1 Introduction

The interface between classical and quantum physics has always been an interesting area, but its importance has nevertheless grown with the current explosive thrusts in nanoscience. Taking devices to the limit of miniaturization (the mesoscale and beyond) where quantum effects become important makes it essential to understand the interplay between the classical properties of the macroscale and the quantum properties of the microscale. This is particularly true in nanomagnetism, where many potential applications require monodisperse, magnetic nanoparticles.

In order to put this review into perspective, let us consider Fig. 1, which presents a scale of size ranging from macroscopic down to nanoscopic sizes. The unit of this scale is the number of magnetic moments in a magnetic system. At macroscopic sizes, a magnetic system is described by magnetic domains (Weiss 1907) [1] that are separated by domain walls. Magnetization reversal occurs via nucleation, propagation, and annihilation of domain walls (see the hysteresis loop on the left in Fig. 1 which was measured on an individual elliptic CoZr particle of 1 μm × 0.8 μm and a thickness of 50 nm [2]). Shape and width of domain walls depend on the material of the magnetic system, on its size, shape and surface, and on its temperature [3,4].

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. Hence, the magnetization might reverse by uniform rotation, curling or other nonuniform modes (see the hysteresis loop in the middle of Fig. 1). For system sizes well below the domain wall width or the exchange length, one must take into account explicitly the magnetic moments (spins) and their couplings. The theoretical description is complicated by the particle's boundaries.

Magnetic molecular clusters (also called molecular nanomagnets or single-molecule magnets) are the final point in the series of smaller and smaller units from bulk matter to atoms (Fig. 1). Up to now, they have been the most promising candidates for observing quantum phenomena because they have a well-defined structure with well-characterized spin ground state and magnetic anisotropy. These molecules can be regularly assembled in large crystals where all molecules often have the same orientation. Hence, macroscopic measurements can give direct access to single-molecule properties.

This review introduces the basic concepts that are needed to understand the quantum phenomena observed in molecular nanomagnets. Most tunneling studies presented here were performed by magnetization measurements on single crystals using an array of micro-SQUIDs [4]. This review concludes by mentioning new trends towards molecular spintronics using junctions and nano-SQUIDs [5].

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, in short Mn₁₂ acetate [6], and an octanuclear iron(III) oxo-hydroxo cluster of formula [Fe₈O₂(OH)₁₂(tacn)₆]⁸⁺ where tacn is a macrocyclic ligand, in short Fe₈ [7]. 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 reversal of the magnetization of about 67 K for Mn₁₂ acetate [8–12] and 25 K for Fe₈ [13].

Thermally activated quantum tunneling of the magnetization has first been evidenced in both systems [14–18]. 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 side [19–26]. 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 [18,27] 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 [28,29]. During the last years, many new molecular nanomagnets were presented (see, for instance, Refs. [30–34]) which show also tunneling at low temperatures. The largest molecular nanomagnet is currently an Mn₈₄ molecule [35] that has a size of a magnetic nanoparticle (Fig. 2) and an Mn₆ has currently a record anisotropy barrier approaching 100 K [36].

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 [Fe₈O₂(OH)₁₂(tacn)₆]⁸⁺ where tacn is a macrocyclic ligand (1,4,7-traiazcyclononane), short Fe₈ (Fig. 3) [7].

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. Spin polarized neutron scattering showed that all Fe ions have a spin 5/2, six spins up and two down [38]. 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. For a nanomagnet like the Fe₈ molecular cluster it has the following Hamiltonian:

S_x, S_y, and S_z are the three components of the spin operator, D and E are the anisotropy constants which were determined via high frequency electron paramagnetic resonance (HF-EPR) (D/k_B ≈ 0.275 K and E/k_B ≈ 0.046 K [13]), and the last term of the Hamiltonian describes the Zeeman energy associated with an applied field H→. This Hamiltonian defines hard, medium, and easy axes of magnetization in x-, y-, and z-directions, respectively (Fig. 4). It has an energy level spectrum with (2S + 1) = 21 values which, to a first approximation, can be labelled by the quantum numbers m = −10, −9, …, 10 choosing the z-axis as quantization axis. The energy spectrum, shown in Fig. 5, can be obtained by using standard diagonalization techniques of the [21 × 21] matrix describing the spin Hamiltonian S = 10. At H→=0, the levels m = ±10 have the lowest energy. When a field H_z is applied, the energy levels with m < −2 increase, while those with m > 2 decrease (Fig. 5). Therefore, energy levels of positive and negative quantum numbers cross at certain fields H_z. It turns out that for Fe₈ the levels cross at fields given by μ₀H_z ≈ n × 0.22 T, with n = 1,2,3,…. The insert of Fig. 5 displays the details at a level crossing where transverse terms containing S_x or S_y 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” Δ, can be tuned by an applied field in the xy-plane (Fig. 4) via the S_xH_x and S_yH_y Zeeman terms (Section 3.2).

The effect of these avoided level crossings can be seen in hysteresis loop measurements (Fig. 6). 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.

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 [40–42]. 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 [43–47]. The tunneling probability P when sweeping the longitudinal field H_z at a constant rate over an avoided energy level crossing (Fig. 7) is given by

Pm,m′=1−exp[−πΔm,m′22ℏgμB|m−m′|μ0ⅆHz/ⅆt].

(2)

Here, m and m′ are the quantum numbers of the avoided level crossing, dH_z/dt is the constant field sweeping rates, g ≈ 2, μ_B the Bohr magneton, and ℏ is Planck's constant.

With the Landau–Zener model in mind, we can now start to understand qualitatively the hysteresis loops (Fig. 6). Let us start at a large negative magnetic field H_z. At very low temperature, all molecules are in the m = −10 ground state (Fig. 5). When the applied field H_z is ramped down to zero, all molecules will stay in the m = −10 ground state. When ramping the field over the Δ_−10,10-region at H_z ≈ 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 [Eq. (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 [48]. When the field H_z is now further increased, 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 Δ_−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 de-excite quickly to the m = 10 state by emitting a phonon. An analogous procedure happens when the applied field reaches the Δ_−10,10−n regions (n = 2, 3, …) until all 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 Δm = 1 or 2, there is a phonon cascade for higher applied fields.

In order to apply quantitatively the Landau–Zener formula [Eq. (2)], we first saturated the crystal of Fe₈ clusters in a field of H_z = −1.4 T, yielding an initial magnetization M_in = −M_s. 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 Δ_−10,10−n [Eq. (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 [48], that is for LZ tunnel probabilities small than few percent. The deviations at lower sweeping rates are mainly due to the hole-digging mechanism [49] which slows down the relaxation (Section 7.2). In other words, the spin has time to interact with its environment at lower sweeping rates. These interactions are not included in the LZ model and it fails to describe the observed tunneling rates. 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 can tune the tunnel splittings Δ_m,m′ via the S_x and S_y 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). Fig. 8 presents a detailed study of the tunnel splitting Δ_±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 (Fig. 4). For small angles φ the tunneling rate oscillates with a period of ∼0.4 T, whereas no oscillations showed for large angles φ [48]. In the latter case, a much stronger increase of Δ_±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 μ₀H_z = n × 0.22 T with n = 0, 1, 2, …. The corresponding tunnel splittings Δ_−10,10−n oscillate with almost the same period of ∼0.4 T (Fig. 8). 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 the Kramers' suppression of tunneling, predicted for half-integer spins [50,51]. A similar strong dependence on the azimuth angle φ was observed for all studied resonances.

3.2.1 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 made by using the path integral formalism [52]. Here [39], the oscillations are explained by constructive or destructive interference of quantum spin phases (Berry phases) of two tunnel paths (instanton trajectories) (Fig. 4). Since our experiments were reported, the Wentzel–Kramers–Brillouin theory has been used independently by Garg [53] and Villain and Fort [54]. The surprise is that although these models [39,53,54] 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. [53] and [54] and then proven in Ref. [55] Some extensions or alternative explications of Garg's result can be found in Refs. [56–60].

The period of oscillation is given by [39]

ΔH=2kBgμB2E(E+D)

(3)

where D and E are defined in Eq. (1). We find a period of oscillation of ΔH = 0.26 T for D = 0.275 K and E = 0.046 K as in Ref. [13]. 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 Section 3.3.

3.2.2 Exact numerical diagonalization

In order to quantitatively reproduce the observed periodicity we included fourth-order terms in the spin Hamiltonian [Eq. (1)] as employed in the simulation of inelastic neutron scattering measurements [61,62] and performed a diagonalization of the [21 × 21] matrix describing the S = 10 system. For the calculation of the tunnel splitting we used D = 0.289 K, E = 0.055 K [Eq. (1)] and the fourth-order terms as defined in [61] 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 [63] and neutron scattering measurements [62].

The calculated tunnel splittings for the states involved in the tunneling process at the resonances n = 0, 1, and 2 are reported in Fig. 9, showing the oscillations as well as the parity effect for odd resonances.

3.2.3 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 and Schilling and Van Hemmen and Sütö [64,65] 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, an ion possessing an odd number of electrons must have a ground state that is at least doubly degenerate, even in the presence of crystal fields and spin–orbit interactions [66].

The predicted spin-parity effect can be observed by measuring the tunnel splitting as a function of the transverse field [67]. 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 3.2.1). 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 [68].

3.3 A classical approach with applications to the quantum regime

Recently, the molecular (or bottom–up) approach has reached the size regime of the classical (or top–down) approach to nanoscale magnetic materials [35]. Indeed, a giant Mn₈₄ SMM was reported with a 4 nm diameter torus structure, exhibiting both magnetization hysteresis and quantum tunneling. The study of such large systems is greatly complicated by the fact that the spin Hilbert space is huge and it is impossible to treat such systems with exact matrix diagonalization methods. However, since some SMMs are now as large as some classical nanoparticles, it raises the interesting possibility that classical models commonly employed to study the latter may be used to obtain a first order understanding for large molecular systems [69]. Indeed, we herein propose the use of the classical Néel–Brown model [70–72] of thermally activated magnetization reversal of a magnetic single-domain particle in order to study large SMMs [69]. This method allows us to determine important parameters that characterize the magnetic properties of the SMM: the energy barrier, the magnetic anisotropy constant, the spin, τ₀, and the crossover temperature from the classical to the quantum regime. The method is particularly useful for SMMs having low-lying energy states and not showing quantum tunneling steps in hysteresis loops. In such systems, electron paramagnetic resonance (EPR) measurements often exhibit only very broad absorption peaks which do not allow the determination of the magnetic anisotropy.

3.3.1 The Néel–Brown model of thermally activated magnetization reversal

The method is based on the Néel–Brown model of thermally activated magnetization reversal of a magnetic single-domain particle which has two equivalent ground states of opposite magnetization separated by an energy barrier due to magnetic anisotropy [70–72]. The system can escape from one state to the other either by thermal activation over the barrier at high temperatures or by quantum tunneling at low temperatures. At sufficiently low temperatures and at zero field, the energy barrier between the two states of opposite magnetization is much too high to observe an escape process. However, the barrier can be lowered by applying a magnetic field in the opposite direction to that of the particle's magnetization. When the applied field is close enough to the reversal field, thermal fluctuations are sufficient to allow the system to overcome the barrier, and the magnetization is reversed.

This stochastic escape process can be studied via the relaxation time method consisting in the measurement of the probability that the magnetization has not reversed after a certain time. In the case of an assembly of identical and isolated particles, it corresponds to measurements of the relaxation of magnetization. According to the Néel–Brown model, the probability that the magnetization has not reversed after a time t is given by

P(t)=ⅇ−t/τ

(4)

and τ (inverse of the reversal rate) can be expressed by an Arrhenius law of the form:

τ(T,H)=τ0ⅇΔE(H)/kBT

(5)

where ΔE(H) is the field dependent energy barrier height and τ₀ is the inverse of the attempt frequency. In most cases, ΔE(H) can be approximated by

ΔE(H)≈E0(1−H/Hc0)α

(6)

where H_c⁰ is the reversal field at zero temperature, E₀ is the barrier height at zero applied field, and α is a constant of the order of unity (for most cases 1.5 ≤ α ≤ 2). In the case of a Stoner–Wohlfarth particle [73,74] with uniaxial anisotropy and the field applied along the easy axis of magnetization, all constants can be determined analytically [70,73]: α = 2, E₀ = KV, and H_c⁰ = 2K/M_s, where K is the uniaxial anisotropy constant, V is the particle volume, and M_s is the saturation magnetization. For SMMs with dominating uniaxial anisotropy: α = 2, E₀ = DS², and H_c⁰ = 2DS/gμ₀μ_B. However, in general, all constants depend slightly on fine details of the magnetic anisotropy and the direction of the applied field H [75,76].

In order to study the field dependence of the relaxation time τ(T, H) and to obtain the parameters of the model, the decay of magnetization has to be studied at many applied fields H and temperatures T. This is experimentally very time consuming and complicated by the fact that the equilibrium magnetization is temperature dependent and difficult to obtain for long relaxation times. In addition, for fast relaxation times the initial magnetization depends on the field sweep rates to apply the field. The number of exploitable decades for τ values is therefore limited for relaxation time measurements.

A more convenient method for studying the magnetization decay is by ramping the applied field at a given rate [77] and measuring the coercive field H_c (the field value to obtain zero magnetization), which is then measured as a function of the field sweep rate and temperature.

The mathematical transformation from a reversal time probability (Eqs. (4) and (5)) to a reversal field probability was first given by Kurkijärvi [78] for the critical current in SQUIDs. Later, Gunther and Barbara calculated similar expressions for magnetic domain wall junctions [79]. A more general calculation was evaluated by Garg [80]. Here, we use a simplified version [77]: the probability density of reversal of a stochastic process is −dP/dt = P/τ and the maximum of the probability density can be derived from d²P/dt² = P(1 + dτ/dt)/τ² = 0. This gives dτ/dt = −1. The application to Eq. (5) leads to ΔE(H)=kBTln[kBT/(τ0(ⅆE/ⅆH)(ⅆH/ⅆT))]. Using Eq. (6) we find Eq. (7). The mean reversal field of an assembly of identical particles or SMMs is approximated by the coercive field H_c:

Hc(T,v)≈Hc0(1−[kTE0ln(cv)]1/α)

(7)

where the field sweeping rate is given by v = dH/dt; H_c⁰ is the coercive field at zero temperature, and c = H_c⁰k_BT/[τ₀αE₀(1 − H_c/H_c⁰)^α−1] [77].

3.3.2 Application to Mn₁₂ SMMs

The method is here applied to an Mn₁₂ SMMs [69] with a spin ground state of S = 10. Fig. 10 shows typical hysteresis loops with a series of quantum steps separated by plateaus [81]. In order to apply the above method, the temperature and field sweep rate dependences of the coercive fields H_c were measured and plotted in Fig. 11. As expected for a thermally activated process, H_c increases with decreasing temperature and increasing field sweep rate. Furthermore, all measurements showed an almost logarithmic dependence of H_c on the field sweep rate. H_c becomes temperature independent below about 0.6 K.

The validity of Eq. (7) was tested by plotting the set of H_c(T, v) values as a function of [T ln(c/v)]^1/2 where c = H_c⁰k_BT/τ₀2E₀(1 − H_c/H_c⁰). If the underlying model is sufficient, all points should collapse onto one straight line by choosing the proper values for the constant τ₀. We found that the data of H_c(T, v) fell on a master curve provided τ₀ = 2.1 × 10⁻⁷ s.

At low temperatures, strong deviation from the master curves is observed. In order to investigate the possibility that these low-temperature deviations are due to an escape from the metastable potential well by tunneling, a common method for classical models is to replace the real temperature T by an effective temperature T^∗(T) in order to restore the scaling plot [82]. In the case of tunneling, T^∗(T) should saturate at low temperatures. Indeed, the ansatz of T^∗(T) as shown in the insert of Fig. 12(b), can restore unequivocally the scaling plot demonstrated by a straight master curve (Fig. 12(b)). The flattening of T^∗ corresponds to a saturation of the escape rate, which is a necessary signature of tunneling. The crossover temperature T_c can be defined as the temperature where the quantum rate equals the thermal one. The insert of Fig. 12(b) gives T_c = 0.97 K. The slopes and the intercepts of the master curves give E₀ = 72.4 K and H_c⁰ = 9.1 T. The E₀ values are in good agreement with those extracted from Arrhenius plots (69 K). This result allows us to estimate the spin ground state using S = 2E₀/(gμ_Bμ₀H_c⁰): S = 11. This differs slightly from S = 10 determined via magnetization measurements. This deviation is due to quantum effects in the thermally activated regime and is considered further below.

Several points should be mentioned: (i) the classical regime of the model corresponds in most SMMs to the thermally activated tunneling regime with tunneling close to the top of the energy barrier. Because all parameters are deduced from this regime, small deviations from the exact values are expected; (ii) the field dependence of the energy barrier can be obtained directly using Eq. (7) and is plotted in Fig. 13; (iii) Eq. (7) is not valid for fields which are close to H = 0 because the model only takes into account the transitions from the metastable to the stable well. However, close to H = 0, transitions between both wells are possible leading to a rounding of the master curve at small fields; (iv) the method can be applied to powder samples with random orientations of the molecules. In this case, α ≈ 1.5, νE₀ = DS² where ν can be calculated [75,76], and the intercept of the master curve gives H_c⁰/2; (v) in the case of a distribution of anisotropies, different parts of the distribution can be probed by applying the method at different M values; (vi) this method is insensitive to small intermolecular interactions when H_c is larger than the typical interaction field; and (vii) the method can be generalized for 1D, 2D, and 3D networks of spins. In this case, Eq. (6) describes a nucleation barrier.

4 Quantum dynamics of a dimer of nanomagnets

We present here a new family of dimers of nanomagnets [83] in which antiferromagnetic coupling between two single-molecule magnets (SMMs) results in quantum behavior different from that of the individual SMMs. Each SMM acts as a bias on its neighbor, shifting the quantum tunneling resonances of the individual SMMs. Hysteresis loop measurements on a single crystal of SMM–dimers established quantum tunneling of the magnetization via entangled states of the dimer. This shows that the dimer really does behave as a quantum-mechanically coupled dimer, and also allows the measurement of the longitudinal and transverse superexchange coupling constants [84]. The experimental evidence for entangled states was confirmed by an electron paramagnetic resonance (EPR) study [85].

The compound [Mn₄O₃Cl₄(O₂CEt)₃(py)₃] crystallizes in the hexagonal space group R3(bar) with two Mn₄ molecules per unit cell lying head-to-head on a crystallographic S₆ symmetry axis [83] (Fig. 14). Each Mn₄ monomer has a ground state spin of S = 9/2, well separated from the first excited state S = 7/2 by a gap of about 300 K [86]. The Mn–Mn distances and the Mn–O–Mn angles are similar and the uniaxial anisotropy constant is expected to be the same for the two dimer systems. These dimers are held together via six C–H···Cl hydrogen bonds between the pyridine (py) rings on one molecule and the Cl ions on the other, and one Cl···Cl Van der Waals interaction. These interactions lead to an antiferromagnetic superexchange interaction between the two Mn₄ units of the [Mn₄]₂ dimer [83]. Dipolar couplings between Mn₄ molecules can be easily calculated and are more than one order of magnitude smaller than the exchange interaction.

Before presenting the measurements, we summarize a simplified spin Hamiltonian describing the [Mn₄]₂ dimer [83]. Each Mn₄ SMM can be modeled as a giant spin of S = 9/2 with Ising-like anisotropy [Eq. (1)]. The corresponding Hamiltonian is given by

Fig. 16 shows typical hysteresis loops (magnetization versus magnetic field scans) with the field applied along the easy axis of magnetization of [Mn₄]₂, that is, parallel to the S₆ axis of the dimer, corresponding to the z-axis of the spin Hamiltonian. These loops display step-like features separated by plateaus. The step heights are temperature independent below ∼0.35 K [83]. The steps are due to resonant quantum tunneling of the magnetization (QTM) between the energy states of the [Mn₄]₂ dimer (see Figs. 15 and 16 for a discussion of 5 tunnel transitions). QTM has been previously observed for most SMMs, but the novelty for [Mn₄]₂ dimers is that the QTM is now the collective behavior of the complete S = 0 dimer of exchange-coupled S = 9/2 Mn₄ quantum systems. This coupling is manifested as an exchange bias of all tunneling transitions, and the resulting hysteresis loop consequently displays unique features, such as the absence for the first time in a SMM of a QTM step at zero field [83].

Even though the five strongest tunneling transitions are observed in Fig. 16, fine structure was not observed. For example, the hysteresis loops do not show the splitting of the (9/2,7/2) states (labelled 5′ and 5″), which we suspected might be due to line broadening. Usually, line broadening in SMMs is caused by dipolar and hyperfine interactions [88], and distributions of anisotropy and exchange parameters. In most SMMs, the zero-field resonance is mainly broadened by dipolar and hyperfine interactions because distributions of anisotropy parameters do not affect the zero-field resonance. For an antiferromagnetically coupled dimer, however, this resonance is shifted to negative fields. Therefore, a distribution of the exchange coupling parameter J_z can further broaden this resonance. In fact, we showed that the latter is the dominant source of broadening [84]. We used the ‘quantum hole-digging’ method (see Section 7.2) [49,88–91] to provide direct experimental evidence for the transitions 5′ and 5″, which established tunneling involving entangled dimer states and allowed us to determine J_xy [84].

5 Resonant photon absorption in Cr₇Ni antiferromagnetic rings

Magnetic molecules are currently considered among the most promising electron spin based quantum systems for the storing and processing of quantum information. For this purpose, ferromagnetic [92] and antiferromagnetic [93,94] systems have attracted an increasing interest [95,96]. In the latter case the quantum hardware is thought of as a collection of coupled molecules, each corresponding to a different qubit. The main advantages would arise from the fact that they are extremely small and almost identical, allowing to obtain, in a single measurement, statistical averages of a large number of qubits. The magnetic properties can be modeled with an outstanding degree of accuracy. And most importantly, the desired physical properties can be engineered chemically.

The suitability of Cr-based antiferromagnetic molecular rings for the qubit implementation has been proposed [95,96]. The substitution of one metal ion in a Cr-based molecular ring with dominant antiferromagnetic couplings allows its level structure and ground state degeneracy to be engineered [97,98]. A Cr₇Ni molecular ring was characterized by means of low-temperature specific-heat and torque-magnetometry measurements, thus determining the microscopic parameters of the corresponding spin Hamiltonian. The energy spectrum and the suppression of the leakage-inducing S-mixing render the Cr₇Ni molecule a suitable candidate for the qubit implementation [95,96,99].

In this section we report the first micro-superconducting quantum interference device (micro-SQUID) [100] studies of the Cr₇Ni molecular ring [101]. Electron paramagnetic resonance (EPR) methods are combined with high-sensitivity magnetization measurements. We found very narrow resonant photon absorption lines which are mainly broadened by hyperfine interactions. Similar measurement was performed on Ni₄ molecules [102] but quantum coherence was not yet directly observed.

The Cr₇Ni molecular ring, is based on a homometallic ring with formula [Cr₈F₈(O₂CCMe₃)₁₆]. The eight chromium(III) ions lie at the corners of a regular octagon [97]. Each edge of the octagon is bridged by one fluoride ion and two pivalate ligands. There is a large cavity at the centre of the ring. If a single chromium(III) ion is replaced by a metal(II) ion, for example nickel(II), this makes the ring anionic and a cation can be incorporated in the cavity. Thus we can make [H₂NMe₂][Cr₇NiF₈(O₂CCMe₃)₁₆] [98]. If crystallized from a mixture of THF and MeCN the Cr₈ and Cr₇Ni compounds are isostructural, crystallizing in the tetragonal space group, P4.

The measurements were made in a dilution cryostat using a 20 μm sized single crystal of Cr₇Ni. The magnetic probe was a micro-SQUID array [4,100] equipped with three coils allowing application of a field in any direction and with sweep rates up to 10 T/s. The electromagnetic radiation was generated by a frequency synthesizer triggered with a nanosecond pulse generator. This setup allows continuous variation of the frequency from 0.1 Hz to 20 GHz, with pulse lengths ∼1 ns to continuous radiation [103]. Using a 50 μm sized gold radio frequency (RF) loop, the RF radiation field was directed in a plane perpendicular to the applied static field μ₀H. The microwave power of the generator could be varied from −80 to 20 dB m (10⁻¹¹–10⁻¹ W). The sample absorbs only a small fraction of the generator power. This fraction is, however, proportional to the microwave power of the generator. The microwave amplitude B_RF can be estimated with the method described in Ref. [100]. We found B_RF ≈ 1 mT at 4 GHz and 15 dB m which is more than 1000 times larger than in our previous work on V₁₅ [100].

Fig. 17a shows magnetization versus applied field curves for several field sweep rates at a cryostat temperature of 0.04 K. The magnetization loops exhibit a clear hysteresis which is characteristic for the phonon-bottleneck regime with a spin–phonon relaxation time to the cryostat of a few seconds [104]. Note that the degeneracy of the Kramers doublet is lifted due to internal transverse fields (mainly the transverse hyperfine fields). In order to quantify the out-of-equilibrium effect, Fig. 17b presents the same data as in Fig. 17a but the magnetization M is converted into a spin temperature T_S using the following equation [19]:

Fig. 18 shows magnetization curves M(H) in the quasi-static regime with a field sweep rate slow enough (0.14 mT/s) to keep the system at equilibrium. During the field sweep, RF pulses were applied to the sample with a pulse length of 1 μs and a period of 4 s between each pulse. Depending on the RF frequency, clear dips are observed which result from resonant absorptions of photons associated with spin transitions between the quantum numbers m_s = 1/2 and −1/2. After each pulse, the magnetization relaxes back to the equilibrium magnetization (see the fine structure in the insert of Fig. 18).

Typical relaxation measurements at a constant applied field after RF pulses of different durations are shown in Fig. 19. The relaxation is exponential with the rate independent of the pulse length. Detailed studies showed that the relaxation rate is dominated by the phonon-bottleneck regime, that is the spin–phonon relaxation time to the cryostat.

The insert of Fig. 19 presents the change of magnetization ΔM between the magnetization before and after the pulse as a function of the pulse length w.ΔM increases linearly with w for short pulses of few tens of ns. It saturates for w ≈ 10 μs and decrease for very long pulses because of cryostat heating effects. Non-resonant photon absorption is also observed for very long pulses.

The resonant photon absorption lines are often taken to estimate a lower bound on the decoherence time of a qubit. We therefore investigated in more detail the line width observed in Fig. 18. Fig. 20(a) presents a typical power dependence of the line width for continuous irradiation at 4.2 GHz. Resonant photon absorption is clearly visible for a generator power larger than −60 dB m (1 nW). The line saturated at about −20 dB m (10 μW). Fig. 20(b) presents the absorption line for the pulsed technique (see Fig. 18) for several pulse lengths and a generator power of 15 dB m (32 mW, B_RF ≈ 1 mT). The resonant photon absorption is clearly visible for pulse lengths longer than 10 ns. Note that the linewidths in Fig. 20(a) are nearly twice as large as those in Fig. 20(b).

In our case of an assembly of identical spins, the line broadening is mainly due to dipolar and hyperfine interactions. The dipolar coupling energy can be estimated with E_dip/k_B ≈ (gμ_BS)²/V ≈ 0.1 mK (S = 1/2 and V = 6.3 nm³) [96]. The hyperfine coupling with the nuclear spins can be obtained by considering the dipolar interaction of one Cr ion (s = 3/2) with the neighboring F nucleus having a nuclear spin I = 1/2. With g_F = + 5.26 and the distance of d = 0.2 nm between F and Cr ions, the interaction energy is about 0.4 mK for each of the eight F nuclear spins [96]. The hyperfine line broadening of all eight F nuclear spins is about 3 mK which corresponds to 5 mT, in good agreement with the observed Gaussian line widths of about σ = 4 mT in Figs. 18 and 20.

Finally, we discuss the possibility of observing Rabi oscillations with the present setup. Due to inhomogeneous broadening only a lower bound of the coherence time τ_c can be estimated from the resonance lines in Fig. 20: τ_c ≈ B_ν/(σν) ≈ 10 ns with B_ν = 0.166 T, σ = 4 mT, and ν = 4.2 GHz. The corresponding number of coherent flips of the spin system is given by N = τ_c/τ_Rabi with τ_Rabi = 2π/(γB_RF) ≈ 40 ns for B_RF ≈ 1 mT. We obtain N ≈ 0.25 showing that there is no hope of seeing Rabi oscillations in the present conditions. In order to get N ≫ 1, it will be necessary to increase further the radiation field B_RF, to reduce substantially the hyperfine broadening by substituting the F ions with OH groups, and to minimize the dipolar coupling by doping the crystal of Cr₇Ni molecules with Cr₈ molecules.

6 Photon-assisted tunneling in single-molecule magnet

It has also been proposed that molecular nanomagnets could be used as quantum computers by implementing Grover's algorithm [92]. For this to occur it is necessary to be able to generate an arbitrary superpositions of eigenstates of these systems. The suggested way to do this was through the use of multifrequency coherent magnetic radiation in the microwave and radiofrequency range. This would first introduce and amplify the desired phase for each m_S state and this information could be finally readout by standard magnetic resonance techniques. In this approach, advantage is taken from the non-equidistance of the m_S levels of the ground multiplet arising from the large axial anisotropy of these systems, which allows coherent populations of the different m_S levels. A theoretical work pointed out that a very accurate control of pulse shape technique, both in amplitude, duration and choice of frequency is needed to fulfill the condition to design quantum computing devices in molecular nanomagnets [105]. In addition to such basic difficulties, we will see below that the total microwave power convoyed onto the samples cannot exceed a critical value above which non-linear effects occur.

In order to investigate the feasibility of the proposed process any preliminary experiment should aim to understand the effects of microwave absorption on the spin dynamics of these systems at low temperature. The measurements were performed using micro-Hall bars [106]. Continuous microwave radiation was generated by a couple of Gunn diodes equipped with calibrated attenuators. With this experimental setup, which has a very good stability of emitted power and a narrow bandwidth (100 kHz), only relative powers at the output of the diode can be known, which will be referred to as ρ in the following. We irradiated the sample using a 6 mm waveguide equipped with infrared filters in order to reduce heating. The circular polarization was maximized around 97%. The study was performed on a 0.1 mm Fe₈ single crystal.

As schematically depicted in Fig. 21, microwave radiation with a frequencies of 115 GHz corresponds to the energy separation between the ground states m_S = ±S and the first excited states m_S = ±(S − 1) of Fe₈ in zero applied magnetic field [13,107]. If the radiation is linearly polarized, the populations of the first excited states (m_S = ±(S − 1)) in both wells will be enhanced equally (equal transition probability for Δm_S = ±1). On the other hand, the use of circular polarization has the advantage to distinguish between Δm_S = +1 (left polarization, σ⁻ photons) or Δm_S = −1 (right polarization, σ⁺ photons) [19], and the population of only one of the two excited states will be enhanced (Fig. 21). An excess of tunneling from one well to the other is then expected. Therefore, circular polarization can help to distinguish between spin–phonons relaxation, and spin–phonons relaxation modified by the absorption of photons. The first equally affects the two sides of the barrier, i.e. the two branches of the hysteresis loop, while the second modifies the population of only one side of the barrier, i.e. one branch of the hysteresis loop. Any difference observed between the two branches of the hysteresis loop, has to be traced back to photon absorption.

Fig. 22a shows the hysteresis loops of a Fe₈ single crystal with the easy axis parallel to the applied field, measured at 60 mK under irradiation. The tunneling transition near zero field is strongly enhanced for a radiation at 115 GHz. This is in agreement with a photon induced population transfer from m_S = 10 to m_S = 9, and agrees with earlier HF-EPR studies showing strong zero field absorption at about 116 GHz [13,107]. Fig. 22a also shows the expected asymmetry of the hysteresis loops in the presence of circularly polarized radiation. In particular, the height of the zero-field step (first tunnel resonance, n = 0), obtained when sweeping the field from negative saturation, is much less affected than when sweeping from positive saturation. This behavior is completely different from phonon-assisted tunneling (Fig. 22b) and clearly establishes that tunneling is assisted by photons for the matching frequency of 115 GHz. The observation of a more symmetric shape of the hysteresis curve at high microwave power can be explained both by the incomplete microwave polarization and by phonon emission, leading to relaxation on both sides of the barrier.

At detailed study showed that at lowest powers, the tunnel probability increases linearly with power, whereas at higher powers a strongly non-linear regime is observed [106]. The latter might be due to multi-spin and coherent photon transitions.

7 Environmental decoherence effects in nanomagnets

At temperatures below 0.36 K, Fe₈ molecular clusters display a clear crossover from thermally activated relaxation to a temperature independent quantum regime, with a pronounced resonance structure of the relaxation time as a function of the external field (Section 3). It was surprising, however, that the observed relaxation of the magnetization in the quantum regime was found to be nonexponential and the resonance width orders of magnitude too large [18,27]. The key to understand this seemingly anomalous behavior involves the hyperfine fields as well as the evolving distribution of the weak dipole fields of the nanomagnets themselves [88]. Both effects were shown to be the main source of decoherence at very low temperature. At higher temperatures, phonons are another source of decoherence.

In the following sections, we focus on the low temperature and low field limits, where phonon-mediated relaxation is astronomically long and can be neglected. In this limit, the m = ±S spin states are coupled due to the tunneling splitting Δ_±S which is about 10⁻⁷ K for Fe₈ (Section 3.2) with S = 10. In order to tunnel between these states, the longitudinal magnetic energy bias ξ = gμ_BSH_local due to the local magnetic field H_local on a molecule must be smaller than Δ_±S, implying a local field smaller than 10⁻⁸ T for Fe₈ clusters. Since the typical intermolecular dipole fields are of the order of 0.05 T, it seems at first that almost all molecules should be blocked from tunneling by a very large energy bias. Prokof'ev and Stamp have proposed a solution to this dilemma by proposing that fast dynamic nuclear fluctuations broaden the resonance, and the gradual adjustment of the dipole fields in the sample caused by the tunneling brings other molecules into resonance and allows continuous relaxation [88]. Some interesting predictions are briefly reviewed in the following section.

7.2 Hole digging method to study dipolar distributions and hyperfine couplings

Motivated by the Prokof'ev–Stamp theory [88], we developed a new technique—which we call the hole digging method—that can be used to observe the time evolution of molecular states in crystals of molecular clusters. It allowed us to measure the statistical distribution of magnetic bias fields in the Fe₈ system that arises from the weak dipole fields of the clusters themselves. A hole can be “dug” into the distribution by depleting the available spins at a given applied field. Our method is based on the simple idea that after a rapid field change, the resulting short time relaxation of the magnetization is directly related to the number of molecules which are in resonance at the given applied field. Prokof'ev and Stamp have suggested that the short time relaxation should follow a t-relaxation law [Eq. (11)]. However, the hole digging method should work with any short time relaxation law—for example, a power law

M(Hz,t)=Min+(Meq(Hz)−Min)(Γshort(Hz)t)α,

(13)

where Γ_short is a characteristic short time relaxation rate that is directly related to the number of molecules which are in resonance at the applied field H_z, and 0 < α < 1 in most cases. α = 0.5 in the Prokof'ev–Stamp theory [Eq. (11)] and Γ_sqrt is directly proportional to P(H_z) [Eq. (12)]. The hole digging method can be divided into three steps (Fig. 23):

• 1. Preparing the initial state. A well-defined initial magnetization state of the crystal of molecular clusters can be achieved by rapidly cooling the sample from high down to low temperatures in a constant applied field H_z⁰. For zero applied field (H_z = 0) or rather large applied fields (H_z > 1 T), one yields the demagnetized or saturated magnetization state of the entire crystal, respectively. One can also quench the sample in a small field of few milliteslas yielding any possible initial magnetization M_in. When the quench is fast (<1 s), the sample's magnetization does not have time to relax, either by thermal or by quantum transitions. This procedure yields a frozen thermal equilibrium distribution, whereas for slow cooling rates the molecule spin states in the crystal might tend to certain dipolar ordered ground state.
• 2. Modifying the initial state—hole digging. After preparing the initial state, a field H_dig is applied during a time t_dig, called “digging field and digging time”, respectively. During the digging time and depending on H_dig, a fraction of the molecular spins tunnel (back and/or fourth); that is, they reverse the direction of magnetization. Note that the field sweeping rate to apply H_dig should be fast enough to minimize the change of the initial state during the field sweep.
• 3. Probing the final state. Finally, a field H_z^probe is applied (Fig. 23) to measure the short time relaxation from which one yields Γ_short [Eq. (13)] which is related to the number of spins which are still free for tunneling after step (2).

The entire procedure is then repeated many times but at other fields H_z^probe yielding Γ_short(H_z, H_dig, t_dig) which is related to the distribution of spins P(H_z, H_dig, t_dig) that are still free for tunneling after the hole digging. For t_dig = 0, this method maps out the initial distribution.

7.3 Intermolecular dipole interaction in Fe₈

We applied the hole digging method to several samples of molecular clusters and quantum spin glasses. The most detailed study has been done on the Fe₈ system. We found the predicted t relaxation [Eq. (11)] in experiments on fully saturated Fe₈ crystals [27,111] and on nonsaturated samples [49]. Fig. 24 displays a detailed study of the dipolar distributions revealing a remarkable structure that is due to next-nearest-neighbor effects [49]. These results are in good agreement with simulations [91,112].

For a saturated initial state, the Prokof'ev–Stamp theory allows one to estimate the tunnel splitting Δ_±S. Using Eqs. (3), (9) and (12) of Ref. [88], along with integration, we find ∫Γsqrtⅆξ=c(ξ0/ED)(Δ±S2/4ℏ), where c is a constant of the order of unity which depends on the sample shape. With E_D = 15 mT, ξ₀ = 0.8 mT, c = 1, and Γ_sqrt [49,89], we find Δ_±10 = 1.2 × 10⁻⁷ K which is close to the result of Δ_±10 = 1.0 × 10⁻⁷ K obtained by using a Landau–Zener method (Section 3.1) [48].

Whereas the hole digging method probes the longitudinal dipolar distribution (H_z direction), the Landau–Zener method can be used to probe the transverse dipolar distribution by measuring the tunnel splittings Δ around a topological quench.

7.4 Hyperfine interaction in Fe₈

The strong influence of nuclear spins on resonant quantum tunneling in the molecular cluster Fe₈ was demonstrated for the first time [89] by comparing the relaxation rate of the standard Fe₈ sample with two isotopic modified samples: (i) ⁵⁶Fe is replaced by ⁵⁷Fe, and (ii) a fraction of ¹H is replaced by ²H. By using the hole digging method, we measured an intrinsic broadening which is driven by the hyperfine fields (Fig. 25). Our measurements are in good agreement with numerical hyperfine calculations [89,91]. For T > 1.5 K, the influence of nuclear spins on the relaxation rate is less important, suggesting that spin–phonon coupling dominates the relaxation rate.

8 Molecular spintronics using single-molecule magnets

A revolution in electronics is in view, with the contemporary evolution of two novel disciplines, spintronics and molecular electronics [113–116]. A fundamental 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 [117]. 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.

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 [113,118,119]. 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 [118–120] and organic materials [121], 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 nanoscale), and the tailoring of new molecules possessing increased conductance and functionalities are driving electronics to its ultimate molecular scale limit [122], and the so-called molecular electronics is now being intensively investigated.

In experiments of molecular electronics, the measuring devices are usually constituted of 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 tunneling 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 [117,123] 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. These systems, combining the advantages of molecular scale with the properties of bulk magnetic materials, look attractive for high-density information storage and also, owing to their long coherence times [124–126], for quantum computing [92,95,96]. Moreover, their molecular nature leads to appealing quantum effects of the static and dynamic magnetic properties. The rich physics behind the magnetic behavior produces interesting effects like negative differential conductance and complete current suppression [127,128], which could be used in electronics. Last but not least, 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 [117]: molecular spin-transistor, molecular spin-valve and spin filter, molecular double-dot devices, and carbon nanotube-based nano-SQUIDs [5]. The main purpose is to fully control the initialization, the manipulation and the readout 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.

A double-dot device (Fig. 26) is one possible way to molecular spintronics [117]. It is a three terminal device, where the current passes through a non-magnetic quantum 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 nano-SQUID [5]. 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 the 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.

9 Conclusion

In conclusion, we presented detailed measurements which demonstrated that 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. The tunneling in this system is remarkable because it does not show up at the lowest orders of perturbation theory.

A new family of supramolecular, antiferromagnetically exchange-coupled dimers of single-molecule magnets (SMMs) has recently been reported [83]. Each SMM acts as a bias on its neighbor, shifting the quantum tunneling resonances of the individual SMMs. Hysteresis loop measurements on a single crystal of SMM-dimers have established quantum tunneling of the magnetization via entangled states of the dimer. This showed that the dimer really does behave as a quantum-mechanically coupled dimer. The transitions are well separated, suggesting long coherence times compared to the time scale of the energy splitting [84]. This result is of great importance if such systems are to be used for quantum computing.

Molecules with small spin have also been studied. For example, time-resolved magnetization measurements were performed on a spin 1/2 molecular complex, so-called V₁₅ [129]. Despite the absence of a barrier, magnetic hysteresis is observed over a time scale of several seconds. A detailed analysis in terms of a dissipative two-level model has been given, in which fluctuations and splittings are of the same energy. Spin–phonon coupling leads to long relaxation times and to a particular “butterfly” hysteresis loop [104,130]. We presented magnetization measurements on a crystal of Cr₇Ni antiferromagnetic rings with a spin 1/2. Irradiation with microwaves at frequencies between 1 and 10 GHz leads to observation of very narrow resonant photon absorption lines which are broadened by hyperfine and spin–spin interactions.

The use of circularly polarized microwaves allowed us to show for the first time the phenomenon of photon-assisted tunneling in magnetism, using a single-molecule magnet Fe₈ [106]. In accordance with the selection rules for EPR spectroscopy [19], circularly polarized radiation promotes the transition m_S = 10 to m_S = 9 with Δm_S = −1, giving an effect of magnetic dichroism at millimeter wavelengths. At lowest powers, the tunnel probability increases linearly with power, whereas at higher powers a strongly non-linear regime is observed. The latter might be due to multi-spin and coherent photon transitions.

Quantum coherence between states of opposite magnetization has been observed on molecules with small spin [124–126] and without barrier. However, for most SMMs, dipole–dipole and hyperfine interactions are source of decoherence. In other words, when a spin has tunneled through the barrier, it experiences a huge modification of its environment (hyperfine and dipolar) which prohibits the back tunneling. Prokof'ev and Stamp suggested three possible strategies to suppress the decoherence [131,132]: (i) Choose a system where the NMR frequencies far exceed the tunnel frequencies making any coupling impossible; (ii) Isotopically purify the sample to remove all nuclear spins; (iii) Apply a transverse field to increase the tunnel rate to frequencies much larger than hyperfine field fluctuations. Several groups are currently working on such proposals.

Concerning the perspectives of the field of single-molecule magnets, we expect that chemistry is going to play a major role through the synthesis of novel larger spin clusters with strong anisotropy. We want to stress that there are already many other molecular nanomagnets (the largest is currently an Mn₈₄, Fig. 2, and an Mn₆ has currently a record anisotropy barriers approaching 100 K [36]), which are possible model systems. We believe that more sophisticated theories are needed which describe the dephasing effects of the environment onto the quantum system. These investigations are important for studying the quantum character of molecular clusters for applications like “quantum computers”. The first implementation of Grover's algorithm with molecular nanomagnets has been proposed [92]. 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 [93–96]. 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 [127,128]. Furthermore, a new nano-SQUID with carbon nanotube Josephson junctions was developed [5], 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 understanding of the electronic and magnetic properties of isolated molecular systems and they will reveal intriguing new physics [117].