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Int. J. Nanotechnology, Vol. x, No. x, xxxx 1

Molecular Nanomagnets: towards molecular

spintronics

Wolfgang Wernsdorfer

Institut Neel, CNRS & Universite 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, haveattracted much interest in recent years both from experimental and theoretical

point of view. These systems are organometallic clusters characterized by a largespin ground state with a predominant uniaxial anisotropy. The quantum nature ofthese systems makes them very appealing for phenomena occurring on the meso-scopic scale, i.e., at the boundary between classical and quantum physics. Belowtheir blocking temperature, they exhibit magnetization hysteresis, the classicalmacroscale property of a magnet, as well as quantum tunneling of magnetizationand quantum phase interference, the properties of a microscale entity. Quan-tum effects are advantageous for some potential applications of single-moleculemagnets, e.g. in providing the quantum superposition of states for quantum com-puting, but are a disadvantage in others such as information storage. It is believedthat single-molecule magnets have a potential for quantum computation, in par-ticular because they are extremely small and almost identical, allowing to obtain,in a single measurement, statistical averages of a larger number of qubits. Thisreview introduces few basic concepts that are needed to understand the quantumphenomena observed in molecular nanomagnets and discusses new trends of the

field of molecular nanomagnets towards molecular spintronics.

Keywords: Single-molecule magnets, molecular nanomagnets, molecular spin-tronics, 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 Wurzburg, Ger-many, in 1966, received his education in Physics in Wurzburg, Lyon, and thenGrenoble, where he is at present Research Director at the Centre National dela Recherche Scientifique. During his PhD in the low-temperature laboratory(CNRS, Grenoble) Wolfgang Wernsdorfer and collaborators developed a uniquedevice (micro-SQUID) for measuring magnetic properties of nanostructures witha billion times higher sensitivity than commercial magnetometers (Bronze Medalfrom CNRS, 1998). His instrument allows observation of the magnetic behaviorof nanomagnets containing less than a thousand magnetic centers, which is stilla world record. Using the unique advantages of this device, Wolfgang Werns-dorfer has studied a variety of peculiar phenomena in depth, such as tunnellingof magnetization in molecular clusters, leading to the Agilent Europhysics Prizein 2002 and the International Olivier Kahn Award in 2006.

Over the years, the innovative approach to such studies combined with the rec-ognized superiority of this micro-SQUID have led to worldwide collaborationwith most other notorious groups working on synthesizing molecular magnetsto investigate single-molecule magnet behavior in more than 350 systems. The

Copyright c 200x Inderscience Enterprises Ltd.


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2 Wolfgang Wernsdorfer

leading work of Wolfgang Wernsdorfer and collaborators in this field is at theheart of todays 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 mag-

nets, which combine the classic macroscale properties of a magnet with the quantum prop-

erties 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 quan-

tum coherence, so that fundamental investigations are shifting from metals to semiconduct-

ing [4, 5], and organic materials [6], which potentially offer best promises for cost, integra-

tion and versatility. For example, organic materials are already used in applications such

as organic light-emitting diodes (OLED), displays and organic transistors. The concomi-

tant 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 high-

density information storage and also, owing to their long coherence times [10, 11, 12],


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Molecular spintronics 3

Figure 1 Representative examples of the peripheral functionalization of the outer organic shell ofthe 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 diameterof 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 conductanceand 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 asso-

ciated to molecular junctions. They are constituted by an inner magnetic core with a sur-

rounding 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 con-

ducting properties. SMMs come in a variety of shapes and sizes and permit selective sub-

stitutions 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 graft-

ing 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 proper-

ties 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


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Figure 2 Scale of size that goes from macroscopic down to nanoscopic sizes. The unit of thisscale is the number of magnetic moments in a magnetic system (roughly corresponding to the numberof 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, andannihilation of domain walls (hysteresis loop on the left). When the system size is of the order ofmagnitude of the domain wall width or the exchange length, the formation of domain walls requirestoo much energy. Therefore, the magnetization remains in the so-called single-domain state, and themagnetization reverse by uniform rotation or nonuniform modes (middle). SMMs are the final pointin the series of smaller and smaller units from bulk matter to atoms and magnetization reverses viaquantum tunneling (right).

Mesoscopic physics

Macroscopic Nanoscopic

Permanent

magnets

Micron

particles

Nanoparticles Clusters Molecular

clusters

Individual

spins

S = 102 0 101 0 108 106 105 104 103 102 10 1

Multi-domain Single-domain Magnetic moments

Nucleation, propagation and

annihilation of domain walls

Uniform rotation

Curling

Resonant tunneling, quantization,

quantum thermodynamics

-1

0

1

- 4 0 - 2 0 0 2 0 4 0

M

/MS

0H(mT)

-1

0

1

- 1 00 0 1 0 0

M

/MS

0H(mT)

-1

0

1

- 1 0 1

M

/MS

0H(T)

Fe8

1K 0.1K

0.7K

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 inter-ference [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 molec-

ular spintronics [1] using junctions [3] and nano-SQUIDs [2].

2 Overview of molecular nanomagnets

Molecular nanomagnets or single-molecule magnets (SMMs) are mainly organic mole-

cules 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 [Fe8O2(OH)12(tacn)6]8+ where tacn is a macrocyclic ligand, short Fe8 [33].

Both systems have a spin ground state ofS = 10and an Ising-type magnetic anisotropy,which stabilizes the spin states with m = 10 and generates an energy barrier for the


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Figure 3 Size scale spanning atomic to nanoscale dimensions. On the far right is shown a high-resolution transmission electron microscopyview along a [110] direction of a typical 3 nm diametercobalt 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 indicatedsmaller Mn nanomagnets, which are drawn to scale. An alternative means of comparison is the Neelvector (N), which is the scale shown. The green arrows indicate the magnitude of the Neel vectors forthe indicated SMMs, which are 7.5, 22, 61, and 168 for Mn4, Mn12, Mn30and Mn84, respectively.

Mn4

Mn30

Mn12

1 10 100 1000N

Quantum world

Molecular (bottom-up) approach

Classical world

Classical (top-down) approach

Mn84

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 quan-

tum numbers |m|, not far below the top of the energy barrier, and the molecules thentunnel 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 relaxationtime 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 re-laxation 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 ma-

gentic nanoparticle (Figure 3). The record anisotropy barriers of 89 K is currently a Mn6SMM [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 [Fe8O2(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.


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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 ofthe ground stateS= 10.

Spin polarized neutron scattering showed that all Fe ions have a spin 5/2, six spins up and

two down [56]. This rationalizes theS = 10 spin ground state that is in agreement withmagnetization measurements.

In principle, a multi-spin Hamiltonian can be derived taking into account of all ex-

change 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 giantspin 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 = DS2z + E

S2x S2y

+ gB0S H (1)

Sx,Sy , andSz are the three components of the spin operator, D and Eare the anisotropyconstants which were determined via high frequency electron paramgnetic resonance (HF-

EPR) (D/kB 0.275 K andE/kB 0.046 K [37]), and the last term of the Hamiltonian

describes the Zeeman energy associated with an applied field H. This Hamiltonian de-fines hard, medium, and easy axes of magnetization in x,y , andz directions, respectively(Figure 5). It has an energy level spectrum with(2S+ 1) = 21 values which, to a firstapproximation, can be labeled by the quantum numbers m = 10,9,..., 10 choosingthez-axis as quantization axis. The energy spectrum, shown in Figure 6, can be obtainedby using standard diagonalisation techniques of the [21 21] matrix describing the spin

Hamiltonian S= 10. At H= 0, the levels m= 10have the lowest energy. When a field

Hz is applied, the energy levels withm < 2increase, while those with m >2 decrease(Figure 6). Therefore, energy levels of positive and negative quantum numbers cross atcertain fieldsHz . It turns out that for Fe8 the levels cross at fields given by 0Hz n0.22 T, with n = 1, 2, 3,.... The inset of Figure 6 displays the details at a level cross-ing where transverse terms containing Sx or Sy spin operators turn the crossing into anavoided level crossing. The spin Sis in resonance between two states when the locallongitudinal field is close to an avoided level crossing. The energy gap, the so-called tun-

nel spitting, can be tuned by an applied field in thexy-plane (Figure 5) via theSxHxandSyHy Zeeman terms (Section 3.2).


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Figure 5 Unit sphere showing degenerate minimaAandBwhich are joined by two tunnel paths(heavy lines). The hard, medium, and easy axes are taken in x-, y-, and z-direction, respectively. Theconstant transverse field Htrans for tunnel splitting measurements is applied in the xy-plane at an

azimuth angle. At zero applied field H= 0, the giant spin reversal results from the interference oftwo quantum spin paths of opposite direction in the easy anisotropy yz-plane. For transverse fields indirection 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 beconverted in an area integral, yielding that destructive interferencethat is a quench of the tunnelingrateoccurs whenever the shaded area isk/S, wherek is an odd integer. The interference effectsdisappear quickly when the transverse field has a component in the y-direction because the tunnelingis then dominated by only one quantum spin path.

Z

Y

X

H

A

B

trans

Easy axis

Medium

axis

Hard

axis

Figure 6 Zeeman diagram of the 21 levels of theS = 10 manifold of Fe8 as a function of thefield applied along the easy axis [equation (1)]. From bottom to top, the levels are labeled withquantum numbersm= 10,9,..., 0. The levels cross at fields given by 0Hz n 0.22 T, withn = 1, 2, 3,.... The insetdisplays the detail at a level crossing where the transverse terms (terms

containingSx or/andSy spin operators) turn the crossing into an avoided level crossing. The largerthe 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.


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Figure 7 Hysteresis loops of a single crystal of Fe8 molecular clusters at different temperatures.The longitudinal field (zdirection) was swept at a constant sweeping rate of 0.014 T/s. The loopsdisplay 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 tempera-ture independent below 0.35 K, demonstrating quantum tunneling at the lowest energy levels

-1

-0.5

0

0. 5

1

-1.2 -0.6 0 0.6 1.2

M

/MS

0 Hz( T )

1K

0.7K

0.5K

0.4, 0.3

and 0.04K

3.1 LandauZener tunneling in Fe8

The nonadiabatic transition between the two states in a two-level system has first

been discussed by Landau, Zener, and Stuckelberg [57, 58, 59]. The original work by

Zener concentrates on the electronic states of a bi-atomic molecule, while Landau and

Stuckelberg considered two atoms that undergo a scattering process. Their solution of

the time-dependent Schrodinger equation of a two-level system could be applied to many

physical systems and it became an important tool for studying tunneling transitions. The

LandauZener model has also been applied to spin tunneling in nanoparticles and clusters

[60, 61, 62, 63, 64]. The tunneling probabilityPwhen sweeping the longitudinal field Hzat a constant rate over an avoided energy level crossing (Figure 8) is given by

Pm,m = 1 exp

2m,m

2gB|m m|0dHz/dt

. (2)

Here,m andm are the quantum numbers of the avoided level crossing, dHz/dt is theconstant field sweeping rates,g 2,B the Bohr magneton, and is Plancks constant.

With the LandauZener model in mind, we can now start to understand qualitatively

the hysteresis loops (Figure 7). Let us start at a large negative magnetic fieldHz . At verylow temperature, all molecules are in the m = 10 ground state (Figure 6). When theapplied fieldHz is ramped down to zero, all molecules will stay in them = 10groundstate. When ramping the field over the 10,10region at Hz 0, there is a LandauZener tunnel probability P10,10 to tunnel from the m = 10 to the m = 10 state.

P10,10 depends on the sweeping rate [equation (2)]; that is, the slower the sweepingrate, the larger the value ofP10,10. This is clearly demonstrated in the hysteresis loopmeasurements 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 = 10state which became a metastable state. The next chance to escape from this state is when

the field reaches the10,9region. There is a LandauZener tunnel probabilityP10,9 totunnel from them= 10to them= 9state. Asm= 9is an excited state, the moleculesin this state relax quickly to the m= 10state by emitting a phonon. A similar mechanismhappens when the applied field reaches the 10,10n regions (n = 2, 3, . . . ) until all


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Figure 8 Detail of the energy level diagram near an avoided level crossing. m and m are thequantum numbers of the energy level. Pm,m is the LandauZener tunnel probability when sweepingthe 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)].

Energy

Magnetic field Hz

| m' >

1 P

1 - P

| m >

| m > | m' >

molecules are in them = 10ground state; that is, the magnetization of all molecules isreversed. As phonon emission can only change the molecule state by m= 1 or 2, thereis a phonon cascade for higher applied fields.

In order to apply quantitatively the LandauZener formula [equation (2)], we first satu-

rated the crystal of Fe8clusters in a field ofHz = 1.4T, yielding an initial magnetizationMin = Ms. Then, we swept the applied field at a constant rate over one of the resonancetransitions and measured the fraction of molecules which reversed their spin. This proce-

dure yields the tunneling rate P10,10n and thus the tunnel splitting 10,10n [equa-tion (2)] withn = 0, 1, 2, . . . .

We first checked the predicted LandauZener sweeping field dependence of the tunnel-

ing 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

LandauZener 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 xyplane adjusts the tunnel splittingsm,m via the Sxand Syspin operators of the Zeeman terms that do not commute with the spin Hamiltonian. This

effect can be demonstrated by using the LandauZener method (Section 3.1). Figure 9

presents a detailed study of the tunnel splitting 10at the tunnel transition between m=10, as a function of transverse fields applied at different angles , defined as the azimuth

angle between the anisotropy hard axis and the transverse field (Figure 5). For small angles the tunneling rate oscillates with a period of0.4 T, whereas no oscillations showedup for large angles [30]. In the latter case, a much stronger increase of10 withtransverse field is observed. The transverse field dependence of the tunneling rate for

different resonance conditions between the state m = 10and (10 n)can be observedby sweeping the longitudinal field around 0Hz = n 0.22 T with n = 0 , 1, 2 , . . . .The corresponding tunnel splittings10,10n oscillate with almost the same period of0.4 T (Figure 9). In addition, comparing quantum transitions betweenm = 10 and(10 n), withn even or odd, revealed a parity (or symmetry) effect that is analogous to


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Figure 9 Measured tunnel splitting as a function of transverse field for (a) several azimuthangles at m = 10 and (b) 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 half-

integer spins. It should also be mentioned that internal dipolar and hyperfine fields hinder a quenchof which is predicted for an isolated spin.

0 0.2 0.4 0.6 0.8 1 1.2 1.40.1

1

10

Tunnelsplitting

(10-7

K)

Magnetic transverse field (T)

n = 0

0

7

20 50 90

(a)

-0.4 -0.2 0 0.2 0 .4 0.6 0.8 1 1.2 1 .40.1

1

10

Tunnelsplitting

(10-7

K)

Magnetic transverse field (T)

n = 0

n = 1

n = 2 0

(b)

the Kramers suppression of tunneling predicted for half-integer spins [66, 67]. A similar

strong dependence on the azimuth angle 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 trajec-

tories) (Figure 5). Since our experiments were reported, the WentzelKramersBrillouintheory 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 beennoted in Refs. [70] and [71] and then proven in Ref. [72] Some extensions or alternative

explications of Gargs result can be found in Refs. [73, 74, 75, 76, 77].

The period of oscillation is given by [69]

H= 2kBgB

2E(E+ D) (3)

where Dand Eare defined in equation (1). We find a period of oscillation ofH= 0.26 Tfor D = 0.275 K and E= 0.046 K as in Ref. [37]. This is somewhat smaller than theexperimental value of0.4 T. We believe that this is due to higher-order terms of the spin

Hamiltonian which are neglected in Gargs calculation. These terms can easily be includedin 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 [2121]


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Figure 10 Calculated tunnel splitting as a function of transverse field for (a) quantum transi-tion betweenm = 10at several azimuth angles and (b) quantum transition between m = 10and(10 n)at = 0 (Section 3.4). The fourth-order terms suppress the oscillations of at large

transverse fields |Hx|.

0 0.4 0.8 1.2 1.6 20.01

0.1

1

10

100

1000

Tunnelsplitting

(10-7

K)

Magnetic tranverse field (T)

0

= 90 5030

20

10

5

(a)

0 0.4 0.8 1.2 1.6 20.01

0.1

1

10

10 0

1000

Tunnelsplitting

(10-7

K)

Magnetic tranverse field (T)

n = 1

(b)

n = 0

n = 2

matrix describing theS = 10system. For the calculation of the tunnel splitting we usedD = 0.289K, E = 0.055K [equation (1)] and the fourth-order terms as defined in [78]withB04 = 0.72 10

6 K,B24 = 1.01 105 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

resonancesn = 0, 1, and 2 are reported in Figure 10, showing the oscillations as well asthe 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 Suto [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 resultof 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 Mn 12clusters with a spin ground state ofS = 10 andS = 19/2 showing oscillations of thetunnel 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].


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Figure 11 Transport experiments on SMMs. a) Schematic using a STM tip to perform trans-port on surface grafted SMMs. b) Schematic of SMM-based molecular transistors, in which agate voltage can modulate transport. c) [Co(TerPy)2] molecular magnet with alkyl spacers, per-

mitting transport in the weakly coupled regime [85]. d) [Co(TerPy)2] molecular magnet with nospacers, showing strong coupling and the Kondo effect [85]. e) Divanadium [(N,N,N-trimethyl-1,4,7-triazacyclononane)2V2(CN)4(-C4N4)] molecular magnet showing the Kondo effect only in thecharged 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, molec-

ular electronics, and quantum computing. The resulting field aims at manipulating spins

and charges in electronic devices containing one or more molecules [1]. The main advan-

tage 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, molec-

ular 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 theo-

retical, 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 elec-

trode 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


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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 elec-

trode. 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 elec-

trodes, 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 withVg and the measurements show Coulomb-blockade diamonds [91].

The experimental realization of this scheme has been achieved using Mn12 with thiol-

containing 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 lig-

ands [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 moretraditional 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 character-

istics 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 ef-

fects. For example, a direct link between shot noise measurements and the detailed mi-

croscopic magnetic structure of SMMs has been proposed [94], allowing the connection

of structural and magnetic parameters to the transport features and therefore a characteri-

zation 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-


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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 orien-

tation 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 cen-

ters [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 vari-

able 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 withthe effectiveS= 1/2state usually attributed to Co2+ ions at low temperatures but a Landfactor g= 2.1is found. This is unexpected for Co2+ ions, characterized by high spin-orbitcoupling 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 mole-

cule 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 trans-

parency [86]. The molecule can be tuned with the gate voltageVg into two differentlycharged states. The neutral state, due to antiferromagnetic coupling between the two mag-

netic centers, hasS= 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 hasa nonzero spin moment. This nicely demonstrates that magnetic molecules with multi-

ple 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 ofVg indicates that not only spinbut also orbital degrees of freedom play an important role on the Kondo resonance of sin-

gle 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]. Thisis 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


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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 magnetiza-tion, 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 ar-row) are partially reflected back. b) Anti-parallel configuration: majority spin-up electrons are onlypartially transmitted by the differently polarized molecule, while the minority spin-down electronspass unaffected. Assuming that the spin-up contribution to the current is larger in the magnetic con-tact, this configuration has higher resistance than that of the previous case. c) Theoretical schematicof a spin-valve configuration with nonmagnetic metal electrodes [8] and d) proposed molecular mag-net 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 trans-

verse 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 forhalf-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 leasttwo magnetic elements (Fig. 12a-b). SVs change their electrical resistance for different

mutual alignments of the magnetizations of the electrodes and of the molecule, analo-

gous 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 fundamen-

tal 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


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Figure 13 Molecular double-dot devices. Magnetic molecules proposed for grafting on sus-pended carbon nanotubes connected to Pd electrodes (form left to right): a C 60 fullerene including arare-earth atom, the Mn12SMM 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 in-sulating layer.

molecule in between two magnetic leads, which can be metallic or semiconducting. The

first experiments sandwitched a C60fullerene 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 mag-

netic 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 po-

larization. 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 mo-

ment 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 bedetected 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-


17/25


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 moleculeand 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 quan-

tum 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 werepossible. In the coming years, chemistry is going to play a major role through the syn-

thesis 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 Grovers algorithm with molecular nanomagnets has been pro-

posed [13]. Antiferromagnetic systems have attracted an increasing interest. In this case

the quantum hardware is thought of as a collection of coupled molecules, each correspond-

ing 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 impactfor 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 chal-

lenge of controlled connection at the single molecule level, molecular self assembly on

nanojunctions obtained by the technique of electromigration was used [3, 16, 17]. Futher-

more, 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-


18/25


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.

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