Spin-glass polyamorphism induced by a magnetic field in LaMnO₃ single crystal

The parent manganite compound LaMnO₃ is studied many decades on from its discovery as the reference material for understanding the unique properties of its solid solutions, multiferroics and colossal magneto-resistance manganites [1, 2]. These materials present an excellent example of solids with a complex coupling between different degrees of freedom, advantageous for many technological applications. However, some important aspects of their behaviour, particularly in the vicinity of phase transitions, remained under debate so far. Among them there is a formation and dynamical response of inhomogeneous magnetic media to external action. This is the key issue in hard-drive application [3] in relevance with tuning the spin arrangements by a magnetic field. Basically, the study of the magnetic states of manganites at low-temperature phase transitions is important for better understanding their ground state. In contrast to conventional magnetic materials, where a formation of inhomogeneous structure under field-induced transition is governed predominantly by an interface energy [4], in manganites, anisotropy and competition of exchange interactions dominate. Eventually, inhomogeneous states with the spin-glass like properties are often manifested. Their morphology and relaxation phenomena are not fully understood [5]. Moreover, the wealth of probed spin models and experimental techniques in hand for their study, allows a deep insight into universal behaviour of amorphous states under external action, which attracts a growing interest [6, 7]. The reason is a plausible theoretical description and available very sensitive magnetic equipment to probe in detail the ac and dc spin dynamics in a broad range of external parameters. The corresponding measurements in other glassy systems are much more difficult to control. The ordering properties of spin glasses have been studied extensively, either experimentally, analytically or numerically [8–10]. It was shown, that randomness and frustration of canonical spin glasses are primarily determined by competing exchange interactions. However, even weak anisotropy introduces noticeable changes in their critical behavior [11–13]. Among others, a special degree of freedom called chirality, is supposed to play a crucial role in the ordering process [14].

The complex oxide LaMnO₃ crystallizes into the orthorhombically distorted perovskite structure, space group P_bnm [15] with a Debye temperature ${\Theta}_{\rm D} = 440$ K. Each trivalent Jahn–Teller ion of manganese Mn³⁺ is centered in octahedron of six oxygen ligands. At high temperatures [16] the magnetic state is ferromagnetically ordered. However, at cooling across $T~\simeq$ 725 K, the Jahn–Teller instability [16] gives rise to cooperative distortion of the MnO₆ octahedra and ordering of the orbitals. Transition from ferromagnetic to paramagnetic state then occurs. The latter state dominates down to T_N  =  140 K, the temperature of transition to the antiferromagnetic, Mott insulator state. The apparent ground state is A-type antiferromagnet with ferromagnetic exchange between ions Mn³⁺ in ab-planes and antiferromagnetic coupling of the planes. Competing exchange interactions are estimated from inelastic neutron scattering [17, 18] as J₁  =  0.83 meV and J₂  =  −0.58 meV, respectively. A weak distortion of crystal structural is known to produce drastic changes in it's ground state [19]. A rich orbital physics [16, 20–22] stands then behind magnetic state formation which remains an open question by now.

Magnetic measurements in zero-field and field-cooling regimes have suggested a spin-glass transition below the magnetic ordering temperature in weakly disordered single crystal LaMnO₃ [23]. Information on magnetic relaxation times and field-evolution of glassy state from dynamic measurements was still needed.

With this motivation in mind, we studied magnetic state of the same LaMnO₃ single crystal by means of dynamic susceptibility measurements. Positional and orbital tilting disorder was produced by small addition of oxygen [24]. X-ray analysis confirmed orthorhombically distorted perovskite structure (space group P_bnm) and high quality of the sample. For details of sample characterization see [23]. A pronounced magnetic anisotropy confirms high quality single crystal in addition to structure analysis by Laue x-ray technique. Low-temperature measurements of longitudinal c-axis dynamic susceptibility were performed in a commercial Quantum Design MPMS XL-5 magnetometer with a driving-field amplitude 2.5 Oe and frequencies $f = \omega/{2\pi} = 0.1$, 1, 10, 88, and 884 Hz, in applied (bias) magnetic field of the strength H_dc up to 3.5 kOe. Repeated measurements were performed during 2 years with perfect reproducibility. Specific heat of the same single crystal was measured in a Quantum Desing PPMS.

The results of the dynamic susceptibility measurements at the driving frequency 884 Hz are illustrated in figure 1. The characteristic cusps in temperature dependence of both in-phase (dispersive) $\chi'$ and out-of-phase (dissipative) $\chi''$ components of dynamic susceptibility are manifested in the temperature range below the ordering transition (estimated from Curie–Weiss plot with $\Theta_{\rm CW} \sim$ 132 K and manifested also by a second cusp in the in-phase component of dynamic susceptibility at higher values of H_dc). No change of the character of temperature dependence of $\chi'$ and $\chi''$ was observed for the frequency change from 0.1 Hz to 884 Hz at the given static magnetic field.

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The routine relaxation analysis of frequency dependencies of cusp temperature T_f (not presented here) was performed. Algebraic divergence [25] of relaxation times

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tau_{\rm max}=\tau_0\cdot\left[\left(T_{\rm f}-T^*\right)/{T^*}\right]^{-z\nu} \label{eq:one} \nonumber \end{align} \tag{ 1 }$$

is as usual observed and asserted to a true phase transition in canonical spin glass at finite temperature. Here, $\tau_0$ is the microscopic trial time, $\tau_{\rm max}=1/({2\pi f})$ refers to driving frequency f [26], z and ν are the dynamic and correlation length exponents, respectively. T_f is the temperature of transition registered at frequency f and is marked by a position of either cusp in dispersion $\chi^{\prime}(T)$ or by an inflection point in absorption $\chi^{\prime\prime}(T)$. This pattern of critical behavior is described in hierarchical approach [27]. Otherwise, at zero-temperature transition, an activated dynamics law

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \ln\left({\mathstrut \frac{\tau_{\rm max}}{\tau_0}}\right)\sim \frac{1}{T}\label{eq:two} \nonumber \end{align} \tag{ 2 }$$

applies within a droplet model [28], when two states are present and related to each other by spin-flip symmetry. In figure 2 (left-hand), the results of the dependence of T_f on driving frequency are illustrated for bias fields 0 and 3 kOe.

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Analysis of the dynamic susceptibility measurements in terms of equation (1) yields the best fitting parameters T^*  =  116.9 K, $\tau_0\sim 10 ^{-12}$ s, $z\nu=6.4$ in the 'low'-field domain ($H\leqslant 1.5$ kOe, upper curve in figure 2, left-hand). Instead, the bottom curve is fitted within activated dynamics approach using Vogel–Fulcher equation (equation (2)). The best fit yields $\tau_0 = 5.6 \cdot 10^{-9}$ s, activation energy E₀  =  318 K. In addition, the time dependence of relaxation rate $W(t)=-({\rm d}/{\rm d}t){\rm ln} M(T)$, related to in-phase and out-of-phase susceptibilities [29, 30] by equation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle W(t)= \frac{2\omega}{\pi}\cdot\frac{\chi^{\prime\prime}(\omega)}{\chi^{\prime}(\omega)} \label{eq:three} \nonumber \end{align} \tag{ 3 }$$

is presented in figure 2, right-hand. It was shown earlier, that dipolar interactions cause the relaxation rate $W(t)$ to decay following a power law [31]. The exponent n depends on the strength of magnetic interactions. The scenario described by $n \geqslant 1$ is interpreted as a clear signature of a spin-glass phase. At field increase, n  >  1 and the behavior of $W(t)$ is attributed to the infinite cluster formation [5]. The presented results therefore confirm glassy character of relaxation process.

The type of magnetic glassy state is usually distinguished by magnetic-field dependence of the transition temperature. In figure 3 a dependence of cusp temperature T_f on applied magnetic field $T_{\rm f}(H^{2/3})$—de Almeida–Thouless (AT) plot is presented. Change of the power law, which describes the field dependence of the cusp temperature from 2/3 to 2 is clearly seen. It means, that inhomogeneous magnetic state experiences transformation. It can be tentatively considered within the spin-glass models in terms of irreversibility lines, initially introduced for Ising [32]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle H_{\rm AT}/\Delta J \sim \left(1-T/T_{\rm f}\right)^{3/2} \label{eq:AT} \nonumber \end{align} \tag{ 4 }$$

and Heisenberg [33] glasses

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle H_{\rm GT}/\Delta J \sim \left(1-T/T_{\rm f}\right)^{1/2}, \label{eq:GT} \nonumber \end{align} \tag{ 5 }$$

where $\Delta J$ is assigned to competing magnetic interactions. Despite restrictions of the models, their agreement with measurements is often observed. In fact, the experimentally obtained T_f follows the H^2/3 dependence in the vicinity of transition temperature T^*(0) (figure 3) in good agreement with equation (4) up to $H_{\rm T} = 1.17$ kOe. Further on increasing the magnetic field it is perfectly described by Gabay–Toulouse (GT) equation (equation (5)). It is demonstrated by $T_{\rm f}(H^2)$ plot of data above H_T (inset in figure 3). Thus, the dynamic study of weakly disturbed LaMnO₃ single crystal demonstrates a change from AT-like to the GT-like behavior at $H_{\rm T}\sim~1.17$ kOe, indicative of polyamorphic transition. In order to confirm the true field-driven transition in a spin-glass state, the temperature dependence of heat capacity was measured on the same sample in a magnetic field. The results are illustrated in figure 4. At low fields the peak in specific heat follows the position of the cusp in the in-phase dynamic susceptibility. On the other hand, the peak in specific heat above H_T only broadens, which can be an evidence of the change in the nature of glassy state.

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In conclusion, temperature dependence of dynamic magnetic susceptibility is studied in applied magnetic field for weakly disordered LaMnO₃ single crystal. It is found, that out-of-phase component appears below the estimated Curie temperature $\Theta\sim 132$ K; the cusps in temperature dependence of in-phase and out-of phase susceptibilities are seen at frequency dependent temperature $T_{\rm f}(\,f)\leqslant T^*(0)$. The frequency dependence of the cusp temperature $T_{\rm f}(\,f)$ is described in a broad field range by algebraic law with an elementary relaxation time, typical of canonical spin glass, $\tau_{0} \sim 10^{-12}$ s. The magnetic-field dependence of cusp temperature T_f is described by power law with exponent 2/3, which is predicted by the theory of spin glass. The deviation from exponent 2/3 in magnetic fields above $H_{\rm T} \sim 1.17$ kOe is observed, which can be tentatively explained in a framework of Gabay–Toulouse model of spin glass by appearance of a transverse magnetization in the presence of antiferromagnetic order. The observed field-induced transition in spin-glass state is similar to pressure-induced transition in structural and metallic glasses, attributed to polyamorphism phenomena ([6, 7, 34–37] and references therein). Observation of novel, spin-glass type of polyamorphism is, therefore, presented here.

We acknowledge Vladimir Gnezdilov for presenting us the samples, Juan Bartolome, Mikhail Bagatsky, Sergey Feodosiev, Gilbert Lonzarich and Siddharth S Saxena for discussions and comments. This work was supported by the State Fund For Fundamental Research (project no. Φ 73-24121) and by Slovak grant agencies under contract nos. APVV-14-0073 and ITMS26220120047.