Accessing antiferromagnetism in metallic thin films through anomalous Hall effect

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SCHOOL OF INDUSTRIAL AND INFORMATION ENGINEERING

Master of Science in Engineering Physics

Accessing Antiferromagnetism in Metallic Thin Films

Through Anomalous Hall Effect

Supervisor:

Prof. Matteo CANTONI

Assistant Supervisor:

Dott. Marco ASA

Thesis by:

Riccardo PAZZOCCO

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Abstract

The unprecedented technological progress of the last few decades is intimately bound to the possibility to store large amount of information in a fast, energy-efficient and non-volatile way.

Among all the solid state technologies under development, antiferromagnet (AFM)-based memories may offer many advantages compared to the currently available magnetic random-access memories: larger storage density, faster information processing, and robustness against external magnetic perturbation could be achieved.

However, the most challenging task to accomplish remains the electrical writing and reading of the information encoded by the magnetic state of the AFM.

This thesis work deals with the design and development of an experimental setup, called spinning current Hall magnetometer, aiming at investigating the relationship among the magnetic ordering in AFMs and their electrical conduction properties. This setup provides a state-of-the-art sensitivity to out-of-plane magnetic moments while being readily and cost-effectively feasible in most academic laboratories.

The technique has been initially validated on the ferromagnetic heterostructure Ta/CoFeB/MgO and on the antiferromagnetic alloy IrMn. Then, the spinning current Hall magnetometer was used to investigate the Pt/Cr heterostructure, comprising antiferromagnetic (Cr) and nonmagnetic (Pt) metals. The existence of a finite magnetic moment induced in Pt by the magnetic proximity interaction with Cr, predicted by

ab-initio calculations, has been proved. This induced moment can be controlled by means

of magnetic field cooling across the Néel temperature of the AFM layer and is stable against external perturbations in the AFM phase.

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The possibility to measure electrically this effect might open new pathways towards the realization of AFM-based memory devices.

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Sommario

Lo sviluppo tecnologico senza precedenti degli ultimi anni è intimamente vincolato alla possibilità di immagazzinare gradi quantità di informazioni in maniera non volatile, rapida e con basso dispendio energetico.

Memorie realizzate con materiali antiferromagnetici (AFM) offrirebbero molti vantaggi rispetto ad altre tecnologie in via di sviluppo. Se comparate alle random-access memories magnetiche attualmente disponibili, esse garantirebbero una maggior densità di memorizzazione, robustezza rispetto a campi magnetici esterni e un’elaborazione più rapida delle informazioni.

Tuttavia, la scrittura e la lettura elettrica dello stato magnetico dell’AFM costituiscono delle sfide tecnologiche fondamentali da affrontare.

Il presente lavoro descrive lo sviluppo di un setup sperimentale, chiamato magnetometro Hall a correnti rotanti, volto ad investigare l’influenza dell’ordinamento magnetico sulle proprietà di conduzione di un AFM. Tale setup fornisce una sensibilità di misura della componente fuori dal piano di momenti magnetici comprabile con lo stato dell’arte pur essendo economicamente e rapidamente realizzabile nella maggior parte dei laboratori accademici.

La tecnica è stata inizialmente testata sulla ben nota eterostruttura ferromagnetica Ta/CoFeB/MgO e sulla lega antiferromagnetica IrMn. Successivamente il magnetometro è stato utilizzato per studiare l’eterostruttura Pt/Cr, composta da un metallo antiferromagnetico (Cr) e da uno non magnetico (Pt). I risultati dimostrano l’esistenza di un momento magnetico indotto nel Pt dall’interazione di prossimità magnetica con il Cr, in accordo con quanto predetto da calcoli ab-initio. Raffreddando il sistema in un campo magnetico esterno attraverso la temperatura di Néel dell’AFM, è possibile controllare tale momento indotto che resulta invece insensibile a campi esterni quando il sistema transisce

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Questi risultati evidenziano come gli effetti di prossimità permettano di identificare l’ordine magnetico nell’AFM, traducendolo in una magnetizzazione indotta nel metallo adiacente.

La possibilità di misurare elettricamente questo effetto può offrire nuovi spunti per la realizzazione di memorie basate sull’impiego di antiferromagneti.

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Ringraziamenti

Molte persone hanno contribuito a rendere quest’esperienza al contempo essenziale alla mia crescita intellettuale e formativa per la mia persona.

In primis desidero ringraziare il Prof. Riccardo Bertacco per avermi concesso l’opportunità di svolgere questo lavoro di tesi presso il suo gruppo di ricerca.

Ringrazio in particolar modo il Prof. Matteo Cantoni per i preziosi consigli, il tempo e l’attenzione senza i quali questo elaborato non esisterebbe nella forma presente.

Un doveroso e sentito ringraziamento va al Dott. Marco Asa per aver messo a mia disposizione la sua esperienza, chiarito pazientemente i miei tanti dubbi e dedicato molto tempo e sforzi alla supervisione di questo elaborato. Desidero sottolineare come egli abbia contribuito, al pari di nessun altro prima d’ora, alla mia formazione.

Ringrazio tutti i componenti del gruppo di lavoro per avermi calorosamente accolto e sostenuto durante questa esperienza.

Grazie a Marco (di nuovo), Sha, Mons, Sara, Daniela per aver condiviso con me la loro esperienza aiutandomi a scegliere con maggior consapevolezza la migliore carriera da intraprendere nel prossimo futuro.

Grazie a Mattia, Livia, Alessia e Giacomo che hanno iniziato con me questa bella esperienza.

Grazie agli amici di una vita (Fede, Zona, Possi, Paniz, Seba, Trevi, Anna, Tome) per tutte le esperienze vissute insieme.

Un grazie speciale alla mia ragazza Ilaria, fonte della mia gioia, per avermi sopportato nei momenti più impegnativi, accompagnandomi in questo lungo percorso ed illuminando ogni giorno con il suo splendido sorriso.

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List of figures

Fig. 1.1: local spin transfer torque in FM/AFM heterostructures ... 5

Fig. 1.2: local spin transfer torque in NM/AFM heterostructures ... 6

Fig. 1.3: AMR and TAMR effects ... 8

Fig. 1.4: state definition and readout in FeRh AFM memory resistor ... 11

Fig. 1.5: CuMnAs memory device. ... 12

Fig. 1.6: SP-STS of Cr[001] surface ... 14

Fig. 2.1: symmetry breaking phase transitions ... 22

Fig. 2.2: spontaneous magnetization in ferromagnetic systems ... 26

Fig. 2.3: magnetic structure in antiferromagnets ... 27

Fig. 2.4: temperature dependence of the magnetic susceptibility of an antiferromagnet ... 30

Fig. 2.5: magnetic phases of IrMn ... 32

Fig. 2.6: wave vector dependent magnetic susceptibility ... 35

Fig. 2.7: schematic illustration of the wave vector dependent magnetic susceptibility ... 37

Fig. 2.8: commensurate and incommensurate SDW ... 38

Fig. 2.9: magnetic anisotropy of the longitudinal resistance in bulk Cr ... 39

Fig. 2.10: typical configuration of a Hall effect measurement ... 41

Fig. 2.11: typical dependence of the Hall resistance on the out of plane applied field for a ferromagnetic sample. ... 45

Fig. 2.12: exchange split energy bands. ... 49

Fig. 2.13: spatial distribution of magnetic moments in the Pt/Cr heterostructure ... 51

Fig. 3.1: four-points and two-points measurement configurations ... 53

Fig. 3.2: Hall measurements contact configuration ... 55

Fig. 3.3: phenomenological model of the self Hall effect ... 58

Fig. 3.4: Van der Pauw contacts configuration ... 60

Fig. 3.5: Delta measurement process ... 61

Fig. 3.6: switching circuit implementation ... 63

Fig. 3.7: control panel of the LabVIEW program implemented to manage the measurements ... 64

Fig. 3.8: complete experimental setup ... 66

Fig. 3.9: Lasse cluster tool ... 67

Fig. 3.10: schematic representation of an MBE apparatus ... 69

Fig. 3.11: schematic representation of the sputtering process ... 70

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Fig. 3.14: photoresist prophile obtained by a inverse lithographic process ... 76

Fig. 3.15: lithographic steps required for the devices fabbrication ... 79

Fig. 4.1: schematic representation of a magnetic tunneling junction ... 81

Fig. 4.2: Hall resistance in CoFeB ... 83

Fig. 4.3: schematic diagram of the exchange bias effect ... 86

Fig. 4.4: diluted antiferromagnet in an external magnetic field ... 87

Fig. 4.5: schematic representation of the magnetic field configurations adopted during the field cooling phases ... 89

Fig. 4.6: longitudinal resistivity IrMn ... 90

Fig. 4.7: (a) transverse resistivity and (b) Hall resistivity of IrMn ... 91

Fig. 4.8: IrMn transverse resistances measured with (a) I = 100 µA and (b) I = 10 µA. ... 93

Fig. 4.9: IrMn Hall resistances measured with I = 100 µA and I = 10 µA ... 94

Fig. 5.1: the epitaxial relation for the Cr/MgO bilayer ... 97

Fig. 5.2: schematic representation of the field applied during the field cooling processes. ... 99

Fig. 5.3: Pt(2 nm)/Cr(100 nm): (a) Hall resistivity and (b) carrier density obtained during the field cooling phases. ... 100

Fig. 5.4: Pt(2 nm)/Cr(100 nm): longitudinal resistivity ... 102

Fig. 5.5: Pt(2 nm)/Cr(100 nm): (a) transverse resistance and (b) Hall resistance measured in the warming phases ... 104

Fig. 5.6: Pt(2 nm)/Cr(25 nm): (a) longitudinal resistivity and (b) carrier density obtained during the field cooling phases. ... 107

Fig. 5.7: Pt(2 nm)/Cr(25 nm): (a) transverse resistance and (b) Hall resistance measured in the warming phases ... 107

Fig. 5.8 Pt(2 nm)/Cr(25 nm): (a) transverse resistance and (b) Hall resistance measured in the warming phases with maximum temperature of 200 K ... 108

Fig. 5.9: Pt(2 nm)-Au(2 nm)/Cr(50 nm): longitudinal resistivity obtained during the field cooling processes. ... 110

Fig. 5.10: Pt(2 nm)-Au(2 nm)/Cr(50 nm): carrier density signals measured during the field cooling phases. ... 110

Fig. 5.11: Hall resistances measured for the Au(2 nm)/Cr(50 nm) and Pt(2 nm)/Cr(50 nm) samples ... 111

Fig. 5.12: (a) pressure dependence of the Néel temperature and the relative longitudinal resistivity anomaly in bulk Cr specimens. (b) Temperature dependence of the longitudinal resistivity for bulk Cr specimens under different applied pressures. ... 113

Fig. 5.13: (a) Néel temperature for Cr films of different thicknesses grown on LiF substrates; (b) temperature dependence of the longitudinal resistivity in 90 nm thick Cr film ... 114

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Fig. 5.16: schematic representation of the magnetic moments in the Pt/Cr heterostructure as a function of the temperature during the cooling (a,b,d,e) and warming (c,f) phases ... 120 Fig. 5.17: schematic representation of the in-plane transverse SDW in Pt/Cr heterostructure ... 121

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List of tables

Tab. 1: summary of the dependencies of the anolaous Hall resistivity on the longitudinal one and on the scattering lifetime for the three mechanisms of the AHE. ... 47 Tab. 2: summary of the contacts configurations required to perform both the Hall and van der Pauw

measurements ... 62 Tab. 3: relays states corresponding to all the contacts configurations required to perform both the Hall and

van der Pauw measurements ... 64 Tab. 4: residual offset and saturation value of the Hall resistance measured with 10 µA and 100 µA in

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1 Introduction

The work presented in this master thesis is focused on the development of a recently proposed technique for the magnetic and electrical characterization of antiferromagnetic thin films and heterostructures and on its application to a scientific case, i.e. Cr thin films for spin electronics applications. Spin electronics applied to antiferromagnets, called antiferromagnet spintronics, is a recent research topic aiming to exploit the peculiar properties of these materials for a more efficient and reliable information processing and storage. In this section, the state of the art in antiferromagnetic spintronics is presented focusing, in particular, on the current phenomena exploited to store and read information in antiferromagnetic systems. In order to underline the advantages and drawbacks of the system developed in this work, some examples of the available techniques to characterize these materials are considered.

1.1 Developments in information processing

Nowadays humanity is experiencing an unprecedented technological progress. The continuously increasing demand for more efficient information elaboration and storage has driven the impressive growth of semiconductor industry and the development of research in solid state physics to identify new interesting phenomena suitable for these purposes. Spintronics, short term for spin electronics, represents one of the most promising candidates for future applications. In classical electronics the information is embodied in the electrical charge degree of freedom. As an example, in dynamic random access memories (DRAM) the binary information is represented by the presence or lack of charge in a capacitor. In spintronic devices, the spin angular momentum of electrons is used as physical entity for data representation.

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magnetic properties in ferromagnetic (FM) metals [1]. This discovery raised the attention of the scientific community on the large variety of effects involving FM systems. In these materials the exchange interaction (see sect. 2.1) forces the magnetic moments of the material to line up along the same direction, resulting in a spontaneous magnetization. As a consequence of that, the electronic states close to the Fermi level, relevant to the electrical conduction, have different characteristics for electrons with opposite spin directions. Hence, a connection between magnetic and transport properties of FM systems exists. This is at the basis of classical spintronic devices as it allows the electrical detection of the magnetization, which is intrinsically non-volatile, and thus suitable for a permanent information storage.

The first widespread application based on these principles is related to the giant magnetoresistance (GMR) effect discovered in the late 1980s by Peter Grünberg and Albert Fert, Nobel Prizes in Physics in 2007 [2]. In a GMR device two FM metal layers are separated by a nonmagnetic one and the resistance of the heterostructure is dependent on the relative orientation of the magnetization of the two FMs. These microelectronic devices, in which high and low resistance states are realized by the interplay between the charge and spin of moving carriers, are called spin-valves. GMR sensors are used, e.g., to read data in magnetic hard disks. The resistance modulation of a GMR heterostructure can be as high as 55% at room temperature [3].

In 1975, Jullière discovered that the resistance of an heterostructure comprising two FM electrodes separated by a nonmagnetic insulator is dependent on the relative direction of the two FM magnetizations [4]. This phenomenon, called tunneling magnetoresistance (TMR), is due to the spin dependency of the tunneling probability across the insulating barrier, which changes when the magnetizations of the two FM electrodes are parallel or antiparallel. The resistance variation found by Jullière was only 14% at 4.2 K. With the improvement of technology, in 2004 Parkin [5] and Yuasa [6] realized Fe/MgO/Fe TMR junctions showing a variation up to 200% at room temperature, whereas a record value of 604% at room temperature has been achieved with the structure CoFeB/MgO/CoFeB by Ikeda in 2008 [7]. Since then, TMR spin valves have replaced GMR structures in the

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based on TMR bit-cells are a commercial reality and a promising counterpart for semiconductor-based RAM.

Antiferromagnets (AFM) represent another broad class of magnetic systems. They have been discovered in 1930’s by Louis Néel. In AFMs the exchange interaction forces the neighbouring spins to align in opposite directions. As a consequence, the magnetic moments in these materials sum up destructively, resulting in a zero net magnetization. This persists also in presence of large magnetic fields as high as 12 T at room temperature [8]. AFMs are currently used in spintronic applications as pinning layers in GMR and TMR devices (sensors, memory cells, ..) exploiting the exchange bias effect [9]. Due to the interaction with a neighboring AFM layer, the magnetization of a FM layer is pinned in a fixed direction. This is useful in GMR and TMR heterostructures since an external magnetic field, as the one produced by a bit stored in a magnetic hard disk, flips only the magnetization of the “free” FM layer with respect to the pinned one. This allows for the electrical readout of the information because the resistance of the spin valve changes according to the relative orientation of the magnetization in the two FM layers.

The magnetic state of FM can be easily controlled with external magnetic fields whereas the lack of any net magnetic moment and the insensitivity to external perturbations makes AFM state both difficult to detect and to impose. This issue was pointed out by Néel itself during its Nobel lecture: “antiferromagnets are extremely interesting from the theoretical viewpoint, but do not seem to have any application”. So far, the “passive role” of AFMs as pinning layers has been their main application in spintronics. However, since few years the idea of the AFM playing an “active” role has been considered. In 2011, a spin-valve-like AFM-based tunnel junction [10] has been proposed. A further improvement has been the realization, in 2013, of a purely AFM-based device [11], without any FM element, providing many advantages if compared to FM-based systems. A larger scalability of AFM-based memories would be afforded by the absence of magnetic stray fields, due to the zero magnetization, preventing cross-talk among neighboring memory elements. Moreover, the information stored in the AFM state would be insensitive to extremely high external magnetic fields (e.g. 12 T at room temperature

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for Cr2O3). Eventually, the magnetization dynamics in AFMs is faster than in FMs [12]

and so higher elaboration frequencies, up to the THz range, could be reached.

All of these outstanding properties justify the rich research activity in recent years aiming to find a reliable and efficient way to write and read the magnetic states of several AFM-based heterostructures [13]. In the following sections, a general introduction to the wide variety of magnetotrasport effects and their applications to write (sect. 1.2.1) and read (sect. 1.2.2) information in AFMs will be presented along with two examples of AFM-based memory devices recently realized (sect. 1.2.3).

1.2 Antiferromagnets based spintronics

1.2.1 Encoding information in antiferromagnets

The magnetic structure in AFMs can be seen as two interpenetrating ferromagnetically ordered sublattices whose magnetizations are staggered (see sect. 2.3). These sum up destructively resulting in the zero macroscopic magnetization of the AFM.

In order to encode information into these materials, it is necessary to control the direction of the staggered magnetization. This is typically done cooling the AFM in an external magnetic field through its Néel temperature, below which the antiferromagnetic phase is stable [14]. The orientation of the staggered magnetization can be varied changing the direction of the field. This method, although reliable and well established, is not adequate for spintronics applications. In a memory array, each bit cell should be locally heated and cooled in a strongly confined magnetic field. This would for sure strongly limit the storage speed and the device scalability.

Clearly, electrical writing is the most appealing methodology from the technological viewpoint. The earliest proposed techniques where based on spin transfer torque (STT), widely employed for the writing of MRAM cells.

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The typical configuration employed for this purpose is shown in Fig. 1.1.

An electrical current flows perpendicular to the plane of the structure composed by a FM and an AFM layer. The magnetization of the FM lies in the xy plane and its direction is identified by the versor p. Due to its net magnetization, the electrical current in the FM is accompanied by a net spin flux, i.e. the number of electrons injected in the underlying AFM with spin parallel or antiparallel to p are different.

When an electron from the FM enters in the AFM, it interacts, by means of the exchange interaction, with the local staggered magnetization of the AFM lining up along its direction. If the two film magnetizations are not collinear, a rotation in the magnetization of the AFM is required to compensate for the angular momentum lost by the electron injected by the FM. This is the phenomenological explanation of the STT effect.

The torque induced on the staggered magnetization Mi of the AFM is Ti = Mi×(Mi×p)

where the index i identifies the two magnetic sublattices. It is important to notice that, from the expression of the torque, when Mi∥p the torque is zero and this configuration is

actually an equilibrium of the system. The fundamental result exposed in [15] is that, for a current density injection above a threshold value (jtr) dependent on the specific material

properties, the Mi∥p state is an unstable equilibrium whereas the stable configuration

corresponds to Mi⊥p. Thus for j > jtr any fluctuation, such as the thermal one, triggers the

Fig. 1.1: local spin transfer torque in FM/AFM heterostructures. The vertical current density (j) injects a net spin flux from the FM (whose magnetization direction is identified by p) into the AFM. If j overcomes a material dependent threshold value, the AFM staggered magnetization (represented by the double-headed arrow) is rotated from the parallel to the orthogonal direction with respect to p.

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from the parallel to the orthogonal direction with respect to the one of the FM [15] as depicted in Fig. 1.1.

Unfortunately, this technique has two main disadvantages:

1) The switching process is not reversible. The STT allows to switch the AFM magnetization from parallel to orthogonal to the FM one, but it is not possible to electrically reverse the operation, i.e. to switch from the perpendicular alignment to the parallel one, without acting on the orientation of the magnetization in the FM layer. 2) A FM layer must be introduced thus the scalability of this AFM-based STT device

would be limited by cross talk effects due to magnetic stray fields.

Introducing the spin orbit coupling, that plays a fundamental role in spintronics, efficient electrical writing of the AFM staggered magnetization can be achieved. In nonmagnetic metals, although there is no net magnetization, the spin orbit interaction links the orbital motion of the conduction electrons with the spin degree of freedom. Consequently, a spin dependency in the scattering processes exists, as explained in [16] and depicted in Fig. 1.2a.

Fig. 1.2: (a) A net spin current js is generated by the charge current jc as a consequence of the dependency

of the scattering probability on the orientation of spin s; (b) switching of the AFM staggered magnetization by means of in plane current that allows to inject a net spin flux from the NM layer by means of the SHE. The reversibility of the process is guaranteed by the possibility to inject a net spin flux polarized in

𝒋_𝒄 𝒋_𝒔 𝒔 a) switch b) switch

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This is called spin Hall effect (SHE) and can be considered as the solid state counterpart of Mott scattering of travelling electrons by heavy nuclei. From a phenomenological point of view, an electron has a different probability to undergo a scattering event that is either right-handed or left-handed with respect to the scattering center depending on its spin polarization (see Fig. 1.2a). As a result, a spin current is induced in the nonmagnetic material without involving any net charge flow. With reference to Fig. 1.2a, for charge current jc flowing in the nonmagnetic metal, the direction of the spin s in the SHE induced

spin current js is given by s = jc×js.

In Fig. 1.2b a typical device geometry comprising a nonmagnetic (NM) layer on top of an AFM is presented. As a result of the SHE, the charge current jc flowing in the plane of the

NM layer injects a net spin current js in the AFM with spin polarization s, where s is the

spin of the conduction electrons. The torque exerted on the staggered magnetization 𝑴_𝒊 of the underlying AFM for an in plane current injected in the NM layer is Ti = Mi×(Mi×(jc×js )) [17].

Contrarily to the STT-induced switching by the perpendicular current in a FM, the SHE-induced torque provides a reversible control of the AFM magnetization by injection of the current in orthogonal directions, as depicted in Fig. 1.2b. These spin orbit induced torques are extremely interesting both from a theoretical and a technological viewpoint as, once properly engineered, a fully electrical writing of the AFM state, not involving any FM layer, could be realized.

1.2.2 Antiferromagnetic state readout

As AFMs are insensitive to external magnetic fields, accessing their magnetic state is far from trivial. Like the writing of the state exposed in the previous section, also the readout of the information stored in the AFM state has been proved to be feasible by means of electrical probes. As stated by Néel during his pioneering studies of these materials, effects occurring in FMs that are evenly dependent on the magnetization should be expected also in AFMs. Indeed, the staggered magnetizations have opposite versus but

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same magnitude and sign. Therefore, if an effect is independent on the versus of the magnetization and determined only by its direction, it should occur also in AFMs.

One of the first electrical techniques used to electrically access the magnetization in FM metals is based on the anisotropic magnetoresistance (AMR) effect, already discovered in 1857 by Lord Kelvin [18]. The origin of this phenomenon lies in the spin orbit interaction that couples the motion of the conduction electrons with the localized magnetic moments. Specifically in 3d transition metals, this determines an anisotropy in the scattering probability depending on the relative orientations of the flowing current and the magnetization (see Fig. 1.3a) [19].

The typical dependence of the longitudinal resistance on the angle between the magnetization and the flowing current in a FM is shown in Fig. 1.3a. The magnetic field is applied in the perpendicular direction with respect to the flowing current. The material possesses a uniaxial magnetic anisotropy with easy axis parallel to the flowing direction. At low fields, this anisotropy dominates and the magnetization stays parallel to the current. At larger fields the magnetization progressively moves and saturates orthogonally to the current inducing a resistance variation. The difference between the two resistance states, Fig. 1.3: (a) dependence of the longitudinal resistance of a FM film as a function of the external field, applied perpendicular to the flowing current j. The high resistance state corresponds to the current flowing parallel to the magnetization M; (b) example of a TAMR junction. Pt and IrMn are, respectively, the NM and the AFM electrical contacts. The NiFe layer is used to control the AFM magnetization by the exchange spring effect. b) TAMR junction j 𝑀 ⊥ 𝑗 𝑀 ⊥ 𝑗 Rlo ng ( W ) H (kA/m) 𝑀 ∥ 𝑗 j M H a)

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resistance is the same for positive and negative saturation of the magnetization, thus the AMR depends only on its direction and not on the versus. As a consequence, this effect can occur also in AFM [20], [21].

Two examples in which AMR has been used to read the state of AFM based memory resistors [22] will be presented in the following section. The main drawback of such devices is that AMR in AFM usually provides small signals. The AMR at room temperature is typically in the order of 0.1% (See section 1.2.3), to be compared with FM that presents values up to 5% at room temperature [23]. A reliable state identification would require an expensive detection electronics, limiting the commercial applications of these concepts.

This has motivated the investigation of antiferromagnetic tunnel junctions, to provide larger resistance variations and easier readout of the AFM state. In these structures, the tunneling anisotropic magnetoresistance (TAMR) effect is the source of the readout signal. TAMR junctions are composed of an AFM electrode and a NM one separated by a thin insulating barrier, as shown in Fig. 1.3b. The anisotropy in the tunneling probability is due to the spin orbit interaction that makes the electronic states to depend on the orientation of the AFM magnetization with respect to the current flowing across the barrier.

In the TAMR junction presented in Fig. 1.3b the exchange spring effect is used to rotate the magnetization of the IrMn AFM film [10]. Applying an external magnetic field, the magnetization of the soft FM NiFe is rotated in the plane. As a result of the exchange interaction at the interface between the IrMn and NiFe, the rotation imposed in the NiFe magnetization by the field drags the magnetization of the nearby AFM film. Due to the anisotropy of the band structure introduced by the spin orbit interaction in IrMn, the tunnelling probability varies depending on the orientation of the spin moments in the AFM with respect to the crystallographic directions. For this reason, the TAMR junctions can operate with only one magnetic electrode, that has the role to drag the AFM spin orientation, whereas the other one can be an isotropic nonmagnetic metal.

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written exploiting the field cooling process previously mentioned. The magnetoresistance originates from the different tunneling probability for electrons with spin parallel or perpendicular to the current, corresponding to AFM spin moments oriented out-of-plane or in-plane. The Néel temperature of this device was TN = 173 K, below room temperature

because a thin (2 nm) IrMn layer was used. As expected for AFM based heterostructures, the stored information is insensitive to external magnetic fields up to 2 T below TN. The

resistance variation at T = 100 K is nearly 2%, much larger than the values achieved by AMR effects. Although, at the actual state-of-the-art of the research in this field, the resistance variation remains small and the writing process requires either magnetic fields or FM layers, TAMR junctions are anyway promising for future applications in the magnetic memory (MRAM) market, thanks to their high scalability, high speed and extreme robustness.

1.2.3 Antiferromagnetic spintronic devices

In the following, to give an overview of the actual panorama, two examples of memory devices based on the previously exposed concepts are presented.

- FeRh: an antiferromagnetic memory resistor [22].

In this device the peculiarity of FeRh to change its magnetic order in different temperature regimes is exploited. At temperatures above 350 K this material is ferromagnetic but gradually undergoes an antiferromagnetic transition as the temperature is decreased. The transition is confirmed by the fact that, at room temperature, it is completely insensitive to external magnetic fields as high as 9 T, as an AFM does.

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The high temperature ferromagnetic state allows to perform a field-cooling-like process as depicted in Fig. 1.4a.

Imposing an in-plane magnetization direction by an external magnetic field, parallel either to the [100] or the [010] directions, in the FM high T regime, the direction of the AFM phase spin orientation deterministically follows, perpendicularly to the FM magnetization. The readout of the stored information is fully electrical and based on the AMR effect, as previously mentioned. Depending on the relative orientation of the current and the staggered magnetization, the longitudinal resistance is enhanced or reduced (Fig. 1.4b). The fundamental disadvantage of this device, although fully based on AFM, is that a thermal writing process is required, inherently limiting the achievable storing speed.

- CuMnAs: full electrical reading and writing of the AFM state [24].

Devices allowing for both electrical writing and reading of the AFM magnetization have been realized exploiting spin orbit torques and AMR concepts in the ternary alloy

a) b)

Fig. 1.4: (a) the field cooling process imposes the magnetization direction in the FM phase of FeRh. Decreasing the temperature, the AFM spin polarization lies orthogonal to the FM phase one. The information is read through a longitudinal resistance measurement at room temperature. (b) AMR signal for the two AFM configurations. Each step corresponds to a longitudinal resistance readout demonstrating the temporal stability of the stored information.

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crystalline structure, in this material a spin orbit torque is induced by the current flowing in the material itself, without the necessity of any other NM layer onto it. Provided that the current density is high enough, this torque allows to electrically control the staggered magnetization of the AFM, forcing the magnetization to lie in the orthogonal direction with respect to the writing current, as depicted in Fig. 1.5a.

Writing current pulses are injected along the red and black arrows and consequently the magnetization is rotated by 90° for the two injection directions. The state readout if performed by measuring the transverse resistance along the remaining electrical contacts which are rotated by 45° relative to the write pulses directions. This geometry allows for maximizing the resistance variation in the two stable configurations of the magnetization due, again, to the AMR effect. The result of subsequent writing and reading cycles are presented in Fig. 1.5b. It is clear that the presented device can be utterly written and read electrically, involving only an AFM layer.

Even though, ideally, all the requirements for a fully AFM based memory cell are satisfied, a widespread technological application of this solution is far from feasible. First of all, the growth of single crystal CuMnAs epilayers has been proved to be quite difficult because of the fine control of stoichiometry and crystal order needed for such applications. This has been done only recently [26] by molecular beam epitaxy which,

a) b)

Fig. 1.5: (a) optical image of the memory device. Four contacts are used for the information storage through current pulses injection in orthogonal directions (red and black arrows). The magnetization results perpendicular to the writing current direction. The remaining electrodes are employed for the information readout by transverse resistance measurement; (b) variation of the transverse resistance after subsequent 50 ms writing pulses of amplitude j = 4·106_A/cm2_{along [010] (red dots) and [100] (black dots) crystal}

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Moreover, each bit cell requires eight electrical contacts and the resistance variations in the two encoding states remain below 0.1%, resulting in a challenging and expensive signal detection. As a result, the proposed solution is not yet competitive with the current semiconductor technologies.

Other AFM metallic alloys, such as Mn2Au [17], show similar properties to those

presented for CuMnAs but the same problematics underlined above persist. IrMn, the most used AFM material, is instead quite easy to grow in a stable AFM phase but the absence of long-range crystal order prevents from exploiting symmetry-based methods for spin texture writing and reading. The opportunity to employ the elemental AFM Cr as fundamental constituent of AFM-based spintronic devices is very appealing. Indeed, it is easy to grow since complex stoichiometries are not required, a good crystalline quality can be achieved on different substrates [27]–[29] and it is relatively abundant on earth. These and other appealing properties justify the interest in studying its intriguing characteristics that constitute the core of this thesis work.

1.3 Accessing antiferromagnetic ordering

In the previous sections, the state of the art in antiferromagnetic spintronics has been briefly presented. Many advantages are foreseen for future applications of these technologies but a reliable and cost-effective solution has not yet been proposed. A fundamental prerequisite for the development of new spintronic devices is the possibility to find a convenient way to inspect the magnetic properties of AFM systems.

The most effective technique for magnetic structure characterization is represented by neutron diffraction. Neutrons are electrically neutral and are not strongly scattered by Coulomb interactions like electrons. Despite the absence of an electrical charge, neutrons possess a magnetic moment. As a result, they are diffracted by the periodic interactions in a magnetically ordered material and allow for detailed description of AFM bulk properties. The magnetic order in the diffraction pattern is identified either by the presence

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it with the crystal structure. This is possible as the scattering cross section for a harmonic component of the magnetic lattice is dependent on the relative angle between the scattering wave vector and the magnetization vector [31]. Therefore, both the periodicity of the magnetic moments and their direction can be identified from the diffraction pattern (see section 2.4.1.2). The main disadvantage of this technique is related to the weakness of the interactions between matter and neutrons. It prevents the identification of pure superficial properties and the small scattering volume makes difficult the measurements on the magnetic thin film structures comprised in spintronic devices.

On the other hand, magnetic surface investigations can be efficiently addressed by spin polarized scanning tunneling spectroscopy (SP-STS). This is a technique based on the scanning tunneling microscopy (STM). The magnetic contrast is obtained scanning a magnetic tip onto a magnetic surface. The electron tunneling probability between the tip and the sample is modulated by the relative orientations of the tip and surface atoms magnetic moments. Thus, a real space image of the superficial moment distribution with atomic resolution is obtained, as shown in Fig. 1.6.

Fig. 1.6: (a) topographic image and (c) line profile (along the blue arrow in (a)) of the Cr [001] surface. Planes with different heights are identified by the topographical contrast. (b) SP-STS image and (d) line profile (along the blue arrow in (b)) obtained with a Cr coated tip of the region investigated in (a). The image contrast is now due to the different in plane orientations of the spins in subsequent planes of the Cr

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Magnetic stray fields produced by FM coated tips might modify the sample surface magnetic properties but, recently, the possibility to effectively exploit AFM coating and avoid cross talks has been proved [32]. The ultimate surface sensitivity of SP-STS prevents from gaining any information about bulk or buried interfaces properties.

Combining the two mentioned techniques a full characterization of an AFM structure can be accomplished. The main drawbacks are related primarily to the complexity and expensiveness of the required setups and secondly to the impossibility to gain information about buried interface effects that represent the fundamental origin of physical phenomena exploited in multilayered structure devices. These techniques are not even suitable to identify effects of magnetic ordering on the electrical conduction since transport measurements are not involved.

The main goal of this thesis work is to develop an efficient and cost-effective method to study the properties of layered AFM systems. As will be explained in the following chapters, magnetic ordering in thin film structures is electrically detected exploiting the magnetic moment induced in a neighboring non-magnetic layer by the antiferromagnet under investigation. One of the main advantages of this technique stems from the fact that it can be implemented with instruments available in most of academic laboratories (current source, voltmeter, …). It can be useful in primary investigation of unknown magnetic systems, for quickly identifying their general features before attempting more time demanding and expensive techniques, such as those previously mentioned. Concurrently, it represents a valuable method to understand physical phenomena influencing the conduction properties of a multilayered structure as a consequence of interfacial interactions between magnetic and nonmagnetic materials. As discussed in section 1.2.2, one of the main goals in spintronics is to electrically detect the information stored in the magnetization configuration. Therefore, efficient measurements of such configuration via interface-induced effects are of fundamental importance both from a theoretical point-of-view and for envisaging future developments of this research field, eventually involving industrial applications to the MRAM market too. The results obtained in this work might be very helpful for a deeper comprehension of the AFM structures and

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1.4 Thesis outlook

In this introduction the field of antiferromagnet spintronics has been presented, mainly focusing on the prospected advantages that the development of such technologies would provide to information elaboration and storage. The fundamental physical phenomena involved in this field have been briefly discussed, in relation to the state of the art, underlying the advantages and the fundamental weaknesses of the devices realized so far. In the last section few examples of the techniques used to characterize AFM materials have been introduced to emphasize the difficulty to access the properties of these materials.

The purpose of this thesis work has been to realize an efficient and cost-effective experimental setup for the characterization of AFMs, useful for the initial investigation of new magnetic structures but also to gain deeper insight into the fundamental relationship between magnetic ordering and transport properties, that represents the basis of any spintronic device.

The setup has been initially validated on an heterostructure comprising a well-known FM alloy, Co0.4Fe0.4B0.2, widely used for magnetic tunneling junctions [7], and a

technologically relevant AFM, Ir20Mn80, universally employed as pinning layer in GMR

and TMR-based devices. This system has been investigated to demonstrate the capability of our setup to access the tiny uncompensated magnetic moment of this antiferromagnetic alloy.

Then, our efforts have been devoted to study the magnetic properties of Cr thin films. The antiferromagnetic order in Cr has been subject of many investigations in the last 50 years and the bulk characteristics are now well established [14]. This is not true for thin films in which many factors such as thickness, strain and interface interactions [33] have been proved to play an essential role, significantly changing the magnetic behavior of thin films with respect to the bulk. In particular, in this project the focus has been on the Pt/Cr heterostructure and, by means of the developed setup, the magnetic moment induced in Pt

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The thesis project has been carried on in the NanoBiotechnology and Spintronics (NaBiS) group, coordinated by prof. R. Bertacco, of Politecnico di Milano, under the supervision of prof. M. Cantoni and Dr. M. Asa. The experimental activity has been carried on at Polifab, the micro and nano fabrication facility of Politecnico di Milano at Polo di Milano Leonardo.

In the following chapters the results obtained during my thesis work will be presented, along with the underlying theoretical background and the experimental methods for the samples growth and characterization.

- Chapter 2 exposes the theoretical basis on which this work has been based, i.e. the physical phenomena involved in ferromagnetic and antiferromagnetic order, with particular attention to Cr and IrMn, a basic introduction to the anomalous Hall effect and the consequences of magnetic proximity of antiferromagnets with Platinum.

- Chapter 3 describes the experimental techniques used for the samples fabrication and characterization, mainly focusing on the developed experimental setup for Hall and Van der Pauw measurements.

- Chapter 4 presents the measurements on the well-known ferromagnetic heterostructure Ta/CoFeB/MgO and those performed on the metallic antiferromagnet IrMn to validate the experimental setup capabilities.

- Chapter 5 exposes the experimental evidences, obtained by means of the developed setup, of a finite magnetic moment induced in Pt as a result of magnetic proximity interaction with the metallic antiferromagnet Cr.

- Finally, Chapter 6 drafts the conclusions of the present work and defines the future perspectives.

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2 Theory

This chapter presents the basic physics involved in antiferromagnetic systems and heterostructures, with particular attention to interface effects with nonmagnetic materials. In section 2.1 the origin of magnetic order in condensed matter and the resulting phase transitions are introduced in a very general context. Sections 2.2 and 2.3 describe the two main types of magnetic structures, ferromagnets and antiferromagnets. The developed concepts are employed to discuss the properties IrMn and Cr (section 2.4), two technologically remarkable antiferromagnetic materials. Section 2.5 introduces the Hall effect as a powerful physical phenomenon to access relevant properties of magnetic solids, both ferromagnetic and antiferromagnetic. Finally, interactions and effects at interfaces between magnetically ordered systems and nonmagnetic metals, specifically Pt and Pd, are considered in section 2.6.

2.1 Magnetic ordering in condensed matter

The theoretical description of magnetic ordering in condensed matter is one of the most complicated and fascinating topics in solid state physics. In a magnetically ordered solid, the microscopic magnetic moments are aligned along a preferential direction. This determines macroscopic behaviours such as ferromagnetism, in which all the moments are parallel, and antiferromagnetism, in which they are antiparallel giving a zero net magnetization. It is worth to notice that the ordered magnetic structures are not determined by the dipolar interactions among localized magnetic moments in solids. Indeed, these would justify only antiferromagnetic ordering and the critical temperature of the phase transition (see Sect. 2.1.1) would be as low as few K [34]. The origin of magnetic ordering is ascribed to a pure electrostatic interaction among the electrons in the solid, combined

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To understand how the spin configuration is related to the spatial distribution of the electronic wave functions, and hence to the coulomb repulsion among them, consider two noninteracting electrons. Let 𝜓_,(𝒓_𝟏) and 𝜓₁(𝒓_𝟐) be the spatial wave functions of the two electrons whose positions are identified by the vectors 𝒓_𝟏and 𝒓_𝟐. The joint state spatial wave function can be written as product of the two single particle states. Moreover, assuming that the Hamiltonian contains no spin dependent terms (as the spin orbit interaction), the spin information is included multiplying the spatial wave function by the spinor state. Defined 𝑺_𝟏 and 𝑺_𝟐 the spin operators for the two electrons, the total spin operator is 𝑺₄₅₄ = 𝑺_𝟏+ 𝑺_𝟐 and its eigenstates can be written as [34]:

𝜒4 =

1 2 ↑_< ↓_> + ↓_< ↑_> ↓_< ↓_>

↑_< ↑_> ( 1 )

𝜒? = 1 2 ↑< ↓> − ↓< ↑> ( 2 )

|↑< and |↓< are the eigenstates of 𝑆<C, and the same holds for 𝑆>C, where z is the spin

quantization axis. Three of these, the triplet states (1), are symmetric with respect to the exchange of the two electrons, whereas the fourth, the singlet state (2), changes sign if the two electrons are interchanged. As for the spin state, also the spatial component can be symmetric or antisymmetric:

𝜓, 𝒓𝟏, 𝒓𝟐 = 1 2 𝜓, 𝒓< 𝜓1 𝒓> − 𝜓,(𝒓>)𝜓1(𝒓<) ( 3 )

𝜓? 𝒓𝟏, 𝒓𝟐 = 1 2 𝜓, 𝒓< 𝜓1 𝒓> + 𝜓,(𝒓>)𝜓1(𝒓<) ( 4 )

𝜓_, and 𝜓_? are antisymmetric and symmetric, respectively, with respect to the exchange of the two electrons. Since the wave function describing a set of fermions must be overall antisymmetric, we must only consider two combinations: 𝜓_? must be multiplied by the singlet state 𝜒_?, or 𝜓_, by the triplet one𝜒₄. As a result, the antisymmetrization requirement introduces a link between the spatial and the spin components of the wave function although no spin dependent term appears in the Hamiltonian. If the interaction between

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the electrons is introduced as a perturbation, albeit this is not sufficient to achieve accurate results, an energy difference between the singlet ad the triplet state results:

𝐸_?− 𝐸₄ = 2 𝜓_,∗ 𝒓_𝟏 𝜓₁∗ 𝒓_𝟐 𝑒>

4𝜋𝜀_K 𝒓_<− 𝒓_> 𝜓, 𝒓𝟐 𝜓1 𝒓𝟏 𝑑𝒓<𝑑𝒓> ( 5 ) This energy difference is called exchange splitting and is a purely quantum term, that can not be interpreted classically as the wave function itself, contrary to its square modulus, has no physical meaning. As mentioned above, this term has a purely electrostatic origin. An important quantity that will be used in the following is the exchange integral, defined as 𝐽 =<_> 𝐸?− 𝐸4 . As it is clear from the definition, if 𝐽 > 0 the ground state of the system

is characterized by spins aligned along the same direction in the triplet configuration and a ferromagnetic configuration is stabilized; if 𝐽 < 0 the ground state is instead characterized by spins antiparallel in the singlet configuration and the magnetic ordering is thus antiferromagnetic, with a zero net magnetic moment.

Following the above discussion, it is possible to introduce the Heisenberg Hamiltonian which models the effect of the exchange interaction. From the quantum theory of the angular momenta, the eigenvalues of 𝑆_<_C and 𝑆_>_C are ± 1/2 (throughout this work, for sake of simplicity all the angular momenta will be expressed in unit of ℏ), thus the eigenvalues of 𝑆₄₅₄> _{are 0 and 2 for the singlet and the triplet state respectively, while the eigenvalues}

of 𝑆_<>_{and 𝑆}

>> are both 3/4.

Therefore, the operator 𝑺_<• 𝑺_> = <_> 𝑆₄₅₄> _{− 𝑆}

<>− 𝑆>> has eigenvalues -3/4 for the singlet

state and 1/4 for the triplet state. Exploiting this property for the two electron system, the Hamiltonian can be written as:

ℋ =1

4 𝐸?+ 3𝐸4 − 𝐸? − 𝐸4 𝑺<• 𝑺> ( 6 ) Apart from the constant energy term, which can be neglected simply changing the energy reference level, the second term models the effect of the exchange interaction as it is dependent on the relative orientation of the electrons spin angular momenta.

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This expression is generalized to a many body system introducing the Heisenberg Hamiltonian:

ℋ = − 𝐽_UV

U,V

𝑺_U • 𝑺_V ( 7 )

𝐽UV identifies the exchange integral between the ith and jth spins and a factor 2 has been

omitted to account for the coupled interactions only once (e.g., in the two electron system

i=1, j=2 and i=2, j=1 correspond the same interaction term). Again if 𝐽_UV > 0 the interaction between the spins favours their parallel alignment, i.e. a ferromagnetic configuration, and vice versa if 𝐽UV < 0 the antiferromagnetic state is favoured. This

Hamiltonian is the starting point for many models aiming to address the problem of magnetic ordering in condensed matter.

2.1.1 Landau theory

The appearance of an ordered phase like a ferromagnetic state always involves a symmetry breaking. Below the critical temperature 𝑇_X,all the magnetic moments in a ferromagnet tend to align along a specific direction (conventionally called “up” or “down” if parallel or antiparallel to a given “up” versus), while above 𝑇_X the system is magnetically isotropic. This asymmetry is not included in the Hamiltonian of the isolated solid (the magnetocrystalline anisotropy only defines the direction but not the versus) and therefore the magnetization process forces the system to break its own symmetry. Usually the parameter that drives the symmetry breaking is the temperature. The thermodynamic reason of this symmetry breaking can be understood considering the Helmholtz free energy, defined as 𝐹 = 𝐸 − 𝑇𝑆, whose minimum determines the equilibrium state of the system. At low temperatures the lowest energy ground state (that minimizes E), which is usually the ordered one, is favoured, whereas as 𝑇 increases the minimum for 𝐹 is found by maximizing 𝑆, and so a disordered phase is favoured due to its larger entropy. Symmetry breaking phase transitions are necessarily sharp because a particular symmetry

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Lev Landau [35] proposed a thermodynamic model able to describe these processes. An order parameter 𝑀 is introduced: it is zero above 𝑇_X whereas it assumes a finite value below it. For a ferromagnet the order parameter coincides with the magnetization. The Helmholtz free energy is written down as a power series in 𝑀. As long as no external magnetic fields are applied, there is no energy difference between spin “up” and “down”. Therefore, all the odd powers in 𝑀 do not provide any contribution:

𝐹 𝑀 = 𝐹_K + 𝑎 𝑇 𝑀>_{+ 𝑏𝑀}\ ( 8 ) 𝐹_K is a constant identifying all the energetic terms not influenced by the order parameter, 𝑏 is a positive constant and 𝑎(𝑇) is assumed to be temperature dependent.

This model describes an appropriate phase transition if 𝑎(𝑇) changes sign at the transition temperature. Therefore, assume 𝑎 𝑇 = 𝑎K 𝑇 − 𝑇X with 𝑎K > 0. Under these

assumptions the evolution of the Helmholtz free energy with the temperature is shown in Fig. 2.1a.

Fig. 2.1: (a) Temperature dependence of the Helmhotz free energy. At any temperature above Tc the unique

minimum is M = 0, below Tc a net magnetization exists (M ≠ 0). (b) Temperature dependence of the

spontaneous magnetization above and below Tc.

T

c

T

M(T) µ (T

c

- T)

1/2 F(M) M T > Tc T < Tc a) b)

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The equilibrium condition corresponds to the minimum of 𝐹. Solving 𝜕𝐹 𝜕𝑀 = 0 two solutions are possible:

𝑀 = 0 𝑜𝑟 𝑀 = ± 𝑎K 𝑇X − 𝑇 2𝑏

( 9 )

The second solution is clearly valid only for 𝑇 ≤ 𝑇X.

Above 𝑇_X the stable solution is 𝑀 = 0, while the other solution is no longer valid (it is an imaginary number). Below 𝑇_X, evaluating 𝜕>_{𝐹 𝜕}>_{𝑀 = 0 it easy to see that the minimum}

of 𝐹 is the second solution, that is 𝑀 ∝ 𝑇_X − 𝑇 . The value of the order parameter among the two degenerate minima is determined at the transition temperature by any thermal fluctuation. The dependence of the parameter 𝑀 in the stable configuration on the temperature 𝑇 is depicted in Fig. 2.1b. Note that the validity of the theory is limited to the neighbourhood of the transition since it is based on a series expansion in 𝑀. It is a mean field theory because it implies that all the spin angular momenta in the solid interact with the same exchange field produced by their neighbours, assumed proportional to the magnetization.

Actually, large fluctuations in the order parameter, not considered in this theory, are present near the critical temperature. Hence, although straightforward to solve, the predictions based on these approximations must be regarded with some caution. During the experimental investigations of symmetry breaking phase transitions, the temperature dependence of the order parameter in the neighbourhood of the critical temperature is assumed to be of the form 𝑇X − 𝑇 c, where 𝛽 is tuned to fit the experimental data.

Theoretical models relying on different approximations usually provide different values for the parameter 𝛽 [34]. As an example, in the model based on the Heisenberg Hamiltonian (7) the value of 𝛽 if found to be 0.367.

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2.2 Ferromagnetic ordering

2.2.1 Weiss model

In section 2.1 the concept of exchange interaction and a general model for phase transitions involving symmetry breaking have been introduced. Now a mean field theory, based on the Heisenberg Hamiltonian and aiming to introduce the general features of the ferromagnetic phase, will be discussed. The magnetic moment 𝝁 associated to the spin 𝑺 of an electron is:

𝝁 = −𝜇_g𝑔_?𝑺 ( 10 )

𝜇_g is the Bohr magneton and gs is the Landè factor that, as long as the magnetic properties

are determined by 3d transition metal ions, is equal to 2 due to the orbital angular momentum quenching [34]. The Hamiltonian to be studied includes the exchange interaction, modelled by the Heisenberg term (7), and the interaction with the external field 𝑩, described by the Zeeman Hamiltonian:

ℋ = − 𝐽UV U,V

𝑺U• 𝑺V+ 𝑔?𝜇g 𝑺V • 𝑩 V

( 11 )

Considering a single electron with spin 𝑺_U, its interaction with all the other moments of the crystal can be written as:

−2𝑺_U• 𝐽_UV𝑺_V

V

( 12 )

the factor 2 appears as for a single spin the coupled interactions are considered only once. The mean field approximation is introduced assuming that all the spins in the lattice interact in the same way with the neighbouring ones. The molecular field perceived by the ith electron as a result of the exchange interaction with the surroundings can be defined as:

𝑩_jk = − 2

𝑔_?𝜇_g 𝐽UV𝑺V

V

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The effect of the exchange interaction is represented by the molecular field that, in a ferromagnet, tends to align all the neighbouring magnetic moments in the same direction. The magnetization corresponds to the average magnetic moment per unit volume:

𝑴 = −𝜇g𝑔?

𝑉 𝑺 ( 14 )

V is the sample volume and 𝑺 is the expectation value of the spin vector. The mean field approximation is included assuming 𝑩_jk proportional to the magnetization, that is averaging the specific interactions among the discrete moments in the ferromagnet:

𝐵jk = 𝜆𝑀 = −𝜆

𝑔_?𝜇_g

𝑉 𝑆oV

V

( 15 )

𝜆 expresses the strength of the exchange interaction and z indicates the magnetization axis. This approximation allows to consider the magnetic moments in the solid as independent and interacting with the same molecular field as in the Langevin model of paramagnetism [34] since 𝑆_oV is the same for every localized magnetic moment throughout the sample. It is worth to notice that in the ferromagnet the ordering field is produced by the same magnetic moments that it tends to align. As a result, the ordered phase at low temperatures can be self-sustaining.

Including eq. (15), the Hamiltonian (11) in the mean field approximation, can be written as:

ℋ = 𝜇_g𝑔_? 𝑺_V• (𝜆𝑴 + 𝑩)

V

( 16 )

The localized magnetic moments can be considered as a perfect gas so its statistical properties are defined by the Boltzmann partition function [35]. From eq. (16) it is possible to see that the whole Hamiltonian of the ferromagnet is given by the sum of non-interacting electron terms. Thus, the average spin in eq. ( 14 ) is the same for all the electrons of the system and can be evaluated exploiting the eigenvalues of the Hamiltonian (16) for the jth_particle.

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Following ref. [34], the magnetization can be found solving simultaneously the equations: p p_q

= 𝐵

?

𝑦

𝑦 =

s_tu_q? gvwp x_ty ( 17 )

𝐵_? is the Brillouin function, 𝑘_g is the Boltzmann’s constant, 𝑀_? the saturation magnetization and 𝑠 is the eigenvalue of 𝑆_o. The equilibrium magnetization can be found solving graphically for M the two equations. In Fig. 2.2 the case for 𝐵 = 0 is depicted.

𝑀 results proportional to y and the slope of the straight line turns out to be proportional to the temperature 𝑇. If 𝑇 > 𝑇_X the only solution is 𝑀 = 0, whereas if 𝑇 < 𝑇_X there are three solutions. The 𝑀 = 0 solution is unstable, whereas the two other solutions with non-zero magnetization are stable. Thus, a spontaneous magnetization exists, with a temperature dependence that turns out to be similar to Fig. 2.1. The phase transition occurs when the tangent in the origin of the Brillouin function is coincident with the straight line in eq. (17). Indeed, when the temperature is decreased below the critical one, the 𝑀 = 0 solution becomes unstable and the 𝑀 ≠ 0 points are the stable ones.

T = Tc

T < Tc

M/Ms

y T > Tc

Fig. 2.2: Spontaneous magnetization in ferromagnetic systems. The magnetization in the ferromagnet is determined by the intersection of the Brillouin function and the straight line (Eq. ( 17 )). Above 𝑇Xthe unique

solution 𝑀 = 0 exists, below it a spontaneous magnetization represents the stable equilibrium state. Bs(y)

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Comparing, for 𝑇 = 𝑇_X, the linear term in eq. (17) with the first order expansionof 𝐵_?, the critical temperature results:

𝑇_X =𝑔?𝜇g𝜆 𝑠 + 1 𝑀?

3𝑘_g ( 18 )

As one might expect, the critical temperature is proportional to the l parameter that determines the strength of the exchange interaction: the larger the molecular field, the stiffer the ordered state against thermal excitations. Therefore, the Weiss model is able to account for the peculiar behavior of ferromagnetic materials, introducing an average exchange interaction among the localized magnetic moments in the crystal. It provides a more quantitative description for the origin of a symmetry braking phase transition than the Landau model introduced in Sect.2.1.1.

2.3 Antiferromagnetic ordering

Antiferromagnets represent another broad class of magnetically ordered elements. The exchange constant 𝐽 in these systems is negative and therefore the nearest neighbours spins tend to align in opposite directions. Very often these materials can be viewed as two interpenetrating ferromagnetically ordered sublattices as shown in Fig. 2.3.

This peculiar alignment justifies the general magnetic properties of these materials: Fig. 2.3: Magnetic structure in antiferromagnets. Schematic representation of the ferromagnetically aligned magnetic sublattices whose superposition determines the antiferromagnetic configuration.

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3. Robustness of the order parameter against external magnetic perturbations. The macroscopic magnetization, resulting from the superposition of the atomic magnetic moments, for antiferromagnetic alignment is zero, as the two sublattices interfere destructively. For the same reason no stray fields are generated. The robustness against external magnetic fields can be understood considering that, lining up along the field direction, the neighbouring spins would not be antiparallel anymore, resulting in a large energy increase due to the strong exchange interaction. These properties, already mentioned in the introduction, suggest that antiferromagnets could be suitable to realize high density and robust magnetic random access memories (MRAM). In fact, the absence of magnetic stray fields would prevent magnetic cross-talk between neighbouring bit cells, which represents the main limit to scalability of ferromagnets-based MRAM.

2.3.1 Mean field theory of localized antiferromagnetism

2.3.1.1 Weiss model

Considering the schematic representation of an antiferromagnet in Fig. 2.3, it is possible to apply the Weiss model to these systems. Now the origin of the molecular field to be considered is the magnetization of each magnetic sublattice. In order to justify the antiparallel alignment, the molecular field acting upon the dipole moments of one sublattice is assumed to be proportional to the magnetization of the other one. Labelling the spin “up” sublattice as “+” and the “down” one as “-” the molecular fields are written as:

𝐵v = − 𝜆 𝑀}

𝐵_} = − 𝜆 𝑀_v

( 19 )

As any magnetic moment tends to align parallel to an external field, the magnetic moments of the + sublattice are aligned in the opposite direction with respect to 𝑀_} by 𝐵_v. For each sublattice the magnetization is therefore given by: