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Int. J. Nanotechnology, Vol. x, No. x, xxxx Molecular Nanomagnets: towards molecular spintronics Wolfgang Wernsdorfer Institut Néel, CNRS & Université J. Fourier, BP 166, 25 rue des Martyrs, 38042 GRENOBLE Cedex 9, France Fax: + 33 476 88 1191 E-mail: [email protected] Abstract: Molecular nanomagnets, often called single-molecule magnets, have attracted much interest in recent years both from experimental and theoretical point of view. These systems are organometallic clusters characterized by a large spin ground state with a predominant uniaxial anisotropy. The quantum nature of these systems makes them very appealing for phenomena occurring on the mesoscopic scale, i.e., at the boundary between classical and quantum physics. Below their blocking temperature, they exhibit magnetization hysteresis, the classical macroscale property of a magnet, as well as quantum tunneling of magnetization and quantum phase interference, the properties of a microscale entity. Quantum effects are advantageous for some potential applications of single-molecule magnets, e.g. in providing the quantum superposition of states for quantum computing, but are a disadvantage in others such as information storage. It is believed that single-molecule magnets have a potential for quantum computation, in particular because they are extremely small and almost identical, allowing to obtain, in a single measurement, statistical averages of a larger number of qubits. This review introduces few basic concepts that are needed to understand the quantum phenomena observed in molecular nanomagnets and discusses new trends of the field of molecular nanomagnets towards molecular spintronics. Keywords: Single-molecule magnets, molecular nanomagnets, molecular spintronics, magnetic hysteresis, resonant quantum tunneling, quantum interference, spin parity effect, decoherence, quantum computation, qubit, exchange-bias, spin-Hamiltonian, micro-SQUID, magnetometer. Biographical notes: Dr. Wolfgang Wernsdorfer, born in Würzburg, Germany, in 1966, received his education in Physics in Würzburg, Lyon, and then Grenoble, where he is at present Research Director at the Centre National de la Recherche Scientifique. During his PhD in the low-temperature laboratory (CNRS, Grenoble) Wolfgang Wernsdorfer and collaborators developed a unique device (micro-SQUID) for measuring magnetic properties of nanostructures with a billion times higher sensitivity than commercial magnetometers (Bronze Medal from CNRS, 1998). His instrument allows observation of the magnetic behavior of nanomagnets containing less than a thousand magnetic centers, which is still a world record. Using the unique advantages of this device, Wolfgang Wernsdorfer has studied a variety of peculiar phenomena in depth, such as tunnelling of magnetization in molecular clusters, leading to the Agilent Europhysics Prize in 2002 and the International Olivier Kahn Award in 2006. Over the years, the innovative approach to such studies combined with the recognized superiority of this micro-SQUID have led to worldwide collaboration with most other notorious groups working on synthesizing molecular magnets to investigate single-molecule magnet behavior in more than 350 systems. The c 200x Inderscience Enterprises Ltd. Copyright ° 1 2 Wolfgang Wernsdorfer leading work of Wolfgang Wernsdorfer and collaborators in this field is at the heart of today’s knowledge on molecular magnetism. 1 Introduction A revolution in electronics is in view, with the contemporary evolution of two novel disciplines, spintronics and molecular electronics. A link between these two fields can be established using molecular magnetic materials and, in particular, single-molecule magnets, which combine the classic macroscale properties of a magnet with the quantum properties of a nanoscale entity. The resulting field, molecular spintronics aims at manipulating spins and charges in electronic devices containing one or more molecules [1, 2, 3]. The contemporary exploitation of electronic charge and spin degrees of freedom is a particularly promising field both at fundamental and applied levels. This discipline, called spintronics, has already seen some of its fundamental results turned into actual devices in a record time of 10 years and it holds great promises for the future [4, 5]. Spintronic systems exploit the fact that the electron current is composed of spin-up and spin-down carriers that carry information encoded in their spin state and interact with magnetic materials differently. Information encoded in spins persists when the device is switched off; it can be manipulated with and without using magnetic fields and can be written using little energy, to cite just a few advantages of this approach. New efforts are now directed towards spintronic devices that preserve and exploit quantum coherence, so that fundamental investigations are shifting from metals to semiconducting [4, 5], and organic materials [6], which potentially offer best promises for cost, integration and versatility. For example, organic materials are already used in applications such as organic light-emitting diodes (OLED), displays and organic transistors. The concomitant trend towards ever-smaller electronic devices (having already reached the nano-scale), and the tailoring of new molecules possessing increased conductance and functionalities are driving electronics to its ultimate molecular-scale limit [7], and the so-called molecular electronics is now being intensively investigated. In experiments of molecular electronics, the measuring devices are usually constituted by two nanoelectrodes and a bridging molecule in between, allowing the measurement of electron transport through single molecules. As the measurement is performed at the molecular level, the observables are connected to molecular orbitals and not to Bloch waves as in bulk materials. Hence, new rules are found for these systems and it becomes possible to probe the quantum properties of the molecule directly. The electron tunnelling processes in the electrode-molecule-electrode system can show the presence of Kondo or Coulomb-blockade effects, depending on the binding strength between the molecule and the electrodes, which can be tuned by selecting the appropriate chemical functional groups. In this context, a new field of molecular spintronics is emerging that combines the concepts and the advantages of spintronics and molecular electronics [1, 8] which requires the creation of molecular devices using one or few magnetic molecules. Compounds of the Single-Molecule Magnets (SMMs) class seem particularly attractive: their magnetization relaxation time is extremely long at low temperature reaching years below 2 K with record anisotropy barriers approaching 100 K [9]. These systems, combining the advantages of molecular scale with the properties of bulk magnetic materials, look attractive for highdensity information storage and also, owing to their long coherence times [10, 11, 12], Molecular spintronics 3 Figure 1 Representative examples of the peripheral functionalization of the outer organic shell of the Mn12 SMM. Different functionalizations used to graft the SMM to surfaces are displayed [1, 3]. All structures are determined by X-ray crystallography, except d, which is a model structure. Solvent molecules have been omitted. The atom color code is reported in the figure, as well as the diameter of the clusters. for quantum computing [13, 14, 15]. Moreover their molecular nature leads to appealing quantum effects of the static and dynamic magnetic properties. The rich physics behind the magnetic behaviour produces interesting effects like negative differential conductance and complete current suppression [16, 17], which could be used in electronics. Another advantage is that the weak spin-orbit and hyperfine interactions in organic molecules is likely to preserve spin-coherence over time and distance much longer than in conventional metals or semiconductors. Last but not least, specific functions (e.g. switchability with light, electric field etc.) could be directly integrated into the molecule. SMMs possess the right chemical characteristics to overcome several problems associated to molecular junctions. They are constituted by an inner magnetic core with a surrounding shell of organic ligands [18] that can be tailored to bind them on surfaces or into junctions [19, 20, 21, 22] (Fig. 1). In order to strengthen magnetic interactions between the magnetic core ions, SMMs often have delocalized bonds, which can enhance their conducting properties. SMMs come in a variety of shapes and sizes and permit selective substitutions of the ligands in order to alter the coupling to the environment [18, 19, 20, 23]. It is also possible to exchange the magnetic ions, thus changing the magnetic properties without modifying the structure and the coupling to the environment [24, 25]. While grafting SMMs on surfaces has already led to important results, even more spectacular results will emerge from the rational design and tuning of single SMM-based junctions. From a physics viewpoint, SMMs are the final point in the series of smaller and smaller units from bulk matter to atoms (Figure 2). They combine the classic macroscale properties of a magnet with the quantum properties of a nanoscale entity. They have crucial advantages over magnetic nanoparticles in that they are perfectly monodisperse and can be 4 Wolfgang Wernsdorfer Figure 2 Scale of size that goes from macroscopic down to nanoscopic sizes. The unit of this scale is the number of magnetic moments in a magnetic system (roughly corresponding to the number of magnetic atoms). At macroscopic sizes, a magnetic system is described by magnetic domains that are separated by domain walls. Magnetization reversal occurs via nucleation, propagation, and annihilation of domain walls (hysteresis loop on the left). When the system size is of the order of magnitude of the domain wall width or the exchange length, the formation of domain walls requires too much energy. Therefore, the magnetization remains in the so-called single-domain state, and the magnetization reverse by uniform rotation or nonuniform modes (middle). SMMs are the final point in the series of smaller and smaller units from bulk matter to atoms and magnetization reverses via quantum tunneling (right). Mesoscopic physics Nanoscopic Macroscopic Permanent magnets Micron particles S = 102 0 101 0 Nanoparticles 108 106 105 Clusters 104 Molecular clusters 103 Individual spins 102 10 1 Multi-domain Single-domain Magnetic moments Nucleation, propagation and annihilation of domain walls Uniform rotation Curling Resonant tunneling, quantization, quantum thermodynamics 1 1 1 0.7K M / MS M / MS M / MS Fe8 0 0 0 1K -1 -40 -1 -20 0 20 µ0 H(mT) 40 0.1K -1 -100 0 100 µ0 H(mT) -1 0 µ 0H ( T ) 1 studied in molecular crystals. They display an impressive array of quantum effects (that are observable up to higher and higher temperatures due to progress in molecular designs), ranging from quantum tunnelling of magnetization [26, 27, 28, 29] to Berry phase interference [30, 31] and quantum coherence [10, 11, 12] with important consequences on the physics of spintronic devices. Although the magnetic properties of SMMs can be affected when they are deposited on surfaces or between leads [23], these systems remain a step ahead of non-molecular nanoparticles, which show large size and anisotropy distributions, for a low structure versatility. This review introduces the basic concepts that are needed to understand the quantum phenomena observed in molecular nanomagnets and shows the new trends towards molecular spintronics [1] using junctions [3] and nano-SQUIDs [2]. 2 Overview of molecular nanomagnets Molecular nanomagnets or single-molecule magnets (SMMs) are mainly organic molecules that have one or several metal centers with unpaired electrons. These polynuclear metal complexes are surrounded by bulky ligands (often organic carboxylate ligands). The most prominent examples are a dodecanuclear mixed-valence manganese-oxo cluster with acetate ligands, short Mn12 acetate [32], and an octanuclear iron(III) oxo-hydroxo cluster of formula [Fe8 O2 (OH)12 (tacn)6 ]8+ where tacn is a macrocyclic ligand, short Fe8 [33]. Both systems have a spin ground state of S = 10 and an Ising-type magnetic anisotropy, which stabilizes the spin states with m = ±10 and generates an energy barrier for the Molecular spintronics 5 Figure 3 Size scale spanning atomic to nanoscale dimensions. On the far right is shown a highresolution transmission electron microscopyview along a [110] direction of a typical 3 nm diameter cobalt nanoparticle exhibiting a face-centered cubic structure and containing about 1000 Co atoms. The Mn84 molecule is a 4.2 nm diameter particle. Also shown for comparison are the indicated smaller Mn nanomagnets, which are drawn to scale. An alternative means of comparison is the Néel vector (N), which is the scale shown. The green arrows indicate the magnitude of the Néel vectors for the indicated SMMs, which are 7.5, 22, 61, and 168 for Mn4 , Mn12 , Mn30 and Mn84 , respectively. Mn4 Mn12 Mn30 Mn84 N 1 10 Quantum world Molecular (bottom-up) approach 100 1000 Classical world Classical (top-down) approach reversal of the magnetization of about 67 K for Mn12 acetate [34, 35, 36] and 25 K for Fe8 [37]. Thermally activated quantum tunneling of the magnetization has first been evidenced in both systems [26, 27, 28, 38, 39]. Theoretical discussion of this assumes that thermal processes (principally phonons) promote the molecules up to high levels with small quantum numbers |m|, not far below the top of the energy barrier, and the molecules then tunnel inelastically to the other [40, 41, 42, 43, 44, 45, 46, 47]. Thus the transition is almost entirely accomplished via thermal transitions and the characteristic relaxation time is strongly temperature-dependent. For Fe8 , however, the relaxation time becomes temperature-independent below 0.36 K [28, 48] showing that a pure tunneling mechanism between the only populated ground states m = ±S = ±10 is responsible for the relaxation of the magnetization. On the other hand in the Mn12 acetate system one sees temperature independent relaxation only for strong applied fields and below about 0.6 K [49, 50]. During the last years, many new molecular nanomagnets were presented (see, for instance, Refs. [51, 52, 53, 54]) which show also tunneling at low temperatures. The largest molecular nanomagnets is currently a Mn84 molecule [55] that has a size of a magentic nanoparticle (Figure 3). The record anisotropy barriers of 89 K is currently a Mn6 SMM [9]. 3 Giant spin model for nanomagnets A magnetic molecule, that behaves like a small nanomagnet, must have a large uniaxial easy axis type magnetic anisotropy and a large ground state spin. A typical example is the octanuclear iron(III) oxo-hydroxo cluster of formula [Fe8 O2 (OH)12 (tacn)6 ]8+ where tacn is a macrocyclic ligand (1,4,7-traiazcyclononane), short Fe8 (Figure 4) [33]. The internal iron(III) ions are octahedrally coordinated to the two oxides and to four hydroxo bridges. The outer iron(III) ions coordinate three nitrogens and three hydroxyls. 6 Wolfgang Wernsdorfer Figure 4 Schematic view of the magnetic core of the Fe8 cluster. The oxygen atoms are black, the nitrogen atoms are gray, and carbon atoms are white. The arrows represent the spin structure of the ground state S = 10. Spin polarized neutron scattering showed that all Fe ions have a spin 5/2, six spins up and two down [56]. This rationalizes the S = 10 spin ground state that is in agreement with magnetization measurements. In principle, a multi-spin Hamiltonian can be derived taking into account of all exchange interactions and the single-ion magnetic anisotropies. However, the Hilbert space is very large (68 ≈ 106 ) and the exchange coupling constants are not well known. A giant spin model is therefore often used that describes in an effective way the ground spin state multiplet. A nanomagnet like the Fe8 molecular cluster has the following Hamiltonian ¡ ¢ ~ ·H ~ H = −DSz2 + E Sx2 − Sy2 + gµB µ0 S (1) Sx , Sy , and Sz are the three components of the spin operator, D and E are the anisotropy constants which were determined via high frequency electron paramgnetic resonance (HFEPR) (D/kB ≈ 0.275 K and E/kB ≈ 0.046 K [37]), and the last term of the Hamiltonian ~ This Hamiltonian dedescribes the Zeeman energy associated with an applied field H. fines hard, medium, and easy axes of magnetization in x, y, and z directions, respectively (Figure 5). It has an energy level spectrum with (2S + 1) = 21 values which, to a first approximation, can be labeled by the quantum numbers m = −10, −9, ..., 10 choosing the z-axis as quantization axis. The energy spectrum, shown in Figure 6, can be obtained by using standard diagonalisation techniques of the [21 × 21] matrix describing the spin ~ = 0, the levels m = ±10 have the lowest energy. When a field Hamiltonian S = 10. At H Hz is applied, the energy levels with m < −2 increase, while those with m > 2 decrease (Figure 6). Therefore, energy levels of positive and negative quantum numbers cross at certain fields Hz . It turns out that for Fe8 the levels cross at fields given by µ0 Hz ≈ n× 0.22 T, with n = 1, 2, 3, .... The inset of Figure 6 displays the details at a level crossing where transverse terms containing Sx or Sy spin operators turn the crossing into an “avoided level crossing”. The spin S is “in resonance” between two states when the local longitudinal field is close to an avoided level crossing. The energy gap, the so-called “tunnel spitting” ∆, can be tuned by an applied field in the xy-plane (Figure 5) via the Sx Hx and Sy Hy Zeeman terms (Section 3.2). Molecular spintronics 7 Figure 5 Unit sphere showing degenerate minima A and B which are joined by two tunnel paths (heavy lines). The hard, medium, and easy axes are taken in x-, y-, and z-direction, respectively. The constant transverse field Htrans for tunnel splitting measurements is applied in the xy-plane at an ~ = 0, the giant spin reversal results from the interference of azimuth angle ϕ. At zero applied field H two quantum spin paths of opposite direction in the easy anisotropy yz-plane. For transverse fields in direction of the hard axis, the two quantum spin paths are in a plane which is parallel to the yz-plane, as indicated in the figure. Using Stokes theorem, it has been shown that the path integrals can be converted in an area integral, yielding that destructive interference—that is a quench of the tunneling rate—occurs whenever the shaded area is kπ/S, where k is an odd integer. The interference effects disappear quickly when the transverse field has a component in the y-direction because the tunneling is then dominated by only one quantum spin path. Z Easy axis A Y ϕ Hard axis X Medium axis Htrans B Figure 6 Zeeman diagram of the 21 levels of the S = 10 manifold of Fe8 as a function of the field applied along the easy axis [equation (1)]. From bottom to top, the levels are labeled with quantum numbers m = ±10, ±9, ..., 0. The levels cross at fields given by µ0 Hz ≈ n× 0.22 T, with n = 1, 2, 3, .... The inset displays the detail at a level crossing where the transverse terms (terms containing Sx or/and Sy spin operators) turn the crossing into an avoided level crossing. The larger the tunnel splitting ∆, the higher the tunnel rate. The effect of these avoided level crossings can be seen in hysteresis loop measurements (Figure 7). When the applied field is near an avoided level crossing, the magnetization relaxes faster, yielding steps separated by plateaus. As the temperature is lowered, there is a decrease in the transition rate due to reduced thermal-assisted tunneling. 8 Wolfgang Wernsdorfer Figure 7 Hysteresis loops of a single crystal of Fe8 molecular clusters at different temperatures. The longitudinal field (z−direction) was swept at a constant sweeping rate of 0.014 T/s. The loops display a series of steps, separated by plateaux. As the temperature is lowered, there is a decrease in the transition rate due to reduced thermal assisted tunneling. The hysteresis loops become temperature independent below 0.35 K, demonstrating quantum tunneling at the lowest energy levels 1 0.7K M / MS 0.5 0.5K 1K 0 -0.5 0.4, 0.3 and 0.04K -1 -1.2 -0.6 0 µ0 Hz ( T ) 0.6 1.2 3.1 Landau–Zener tunneling in Fe8 The nonadiabatic transition between the two states in a two-level system has first been discussed by Landau, Zener, and Stückelberg [57, 58, 59]. The original work by Zener concentrates on the electronic states of a bi-atomic molecule, while Landau and Stückelberg considered two atoms that undergo a scattering process. Their solution of the time-dependent Schrödinger equation of a two-level system could be applied to many physical systems and it became an important tool for studying tunneling transitions. The Landau–Zener model has also been applied to spin tunneling in nanoparticles and clusters [60, 61, 62, 63, 64]. The tunneling probability P when sweeping the longitudinal field Hz at a constant rate over an avoided energy level crossing (Figure 8) is given by # " π∆2m,m0 Pm,m0 = 1 − exp − . (2) 2~gµB |m − m0 |µ0 dHz /dt Here, m and m0 are the quantum numbers of the avoided level crossing, dHz /dt is the constant field sweeping rates, g ≈ 2, µB the Bohr magneton, and ~ is Planck’s constant. With the Landau–Zener model in mind, we can now start to understand qualitatively the hysteresis loops (Figure 7). Let us start at a large negative magnetic field Hz . At very low temperature, all molecules are in the m = −10 ground state (Figure 6). When the applied field Hz is ramped down to zero, all molecules will stay in the m = −10 ground state. When ramping the field over the ∆−10,10 –region at Hz ≈ 0, there is a Landau– Zener tunnel probability P−10,10 to tunnel from the m = −10 to the m = 10 state. P−10,10 depends on the sweeping rate [equation (2)]; that is, the slower the sweeping rate, the larger the value of P−10,10 . This is clearly demonstrated in the hysteresis loop measurements showing larger steps for slower sweeping rates [30, 31]. When the field Hz is now increased further, there is a remaining fraction of molecules in the m = −10 state which became a metastable state. The next chance to escape from this state is when the field reaches the ∆−10,9 region. There is a Landau–Zener tunnel probability P−10,9 to tunnel from the m = −10 to the m = 9 state. As m = 9 is an excited state, the molecules in this state relax quickly to the m = 10 state by emitting a phonon. A similar mechanism happens when the applied field reaches the ∆−10,10−n regions (n = 2, 3, . . . ) until all Molecular spintronics 9 Figure 8 Detail of the energy level diagram near an avoided level crossing. m and m0 are the quantum numbers of the energy level. Pm,m0 is the Landau–Zener tunnel probability when sweeping the applied field from the left to the right over the anticrossing. The greater the gap ∆ and the slower the sweeping rate, the higher is the tunnel rate [equation (2)]. |m> | m' > Energy 1 - P ∆ 1 P |m> | m' > Magnetic field H z molecules are in the m = 10 ground state; that is, the magnetization of all molecules is reversed. As phonon emission can only change the molecule state by ∆m = 1 or 2, there is a phonon cascade for higher applied fields. In order to apply quantitatively the Landau–Zener formula [equation (2)], we first saturated the crystal of Fe8 clusters in a field of Hz = −1.4 T, yielding an initial magnetization Min = −Ms . Then, we swept the applied field at a constant rate over one of the resonance transitions and measured the fraction of molecules which reversed their spin. This procedure yields the tunneling rate P−10,10−n and thus the tunnel splitting ∆−10,10−n [equation (2)] with n = 0, 1, 2, . . . . We first checked the predicted Landau–Zener sweeping field dependence of the tunneling rate. We found a good agreement for sweeping rates between 10 and 0.001 T/s [30]. The deviations at lower sweeping rates are mainly due to the hole-digging mechanism [65] which slows down the relaxation. Our measurements showed for the first time that the Landau–Zener method is particularly adapted for molecular clusters because it works even in the presence of dipolar fields which spread the resonance transition provided that the field sweeping rate is not too small. 3.2 Oscillations of tunnel splitting An applied field in the xy−plane adjusts the tunnel splittings ∆m,m0 via the Sx and Sy spin operators of the Zeeman terms that do not commute with the spin Hamiltonian. This effect can be demonstrated by using the Landau–Zener method (Section 3.1). Figure 9 presents a detailed study of the tunnel splitting ∆±10 at the tunnel transition between m = ±10, as a function of transverse fields applied at different angles ϕ, defined as the azimuth angle between the anisotropy hard axis and the transverse field (Figure 5). For small angles ϕ the tunneling rate oscillates with a period of ∼0.4 T, whereas no oscillations showed up for large angles ϕ [30]. In the latter case, a much stronger increase of ∆±10 with transverse field is observed. The transverse field dependence of the tunneling rate for different resonance conditions between the state m = −10 and (10 − n) can be observed by sweeping the longitudinal field around µ0 Hz = n × 0.22 T with n = 0, 1, 2, . . . . The corresponding tunnel splittings ∆−10,10−n oscillate with almost the same period of ∼0.4 T (Figure 9). In addition, comparing quantum transitions between m = −10 and (10 − n), with n even or odd, revealed a parity (or symmetry) effect that is analogous to 10 Wolfgang Wernsdorfer Tunnel splitting ∆(10- ϕ ≈ 50° (a) ϕ ≈ 20° ϕ ≈ 7° 1 0.1 0 (b) K) ϕ ≈ 90° 7 7 10 Tunnel splitting ∆(10- K) Figure 9 Measured tunnel splitting ∆ as a function of transverse field for (a) several azimuth angles ϕ at m = ±10 and (b) ϕ ≈ 0◦ , as well as for quantum transition between m = −10 and (10 − n). Note the parity effect that is analogous to the suppression of tunneling predicted for halfinteger spins. It should also be mentioned that internal dipolar and hyperfine fields hinder a quench of ∆ which is predicted for an isolated spin. ϕ ≈ 0° n=0 0.2 0.4 0.6 0.8 1 1.2 Magnetic transverse field (T) 1.4 n=2 10 ϕ ≈ 0° n=1 1 n=0 0.1 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Magnetic transverse field (T) the Kramers’ suppression of tunneling predicted for half-integer spins [66, 67]. A similar strong dependence on the azimuth angle ϕ was observed for all studied resonances. 3.3 Semiclassical descriptions Before showing that the above results can be derived by an exact numerical calculation using the quantum operator formalism, it is useful to discuss semiclassical models. The original prediction of oscillation of the tunnel splitting was done by using the path integral formalism [68]. Here [69], the oscillations are explained by constructive or destructive interference of quantum spin phases (Berry phases) of two tunnel paths (instanton trajectories) (Figure 5). Since our experiments were reported, the Wentzel–Kramers–Brillouin theory has been used independently by Garg [70] and Villain and Fort [71]. The surprise is that although these models [69, 70, 71] are derived semiclassically, and should have higher-order corrections in 1/S, they appear to be exact as written! This has first been noted in Refs. [70] and [71] and then proven in Ref. [72] Some extensions or alternative explications of Garg’s result can be found in Refs. [73, 74, 75, 76, 77]. The period of oscillation is given by [69] 2kB p ∆H = 2E(E + D) (3) gµB where D and E are defined in equation (1). We find a period of oscillation of ∆H = 0.26 T for D = 0.275 K and E = 0.046 K as in Ref. [37]. This is somewhat smaller than the experimental value of ∼0.4 T. We believe that this is due to higher-order terms of the spin Hamiltonian which are neglected in Garg’s calculation. These terms can easily be included in the operator formalism as shown in the next subsection. 3.4 Exact numerical diagonalization In order to quantitatively reproduce the observed periodicity we included fourth-order terms in the spin Hamiltonian [equation (1)] as employed in the simulation of inelastic neutron scattering measurements [78, 79] and performed a diagonalization of the [21 × 21] Molecular spintronics 11 Figure 10 Calculated tunnel splitting ∆ as a function of transverse field for (a) quantum transition between m = ±10 at several azimuth angles ϕ and (b) quantum transition between m = −10 and (10 − n) at ϕ = 0◦ (Section 3.4). The fourth-order terms suppress the oscillations of ∆ at large transverse fields |Hx |. 1000 1000 30° 20° 10 10° 5° 1 0° 0.1 0.01 0 (b) K) (a) 7 100 50° Tunnel splitting ∆ (10- Tunnel splitting ∆ (10- 7 K) ϕ = 90° 0.4 0.8 1.2 1.6 Magnetic tranverse field (T) 2 100 10 1 n=2 n=1 n=0 0.1 0.01 0 0.4 0.8 1.2 1.6 Magnetic tranverse field (T) 2 matrix describing the S = 10 system. For the calculation of the tunnel splitting we used D = 0.289 K, E = 0.055 K [equation (1)] and the fourth-order terms as defined in [78] with B40 = 0.72× 10−6 K, B42 = 1.01× 10−5 K, B44 = −0.43× 10−4 K, which are close to the values obtained by EPR measurements [80] and neutron scattering measurements [79]. The calculated tunnel splittings for the states involved in the tunneling process at the resonances n = 0, 1, and 2 are reported in Figure 10, showing the oscillations as well as the parity effect for odd resonances. 3.5 Spin-parity effect The spin-parity effect is among the most interesting quantum phenomena that can be studied at the mesoscopic level in SMMs. It predicts that quantum tunneling is suppressed at zero applied field if the total spin of the magnetic system is half-integer but is allowed in integer spin systems. Enz, Schilling, Van Hemmen and Süto [81, 82] were the first to suggest the absence of tunneling as a consequence of Kramers degeneracy. The Kramers theorem asserts that no matter how unsymmetric the crystal field is, an ion possessing an odd number of electrons must have a ground state which is at least doubly degenerate, even in the presence of crystal fields and spin-orbit interactions [83]. The predicted spin parity effect can be observed by measuring the tunnel splitting as a function of transverse field [84]. An integer spin system is rather insensitive to small transverse fields whereas a half-integer spin systems is much more sensitive. However, a half-integer spin system will also undergo tunneling at zero external field as a result of environmental degrees of freedom such as hyperfine and dipolar couplings or small intermolecular exchange interaction. The nicest observation of the spin parity effect has been seen for two molecular Mn12 clusters with a spin ground state of S = 10 and S = 19/2 showing oscillations of the tunnel probability as a function of a transverse field being due to topological quantum phase interference of two tunnel paths of opposite windings (Section 3.3). Spin-parity dependent tunneling was established for the first time in these compounds by comparing the quantum phase interference of integer and half-integer spin systems [31]. 12 Wolfgang Wernsdorfer Figure 11 Transport experiments on SMMs. a) Schematic using a STM tip to perform transport on surface grafted SMMs. b) Schematic of SMM-based molecular transistors, in which a gate voltage can modulate transport. c) [Co(TerPy)2 ] molecular magnet with alkyl spacers, permitting transport in the weakly coupled regime [85]. d) [Co(TerPy)2 ] molecular magnet with no spacers, showing strong coupling and the Kondo effect [85]. e) Divanadium [(N,N’,N”-trimethyl1,4,7-triazacyclononane)2 V2 (CN)4 (-C4 N4 )] molecular magnet showing the Kondo effect only in the charged state [86]. The color code is the same as in Fig. 1, except for Co atoms (green) and V atoms (Orange). 4 Molecular spintronics using single-molecule magnets Molecular spintronics combines the ideas of three novel disciplines, spintronics, molecular electronics, and quantum computing. The resulting field aims at manipulating spins and charges in electronic devices containing one or more molecules [1]. The main advantage is that the weak spin-orbit and hyperfine interactions in organic molecules is likely to preserve spin-coherence over time and distance much longer than in conventional metals or semiconductors. In addition, specific functions (e.g. switchability with light, electric field etc.) could be directly integrated into the molecule. In order to lay the foundation of molecular spintronics, several molecular devices have been proposed [1]: molecular spin-transistor, molecular spin-valve and spin filter, molecular double-dot devices, and carbon nanotube-based nano-SQUIDs [2]. The main purpose is to fully control the initialization, the manipulation and the read-out of the spin states of the molecule and to perform basic quantum operations. The main targets for the coming years concern fundamental science as many issues, experimental, technological and theoretical, must be addressed before applications, for instance in quantum electronics, can be realistically considered. 4.1 Molecular spin-transistor The first scheme we consider is a magnetic molecule attached between two non-magnetic electrodes. One possibility is to use a scanning tunneling microscope tip as the first electrode and the conducting substrate as the second one (Fig. 11a). So far, only few atoms on surfaces have been probed in this way, revealing interesting Kondo effects [87] and single-atom magnetic anisotropies [88]. The next scientific step is to pass from atoms to molecules in order to observe richer physics and to modify the properties of the magnetic Molecular spintronics 13 objects. Although isolated SMMs on gold have been obtained [19, 20, 21, 22], the rather drastic experimental requirements, i.e. very low temperatures and high magnetic fields, have not yet been achieved. The first theoretical work predicted that quantum tunneling of the magnetization is detectable via the electric current flowing through the molecule [89], allowing therefore the readout of the quantum dynamics of a single molecule. Another possibility concerns break-junction devices [90], which integrate a gate electrode. Such a three-terminal transport device, called a molecular spin-transistor, is a single electron transistor with nonmagnetic electrodes and a single magnetic molecule as the island. The current passes through the magnetic molecule via the source and drain electrodes, and the electronic transport properties are tuned via a gate voltage Vg (Fig. 11b). Similarly to molecular electronics, weak- and strong-coupling regimes can be distinguished, depending on the coupling between molecule and electrodes. In the weak-coupling limit charging effects dominate the transport. Transport takes place when a molecular orbital is in resonance with the Fermi energy of the leads and electrons can then tunnel through the energy barrier into the molecular level and out into the drain electrode. The resonance condition is obtained by shifting the energy levels with Vg and the measurements show Coulomb-blockade diamonds [91]. The experimental realization of this scheme has been achieved using Mn12 with thiolcontaining ligands (Fig. 11b), which bind the SMM to the gold electrodes with strong and reliable covalent bonds [16]. An alternative route is to use short but weak-binding ligands [17]: in both cases, the peripheral groups act as tunnel barriers and help conserving the magnetic properties of the SMM in the junction. As the electron transfer involves the charging of the molecule, we must consider, in addition to the neutral state, the magnetic properties of the negatively- and positively-charged species. This introduces an important difference with respect to the homologous measurements on diamagnetic molecules, where the assumption is often made that charging of the molecule does not significantly alter the internal degrees of freedom [92]. Because crystals of the charged species can be obtained, SMMs permit direct comparison between spectroscopic transport measurements and more traditional characterization methods. In particular, magnetization measurements, electron paramagnetic resonance, and neutron spectroscopy can provide energy level spacings and anisotropy parameters. In the case of Mn12 , positively charged clusters possess a lower anisotropy barrier [93]. As revealed by the first Coulomb-blockade measurements, the presence of these states is fundamental to explain transport through the clusters [16, 17]. Negative differential conductance was found that might be due to the magnetic characteristics of SMMs. Studies in magnetic field showed a first evidence of the spin transistor properties [17]. Degeneracy at zero field and nonlinear behavior of the excitations as a function of field are typical of tunneling via a magnetic molecule. In these first studies, the lack of a hysteretic response can be due, besides environmental effects [23], to the alternation of the molecules during the grafting procedure, to the population of excited states with lower energy barriers, or might also be induced by the source-drain voltage scan performed at each field value. Theoretical investigations in the weak-coupling regime predict many interesting effects. For example, a direct link between shot noise measurements and the detailed microscopic magnetic structure of SMMs has been proposed [94], allowing the connection of structural and magnetic parameters to the transport features and therefore a characterization of SMMs using transport measurements. This opens the way to rational design of SMMs for spintronics and to test the physical properties of related compounds. The first step in this direction has already been made by comparing the expected response of chem- 14 Wolfgang Wernsdorfer ically related SMMs [95]. Note that this direct link cannot be established for nanoparticles or quantum-dots (QDs) because they do not posses a unique chemical structure. A complete theoretical analysis as a function of the angle between the easy axis of magnetization and the magnetic field showed that the response persists whatever the orientation of the SMM in the junction and that even films of SMMs should retain many salient properties of single-molecule devices [96, 97]. For strong electronic coupling between the molecule and the leads, higher-order tunnel processes become important, leading to the Kondo effect [98, 99, 100, 101]. This regime has been attained using paramagnetic molecules containing one [85] or two magnetic centers [86], but remains elusive for SMMs. The first mononuclear magnetic molecule investigated (Fig. 11c) is a Co2+ ion bound by two terpyridine ligands, TerPy, attached to the electrodes with chemical groups of variable length [85]. The system with the longer alkyl spacer, due to a lower transparency of the barrier, displays Coulomb blockade diamonds, which are characteristic for the weak coupling regime, but no Kondo peak. Experiments conducted as a function of magnetic field reveal the presence of excited states connected to spin excitations, in agreement with the effective S = 1/2 state usually attributed to Co2+ ions at low temperatures but a Land factor g = 2.1 is found. This is unexpected for Co2+ ions, characterized by high spin-orbit coupling and magnetic anisotropy, and this point needs further investigation. The same complex with the thiol directly connected to the TerPy ligand (Fig. 11d) shows strong coupling to the electrodes, with exceptionally high Kondo temperatures around 25 K [85]. Additional physical effects of considerable interest were obtained using a simple molecule containing two magnetic centers [86]. This molecule, the divanadium molecule (Fig. 11e), was again directly grafted to the electrodes, so as to have the highest possible transparency [86]. The molecule can be tuned with the gate voltage Vg into two differently charged states. The neutral state, due to antiferromagnetic coupling between the two magnetic centers, has S = 0, while the positively charged state has S = 1/2. Kondo features are found, as expected [98, 99, 100, 101], only for the state in which the molecule has a nonzero spin moment. This nicely demonstrates that magnetic molecules with multiple centers and antiferromagnetic interactions permit to switch the Kondo effect on and off, depending on their charge state. The Kondo temperature is again exceptionally high, exceeding 30 K, and its characterization as a function of Vg indicates that not only spin but also orbital degrees of freedom play an important role on the Kondo resonance of single molecules. Molecular magnets, in which spin-orbit interaction can be tuned without altering the structure [25], are appealing to investigate further this physics. The Kondo temperatures observed in the two cases [85, 86] are much higher than those obtained for QDs and carbon nanotubes [98, 99, 100, 101], and are extremely encouraging. The study of the superparamagnetic transition of SMMs while in the Kondo regime thus seems achievable, possibly leading to an interesting interplay of the two effects. In order to observe the Kondo regime one might start with small SMMs [25, 102], with core states more affected by the proximity of the leads and use short and strongly bridging ligands to connect SMMs to the electrodes [19, 85]. Theoretical investigations have explored the rich physics of this regime [94, 103, 104], revealing that the Kondo effect should even be visible in SMMs with S > 1/2 [94]. This is in contrast to expectations for a system with an anisotropy barrier, where the blocked spin should hinder cotunneling processes. However in SMMs, the presence of a transverse anisotropy induces a Kondo resonance peak [94]. The observation of this new physical phenomena should be possible because of the tunability of SMMs, allowing a rational Molecular spintronics 15 Figure 12 Spin-valves based on molecular magnets. Yellow arrows represent the magnetization. a) Parallel configuration of the magnetic source electrode (copper color) and molecular magnetization, with diamagnetic drain electrode (golden color). Spin-up majority carriers (thick green arrow) are not affected by the molecular magnetization, while the spin-down minority carriers (thin blue arrow) are partially reflected back. b) Anti-parallel configuration: majority spin-up electrons are only partially transmitted by the differently polarized molecule, while the minority spin-down electrons pass unaffected. Assuming that the spin-up contribution to the current is larger in the magnetic contact, this configuration has higher resistance than that of the previous case. c) Theoretical schematic of a spin-valve configuration with nonmagnetic metal electrodes [8] and d) proposed molecular magnet between gold electrodes: a conjugated molecule bridges the cobaltocene (red) and ferrocene (blue) moieties [106]. choice of the physical parameters governing the tunneling process: low symmetry transverse terms are particularly useful, because selection rules apply for high symmetry terms. The first theoretical predictions argued that the Kondo effect should be present only for half-integer spin molecules. However the particular quantum properties of SMMs allow for the Kondo effect even for integer spins. In addition, the presence of the so-called Berry-phase interference [30, 31, 69], a geometrical quantum phase effect, can produce not only one Kondo resonance peak, but a series of peaks as a function of applied magnetic field [105]. These predictions demonstrate how the molecular nature of SMMs and the quantum effects they exhibit differentiate them from inorganic QDs and nanoparticles and should permit the observation of otherwise prohibited phenomena. 4.2 Molecular spin-valve A molecular spin-valve (SV) [8] is similar to a spin transistor but contains at least two magnetic elements (Fig. 12a-b). SVs change their electrical resistance for different mutual alignments of the magnetizations of the electrodes and of the molecule, analogous to a polarizer-analyzer setup. Non-molecular devices are already used in hard disc drives, owing to the giant- and tunnel-magnetoresistance effects. As good efficiency has already been demonstrated for organic materials [6], molecular SVs are actively sought after [107, 108]. As only few examples of molecular SVs exist [109, 110], the fundamental physics behind these devices remains largely unexplored and will likely be the focus of considerable attention in the near future. The simplest SV consists of a diamagnetic 16 Wolfgang Wernsdorfer Figure 13 Molecular double-dot devices. Magnetic molecules proposed for grafting on suspended carbon nanotubes connected to Pd electrodes (form left to right): a C60 fullerene including a rare-earth atom, the Mn12 SMM and the rare-earth-based double-decker [Tb(phtalocyanine)2 ] SMM. The gate voltage of the double-dot device is obtained by a doped Si substrate covered by a SiO2 insulating layer. molecule in between two magnetic leads, which can be metallic or semiconducting. The first experiments sandwitched a C60 fullerene between Ni electrodes, showing a very large negative magnetoresistance effect [109]. Another interesting possibility is to use carbon nanotubes connected with magnetic halfmetallic electrodes transforming spin information into large electrical signals [106]. A SMM-based SV can have one or two magnetic electrodes (Fig. 12a-b), or the molecule can possess two magnetic centers in between two non-magnetic leads (Fig. 12c-d), in a scheme reminiscent of early theoretical models of SVs [8]. Molecules with two magnetic centers connected by a molecular spacer are well-known in molecular magnetism and a double metallocene junction has been theoretically studied [106]. This seems a good choice, as the metallocenes leave the d-electrons of the metals largely unperturbed. Theory indicates that, when using SMMs, the contemporary presence, at high bias, of large currents and slow relaxation will individuate a physically interesting regime [111, 112]. Only spins parallel to the molecular magnetization can flow through the SMM and the current will display, for a time equivalent to the relaxation time, a very high spin polarization. For large currents this process can lead to a selective drain of spins with one orientation from the source electrode, thus transferring a large amount of magnetic moment from one lead to the other. This phenomenon, due to a sole SMM, has been named giant spin amplification [111] and offers a convenient way to read the magnetic state of the molecule. The switching of the device seems more complicated, at first sight, involving a two-step process that includes the application of a magnetic field and the variation of the bias voltage. However, it has recently been suggested that the spin-polarized current itself can be sufficient to switch the magnetization of a SMM [113]. The switching can be detected in the current as a step if both leads are magnetic and have parallel magnetization or as a sharp peak for the anti-parallel configuration. 4.3 Molecular multi-dot devices A double-dot devices (Figure 13) is one possible route for molecular spintronics [1]. It is a three terminal device, where the current passes through a non-magnetic quan- Molecular spintronics 17 tum conductor (quantum wire, nanotube, molecule, or quantum dot (QD)). The magnetic molecule is only weakly coupled to the non-magnetic conductor but its spin can influence the transport properties, permitting readout of the spin state with minimal back-action. Several mechanisms can be exploited to couple the two systems. One appealing way is to use a carbon nanotube as a detector of the magnetic flux variation, possibly using the nanoSQUID [2]. Other possibilities involve the indirect detection of the spin state through electrometry. Indeed, a non-magnetic quantum conductor at low temperatures behaves as a QD for which charging processes become quantized, giving rise to Coulomb blockade and Kondo effect depending on the coupling to the leads. Any slight change in the electrostatic environment (controlled by the gate) can induce a shift of the Coulomb diamonds of the device, leading to a conductivity variation of the QD at constant gate voltage. QDs are therefore accurate electrometers. When the QD is coupled, even weakly, with a magnetic object, due to the Zeeman energy the spin flip at non-zero field induces a change of the electrostatic environment of the QD. This effect, called magneto-Coulomb effect, enables therefore to detect the magnetization reversal of the molecule. Another route is weak exchange or dipole coupling between the magnetic molecule and the QD. It is interesting to probe these effects as a function of the number of trapped electrons because odd or even number of electrons should lead to different couplings. The main advantage of these schemes is that the coupling to the leads and the injected current does not alter the magnetic properties of the molecule. Because coupling is small, these devices might allow a non-destructive readout of the spin states. 5 Conclusion In conclusion, molecular nanomagnets offer a unique opportunity to explore the quantum dynamics of a large but finite spin. We focused our discussion on the Fe8 molecular nanomagnet because it is the first system where studies in the pure quantum regime were possible. In the coming years, chemistry is going to play a major role through the synthesis of novel larger spin clusters with strong anisotropy [9]. The unique properties of SMMs will soon lead to design the molecules for specific transport characteristics using the flexibility of supramolecular chemistry. Important investigations concern the studies of the quantum character of molecular clusters for applications like quantum computers. The first implementation of Grover’s algorithm with molecular nanomagnets has been proposed [13]. Antiferromagnetic systems have attracted an increasing interest. In this case the quantum hardware is thought of as a collection of coupled molecules, each corresponding to a different qubit [14, 15, 114, 115]. In order to explore these possibilities, new and very precise setups are currently built and new methods and strategies are developed. The field of molecular nanomagnets evolves towards molecular electronics and spintronics, which are both rapidly emerging fields of nanoelectronics with a strong potential impact for the realization of new functions and devices helpful for information storage as well as quantum information. New projects aim at the merging of the two fields by the realization of molecular junctions that involve a molecular nanomagnet. In order to tackle the challenge of controlled connection at the single molecule level, molecular self assembly on nanojunctions obtained by the technique of electromigration was used [3, 16, 17]. Futhermore, a new nano-SQUID with carbon nanotube Josephson junctions was developed [2], which should be sensitive enough to study individual magnetic molecules that are attached to the carbon nanotube. Such techniques will lead to enormous progress in the understand- 18 Wolfgang Wernsdorfer ing of the electronic and magnetic properties of isolated molecular systems and they will reveal intriguing new physics [1]. The author is indebted to F. Balestro, N. Bendiab, L. Bogani, E. Bonet, J.-P. Cleuziou, E. Eyraud, D. Lepoittevin, L. Marty, C. Thirion. This work is partially financed by STEP MolSpinQIP, ERC-Advanced Grant MolNanoSpin, ANR Pnano MolNanoSpin. References 1 Bogani, L. and Wernsdorfer, W. (2008) ‘Molecular spintronics using singlemolecule magnets’, Nat. 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