RABAT
Faculté des sciences, 4 Avenue Ibn Battouta B.P. 1014 RP, Rabat-Maroc Tel +212 (0) 37 77 18 34/35/38, Fax: +212 (0) 37 77 42 61, http:/www.fsr.ac.ma
N° d’ordre 2848
THÈSE DE DOCTORAT Présentée par Omar El RHAZOUANI Discipline : Physique Informatique
Spécialité : Physique Informatique
Monte Carlo simulation study of magnetic properties of
Cr-based double perovskites as spintronic materials with the
highest Curie temperature
Soutenue le 12 Mars 2016Devant le jury Président :
Pr. Hamid EZ-ZAHRAOUY PES, Faculté des sciences de Rabat Examinateurs :
Pr. Lahoucine BAHMAD PES, Faculté des sciences de Rabat Pr. Noureddine BENAYAD PES, Faculté des sciences de Ain hoc, Casablanca
Pr. Abdelilah BENYOUSSEF Professeur Membre résident de l’Académie Hassan II des sciences et techniques Rabat
Pr. Youssef El AMRAOUI PES, Faculté des sciences de Rabat Pr. Abdellah EL KENZ PES, Faculté des sciences de Rabat Pr. Mohammed LOULIDI PES, Faculté des sciences de Rabat
Dedication
To
My mother
To my brothers and sister
To all my professors and
teachers
This thesis has been carried out at the Laboratory of Magnetism and High Energy Physics (LMPHE), Faculty of Science, Mohammed V University – Rabat, under the supervision of prof. AbdelilahBenyoussef and Prof. Abdallah EL Kenz.
First and foremost, praises and thanks go to Allah, the Almighty, for his unlimited and uncountable blessings in my whole life and throughout my research work especially.
On this page, I would like to express my very great appreciation, deep gratitude and sincere thanks to my first supervisor, prof. Abdelilah Benyoussef, and my second supervisor, Prof. Abdallah EL Kenz, for giving me this opportunity to learn from their valuable expertise and to have the fruitful discussions with them. They always had time to answer my questions and they patiently provided the vision, encouragement and advise necessary for me to proceed throughout my research period. It is my privilege to be their student.
Besides my supervisor, I would like also to offer my thanks to the thesis committee for reviewing my thesis and giving their insightful and useful comments.
I would like to thank the president of the thesis committee Prof. Hamid EZ-Zahraouy from Faculty of Science Rabat, for reviewing this PhD thesis.
I would like also to offer my thanks to Prof. Noureddine Benayad from Faculty of Science Ain ChocK Casablanca, for reviewing and reporting my PhD thesis.
I would like to thank Prof. Lahoucine Bahmad from Faculty of Science Rabat, for reviewing and reporting this thesis.
I would like also to offer my thanks to Prof. Youssef EL Amraoui from Faculty of Science Rabat, for reporting this thesis.
I would also like to extend my thanks to Prof. Mohamed Loulidi from Faculty of Science Rabat, for reviewing and reporting my thesis.
I would like also to offer my special thanks to all the professors of Laboratory of Magnetism and High Energy Physics for their valuable suggestions and discussions.
Special thanks should be given to my mother, my brothers and sister, for their support, patience and encouragement throughout my study.
I
In this entire work, we study the magnetic properties of Cr-based Double Perovskites (DPs) that have the highest Curie temperature measured until now in the class of magnetic oxides. This class of DP materials holds a lot of promise to realize new electronic devices in the domain of spintronics. First, we have studied phase diagrams and magnetic properties of the DP Sr2CrReO6 by
using different methods of statistical physics, such as mean field approximation (MFA), effective field theory (EFT) and Monte Carlo simulation (MCS) in the framework of Ising model. Critical exponents have been computed by MCS and compared with the universality class of the 3-dimensional Ising Model. We have shown that MCS provides results better than MFA and EFT methods. We have investigated the antisite defect in the DP Sr2CrReO6 by using MCS. This
simulation has shown that the antisite disorder, which is conceived as defect, can be transformed via MCS into a useful tool to explain the role of transition metals, namely Cr and Re, in the stability and the magnetic performance of the compound Sr2CrReO6. We also performed a MCS to study the effect of
Re-substitution by Os, in a wide range of doping rates, in the compound
Sr2CrReO6, which can be considered as a junction between the two compounds
Sr2CrReO6 and Sr2CrOsO6. It has been found that Re-substitution in Sr2CrReO6
by only 10% of Os-concentration increases the reduced TC considerably which
shows the positive influence of Os-atoms (that refers to the 5d-band filling) on the magnetic performance. At a certain concentration of Os-atoms, we have noticed a competition between ferromagnetic double exchange and antiferromagnetic super exchange interactions that occurs at the half-metal-insulator transition due to the 5d-band filling which affects the stability of the material. Finally, a MCS investigation has been conducted to better understand the complexity of correlated interactions characterizing the promising DP
Sr2CrIrO6. This study has found two new factors behind the unusual behavior of
the magnetization, which are the interaction strength between 5d-element atoms and crystal field strength of 5d-element atoms. Magnetic hysteresis cycle has been studied in relation to different exchange coupling values. Effects of Ir-substitution doping by Os and Re on the magnetic behavior have been also investigated.
Keywords: Double Perovskite; Monte Carlo simulation; Mean field
II
Dans cette thèse, Nous avons étudiez les propriétés magnétiques des Doubles Pérovskites (DPs) à base de Cr qui sont caractérisés par la température de Curie la plus élevée mesurée jusqu’à aujourd’hui dans la classe des oxydes magnétiques. D’abord, nous avons étudié les diagrammes de phase et les propriétés magnétiques de la DP Sr2CrReO6 en utilisant
l’approximation du champ moyen, la théorie du champ effectif et la simulation Monte Carlo (MCS) dans le cadre du model d’Ising. Les exposants critiques ont été calculés et comparés avec la classe d’universalité du model d’Ising 3D. Nous avons étudié également l’effet du désordre d’antisite dans la DP
Sr2CrReO6 par le billet de la MCS. Cette simulation a permis de transformer le
concept du désordre d’antisite conçu comme défaut de structure en un outil permettant d’interpréter le rôle des métaux de transition (Cr et Re) dans la stabilité et la performance magnétique du composé Sr2CrReO6. Ensuite, nous
avons mené une étude par MCS pour étudier la jonction entre les deux composés Sr2CrReO6 et Sr2CrOsO6 en explorantl’effet de la substitution de Re
par Os dans le composé Sr2CrReO6 dans un large spectre de concentration de
dopage. Finalement, nous avons étudié par MCS la DP Sr2CrIrO6 pour bien
comprendre la complexité de ses interactions étroitement corrélées. Cette étude a permis de trouver deux nouveaux facteurs responsables du comportement non-monotonique de l’aimantation qui sont : - l’amplitude de l’interaction entre les atomes de l’élément 5d (Ir dans ce cas) - et l’amplitude de leur champ cristallin. Nous avons étudié le cycle d’hystérésis en fonction de différentes valeurs de couplages d’échange et l’effet du dopage par substitution de Ir par Os et Re dans la DP Sr2CrIrO6 sur les propriétés magnétiques.
Mots-clefs: Double Perovskite; Simulation Monte Carlo; Approximation du
champ moyen; théorie du champ effectif; désordre d’antisite; dopage par substitution.
III
L’objectif principal de cette thèse est d’enrichir le savoir théorique à l’échelle atomique des propriétés magnétiques des Doubles Pérovskites (DPs) à base de Cr avec la température de Curie la plus élevée mesurée jusqu’à aujourd’hui, dans le cadre de la physique statistique. Ainsi, Nous avons utilisé des méthodes statistiques, comme l’approximation du champ moyen (MFA), la théorie du champ effectif (EFT), et la simulation Monte Carlo (MCS), dans le cadre du model d’Ising tridimensionnel, pour développer un savoir théorique détaillé du comportement magnétique et des effets de quelques phénomènes comme le défaut d’antisite, le dopage par substitution et le cycle d’hystérésis dans les DPs Sr2CrReO6 et Sr2CrIrO6.
Dans le domaine de la matière condensée, les nouvelles méthodes de modélisation des matériaux et de simulation de leurs propriétés magnétiques, à différents échelles, ont prouvé leur efficacité dans la caractérisation de nouvelles générations de matériaux avec des structures complexes et dans la prédiction de nouveaux matériaux qui peuvent jouer un rôle majeur dans les applications de l’avenir. Vue la grande précision des résultats qu’offrent ces méthodes, elles deviennent de plus en plus utilisées dans le domaine de la recherche scientifique pour prédire de nouvelle structures de matériaux et pour l’interprétation des résultats expérimentaux. MCS, MFA et EFT font partie intégrante de cette catégorie de méthodes. Spécialement la méthode MCS tiens sa puissance de son développement au cours de longues années de recherche intense dans deux domaines majeurs, le premier concerne les algorithmes numériques qui deviennent de plus en plus puissants, et le deuxième concerne les machines de calcule en terme de capacité de stockage qui ne cesse d’augmenter et la vitesse de traitement des données qui deviennent de plus en plus rapide.
Pour bien comprendre le comportement magnétique de cette classe de matériaux et donner une contribution scientifique au savoir théorique à propos de ces oxydes magnétiques, on s’est proposé de répondre aux questions suivantes :
o Quelle est le comportement magnétique de la DP Sr2CrReO6 par
rapport aux différents couplages d’interaction et par rapport aux champs cristallins dominant dans sa structure ? parmi les méthodes : MCS, MFA et EFT, quelle est la méthode la plus appropriée pour étudier ce type de structure ?
IV
o Puisque la DP Sr2CrOsO6 a une température de transition plus
élevée que Sr2CrReO6, quel est le comportement magnétique de
ce dernier vis-à-vis le dopage par substitution du Re par Os ? o La DP Sr2CrIrO6 devra présenter la température de transition la
plus élevée jamais mesurée dans la classe des oxydes magnétiques. Quel est le comportement de ce composé par rapport aux différents couplages d’interaction et par rapport aux champs cristallins dominant dans sa structure ? quelles sont les propriétés magnétiques de ce matériau ? quel est son comportement magnétique vis-à-vis le dopage par substitution de l’Ir par le Re et par l’Os ?
Ainsi, cette thèse est structurée de façon à répondre à toutes ces questions dans les chapitres de 1 à 4 dans la troisième partie. Nous rapportons les résultats existants dans la littérature, nous interprétons à notre façon les résultats qui sont sujet de débats actuel, et nous donnons une contribution détaillée au savoir théorique des phénomènes étudiés.
Dans la première partie de cette thèse, nous décrivons les DPs de façon générale. Nous cadrons cette classe d’oxydes magnétiques dans le cadre de la spintronique qui est une nouvelle science basée la corrélation entre la direction des spins des électrons de conduction et de leur charge. Nous présentons la structure cristalline et électronique de la classe des DPs à base de Cr. Nous regroupons les observations expérimentales et théoriques phare au sein de classe. Nous décrivons le désordre d’antisite et évoquons ses effets sur la performance des matéraux. Nous définissons le dopage par substitution et donnant quelques exemples de dopage dans cette même classe de matériaux avec l’impact sur la performance qu’introduit ce type de dopage.
Dans la deuxième partie, nous présentons le model d’Ising utilisé dans tout ce travail. Ce modèle a été développé dans les quatre chapitres pour le rendre adaptable au type de problème abordé. Nous décrivons aussi dans cette partie chaque méthode de calcul (MFA, EFT et MCS) à part.
Chapitre 1 dans la troisième parti est consacré à l’étude des propriétés magnétiques de la DP Sr2CrReO6 par rapport aux différent couplages
d’interaction et par rapport aux champs cristallins dominant dans sa structure. L’étude a été faite par les méthodes : MFA, EFT et MCS, dont les résultats concernant les diagrammes de phases ont été comparés. Les exposants
V
Chapitre 2 discute les effets du désordre d’antisite observés par la simulation Monte Carlo sur le comportement magnétique de la DP Sr2CrReO6.
La simulation effectuée dans le cadre du premier chapitre a été reprise et développée avec deux modifications pour d’abord simuler le désordre créé par l’excès des atomes Cr (Sr2Cr1+xRe1-xO6) puis pour simuler le désordre créé par
l’excès des atomes Re (Sr2Cr1-xRe1+xO6).
Chapitre 3 concerne l’étude de l’impact du dopage par substitution du Re par Os dans la DP Sr2CrReO6. Ainsi, une MCS a été développée dans le cadre
du modèle d’Ising pour étudier le magnétisme lors de ce type de dopage dans un large spectre de concentration du dopant « Os » ce qui constitue en quelque sorte l’étude de la jonction entre les deux DPs Sr2CrReO6 et Sr2CrOsO6.
Chapitre 4 focalise sur la DP Sr2CrIrO6 en l’étudiant par la MCS pour
décrire son comportement magnétique par rapport aux différent paramètres de l’Hamiltonien. Une étude du cycle d’hystérésis magnétique a été développée par rapport aux différentes valeurs de couplages d’interactions. Le dopage par substitution de l’Ir a été étudié dans le cas de la substitution par l’Os “Sr2CrIrxOs1-xO6” qui a ramené à des résultats attendus, et dans le cas de la
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Figure 1.1: MRAMs: a) schematic drawing illustrating the principles of MRAM
functioning,b) 4MB MRAM from Motorola, c) 16MB MRAM prototype from
IBM-Infineon, c) STT-MRAM (Spin-transfer torque -MRAM) 1MB test chip from Toshiba. ... 9
Figure 1.2: Ideal cubic unit cell of a perovskite ABO3. A in the corners are
alkaline earth metal cations, B in the center is a transition metal cation and O in the center of faces are the oxygen anions. ... 10
Figure 1.3: The ideal structure of a double perovskite A2BB’O3. Due to visibility
concerns, A-cations in the center of each cell are note illustrated. B and B’ are the transition metal cations and O surrounding B and B’ are the oxygen anions. ... 11
Figure 1.4: illustration of the G type antiferromagnetic coupling that occurs
between the two sublattices B and B’ of a double perovskite with the formula A2BB’O6. ... 14 Figure 1.5: the electronic configurations of the B - B’ cations in the formula
A2BB’Oare illustrated for: a) CrIII - ReII , b) CrIII - OsIII , c) CrIII - IrIV. ... 14 Figure 1.6: Measured and calculated Tc vs. the number of valence electrons.
The names of the elements BB′ in the DPs A2BB′O6 appear near the data
points. The value in parenthesis (CrIr) is the calculated Tc because the
measured one is not yet available [21]. ... 15
Figure 1.7: Total (black) and partial (colored) density of states of Sr2CrReO6.
The number of valence electrons from Cr(3d) and Re(5d) is NV = 5. The total
magnetic moment is Mtot = 1.0 µB per f.u. Symmetry labels of the parent states
are included [21]. ... 18
Figure 1.8: Total (black) and partial (colored) density of states of Sr2CrOsO6.
The number of valence electrons from Cr(3d) and Os(5d) is NV = 6. The total
magnetic moment is Mtot = 0.0 µB per f.u. Symmetry labels of the parent states
are included [21]. ... 19
Figure 1.9: Total (black) and partial (colored) density of states of Sr2CrIrO6. The
number of valence electrons from Cr(3d) and Ir(5d) is NV = 7. The total
magnetic moment is Mtot = 1.0 µB per f.u. Symmetry labels of the parent states
VII
Figure 1.11: Magnetic (a) and antiphase domain (b), (c) structures in Ba
2FeMoO6. (a) Fresnel image: magnetic domain walls are represented by bright and dark
lines. (b) Magnetization distribution map recovered by the transport-of-intensity equation method. (c) Antiphase boundaries are represented by dark lines and areas. Local magnetization is represented by the small red arrows [66, 83]. ... 24
Figure 1.12: (a) Atomic structure shown in a HREM image [84], (b) antiphase
boundary (APB) recovered by Fourier reconstruction, (c) Schematized atomic structure at the APB. (d) 3D Slices of Fe-Mo cubic lattices [66, 85]. ... 25
Figure 1.13: 1D ordered and disordered Sr2FeMoO6 at antiphase boundary
[84]. F: ferromagnetic interaction, AF: antiferromagnetic interaction. ... 26
Figure 1.14: a): Magnetization recovered at 0.5T for single crystals of
Sr2FeMoO6 [86] with an antisite disorder of about 8%; in the inset: Hysteresis
cycle at 5K. b): Hysteresis cycle curves recovered at 5K for polycrystalline Sr2FeMoO6 series [87]; in the Inset: Saturation magnetization at 5K with varying
disorder. The middle line represents Ms / ξ= 4 µB/f.u. ... 27 Figure 1.15: Magnetoresistance versus external magnetic field at 5K and 300K
for Sr2FeMoO6 samples for antisite disorder rates decreased from A to F [89] ... 28
Figure 1.16: a,b)Band structures of the metallic spin direction of Sr2FeReO6(a)
and Sr2FeMoO6(b); c,d) black triangles represent the saturation magnetization
measured at 5 K and white triangles represent TC measured at magnetic field of
0.1T versus the amount of doping in Sr2Fe1−xCrxReO6(c) and
Sr2Fe1−xZnxReO6(d)[32, 98]. ... 31 Figure 1.17: (a) Intensity patterns obtained by X-ray powder Diffraction of
Sr2FeRe1−xFexO6, (b) magnetization of Sr2FeRe1−xGaxO6, (c) resistivity
measurement of Sr2FeRe1−xFexO6, (d) magnetization of Sr2FeRe1−xFexO6 [32].
... 33
Figure 2.1: The exponential growth of computing power predicted by Moore’s
law, which is closely related to various technologies: electromechanical between 1900 and 1935 , relays between 1934 and 1940 , vacuum tubes between 1940 and 1960 , transistors between 1960 and 1970 , and integrated circuits since 1970. ... 36
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Figure 3.1: 3D-illustration of the crystalline structure and the coupling
interactions taken into account in the Ising model. (a) First finite cluster centered on a chromium atom. (b) Second finite cluster centered on a rhenium atom.... 62
Figure 3.2: Ground state phase diagrams depending on the reduced exchange
couplings r = J /J and r = J /J for the crystal fields ∆ /|J | = ∆ /|J | = 0.25. a) J < 0, b) J > 0 ... 63
Figure 3.3: Total magnetic order parameter (M = |m − m |/2) as a function of
the reduced temperature T/ |J | for the reduced exchange couplings (r2=r3=0.5)
and the crystal fields (∆ /|J | = ∆ /|J | = 0.25) (J1 negative). ... 66 Figure 3.4: a-b) Reduced transition temperature T / |J | versus the reduced
exchange coupling r2 (results of MFA method). a’-b’) Total magnetic order
parameter of the system versus the reduced exchange coupling r2 at T/ |J | =
1. a-a’) Illustration of the transition from (1/2,-1) phase to (3/2,-1) phase. b-b’) Illustration of the transition from (1/2, 0) phase to (3/2, 0) phase. ∆ /|J | = ∆ /|J | = 0.25. ... 67
Figure 3.5: a-b) Reduced transition temperature T / |J | versus the reduced
exchange coupling r3 (results of MFA method). a’-b’) Total magnetic order
parameter of the system versus the reduced exchange coupling r3 at
temperature T = 1. a-a’) Illustration of the transition from (3/2, 0) phase to (3/2,-1) phase. b-b’) Illustration of the transition from (1/2, 0) phase to (1/2,-(3/2,-1) phase. ∆ /|J | = ∆ /|J | = 0.25. ... 68
Figure 3.6: Reduced transition temperature T / |J | versus the reduced
exchange coupling r2 (for r3=0.5) (a) and r3 (for r2=0.5) (b) recovered by the
three methods MCS, EFT, and MFA. Dashed lines are first order transition lines. Crystal fields are taken equal to ∆ /|J | = ∆ /|J | = 0.25. ... 73
Figure 3.7: Magnetic order parameter (a) and magnetic susceptibility (b) versus
the reduced temperature for various sizes L of the system ranging from L =4 to L =32 for values of crystal field (∆ /|J | = ∆ /|J | = 0.25). ... 76
Figure 3.8: Magnetic order parameter (a) and magnetic susceptibility (b) versus
the reduced temperature for various sizes L of the system ranging from L =4 to L =32 for values of crystal field (∆ /|J | = ∆ /|J | = 2 ... 77
Figure 3.9: Magnetic order parameter (a) and magnetic susceptibility (b) versus
the reduced temperature for various sizes L of the system ranging from L =4 to L =32 for values of crystal field (∆ /|J | = ∆ /|J | = 4). ... 78
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from (1/2,0) phase to (3/2,0) phase. ... 79
Figure 3.11: Reduced transition temperature T / |J | versus the reduced
exchange coupling r3 (Monte Carlo simulation results). a) Illustration of the
transition from (3/2,0) phase to (3/2,-1) phase. b) Illustration of the transition from (1/2,0) phase to (1/2,-1) phase. ... 79
Figure 3.12: Derivatives ( in (a), (c), and (e)) and ( ( ) in (b), (d), and (f)) versus the reduced temperature T / |J |, for ∆ /|J | = ∆ /|J | = 0.25 in (a), (b); ∆ /|J | = ∆ /|J | = 2 in (c), (d); ∆ /|J | = ∆ /|J | = 4 in (e), (f). ... 81
Figure 4.1: Partial magnetization and total magnetic order parameter as a
function of the reduced temperature for excess rates ranging from 5% to 50%.
(a) Magnetization of sublattice for excess, (a’) magnetization of
Cr-sublattice for Re-excess. (b) Magnetization of Re-Cr-sublattice for Cr-excess, (b’) Magnetization of Re-sublattice for Re-excess. (c) Total magnetic order parameter for Cr-excess, (c’) total magnetic order parameter for Re-excess. . 90
Figure 4.2: Internal energy as a function of the reduced temperature for excess
rates ranging from 5% to 50%. (a) Internal energy corresponding to the disorder created by Cr-excess. (b) Internal energy corresponding to the disorder created by Re-excess. ... 91
Figure 4.3: Magnetic susceptibility as a function of the reduced temperature for
excess rates ranging from 5% to 50%. (a) Susceptibility corresponding to disorder created by Cr-excess. (b) Susceptibility corresponding to disorder created by Re-excess. ... 92
Figure 4.4: Magnetic susceptibility as a function of the reduced temperature for
reduced crystal field of Rhenium ∆ /|J | ranging from 1 to 4. (a) Susceptibility corresponding to the disorder created by 10% of Cr-excess. (b) Susceptibility corresponding to the disorder created by 25% of Cr-excess. (c) Susceptibility corresponding to the disorder created by 40% of Cr-excess. ... 93
Figure 5.1: Magnetization as a function of reduced temperature T/|J1|; (a)
magnetization of Chromium sublattice; (b) magnetization of Rhenium sublattice;
(c) magnetization of Osmium sublattice; (d) total magnetic order parameter for
Os-concentrations ranging from 10% to 90%. Reduced crystal fields are taken uniform △Cr/|J1|= △Re/|J1|= △Os/|J1|= 1. ... 101
X
Figure 5.3: Internal energy as a function of reduced temperature T/|J1| for
various Os-concentrations ranging from 10% to 90% for uniform crystal fields (△Cr/|J1|= △Re/|J1|= △Os /|J1|= 1 ... 103 Figure 5.4: Specific heat as a function of reduced temperature T/|J1| for
various Os-concentrations ranging from 10% to 90% for uniform crystal fields (△Cr /|J1|= △Re /|J1|= △Os /|J1|= 1). ... 103 Figure 5.5: (a) Magnetic susceptibility, (b) specific heat as a function of
reduced temperature T/|J1| for various system sizes ranging from L =16 to
L =32. Os-concentration is fixed at 50%. Crystal fields values are taken uniform (△Cr/|J1|= △Re/|J1|= △Os/|J1|= 1 ... 104 Figure 5.6: Magnetic susceptibility as a function of reduced temperature
T/|J1| for various reduced crystal field △Os/|J1| ranging from 1 to 4.
Os-concentration fixed at (a) 20%, (b) 40%, (c) 60% and (d) 80%. Crystal fields of Chromium and Rhenium are taken uniform (△Cr /|J1|= △Re/|J1|= 1). ... 105 Figure 6.1: a) Magnetization and magnetic susceptibility vs. the reduced
temperature T/|J1| for the reduced exchange couplings: J2 /|J1| = J3 /|J1| =1. b)
Magnetization vs. the reduced temperature T /|J1| for the reduced exchange
couplings: J2 /|J1| fixed at 1 and J3 /|J1| ranging from 1 to 1.6. Reduced crystal
fields (△Cr/|J1|, △Ir/|J1|) and the reduced external field h/|J1| are taken null... 112 Figure 6.2: a) Internal energy and specific heat vs. the reduced temperature T
/|J1| for the reduced exchange couplings: J2 /|J1| = J3 /|J1| =1. b) Internal energy
vs. the reduced temperature T /|J1| for the reduced exchange couplings: J2 /|J1|
fixed at 1 and J3 /|J1| ranging from 1 to 1.6. Reduced crystal fields (△Cr/|J1|,
△Ir/|J1|) and the reduced external field h/|J1| are taken null. ... 113 Figure 6.3: a) Magnetization vs. the reduced temperature T/|J1|, b) magnetic
susceptibility vs. the reduced temperature T/|J1|, for reduced crystal field △Ir /|J1|
ranging from 0.25 to 2.25. Reduced crystal field △Cr/|J1| is fixed at 0.25.
Reduced exchange couplings are taken: J2 /|J1| = J3 /|J1| =1. Reduced external
field h/|J1| is taken null ... 114 Figure 6.4: Magnetization vs. magnetic field loops for Sr2CrIrO6 DP for the
reduced exchange couplings: J2 /|J1| =5, J3 /|J1| =1. Reduced crystal fields
(△Cr/|J1|, △Ir/|J1|) are taken null. Data are calculated at the reduced
temperatures: T/|J1| =10 (a), T/|J1| =20 (b), T/|J1| =30 (c), T/|J1| =40 (d). ... 115 Figure 6.5: Magnetization vs. magnetic field loops for Sr2CrIrO6 DP for the
XI
Figure 6.6: Magnetic field dependence of the magnetization of Sr2CrIrO6 DP, a)
at the reduced temperature T/|J1| =20 for the reduced exchange couplings ( J2
/|J1| =5, J3 /|J1| =1) and ( J2 /|J1| =1, J3 /|J1| =5), and b) at the reduced
temperature T/|J1| =60 for the reduced exchange couplings ( J2 /|J1| = J3 /|J1|
=1), ( J2 /|J1| =5, J3 /|J1| =1) and ( J2 /|J1| =1, J3 /|J1| =5). Reduced crystal fields
(△Cr/|J1|, △Ir/|J1|) are taken null. ... 116 Figure 6.7: Magnetization (a-a’) and magnetic susceptibility (b-b’) vs. the
reduced temperature T/|J1| of Sr2CrIrxOs1-xO6 (a-b) and Sr2CrIrxRe1-xO6 (a’-b’)
for x ranging from 0.1 to 0.5. Reduced exchange couplings are taken uniform: J2 /|J1| = J3 /|J1| =1. Reduced crystal fields (△Cr/|J1|, △Os/|J1|, △Re/|J1|, △Ir/|J1|)
and the reduced external field h/|J1| are taken null. ... 117
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Table 1.1: This table collects the pertinent experimental and theoretical data for
some known DPs Sr2BB′O6 and for Ba2MnReO6 as well as Ba2FeMoO6. NV is
the total number of valence electrons supplied by B and B′. symm. Column presents the space group symmetry, where I4/m and Fm3m denote the tetragonal and the cubic symmetry respectively. Type column gives the type of the calculated magnetic structure, where HMF stands for half-metallic ferrimagnet, MIN for magnetic insulator, and MHI for Mott-Hubbard insulator. The total calculated magnetic moment M is given in µB per formula unit. LB
and LB′ denote the local moment of B and B’ respectively. 6LO denotes the
induced moments of the 6O atoms that construct the octahedral around B and B’. LB, LB′ and 6LO are all in units of µB. The calculated Curie temperatures,
T , and the experimental ones, T , are given in K [21]. ... 17
Table 1.2: Ferro/ferromagnetic insulating Double Perovskites [66]. ... 21
Table 1.3: antiferromagnetic insulating Double Perovskites [66]. ... 22
Table 2.1: List of the top ten algorithms of the 20th century [103], [104]. ... 35
Table 3.1: Critical exponents values and the critical temperature vs. the reduced crystal fields and the universality class for the three dimensional Ising model ... 82
Table 5.1: Monte Carlo Simulation scenario describes the composition of the lattice and the system states during the simulation. ... 99
XIII
2D Two Dimensional
3D Three Dimensional
AF Anti-Ferromagnetic
APB Anti-Phase Boundary
DE Double Exchange
DFT Density Functional Theory
DOS Densities Of States
DP Double Perovskite
DRAM Dynamic Random Access Memory
EFT Effective Field Theory
EFRG Effective Field Renormalization Group
F Ferromagnetic
FCC Face Centered Cubic
Fi Ferrimagnetic
Fo Ferromagnetic
GMR Giant Magneto-Resistance
HDD Hard Disc Drives
HMF Half-Metallic Ferrimagnetism
HREM High-Resolution Electron Microscopy
IT Information Technology
MCMC Markov Chain Monte Carlo
MCS Monte Carlo Simulation
MFA Mean Field Approximation
MHI Mott-Hubbard Insulator
MIN Magnetic Insulator
MRAM Magnetic Random Access Memory
MTJ Magnetic Tunnel Junction
NN Nearest-Neighbors
NNN Next Nearest-Neighbors
SE Super Exchange
SEM Scanning Electron Microscopy
SOC Spin-Orbit Coupling
SRAM Static Random Access Memory
STT-MRAM Spin-transfer torque -MRAM
XMCD X-Ray Magnetic Circular Dichroism
XIV
Abstract ... I Résumé ... II Résumé détaillé ... III List of Figures ... VI List of Tables ... XII List of Abbreviations ... XIII Contents ... XIV
General Introduction ... 1
Part I: Double Perovskits: Basic Survey 1. Double Perovskites: Basic Survey ... 6
1.1. Generality ... 6
1.2. Spintronics ... 7
1.3. Crystalline structure ... 10
1.4. Electronic structure ... 13
1.5. Cr-based double perovskites ... 15
1.6. Magnetic order in double perovskites ... 20
1.6.1. Ferro/ferrimagnetic double perovskites ... 21
1.6.2. Antiferromagnetic compounds ... 21
1.6.3. Spin glass compounds ... 22
1.7. Antisite disorder ... 23
1.7.1. Spatial correlation of antisite disorder ... 24
1.7.2. Antisite effect on magnetization ... 26
1.7.3. Antisite effect on spin-polarization ... 27
1.7.4. Antisite effect on transport properties... 28
1.8. Substitution doping ... 29
1.8.1. Doping on A site ... 29
1.8.2. Doping on B site ... 30
1.8.3. Doping on B’ site ... 32
Part II: Model and Methods 2. Model and Methods ... 34
2.1. Introduction ... 34
2.2. Ising Model ... 37
2.3. Monte Carlo simulation ... 39
2.3.1. Introduction ... 40
2.3.2. Monte Carlo simulation concept ... 42
2.3.3. Metropolis algorithm ... 47
2.3.4. Equilibrium and measurements ... 51
2.4. Mean Field Approximation ... 53
XV
3. Chapter 1: Magnetic properties of Sr2CrReO6 double perovskite: Mean
Field Approximation, Effective Field Theory and Monte Carlo Simulation
... 59
3.1. Introduction ... 59
3.2. Structure and model ... 61
3.3. Ground state phase diagrams ... 64
3.4. Mean Field Approximation analysis ... 64
3.5. Effective Field Theory analysis ... 69
3.6. Monte Carlo Simulation ... 74
3.7. Conclusion ... 83
4. Chapter 2: Antisite disorder study by Monte Carlo Simulation of the double perovskite Sr2CrReO6 ... 84
4.1. Introduction ... 84
4.2. Antisite Ising Model ... 85
4.3. Simulation process ... 86
4.4. Results and discussion ... 86
4.5. Conclusion ... 94
5. Chapter 3: Investigation by Monte Carlo Simulation of the diluted Double Perovskite Sr2CrRexOs1-xO6 ... 96
5.1. Introduction ... 96
5.2. Model and method ... 97
5.3. Results and discussion ... 99
5.4. Conclusion ... 106
6. Chapter 4: Magnetism, hysteresis cycle, and Ir-substitution doping of Sr2CrIrO6 double perovskite: a Monte Carlo Simulation ... 107
6.1. Introduction ... 107
6.2. Model and method ... 109
6.3. Results and discussion ... 111
6.4. Conclusion ... 119
Thesis summary ... 120
Appendix ... 123
1
General Introduction
The broad aim of this thesis is to develop a theoretical understanding at the atomic scale in the statistical physics background of the magnetic properties of Cr-based double perovskites (DPs) with the highest Curie temperature measured until now in the magnetic oxides. Thus, we have used statistical methods, such as mean field approximation (MFA) and effective field theory (EFT), and performed Monte Carlo simulations (MCS) to develop a detailed theoretical knowledge of the magnetic behavior and the effects of some phenomena such as antisite defect, substitution doping and magnetic hysteresis on the DPs Sr2CrReO6 and Sr2CrIrO6 in the framework of the three-dimensional Ising model.
Magnetic materials, which have a Half-metallic character in one spin channel and insulating character in the other, continue to attract considerable attention due to their potential application possibility for advanced sensor and memory applications. Half-metallic nature of these materials involves the carrier scattering to be spin-dependent, which leads to a strong influence of the resistance by low magnetic fields. Most of the known half-metallic magnets have generally low range of operating temperatures (transition temperature “TC”
close to the room temperature), which open the need for additional materials with high transition temperatures. Cr-based DPs having the general formula
Sr2CrB′O6 offer the much attractive possibility in this respect because they
display transition temperatures highest ever known in the magnetic oxide family. The high degree of spin polarization in magnetoresistive materials such as the here studied Cr-based DPs is believed to result from the half-metallic character of these materials. Giant magnetoresistance (GMR) is known to be closely related to the high spin polarized materials, which make materials with high spin polarization essential in spintronic applications. However, various families of half-metallic compounds that have ferro/ferrimagnetism have not shown large values of tunneling magnetoresistance at room temperature, because of their low ordering temperatures. Therefore, developing the understanding of half-metallic ferromagnetic materials, such as Cr-based DPs, with high transition temperatures is important in the field of spintronics.
The key notion, in spintronics materials and devices, is the strong correlation between the spin direction (spin-up or spin-down) and charge of conduction carriers. High spin-polarized conduction carriers at room temperature were found in few ferro/ferrimagnetic DPs with interesting properties promising their use in spintronic applications. The chemical
2
composition and crystal structure of DPs provide a large variety of possibilities to find other highly spin-polarized materials with high transition temperature, and large magnetization suitable for spintronics applications. Interesting novel applications such as advanced sensors and magnetic memories have been performed from the manipulation of the spin of electrons in information processing. Highly sensitive magnetic field sensors in recording read heads for magnetic disk drives is one of these important applications. Likewise, spintronics engineering can perform potential suitable materials for novel solid-state memories. Perovskites are also considered the future of solar cells, as their distinctive structure makes them perfect for enabling low-cost and efficient photovoltaics. They are also predicted to play a role in next-generations of electric vehicle batteries, sensors, displays, lasers and much more.
In field of condensed matter, the new characterization tools based on modeling structures of materials and simulating their physical properties, in different scales, have proven their efficiency in understanding the complex structure of new synthetized materials and predicting materials with outstanding properties that can be used in new applications. Thanks to the high accuracy in describing the materials properties, these tools become today increasingly used in the development of new materials and for interpretation of experimental results. Besides this, these modeling tools are able to go beyond the limits of manageable mathematics and experimental capability. MCS and effective field approaches belong to this category of tools. MCS, especially, derives its power from decades of intense research based essentially on two equally important developments that are the powerful numerical algorithms and the hardware capacities in term of storage and computational facilities of modern computing systems that known an increasing evolution thanks to the continuous down-scaling of microelectronic devices.
In order to develop a detailed theoretical knowledge of Cr-based DPs, the need of understanding their magnetic properties brings up the following broad issues:
o How does the DP Sr2CrReO6 behave magnetically versus the different
exchange couplings and crystal fields prevailing in its structure? Which method, among MCS, MFA, and EFT, is the most appropriate to study the magnetic properties of such a material?
o What are the antisite disorder effects that can be recovered from the MCS, on the magnetic behavior of Sr2CrReO6 DP?
o Since the DP Sr2CrOsO6 has a transition temperature higher than
Sr2CrReO6 DP, how does the latter DP change its magnetic behavior
3
o The hypothetical compound Sr2CrIrO6 should exhibit the highest TC ever
reported for half-metallic ferromagnets. How does this compound behave magnetically versus the different exchange couplings and crystal field prevailing in its structure? What are the hysteresis properties of this DP? And how does it behave regarding different rates of Ir-substitution doping by Os and Re?
This thesis tries, in the chapters 1 to 4, to answer these issues. We thus throw some light on existing experimental and theoretical data, make predictions where experiments on some phenomena do not yet exist, and try to provide interpretations and a theoretical understanding of the phenomena in consideration.
In the first part of this thesis, we focus on double perovskites with a general basic survey. We present DPs in the spintronics field that is a modern solid-state science focused on both, the intrinsic spin of the electron and its associated magnetic moment in addition to its fundamental electronic charge. We present the general crystalline and electronic structure of DPs. Then, we summarize the key experimental and theoretical observations on the Cr-based DPs, we emphasize the effects of antisite disorder and substitution doping, and also describe past efforts at simulating such materials. This includes discussions of the spin- and site-resolved densities of states (DOS) of the here studied Cr-based DPs.
In the second part, we present and discuss the Ising model Hamiltonian used in this entire work, because studying the magnetic properties in this work is based on reducing the studied systems into simple models consisting of discrete variables that represent magnetic dipole moments of atomic spins that can take only two directions (up- or down-direction) and therefore, the more appropriate models that can describe the magnetic properties in consideration are the ones based on discrete spins such as Ising model. Then, we describe MFA, and EFT methods for a simple Ising model, and we emphasize the general ideas behind MCS method. We also discuss some details of the application of MCS combined with Metropolis algorithm on a simple Ising model. Chapter 1 in the third part is devoted to the theoretical study of phase diagrams and magnetic properties of the DP Sr2CrReO6 using different methods
of statistical physics, such as MFA, EFT and MCS in the framework of Ising model. The model has been constructed to describe the compound as a mixed spin system (3/2, -1). These three methods have been thus performed to analyze the phase diagrams and study the magnetic properties and critical exponents of the DP Sr2CrReO6. We presented first the compound structure
4
structure. Then, we established the ground state phase diagrams. Next, we presented the application of MFA method followed by the formulation of EFT which is more accurate than the standard mean-field theory [127]. We used the differential operator as a mathematical tool [16] in the Callen identity [157] that is the basis of the EFT method. Next, we performed a MCS combined with the Metropolis algorithm. We presented Phase diagrams depending on the different exchange couplings with analysis of crystal field effects on these diagrams. We computed the magnetic susceptibility and the critical exponents using MCS. We concluded with a comparison between the critical exponents recovered by MCS and the universality class of the 3D-Ising model.
Chapter 2 discusses the antisite disorder study of the DP Sr2CrReO6 by
using a Monte Carlo simulation, that was resumed from the study in chapter 1 and performed with two modifications to make it suitable first to simulate the effect of the antisite disorder created by excess of Cr-atoms in Re-sublattice (Sr2Cr1+xRe1-xO6) and then to simulate the effect of the same disorder created
by excess of Re-atoms in Cr-sublattice (Sr2Cr1-xRe1+xO6). We thus detail the
Ising model of this system for both cases of the disorder treated. We describe the simulation process for both cases. Then, we continued by the presentation and the discussion of the recovered results and we concluded with summarizing the most important results of this work and drawing conclusions.
Chapter 3 is concerned with the impact of Re-substitution doping by Os in the DP Sr2CrReO6. We have therefore focused the investigation on the DP
Sr2CrReO6 with a random dilution by Os in site B' (RexOs1-x). MCS combined
with Metropolis algorithm in the framework of Ising model have been performed to study the energetic and magnetic behavior of Re-substitution by Os in the range (0. 1≤ x ≤0.9). We report the size effect for x = 0.5 and the effect of rising Os-crystal field △Os on the magnetic properties, correlated with the
Os-concentration.
Chapter 4 focused on the study conducted by using a MCS that has been performed in the framework of Ising model to study the magnetism, the hysteresis, and the effect of Ir-substitution doping in the promising Sr2CrIrO6
DP. Thus, we studied the magnetization and the internal energy versus different exchange coupling values. Next, we investigated the effects of iridium crystal field on the magnetization and the magnetic susceptibility. Then, we examined the magnetic hysteresis regarding different exchange coupling values at different temperatures. We also explored the effects of Ir(5d4)-substitution doping on the magnetic behavior for two cases: doping by Os(5d3) “Sr
2CrIrxOs 1-xO6”and by Re(5d2) “Sr2CrIrxRe1-xO6” in the range (0.1≤x≤ 0.5).
5
To summarize, this thesis considers the exploration of magnetic properties of Cr-based DPs that have the highest Curie temperature measured until now in the magnetic oxides. We report on the magnetic properties of
Sr2CrReO6 DP and clarify the difference between MFA, EFT and MCS results.
We study, by using a MCS, the impact of antisite disorder on the DP
Sr2CrReO6, explore the junction between the DPs Sr2CrReO6 and Sr2CrOsO6 by
studying the impact of Re-substitution doping by Os in Sr2CrReO6 DP, and
provide some interpretations of some phenomena such as the sharp drop of the partial magnetization due to the antisite defect in Sr2CrReO6 DP, and the
unusual non-monotonic behavior of the magnetization related to Sr2CrIrO6 DP.
6
1. Double Perovskites: Basic Survey
1.1. Generality :Perovskite materials are a class of magnetic oxides that have been gaining importance in recent years as spintronics materials. Their name originates from the mineral CaTiO3 that has a similar crystalline structure. This
mineral was first described in 1830 by the geologist Gustav Rose who named it after the russian mineralogist Count Lev Aleksevich von Perovski.
Several phenomena can be hosted by transition-metal oxides with the perovskite structure, like: high-temperature superconductivity [3], colossal magnetoresistance [4, 5], multiferroicity [6], metal-insulator transition [7, 8], half-metallicity [9]. This richness of behaviors has been further expanded in recent years with recent observations in novel perovskite oxides, such as: localized-like transport [10, 11, 12], non-Fermi-liquid behavior [13], and spin-liquid ground state [14]. Moreover, unconventional electronic structures have shown novel phases that have been theoretically described: axionic insulators and Weyl semimetals [19], Mott insulators with orbital mediated exchange coupling in Kitaev-type models [15], topological Mott insulators [17], layered quantum spin Hall systems [16], spin liquids [18], and novel high-temperature superconductors [20]. In the class of Cr-based double perovskites (DPs), which display transition temperatures higher than all the magnetic oxides, the hypothetical compound Sr2CrIrO6 should exhibit the highest TC ever reported for
half-metallic ferromagnets [21]. Perovskites are also considered the future of solar cells, as their distinctive structure makes them perfect for enabling low-cost and efficient photovoltaics. They are also predicted to play a role in next-generations of electric vehicle batteries, sensors, displays, lasers and much more.
Half-metals are highly desired materials in spintronics, where, charge carriers that have one of the two possible polarization states, are the only ones contributing to the conduction. The first well known half-metal in the class of double perovskites is the most studied DP Sr2FeMoO6 [5]. While the halfmetallic
ferrimagnetic double perovskites can have a high Curie temperature “TC“ [22],
simple perovskites as the half-metallic ferromagnetic manganites TC in the
highest case still close to room-temperature. Ferrimagnetism in the double perovskites has been predicted to be kinetic energy driven [23, 24, 25]. Briefly , this mechanism result from the hybridization of the exchange split 3d-orbitals (3d5 in Fe-based DPs, where spin majority orbitals fully occupied; or 3d3 in Cr-based DPs, which are the topic of this work, where only t2g are fully occupied),
7
gain, which lead to a shift of the bare energy levels at the 5d-metal site, and a strong tendency to half-metallic behavior. This mechanism is operative for the Fe3+ -based and Cr3+-based DPs, where all 3d majority spin states resp. all t2g
majority spin sates are fully occupied and represent localized spins. Strong correlation between this mechanism and the half metallic behavior was reported by band-structure calculations [5, 9, 23, 26, 27, 28], as the spin-polarized conduction carriers mediate an antiferromagnetic order between the 3d and the 5d transition metal ions, and a ferromagnetic order between the 3d transition metal ions.
In spintronics materials and devices, the key notion is a strong integration between the spin direction (spin-up or spin-down) and charge of conduction carriers. High spin-polarized conduction electrons at room temperature were found in few ferromagnetic or ferrimagnetic transition metal complex perovskites with interesting properties to use in spintronic applications. The chemical composition of DPs offers a large variety of possibilities to find other spin-polarized materials with high Curie temperature, and large magnetization suitable for spintronics applications.
1.2. Spintronics:
Spintronics, or spin-transport electronics, is a modern solid-state science focused on both, the intrinsic spin of the electron and its associated magnetic moment in addition to its fundamental electronic charge, for the purpose of producing modern devices exploiting this technology. Basis discoveries concerning spin-dependent electron-transport phenomena in solid-state devices were at the origin of this technology. In the modern electronics industry, spintronics is the key to revolutionize the market of electronic devices. High storage density, low energy consumption, nonvolatility of data storage and high speed of data processing are some interesting properties offered by this technology [29]. Today the transport of electrical charge electrons in a semiconductor like silicon is the main function in which are based spintronics devices like in the conventional electronic devices. However, exploiting the spin of the electron rather than its charge is the big challenge taken up by researchers to create remarkable new generation of spintronics devices smaller, robust and more versatile than those present currently in the market exploiting silicon chips and circuit elements [30].
The operating mode of spintronics devices is based on the following process: first the input data “the information” is stored, then written into spins as a particular spin orientation spin-up (S↑) or spin-down (S↓), next the spins, being the own spins of mobile carriers, carry the information along a conductor,
8
and finally the output information is read at the terminal [31]. Interesting novel applications such as advanced sensors and magnetic memories have been performed from the manipulation of the spin of electrons in information processing. Highly sensitive magnetic field sensors in recording read heads for magnetic disk drives is one of these important applications. These sensors, well known as “spin-valve” sensors, were predicted in the early 1990s after the
discovery of the twin phenomena of giant magnetoresistance (GMR) and
oscillatory interlayer magnetic coupling in magnetic multilayers. More recently, engineering concepts in conjunction with the phenomenon of spin-dependent tunneling have given rise of development of even more sensitive recording read heads. Since about 2007, have completely replaced sensors based on the phenomenon of GMR have been replaced by the magnetic tunnel junction (MTJ) read heads. New sensing materials with a complex set of intertwined properties are required for further advances in such recording read heads. Likewise, spintronics engineering can perform potential suitable materials for novel solid-state memories. In the mid-1990s, it was proposed a MTJ based magnetic random access memory (MRAM) which is exploiting the principle of non-volatile memory. In 1999, IBM has proposed the first MTJ based MRAM. The magnetoresistive memory “MRAM” is formed from two ferromagnetic plates separated by a thin insulating layer (figure 1.1.a). A plate is a permanent magnet, while the other can change its magnetization by an external field to store memory. This is the simplest configuration of a MRAM called “spin valve” cell. Such cells are made in a grid to build the memory device. Reading process is accomplished by measuring the electrical resistance of the cell. Motorola has manufactured the first 4MB of memory in 2003 (figure 1.1.b). A very high density MRAM prototype of 16MB was manufactured by IBM-Infineon in June 2004 (figure 1.1.c). Development of this technology focused on writing cycles rapidity and on nano-dimension of elements. In 2011, PTB-Germany announces having achieved a below 500 ps (2GBit/s) write cycle. MRAM memory is fast, high-density and non-volatile and can replace the flash memory (that currently dominate the data storage market) and all kinds of memories used today in a single chip. Quite recently (march 2015), Toshiba presented a new STT-MRAM (Spin-transfer torque -MRAM) 1MB test chip (figure 1.1.d) that provides speed performance capable of 3.3-ns access to in-cache memory. Consumption of that new chip is about 80% less than conventional SRAM as embedded memory which makes it the best power-performing embedded memory [36].
Currently, the scientific and the industrial community have a vested interest in replacing the conventional charge based memories such as DRAM, e-DRAM and even SRAM (Dynamic and Static Random Access Memory) by the wide variety of possibilities offered by the potential of MRAM for high density
9
applications. The conventional charge based memories have difficulties scaling to dimensions below 10–15 nm [32]. Just like sensors, the MRAM require new materials with the needed specific properties for this application. High tunneling magnetoresistance is the first property needed in the MRAM; in addition, the MTJ memory element would be able to switch its magnetic state using tiny currents passed through it, taking advantage of the phenomenon of spin-momentum transfer. Moreover, the more important property of these MTJ elements is the stability against thermal fluctuations, as their memory, namely the direction of magnetization of the electrode in which the data is stored, could be sustained for a decade or longer [32]. Find materials with these stringent requirements is a major challenge.
Figure 1.1: MRAMs: a) schematic drawing illustrating the principles of MRAM
functioning,b) 4MB MRAM from Motorola, c) 16MB MRAM prototype from
IBM-Infineon, d) STT-MRAM (Spin-transfer torque -MRAM) 1MB test chip from Toshiba.
In this work, properties of Cr-based DPs were explored theoretically to assess their performance for spintronics applications. The more important features that attract the interest in this class of spintronics materials are the high ordering temperature and the high spin polarization (100% in most cases). In spintronics applications, only Heusler compounds could be more competitive
10
than the DPs, because Heusler compounds can have Curie temperatures higher than 1000K (Co2FeSi was reported to have TC greater than 1120 K [35]).
1.3. Crystalline structure :
The cubic structure is the typical structure of perovskite materials. However, there are a large number of exceptions that have the cubic structure slightly distorted. Chemical composition of oxides with perovskite structure is generally composed of an alkaline earth metal cation (A), a tetravalent transition metal cation (B) and oxide anions. This composition (AIIBIVO)is very similar to
the composition of the main compound CaTiO3 that has an orthorhombic
structure. However, other compositions such as AIIIBIIIO or AIBVO have been
known for a long time [1, 2].
Figure 1.2: Ideal cubic unit cell of a perovskite ABO3. A in the corners are
alkaline earth metal cations, B in the center is a transition metal cation and O in the center of faces are the oxygen anions.
11
Figure 1.3: The ideal structure of a double perovskite A2BB’O3. Due to visibility
concerns, A-cations in the center of each cell are note illustrated. B and B’ are the transition metal cations and O surrounding B and B’ are the oxygen anions.
An ideal perovskite ABO3 has a cubic unit cell, with the parameter
a ≈ 3,9 Å (figure 1.2). Alkaline earth metal cations A are located at the corners of the cube, transition metal cations B are in the center of the cube (octahedrally coordinated) and the oxygen anions are in the center of each face of the cube. DPs have generally the formula A2BB’O6, as simple perovskites, A is an alkaline
earth metal or a lanthanide. In the B/B’ sites there are two transition metals surrounded by the oxygen anions to form the octahedra BO6 and B'O6 attached
by the corners (figure 1.3). Atoms in the sites B and B’ are each placed in a Face Centered Cubic (FCC) structure forming them both a rock-salt structure.
Under the arrangement of the octahedra in the crystal, it is possible to have three possible situations: random arrangement, ordered arrangement by alternating layers BO6 / B’O6 /BO6 and ordered arrangement by tridimensional
alternating the octahedral (i.e. Each octahedra BO6 has as neighbors only the
octahedra B’O6, and vice-versa.). The third arrangement, which is illustrated in
(figure 1.3), was reported to be the ideal one for several DPs, and is the composition of the here studied Cr-based DPs. The actual structure of this kind of composition lies between the random arrangement and the tridimensional altering arrangement, according to the disorder of cations B and B’ (the antisite disorder). The structure of DPs in the crystallographic descriptions is note a
12
2 ) that most accurately reflects the slight distortion of the structure [5, 33, 34]. However, in this work, the centered quadratic unit cell has imposed many exchange couplings. Therefore, during the modeling process, to minimize the number of couplings, the FCC unit cell was the suitable structure for the simulation which has given rise to consistent results.
Perovskite composition has avery compact structure that does not allow
the formation of interstitial compositions. By contrast, several substitutions on the sites A and B are possible. Thus, each composition obtained by these substitutions gives rise to a distorted perovskite structure according to the size of the cations occupying the sites A and B.
Crystal structure of a DP with the formula A2BB’O6 is very flexible, as A,
B and B’ ions can be varied leading to the large number of known DP compounds. Most perovskites compounds do not have the ideal cubic structure and present distortions. The distortion has mainly three: + size effects, + deviations from the ideal composition and + the Jahn-Teller effect. A single effect can rarely affect the structure of of a complex perovskite compound. Therefore, in most cases, the distortion is assigned to several factors. Goldschmidt, by his pioneering work [37], was the first to describe the ionic size effect on the structure of simple perovskites AMO3. For an ideal cubic structure,
there is a geometrical relation between the cell axis and the ionic radii, rA, rB, rB’,
and rO in A2BB’O6:
= √2( + ) = 2 + (1) Goldschmidt’s tolerance factor (t), which is the ratio of the two expressions to the cell length in Eq. (1), give information of the degree of distortion [38] and can be described as:
=
( )√ (2)
Ideal cubic perovskite compound has a tolerance factor (t ≈ 1). Transition temperature TC of ferromagnetic (ferromagnetic) DPs was reported to be closely
related to the tolerance factor (t) [39]. TC is maximal for 0.99 < t < 1.05 where
the cubic crystal structure occurs; 0,75 < t < 0,96 is leading to an orthorhombic distortion with the lower TC and 0,96 < t < 0,99 to a rhomboedric distortion. For
reasons of modeling and to make the simulation of the here studied Cr-based DP compounds less complex, structural distortions have not been considered, and so the tolerance factor was taken t = 1 which corresponds to the ideal case where the structure is perfectly cubic.
13
1.4. Electronic structure:
Transition metals are chemical elements belonging to the d-block present in the center of the long form of the periodic table. In the d-block, atoms of elements have between 1 and 10 d electrons. Cr-based DPs covered in this study are: Sr2BB’O6 (B=Cr; B’= Re, Os, Ir) where the transition metal in site B is
Chromium, which belong to the (3d) Transition Series (Sc-Zn) , with the electronic configuration 3d54s1; in the site B’, there are: Rhenium, Osmium and Iridium that belong to the (5d) Transition Series (Lu-Hg) have respectively the electronic configurations: 5d56s2, 5d66s2, 5d66s2. In the molecule, according to Lewis structures [40] these metals have the electronic configurations: Cr(3d3), Re(5d2), Os(5d3) and Ir(5d4). Consequently, their spins are: Cr ( = ± , ± ), Re( = ±1, 0), Os( = ± , ± ), Ir( = ±2, ±1, 0). In several researches, the G type antiferromagnetic configuration (figure 1.4) was proposed for this category of DPs [5, 23, 41, 42]. Thus, atoms of the Cr-sublattice are coupled with the 5d-sublattice by an antiferromagnetic exchange coupling (this interaction is known to be a super exchange (SE) interaction), which means that Cr-atoms are coupled with each other’s by a ferromagnetic coupling (this mechanism is known to be a double exchange (DE) interaction, sometimes also kinetic exchange) just like the 5d-atoms which are also coupled with each other’s by a ferromagnetic coupling (DE interaction). The antiferromagnetic coupling between the two sublattices gives rise to a half-metallic ferrimagnetic behavior in Sr2CrReO6 and Sr2CrIrO6, and an insulating antiferromagnetic behavior in
Sr2CrOsO6 (figure 1.5). The last one is known to be ferrimagnetic only thanks to
the large Spin-Orbit Coupling (SOC) of Os. Recent studies evoke that the net moment in Sr2CrOsO6 was observed by Density Functional Theory (DFT)
calculation only when SOC is included [43, 44] while the same SOC seems without influence in X-ray magnetic circular dichroism (XMCD) experiments [45].
14
Figure 1.4: illustration of the G type antiferromagnetic coupling that occurs
between the two sublattices B and B’ of a double perovskite with the formula A2BB’O6.
Figure 1.5: the electronic configurations of the B - B’ cations in the formula
15
1.5. Cr-based double perovskites:
Technological advancement in spintronics requires oxides with high magnetic transition temperature (TC). In this connection, the much discussed class of
compounds are the DPs with the formula A2BB’O6. This family of compounds is
becoming increasingly important following the observation of large magnetoresistance and an unusual origin of magnetism with a quite high ferromagnetic TC of about 410K in Sr2FeMoO6 [5]. Since then, the challenge
faced up by researchers was found new materials with higher TC, or enhance
the performance of materials by doping to boost the TC further. Several
attempts have been made in Sr2FeMoO6 by La doping to increase the TC.
Following this path, some partial success has been achieved [46], by changing choices of B and B’ ions TC was noticed to be boosted more efficiently.
However, the microscopic raisons of this increase are still being discussed. A collection of measured and calculated transition temperatures shows a correlation with the number of valence electrons among different double perovskite compounds and their corresponding TC (figure 1.6) [21].
Figure 1.6: Measured and calculated TC vs. the number of valence electrons.
The names of the elements BB′ in the DPs A2BB′O6 appear near the data
points. The value in parenthesis (CrIr) is the calculated TC because the
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