$ is the frequency of the resonator and $\mu_B$ is the Bohr magneton. Prominent broad resonances are centered around $g = 2.1$ and $g = 4 - 9$.
• Figure 2: Frequency and Temperature dependence of the spin resonance signal: (a) Decay rate of the resonator measured as a function of magnetic field for $\lambda/4$, 3$\lambda/4$ and $5\lambda/4$ modes of the same resonator. The corresponding frequencies are $f$, $3f$ and $5f$. An extracted g-factor $\left( g = h f/(\mu_B B_0) \right)$ is used as the x-axis. The response is qualitatively similar across the three modes when plotted as a function of g. (b) A multipeak Gaussian model with three major components gives a good fit to the signal measured at the lowest temperature ($\sim 18\,mK$). We denote these three major components as R1 ($g = 2.13 \pm 0.01$), R2 ($g = 5.1 \pm 0.9$) and R3 ($g = 6.6 \pm 0.5$). (c) Temperature dependence of the decay rate vs g-factor as a function of sample temperature from $18\,mK$ to $2\,K$. (d) Two narrow resonances at $g = 1.95$ and $g = 2.3$ are observed on top of R1. (e) A third narrow resonance at $g=4.1$ has a very different phenomenology: suppressed for $T \lesssim 800\,mK$, but present at higher temperatures.
• Figure 3: Crystal-field model and c-axis displacement of Tb$^{3+}$ ions (a) The hierarchy of energy scales that govern the ground state of a single Tb$^{3+}$ ($4f^8,\, L = 3, \, S=3$) ion, neglecting coupling to other Tb$^{3+}$ sites. A large spin-orbit coupling from the high-Z Tb nucleus implies $J = L+S$ is a good quantum number and the atomic term $^7F_6$ with $J = 6$ is the ground state. $J=0,...,5$ states are separated by $\sim 3000\,K$ from the ground state. The $^7F_6$ state with $J=6$ is a collection of $2J+1 = 13$ levels. The crystal field from nearby ions further splits these into a doublet ground state separated from the lowest-lying crystal field excited level at $10\,K$. We apply an external magnetic field to finally split this doublet, adjusting the field so the splitting matches the resonator frequency. Note that the splittings are not drawn to scale.(b) $g$-factor calculated using a simple crystal-field model that mixes the $m_J=-6,...,6$ components of the $J=6$ moment, yielding a ground-state doublet built from either $m_J = \pm 1,\pm 2$ states (solid line) or $m_J = \pm 4,\pm 5$ states (dash-dot line). The mixing angle $\theta$ parametrizes the superposition between the two constituent $m_J$ states in each of these scenarios. For either assignment of the constituents of the ground-state doublet, the two $g$-factors for R2 and R3 as extracted in the previous figure can be matched by setting two different values of the mixing angle, as represented by red circles ($m_J = \pm 1,\pm 2$) and teal squares ($m_J = \pm 4,\pm 5$). (c) Displacement of $1/3$ Tb ions along the c-axis leads to a distortion of the triangular lattice and hence the local crystal field environment. As a result, two flavors of Tb ions exist, each with a distinct crystal-field environment. (d) Amplitude ratio of the two Tb-moment peaks R2 ($g = 5.1 \pm 0.9$) and R3 ($g = 6.6 \pm 0.5$), extracted from the Gaussian fit model. At low temperatures, the amplitude ratio is close to 2. This suggests two classes of Tb moments with distinct g-factors are present in the film in a 2:1 ratio of concentration.
• Figure 4: Magnetic susceptibility from $20\,mK-2\,K$ (a) Magnetic susceptibility inferred by calculating the area under the decay rate vs magnetic field curve. Since the resonator is also sensitive to signals from the substrate, we fabricate a second resonator on the same chip in a region where the TbInO$_3$ has been removed by ion milling. The susceptibility is reported on the same y-axis scale with TbInO$_3$ plus substrate (purple squares) and substrate alone (orange diamonds) at temperatures from $20\,mK-2\,K$. (b) The difference between the two values at each temperature is our best measure of the susceptibility of the TbInO$_3$ film. We observe a non-saturating susceptibility which is well-described by a naive Curie-Weiss law down to $50\,mK$, below the Curie-Weiss scale from that same fit, and 220 times below the Curie-Weiss scale extracted from higher-temperature data. Susceptibility may start to saturate between $20$ and $50\,mK$.
• Figure S1: (a) Design file for the resonator device used in the main text. The black outer square denotes the boundary of a $10\,mm\times10\,mm$ chip. The region denote by the orange hatched rectangle is ion-milled to remove the TbInO$_3$ film as grown on the YSZ substrate. Two microwave feedlines with identical resonators are patterned on the same chip. The top device is used for TbInO$_3$/YSZ measurements while the bottom device is used to determine the YSZ substrate contribution. Black scale bar at bottom right is $1000\,µm$.(b) Etch mask for a CPW resonator coupled to a microwave feedline on the left. The region in dark blue is etched using reactive-ion etching. (c) Optical image of a CPW resonator used for measurements described in the main text. The superconducting NbTiN film has a golden-brown color. White scale bar is $500\,µm$.