United States

The achievement of quantum error correction is a fundamental step towards the implementation of a scalable platform for quantum computation, in which logical units are made robust against errors. Here we show that magnetic molecules consisting of weakly interacting and individually addressable spins ½ centers are a very promising platform to implement repetition codes for quantum error-correction, where several spins ½ are used to encode an error protected logical qubit. In particular, we demonstrate that these challenging requirements can be fulfilled by the [Er-Ce-Er] molecular trinuclear coordination compound. We show that this complex can be exploited to efficiently implement the three-qubit quantum error correction code protecting against pure dephasing, the most important error in magnetic molecules [1]. The system is characterized by combining magnetometry, low-temperature specific heat and electron paramagnetic resonance measurements on both the trimer and the properly synthesized [LuCeLu] and [ErLaEr] analogues, which contain only one of the two types of magnetic ions. The resulting large and distinct g values, together with weak but non-negligible dipolar interactions, enable us to selectively address different transitions by a proper sequence of microwave pulses, implementing the three-qubit repetition code. The parameters obtained from the experimental characterization are finally used to perform thorough numerical simulations of the system dynamics, including the effect of pure dephasing in a Lindblad formalism. These show a remarkable gain of the quantum error correction procedure, thus proving the efficiency of the device in suppressing pure dephasing.References: [1] E. Macaluso, M. Rubin, D. Aguilà, A. Chiesa, L. Barrios, J. I. Martínez, P. J. Alonso, O. Roubeau, F. Luis, G. Aromí and S. Carretta, submitted.

MMM 2020

A Heterometallic [LnLn’Ln] Lanthanide Complex as a Qubit with Embedded Quantum Error Correction

The 65th Annual Conference on Magnetism and Magnetic Materials (MMM 2020) was held in a virtual form. The COVID-19 pandemic, with its associated uncertainty about international travel and the ability to gather in large groups, meant that the Steering Committee decided to cancel the face-to-face meeting that had been planned in Palm Beach, Florida November 2-6, 2020.

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technical paper

Exchange-coupled systems employing ferrimagnets (FI) provide high tunability, interfacial exchange interaction, and zero magnetic moment at the compensation temperature [1], which is highly beneficial for many applications such as spin valves [2] and magnetic tunnel junction devices [3]. Micromagnetic simulations showed that a partial domain wall is formed at the FI/ferromagnet(FM) interface layer during the reversal of the FM. The minor loop becomes fully reversible if this domain wall generates a hard-axis field that overcomes the anisotropy field of the FM [4]. In this study, the magnetic reversal process of exchange-coupled bilayer systems, consisting of a FI TbFeCo alloy layer and a FM [Co/Ni/Pt]N multilayer (ML), was investigated [5]. In particular, minor loop studies, probing solely the reversal characteristics of the softer FM layer, reveal two distinct reversal mechanisms, which depend strongly on the thickness of the FM layer. For thick layers (N=9, 10.8 nm), irreversible switching of the macroscopic minor loop is observed. The underlying microscopic origin of this reversal process was studied in detail by high-resolution magnetic force microscopy, showing that the reversal is triggered by in-plane domain walls propagating through the FM layer. In contrast, thin FM layers (N=5, 6.0 nm) show a hysteresis-free reversal, which was already reported in previous work [6] and is nucleation-dominated due to grain-to-grain variations in magnetic anisotropy of the Co/Ni/Pt ML and an inhomogeneous exchange coupling with the magnetically hard TbFeCo layer. The results were confirmed by micromagnetic simulations.References: [1] B. Hebler, S. Böttger, M. Albrecht et al., Phys. Rev. B93, 184423 (2016) [2] F. Radu, R. Abrudan, H. Zabel et al., Nat. Commun. 3, 715 (2012) [3] J. S. Moodera, L. R. Kinder, R. Meservey et al., Phys. Rev. Lett. 74, 3273 (1995) [4] C. Vogler, M. Heigl, D. Suess et al., arXiv:2004.04419 (2020) [5] M. Heigl, C. Vogler, M. Albrecht et al., arXiv:2007.00333 (2020) [6] X. Zhao, A.-O. Mandru, H. J. Hug et al., ACS Appl. Nano Mater. 2, 12, 7478 (2019)

Microscopic Origin of Magnetization Reversal in Exchange-coupled Ferro-/Ferrimagnetic Bilayers

Exchange bias is essential to the operation of magnetic read sensors. Recently it has been discovered that in single grain Co/IrMn3 bilayers the exchange bias is caused by a small statistical imbalance in the number of Mn atoms in each sublattice [1]. In realistic devices the IrMn is split into multiple non interacting grains, each of which contributes separately to the exchange bias. Here, we have developed a model of a multigranular IrMn/CoFe bilayer using an atomistic spin model [2]. The properties of the disordered γ-IrMn3 AFM are modelled including localized magnetocrystalline anisotropies and Heisenberg exchange, reproducing key experimental results including the Néel temperature and magnetic ground state. For an atomically flat interface the polygranular system qualitatively reproduces the experimentally observed temperature and grain size dependence of the exchange bias. A problem in maximising the exchange bias is the training effect. There are two known contributions to training: thermal and athermal. Thermal training arises due to the AFM spins undergoing thermally activated transitions during the magnetization reversal process of the FM layer [3]. However, the origin of athermal training is still a widely disputed problem due to the difficulty in experimentally probing the rearrangement of AFM spins. For an atomically flat interface no athermal training was found as the switching of the interfacial spins was found to be completely reversible. By increasing the intermixing between the ferromagnetic and antiferromagnetic layers at the interface we found the appearance of thermal training due to metastable spin states in the interfacial Mn spins which reorient during the first hysteresis cycle. We find a strong dependence on the amount of training on the interface mixing as shown in Fig. 1. These results represent a significant step towards a full understanding of the exchange bias effect in nanoscale devices.References: [1] S. Jenkins, W. J. Fan, R. Gaina, R. Chantrell, T. Klemmer, and R.F.L. Evans. Uncovering the mystery of delocalised pinned interface spins responsible for exchange bias. arXiv:1610.08236 [2] R. F.L. Evans, et al, J Phys Cond. Matter, 26, (2014) [3] X. P. Qiu et al, Phys. Rev. Lett., 101:147207 (2008).

Atomistic Spin Dynamics simulations of exchange bias in polygranular IrMn/CoFe bilayers and the origin of the athermal training effect

Exchange bias is usually the product of an uncompensated spin plane at the interface between an anti-ferromagnet and a ferromagnet. First discovered in 1956, the effect is one of the most studied phenomena in magnetism. [1] However, in the past ten years, numerous observations have been made of exchange bias at molecule-metal interfaces. [2,3] This is unusual, since van der Waals crystals of organic molecules are not expected to exhibit robust magnetic order or strong exchange interactions. There is not yet a universally accepted mechanism for molecular exchange bias, and debate has raged about whether these effects are truly exchange bias, or simply the result of increased interfacial anisotropy.[4] Furthermore, it is not yet clear whether molecular exchange bias occurs only in anti-ferromagnetic stacks of molecular magnets, or whether hybridization and charge transfer also play a role. Here, we present a detailed study of molecular exchange bias at the interface between Co and the non-magnetic molecule C60.[5] We observe the largest bias field and magnetic hardening ever observed in a hybrid molecular system, and provide density functional theory and micro-magnetic models which strongly suggest an exchange spring like mechanism arising from broken interfacial symmetry. We provide a formalism for a new form of interfacial anisotropy, dubbed π-anisotropy, which we explain as the result of asymmetric p-d hybrid interfacial dipoles. While this effect is still confined to low temperature, we explain a development pathway towards a room temperature manifestation of the effect.References: [1] W H Meiklejohn and C P Bean, Phys. Rev. 102, 1413, 1956. [2] M Gruber et al, Nat Mat, 14, 981-984, 2015. [3] J Jo et al, ACS Nano, 13, 1, 894-903, 2019. [4] S Boukari et al, Nano Lett. 18, 8, 4659 – 4663, 2018. [5] T Moorsom et al, Phys Rev B. 101, 060408(R), 2020.

π – Anisotropy: A Magneto-Electric Explanation for Molecular Exchange Bias

Exchange bias (EB) plays an important role in spintronic devices. It has been thoroughly studied between inorganic anti-ferromagnets and metallic ferromagnetic (FM) layers but rather surprisingly, similar EB effects have been reported between metallic ferromagnets and organic molecular (OM) layers. In this case, EB is considered “spinterface” induced magnetic effect and could play big role in the next generation of environment-friendly electronic devices. Molecular EB effect has been first observed in 2015 [1] between cobalt films and manganese phthalocyanine, then expanded and reproduced with different metal (Zn, Fe, Co) phthalocyanine [2] and metal octa-ethyl porphyrin molecules [3]. However, fundamentals of molecular EB and its origins is still not understood and stays controversial. In this work, we investigate the possibility of molecular EB in cobalt/metal tetra-phenyl porphyrin (M=Zn and Ni) hybrid structures. Careful SQUID magnetometer measurements show the absence of EB in “freshly deposited” hybrid samples. Surprisingly, EB appears within few hours and gradually increases with time. Additionally, new, and similar “freshly deposited” samples kept under ultra-high vacuum (UHV) for the same time span did not show EB when measured directly after removal from the UHV chamber. This indicates that EB effect has its origin from the exposure of samples to air. In the light of these unexpected results, atomic force microscopy and X-ray photoelectron spectroscopy have been performed to identify and understand the reasons of our observations. The experimental evidences strongly suggest that EB in FM/OM systems is not an intrinsic effect and originates from air-driven oxidation of cobalt films transforming part of the metallic cobalt into CoOx that is well known to produce EB effects.References: [1] M. Gruber, F. Ibrahim, S. Boukary et al. Nature Materials., Vol 14, p. 981 (2015). [2] S. Boukary, H. Jabbar, F. Schleicher et al. Nano Letters., Vol 18, p. 4659 (2018). [3] J.Jo, J. Byun, I. Oh et al. ACS Nano., Vol 13, p. 894 (2019).

Is Exchange Bias at the Hybrid Organic/Ferromagnet Interface an Intrinsic Effect?

The exchange bias effect, which manifests itself as a shift of the hysteresis loop of a ferromagnet when it is adjacent to an antiferromagnet, is a key feature used in such technological spheres as magnetic recording and storage as well as magnetic sensorics [1]. However, technological applications require materials whose performance is sustainable in a broad temperature range. A perfect candidate for this role is the antiferromagnetic Ni-Mn, multilayered structures based on which are capable of exhibiting exchange bias up to temperatures of above 700 K [2]. It is also a much more affordable option when compared to currently used Ir-Mn and Pt-Mn, since it doesn’t consist precious metals. Although very promising, formation of the ordered q-NiMn requires post-deposition annealing. In this work we show how the emergence of this phase leads to the emergence of exchange bias, depending on the initial condition of the Ni-Mn layer. Moreover, we show that the irreversible disappearance of exchange bias is associated with structural transformations occurring in the films at higher temperatures. In this work we study magnetron-sputtered Ni-Mn (20 nm) and Ni-Mn (20 nm)/Fe-Ni (40 nm) films with and without a 5 nm thick buffer Fe-Ni layer. Films were deposited on Corning glass substrates; 5 nm thick Ta layers were used as seed and capping layers. The composition of the Ni-Mn layer was varied from 10 to 50 at. % of Ni. Structural and magnetic properties of the films were studied by means of X-ray diffractometry (XRD) and vibrating sample magnetometry, accordingly. XRD measurements included a variety of geometries, such as grazing incidence (GIXRD), θ-2θ, and rocking curves (ω-scans). Structural and magnetic studies were executed in a high-temperature chamber, which allowed in-situ observation of corresponding evolution of structural and magnetic properties of films. This work was financially supported by the Russian Science Foundation, project No. 18-72-10044.References: 1. F. Radu, H. Zabel, Magnetic heterostructures, Springer, 97 (2008). 2. V.O. Vas'kovskiy et al., J. Alloys Compd., 777, 264 (2019).

Emrgence and Disappearance of Exchange Bias in Ni-Mn/Fe-Ni Thin Films Due to Structural Transformations

Rubber bands can be found in many children’s toys. For example, one winds up a rubber band and when released it drives an airplane or a racing car. If instead we wind up a tiny magnetic structure, where the magnetization twists and stores energy, then that energy will be released upon unwinding. In this theoretical work we consider a very soft, low-exchange film (80-400 atomic layers of Cu-doped Permalloy [1]), which is pinned at the bottom and coupled to a high-exchange ferromagnetic film on top. We apply a magnetic field which is rotated in the plane of the film. This generates a slow twisting of the soft material with the high-exchange ferromagnet acting as a handle as is seen in Fig. 1. We use both a local self-consistent mean-field theory [2] as well as a dynamic approach with the Landau-Lifshitz equation, with results matching well for both. With these models we have been able to generate multiple twists (over 10) in these films. At some point an instability occurs that prevents additional twists. Often the instability causes one or more twists to be released from the structure. The tightness of the twists as well as how many twists are formed before snapping depends on the length of the structure, the strength of the applied field, the size of the of the ferromagnetic cap, the exchange coupling strength, and the saturation magnetization. For example, increasing the exchange coupling decreases the number of twists that are stable in a given structure. The number of allowed twists also increases linearly with the length of the sample. Analytic energy minimization calculations, using a constant turn angle approximation, also allow us to estimate how many twists will fit into a given structure, matching reasonably well with numerical outcomes. The more twists in a given structure, the more energy it can hold. Once the twist is released or snapped, the system will unwind with frequencies in the gigahertz range. We study the frequency distribution as a function of the number of twists. Based on energy storage and this frequency generation we discuss potential applications.References: [1] N. Sorensen, R. E. Camley and Z. Celinski, J.Magn Magn Mat, vol. 477, p. 344, (2019). [2] R. E. Camley, Phys Rev B, vol. 35, p. 3608, (1987).

Mini Magnetic Propellers: Stability Considerations in Twisted Magnetic Structures

We analyze in this work the low temperature magnetic behavior of a series of Co/Ag multilayers formed by samples having reduced Co content per period (either one monolayer or half a monolayer) separated by several Ag monolayers (either 8 or 16). The multilayers were grown by molecular beam epitaxy on a clean MgO (001) surface, alternating Ag and Co deposition up to complete either 32 or 64 periods (in order to maintain constant in the series the nominal number of Co atoms per unit surface of the sample) and protected by a 3nm Ag cap layer. The samples were studied by X-ray reflectivity and diffraction, TEM, SQUID magnetometry, and ac susceptometry. From the results of our measurements we identified the following facts: 1) the occurrence in all the samples of stacking sequences corresponding to the nominal ones 2) the spatial discontinuity of the Co layers that corresponded to a quasi-monodisperse in-plane distribution of Co particles embedded in a Ag(001) matrix; 3) the observation of hysteresis at temperatures (always below 20 K) that depended on both the number of Co and Ag monolayers per period; 4) an ac field frequency dependence of the real part of the susceptibility, χ’, maxima temperatures corresponding to a Vogel-Fulcher behavior with a frequency shift parameter of the order in all the samples of 4 x 10-2; 5) the collapse (Figure 1 ), for each sample, of the experimental data corresponding to the ac field frequency, f, and temperature, T, variations of the imaginary part of the ac susceptibility, χ’’, in a single curve, G(x) with x = f[(T/Tg) − 1]−Ζν (where Tg and Ζν are the critical temperature and exponent, respectively), and verifying T χ’’ (f, T) = [(T/Tg) – 1]β G(x) [1]. Taken together, these results allow us to conclude that our three multilayers experience a phase transition (of the paramagnetic to superspin glass type) at the corresponding critical temperature and that the exponent β in the universal dynamic scaling law takes values dependent on the number of Co monolayers per period.References: [1] P. Beauvillain, C. Dupas, J.P. Renard, and P. Veillet, Physical Review B 29 (7), 4086 (1984).

Critical Magnetic Behavior in [Ag8/Co0.5]x64, [Ag16/Co1]x32 and [Ag8/Co1]x32 Epitaxial Multilayers

The study of magnetic systems, spin glasses, and their states, is an actively developed field of statistical physics. Spin glass models are studied in various scientific contexts, including experimental physics, condensed matter physics, theoretical physics, mathematical statistical physics and, more recently, probability theory [1 - 3]. The problem of finding the ground states (GS), i.e. states with the lowest energy is a very difficult task due to the macroscopic degeneracy and frustrations that are exist in various realistic spin glass models. In fact, it’s well known that finding the GS, of a spin glass in a three-dimensional lattice is an NP-complete problem, which means that this challenge is at least as difficult as the hardest problems of practical interest. The system of interest is the Edwards-Anderson (EA) spin glass model [4] in two-dimensional (2D) lattice, with bimodal distribution of bonds. In present work, for simulation we used a combination of our Hybrid Multispin Method (HMM) [5] and Long Short-Term Memory (LSTM) - an artificial recurrent neural network (RNN) architecture [6] with Boltzmann distribution. HMM’s advantages are the ability to increase the number of simultaneously gathered statistical-thermodynamic parameters without noticeable speed and efficiency loss, and the ability to search for configurations of the ground and low-energy states. The usage of machine learning in statistical physics began relatively recently but is developing rapidly. Neural networks have become popular tool due to the high speed of their training and accuracy of predictions. LSTM with Boltzmann distribution is useful for certain types of prediction that require the network to retain information for longer periods of time that traditional RNNs encounter. In our research, MC algorithm generated a training data set which is configurations of SG system. The trained predicted system configurations, which are using for the next computing cycle for HMM. Such iterative approach allows us to efficiently simulate the spin-glass systems even in the low-temperature regime, avoiding the critical slowing down that plague of usual MC simulation. This work was supported by the state task of the MSHE of Russia No. 0657-2020-0005.References: [1] A. Callison et al., New J. Phys., Vol. 21, 123022 (2019) [2] B. McNaughton et al., Phys. Rev. E, Vol. 101, 053312 (2020) [3] L. A. Fernandez, E. Marinari, V. Martin-Mayor, I. Paga, and J. J. Ruiz-Lorenzo, Phys. Rev. Vol. 100, 184412 (2019) [4] S. Edwards, P. Anderson, Phys. F: Metal Phys., Vol. 5 (5), p. 965 (1975) [5] A.G. Makarov et al., JETP Letters., Vol. 110 (10), pp. 702-706 (2019) [6] S. Hochreiter, J. Schmidhuber, Neural computation., Vol. 9(8), pp. 1735-1780 (1997)

Investigation of the spin glass density of states using a combination of Monte Carlo method and neural network

We are studying the effect that illumination by coherent resonant x-rays may have on magnetic domain memory (MDM) in a [Co / Pd] / IrMn multilayers [1-3]. MDM is the ability of the magnetic domains to retain their exact same domain topology upon field cycling. Earlier studies have suggested that under higher doses of x-ray illumination, the material may lose its existing MDM. To investigate this possible effect we have used both x-ray resonant magnetic scattering (XRMS) along with magneto-transport measurements [4,5] to track the exchange bias while the sample is illuminated with x-rays, as illustrated in Figure 1. Fig.1a shows magneto-transport data collected on the [Co/Pd] / IrMn multilayer from which we hope to extract information about exchange bias. A loss of exchange bias would indicate that the x-rays illumination dose may alter the strength of the internal exchange couplings between the ferromagnetic and antiferromagnetic layers. Ultimately it would alter the amount of MDM, which is measured via XRMS as illustrated in Fig. 1b, and represented in a correlation map such as Fig.1c.References: [1] K. Chesnel, A. Safsten, M. Rytting, and E.E. Fullerton, Nature Communications 7, (2016). [2] K. Chesnel, B. Wilcken, M. Rytting, S.D. Kevan, and E.E. Fullerton, New Journal of Physics 15, 023016 (2013). [3] K. Chesnel, J. Nelson, B. Wilcken, and S.D. Kevan, Journal of Synchrotron Radiation 19, 293 (2012). [4] C. Hurd, Hall Effect in Metals and Alloys (Springer, 2012). [5] L.J. van der Pauw, A Method of Measuring Specific Resistivity and Hall Effect of Discs of Arbitrary Shape (1958).

The effect of x-ray illumination on magnetic domain memory in [Co/Pd] / IrMn multilayers

One of the most important goals in the current research in molecular magnetism is to reach a deep understanding of the relaxation dynamics of Molecular NanoMagnets (MNMs), being this aspect of paramount importance in the design of new systems acting as classical [1, 2] or quantum bits [3]. Despite being the main source of relaxation in MNMs, very limited experimental investigations on their phonons have been done so far, yielding no information about their dispersions. Indeed, the key to construct a reliable model of phonon-induced relaxation dynamics in MNMs is to have access to phonon dispersions and eigenvectors, which are necessary for a quantitative evaluation of spin-phonon coupling coefficients. Here we exploit the four-Dimensional Inelastic Neutron Scattering (4D-INS) technique [4] to directly measure for the first time phonon dispersions in a prototypical molecular qubit, [VO(acac)2]. Vanadyl-based molecules are in fact currently emerging as archetypes of a new generation of molecular qubits with long coherence times even at high temperatures [3]. 4D-INS is a very powerful technique to study phonons, because it enables a direct access to phonon eigenvalues and eigenvectors, by measuring the four-dimensional scattering function in large portions of the reciprocal space. We detected both acoustic and optical branches in single crystals of [VO(acac)2] along different directions in the reciprocal space and we compared the results of this challenging experiment with state-of-the-art density functional theory (DFT) calculations (Fig.1). Simulations reproduce important features of the 4D-INS data, such as the presence of low-lying optical modes and anti-crossings between acoustic and optical phonons, providing insights on their effect on spin-phonon couplings [5].References: [1] C.A.P. Goodwin, F. Ortu, D. Reta, et al., Nature Vol. 548, p. 439, (2017). [2] F.-S. Guo, B. M. Day, Y.-C. Chen, et al., Science Vol. 362, p. 1400, (2018). [3] M. Atzori, L. Tesi, E. Morra, et al., J. Am. Chem. Soc. Vol. 138, p. 2154, (2016). [4] E. Garlatti, A. Chiesa, T. Guidi, et al., Eur. J. Inorg. Chem., p. 1106, (2019). [5] E. Garlatti, L. Tesi, A. Lunghi, et al., Nat. Commun. Vol. 11, p. 1751 (2020).

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A Heterometallic [LnLn’Ln] Lanthanide Complex as a Qubit with Embedded Quantum Error Correction

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Microscopic Origin of Magnetization Reversal in Exchange-coupled Ferro-/Ferrimagnetic Bilayers

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