Magnetism of Manganites, Semiconductors and Spin Glasses

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Magnetism of manganites, semiconductors and spin glasses

Roland Mathieu Dissertation for the Degree of Doctor of Philosophy in Physics presented at Uppsala University in 2002

Abstract

Mathieu, R. 2002. Magnetism of manganites, semiconductors and spin glasses. Acta Universitatis Upsaliensis. Uppsala Dissertations from the Faculty of Science and Technology 38. 115 pp. Uppsala. ISBN 91-554-5339-2. Magnetic and electrical properties of selected compounds containing manganese (Mn) are investigated by SQUID magnetometry and transport measurements. (Ga,Mn)As is a magnetic semiconductor obtained from GaAs by substituting Ga3+ for Mn2+. Mn acts in the alloy as a magnetic impurity, as well as a hole dopant. A carrier mediated ferromagnetic interaction is observed in (Ga,Mn)As single layers, as well as in (Ga,Mn)As/GaAs superlattices. The magnetic and electrical properties of these structures are controlled by the amount of holes, and thus by the amount of compensating defects such as AsGa antisites. Magnetic inhomogeneity appears for thin layers as well as for layers containing large concentration of Manganese. In non magnetic metallic elements containing a small amount of manganese impurities, a magnetic interaction develops, oscillating in sign with the distance between Mn atoms. Due to random distribution of manganese in a Ag(Mn) alloy, competing ferromagnetic and antiferromagnetic interaction appears, yielding magnetic frustration and the appearance of a spin glass phase at low temperature. These disordered systems show aging, chaos and memory phenomena, which are investigated in the three dimensional Ag(Mn) and Fe0.5Mn0.5TiO3 spin glasses using time dependent magnetization measurements. 3+ 2+ Perovskite manganites of type (R1−xAx )MnO3 show colossal magnetoresistive effects (CMR). For an optimum doping x, a ferromagnetic order is established, and large changes of their electrical resistance with an applied magnetic field are observed; a magnetoresistance which can be tailored by adding oriented grain boundaries in thin films of these materials. The Manganese appears in the system as Mn3+ and Mn4+, and both ferromagnetic and antiferromagnetic interaction is mediated by the charge carriers along the Mn-O-Mn bonds of the perovskite structure. Depending on the cations forming the manganite, and their relative amount, glassy dynamics may appear, yielding aging and memory features similar to those observed in spin glasses.

Keywords: Magnetic order, magnetic disorder, transport, colossal magnetoresistive materials, diluted magnetic semiconductors, spin glasses, manganese, thin ﬁlms and superlattices.

Roland Mathieu, Department of Materials Science, Uppsala University, Box 534, SE- 751 21 Uppsala, Sweden. [email protected]

c Roland Mathieu 2002

ISSN 1104-2516 ISBN 91-554-5339-2

Printed in Sweden by Eklundshofs Graﬁska AB, Uppsala 2002 Les savants sont des gens, qui sur la route des choses inconnues, s’embourbent un peu plus loin que les autres. Alphonse Karr 4 Contents

1 Introduction 7

2 Fundamentals 9 2.1 Magnetization and electrical resistance ...... 9 2.2 Phase transitions and order parameter ...... 11 2.3 Experiments ...... 12 2.3.1 Magnetization measurements ...... 13 2.3.2 Transport measurements ...... 18 Bibliography of Chapter 2 ...... 21

3 Diluted magnetic semiconductors: (Ga,Mn)As 23 3.1 Ferromagnetism and hole concentration ...... 23 3.2 Antisites ...... 25 3.3 More on the antisites ...... 28 3.4 GaMnAs/GaAs superlattices ...... 30 3.4.1 Magnetic properties ...... 31 3.4.2 Transport properties ...... 33 3.5 What is next? ...... 34 Bibliography of Chapter 3 ...... 35

4 Spin glasses 39 4.1 Fundamentals ...... 40 4.1.1 Equilibrium properties ...... 40 4.1.2 Droplet model ...... 43 4.1.3 Non-equilibrium properties ...... 43 4.1.4 Linear response and superposition ...... 45 4.1.5 Chaos and overlap length ...... 47 4.2 Dc-memory experiments ...... 50 4.2.1 Experimental procedure ...... 50 4.2.2 Memory and superposition ...... 52 4.2.3 Chaos ...... 54 4.2.4 Comparison between Ising and Heisenberg spin glasses ...... 55 4.2.5 Dc-memory and not so conventional spin glasses .... 59 Bibliography of Chapter 4 ...... 63

5 6 CONTENTS

5 Colossal magnetoresistive materials 67 5.1 Orbital order and phase diagram ...... 69 5.1.1 Orbital structure ...... 69 5.1.2 Magnetic interaction ...... 69 5.1.3 Bandwidth considerations ...... 71 5.1.4 Phase diagram ...... 72 5.2 CMR ferromagnets ...... 74 5.2.1 Epitaxial ﬁlms, grain boundaries ...... 75 5.2.2 Artiﬁcial grain boundaries ...... 76 5.3 Frustrated CMR ferromagnets ...... 84 5.3.1 Frustration in Nd0.7Sr0.3MnO3 ...... 84 5.3.2 Reentrant ferromagnets ...... 85 5.4 Short range CMR ferromagnets ...... 90 5.4.1 Dynamics of Y0.7Ca0.3MnO3 ...... 90 5.4.2 (La,Y)0.7Ca0.3MnO3 and Nd0.7Sr0.3(Mn,Mg)O3 ..... 93 5.5 Charge ordered antiferromagnets ...... 96 5.6 What is next? ...... 102 Bibliography of Chapter 5 ...... 103

List of Publications 107

List of Figures 111

Acknowledgments 115 Chapter 1

Introduction

n the periodic classiﬁcation of the elements, there is a group of metals referred Ito as transition metals. This group occupies the central part of the periodic table, from the third column (Scandium) to the twelfth (Ununbium1). Transi- tion metals use their valence electrons to form compounds with other elements. Iron (Fe), Nickel (Ni) and Cobalt (Co) are transition metals which exhibit ferromagnetic properties. They also carry a magnetic moment when introduced as an impurity in a noble metal or as components in a compound. The latter is also true for other transition metals, as for example Chromium (Cr), Vanadium (V) or Manganese (Mn). E.g., the Ag(Mn) alloy is an archetypical random magnet. Manganese also brings spins (magnetic moments) and holes (charge carriers) to the (Ga,Mn)As diluted magnetic semiconductor. (Fe,Mn)TiO3 is a mixture of two antiferromagnetic ilmenites which shows magnetic frustration, a phe- 3+ 2+ nomenon quite common in so called perovskite manganites (T ,D )MnO3, in which the amount and distribution of the manganese ions is a key parameter. Magnetic, as well as electrical properties of manganese doped compounds or structures similar to the ones mentioned above will be investigated in this thesis. Studying the history of the discovery of manganese, one ﬁnds that it is quite legitimate to study the magnetic properties of manganese doped compounds in Uppsala. Magnesium (Mg) and Manganese (Mn) were abundant in oxide and carbonate ores in a region of the Greek Thessaly called Magnesia. They there- fore became referred as Magnes Lithos, or stones from Magnesia. The region also contained large amounts of iron oxides (magnetite, or lodestone, for example), so that the ores were magnetized. Magnesium (or the mineral MgCO3) of light color, was named Magnesia Alba, or white Magnesia. Manganese (or the mineral MnO2), instead was much darker, and was, to distinguish it from magnesium, called Magnesia Nigra, i.e. black Magnesia, which later became Manganesium. Manganese eventually got its actual name2 when manganese

1Obtained from the fusion of Zinc and Lead. 2It actually became Mangan, and could have been called Manganium. See “Jordens Grund¨amnen och deras uppt¨ackt” by P. Enghag. Thank you Olle and Nippe for your help!

7 8 CHAPTER 1. INTRODUCTION

3 metal was obtained by chemical reduction of the pyrolusite (MnO2), by Jo- han Gottlieb Gahn (1745-1818), during the year 1774. Gahn was a mineralogist and crystallographer working in the copper mines of Falun4, Sweden, and assis- tant to Torbern Olof Bergman (1735-1784), Professor of Chemistry at Uppsala University. This thesis mainly contains results of magnetization and electrical resistance measurements performed on various manganese doped compounds in Uppsala, in the magnetic part of the Solid State Physics Department. The second chapter recalls fundamental statistical and thermodynamical quantities connected to magnetism. The third chapter is dedicated to diluted magnetic semiconductors. These compounds show some ferromagnetic order at low temperature, as well as some magnetic inhomogeneity. The fourth chapter will describe the magnetic properties of even more disordered systems, referred to as spin glasses. In the ﬁfth and last chapter, colossal magnetoresistance materials, which can appear as a ferromagnetic metal, an antiferromagnetic insulator, as well as a disordered and frustrated spin glass, will be studied.

The following discussion is based on the articles listed at the end of the thesis.

3 MnO2 is the most common compound of manganese, and makes up about 0.14% of the Earth’s crust (in comparison MgO makes 35%). Manganese metal is still nowadays obtained from pyrolusite. 4A small town close to Borl¨ange according to P. Nordblad. Chapter 2

Fundamentals

n this chapter we recall some fundamental statistical and thermodynamical Iquantities employed to describe magnetic conﬁgurations of materials. Ex- cellent texts on these subjects can be found in the literature, for example in “Physics of Ferromagnetism” by S. Chikazumi [1] or “Ferromagnetism” by R. Bozorth [2].

2.1 Magnetization and electrical resistance

A system is characterized by its energy. In order to describe the magnetic configuration of a system, Heisenberg proposed to consider the following Hamiltoni- an [3]: → → HJ = − Jij Si · Sj (2.1) i,j → → where Jij is the exchange coupling constant between the spins Si and Sj distributed on a regular lattice. Only nearest neighbors are included in the sum- mation. The magnetic properties of the system is thus dependent on the sign and strength of the interaction between spins: If Jij = J>0 ferromagnetic orientation of the spins is favored and at low temperatures all spins are aligned. If Jij = J<0 the low temperature phase is antiferromagnetic with the spins antiparallel to their neighbors. In the case of spin glasses, there is a distribution of coupling constants yielding i,j Jij =0. Experimentally, one has to apply a small magnetic field in order to probe a system and measure its magnetization. Considering the energy supplied by the magnetic field to the system, also known as Zeeman energy, the Hamiltonian becomes: → → H = HJ − µ0 H · M (2.2) where H is the applied magnetic field and M the total magnetization of the spin configuration considered: → → M= Si (2.3) i

9 10 CHAPTER 2. FUNDAMENTALS

Ising proposed a Hamiltonian [4] similar to that of Eq. (2.1), including uniaxial anisotropy: HI = − JijSiSj (2.4) i,j In this case, the spins are oriented along one direction, usually z, so that → → Si= Sz z . Both HJ or HI can be used to study a physical system, depending for example on its anisotropy, or the type of interaction. Experimentally, one can measure the magnetic susceptibility, which describes the response of a system to a magnetic ﬁeld. One can deﬁne this parameter as follows: An important thermodynamical quantity is the partition function Z, which contains all information on a system. Z can be used to calculate all macroscopic thermodynamics quantities, e.g. the free energy of a system:

F = U − TS (2.5)

U is the internal energy, and S the entropy of the system. The partition function can be written: Z(T,H)= exp(−H/kBT ) (2.6) states where T is the temperature and H the magnetic ﬁeld and the free energy is then accessible through:

F (T,H)=−kBT ln Z(T,H) (2.7)

The macroscopic magnetization of the system , i.e. the statistical- mechanical average of the particular spin state defined by Eq. (2.3), is defined as (M is parallel to H): 1 ∂F M = − (2.8) µ0V ∂H T where < ... > denotes the statistical average and V the volume of the system. The magnetization thus reflects the change in energy associated with the presence of an applied magnetic field. One can of course characterize the way the system responds to this perturbation by considering the variation of the magnetization with the magnetic field, yielding: ∂M 1 ∂2F − χ = = 2 (2.9) ∂H T µ0V ∂H T where χ is the magnetic susceptibility. As we will illustrate in the following, one can use a small dc magnetic field, as well as an oscillatory ac-field to investigate the magnetic state of a system. In many systems, the magnetic properties are closely linked to the electrical ones. A quantity that is relatively easy to measure is the electrical resistance. As written by my thesis opponent [7], the components Ei of the electric field 2.2. PHASE TRANSITIONS AND ORDER PARAMETER 11 inside a magnetic mono-domain conductor can be related to the current density Ji through: Ei = ρij Jj (2.10) j defining a resistivity tensor with matrix elements ρij , which depend on the orientation of the current and the magnetization. In the case of thin films, or thin superlattices, it is interesting to measure both ρxx and ρxy. For example, applying the magnetic field H in the film plane, one accesses the resistivity ρxx. If the field is applied out-of-plane, one can measure the Hall resistivity ρxy, which yields information on the density of carriers in semiconductors for example. Since many systems have a resistivity depending on the magnetic scattering, it is of interest to measure the resistivity in zero applied field, as well as in a field of some magnitude. The magnetoresistance is defined from ρxx as: ρ (H =0,T) − ρ (H, T) MR(H, T)= xx xx (2.11) ρxx(H =0,T)

We will see that ρxx(H, T) depends on the angle between the applied current and magnetic ﬁeld, as ρxx(H, T)=ρxx(H,T ,θ), where θ is the angle between H and I. The Hall resistivity can be written [7] as:

ρxy = RoH +4πMRs (2.12) where Ro is the ordinary Hall coeﬃcient, and Rs the extraordinary (or spontaneous) Hall coeﬃcient1. The extraordinary component is proportional to the magnetization of the sample.

2.2 Phase transitions and order parameter

A system undergoes a “phase transition” when evolving from one phase to another phase having different physical properties. Phase transitions are connected to spontaneous symmetry breaking, and the appearance of discontinui- ties in the physical properties of the system. Usually the phase having the lowest symmetry has the lowest entropy, so that one refers to this phase as the “ordered phase”. One can find a quantity which is not conserved by the symmetry of the system; this quantity is non zero when the symmetry is broken, i.e. in the ordered phase. This quantity is called the “order parameter”, and characterizes the symmetry breaking. If we take a simple Ising system as example, with a Hamiltonian defined ± as in Eq. (2.4), with Jij=J and Si= 1. M = i Si , then represents the magnetization of the system and is the order parameter: Without magnetic field, at high temperature, the system obeys the symmetry: Si → -Si, and the magnetization of the system is zero. At a critical temperature, denoted Tc, the system gains energy by breaking the Si → -Si symmetry, and evolves into one

1One usually uses the magnetic induction B instead of H, but in the Hall geometry, where the field is applied perpendicular to the film, the demagnetizing factor N becomes 1 so that B = H +4πM(1 − N)=H. 12 CHAPTER 2. FUNDAMENTALS of the two possible ordered states, where the spins are all equal to 1 or all equal to -1. M is non-zero, and equals in absolute value the so called saturation magnetization, MS. If M gradually decreases to zero at Tc, a “continuous phase transition” occurs. Recalling the definition of the magnetization [i.e. Eq. (2.8)], this means that the first derivative of the free energy is continuous at the transition. The transition is of “second order” if the second derivative of F shows discontinu- ities at Tc. A “first order phase transition” refers instead to the case where the first derivative of the energy is discontinuous at Tc. It is convenient to define a reduced temperature as: T − T = c (2.13) Tc Close to the critical temperature, locally ordered regions appear with a size or correlation length ξ. In a second order phase transition, ξ diverges at Tc, and can be described by a power law of the reduced temperature , involving an exponent which is referred to as a “critical exponent”. When studying spin glasses in the next chapters, we will use such power laws and corresponding exponents to describe certain physical properties close to the glass transition temperature of disordered systems.

2.3 Experiments

In the following chapters, we will report results of magnetization measurements performed in commercial as well as non-commercial [5] squid magnetometers. Squids [6], or Superconducting QUantum Interference Devices, allow measurements of the total magnetic moment of a system. The squid sensor mainly consists of a superconducting ring, interrupted by a thin insulating layer called weak link. The sensor and its pickup system is sensitive to changes of magnetic fluxes, and thus to the changes of the magnetization of a magnetic material with temperature, magnetic field, or time. Many different units are used to measure the magnetic moment and magnetization of a magnetic material. The most common units for magnetic moment are emu in [cgs] and Am2 in [SI], related as 1 Am2=1000 emu. The natural unit for the magnetization is thus emu/g or emu/cm−3 (one estimates the weight of a powder or of a sintered polycrystal, and the volume of a single crystal or of a thin film), or A/m in [SI]. If one can estimate the number of atoms in the sample, one can also calculate the magnetic moment per atom, in µB. The magnetic field can also be expressed in different units: in Oe or Gauss in [cgs] or in A/m in [SI]. 1 Oe is ∼ 80 A/m. Both sets of units will be used in the following chapters. One can, as mentioned above use a dc or an ac excitation to perform the measurements in a squid. In our non-commercial systems, the shortest time measurable in dc is of the order of 0.1 s, while the observation time can decrease to 10−6 s in ac by increasing the frequency of the oscillations of the field. Other techniques come closer to the microscopic flipping time of a single spin −13 −9 τ0=10 s: Ferromagnetic resonance (FMR) can reach 10 s, and neutron 2.3. EXPERIMENTS 13 scattering experiments down to 10−12s. But, we will see in the following that the time windows accessible by squid magnetometry are useful to investigate the static or dynamic magnetic properties of most compounds. We will now describe typical measurements that can be performed in a squid magnetometer.

2.3.1 Magnetization measurements Field dependence

In order to characterize ferromagnetic materials, one usually measures the magnetic field dependence of the magnetization at a constant temperature, making a so called hysteresis measurement. It is often interesting in the case of ferromagnets, to estimate the coercive field, as well as the saturation and the remanent magnetization. In this type of measurements, the magnetic field is increased from zero to a large value Hmax, and then decreased to a large negative (usually -Hmax) value, and finally increased again to Hmax. Figure 2.1 illustrates a typical hysteresis measurement on a thin film of nickel [8]. The saturation (MS) and remanent (MR) magnetization, as well as the coercive field Hc are identified on the curve. The saturation magnetization is estimated at large magnetic fields, when all spins are parallel to each other. The shape of the observed hysteresis curve is here rather square, because the magnetic field is applied in the plane of the film, yielding a near zero demagnetizing factor N.

M R M s M

H c

Figure 2.1: Hysteresis measurement on a thin ﬁlm of Nickel. T =10K.

The shape of the measured curve will depend on the shape of the sample (as M ∝ 1/N )2. The magnitude of the coercive ﬁeld or coercivity, and re-

2 One can also correct the data by plotting M vs. Hi=H − NM; Hi is the intrinsic 14 CHAPTER 2. FUNDAMENTALS manent magnetization will mainly depend on the quality of the material and its magnetic anisotropy. Defects can pin domains so that they will not be able to follow the magnetic ﬁeld reversal, increasing the coercivity3. Another important factor is the magnetic anisotropy inherent to ferromagnets. Due to spin-orbit coupling, the crystallographic symmetry of the lattice of a system induces preferred directions for the magnetization. This is illustrated in Fig. 2.2 which in the main frame displays the magnetic ﬁeld dependence of a ferromagnetically coupled Fe/V superlattice.

1.2

[100] 1

1 0.8 [110] S

0.6 S M/M 0.5 M/M

0.4 NV=13 0 0.2 0 5000 10000 NV=11 H (Oe) 0 0 200 400 600 800 1000 H (Oe)

Figure 2.2: M vs H for a ferromagnetically coupled Fe/V [001] superlattice. The measurements are performed applying the ﬁeld along two diﬀerent crystallographic directions, [100] and [110]. The insert shows the corresponding results for an antiferromagnetically coupled superlattice; T =10 K.

For this ferromagnetic film, as for Fe, [100] is an easy direction of magnetization, so that only a small magnetic field is required to reach saturation. [110] is instead a hard direction, and a field of about 400 Oe is necessary to saturate the film. The area enclosed by the two curves corresponds to an energy, which is referred to as the anisotropy energy Ea, proportional in this simple case of 3 cubic anisotropy to the anisotropy constant K (or K1, ∼ 55 kJ/m for Fe). As seen in the insert, no such anisotropy is observed for an antiferromagnetically coupled superlattice measured with an applied field along [100] and [110]4.

magnetic field. 3We will see later that defects - in a small amount - can also lower the coercivity, by acting as nucleation centers during the reversal. 4In this case, we have an “artificial” antiferromagnetic coupling, which can be overcome by application of a magnetic field. See ref. [10]. 2.3. EXPERIMENTS 15

Temperature dependence

One can perform hysteresis measurements at many diﬀerent temperatures, and for example follow the evolution of the saturation magnetization. One can also directly measure the variation of the magnetization with the temperature. In ac susceptibility measurements the magnetization is probed by a small sinusoidal ﬁeld of frequency f. This dynamical susceptibility χ has two components: one component is in-phase (χ) with the excitation while the other one is a dissi- pative out-of-phase (χ) component. χ can be measured on cooling or heating the sample. We will show in the next chapters some typical ac-susceptibility curves. In the case of dc measurements, one usually employs the following protocols:

5 FC

4 1

ZFC 0.8 3 0.6 IRM M (arb. units) 0.4 2

M (arb. units) 0.2 TRM 0 1 20 25 30 35 40 45 T(K) 0 20 25 30 35 40 45 T(K)

Figure 2.3: Temperature dependence of the ZFC, FC, TRM and IRM magnetization for a AgMn spin glass. H=1 Oe is employed.

• Zero-field-cooled (ZFC) magnetization: the sample is, as the name indicates, cooled in zero field. A small magnetic field, necessary to probe the system is applied at the lowest temperature, and the magnetization recorded on heating. • Field-cooled (FC) magnetization: the sample is cooled in a small field down to the lowest temperature, while the magnetization is recorded. One can also collect the magnetization on re-heating. • Thermo-remanent (TRM) magnetization: the sample is cooled in a small field. The field is removed at the lowest temperature, and the magnetization is recorded in zero field on re-heating. 16 CHAPTER 2. FUNDAMENTALS

• Isothermal remanent (IRM) magnetization: The sample is cooled in zero magnetic field and the magnetization measured in zero field on re-heating. During the cooling, a halt is made and a small field is applied at a constant temperature. After some duration, the field is then switched back to zero and the cooling resumed. The results are illustrated in Fig. 2.3, which shows the temperature dependence of the ZFC, FC, TRM and IRM for a AgMn spin glass. The IRM is recorded on re-heating after a 10000s stop in the cooling at 27 K. We will see later that useful information can be obtained from such measurements.

1.2

0.8 ZFC

0.4

FC 0 M/h (arb. units) −0.4 IRM

−0.8 TRM

−1.2 0 1 2 3 4 10 10 10 10 10 t(s)

Figure 2.4: ZFC, FC, TRM and IRM relaxation curves measured at Tm=27K on a AgMn spin glass. In the case of the IRM, a magnetic ﬁeld is applied for 100s before collecting the data in zero magnetic ﬁeld.

Usually, at high temperature, most materials are paramagnetic, and rather insensitive to a magnetic field (i.e. with a small susceptibility). The magnetization usually follows a characteristic 1/(T − θ) behavior, known as the Curie-Weiss law, which contains information on the nature of the interactions in the system. At lower temperatures, magnetic ordering takes place, and the magnitude of the magnetization depends on the spin configuration. Typical spin arrangements are ferromagnetic (FM) with all spins parallel to each other, as well as arrangements with antiparallel spins. In the latter case, one can divide the system into two sublattices with opposite magnetization directions; if both sublattices have equal magnetization, one speaks about antiferromagnetism (AFM); if not of ferrimagnetism (FI). The susceptibility is usually positive, but some compounds, such as superconductors expelling any magnetic field, show a diamagnetic (DM) behavior. In the case of ferroma- 2.3. EXPERIMENTS 17 gnets, the temperature dependence of the ZFC and FC magnetization is usually recorded in a small magnetic field, in order to determine the Curie temperature Tc, at which the system becomes ferromagnetic. The temperature dependence, or even the time dependence, as seen below, of the TRM and IRM magnetization is usually of greater interest in the case of spin glasses and disordered systems.

Time dependence

In the case of magnetically disordered systems such as spin glasses, it is interesting to monitor the evolution of the magnetization at a constant temperature, after diﬀerent coolings protocols. If a spin glass is cooled from a high temperature in the paramagnetic state down to a constant temperature in the frustrated phase, the spin conﬁguration rearranges toward the equilibrium state for this temperature. As a consequence, the response of the system depends on the time it has been relaxing at the constant temperature. We will study such phenomena in detail in the next chapters. One can perform the corresponding

1.006 NV=13 NV=11 H ⊥ I 1.06 1.003 0 ρ /

ρ 1

0.997 NFe=6

sat H // I ρ 1.04 0.994 / −0.2 −0.1 0 0.1 0.2 ρ H (T)

1.02 NFe=3

1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 H (T)

Figure 2.5: R versus H measurements for antiferromagnetically coupled (main frame) and ferromagnetically coupled (insert) Fe/V superlattices measurements to the temperature dependent ones, i.e. record the ZFC, FC, TRM and IRM magnetization versus time. Figure 2.4 shows the result of such measurements. For example, in the case of a ZFC relaxation measurement, the sample is cooled in zero magnetic ﬁeld down to the measurement temperature. 5 After a wait time tw , a small magnetic ﬁeld is applied and the magnetization recorded versus time. In the case of the IRM, as in the temperature dependent

5We will see later that this wait time is very important. 18 CHAPTER 2. FUNDAMENTALS measurement, the cooling and measurement fields are zero, but a small magnetic field is applied for some duration before starting collecting the magnetization. One can also “perturb” the system by performing so called temperature and field cyclings before recording the data. In this kind of measurements, a fast field switching is necessary, as well as a very good temperature control. Those measurements are usually performed in non-commercial squids [5].

0.19

0.17 H ⊥ I .cm) Ω (

ρ 0.15

H // I 0.13 −0.1 −0.05 0 0.05 0.1 H (T) −5 x 10 8 H ⊥ I .cm)

Ω θ =45o

( H,I

ρ 7.6

H // I 7.2 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 H (T)

Figure 2.6: R vs H for thin films of colossal magnetoresistive materials showing hysteresis. The magnetic field is applied in different orientations compared to the current fed through the films.

2.3.2 Transport measurements The electrical resistance can be accurately measured using a four probe method. The sample is mounted inside a VTI or variable temperature insert, dipped into liquid helium. As for the commercial squid magnetometers, a superconducting magnet sits in the liquid helium, so that large magnetic fields can be generated. Thus, as for the magnetization, one can measure the resistance of a material versus temperature, magnetic field, or time. It is very interesting to investigate the transport properties of thin films, since one can vary the angle between the in-plane applied magnetic field and the applied current or the preferred direction of the magnetization. Figure 2.5 illustrates the magnetic field dependence of the resistance of the ferromagnetically and antiferromagnetically coupled Fe/V superlattices depicted in Fig. 2.2. As seen in 2.3. EXPERIMENTS 19 the figure, in the case of the antiferromagnetically coupled superlattices, the resistance decreases with increasing magnetic field. In small fields the Fe layers are antiparallel to each other and the scattering of the electrons crossing the superlattice is large, but with increasing magnetic fields, the magnetizations of the individual Fe layers align more and more along each other, and the scattering decreases. Once the magnetizations are parallel and the system saturated, the resistance becomes constant. This large change of resistance with magnetic field is referred as the Giant Magnetoresistance [9] or GMR. We will see later that one can also observe a Colossal Magnetoresistance (CMR) effect. Since the GMR effect is used in industry for sensor applications, it is convenient to change the definition of the magnetoresistance and consider instead MR =[R(H =0)− R(H = Hsat)]/R(H = Hsat) to increase the performance of the sensor; Hsat refers to the saturation field, at which the magnetization and resistance become constant. The ferromagnetically coupled Fe/V superlattice exhibits instead, as shown in the insert of the figure, a so called Anisotropic Magnetoresistance (AMR) [7], in which the variation of the resistance with the applied magnetic field depends on the direction between the in plane magnetic field and the direction of the current fed through the superlattice. The anisotropy in resistance is connected to the magnetic anisotropy of the ferromagnetic material. In this kind of superlattices, it is possible to observe both GMR and AMR effects at the same time [10]. One can also observe hysteresis features, as shown in Fig. 2.6 for patterned CMR films [11]. The figure (lower panel) also illustrates how the magnetoresistance curves change when the field is applied along or perpendicular to a preferred direction (easy axis) of magnetization. When the magnetic field is applied perpendicular to the easy direction of magnetization, the rotation of the domains during the field reversal is totally reversible, yielding the reversible magnetoresistance. 20 CHAPTER 2. FUNDAMENTALS Bibliography

[1] S. Chikazumi, “Physic of Ferromagnetism”, 2nd ed., Oxford University press, Oxford (1997).

[2] R. M. Bozorth, “Ferromagnetism”, van Nostrand, Princeton (1951).

[3] W. Heisenberg, “Zur Theorie des Ferromagnetimus”, Z. Phys. 49, 619 (1928).

[4] E. Ising, “Beitrag zur Theorie des Ferromagnetimus”, Z. Phys. 31, 253 (1925).

[5] J. Magnusson, C. Djurberg, P. Granberg, and P. Nordblad, Rev. Sci. In- strum. 68, 3761 (1997).

[6] J. Clarke, Proceedings of the IEEE 77, 1208 (1989).

[7] I. A. Campbell and A. Fert, “Transport properties of ferromagnets”, Fer- romagnetic materials, Vol. 3, edited by E. P. Wohlfarth (1982).

[8] A. Broddefalk, unpublished.

[9] B. Dieny, J. Magn. Magn. Mater. 136, 335 (1994).

[10] A. Broddefalk, R. Mathieu, P. Nordblad, P. Blomquist, R. W¨appling, J. Lu, and E. Olsson, unpublished, cond-mat/0111255.

[11] Biepitaxial and bicrystal ﬁlms of CMR materials; see Chapter 5.

21 22 BIBLIOGRAPHY Chapter 3

Diluted magnetic semiconductors: (Ga,Mn)As

he interest in magnetic semiconductors rests on fundamental physics as Twell as in their potential application in spin electronics or even quantum computing1 [1, 2, 3]. (Ga,Mn)As is a III-V diluted magnetic semiconductor (DMS) obtained by substituting Ga for Mn. The discovery of ferromagnetism in Ga(1−x)MnxAs is recent [4], and dates from 1996. Ohno et al. [5] have later 2+ reported Curie temperatures Tc as high as 110 K for x=0.053. Mn replaces Ga3+, and is thus a p-dopant, bringing both magnetic moments and holes to GaAs. Mn has a limited solubility in bulk GaAs and one has to grow thin ﬁlms by means of low temperature molecular beam epitaxy (MBE) [4, 6] in order to dope GaAs over the solubility limit of Mn without introducing impurity phases such as MnAs [4]. The maximum Mn doping obtained (currently) by MBE is about 7%, i.e. x=0.07 [4].

3.1 Ferromagnetism and hole concentration

Single layers of Ga(1−x)MnxAs (GaMnAs) have been grown by MBE, with x=0.02, 0.055, and 0.07. As seen in Fig. 3.1, all three samples are ferromagnetic, with transition temperatures Tc of 38, 58, and 39 K, respectively. The appearance of ferromagnetism in GaMnAs is related to the magnetic exchange interaction between charge carriers (holes) and magnetic moments [7] supplied by the Mn atoms. The magnitude of Tc crucially depends on both the amount of Mn and the concentration of holes [7]. Tc increases as p increases, up to x ∼ 0.055, and decrease above this percentage, as if the carrier mediated interaction was hindered at too large Mn concentrations. The exact nature of the magnetic interaction is still under discussion. The most popular theories include

1Check also the picture gallery from http://www.quantware.ups-tlse.fr/pictures.html.

23 24CHAPTER 3. DILUTED MAGNETIC SEMICONDUCTORS: (GA,MN)AS hole (free carrier) mediated ordering of the local Mn spins via the Ruderman- Kittel-Kasuya-Yosida (RKKY) interaction [7], or the competition between

25 H=20 Oe

7% 15 5.5% M(kA/m) 10 2%

0 0 20 40 60 80 T (K)

Figure 3.1: Temperature dependence of the zero-field cooled and field cooled magnetizations for different single layers of GaMnAs recorded in H=20 Oe.

indirect exchange mechanisms such as double and super exchange [8]. We will describe these interaction mechanisms further in the next chapters, since they can be used to describe the magnetic and electrical properties of the colossal magnetoresistive materials. By Hall measurements, Ohno et al. [5] could estimate p ∼ 3.5 × 1020 cm−3 for x=0.053. Similar measurements on the present x=0.055 single layer [9] yielded p ∼ 2 × 1020 cm−3. The structure of GaMnAs is, as GaAs, a Zinc- Blende structure, with four Ga atoms in the unit cell of dimension (5.68 × 10−8 cm)3. A doping of x=0.055 thus corresponds to a Mn concentration of [Mn] ∼ 1.2 × 1021 cm−3, and thus to p=1.2 × 1021 cm−3, since Mn is divalent and replaces the trivalent Ga. This p value is one order of magnitude larger than the experimentally determined ones. The diﬀerence in hole concentration and Tc between the two above discussed samples with similar doping, and the p value an order of magnitude lower than the expected one, suggests that the properties of the materials depend strongly on the conditions of preparation. The quality of the samples, and thus the amount of defects in the structure such as interstitials, vacancies, or antisites, will be diﬀerent between samples, yielding variations of p and Tc. 3.2. ANTISITES 25 3.2 Antisites

Raman scattering experiments [10], as well as cross-sectional scanning tunneling microscopy [11] reveal that the most abundant defects are the antisites, i.e. As located on the Ga sublattice (denoted AsGa). In both cases, a concentration of 3 to 6 × 1019 cm−3 is obtained for samples grown at low temperatures. In addition, Liu et al. [12] have shown, by monitoring the change in lattice parameter of GaAs induced by the presence of antisites, that the amount of antisites increases with decreasing growth temperature. The factor of 10 between the expected and experimentally measured p could be related to the presence of donor AsGa antisites, compensating the holes from the Mn acceptor; each antisite carries two electrons and thus compensates 2 holes. Fig. 3.2 shows the

12 H=20 Oe

8 10% M (kA/m)

0 0 20 40 60 80 T (K)

Figure 3.2: Temperature dependence of the zero-field cooled and field cooled magnetization of two single layers with higher Mn percentage, recorded in H=20 Oe. results of magnetization measurements performed on single layers of GaMnAs with higher percentage of Mn, up to 9 and 10 %. Those samples cannot be obtained by conventional MBE, since one has to control with accurary the flow of the As (As2) gas, alternatively shuttering the As, Ga, and Mn sources. This modified MBE is known as migration enhanced epitaxy (MEE) [13]. The two samples were grown by MEE [14], at a lower temperature than the MBE ones, to avoid the formation of MnAs. As seen in the figure, both compounds show some onset of ferromagnetism. In the case of the single layer with 10 % Mn, the magnetic ordering already appears just below 80 K. The structural quality of the two single layers is very good [14], and the broad magnetization curves could then be related to a low hole concentration, since one might expect a larger concentration of antisites for MEE layers grown at very low tempera- 26CHAPTER 3. DILUTED MAGNETIC SEMICONDUCTORS: (GA,MN)AS ture. In addition, it has been shown numerically that the random distribution of the Mn on the Ga sites will affect the magnetic configuration of GaMnAs [15, 16]. The holes are antiferromagnetically coupled to the Mn spins [17], and prefer regions of the sample with high local Mn concentration [16], where their energy is lower. The holes polarize the nearby Mn, so that the Mn-rich regions become magnetically ordered at higher temperatures than the regions with lower Mn density, where the holes have lower probability of presence. Thus parts of the sample become ferromagnetic at higher temperature than others, yielding a broad magnetization vs. temperature curve [16], similar to the one observed for the single layers with 9 and 10% Mn (c.f. Fig. 3.2 and [14]). Si- milar broad features can also be expected in very thin layers of GaMnAs, where magnetic inhomogeneity is likely to appear. At the top of Fig. 3.3, we show the FC magnetization of very thin single layers having the same Mn percentage.

3 250 Å 200 Å 2.5 150Å

2 ) b µ 1.5 M (

0.5

0 0 20 40 60 80 100 T (K)

2.5

2.0

1.5

Mn 1.0 , S h s 0.5

0.0

-0.5 0 0.2 0.4 0.6 0.8

kBT/ J

Figure 3.3: Top: Temperature dependence of the ﬁeld cooled magnetization for thin single layers with a Mn percentage of 5%, recorded in H=20 Oe. Bottom: the average Mn spin SMn and average spin per hole sh for a GaMnAs layer as calculated by Berciu et al. The curves correspond, in increasing order of Tc, to ordered, weakly disordered, moderately disordered and completely random distributions of Mn. The ﬁgure is taken from Berciu et al. [16]. 3.2. ANTISITES 27

The curves are very similar to those obtained by Berciu and Bhatt [16], shown at the bottom of Fig. 3.3, for more and more disordered Mn distributions; the total magnetization of the system has a temperature dependence similar to that of the average Mn spin SMn (p=10% of [Mn]). If we thus consider that in thin layers, we have a collection of regions with different coupling strengths depending on their local Mn composition, it is difficult to define a Tc for the system. Instead, one can define TFM which corresponds to the temperature where some magnetic ordering first appears in the sample. In the figure above, one would obtain TFM ∼ 80 K for the thinner layer. In the case of the thicker layers, the sample is more homogeneous and TFM ∼ Tc. Recent experiments involving successive annealing of GaMnAs layers [18] have shown that the number of antisites can gradually be decreased by consecu- tive annealing, gradually increasing p and Tc. In their study, Takashi et al. found that Tc is increased from 70 to 110 K, thus reaching the highest Tc value earlier reported by Ohno [5] for GaMnAs. Since mean field theories [7] predict a Tc of 300 K for a Mn percentage of 10%, it would be of great interest to repeat the successive annealing procedure on the present 10% single layer! [19] Also, it has been predicted theoretically that the AsGa antisites of GaMnAs could undergo a structural transition [20] to an As intersititial - Ga vacancy (Asi-VGa) pair upon illumation [21]. This indicates an alternative method to reduce the hole compensation by the AsGa antisites, and increase Tc. Interes- tingly, in that case, the (Asi-VGa) pair is only metastable, and the AsGa defect can be created again by annealing.

2 ) B 1 µ ) B µ Ms ( 0

1 M ( −1

−2

−500 0 500 1000 1500 2000 H (Oe) 0 0 2 4 6 8 Mn %

Figure 3.4: Mn concentration dependence of the saturation magnetization evaluated from hysteresis measurements at T =10 K. The hysteresis curve obtained for the sample with x=0.055 is shown in insert. 28CHAPTER 3. DILUTED MAGNETIC SEMICONDUCTORS: (GA,MN)AS 3.3 More on the antisites

It has been shown theoretically that AsGa antisites strongly inﬂuence the magnetic properties of GaMnAs. It was found in the study by Sanvito et al. [15], that not only the total amount of antisites, but also their position with respect to Mn ions was relevant to the magnetic interaction. There is for example a large discrepancy between the theoretical and experimental results concerning the saturation magnetization of GaMnAs single layers.

(Ga0.96Mn0.04)As 5

1% AsGa

1.25%

1.5%

0 1.75%

Energy [mRy/Mn atom] 2%

4% −5

−10 1

Mtot /|Mloc|

Figure 3.5: Calculated total energy as function of Mtot/|Mloc| for diﬀerent antisite concentrations.

First principles calculations for ferromagnetically coupled Mn impurities 2+ yield moments of 4-5 µB per Mn atom, which could be expected for Mn 3.3. MORE ON THE ANTISITES 29 and S=5/2. Experimentally, as shown in Fig. 3.4, the saturation magnetization at low temperature corresponds to a moment of 2.2 µB per Mn atom at maximum. As seen in insert, the saturation magnetization is experimentally evaluated in a moderate field. The magnetization always increases with magnetic field and never really saturates,2 but the effect is small and cannot account for the present discrepancy. Recently, Korzhavyi et al. [22] could conciliate theory and experiments, by performing total energy calculations [23] in which the GaMnAs system was allowed to form a ferromagnetic state with some local moment disorder, i.e. to form a state intermediate between a saturated ferromagnetic state and a completely paramagnetic one, with only part of the Mn atoms pointing along the global magnetization direction. The Mn atoms have random spin up or spin down orientations of their local spin moments according to the so-called disordered local moment (DLM) model [24]. Fig. 3.5 shows the variation of the total energy of a Ga0.96Mn0.04As system with increasing concentration of AsGa antisites y, thus considering a (Ga0.96−yMn0.04Asy)As system.

Local moment /Mn atom] B

µ Total moment Spin moment [

0 01234

Concentration of AsGa [%]

Figure 3.6: Calculated Mtot and Mloc dependences on antisite concentrations.

2This explains why Hall measurements on GaMnAs are performed at low temperatures in very large magnetic fields (∼ 27 Tesla), where the magnetization and thus the extraordinary component of the Hall coefficient saturates; see reference [5]. In smaller fields, ρxy is proportional to the sample magnetization and thus allows to detect the appearance of ferromagnetism in very thin layers of GaMnAs, having a too small magnetic signal to be probed in a squid (for example). One thus accesses TFM by Hall measurements, which, as mentioned earlier, might differ from Tc. 30CHAPTER 3. DILUTED MAGNETIC SEMICONDUCTORS: (GA,MN)AS

The energy is calculated as function of the “degree” of ferromagnetism |Mtot/Mloc|, where Mtot is the total net magnetization and Mloc the value of the local moment per Mn atom. |Mtot/Mloc|=0 means equal amount of spins up and spins down, i.e. a paramagnetic DLM state with no net moment, while |Mtot/Mloc|=1 corresponds to a fully saturated ferromagnetic state. 0 < |Mtot/Mloc| < 1 thus describes a partial ferromagnetic order. As seen in the figure, without antisites, the system is an ideal ferromagnet (lowest energy for |Mtot/Mloc|=1), while it is paramagnetic (lowest energy for |Mtot/Mloc|=0) for concentrations of antisites above 2%. 2% corresponds to the compensation of all the holes in the system, since as mentioned above, each antisite carries 2 electrons and thus can compensate 2 holes; [Mn] = 4% = p. For intermediate concentrations of antisites, there is an energy minimum for values of |Mtot/Mloc| between 0 and 1, i.e. for DLM states with only partial ferromagnetic order. The values of Mtot and Mloc corresponding to such a state can be computed separately, and are shown in Fig. 3.6. One can see that if the local magnetization remains roughly [22] constant, the net moment decreases with increasing AsGa concentration, and disappears at full compensation (AsGa=2%). With a concentration of antisites between 1 and 2 %, the net moment becomes of the order of the saturation magnetization experimentally measured [22]. In the same study, Korzhavyi et al. also calculated the density of states of their system, with and without antisites, and could explain the formation of a DLM state by a gain in energy associated with the recom- bination of the two electrons from the antisites [22] with the Mn holes. In the ferromagnetic state, only one of the AsGa electrons can lower its energy from its impurity level, while in the DLM case, both spin up and spin down electrons can occupy Mn states of the valence band of lower energy. The total energy is thus lower in the DLM case, and the state stabilized. From Fig. 3.5, one can also estimate the total energy difference between the disordered (DLM) and the partially ferromagnetically ordered magnetic configurations [22]. This difference is proportionnal to Tc, and as seen in the figure, the energy difference between the two magnetic states is smaller and smaller, indicating that Tc becomes smaller and smaller. Without antisites, the energy difference (see Fig. 3.5) amounts to ∼ 3.25 mRy, corresponding to 170 K, which is very high compared the experimental values, which vary between ∼ 50 K (the present study) and 80 K for Ohno [5]. Adding 1.25 % of antisites, Tc amounts to ∼ 70 K, while it decreases to ∼ 35 K for 1.5 % antisites, being much closer to the experimental estimate.

3.4 GaMnAs/GaAs superlattices

Low dimensional structures like trilayers [25], superlattices [26], quantum wires [27] and quantum dots [28] made up from ferromagnetic semiconductor materials are particularly interesting for application in the above mentioned spin- tronics [3, 29] and quantum computing [30, 3]. The superlattice structures are 3.4. GAMNAS/GAAS SUPERLATTICES 31

2 2 8/4 10/4 1.5 1.5 ) ) B B µ 1 µ 1 M ( M ( 0.5 0.5

0 0 0 20 40 60 80 0 20 40 60 80 T (K) T (K) 2 12/4 1.5 ) B

µ 1 7%Mn H=20 Oe M ( 0.5

0 0 20 40 60 80 T (K)

Figure 3.7: Temperature dependence of the ZFC and FC magnetizations for GaM- nAs/GaAs superlattices with respectively 8/4, 10/4 and 12/4 monolayers of each compound. H=20 Oe. composed of a certain number of GaMnAs layers separated by GaAs layers. Thus an electronic device based on such structures would include both ferromagnetic GaMnAs layers and non-magnetic GaAs layers, yielding additional spin dependent eﬀects [25, 27, 29, 26, 30].

3.4.1 Magnetic properties GaMnAs/GaAs superlattices (SL) can be grown by MBE [9], with similar percentages of Mn as the single layers. They are also sensitive to defects, such as antisites, and since as mentioned in the above, a lower growth temperature yields a higher concentration of antisites [12], only the properties of superlattices originating from the same batch (i.e. grown in the same conditions) will be considered. Three superlattices with 7% of Mn were grown by MBE [31], with the compositions: 8/4, 10/4, and 12/4. 8/4 refers to a block of 8 monolayers of GaMnAs grown on 4 monolayers of GaAs, repeated 100 times on a GaAs substrate. As seen in Fig. 3.7, the three multilayers are ferromagnetic, with a Tc around 60 K. The shape of the M vs. T curves is similar to those of the single layers of GaMnAs. A “digital alloy” picture [32] implies that the superlattices will exhibit similar magnetic properties to those of single layers with the same average Mn concentration. The 8/4, 10/4 and 12/4 all have average Mn percentages around 5% (8/4: 8/(8+4)×7% ∼ 4.67%; 12/4: 12/(12+4)×7% ∼ 32CHAPTER 3. DILUTED MAGNETIC SEMICONDUCTORS: (GA,MN)AS

4 50

n=1 3 (K) ) c

B 25 T µ n=5 M ( 2 0 0 3 6 9 n=9 1 n (ML) n=3 (a) 0 n=7 2.5 (b) 2 n=4 ) B

µ 1.5

M ( n=8 1

n=6 0.5

n=10 0 0 10 20 30 40 50 60 T(K)

Figure 3.8: Temperature dependence of the FC magnetization for GaMnAs/GaAs superlattices with a ﬁxed number of GaMnAs monolayers m and a varying number of GaAs monolayers n; H=20 Oe. (a) n=1, 3, 5, 7 and 9 (b) n=4, 6, 8 and 10. The insert shows the variation of Tc with n for the superlattices presented in (a).

5.25%), and have a Tc close to that of a single layer with 5.5% Mn; see Fig. 3.1. The temperature onset of ferromagnetic behavior also increases from 8/4 to 12/4, following the average Mn concentration, as observed for single layers of (Ga,Mn)As [4]. The layers of the superlattices are well deﬁned [9], and show signs of interlayer exchange interaction. It is oscillatory in character, similar to the RKKY-like exchange observed in metallic multilayers consisting of magnetic materials sandwiched between non-magnetic ones [33, 34]. Figure 3.8 shows this oscillatory behavior for two series of superlattices with a Mn percentage of 4% with a constant thickness of GaMnAs, and a varying thickness of GaAs. As seen in the ﬁgure, the Curie temperature of the superlattices (and thus the coupling) does not vary monotonously with the GaAs thickness. The insert shows the oscillations of Tc with the GaAs thickness. The oscillations are very fast, with a period T of two monolayers. If one assumes an RKKY-like coupling [35], the interlayer exchange should oscillate with a period inversely proportional to the third root of the hole concentration3. A period of T =2ML thus corresponds to a hole concentration of p=1021cm−3, which is close to the

3 2 1/3 T = π/kF; kF =(3π p) for a free electron gas; see [35]. 3.4. GAMNAS/GAAS SUPERLATTICES 33

2 1 10 1 ρ (Ω.cm) MR 0.5% 0 7% 10 0 0.5 10 0.5

0 −2 0 50 100 150 0 50 100 150 10 0 −3 0 50 100 150 0 50 100 150 x 10 3 1 2 10 1 H=0 2% 9%

2.5 0.5 0 10 0.5 H=6T

2 0 −2 0 50 100 150 0 50 100 150 10 0 −3 0 50 100 150 0 50 100 150 x 10 12 1 5.5% 0.15 1 10% 10 0.1 0.5 0.5 8 0.05 6 0 0 50 100 150 0 50 100 150 0 0 T(K) 0 50 100 150 0 50 100 150 T(K) T(K) T(K)

Figure 3.9: Zero magnetic ﬁeld resistivity and magnetoresistance for thick single layers of GaMnAs containing 0.5, 2, 5.5, 7, 9 and 10 % of Mn; see legend. concentration of Mn, but as mentioned at the beginning of the chapter, almost ten times larger than the actual concentration of holes in the system. One will thus need to take into account the ﬁnite thickness of the GaMnAs, as well as the GaAs layers to describe the exchange in these superlattices [36].

3.4.2 Transport properties

Also, in agreement with the digital alloy picture, one observes that the electrical resistance, as well as the magnetoresistance is similar in the superlattices with 8/4, 10/4 and 12/4 to that of a single layer of GaMnAs with similar a- verage Mn content. Figure 3.9 shows the temperature dependence of these two quantities for thick single layers of GaMnAs with different Mn percentages (the ones depicted in Fig. 3.1 and Fig. 3.2). As seen in the figures, the zero-field resistivity shows an insulator-to-metal transition close to the paramagnetic- to-ferromagnetic transition temperature of the layers. The magnitude of the resistance decreases by the application of a magnetic field. As seen in the figure, the magnetoresistance is larger close to Tc. We will see in the last chapter that simultaneous insulator-to-metal and paramagnetic-to-ferromagnetic phase transitions are common in CMR ferromagnets governed by double exchange interaction; an observation which certainly motivated Akai [8] in his calculations. The resistance curves of the superlattices show similar properties, as seen in Fig. 3.10. The resistivity in zero magnetic field and in 6 Tesla are very close to those of the single layer with 5.5% Mn, with additional resistivity arising from the GaAs layers. 34CHAPTER 3. DILUTED MAGNETIC SEMICONDUCTORS: (GA,MN)AS

ρ (Ω.cm) MR 0.04 1 8/4 0.8 H=0 0.03 0.6

0.4 0.02 H=6 T 0.2 (a) (b) 0.01 0 0 50 100 150 0 50 100 150 T(K) T(K) 0.15 H=0

0.1 .cm) Ω

( 12/4 ρ

0.05 10/4

8/4

0 0 50 100 150 T(K)

Figure 3.10: Top: zero magnetic ﬁeld resistivity and magnetoresistance for a Ga- MnAs/GaAs (8/4) superlattice. Bottom: zero magnetic ﬁeld resistance of the superlattices with 8/4, 10/4 and 12/4 composition.

3.5 What is next?

One can produce hetero-structures of magnetic and non magnetic semiconductors in lower and lower dimensions, so one can safely predict that a lot will happen in the research on DMS, especially in the superlattice, trilayer and quantum dot branches. Some DMS formed with other elements are being investigated. For example, (Ga,Cr)As shows promising properties theoretically [37, 38], but has not yet been obtained ferromagnetic experimentally [39]. The same holds for (Ga,Mn)N. Dietl et al. [40] predict from the results of their calculations an extremely large Tc of 1400 K for Ga0.91Mn0.09N. However, so far, only paramagnetic (Ga,Mn)N has been obtained experimentally [41]. Bibliography

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[41] M. Zajac, J. Gosk, M. Kaminska, A. Twardowski, T. Szysko, and S. Posi- adlo, Appl. Phys. Lett. 79, 2432 (2001). 38 BIBLIOGRAPHY Chapter 4

Spin glasses

s J. -P. Bouchaud recently mentioned [1], “spin glasses are totally use- Aless pieces of material”. At low temperature, the magnetic moments of these materials are frozen in an arbitrary state, with no long range order. Spin glasses thus consist of an ensemble of disordered spins, and represent a model system for the statistical mechanics of a system with quenched randomness.

+ + +

_ C E + A _ _ ++ _

+ + _ D F + B + + _ _ _ _ ? ? ?

Figure 4.1: Examples of magnetic frustration on square and triangular lattices. A, C, and E show no frustration, while B, D and F show frustration associated to the disorder of the interactions (B,D) as well as to the geometry of the lattice (F).

Because of the spin disorder, the spins are subject to interactions of different sign, positive: ferromagnetic or negative: antiferromagnetic, so that a particular spin will receive conflicting information on how to order from its neighbors, and it might not be possible for the system to choose a certain spin configuration and minimize its energy. The phenomenon is referred to as frustration [2].

39 40 CHAPTER 4. SPIN GLASSES

Figure 4.1 shows some examples of frustration. Example A represents a square lattice without frustration, since all positive and negative interactions can be satisﬁed: The spin on the upper left can couple ferromagnetically with the spins on the upper right and lower left, while the spin on the lower right cou- ples antiferromagnetically with them. In example B, some frustration appears, since there is not an even number of positive and negative interaction. Here, the frustration originates from the disorder of the interactions. Frustration can also originate from the lattice, and is then referred to as geometrical frustration. Examples C, D, E and F of Fig. 4.1 present diﬀerent spin arrangements on triangular lattices. In C, there is no frustration, but it appears in its counterpart D, as in the square lattice. In E, there is no magnetic disorder, since all sites are occupied and there are only positive interactions between spins, and no frustration. But in its counterpart F, the system is geometrically frustrated. This situation is typical for compounds with a so called Kagome lattice [3] and pyrochlores [4]. There is also frustration associated with the disorder in the position of the spins, as well as with the geometry of the interactions [5].

4.1 Fundamentals

4.1.1 Equilibrium properties Since in zero magnetic field, the net magnetic moment of the spin glass phase is zero, the magnetization can not be chosen as order parameter. Edwards and Anderson proposed [6] instead the statistical mechanics order parameter q, defined as: 2 q =[Si ] (4.1) where Si is an Ising-like atomic magnetic moment. ... denotes thermal averaging and [...] the averaging over interaction strengths Jij . q=0 at temperatures above the spin glass temperature Tg, while it below Tg follows the temperature dependence: T − T β q ∝ g = −β (4.2) Tg where β is a critical exponent. One can also define the spatial correlation function [7]: 2 G(rij)=[(SiSj−SiSj) ] (4.3) where rij is the distance between two magnetic moments. Close to Tg, the spatial correlations decay to zero at length scales rij larger than the correlation length ξ, as: − − − r G(r ) ∝ r (d 2 η)exp − ij (4.4) ij ij ξ where d is the dimension of the system and η a critical exponent. At Tg, the correlation length diverges as: ξ ∝||−ν (4.5) 4.1. FUNDAMENTALS 41 where ν is a critical exponent. The conventional critical slowing down theory predicts that the time necessary to reach equilibrium becomes longer and longer when approaching Tg, and the relaxation time τ diverges at Tg, as:

τ ∝ ξz (4.6) where z is a critical exponent.

−3 x 10 1.2

0.8 ω/2π=0.17 Hz

) (arb. units) ω/2π=170 Hz ω 0.4 ( χ′

0 −5 x 10 6 2 0

) −2 τ

4 ln( −4 ω/2π=170 Hz −6 −8 −3 −2.5 −2 ) (arb. units)

ω 2 ( ln(ε) χ′′ ω/2π=0.17 Hz Tf (ω)

0 14 16 18 20 22 24 26 28 30 T (K)

Figure 4.2: Temperature dependence of the ac-susceptibility of Fe0.5Mn0.5TiO3 for diﬀerent frequencies: ω/2π=0.17, 0.51, 1.7, 5.1, 17, 51, and 170 Hz. h=0.1 Oe. The insert shows the scaling of τ with according to Eq. (4.8).

In experiments, one can define a “freezing” temperature Tf , so that τ(Tf ) is equal to the experimental observation time tobs=1/ω. For temperatures T above Tf , tobs >τ(T ) and the system is in true thermal equilibrium. For lower temperatures, the system is out of equilibrium, and an imaginary part, as well as a frequency dependence appear in the ac-susceptibility χ(ω)=χ(ω)+χ(ω), ω being the angular frequency. Fig. 4.2 shows the temperature dependence of the two components of the ac-susceptibility for the 3d Ising spin glass Fe0.5Mn0.5TiO3 for different frequencies f = ω/2π. The onset of the non- equilibrium dynamics defines the freezing temperature Tf (ω) (marked in the 42 CHAPTER 4. SPIN GLASSES

ﬁgure in χ(T)) which, according to Eqs. 4.5 and 4.6, scales with ω as [8]: T (ω) − T zν ω ∝ f g (4.7) Tg more commonly written as:

− τ T (ω) − T zν = f g = −zν (4.8) τ0 Tg

−13 where τ0 is the microscopic spin flip time, of the order of 10 s, and =(Tf (ω)− Tg)/Tg. ln(τ) is thus proportional to ln(), with a proportionality constant of −zν. One can then iteratively find values of Tg, τ0, and zν satisfying the proportionality between the two quantities. In the insert of Fig. 4.2, ln(τ) is linear −13 with ln() for Tg=21.1 K, τ0=3.1× 10 s, and zν=10.6. The latter two values are characteristic for a 3d Ising spin glass. Activated dynamics could also govern the dynamics, still yielding a finite phase transition. In this case, the slowing down of the relaxation should obey [9]: − τ 1 T (ω) − T ψν ln( )= f g (4.9) τ0 Tf (ω) Tg where ψν is a critical exponent. Without a finite phase transition temperature (Tg=0), the slowing down is rather described by a generalized Arrhenius law:

τ −x ln( ) ∝ Tf (ω) (4.10) τ0 where x is a critical exponent. The spin glass dynamics can be described by the spin-spin autocorrelation function, written as [6]:

q(t)=[Si(0)Si(t)] (4.11) which, for equilibrium dynamics is related to the time dependent magnetization m(t) by the ﬂuctuation-dissipation theorem [10]. The response to a weak applied magnetic ﬁeld h is :

m(t) 1 − q(t) = χ(t) ∝ (4.12) h T where χ(t) is the time dependent zero field susceptibility. χ(t)isthusadi- rect measure of the spin glass dynamics, and can be experimentally obtained by probing the system by either a small ac-field [ac-susceptibility χ(ω)] or a small dc-field [time dependent magnetization m(t) measurements]. Eq. (4.12) only holds if all the time dependent quantities are representative of equilibrium conditions. The zero field spin glass is always out of equilibrium at temperatures below Tg, but on short times scales, quasi-equilibrium conditions are reached and Eq. (4.12) is satisfied [11, 12]. The time dependent experimental 4.1. FUNDAMENTALS 43 susceptibilities χ(ω)=χ(ω)+χ(ω) and χ(t) are related through: [13]

m(t) χ(ω) ∼ = χ(t) h (4.13) π dm(t) π χ(ω) ∼− = − S(t) 2 hd ln(t) 2 where ω=1/t and 1 dm(t) S(t)= (4.14) h d ln(t) is the relaxation rate.

4.1.2 Droplet model In the droplet model of spin glasses [9, 14], the equilibrium dynamics is governed by low energy excitations from the ground state. There is a two fold degenerate ground state in zero magnetic field where the states are related by a global flip. The spontaneous excitations from the ground state are collections of coherently flipped spins or droplets. If a weak magnetic field - weak enough not to affect the system - is applied, the equilibrium system is magnetized via polarization of droplets of ever increasing size. Droplets of size L contain ∼ Ld spins, [14] where d is the spatial dimension, each of them have spins “up” or “down” with equal probability, so that the fluctuation in the number of “up” and “down” spins is of the order of Ld/2, and thus each droplet carries a magnetic moment proportional to Ld/2. In zero applied magnetic field, the directions of these magnetic moments are randomly distributed and the sample magnetization is zero. When applying a magnetic field, the droplets having their moments aligned with the field will be energe- tically favored, giving rise to a magnetization of the sample. We consider fields low enough to probe the system without affecting its dynamics. Fisher and Huse [14] suggest that the free energy cost for creating a droplet of size L scales as: F ∝ YLθ (4.15) where Y (T ) is the stiffness modulus and θ the stiffness exponent, limited by 0 ≤ θ ≤ (d − 1)/2 for 3d spin glasses. If θ is positive, no arbitrary large excitations exist at any temperature and the phase is stable. θ has recently been determined [15] experimentally for Fe0.5Mn0.5TiO3, and amounts to 0.2, which agrees with earlier numerical studies [7].

4.1.3 Non-equilibrium properties

After a quench from a temperature above Tg to the spin glass phase, a spin glass system is in non-equilibrium. The spin conﬁguration, if mapped to the ground state Ψ, consists of domains of ground state order separated by domain walls from the spin reversal Ψ.¯ The relaxation toward the ground state Ψ is governed by motion of the domain walls separating the two states through 44 CHAPTER 4. SPIN GLASSES droplet excitations.

Domain growth and aging

At a constant temperature in the spin glass phase, our system rearranges its spin configuration to reduce the total domain wall energy (in the ground state, all walls are removed). The movement of a domain wall a distance L is equiva- lent to a flip of a droplet of size L near the domain wall. In the droplet model, an energy barrier must be overcome to flip a droplet. The barriers are relatively large and thus difficult to surmount by thermal activation, yielding the slow dynamics. According to Fisher and Huse [14], a characteristic barrier height is: B ∝ ∆Lψ (4.16) where ∆(T ) is the barrier stiffness and ψ the barrier exponent, limited by θ ≤ ψ ≤ (d − 1). The reversal of a droplet of size L thus requires that an energy barrier of size B(L) is surmounted. Activated dynamics yields: t B ∝ T ln (4.17) τ0 So that barriers of height B will be surmounted at time t, and larger and larger energy barriers can be surmounted as time evolves. Thus, the characteristic length scale of the thermally active droplets becomes of the order: [9, 14]

1 T ln( t ) ψ L ∝ τ0 (4.18) ∆(T )

If one in an experiment, applies a small (dc) magnetic ﬁeld at tobs=0, the system is magnetized by polarization of droplets of sizes increasing with tobs according to Eq. (4.18). After a wait time tw, the droplet excitations of length L take place within a domain structure with domains of a size R growing with the age of the system ta=tobs+tw as:

1 T ln( ta ) ψ R ∝ τ0 (4.19) ∆(T )

At short time scales, tobs << tw, the thermally active excitations are small, L<

1And rarely at the domain walls. 4.1. FUNDAMENTALS 45

(a) Tm=27K; h=0.1 Oe 7.5

5 M/h(t=0.3s) − 2.5 M/h(t) 0 2 (b) 1.5

1 S(t)

0.5

0 −1 0 1 2 3 4 10 10 10 10 10 10 t (s)

Figure 4.3: ZFC relaxation recorded after tw=1000 s on a Ag(11 at % Mn) spin glass. (a) shows the time dependence of M/h, while (b) shows the corresponding relaxation rate S(t)=1/hdm(t)/d ln(t).

Figure 4.3 shows the result of a ZFC relaxation experiment on a Ag(11 at % Mn) spin glass. As seen in the figure, there is an inflection point in the magnetization vs. log(t) curve [16, 17], marking the separation between the short (quasi-equilibrium) and long (non-equilibrium) time scales. As a result, the relaxation rate exhibits a maximum close to the wait time employed in the relaxation measurement [11]. The relaxation curve and its associated relaxation rate are thus characteristic of the magnetic configuration established after the quench, and can thus be used to investigate the dynamical properties of a glassy system. We will see in the following that, to learn more about the system, one can perturb it further by imposing shifts and cyclings of temperature and magnetic field prior to the relaxation measurement.

4.1.4 Linear response and superposition The ZFC magnetization of a spin glass exhibits a linear response [18] to small magnetic fields, i.e. the response of the system is not affected by the probing field. At constant temperature in the spin glass phase, the response to a small field h applied after a wait time tw can be written as:

MZFC(tw,t)=hp(tw,t) (4.20) 2 where p(tw,t) is the response function . In a regime of linear response, the principle of superposition applies; the superposition implies that the response 2The response function is also linear, at small enough field changes, h, in a superposed magnetic field H. However, p depends on the magnitude of H. In non equilibrium regions 46 CHAPTER 4. SPIN GLASSES of a system to series of field changes is the sum of all individual responses, as [19]: n M(t)= hip(twi,t+(twn − twi)) (4.21) i=1 where hi represents a field change (from any initial state) made after twi at a constant temperature in the spin glass phase.

12 0 ZFC

10 −2 t ↑ −4 e 8 −6 t ↑ 6 IRM e −8 M (arb. units) 4

0 1 10 100 1000 10000 t (s) 2 ZFC+IRM

−2 t =3s e t =30s −4 e

M (arb. units) t =100s e t =300s e t =1000s −6 e t =3000s e t =10000s e −8 1 100 10000 t +te (s)

Figure 4.4: Top: ZFC (main frame) and IRM (insert) relaxation for diﬀerent te for Ag(11 at % Mn); Tm=29K, h=0.5 Oe. Bottom: the ZFC+IRM sum plotted versus t + te.

t>>tw, p(H, tw,t) decreases with increasing ﬁeld; on short times scales t<

In particular, one can show that the relaxation of the ZFC, FC and TRM magnetization is connected via: [18]:

MZFC(tw,t)=MFC(0,t+ tw) − MTRM(tw,t) (4.22)

The usefulness of this relation will be exemplified in the next sections. It should be noted that MFC(h) is non linear at all field strengths in the spin glass phase [20], i.e. MFC/h and MTRM are always (weakly) field dependent, in contrast to the field independent response function p(t, tw). Another interesting relation is [19]: MZFC(tw,t+ te)=MZFC(tw + te,t)+MIRM(tw,t) (4.23) where te is the duration of the field pulse in the IRM measurements. In this relation, the positive field change occurring at t=tw +te in the ZFC magnetization experiment is canceled by the negative one in the IRM occurring at a time te after tw, i.e. at t=tw + te. This relation was recently confirmed numerically by Yoshino et al. [21], and employed to investigate the isothermal aging of spin glasses within the droplet scenario in terms of the time dependent length scale L(t). Figure 4.4 shows the results of corresponding measurements on a Ag(11 at % Mn) spin glass. The single ZFC and IRM curves are shown in (a), and as seen in (b), the sum of the curves becomes independent of te, as predicted by Eq. (4.23), yielding the collapse of the data utilized in ref. [21].

4.1.5 Chaos and overlap length Bray and Moore [22] predicted that the spin glass phase is chaotic, in the sense that the equilibrium spin configuration at a specific temperature is unstable to any temperature change (or changes in the distribution of the interaction), so that the system must rearrange after such a perturbation; this phenomenon is referred to as “temperature chaos”. However, one also defines an “overlap length” inside which the equilibrium configurations at two temperatures T and T +∆T are indistinguishable. The out-of-phase component of the ac- susceptibility χ(ω) gives a measure that probes the number of active droplets with relaxation time ∼ 1/ω [on a length scale L(1/ω)]. The low frequency χ(ω) decreases with increasing time at a constant temperature. This reflects the aging process, and the fact that the number of droplets of relaxation time 1/ω decays with time as equilibrium domains are growing. The temperature chaos scenario can be nicely illustrated [23] in low frequency [i.e. larger L(1/ω)] ac-susceptibility experiments [24, 25]. Figure 4.5 shows the out-of-phase component of the ac-susceptibility measured on a Cu(13.5 at % Mn) spin glass on heating after specific cooling protocols. χ(ω, T) is cooling rate dependent, and decays in magnitude during a stop at constant temperature. On resuming the cooling, χ(ω, T) recovers its cooling rate governed level, it is rejuvenated. Rejuvenation also occurs on heating. The curve plotted in Fig. 4.5 using filled markers was measured after a 10000 s halt at 40 K performed while cooling the system from above Tg. One can see that the spin state equilibrated during the aging at 40 K is retrieved when re-heating the system to 40 K. The systems remembers its ages, keeping 48 CHAPTER 4. SPIN GLASSES

300 Ref. Ts = 40K 250 Ts1 = 50K; Ts2 = 40K

200

150 " (arb. units) χ

100

0 30 40 50 60 70 80 T (K)

Figure 4.5: Temperature dependence of the out-of-phase component of the ac- susceptibility of Cu(13.5 at % Mn). Single and double memory experiments are illustrated. The sample is continuously cooled from above Tg to a low temperature in the spin glass phase and χ(T ) is recorded on continuous re-heating (Ref.). In the single (resp. double) memory experiment, the cooling was additionally halted at TS =40 K (resp. TS1=40 K and TS2=50 K) for 10000 s. a memory of the equilibration at constant temperature. On further re-heating, the system is rejuvenated, and the susceptibility curve merges with the reference one3. The system thus shows both signatures of memory (dip) and chaos (rejuvenation). Looking now at the curve plotted using open markers in Fig. 4.5, measured after two halts of 10000 s at 50 and 40 K, one can see that both equilibrations are retrieved, unmasking the two distinct length scales characteristic of the two aging processes. Since the two stop temperatures are well separated, the dip present at 40K in the susceptibility curve, as a result of the equilibration during the halt, is the same whether or not a first equilibration has occurred at 50K [27]. Similar memory features can be observed using a dc magnetic field as we will see in the next section. Chaos and memory features can also be observed in the relaxation of the dc magnetization. One can indeed, after a first wait time tw1 (= tw in the regular ZFC relaxation experiment case) at the measurement temperature Tm, shift the temperature ±∆T for a duration tw2, and record the evolution of the magnetization immediately after returning to Tm. The result of such experiments, referred as temperature cyclings [26], is illustrated in Fig. 4.6, which shows dif-

3The system even appears “younger”; a ﬁnite cooling rate is employed in the experiments, so that the system ages during the cooling. 4.1. FUNDAMENTALS 49

0.6 0.5 (a) (a) ∆T=∞ 0.5 ∆T= ∞ ∆T= 0 0.4 ∆ ∆ T= 0 T= -0.8 K ∆T= -3 K 0.4 ∆T= -2 K ∆ 0.3 T= -10 K 0.3 0.2 S (arb. units)

0.2 S (arb. units) 0.1 0.1

0 0 0.6 0.5 (b) ∆T= ∞ (b) ∆ ∆T= ∞ 0.5 T= 0 0.4 ∆ ∆T= 0 T= +0.8 K ∆ 0.4 ∆T= +2 K T= +3 K 0.3 ∆T= +10 K 0.3 0.2 S (arb. units) S (arb. units) 0.2 0.1 0.1 T =40 K T =11K m m 0 0 -101234 -101234 log t log t 10 10

Figure 4.6: Temperature cycling experiments on (Fe0.8Ni0.2)75P16B6Al3. Negative (a) and positive (b) cyclings are performed below (left) and above (right) the spin glass transition temperature. tw1=3000 s, tw2=100 s.

ferent relaxation curves for the reentrant ferromagnet (Fe0.8Ni0.2)75P16B6Al3. (Fe0.8Ni0.2)75P16B6Al3 is an interesting system, since the magnetic interaction is essentially ferromagnetic, but antiferromagnetic interaction is also present, yielding some frustration. As a result, the compound is a frustrated ferromagnet below 92 K, and shows a transition to a spin glass phase at low temperatures, below Tg=15 K [28]. We will see in the next chapter that CMR materials often behave like reentrant ferromagnets. Below Tg, (Fe0.8Ni0.2)75P16B6Al3 behaves like an ordinary spin glass. As seen on the left side of Fig. 4.6, negative temperature cyclings have little inﬂuence on the relaxation rate, while positive ones reinitialize the system: the peak present in the reference curve is not observed after the positive cycling (for ∆T = 0.5 K), and the relaxation rate in this case is instead similar to that of a relaxation curve measured immediately on reaching Tm after the initial cooling from 4 above Tg, labeled ∆T = ∞ in the plot . In the ferromagnetic phase, as seen on the right side of Fig. 4.6, the reinitialization occurs irrespective of the sign of ∆T . The temperature perturbation destroys the domain structure established at Tm, which thus appears less robust to any spin rearrangements than that of the spin glass phase [29]. In the previous example, the sample remained at the “cycling temperature” or T +∆T for 100 s only. Of course, if the systems remains at a lower temperature for longer and longer time, correlations will develop on longer and

4 ∆T = ∞ thus means tw=0 s. However the corresponding relaxation rates do not have a maximum at 0 s, but at around 100 s, which is an eﬀective wait time governed by cooling and heating rates, as well as other experimental factors. 50 CHAPTER 4. SPIN GLASSES

10 Ref. (tw2=0) tw2=30s 7.5 tw2=300s tw2=3000s 5 tw2=30000s M/h

2.5

0 1.5

1 S(t) 0.5

0 0 1 2 3 4 10 10 10 10 10 t (s)

Figure 4.7: Negative temperature cyclings on Ag(11 at % Mn); a fixed tw1=3000 s is employed while tw2 is gradually increased from 30 s to 30000 s. Tm=27 K, ∆T =-1 K. longer length scales, and some reinitialization will eventually occur after a negative temperature cycling [25, 30]. This behavior is illustrated by temperature cycling experiments on a Ag(11 at % Mn) spin glass in Fig. 4.7. As seen in this figure, as tw2 increases, a second maximum appears at short time scales in the relaxation rate, so that the two different length scales characteristic of the two aging processes are evidenced. For the largest tw2, the system is almost completely reinitialized and has only little memory of the first aging at the measurement temperature. Yoshino et al. [31] proposed a “ghost domain” picture, in which the systems keeps a memory of the larger domains equilibrated during the first aging, although their finer structure is affected by the second aging.

4.2 Dc-memory experiments

4.2.1 Experimental procedure Temperature stops are performed in magnetization vs. temperature experiments. For example in the ZFC case, a spin glass is cooled in zero magnetic 4.2. DC-MEMORY EXPERIMENTS 51

field from a temperature Tref above Tg down to Ts in the spin glass phase. A stop is made during ts, without any field change. After ts, the cooling is resumed down to the lowest temperature. MZFC is recorded on re-heating to Tref in a small field. The corresponding experiments can be performed in the FC and TRM case, where instead the field keeps its non-zero value during the cooling and stop at Ts. These experiments are also referred to as field stop experiments. The top of Fig. 4.8 illustrates the results of such experiments on a Ag(11 at % Mn) spin glass [32]. As seen in the figure, the TRM curve

6 h=0.1 Oe 5 FC

4 ZFC 3

M/h (arb. units) 2 TRM 1

0 3 0.5 0.4

0.3 ∆FC−∆ZFC 2 TRM 0.2 0.1 M/h (arb. units) 0 ∆TRM FC−ZFC −0.1 M/h (arb. units) 1 20 25 30 35 40 T(K)

0 20 25 30 35 40 45 T(K)

Figure 4.8: Top: ZFC, FC, and TRM magnetization recorded for Ag(11 at % Mn) in H=0.1 Oe after direct cooling (open symbols), and after a field stop of ts=10000 satTs=27 K (filled symbols). The FC relaxation is small and in this case, the difference between the direct cooling and the field stop experiment is very small (c.f. [20]). Bottom: TRM and FC-ZFC magnetization curves (main frame), as well as the corresponding excess magnetization (insert). obtained after a field stop lies above the reference curve obtained after direct cooling, showing that a considerable reinforcement of the spin structure has occurred during the stop at constant temperature. Correspondingly in the ZFC case, but in this case, the curve obtained after a field stop lies below the reference one. When the system is left unperturbed at constant temperature Ts, 52 CHAPTER 4. SPIN GLASSES it rearranges its spin configuration toward the equilibrium one5. The equilibrium state achieved during the stop becomes frozen in on further cooling and is retrieved on re-heating. The equilibration of the spin configuration as Ts corresponds to the aging phenomenon. The system thus remembers its age, and the observed phenomenon is referred to as memory effects (see also [33]), as mentioned above for ac-experiments.

1 ZFC 0.5

M/h (arb. units) −0.5 TRM ∆ (a) −1

0 (b) 0 FC −0.02 −0.02 ZFC+TRM −0.04 FC

M/h (arb. units) −0.04 ∆ 0 2 4 10 10 10 −1 0 1 2 3 4 10 10 10 10 10 10 t(s)

Figure 4.9: ZFC, TRM (a) and FC (b) relaxations of a Ag(11 at % Mn) spin glass at Tm=27 K, h=0.1 Oe for diﬀerent waiting times: tw=0 s (ﬁlled symbols), tw=1000 4 s (dotted lines) and tw=10 s (open symbols). The insert shows the agreement between the ZFC+TRM and the FC relaxations (tw=0 s).

4.2.2 Memory and superposition The principle of superposition can be checked either by varying the observation time at a constant temperature (relaxation experiments) or keeping a constant observation time, and varying the temperature, so that the superposition can be observed with the dc-method. The temperature dependence of the ZFC, FC and TRM magnetization of a Ag(11 at % Mn) spin glass obtained after direct cooling to the lowest temperature are shown at the top of Fig. 4.8. The TRM curve is reported in the main frame of the bottom of Fig. 4.8, as well as the difference between the FC and ZFC curves. One easily sees that as in Eq. (4.22), MZFC = MFC − MTRM. ZFC, FC and TRM curves obtained after a temperature stop are also shown at the top of Fig. 4.8. The different curves are subtracted from their corresponding references, yielding ∆ZFC, ∆FC and ∆TRM curves. The insert shows ∆TRM and ∆FC - ∆ZFC, which virtually coincide, showing as in Eq. (4.22) that the excess magnetization gained due to the field stop in TRM and FC corresponds to the magnetization “lost” in

5For this temperature, as seen in the following. 4.2. DC-MEMORY EXPERIMENTS 53 the ZFC. Conventional magnetic relaxation experiments can be performed for comparison: Fig. 4.9(a) and 4.9(b) show the ZFC, FC and TRM relaxation curves obtained after diﬀerent wait times. Subtracting the diﬀerent curves for the same wait times, one observes again, as seen in insert, the validity of the principle of superposition, so that the FC and ZFC+TRM curves coincide.

3 TRM +FS(t=10000 s) TRM +FS(t=0 s) 2 TRM ref. TRM +ZFS(t=10000 s) TRM +ZFS(t=0 s) 1 M (arb. units) (a) TRM 0 0.6 IRM +∆ H(t=0 s) IRM +∆ H(t=10000 s) 0.4 IRM ref.

0.2 M (arb. units) (b) IRM 0 0.6 (c) ∆ IRM, ∆TRM 0.06 0.04 0.4 ∆TRMZFS 0.02 0 ∆IRM −0.02 0.2 20 30 40 M (arb. units) ∆TRMFS 0 20 25 30 35 40 T(K)

Figure 4.10: TRM (a) and IRM (b) magnetization of Ag(11 at % Mn) measured after using different cooling protocols; Ts=27K, H=0.1 Oe. (c) displays the difference between pairs of curves shown in (a) and (b); see main text. The inset shows the difference (∆TRMZFS - ∆IRM) and ∆TRMFS (which is marked by the same symbol as in the main frame).

One can further explore the consequences of the dc-memory effect by performing zero field stops in TRM experiments [32]. In this case, the field is removed during the stop at Ts for a duration ts. The counterpart to this experiment is an IRM measurement in which the magnetic field is turned on at Ts for a duration ts. Fig. 4.10 presents the temperature dependence of the (a) TRM and (b) IRM magnetization. Both were measured on reheating after the fol- 54 CHAPTER 4. SPIN GLASSES lowing cooling protocols: 1 The sample is cooled in constant field H0 = 0.1 Oe (TRM) or zero field (IRM) from a reference temperature above Tg to a stop temperature Ts

4.2.3 Chaos The excess magnetization observed in Fig. 4.8 and the insert of Fig. 4.9(a) is significant only close to Ts, and fades away for lower and higher temperatures. The influence of a stop at constant temperature during cooling is limited to a restricted temperature range around Ts, and the width of this region may be assigned to the overlap between the spin configuration attained at T and the corresponding state at a neighboring temperature T +∆T . The two concepts that explain the width are then: A chaotic nature of the spin glass equilibrium configuration and an overlap on short length scales between the equilibrium configurations at T and T +∆T . To further explore this phenomenon, one can perform more than one stop during the initial cooling to the lowest temperature [33]. Fig. 4.11 shows the results of two field stops of ts=3000 s in the ZFC magnetization of the Fe0.5Mn0.5TiO3 spin glass. The main frame shows the difference plots corresponding to two single fields stop at two different temperatures (simple line) and to their sum (filled diamonds), as well as a double field stop experiment (open diamonds). One can see that the curve of single and double stops experiments virtually coincide around the two stop temperatures. It indicates 1) that the relevant spin configuration at the higher temperature is not affected by the second stop at lower temperature and reciprocally 2) that the aging at the lower temperature is not influenced by the first equilibration at 4.2. DC-MEMORY EXPERIMENTS 55

0.2

−0.2 12 M (arb. units) ∆ −0.4 8

4 M (arb. units) −0.6 0 12 16 20 24 28 Ts=17K Ts=21K T (K) −0.8 14 16 18 20 22 24 26 28 30 T (K)

Figure 4.11: Difference plots corresponding to single-stops of 3000 s at Ts=17 K and Ts=21 K (simple lines), and double-stops of 3000 s both at Ts1=21 K and Ts2=17 K (open diamonds) in Fe0.5Mn0.5TiO3. The sum of the two single-stop curves is added for comparison (filled diamonds). The insert shows the original ZFC magnetization vs. temperature curves. the higher one. We thus observe memory and chaotic features: During the first stop, the system reaches a characteristic age - or length scale. This age is conserved on further cooling, even if the system is again aged at lower temperature toward its equilibrium state at this temperature. The state equilibrated at the higher temperature thus survives the spin re-configuration occurring at lower temperature on shorter length scales, and the system remembers its initial high temperature state on re-heating. Of course, if the stop at lower temperature is performed for a much longer time, allowing the correlations to develop on longer length scales, a (partial) re-initialization of the spin configuration would occur, unmasking the two different length scales characteristic of the two a- ging processes [25]. Corresponding results can be obtained by performing ZFC relaxation experiments using the same cooling protocols [33].

4.2.4 Comparison between Ising and Heisenberg spin glasses Let us now compare the dynamical properties of two diﬀerent spin glasses, an alloy of Ag(11 at % Mn) and a single-crystal of Fe0.5Mn0.5TiO3, using the dc-method, as well as some complementary relaxation measurements. The two systems have diﬀerent magnetic interaction mechanisms. In a Ag(Mn) alloy, as in many metallic elements containing a small amount of Mn (or another transition metal) impurities like Cu(Mn) [or Au(Fe)], the interaction is 56 CHAPTER 4. SPIN GLASSES

0.2

0 AgMn

−0.2 100 Ising 80

−dip 0.4 60

ZFC 40 ZFC M

M / −0.6 Ising ∆ 20 ZFC+FS(Ts/Tg=0.82) 0 −0.8 −20 10 15 20 25 30 T(K) −1

Ts/Tg ∼ 0.82 −1.2 0.6 0.8 1 1.2 1.4 T/Tg

Figure 4.12: Results of temperature stops in the Ag(11 at % Mn) and Fe0.5Mn0.5TiO3 spin glasses; Ts/Tg=0.8, ts=3000 s. H is chosen to yield a linear response of the systems. The insert depicts the results before substraction for the Ising system. of RKKY-type, in which the charge carriers mediate the interaction between Mn moments; this interaction is of rather long range, decaying with the distance as r−3, and oscillating in sign. Due to the random distribution of the Mn atoms, both positive (ferromagnetic) and negative (antiferromagnetic) interaction appear in the alloy, yielding the observed frustration. The system is rather isotropic and considered as a model 3d Heisenberg spin glass [25]. In the case of single crystals of Fe0.5Mn0.5TiO3, the magnetic interaction is of short range, limited to nearest and next nearest neighbors. FeTiO3 and MnTiO3 are antiferromagnetic at low temperatures. In FeTiO3, the (a,b) planes of the hexagonal structure are ferromagnetic, but antiferromagnetically coupled to each other. In MnTiO3, the Mn moments are antiferromagnetically coupled within the planes, which are also antiferromagnetically coupled to each other (c.f. the diﬀerent types of antiferromagnetic arrangements in the next chapter). Thus in Fe0.5Mn0.5TiO3, the competition between ferromagnetic and antiferromagnetic interaction within the (a,b) planes yields the frustration between spins and a spin glass phase transition. As shown by A. Ito et al. [34], the system has a strong uniaxial anisotropy, and is a model system for a 3d Ising spin glass. Figure 4.12 shows the results of a 3000 s stop at Ts/Tg ∼ 0.82 (Ts=27 K for Ag(11 at % Mn) and Ts=18 K for the Ising system) performed during the cooling of the two spin glasses. Similar cooling and heating rates were used in the experiments. As seen in the ﬁgure, the “memory dip” of Ag(11 at % Mn) is 4.2. DC-MEMORY EXPERIMENTS 57

1.4

Ta=29.9K, Tm=30K

1 Ta=30K = Tm S(t)

0.6 Ta=30.1K, Tm=30K

0.2 −1 0 1 2 3 4 10 10 10 10 10 10 t(s)

Figure 4.13: Relaxation rates for Ag(11 at % Mn) after temperature shifts of ±0.1 K; tw=1000 s. less broad than the Ising one. As we have mentioned above, the aging occurring at Ts also affects the neighboring temperatures, and the width of the affected temperature range may be related to the overlap between the spin configuration attained at Ts and the corresponding state at the neighboring temperatures. One can thus speculate that the Ising spin glass has a larger overlap length than Ag(11 at % Mn). One thus expects Ag(11 at % Mn) to be more “chaotic” than the Ising sample. Jönsson et al. have investigated the temperature chaos of Ag(11 at % Mn) by recording the ZFC relaxation of Ag(11 at % Mn) after temperature shifts of different magnitude and duration at temperatures below and above the measurement temperature [35]. Figure 4.13 shows the effect of such temperature shifts on the relaxation rate curve. In the reference measurement (thicker line), the system is aged at Ta=30 K for tw=1000 s, and the ZFC magnetization is recorded versus time at the same temperature, i.e. Tm=30 K. The obtained S(t) thus shows the typical maximum close to the wait time. The two other curves describe the results of a positive and negative temperature shift of 0.1 K: In a positive (resp. negative) temperature shift, the sample is aged at Ta=29.9 K (resp. 30.1 K) for tw=1000 s, and the relaxation is collected at Tm=30 K. The magnitude of the temperature shift ∆T is defined as Tm −Ta or (Tm −Ta)/Tg. In all cases, the relaxation rate still shows a maximum, but it is shifted to lower (resp. higher) time scales. The new position of the maximum of S(t) defines an effective time teff . For small temperature shifts, the aging at Ta and Tm is accumulative, so that it seems that the system has aged for teff at Tm; teff is thus an effective age of the system. The process is then reversible, so that there is a symmetry 58 CHAPTER 4. SPIN GLASSES

Ising 3

w AgMn /t eff

t 2

0 −0.06 −0.03 0 0.03 0.06

(Tm −Ta)/Tg

Figure 4.14: teﬀ /tw as a function of the magnitude of the temperature shift in the Ag(11 at % Mn) and Ising spin glasses, for diﬀerent wait times.

between positive and negative temperature shifts: If one ages the system at Ta -0.0062 (i.e. Ta >Tm+0.2 K), the effective times become shorter, and the system appears younger and younger as ∆T increases. In other words, the systems is being reinitialized when the difference between Ta and Tm over- comes the temperature range of overlap between the equilibrium states at Ta and Tm. In the case of the Ising system, it appears that larger ∆T are required to reach non-accumulative effects [36]. These results are in agreement with the already observed broader “dips” in the dc-memory of the Ising system. The Ising spin glass is indeed expected to appear ’less’ chaotic than a Heisenberg spin glass due to a smaller chaos exponent, i.e. larger overlap length [35]. The critical exponents associated with the spin glass transition of the two materials are found to be rather different [15], supporting the existence of different universality classes for spin glasses [37]. 4.2. DC-MEMORY EXPERIMENTS 59

4.2.5 Dc-memory and not so conventional spin glasses Manganites show, as is extensively discussed in the next chapter, rich and complex phase diagrams. Another example of a frustrated perovskite system is (La,Sr)CoO3 [38], where La0.95Sr0.05CoO3 has a low-temperature spin-glass phase [39], evidencing magnetic disorder due to competing ferromagnetic and antiferromagnetic interaction in the system. We will see in the following that it is quite common to observe such eﬀects also in CMR manganite perovskites. A dc-memory eﬀect similar to that of Ag(11 at % Mn) is observed in La0.95Sr0.05-

UHI V V V V V HPXJ

WUP 0

+ *7V . D

UHI . . .

HPXJ

WUP 0

E + *WV V 7 .

Figure 4.15: (a) Zero ﬁeld stops in the TRM magnetization of La0.95Sr0.05CoO3 for logarithmically spaced stop times. (b) shows the results of single, double and triple memory experiments.

CoO3 [39]. Figure 4.15(a) shows the results of zero field stops performed in TRM measurements on a polycrystal La0.95Sr0.05CoO3 sample [39]. Stops of different “logarithmically spaced” duration (see legend in the figure) have been employed. The corresponding TRMZFS curves consequently show almost equidistant separation at low temperatures, illustrating the logarithmic spin- 60 CHAPTER 4. SPIN GLASSES glass like nature of the relaxation in the sample. To illustrate further the feasibility of the method, double and even triple zero field stop experiments are performed, and as seen in Fig. 4.15(b), a superposition of the different “memories” occur. Similar effects are not only expected to occur in systems with strongly correlated (aging) dynamics, but in all systems with strongly temperature dependent and widely distributed relaxation times, such as non- interacting nanoparticles systems [40].

−2 Ts=73K M/H ∆ −4 Ts=76K

Ts=79K −6

Ts=82K −8 2

−2 M/H ∆ −4

−6 Ts=82,79,76,73K −8 60 65 70 75 80 85 90 95 T(K)

Figure 4.16: Results of single and quadruple stops performed on a superconductor showing a glassy PME state. The “four stop” curve actually corresponds to the sum of the four single stops; not included.

A melt-cast prepared Bi2Sr2CaCu2O8 high-Tc polycrystal exhibiting the so- called “paramagnetic Meissner effect” (PME) has been found to possess a spin glass like aging phenomenon at temperatures just below Tc ∼ 85 K [41]. The FC magnetization of this sample exhibits a positive magnetization in weak fields (coining the name PME). The ZFC magnetization is dominated by a shielding effect at low temperatures, but close to Tc the field penetrates the material and the spontaneous moments causing the PME are exposed to the field. 4.2. DC-MEMORY EXPERIMENTS 61

Rejuvenation, chaos, and memory effects in this temperature range have been reported recently [42]; we here report results from dc-memory experiments with multiple stops. The results of temperature stops (∆M=MZFC(T )- MZFC(T,Ts), where MZFC(T ) is collected after direct cooling and MZFC(T,Ts) after a stop at Ts) performed in ZFC experiments are shown in Fig. 4.16. In these materials the relaxation is very small, and the signal is slightly more noisy than in the measurements of spin glasses. In the figure, four stops are performed, and in all cases the effect of the stop is retrieved, i.e. as in spin glasses, the states equilibrated at the four temperatures are retrieved. The memory “dips” are in this case quite sharp, comparable in width to those of the Ag(11 at % Mn) Heisenberg spin glass. It is interesting to note that the chi- ral glass model proposed by H. Kawamura [43] can be used to describe both the glassy phase of the PME and the Heisenberg spin glass [44]. A small amount of anisotropy may indeed exist in Heisenberg spin glasses, due to dipolar interaction and the scattering of the charge carriers by non magnetic impurities (Dzaloshinsky-Moriya interaction). In this case, the presence of frustration will induce a spin chirality [45], and as for the PME sample, the frustration will be affected by the chirality of the non co-planar spin configuration [43]. 62 CHAPTER 4. SPIN GLASSES Bibliography

[1] J. -P. Bouchaud, V. Dupuis, J. Hammann, and E. Vincent, Phys. Rev. B 65, 024439 (2001). [2] G. Toulouse, Commun. Phys. 2, 115 (1977). [3] V. Dupuis, E. Vincent, J. Hammann, J. E. Greedan, and A. S. Wills, cond-mat/0109242. [4] Y. Taguchi, Y. Oohara, H. Yoshizawa, N. Nagaosa and Y. Tokura, Science 291, 2573 (2001). [5] D. Petit, in “Nature de la phase basse temperature des verres de spin Heisenberg en dimension 3”, PhD thesis, Orsay, France (2002). [6] S. F. Edwards and P. W. Anderson, J. Phys. F 5, 965 (1975). [7] K. H. Fisher and J. A. Hertz in “Spin Glasses”, edited by D. Edwards and D. Melville, Cambridge University Press (1991). [8] P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977). [9] D. S. Fisher and D. A. Huse, Phys. Rev. B 38, 373 (1988). [10] K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801 (1986). [11] J. -O. Andersson, J. Mattsson, and P. Svedlindh, Phys. Rev. B 46, 8297 (1992). [12] M. Ocio, H. Bouchiat, and P. Monod, J. Magn. Magn. Mater. 54-57,11 (1986). [13] L. Lundgren, P. Svedlindh, and O. Beckman, J. Magn. Magn. Mater. 25, 33 (1981). [14] D. S. Fisher and D. A. Huse, Phys. Rev. B 38, 386 (1988). [15] P. E. J¨onsson, H. Yoshino, P. Nordblad, H. Aruga Katori, and A. Ito, cond-mat/0112389. [16] L. Lundgren, P. Svedlindh, P. Nordblad, and O. Beckman, Phys. Rev. Lett. 51, 911 (1983).

63 64 BIBLIOGRAPHY

[17] See also H. Yoshino, K. Hukushima, and H. Takayama, cond-mat/0202110. [18] C. Djurberg, J. Mattsson, and P. Nordblad, Europhys. Lett. 29, 163 (1995). [19] L. Lundgren, P. Nordblad, and L. Sandlund, Europhys. Lett. 1, 529 (1986). [20] T. Jonsson, K. Jonason, and P. Nordblad, Phys. Rev. B 59, 9402 (1999). [21] H. Yoshino, K. Hukushima, and H. Takayama, cond-mat/0203267. [22] A. J. Bray and M. A. Moore, Phys. Rev. Lett 58, 57 (1987). [23] P. Nordblad and P. Svedlindh, in “Spin-glasses and Random Fields”, Edited by A. P. Young, World Scientific, pp. 1-27 (1997). [24] K. Jonason, E. Vincent, J. Hammann, J. -P. Bouchaud, and P. Nordblad, Phys. Rev. Lett. 81, 3243 (1998). [25] T. Jonsson, K. Jonason, P. Jönsson, and P. Nordblad, Phys. Rev. B 59, 8770 (1999). [26] P. Nordblad, in “Dynamical Properties of Unconventional Magnetic Sys- tems”, edited by A.T. Skjeltorp and D. Sherrington, Kluwer, Dordrecht, pp 343-366 (1998). [27] K. Jonason, P. Nordblad, E. Vincent, J. Hamman, and J. -P. Bouchaud, Eur. Phys. J. B 13, 99 (2000). [28] K. Jonason, J. Mattsson, and P. Nordblad, Phys. Rev. B 53, 6507 (1996). [29] K. Jonason, J. Mattsson, and P. Nordblad, Phys. Rev. Lett. 77, 2562 (1996). [30] Artistic 3D plots can be found in P. Granberg, L. Lundgren, and P. Nord- blad, J. Magn. Magn. Mater. 92, 228 (1990). [31] H. Yoshino, A. Lemaitre, and J. -P. Bouchaud, Eur. J. Phys. B 20, 367 (2001). [32] R. Mathieu, P. Jönsson, D. N. H. Nam, and P. Nordblad, Phys. Rev. B 63, 092401 (2001). [33] R. Mathieu, P. E. Jönsson, P. Nordblad, H. Aruga Katori, and A. Ito, Phys. Rev. B 65, 012411 (2002). [34] A. Ito, H. Aruga, E. Torikai, M. Kikuchi, Y. Syono, and H. Takei, Phys. Rev. Lett. 57, 483 (1986). [35] P. E. Jönsson, H. Yoshino, and P. Nordblad, cond-mat/0203444. [36] P. E. Jönsson, R. Mathieu, H. Yoshino, and P. Nordblad, unpublished. [37] H. Kawamura, Phys. Rev. Lett. 80, 5421 (1998). BIBLIOGRAPHY 65

[38] M. Itoh, I. Natori, S. Kubota, and K. Motoya, J. Phys. Soc. Jpn. 63, 1486 (1994); J. Magn. Magn. Mater. 140-144, 1811 (1995). [39] D. N. H. Nam, R. Mathieu, P. Nordblad, N. V. Khiem and N. X. Phuc, Phys. Rev. B 62, 8989 (2000). [40] See the excellent J. L. Garc´ıa-Palacios, Adv. Chem. Phys. 112, 1 (2000). [41] J. Magnusson, M. Björnander, L. Pust, P. Svedlindh, P. Nordblad, and T. Lundström Phys. Rev. B 52, 7675 (1995); E. L. Papadopoulou, P. Nordblad, P. Svedlindh, R. Schöneberger, and R. Gross, Phys. Rev. Lett. 82, 173 (1999). [42] E. L. Papadopoulou and P. Nordblad, Eur. Phys. J. B 22, 187 (2001).

[43] H. Kawamura, Phys. Rev. Lett. 68, 3785 (1992).

[44] M. Matsumoto, K. Hukushima, and H. Takayama, cond-mat/0204225

[45] J. Villain, J. Phys. C: Solid State Phys. 10, 4793 (1977). 66 BIBLIOGRAPHY Chapter 5

Colossal magnetoresistive materials

he discovery of colossal magnetoresistance [1, 2] (CMR) in hole-doped man- Tganese oxides R1−xAxMnO3 (R: trivalent rare earth, A: divalent alkaline- earth) has revived the interest in this complex magnetic system [3, 1]. In particular, for dopings x ∼0.3, the CMR materials show simultaneous paramagnetic- to-ferromagnetic and insulator-to-metal transitions [1, 2, 4]. At a temperature close to the transitions temperature, the electrical resistance is very much affected by an applied magnetic ﬁeld, yielding large (or colossal) magnetoresistance eﬀects. We will in the following study the magnetic and electrical properties of manganites with x ∼0.3, as well as the properties of CMR materials in other parts of the phase diagram.

ABO3

A sites: R, A

B sites: Mn

Figure 5.1: Schematic view of the ABO3 perovskite structure. The Mn atoms occupy the center of the cells.

67 68 CHAPTER 5. COLOSSAL MAGNETORESISTIVE MATERIALS

The presence of both divalent and trivalent ions on the A-site of the ABO3 perovskite structure of these compounds yields a mixed valence of Mn, which appears in the system both as Mn3+ and Mn4+. The magnetic and electrical properties of these CMR manganites essentially depend on the hole doping x. Figure 5.1 shows a representation of the perovskite structure. For an ideal perovskite, the cell is cubic, so that the distance between the atom on the A- site and O is the same as the one between the atom on the B-site and O. It thus means,√ if one looks at the figure, that the A-O-A diagonal has the same length as 2 of an edge of the√ cube, i.e. O-B-O. This equality can be expressed as: 1/2rA + rO +1/2rA = 2(1/2rO + rMn +1/2rO) where the ri are the ionic 3+ radii of the different atoms. A typical perovskite is LaMnO3, where the La and Mn3+ ions occupy the A and B sites, respectively. Since in the case of CMR materials some of the ions on the A-site are substituted by cations of a different size, one sometimes uses the so called tolerance factor t: r + r t = √ O (5.1) 2(rMn + rO) where r refers to the average ionic radius of cations occupying A sites, and 3+ 4+ rMn the average ionic radius of Mn and Mn .

0.6 FC

M (emu/g) (x 1/3) 0.8

0.6 ρ (Ω.cm) 0.4 0.4 MR H ↑ 0.2

0 0 100 200 300 T(K) 0.2

ZFC H ↑

0 0 50 100 150 200 250 300 T(K)

Figure 5.2: Temperature dependence of the magnetization and electrical resistivity, for Nd0.7Sr0.3MnO3. The resistivity is measured in an applied ﬁeld of H=0, 1 and 5 Tesla. The insert shows the temperature dependence of the corresponding magnetoresistance curves, as deﬁned in Eq. (2.11). 5.1. ORBITAL ORDER AND PHASE DIAGRAM 69

The tolerance factor t thus represents the microscopic distortion from the ideal cubic perovskite structure, for which t=1 as mentioned above. Figure 5.2 shows the temperature dependence of the magnetization and electrical resistance of a polycrystalline Nd0.7Sr0.3MnO3 CMR manganite. As mentioned above, for this doping, the system undergoes simultaneous ferromagnetic and metallic transitions, close to Tc=240K. The magnetoresistance eﬀect is strongest close to Tc, but one can see from the ﬁgure that the magnetoresistance again increases at low temperature. We will see in the next sections that this low temperature tail is related to an additional scattering due to the presence of grain boundaries in the material.

5.1 Orbital order and phase diagram

5.1.1 Orbital structure

In the ABO3 perovskite structure, the Mn atoms lie on the B-sites, at the center of an octahedron defined by its 6 neighboring oxygen atoms; see Fig. 5.1. In such an environment, the five times degenerated 3d-orbitals of the Mn3+ atoms will split into two different energy states [5]. The different orbitals involved have different shapes, and thus different level of interaction with the surrounding O2− ions [5]. Fig. 5.3 illustrates the splitting of the 3d energy levels, and a schematic representation of the different orbitals. The state with lower energy contains the dxy, dyz and dzx orbitals, while the higher one is composed of the dx2−y2 and d3z2−r2 ones. The triplet is referred as the eg 3+ state, while the doublet is called t2g state. The electronic structure of Mn 3 1 3 4+ is thus t2geg, while it is t2g for Mn . The orbital degeneracy of the eg state is likely to yield a so-called cooperative Jahn-Teller distortion [6, 7] of the O6 octahedron, i.e. a distortion lowering the symmetry of the crystal and lifting the orbital degeneracy. This phenomenon is referred to as orbital ordering [8, 12] and Mn3+ as a Jahn-Teller [6] ion. The distortion lowers the energy of the eg electron, binding it tighter to its local site. It might be enlightening in this case to evaluate the tolerance factor defined in Eq. (5.1). We will see later, that one can relate the evolution of the magnetic properties with the hole doping of CMR materials to their orbital order. For example, Goodenough [5] predicted that the magnitude and sign of the magnetic interaction between the Mn ions depend on the orbitals occupied by the electrons.

5.1.2 Magnetic interaction The CMR eﬀect is attributed to an indirect magnetic interaction between Mn4+ and Mn3+ spins, known as double exchange [9]. In this picture, ferromagnetism (and conductivity) arises from hopping of the itinerant carrier from Mn3+ to 4+ Mn . The itinerant eg electrons have their spins parallel to the localized 3 t2g (S=3/2) spins via a strong Hund coupling, and the electron hops easily along the (Mn-O-Mn) bonds between pairs of ferromagnetically ordered Mn3+ 3 1 4+ 3 (t2geg) and Mn (t2g) ions, while its motion is hindered between disordered 70 CHAPTER 5. COLOSSAL MAGNETORESISTIVE MATERIALS

y y z - + - + ++ x - - y x - + + -

2 d x2-y d 3z22-r dxy

Figure 5.3: Schematic description of the ﬁeld splitting of the degenerated 3d- orbitals, as well as a typical representation of the dx2−y2 , d3z2−r2 , and dxy orbitals. dzx and dyz are similar to dxy.

spins. The motion of the eg electrons is thus dependent on the relative spin orientation of the localized t2g moments, which is describe by the so-called transfer integral: [10]:

Tij ∝ cos(θij /2) (5.2)

where θij is the angle between moments i and j; Tij has of course a maximum when i and j are parallel to each other, and minimum when they are antiparallel. The process involves two simultaneous hopping stages: Mn3+ → O2− and O2− → Mn4+, justifying the name of “double” exchange. The model can be modiﬁed to take into account lattice polarons, formed on 3+ localization of an eg electron on Mn , as proposed by Millis et al. [11], to explain the large resistivity of the paramagnetic regime. One can also take into account antiferromagnetic interaction between the Mn atoms [10], connected to the superexchange interaction. In this case, the eg electron consecutively hops from Mn3+ → O2− and then from O2− → Mn4+. We will see in the following sections that changes in the structure, and thus in the angles or distances between Mn spins, aﬀect the strength of both the ferromagnetic double exchange and the antiferromagnetic superexchange interaction. 5.1. ORBITAL ORDER AND PHASE DIAGRAM 71

5.1.3 Bandwidth considerations

Using the double exchange model, Kubo and Ohata [13] obtained a half metallic ferromagnetic ground state, with well separated spin-up and spin-down bands. In the ferromagnetic case, the width of the spin-down band was broad, compared to the spin-up one, which was narrow. The width of the spin-down band is referred as the bandwidth W . A larger bandwidth yields a stronger ferromagnetic state [14]. As observed by Arulraj et al. [15], the bandwidth increases with the (average) size of the cations on the A-site of the perovskite structure. Kajimoto et al. [16] could establish the phase diagram shown in

R1-xAxMnO3 (R, A)

(La, Sr)

Metal (Pr, Sr) A C G F (Nd, Sr)

Band width A (La, Ca)

Insulator CxE1-x CE Insulator (Pr, Ca) 0.0 0.2 0.4 0.6 0.8 1.0 Hole concentration x

Figure 5.4: Phase diagram describing the magnetic and electrical properties of CMR manganites as a function of doping x and bandwidth W . The different phases are identified by capital letters. F refers to a ferromagnetic arrangement, while A, C, CE and G reflects different antiferromagnetic configurations; see main text. Figure taken from Kajimoto et al. [16].

Fig. 5.4, predicting the main phases of diﬀerent compounds, depending on their bandwidth. For example, the main phase of (La,Sr)MnO3 of high bandwidth will be a ferromagnetic metal (from 0.15

Figure 5.5: Phase diagram describing the diﬀerent magnetic structures of (Nd,Sr)MnO3 as a function of the Sr doping. Schematic representations of the orbital order are added for each phase; ﬁgure taken from Okuda et al. [17].

5.1.4 Phase diagram

We can follow the eﬀect of hole doping1 on the magnetic and electrical properties of CMR materials. In Fig. 5.5, we show the phase diagram of Nd1−xSrxMnO3 of intermediate bandwidth. The orbital order of each phase is also illustrated [17]. At high temperatures, for all doping levels, the system is a paramagnetic insulator, denoted by ”P” on the diagram. At low temperatures, it undergoes the following transitions:

1In the overdoped regime, i.e. x>0.5, one could speak instead of electron doping. 5.1. ORBITAL ORDER AND PHASE DIAGRAM 73

Figure 5.6: Schematic representation of the CE-type ordering of charges, orbitals and spins. o and ∞ describe the orbitals of the Mn4+ and Mn3+ ions respectively. The arrows denote the orientation of the spins of the diﬀerent ions. The “zig-zag” chains are shown in thicker lines. Figure taken from D. Khomskii [8]. exchange, the state remains insulating.

0.1

0.48 magnetic structure is thus re-established. In this case, the c axis (along z)is compressed and becomes smaller than a and b (x,y), and the dx2−y2 orbitals order is favorable, as shown in Fig. 5.5. The hopping of electrons is still possible within each (x,y) plane, but diﬃcult perpendicular to the plane, due to the shape of dx2−y2 orbitals, which lack overlap along the z direction. The state is thus metallic, but highly anisotropic and quasi two dimensional [19].

A special doping: x ∼ 0.5 - insulating CE-type antiferromagnetism and charge ordering. CE-type refers to “charge-exchange-type”. As in the 0.48 < x<0.62 range, the d3x2−r2 and d3y2−r2 orbitals order in the (a,b) plane. But for this doping, there are as many Mn3+ as Mn4+ ions, and the charges order, 74 CHAPTER 5. COLOSSAL MAGNETORESISTIVE MATERIALS forming a chessboard arrangement of Mn3+ as Mn4+ ions. Fig. 5.5, as well as Fig. 5.6 illustrate the obtained structure. Since all Mn3+ ions are separated by Mn4+ ions in the (a,b) plane, it takes two times a and two times b to obtain the A-type structure, so that the magnetic unit cell of the Mn3+ sublattice is doubled in both direction. According to Goodenough’s results [5, 18], when Mn3+ have orbitals directed toward Mn4+, they are ferromagnetically coupled to Mn4+, while they are antiferromagnetically coupled to Mn4+ when directed away from them (see Fig. 5.6). As a result, the d3x2−r2 and d3y2−r2 orbitals order in a so called CE-type antiferromagnetic structure [20], composed of ferromagnetic “zig-zag” chains, antiferromagnetically coupled to each other within the (a,b) planes, as well as between those planes. The state is thus insulating, although the electrons can hop along the chains. One sometimes refers to “diagonal” charge order in the CE-type case. In the A-type structure, between x=0.48 and 0.5, “parallel” charge ordering occurs, with charges ordered in stripes. Within each Mn4+ stripe, the double exchange is stronger than within the diagonal zig-zag chains, yielding the observed metallicity [16].

0.62

0.8

Depending on the doping, one can thus obtain ferromagnetic or antiferromagnetic metallic compounds, as well as antiferromagnetic insulators. Ferro- magnetic insulators are less common, since the occurrence of ferromagnetism is associated with the movement of free carriers in the lattice, but can be obtained for some partial orbital ordering cases [8]. A large amount of theoretical work has been devoted to disclose the orbital eﬀects on the magnetic states of CMR manganites. The results include a thermodynamic phase separation in the materials, with mixed-phase states of diﬀerent magnetic and electrical properties. Many details can be found in ref. [21], which comprehensively and clearly reviews the topic.

5.2 CMR ferromagnets

For doping levels close to x=0.3 and large enough bandwidth, the CMR manganites are essentially ferromagnetic, with relatively large magnetoresistance 5.2. CMR FERROMAGNETS 75 close to their Curie temperature. It is in this context interesting to investigate thin ﬁlms of manganites, the physical properties of which can be tailored in many ways.

5.2.1 Epitaxial ﬁlms, grain boundaries

The temperature dependence of the magnetization of an epitaxial film of La0.7- Sr0.3Mn03 is shown in Fig. 5.7. The La0.7Sr0.3Mn03 film is ferromagnetic with a Tc close to 360 K. As seen in insert, the film has a low coercivity, and shows a rather square formed hysteresis curve, indicative of high film quality. If one measures the electrical resistivity of this film in different magnetic fields, it will show sizable magnetoresistance in low fields only close to Tc. One can add some spin disorder and thus enhance the low field magnetoresistance by introducing grain boundaries in the structure. In addition, in contrast to polycrystalline samples, one can orient the grain boundaries, and investigate their effect on the magnetic and electrical properties of the films.

350 500 500 T=5K 300 250 250 0 0 250 M (kA/m) 2DA −250 −250 GBFEF(a) (b) −500 −500 200 −200 −100 0 100 200 −40 −20 0 20 40 H (kA/m) 150 FC M (kA/m)

100

ZFC 50 H=4 kA/m 0 0 50 100 150 200 250 300 350 400 T (K)

Figure 5.7: Main frame: Temperature dependence of the ZFC and FC magnetization of an epitaxial ﬁlm of La0.7Sr0.3Mn03 of 200 nm of thickness. H=50 Oe (or 4 kA/m). The insert (a) shows a hysteresis measurement at low temperature, enlarged in (b).

As an example, Fig. 5.8 shows hysteresis curves of three La0.7Sr0.3Mn03 films, having an increasing number of grain boundaries. EF refers to the epitaxial film discussed above, GBF to a film containing irregular grain boundaries, while 2DA is a two dimensional array containing many oriented grain boundaries. In the case of EF, the hysteresis curve is rather square shaped, 76 CHAPTER 5. COLOSSAL MAGNETORESISTIVE MATERIALS

400 200 0

M(kA/m) −200 EF −400 400 200 0 GB↑ M(kA/m) −200 GBF −400 400 200 0 M(kA/m) −200 2DA −400 −100 −50 0 50 100 H(kA/m)

Figure 5.8: Hysteresis curves for thin films of La0.7Sr0.3Mn03 containing larger and larger amounts of grain boundaries; T =5 K. as in a sample with no or few defects. GBF contains some amount of grain boundaries; as a result, the hysteresis curve is more inclined. Also, one notices that the addition of defects in the form of grain boundaries promote the nucleation of reversed domains, thereby reducing the coercivity. Adding more boundaries, as is the case of 2DA, the hysteresis curve becomes even more inclined, but the coercivity increases, indicating a pinning controlled mechanism for the coercivity in this sample. These general characteristics remain at higher temperatures. We will see that the addition of grain boundaries in thin films of CMR materials yields paramount effects on their (magneto)resistive properties.

5.2.2 Artificial grain boundaries Grain boundaries (GB) can be introduced in epitaxial films of CMR material by altering their substrates before deposition. For example, Ziese et al [22] imprinted sharp steps onto their single-crystalline substrate by ion-beam etching. Depositing buffer and seed layers on their substrates, Lee et al [23] could fabricate c-axis oriented biepitaxial CMR films. Well defined GB associated with a 45o rotation of the (a,b) plane in the middle of the film were obtained. Similarly, using bicrystal substrates, i.e. two single-crystals of the same material glued together with a misorientation angle, Steenbeck et al [24] obtained bicrystal CMR films with single artificial GB. The magnetic and elec- 5.2. CMR FERROMAGNETS 77 trical properties of biepitaxial and bicrystal films will be discussed in detail in the following.

Biepitaxial films ˚ A c-axis oriented La0.7Sr0.3Mn03 biepitaxial film (aLa0.7Sr0.3Mn03 =3.82 A) is ˚ grown on a SrTiO3 substrate (aSrTiO3 =3.905 A), using buffer and seed layers ˚ ˚ of MgO (aMgO=4.21 A) and CeO2 (aCeO2 =5.41 A), respectively. The seed layer was etched into a chess board pattern with squares or fields of a size of 2 8x8µm . The chess board fields, where the SrTiO3 surface is disclosed, initiate a45o in-plane rotated growth of the buffer layer2. As shown in Fig. 5.9, the La0.7Sr0.3Mn03 film inherits the orientation of the buffer layer, yielding the formation of a 500x500 array of well defined 45o grain boundaries.

b b

a CMR film c CMR film c -a b b buffer buffer layer -a layer c c b a seed layer -a c b b substrate -a a substrate c c grain boundaries

CMR film LSMO

buffer layer CeO seed layer MgO

STO substrate

Figure 5.9: Schematic representation of the diﬀerent layers of the biepitaxial structure; see main text.

The array is ferromagnetic with Tc ∼ 360 K, as an epitaxial ﬁlm of La0.7Sr0.3Mn03; see [25] and Fig. 5.7. As seen in the main frame of Fig. 5.10(a), its resistivity is dominated by the grain boundaries, showing a broad maximum [26] well below √ 2 a × ˚ The diagonal of the CeO2 lattice is thus CeO2 =5.41 2 = 7.65 A, which is about twice the edge length of both SrTiO3 and La0.7Sr0.3Mn03 78 CHAPTER 5. COLOSSAL MAGNETORESISTIVE MATERIALS

Tc, and no significant features around this temperature. The low-field magnetoresistance (MR) of the array, shown in the upper insert of the figure, exhibits a low-temperature tail below Tc. The behavior is very different from that of the low-field MR curve of an epitaxial film of the same material (lower insert of the figure), and suggests a different transport mechanism, related to the GB.

0.04 30 Biepitaxial film Biepitaxial film 20

0.03 MR (%) 10

0 .cm) 0 200 400 Ω

( T(K) 2

ρ Epitaxial film 1.5 0.02 1

MR (%) 0.5 0 (a) 0 200 T(K) 400 0.01 0 100 200 300 400 T(K)

(b) 1

0.9 T=300K

) 0.06 1 (H=0) 0.8 − ρ

/ 0.04 ρ b (T 0.02 ≈ Lee et al 0 0.7 0 100 200 300 T(K)

0.6 T=10K

0.5 −6 −4 −2 0 2 4 6 µ0H(T)

Figure 5.10: (a) Main frame: Temperature dependence of the zero field resistivity of the biepitaxial film. Upper insert: corresponding temperature dependence of the low-field (µ0H=0.1 T) MR. The low field MR of an epitaxial film is added in the lower insert for comparison. (b) High field MR for the biepitaxial film, at low temperatures (T =10 K) and around room temperature (T =300 K). The insert shows the temperature dependence of the slope of the high field magnetoconductivity. 5.2. CMR FERROMAGNETS 79

The temperature dependence of the low-field magnetoresistance is similar to that of CMR-based magnetic tunnel junctions [27]. Also ρ(H) measurements performed at low temperatures [25] in fields ≤ 0.1 T show similarities to results obtained for such structures [27]. This suggests that the low-field magnetoresistance observed in the GB film and its temperature dependence are associated with a spin polarized tunneling process. Lyu et al. proposed a model considering the effects of a temperature dependent spin polarization on the tunneling magnetoresistance [28], and obtained results similar to those of the array. The low field MR is also observed in ρ(H) measurements performed at low temperatures, as seen in Fig. 5.10 (b). One also sees that for higher fields, the normalized resistance continuously decreases with increasing magnetic field. Linear fits of the high-field regime of the normalized resistance and conductance show that surprisingly the conductance, rather than the resistance, shows a linear high-field regime. This corresponds with the model proposed by Lee et al. for polycrystalline CMR samples, based on second-order tunneling through interfacial sites [29]. In this model, the eg electrons first tunnel from grain 1 to a state in the GB interface and then into grain 2. A tunneling junction can be modeled as a resistor [30], with a resistance Rj=1/Gj, where Gj is the tunneling conductance. Using the transfer integral defined in Eq. (5.2): → → T12 ∝ 1+ s1 · s2 (5.3) → → for itinerant eg electrons between localized t2g moments (s1 and s2 are the normalized spin moments in grain/electrode 1 and 2), the conductivity Gj in a two-step tunneling is given by:

2 2 → → → → Gj ∼ T1j Tj2 = (1+ s1 · sj) · (1+ sj · s2) (5.4) → where sj is the normalized grain boundary spin moment and ... denotes thermal average. For large enough magnetic fields, having saturated the ma- → → gnetization of the two La0.7Sr0.3Mn03 electrodes s1=s2, one obtains: → Gj ∼sj∝χjH (5.5) where χj is the susceptibility of the boundary region. The conductance is thus linear with the magnetic field for large fields, as here observed experimentally. The temperature dependence of the high field slope of the magnetoconductivity b(T )=dG/µ0G0dH of the array is shown in the insert of Fig. 5.10(b), as well as the results obtained by Lee et al. for polycrystals [29]. The similarity between the two curves shows the intrinsic nature of the magnetism close to an interface. When studying the transport properties of GB films, Evetts et al. considered the local field Hj acting on the GB [26], rather than the applied field H.We can in a similar way add a geometry dependent term to Eq. (5.5), and obtain: → Gj ∼sj∝χj(H + f(φ)Me) (5.6) where f(φ) is a geometric factor and Me the saturation magnetization of the La0.7Sr0.3Mn03 electrode. f(φ) depends on the orientation of the applied field 80 CHAPTER 5. COLOSSAL MAGNETORESISTIVE MATERIALS with respect to the GB array. It thus adds an orientation dependent term to the conductivity, suitable to explain anisotropic magnetoresistive (AMR) effects [25].

Bicrystal films In the above, we have studied the magnetic and electrical properties of a 500x500 array of 45o oriented GB. In the case of patterned bicrystal films, one can investigate the properties of a single GB. Two c-axis oriented La0.7Sr0.3Mn03 films were grown on a LaAl03 bicrystal substrates, and patterned into a 6 µm meander containing 100 GB [31]. A schematic view of the bicrystals is shown in Fig. 5.11. As indicated in the figure, the orientation of the (a, b) plane is LSMO a

V1 b I

6 µm a V2 b

Figure 5.11: Schematic representation of the patterned bicrystal film. The current I is fed along the meander, and the voltage ∆V =V 1-V 2 measured over a single GB. different on each side of the bicrystal. We here study crystals with misorientation angles of 18.4o (sample 1) and 8.2o (sample 2). The temperature dependence of the low field magnetoresistance [32] of sample 1 is presented in the main frame of Fig. 5.12. As the biepitaxial film, it shows a low temperature tail, related to the presence of the GB. Due to shape anisotropy of the meander, the magnetization lies in plane, and perpendicular to the GB: As seen in the upper insert of the figure, when applying the magnetic field along the GB, i.e. perpendicular to the current, a reversible magnetoresistance is observed, related to the reversible rotation of the domain magnetization. When applying the magnetic field parallel to the current instead, two hysteretic peaks appear, at fields close to the coercive field of the epitaxial material. As observed by Isaac et al., the magnitude of the low field MR increases with the degree of structural disorder associated with the GB, and thus with the misorientation angle [33]. Indeed, comparing Fig. 5.10(b) and the lower insert of Fig. 5.12, one can see that the low field MR is lower for the bicrystal film (18.4 0 GB) than for the biepitaxial one (45 0 GB). The resistivity is still associated with spin polarized tunneling through interfacial sites, and as for the biepitaxial array, the magnetoconductivity of the bicrystal shows a 5.2. CMR FERROMAGNETS 81 high field linear regime [32], as predicted by Eq. (5.5). Figure 5.13(a) presents

8 76 H ⊥ I 7 ) Ω 72 H // I H // I R( 6 T=80K 68 5 −0.3 0 0.3 µ0H(T) 1 4 T=10K MR (%)

(H=0) 0.9 ρ

3 / ρ H ⊥ I 2 0.8 −6 −3 0 3 6 µ H(T) 1 0

0 0 100 200 300 400 T (K)

Figure 5.12: Main frame: temperature dependence of the MR for sample 1. The field is applied perpendicular (H ⊥ I) and parallel (HI) to the current. Upper insert: low field R(H) for these field orientations; T =80 K. Lower insert: high field MR; T =10 K. hysteresis measurements of the dc resistance performed on sample 2 (lower GB angle) for two different magnetic field orientations. When H GB, sharp switches occur around the coercive field of the La0.7Sr0.3Mn03 electrode [25], and multiple steps appear. Similar results were obtained by Steenbeck et al. after annealing their sample and improving the GB definition [34]; in tunneling structures [35, 27] with multi-domain configurations, two separated MR peaks with similar multiple steps are observed. When instead H ⊥ GB, the curve is more smooth, and antiparallel domains across the GB seem to form more easily. In a low field, as seen in the insert of Fig. 5.13, which shows the variation of the resistance as a function of the angle between the applied field and the GB, the resistance switches to a high resistance state when the magnetic field is close to being perpendicular to the GB. The MR thus depends on the domain configurations neighboring the GB. One way to investigate the magnetic microstructure of the GB region is to study the low field dependence of the resistivity noise [31]. Fig. 5.13(b) and (c) show the low field dependence of the 2 noise level SV /V recorded at constant temperature for the frequency f=333 Hz. The noise level exhibits a similar field dependence as the dc resistance and suggests that the low field noise is of magnetic origin, related to the GB and the multi-domain state around it. The noise spectrum (over a broad range of frequencies) of polycrystalline, as well as epitaxial CMR material, usually exhibits an 1/f character, described 82 CHAPTER 5. COLOSSAL MAGNETORESISTIVE MATERIALS

580 550

(a) ) Ω R ( 570 545

H ⊥ gb 540 ) 0 90 180 270 360 Ω 560 θ (deg.) R (

550 T=10K H // gb 540 −0.1 −0.075 −0.05 −0.025 0 0.025 0.05 0.075 0.1 µ H (T) −13 0 −13 x 10 x 10 1.3 555 1.3 (b) (c) 5 3 ) 1 ) − 1.2 550 1 1.2 4 − R ( (Hz Ω 2 (Hz ) 2 9 / V / V V 7 V S 1.1 545 1.1 S 1 6 T=10K 2 8 1 540 1 −0.3 −0.2 −0.1 0 0.1 535 540 545 550 555 µ H (T) 0 R (Ω)

Figure 5.13: (a) Dc resistance vs. magnetic field for different magnetic field orientations with respect to the GB; T =10 K. The insert shows the angular dependence of the dc resistance for µ0H=250 Oe; θ refers to the angle between the applied field and the GB. (b) Magnetic field dependence of the noise level for low fields, f=333 Hz and HGB; the corresponding dc resistance is added for comparison. In (c), the noise levels are plotted vs. the dc resistance; for clarity the measurement points are numbered from 1 to 9, and the first point (corresponding to H=0) is denoted with a bigger marker. for metal samples as [36]:

S (f,T) α 1 V = (5.7) V 2(T ) N f where α is the Hooge parameter and N is the number of charge carriers in the material. As illustrated in Fig. 5.14, the noise spectrum of sample 2 exhibits a clear 1/f dependence for zero magnetic field. For a small field (H=-0.02T), an additional contribution appears, and the spectrum can be well fitted by adding 5.2. CMR FERROMAGNETS 83 a Lorentzian [37] contribution to the 1/f noise: ωτ fS (f)=C + C (5.8) V 1 2 ω2τ 2 +1 with ω=2πf, and where C1 and C2 are constants related to the magnitude of the 1/f and Lorentzian noise contributions, respectively. τ is a characteristic time of the random processes characterized by a Lorentzian spectrum. The insert shows the good agreement when fitting the experimental data to Eq. (5.8). The dislocations around the GB are likely to act as pinning centers for domain walls, and the results indicate that the Lorentzian distribution may originate from thermally activated domain-wall motion in the multidomain state around the GB. For thermally activated processes, one can calculate the corresponding energy barrier from: Eb/kB T τ = τ0e (5.9) −12 At T =10 K, using τ0=5.10 s for domain wall fluctuations [38], the average is 170 K. Comparing now with Eb = K1V gives an estimate of the domain size corresponding to this energy. Using the anisotropy constant 3 K1 determined at low temperatures for La0.7Sr0.3Mn03, |K1|≈10 kJ/m [39], yields a fluctuating length of 60 A,˚ which is of the order of magnitude of the expected domain wall width [40].

−11 −10 10 x 10 3

µ H= −0 .02T 2 0 2 / V −12 v 10

H=0 f . S ) µ0 1 − (Hz

2 1 −13 101 102 103 104 105 / V 10 v f (Hz) S

−14 10

T=10K −15 10 1 2 3 4 5 10 10 10 10 10 f (Hz)

Figure 5.14: Frequency dependence of the excess noise for two different fields, 2 HGB; T =10 K. In insert, f.SV /V is plotted vs. frequency for H=-0.02 T. A fit of the experimental data to Eq. (5.8) is shown as a solid line.

Looking again at Fig. 5.13(c), one notices that the noise level systematically switches before the dc resistance. For example, the noise level ﬁrst switches 84 CHAPTER 5. COLOSSAL MAGNETORESISTIVE MATERIALS from 2 to 3, and then the resistance, from 4 to 5. This shows that the domain wall motion between pinning sites is a precursor of larger changes in the domain conﬁguration occurring when the resistance switches between low and high resistance states. The behavior is observed both for H GB and H ⊥ GB [31].

5.3 Frustrated CMR ferromagnets

5.3.1 Frustration in Nd0.7Sr0.3MnO3 Even if, at a doping close to x=0.3, the ferromagnetic double exchange interaction dominates, there is also some antiferromagnetic superexchange interaction between the Mn ions in the material. For example, while Nd0.7Sr0.3MnO3 has magnetization curves typical of that of a long range ferromagnet [41], with Tc=235 K, glassy-like dynamics is observed in the relaxation of its magnetization. One can see in Fig. 5.15 that the magnetization of the compound slowly relaxes with time without reaching equilibrium. An aging phenomenon is re- vealed from the difference between the M(t) curves, collected after different wait times tw. The relaxation rate S(t), plotted vs log(t) in the figure indeed shows a characteristic maximum at an observation time close to tw,asdoesS(t) of the archetypical spin glasses (c.f. earlier). A similar behavior is observed at all temperatures below Tc.

950 1450

Tm=200 K, h=0.5 Oe Tm=30 K, h=0.5 Oe 900 1400

850 1350 M (arb. units) M (arb. units) 800 1300 100 s 100 s 1000 s 1000 s 10000 s 10000 s (a2) (a1) 750 1250

50 50

40 40

S(t) 30 30

20 S(t) 20

10 10

(b1) (b2) 0 0 -101234 -101234 log(t) log(t)

Figure 5.15: Relaxation of the ZFC magnetization for Nd0.7Sr0.3MnO3, for low temperatures (Tm=30 K) and high temperatures (Tm=200 K) in a small magnetic ﬁeld. 5.3. FRUSTRATED CMR FERROMAGNETS 85

T =40 K, t =103 s 50 m w

40 0.2 Oe 0.4 Oe S(t) 30 0.6 Oe 0.8 Oe 1.0 Oe 20 1.2 Oe 1.4 Oe 2.0 Oe 10

0 -1 0 1 2 3 4 5 log(t)

Figure 5.16: S(t)ofNd0.7Sr0.3Mn03 for diﬀerent magnetic ﬁeld strength. Tm=40 K; tw=1000 s.

As for reentrant ferromagnets [42], the aging of the system is observed only in vanishingly small fields. As seen in Fig. 5.16, the relaxation rate is a lot affected by the applied magnetic field, and in the field range investigated, the response is non linear. For magnetic fields larger than h=1.6 Oe, the aging states are destroyed (this also happens at “high” fields in spin glasses), and the relaxation becomes independent of the wait time. In addition, as seen in Fig. 5.17, the maximum of S(t)attw is suppressed by both negative and positive temperature cyclings, and even disappears for the larger ∆T . The spin configuration equilibrated during the first wait time is thus re-organized by both negative and positive cyclings, revealing the chaotic nature of the material. In conventional spin glasses, negative temperature cyclings do not reinitialize the system (see the previous chapter). The symmetry of the effect of negative and positive temperature cyclings is again very similar to the behavior of reentrant ferromagnets [43], i.e. ferromagnets with some degree of frustration. In the 3 case of Nd0.7Sr0.3MnO3, no reentrant spin glass phase transition is observed down to the lowest temperature (5 K). We will see below that transitions from a frustrated ferromagnetic state to a spin glass phase can be observed in other manganites.

5.3.2 Reentrant ferromagnets A possible reason for the appearance of both ferromagnetic and antiferromagnetic interaction in manganites is linked to the overlap of both the 2pπ and 2pσ orbitals of the oxygen with those from the electrons of the Mn ions along the Mn-O-Mn bonds [44]. The double exchange hopping integral is thus controlled by the eg(Mn)−2pσ(O)−eg(Mn) orbitals, while the superexchange one

3One also speaks about reentrant spin glasses instead of reentrant ferromagnets 86 CHAPTER 5. COLOSSAL MAGNETORESISTIVE MATERIALS

T =40 K, h=0.5 Oe 35 m 0 K 30 -0.1 K -0.5K S(t) -1 K 25 -2 K -5 K 20

15 (a) 10

+0 K 35 +1 K +2 K +3 K +5 K

S(t) 30

(b) 15 -101234 log(t)

Figure 5.17: (a) Negative and (b) positive temperature cyclings performed on Nd0.7Sr0.3Mn03 at Tm=40 K; tw=1000 s.

follows the overlap of t2g(Mn) − 2pπ(O) − t2g(Mn). Due to the nature of the dpπ hybridization, the AFM interaction is expected to be less influenced by changes of the Mn-O-Mn bond angle than the ferromagnetic one. One can check this by substituting the trivalent cation on the A-site with a smaller one. The substitution will yield a bond angle becoming smaller and smaller as the concentration of the dopant is increased [44]. The 4 effect of such substitution has been investigated [45] on the La0.96K0.04MnO3 compound, which is a regular ferromagnet with Tc ∼ 270 K. La (rLa3+ =1.216 A)˚ is gradually replaced by the smaller trivalent ion [46] Nd (rNd3+ =1.163 A)˚ as La0.96−yNdyK0.04MnO3, with y=0.1, 0.2, 0.3 and 0.4. As the content of Nd increases, the average size of the ion occupying the A-site decreases, and x-ray diffraction (XRD) experiments show that the average bond angle Mn-O-Mn decreases as well; see Table 5.1. One thus expects that the ferromagnetic interaction should weaken as the amount of Nd increases, while the antiferromagnetic interaction should remain rather unaffected, so that, eventually the two types of interaction will compete in strength, yielding frustration. Figure 5.18 shows

4In this compound, we do not have a trivalent and a divalent ion on the A-site, but a trivalent and a monovalent one. 5.3. FRUSTRATED CMR FERROMAGNETS 87 the temperature dependence of the ac-susceptibility for the undoped system, as well as for the compounds substituted with y=0.2 and y=0.4. As mentioned above, the compound without Nd is a typical ferromagnet.

0.6 Y=0 h=1 Oe /Oe)

3 0.4

0.2 (emu/cm χ

0 0.6 Y=0.2 /Oe) 3 f=1000Hz 0.4 f=125Hz (emu/cm

χ 0.2

0 0.6 0.015 Y=0.4 h=1 Oe f=125Hz /Oe) 3 χ’ 0.01

0.4 ’’ χ H ↑ 0.005

(emu/cm 0.2 χ 0 χ’’ (x 20) 0 100 200 0 T(K) 0 50 100 150 200 250 300 T (K)

Figure 5.18: Eﬀect of the Nd doping on χ for La0.96−yNdyK0.04MnO3. For undoped system (y=0), only the in-phase component of the ac-susceptibility is shown. For y=0.2 and y=0.4, the out-of phase component is added (× 20), and the results are shown for two frequencies. The insert shows the out-of-phase component of the ac- susceptibility for y=0.4, with a dc-magnetic ﬁeld of 0, 0.1, and 1 Tesla superimposed. χ is, for clarity multiplied by a factor of 1, 7 and 10 respectively; f= 125 Hz.

As Nd is added, Tc is lowered, indicating a decrease of the strength of the ferromagnetic interaction, and, as seen for the samples with y=0.2 and y=0.4, a knee appears in the in-phase component of the susceptibility at low temperatures, accompanied by a large frequency dependent peak in the out-of- phase component, indicative of magnetic disorder at lower temperatures. The electrical resistance of the compounds is measured in zero magnetic field, and in a relatively large field of H=6 Tesla, in order to estimate the large field magnetoresistance MR =[R(H =0)− R(H =6T )]/R(H = 0) of the material. Fig. 5.19 shows the results of the measurements: while the y=0 compound 88 CHAPTER 5. COLOSSAL MAGNETORESISTIVE MATERIALS

(a) y=0 104 y=0.1 y=0.2 y=0.3 y=0.4 102 cm] Ω [ ρ

100

10-2 80 (b)

60 [%] o 40 )/ R H -R o

(R 20

0 0 100 200 300 T (K)

Figure 5.19: Zero magnetic ﬁeld resistivity and high ﬁeld magnetoresistance for all La0.96−yNdyK0.04MnO3 samples.

Table 5.1: Structural data obtained from XRD for the diﬀerent compositions of La0.96−yNdyK0.04MnO3: average A-site ionic radius (A),˚ average bond length Mn-O (A),˚ bond angle Mn-O-Mn (degrees); is calculated from [46]. The Curie temperature [K] obtained from magnetization measurements is added.

y Struct. Mn-O Mn-O-Mn Tc 0 R-3C 1.229 2.080 160.73 252 0.1 (R-3C,Pnma) 1.224 (1.970,1.973) (159.75,160.11) 240 0.2 Pnma 1.218 1.977 158.23 182 0.3 Pnma 1.213 1.976 157.98 165 0.4 Pnma 1.208 1.972 157.97 167 5.3. FRUSTRATED CMR FERROMAGNETS 89

7 . W : V W : V W : V

G0 W GORJ W >DX@

6 W +

7 . D E + 2H + 2H

7 .

+ 2H + 2H G0 W GORJ W >DX@

+ 2H 7 . + 2H

6 W + + 2H

W V

Figure 5.20: ZFC relaxation measurements on La0.96−yNdyK0.04MnO3 at Tm=110 K and Tm=50 K (a) for different wait times (h=0.1) Oe and (b) for different magnetic fields; tw=1000 s. shows an overall metallic behavior, with a typical metal-insulator transition close to Tc, the resistivity of the samples increases rapidly with the Nd content. For y=0.4, the material is essentially an insulator, and its resistivity at low temperature is more than 6 decades larger than that of the undoped compound. One also notices that there is a bump in all curves at temperatures below Tc. Furthermore, a second peak appears in the magnetoresistance of the compounds for larger doping levels, at the same temperature as the resistivity bump, and the frequency dependent peak observed in the ac-susceptibility (c.f. Fig. 5.18), suggesting a common origin of all these features. By application of a large magnetic field, the spin disorder scattering induced by the magnetic disorder is removed, creating a new magnetoresistance peak. As shown in the insert of Fig. 5.18, the frequency dependent peak of the out-of-phase compo- 90 CHAPTER 5. COLOSSAL MAGNETORESISTIVE MATERIALS nent of the susceptibility remains when a dc-magnetic field is superposed to the ac-one, but progressively decreases as the strength of the dc-field increases. The magnetic disorder is likely to originate from the competition between the ferromagnetic and the antiferromagnetic interaction, rather than from some canting of the moments or the ordering of the charges and orbitals of the Mn3+ electrons [47]. The ZFC relaxation of the magnetization is recorded for the y=0.4 sample after different wait times tw at temperatures above (110 K) and below (50 K) the frequency dependent peak. Figure 5.20(a) shows the corresponding relaxation rate S(t), which as for Nd0.7Sr0.3MnO3, evidences glassy dynamics. As seen in Fig. 5.20(b), while the glassy state is rather robust to magnetic fields, the relaxation rate is very much affected by the increase of the magnetic field. For H=1 Oe, the aging features are suppressed, as in a reentrant ferromagnet [42]. While in Nd0.7Sr0.3MnO3, the traces of aging are suppressed for H ∼ 2 Oe at all temperatures [41], it here remains to higher fields for the temperatures below the ac-peak, indicating a true reentrant spin glass phase, with a closely linear response to our small probing field.

5.4 Short range CMR ferromagnets

5.4.1 Dynamics of Y0.7Ca0.3MnO3

3+ Y is even smaller than Nd, with rY3+ = 1.075 A,˚ and polycrystalline Y0.7Ca0.3- MnO3 only shows some short range ferromagnetism as evidenced by analysis of the Curie Weiss behavior and Arrot plots [48]. The susceptibility deviates from a Curie Weiss behavior below 70 K, suggesting an establishment of ferromagnetic correlations. However, Arrot plots (M 2 vs. H/M) show no indication of spontaneous magnetization. The ferromagnetism appearing below 70 K is spatially confined, i.e., only ferromagnetically ordered clusters appear at low temperatures. Similar short range correlations have been reported in neutron scattering experiments [49] on (La,Tb)0.67Ca0.33MnO3. In addition, the ZFC magnetization shows a cusp at T ∼ 30 K. Relaxation measurements performed below and above this temperature show aging features [48] similar to those of Nd0.7Sr0.3MnO3 and La0.96−yNdyK0.04MnO3, and thus evidence the existence of non equilibrium glassy dynamics below 70 K, which could be attributed to random dipolar interaction between the ferromagnetic clusters. Figure 5.21 shows the results of temperature cyclings experiments. Y0.7- Ca0.3MnO3 does not behave like Nd0.7Sr0.3MnO3 and La0.96−yNdyK0.04MnO3: here the relaxation curve is rather unaffected by the negative temperature cyclings, indicating that the spin configuration is frozen in as the temperature is lowered. A large reinitialization is observed only for the positive cyclings. Y0.7Ca0.3MnO3 thus behaves like an ordinary spin glass rather than a reentrant system. It is surprising that slow dynamics and aging remains at temperatures above the cusp in the susceptibility. Figure 5.22 depicts the temperature dependence of the ac-susceptibility of Y0.7Ca0.3MnO3, recorded for different frequencies (a1,a2) and with superposed dc-field (b1,b2). The onset of ferro- 5.4. SHORT RANGE CMR FERROMAGNETS 91

7 . P

6 W DUEXQLWV

7 . P

6 W DUEXQLWV UHI 67 LQI E W V

Figure 5.21: Negative and positive temperature cyclings performed above and below the cusp of the ZFC magnetization of Y0.7Ca0.3MnO3. magnetism, around 70 K is frequency independent, but the cusp around 30 K has a large frequency dependence. These data can thus be analyzed [48] according to dynamic scaling laws (c.f. earlier). The best results are obtained using activated dynamics and a ﬁnite critical temperature of Tg=28.9 K, and 5 seem to indicate a true phase transition . Since Y0.7Ca0.3MnO3 exhibits aging and memory (c.f. temperature cyclings) features reminiscent of ordinary spin glasses, memory eﬀects are investigated below and above 30 K using the relaxation of the out-of-phase component of the ac susceptibility at 27 K and 45 K (see insert of Fig. 5.23). The sample is cooled from above 70 K to the lowest temperature, and re-heated to 70 K. The ac-susceptibility is recorded during the temperature cycling, yielding two reference curves. The sample is again cooled from above 70 K down to 45 K, and kept at this constant temperature for 10000 s. The cooling is resumed down to 27K, and the material is again left for 10000 s at constant temperature. The cooling continues down to the lowest temperature. The ac-susceptibility is recorded during the cooling, and

5But not to a conventional spin glass phase; the relaxation time obtained in the scaling is too large to represent the ﬂipping time of a single spin; see ref. [48]. 92 CHAPTER 5. COLOSSAL MAGNETORESISTIVE MATERIALS

400 h=0.01 Oe χ’ (b1) χ’’

300

200 (arb. units)

χ H ↑

100 h=0.01 Oe (a1) f=0.51Hz 0 10 χ’’ (b2) h=1 Oe χ’’ 8 f=125Hz

6 f ↓

4 H ↑ (arb. units)

χ 2 (a2) 0 0 20 40 60 80 0 20 40 60 80 T(K) T(K)

Figure 5.22: Temperature dependence of the ac-susceptibility of Y0.7Ca0.3MnO3. (a1): in-phase component for different frequencies: 510, 170, 51, 17, 5.1, and 1.7 Hz. (a2): out-of-phase component for the same frequencies; the frequency decreases between different curves as indicated by the arrow. (b1): out-of-phase component at one single frequency, measured with a superimposed dc-field H= 0, 1, 2, and 5 Oe; the magnitude of the dc-field increases between the curves as indicated by the arrow. (b2): Idem as (b1) with larger fields superimposed: H= 0, 10, 100, 300, 1000, and 10 000 Oe. yield the curve marked “cooling” in Fig. 5.23(a). One can see that at both halt temperatures, the system has relaxed toward its equilibrium state, so that the susceptibility has decayed in magnitude. The equilibrium state became frozen in on lowering the temperature immediately after the halts, and the achieved spin state is retrieved on reheating the system, as shown by the susceptibility curve measured on re-heating to 70 K [“heating” in Fig. 5.23(a)]. The memory dip appears even more clearly when subtracting the reference curves, as seen in Fig. 5.23(b). The relaxation of the susceptibility shown in the insert of the figure is larger than the one obtained in the memory experiment, since the relaxation curves were obtained after a direct (and fast) cooling to the measurement temperature. In both cases, the relaxation is comparably smaller at 45 K. Surprisingly, Y0.7Ca0.3MnO3, in which ferromagnetic clusters appear at low temperature, shows large memory effects very similar to those of spin glasses. Additional information on the spatial correlation of the interaction is 5.4. SHORT RANGE CMR FERROMAGNETS 93

400 ref. cooling 350 ref. heating cooling 300 heating 250

200

arb. units 150 χ′′ 100

50 (a) 0

−50 ref −100 45K χ′′ (t=0.3s)

−

χ′′

−

χ′′ −200 27K −100 χ′′

−300 (b) 0 5000 10000 t(s) −150 20 30 40 50 60 70 80 T(K)

Figure 5.23: Results of a double ac-memory experiment on Y0.7Ca0.3MnO3; (a) shows the temperature dependence of χ measured on cooling and re-heating (reference curves) and on imprinting memories of two temperature stops (at 45 and 27 K) during the cooling. (b) shows the diﬀerence plots of the respective curves. The relaxation of χ with time is added in the insert; h=0.01 Oe, f=0.51 Hz.

needed to describe the magnetic state of Y0.7Ca0.3MnO3 in more details. We will see in the following that neutron scattering experiments can yield such information.

5.4.2 (La,Y)0.7Ca0.3MnO3 and Nd0.7Sr0.3(Mn,Mg)O3

In the polycrystalline (La,Y)0.7Ca0.3MnO3 system, La is gradually replaced by Y of smaller size. As in La0.96K0.04MnO3, the ferromagnetic interaction decreases with the average radius of the cation of the A-site. As seen in Fig. 5.24, the undoped compounds with x=0 (x refers to the amount of Y) is a typical ferromagnet, while for x=0.1, the susceptibility is similar to that of a reen- 94 CHAPTER 5. COLOSSAL MAGNETORESISTIVE MATERIALS

0.1 −3 x 10 h=1Oe 1 χ′′ (emu/g/Oe) 0.08 0.5 f ↑

0.06 0 0 20 40 60 T(K)

(emu/g/Oe) 0.04 χ′

x=0.2 0.02 x=0.1 x=0

0 0 50 100 150 200 250 300 350 400 T(K)

Figure 5.24: Temperature dependence of the in-phase component of the ac- susceptibility for (La1−xYx)0.7Ca0.3MnO3. h=1 Oe and f=125 Hz. The frequency dependence of the out-of-phase component for the x=0.2 sample is added in insert for comparison; f=15, 125 and 1000 Hz.

trant ferromagnet, as discussed for La0.96K0.04MnO3.Forx=0.2, the system does not seem to establish long range order, just as the magnetically inho- mogeneous Y0.7Ca0.3MnO3 manganite. A similar behavior is observed when substituting Mn for the non-magnetic Mg, as in Nd0.7Sr0.3(Mn,Mg)O3 [50]. In the case of the latter system, the appearance of glassy features could tenta- tively be related to the “dilution” of the ferromagnetic interaction occurring in the material when replacing parts of the Mn-O-Mn bonds by the Mn-O-Mg ones, containing non magnetic Mg ions. In order to propose a more elaborate scenario, one needs to know the exact position occupied by the Mg atoms. In- deed, if Mg (Mg2+) is smaller than the other cations on the A-sites, and has a size similar to that of Mn (rMg2+ = 0.72A),˚ it has also the same valence as Sr (Sr2+), which could affect the substitution of Mn [51]. This substitution, as well as the magnetic properties of the system can be investigated by performing neutron scattering experiments. Typical results are shown in Fig. 5.25. The intensity of the scattering is plotted as a function of twice the scattering angle θ. The peaks observed in the upper panel of the figure corresponds to Bragg reflections characteristic of the lattice of the sample. This spectrum was collected at a temperature above the Curie temperature of the compound (Tc ∼ 200 K). The lower panel shows the results of the measurement performed at a very low temperature, much lower than the Curie temperature of the system, and thus at which the material is ferromagnetic. One can see that the intensity of the peaks have increased, since some additional scattering, 5.4. SHORT RANGE CMR FERROMAGNETS 95 connected to the ordered moments of the different atoms, has appeared (see for example around 2θ∼ 37 and 53 degrees); the magnetic contribution is always largest at low angles. By fitting the spectrum using the Rietveld technique, one can derive the structural and magnetic configuration of the compound.

4 x 10 4 T=240K 3

1 Intensity (arb. units)

0 4 x 10 4 3 Mn T=1.4K )

B 2

3 µ M( 1 Nd 2 0 0 100 200 T(K) 1 Intensity (arb. units)

0 11 19 27 35 43 51 59 67 75 83 91 2θ (deg.)

Figure 5.25: Neutron spectra for Nd0.7Sr0.3MnO3 doped with 10% of Mg, at high (top) and low (bottom) temperatures. The insert shows the temperature dependence of the magnetic moments of Mn and Nd estimated from reﬁnements of the neutron scattering spectra; typical error-bars are indicated.

If neutron scattering spectra are collected at different temperatures, one can estimate the moment on the different sites or atoms. The insert of Fig. 5.25 shows the temperature dependence of the moments of the Mn and Nd ions, obtained from the refinement of the different spectra. The moment on the Mn atoms has a temperature dependence similar to that of the total magnetization of the sample [50]. In the case of Nd, a smaller moment is detected, likely to be induced by the ferromagnetic Mn ions, as proposed by Millange et al. [52]. Neutrons are also useful to investigate magnetic inhomogeneities, for example in the case of charge ordered manganites, in which orbital effects influence the magnetic interaction. 96 CHAPTER 5. COLOSSAL MAGNETORESISTIVE MATERIALS 5.5 Charge ordered antiferromagnets

We have seen in the previous sections that it was common to obtain CMR manganites with some degree of magnetic inhomogeneity. In the the x ∼ 0.5 doping (R0.5A0.5MnO3), the magnetic interaction is aﬀected by the ordering of the Mn3+ and Mn4+ charges, or charge ordering [53, 54]. This charge ordering (CO) is accompanied by the ordering of the eg electron orbitals on the Mn3+ sites, also known as orbital ordering (OO) [54]. We now study the ma-

0.04 −3 (a) Nd Ca MnO x 10 0.5 0.5 3 4 −5 ∆FC ∆M=3.4x10 µ /f.u FC B 0.03 2 ∆ZFC M/H (SI) ∆

M/H (SI) 0 100 200 0.02 T (K)

ZFC 0.01

0.12 (b) Gd0.5Ca0.5MnO3 0.03 f=125 Hz

0.08 (SI) 0.02 FC χ′ M/H (SI) 0.04 0.01 ZFC 0 100 200 300 T (K)

0 0 50 100 150 200 250 300 T (K)

Figure 5.26: Temperature dependence of the ZFC (markers) and FC (simple line) magnetization of (a) Nd0.5Ca0.5MnO3 and (b) Gd0.5Ca0.5MnO3; H 20 Oe. For both samples, the magnetization is recorded on reheating after fast (solid line) and slow (dotted line) cooling down to 5 K. The upper insert shows the diﬀerence plots of the FC and ZFC magnetization curves, using the same symbols as in the main frame. The value of ∆M at T =50K is indicated in µB/f.u. for comparison. The lower insert shows the cooling-rate dependence of the ac susceptibility of Nd0.5Ca0.5MnO3, recorded vs temperature using a small ac-ﬁeld, after fast (solid line) and slow (dotted line) cooling. h=20 Oe, f=125 Hz.

gnetic properties of single crystals of Nd0.5Ca0.5MnO3 and Gd0.5Ca0.5MnO3 [55]. The average cation size on the A-site is different: =1.172 A˚ for Nd0.5Ca0.5MnO3 and =1.127 A˚ for Gd0.5Ca0.5MnO3 (rNd3+ =1.163 A,˚ Gd is small, with rGd3+ =1.107 A˚ and rCa3+ =1.18 A).˚ As a result, the apparent 5.5. CHARGE ORDERED ANTIFERROMAGNETS 97 electron bandwidth is much smaller in the case of Gd0.5Ca0.5MnO3 [15]. The magnetic and electrical properties of the two crystals are also quite different: Nd0.5Ca0.5MnO3 undergoes a CO transition [56] at TCO=245 K with partial orbital ordering and magnetic correlations of short range. At lower temperatures, the OO increases and a long range CE-type (c.f. above) antiferromagnetic state is established at TN =145 K. An insulator-to-metal transition occurs around this temperature in intermediate magnetic fields [56]. In comparison, Gd0.5Ca0.5MnO3 also shows charge ordering at TCO=260 K, but no long-range antiferromagnetic state is established at low temperature. The magnetic ordering developing at low temperatures in Gd0.5Ca0.5MnO3 seems to remain of short range, as observed for the very similar Y0.5Ca0.5MnO3 [57]. It also remains insulating at all temperatures, even in large magnetic fields [15]. The magnetic properties of the two crystals are illustrated in Fig. 5.26. In Both cases, some irreversibility appears at low temperatures and the ZFC and FC curves deviate from each other; at TN in the case of Nd0.5Ca0.5MnO3. The low temperature increase of the magnetization is attributed to paramagnetic Nd and Gd ions [52]. A Curie Weiss analysis of 1/M for Nd0.5Ca0.5MnO3 can 6 be performed at low temperatures, yielding an effective moment of p=3.4µB, close to the value 3.6 expected for those J=9/2 ions. The low temperature “tail” of the magnetization is also similar to the temperature dependence of moment induced on Nd by the ferromagnetic phase of Nd0.7Sr0.3(Mn,Mg)O3 (c.f. the insert of Fig. 5.25); see also ref. [52]. In Fig. 5.26, the ZFC and FC magnetizations curves of Nd0.5Ca0.5MnO3 and measured on re-heating after cooling to the lowest temperature are shown, using two different cooling rates; a fast (∼60K/min) and a slower (∼ 3K/min), see the caption of the figure. In the case of fast cooling or quench to low temperatures, an excess magnetization ∆M appears below TN , in both the ZFC and FC magnetization curves. Differ- ence plots of the FC and ZFC curves, using the same symbols as in the main frame are added in the upper insert. This excess magnetization ∆M=M(fast cooling)-M(slow cooling) actually appears slightly below TN , around T =130 K (not observed in the case of Gd0.5Ca0.5MnO3). The same measurements were repeated for different orientation of the crystals, yielding similar results [55]. The temperature dependence of the ac-susceptibility of Nd0.5Ca0.5MnO3 and Gd0.5Ca0.5MnO3, recorded in a small oscillatory field, was also measured, as well revealing some excess magnetization in the case of Nd0.5Ca0.5MnO3, as shown in the lower insert of Fig. 5.26. This excess magnetization is thus not driven by the (rather small, H=20 Oe) magnetic field employed in the dc measurements. In our weak probing field of H=20 Oe=1.6 kA/m, ∆M amounts to −5 2.9×10 µB/f.u at T =35 K. M vs H measurements up to higher fields (H=4000 kA/m) recorded after fast and slow cooling to 35 K of Nd0.5Ca0.5MnO3 are shown in Fig. 5.27. In both cases, a linear field dependence of the magnetization is observed, reflecting the low temperature antiferromagnetic order. The difference plot of the two curves (in insert) reveals that the earlier observed excess magnetization corresponds to a weak spontaneous moment of 4.4×10−4

6 2 2 From C = n(µB) µ0p /3kB 98 CHAPTER 5. COLOSSAL MAGNETORESISTIVE MATERIALS

0.5 −3 x 10

2 /f.u)

0.4 b µ

M ( 1 ∆

4.4e−4 µb/f.u 0.3 0

/f.u) 0 400 800 b

µ H (kA/m) M ( 0.2

0.1

Tm=35K 0 0 1000 2000 3000 4000 H (kA/m)

Figure 5.27: M vs. H up to high magnetic ﬁelds recorded after fast (solid line) and slow (dotted line) cooling of Nd0.5Ca0.5MnO3 to T =35 K. The inset shows the corresponding M(fast cooling)-M(slow cooling) diﬀerence plot. M and ∆M are expressed in µB/f.u.

µB/f.u, superposed on a small excess susceptibility. The small moment reflects the presence of uncompensated spins in the antiferromagnetic state. The excess magnetization observed in Nd0.5Ca0.5MnO3 relaxes with time. FC relaxation experiments, in which the FC magnetization is recorded versus time during 10000 s at 35 K after different thermal protocols (A, B, C, and D) are performed. The obtained relaxation curves are plotted in Fig. 5.28. The curves are labeled according to the thermal protocol employed. A: the sample is rapidly cooled (fast cooling rate) to 35 K in H=20 Oe, and after achieving temperature stability (20 s), the magnetization is recorded versus time during 10000 s. B: the sample is rapidly cooled to the lowest temperature (5 K) and the FC relaxation collected after reheating to 35 K. The temperature dependence of the magnetization is recorded during the reheating to 35 K and above, as illustrated in the insert of the figure. C: the sample is rapidly cooled to 35 K, from where the cooling proceeds with a slower rate and the magnetization is recorded on cooling and reheating to 35 K, where the FC relaxation is collected during 10000 s. As in B, the magnetization is further recorded during the reheating to room temperature. D is similar to C, but employing a slower cooling rate when initially cooling to 35 K. Curves A and B are virtually identical, showing that the cooling rate through TN controls the relaxation. As expected, the relaxation diminishes when the effective cooling slows down, from experiments A,B to experiment D. 5.5. CHARGE ORDERED ANTIFERROMAGNETS 99

−3 x 10 1.2 0.04 M/H 0.03

0.8 B A B 0.02 C M(t=20s)/H (SI)

− 0.01

M 0 100 200 300 0.4 T(K)

0 1 2 3 4 10 10 10 10 t (s)

Figure 5.28: FC relaxation curves for Nd0.5Ca0.5MnO3 obtained after diﬀerent cooling protocols A, B, C and D; see main text. Tm=35 K, H= 20 Oe. The insert shows the temperature dependence of FC magnetization in experiment B.

The relaxation at 35 K reflects the evolution of the domain configuration of the AFM state, possibly via the large increase in orthorhombic distortion [15] and c-axis contraction [56] occurring between TCO and TN upon cooling. The observed spontaneous moment could then be related to the presence of defects in the low-temperature antiferromagnetic arrangement or an antiferromagnetic domain state. Uncompensated spins at domain walls would give rise to an excess magnetization. Neutron powder diffraction studies on a similar charge and orbital ordered CE-type AFM manganite [58] indicate some magnetic disorder in the Mn3+ sublattice, associated with domain boundaries breaking the long-range orbital ordering. Recent x-ray scattering results [59] reveal a partial orbital ordering of the low-temperature phase, leading to an orbital domain state. The here observed cooling rate dependence of the magnetization of Nd0.5Ca0.5MnO3 below TN could thus be related to intrinsic inhomogeneities of the CE-type structure and the nucleation or rearrangement of orbital domains and domain walls to accommodate the large contraction of the structure occurring upon cooling. The cooling rate thus determines the time allowed to the system to accommodate the structural modifications governed by the temperature. Relaxation of the ZFC magnetization and resistance of Nd0.5Ca0.5MnO3 has also been observed in large magnetic fields [60]. One can perturb the CE-type antiferromagnetic state by introducing impurities in the structure, for example, by replacing some of the Mn cations by Cr or Ru [61]. It has also been shown that photo-illumination could perturb the orbital ordering of a CO manganite [62] and induce FM-like correlations. 100 CHAPTER 5. COLOSSAL MAGNETORESISTIVE MATERIALS

Another interesting observation is that resistivity of charge-ordered CMR materials is aﬀected by the presence of magnetic inhomogeneities at low temperatures: In Nd0.5Ca0.5MnO3, the low temperature antiferromagnetic state becomes weaker, so that the application of a large magnetic ﬁeld induces an insulator-metal transition near TN . In the case of Gd0.5Ca0.5MnO3, no long range antiferromagnetism is established at any temperature, and neither a cooling rate dependence nor an insulator-metal transition is observed.

17 Tm=21K

16.9

16.8 16 /H (arb. units) Tm=21K ZFC 16.7 M −1 1 3 5 10 10 10 10 t (s)

Tm=28K

/H (arb. units) 15 ZFC M Tm=35K

Tm=42K 14

Tm=50K

13 −1 0 1 2 3 4 5 10 10 10 10 10 10 10 t (s)

Figure 5.29: ZFC relaxation curves of Nd0.5Ca0.5MnO3 recorded at diﬀerent temperatures; a wait time of 1000 s is employed. The insert shows ZFC relaxation curves collected at Tm=21 K for diﬀerent wait times.

Another indication of this correlation is that Tokunaga et al. [63] obtained high quality single crystals of Nd0.5Ca0.5MnO3, which remained insulator at all temperatures. A magnetic ﬁeld larger than 20 Tesla was necessary to induce a metal-insulator transition. It is also mentioned7 in ref. 18 of [60] and in the caption of the third ﬁgure of [15], that the low temperature resistivity critically

7Hard to find 5.5. CHARGE ORDERED ANTIFERROMAGNETS 101 depends on the preparation conditions, indicating again a connection between the occurrence of metallicity and inhomogeneities. In order to investigate further the puzzling time-dependence of the magnetization observed in Nd0.5Ca0.5MnO3, ZFC relaxation experiments are performed. Figure 5.29 shows MZFC(T )/H vs. log(t) for different temperatures; H=2 Oe is employed. Surprisingly, although the irreversibility appears at TN ∼ 140 K, no significant relaxation is observed above T ∼ 55 K; below this temperature, a logarithmic relaxation is observed. In the dilute Ising an- tiferromagnet Fe0.7Mg0.3Cl2 [64], an antiferromagnetic domain state appears at TN ∼ 15 K, and an excess magnetization similar to the one observed in Nd0.5Ca0.5MnO3 appears below this temperature. But for Fe0.7Mg0.3Cl2,a logarithmic relaxation associated with ∆M is observed already at TN . In the case of Nd0.5Ca0.5MnO3, the relaxation increases with decreasing temperature (as in Fe0.7Mg0.3Cl2), but remains small. At T =21 K, ∆MZFC/H = MZFC(t = 10000s)/H-MZFC(t =0.5s)/H amounts to 0.18, which represents about 1% of MZFC at this temperature. Measurements performed at this temperature with H=1 Oe and H=3 Oe essentially yield the same results, con- firming the linear response of the system.

0.025 6

0.02 /H 4 ZFC M −

0.015 /H FC 2 M S (arb. units) 0.01 0 0 40 80 120 160 200 T (K) 0.005

0 0 25 50 75 100 T (K)

Figure 5.30: Temperature dependence of the relaxation rate, derived for the relaxation experiments shown in Fig. 5.29. The insert recalls the temperature dependence of the irreversibility for comparison.

The insert shows relaxation curves obtained at T =21 K for two different wait times. The curves are virtually identical, indicating that there is no col- lective behavior as in glassy systems. Fig. 5.30 shows the temperature dependence of S(T ), estimated from the slopes of the relaxation curves. As mentioned above, S is only significant below 55 K, and then increases rapidly with decreas- 102 CHAPTER 5. COLOSSAL MAGNETORESISTIVE MATERIALS ing temperature. Interestingly, the earlier observed cooling rate dependence in Nd0.5Ca0.5MnO3 (c.f. Fig. 5.26) yields an excess magnetization which appears at TN , and has a maximum around T ∼ 50 K in any orientation of H. This could indicate that the two observations are related. The excess magnetization is associated with defects in the AFM structure and the rearrangement of orbital domains [55]. The magnetic relaxation could then be related to the rearrangement of the domain state and the relaxation of the associated stresses [65]. As shown by Vogt et al. [56], the bond valence sums, calculated from neutron diffraction data, for the two symmetrically independent Mn atoms demonstrate the appearance of charge ordering in Nd0.5Ca0.5MnO3, but it is only at around 50 K (where we obtain the maximum of the excess magnetization and a sizable magnetic relaxation) that one of the Mn attains a bond valence sum of 4 while the other is around 3.

5.6 What is next?

We have seen that CMR manganites can exhibit essentially any kind of magnetic interaction. For use in applications, the most interesting phase is the ferromagnetic one; many groups are thus trying to increase the Curie temperature of the compounds. Also, new materials are appearing. For example, Taguchi et al. [66] have shown that one could tailor the electronic properties of geometrically frustrated pyrochlores such as Nd2Mo2O7. Also, Kobayashi et al. [67] have obtained large room temperature (and above) colossal magnetoresistance effects in the so-called double perovskite (ABO3 → A2BB’O6) Sr2FeMoO6, which has an impressively large Tc of 420 K. It would be of interest to prepare biepitaxial and bicrystal films of Sr2FeMoO6 to see if the low field magnetoresistance in this case can sustain up to room temperature [68]. Dou- ble perovskites are of course good candidates for use in tunneling devices [69] involving the tunneling of electrons between ferromagnetic electrodes (TMR). Bibliography

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This thesis is based on the collection of articles given below. Each article will be referred to in the list of ﬁgures by its capital Roman numeral. Most of the articles can be found in the Los Alamos preprint archive at http : //arXiv.org/find/cond−mat/1/au :+Mathieu R/0/1/0/all/0/1. Only published or accepted articles are included in the list.

I. Ferromagnetism and frustration in Nd0.7Sr0.3MnO3 D. N. H. Nam, R. Mathieu, P. Nordblad, N. V. Khiem, and N. X. Phuc, Phys. Rev. B 62, 1027 (2000).

II. Reentrant spin glass transition in La0.96−yNdyK0.04MnO3: origin and eﬀects on the colossal magnetoresistivity R. Mathieu, P. Svedlindh, and P. Nordblad Europhys. Lett. 52, 441 (2000).

III. Grain boundary eﬀects on magnetotransport in bi-epitaxial ﬁlms of La0.7Sr0.3MnO3 R. Mathieu, P. Svedlindh, R. A. Chakalov, and Z. G. Ivanov Phys. Rev. B 62, 3333 (2000)).

IV. Spin glass dynamics of La0.95Sr0.05CoO3 D. N. H. Nam, R. Mathieu, P. Nordblad, N. V. Khiem and N. X. Phuc Phys. Rev. B 62, 8989 (2000).

V. Structural and magnetic properties of GaMnAs layers with high Mn content grown by Migration Enhanced Epitaxy on GaAs(100) substrates J. Sadowski, R. Mathieu, P. Svedlindh, J. Z. Domagala, J. Bak Mis- iuk, K Swiatek, M. Karlsteen, J. Kanski, L. Ilver, H. Asklund, and U. Sodervall Appl. Phys. Lett. 78, 3271 (2001).

VI. Magnetic contribution to the resistivity noise in a La0.7Sr0.3MnO3 ﬁlm grain boundary R. Mathieu, P. Svedlindh, R. Gurnasson, and Z. G. Ivanov Phys. Rev. B 63, 132407 (2001).

107 VII. Short range ferromagnetism and spin glass state in Y0.7Ca0.3MnO3 R. Mathieu, P. Nordblad, D. N. H. Nam, N. X. Phuc, and N. V. Khiem Phys. Rev. B 63, 174405 (2001).

VIII. Synthesis, crystal structure, and magnetic characterization of the double perovskite Ba2MnWO6 A. K. Azad, S. A. Ivanov, S.- G. Eriksson, J. Eriksen, H. Rundl¨of, R. Mathieu, and P. Svedlindh Mater. Res. Bull. 36, 2215 (2001).

IX. Nuclear and magnetic structure of Ca2MnWO6 - A neutron powder diﬀraction study A. K. Azad, S. A. Ivanov, S.- G. Eriksson, J. Eriksen, H. Rundl¨of, R. Mathieu, and P. Svedlindh Mater. Res. Bull. 36, 2485 (2001).

X. Memory and superposition in a spin glass R. Mathieu, P. J¨onsson, D. N. H. Nam, and P. Nordblad Phys. Rev. B 63, 92401 (2001).

XI. Structural and magnetic properties of the double perovskite Sr2MnWO6 A. K. Azad, S. A. Ivanov, S.- G. Eriksson, J. Eriksen, H. Rundl¨of, R. Mathieu, and P. Svedlindh J. Magn. Magn. Mater. 237, 124 (2001).

XII. Memory and chaos in an Ising spin glass R. Mathieu, P. E. J¨onsson, P. Nordblad, H. Aruga Katori, and A. Ito Phys. Rev. B 65, 012411 (2002).

XIII. Cooling rate dependence of the antiferromagnetic domain structure of a single crystalline charge ordered manganite R. Mathieu, P. Nordblad, A. R. Raju, and C. N. R. Rao Phys. Rev. B 65, 132416 (2002).

XIV. Defect induced magnetic structure in (Ga1−xMnx)As ﬁlms P. A. Korzhavyi, E. A. Smirnova, I. A. Abrikosov, L. Bergqvist, P. Mohn, R. Mathieu, P. Svedlindh, J. Sadowski, E. I. Isaev, Y. K. Vekilov, and O. Eriksson Phys. Rev. Lett. 88, 187202 (2002).

XV. Interlayer exchange coupling and giant magnetoresistance in Fe/V (001) superlattices A. Broddefalk, R. Mathieu, P. Nordblad, P. Blomquist, R. W¨appling, J. Lu, and E. Olsson Phys. Rev. B (accepted). Conference papers:

XVI. Colossal magnetoresistance of La0.96−yNdyK0.04MnO3 R. Mathieu, Y. Guo, P. Svedlindh, P. Nordblad, and R. W¨appling LT22, Helsinki, Finland Physica B 284-288, 1432 (2000).

XVII. Magnetotransport in a bi-crystal ﬁlm of La0.7Sr0.3MnO3 R. Mathieu, P. Svedlindh, R. A. Chakalov, and Z. G. Ivanov ICM 2000, Recife, Brazil J. Magn. Magn. Mater. 226, 786 (2001).

XVIII. Magnetic ageing and non-equilibrium dynamics in Y0.7Ca0.3MnO3 D. N. H. Nam, R. Mathieu, P. Nordblad, N. X. Phuc, and N. V. Khiem ICM 2000, Recife, Brazil J. Magn. Magn. Mater. 226, 1335 (2001).

XIX. Eﬀects of Mg doping in Nd0.7Sr0.3Mn1−yMgyO3 D. N. H. Nam, R. Mathieu, P. Nordblad, N. V. Khiem, and N. X. Phuc ICM 2000, Recife, Brazil J. Magn. Magn. Mater. 226, 1340 (2001).

XX. Properties of GaMnAs layers grown by migration enhanced epitaxy at very low substrate temperatures J. Sadowski, R. Mathieu, P. Svedlindh, M. Karlsteen, J. Kanski, L. Ilver, H. Asklund, K. Switek, J. Z. Domagala , J. Bk-Misiuk, and D. Maude PASPS 2000, Sendai, Japan Physica E 10, 181 (2001).

All figures included in this thesis, but figures 3.3 (bottom), 4.6, 5.4, 5.5 and 5.6, are taken, as is or modified, from articles co-authored by the author, as well as from unpublished data. All published figures are reprinted with permission of the authors and from the different journals in which they have appeared. The figures are listed in the next pages. The roman number of the paper they refer to is included.

List of Figures

2.1 Hysteresis measurement on a thin ﬁlm of Nickel...... 13 2.2 M vs. H for Fe/V superlattice. Adapted from paper XV .... 14 2.3 M vs. T for Ag(11 at % Mn). Adapted from paper X ...... 15 2.4 Relaxation curves at a constant temperature for Ag(11 at % Mn). 16 2.5 R vs. H for Fe/V superlattices. Adapted from paper XV .... 17 2.6 R vs. H measurements showing hysteresis. Adapted from papers III and XVII ...... 18

3.1 M vs. T for single layers of (Ga,Mn)As. Adapted from paper V 24 3.2 M vs. T for (Ga,Mn)As layers with Mn > 8%. Adapted from paper V ...... 25 3.3 Top: M vs. T for thin single layers of (Ga,Mn)As. Bottom: Temperature dependence of SMn and sh as calculated by Berciu et al. in “M. Berciu and R. N. Bhatt, Phys. Rev. Lett. 87, 107203 (2001)”...... 26 3.4 Mn concentration dependence of MS of (Ga,Mn)As layers. Taken from paper XIV ...... 27 3.5 Calculated (by P. Korzhavyi) total energy as function of Mtot/|Mloc| for diﬀerent antisites concentrations. Taken from paper XIV .. 28 3.6 Calculated (by P. Korzhavyi) Mtot and Mloc dependences on antisite concentrations. Taken from paper XIV ...... 29 3.7 M vs. T for (Ga,Mn)As/GaAs superlattices...... 31 3.8 M vs. T for (Ga,Mn)As/GaAs superlattices with a varying number of GaAs monolayers...... 32 3.9 Resistivity and magnetoresistance for thick layers of (Ga,Mn)As. 33 3.10 Resistivity and magnetoresistance of (Ga,Mn)As/GaAs superlattices...... 34

4.1 Examples of frustration in ensembles of spins...... 39 4.2 Temperature dependence of the ac-susceptibility of Fe0.5Mn0.5Ti- O3 for diﬀerent frequencies...... 41 4.3 Wait time dependence of the ZFC relaxation in Ag(11 at % Mn). 45 4.4 ZFC and IRM relaxation curves for diﬀerent te; the ZFC+IRM sum is plotted versus t + te...... 46 4.5 Results of single and double ac-memory experiments on Cu(13.5 at % Mn)...... 48

111 4.6 Negative and positive temperature cycling experiments on (Fe0.8- Ni0.2)75P16B6Al3, as shown in “K. Jonason and P. Nordblad, Eur. Phys. J. B 10, 23 (1999)”...... 49 4.7 Negative temperature cyclings on Ag(11 at % Mn) for different tw2...... 50 4.8 Temperature dependence of the magnetization of Ag(11 at % Mn) recorded after direct cooling, and after a temperature stop. Adapted from paper X ...... 51 4.9 Relaxation curves for different wait times in Ag(11 at % Mn). . 52 4.10 Temperature dependence of magnetizations of Ag(11 at % Mn) measured after using different cooling protocols. Taken from paper X ...... 53 4.11 Difference plots corresponding to single-stop and double-stop experiments for the Fe0.5Mn0.5TiO3 spin glass. Taken from paper XII 55 4.12 Results of single temperature stop experiments in the Ag(11 at % Mn) and Fe0.5Mn0.5TiO3 spin glasses...... 56 4.13 Relaxation rate after temperature shifts of different signs in Fe0.5Mn0.5TiO3...... 57 4.14 teff /tw as a function of the magnitude of the temperature shift in the Ag(11 at % Mn) and Fe0.5Mn0.5TiO3 spin glasses. .... 58 4.15 Results of zero field stop experiments in the TRM magnetization of La0.95Sr0.05CoO3. Taken from paper IV ...... 59 4.16 Results of temperature stops experiments on a Bi2Sr2CaCu2O8 superconductor showing a glassy PME state...... 60

5.1 Schematic view of the ABO3 perovskite structure...... 67 5.2 Temperature dependence of the magnetization and electrical resistivity, for Nd0.7Sr0.3MnO3...... 68 5.3 Schematic description of the field splitting of the degenerated 3d-orbitals...... 70 5.4 Magnetic and electrical properties of CMR manganites as a function of doping x and bandwidth W , as shown in “R. Kajimoto, H. Yoshizawa, Y. Tomioka, and Y. Tokura, cond-mat/0110170 ”. 71 5.5 Magnetic configurations of (Nd,Sr)MnO3 as a function of the Sr doping, as shown in “T. Okuda, T. Kimura, H. Kuwahara, Y. Tomioka, A. Asamitsu, Y. Okimoto, E. Saitoh, and Y. Tokura, Mater. Sci. Eng. B 63, 163170 (1999)”...... 72 5.6 Schematic representation of the CE-type ordering of charges, orbitals and spins, as drawn by “D. I. Khomskii, cond-mat/0104517 ”. 73 5.7 M vs. T for an epitaxial film of La0.7Sr0.3Mn03; the insert shows an hysteresis measurement. Adapted from paper III ...... 75 5.8 Hysteresis curves for thin films of La0.7Sr0.3Mn03 containing larger and larger amounts of grain boundaries. Taken from paper III ...... 76 5.9 Schematic representation of the different layers of a La0.7Sr0.3Mn03 biepitaxial structure. Taken from paper III ...... 77 5.10 Temperature dependence of the resistivity and magnetoresistance of a biepitaxial film of La0.7Sr0.3Mn03. Adapted from paper III ...... 78 5.11 Schematic representation of a patterned bicrystal film...... 80 5.12 Temperature dependence of the magnetoresistance for bicrystal films of La0.7Sr0.3Mn03. Adapted from paper XVII ...... 81 5.13 Dc resistance and noise level vs. magnetic field in a bicrystal film of La0.7Sr0.3Mn03. Adapted from paper VI ...... 82 5.14 Frequency dependence of the excess noise in a bicrystal film of La0.7Sr0.3Mn03. Adapted from paper VI ...... 83 5.15 ZFC relaxation curves for Nd0.7Sr0.3Mn03, measured at low and high temperatures. Adapted from paper I ...... 84 5.16 Relaxation rate for different magnetic field strength in Nd0.7Sr0.3- Mn03. Adapted from paper I ...... 85 5.17 Negative and positive temperature cyclings performed on Nd0.7- Sr0.3Mn03. Adapted from paper I ...... 86 5.18 Ac-susceptibility vs. temperature for different dopings of La0.96- K0.04MnO3. Adapted from paper II ...... 87 5.19 Zero magnetic field resistivity and high field magnetoresistance for La0.96K0.04MnO3. Taken from paper II ...... 88 5.20 ZFC relaxation curves of La0.96K0.04MnO3 for different wait times and magnetic fields. Adapted from paper II ...... 89 5.21 Negative and positive temperature cyclings performed on Y0.7- Ca0.3MnO3. Adapted from paper VII ...... 91 5.22 Temperature dependence of the ac-susceptibility of Y0.7Ca0.3Mn- O3 for different frequencies and different superimposed dc-fields. Adapted from paper VII ...... 92 5.23 Results of double ac-memory experiments on Y0.7Ca0.3MnO3. Taken from paper VII ...... 93 5.24 Temperature dependence of ac-susceptibity for (La,Y)0.7Ca0.3- MnO3...... 94 5.25 Neutron spectra for Nd0.7Sr0.3(Mn,Mg)O3...... 95 5.26 Temperature dependence of the ZFC and FC magnetization of Nd0.5Ca0.5MnO3 and Gd0.5Ca0.5MnO3. Adapted from paper XIII 96 5.27 M vs. H for Nd0.5Ca0.5MnO3 recorded after slow and fast cooling. Taken from paper XIII ...... 98 5.28 FC relaxation curves for Nd0.5Ca0.5MnO3 obtained after different cooling protocols. Adapted from paper XIII ...... 99 5.29 ZFC relaxation curves of Nd0.5Ca0.5MnO3 recorded at different temperatures...... 100 5.30 Temperature dependence of the relaxation rate derived for the relaxation experiments shown in the previous figure...... 101

Acknowledgments

irst, I would like to thank Professor Per Nordblad for his excellent super- Fvision. Thank you for always finding time to discuss, or more correctly, for always finding time to explain things to me. I also thank Professors C. G. Granqvist and P. Svedlindh for accepting me as a PhD student in the Solid State Physics group. None of the low temperature measurements I have performed would have been possible without liquid helium, so thank you Mr. Erland Falk for producing the coldest helium. I would like to thank all the people I have been working with, and mainly: Dr. D. N. H. Nam, Dr. N. V. Khiem and Prof. N. X. Phuc (Hanoi), Prof. C. N. R. Rao (Bangalore), Dr. R. Chakalov and Prof. Z. Ivanov (Göteborg), Dr.-to-be T. Eriksson, Dr.-to-be M. Vennström, J. Karjalainen, Prof. Y. Andersson and Prof. R. Tellgren (Uppsala), Dr.-to-be M. Valkepää, Dr.-to-be Md. A. K. Azad, Dr. S. Ivanov and Prof. S. Eriksson (Studsvik - Göteborg), Dr. J. Sadowski and Dr. J. Kanski (Göteborg-Lund), Dr. P. Korzhavi, Dr.-to-be L. Bergqvist, Prof. O. Eriksson, Prof. I. Abrikosov (Uppsala) and Prof. P. Mohn (Uppsala - Vienna), Dr. H. Aruga Katori and Prof. A. Ito (Tokyo), Dr. P. Blomqvist and Prof. R. Wäppling (Uppsala), Dr. J. Lu and Prof. E. Olsson (Uppsala - Göteborg), as well as E. Bilius, S. Zhao and B. Götesson (Uppsala). The travel support for myself and P. Jönsson to Japan from “Anna Maria Lundins Stipendiefond” is gratefully acknowledged. Many thanks to our kind hosts: Dr. H. Yoshino and Prof. H. Kawamura (Osaka), Dr. K. Hukushima and Prof. H. Takayama (Tokyo), Prof. A. Asamitsu and Prof. Y. Tokura (Tokyo), and our dear friend Prof. T. Sato (Keio). I also thank everyone from the Solid State Physics department as well as some former members of the group: Arvid, Kristian, Tomas, Per G., Laszlo, Evie, Yang, Tuquabo, Mghendi, Björn, Monica and José, Ralph and Tanja, Juan, Mikkel, Feroz, José and Pilar. Thank you Ingrid Ring˚ard for always finding time to help me. I thank my family for always supporting me and of course thank you Petra for bringing me to Sweden! This work has partially been financed by the Swedish Research Council. Uppsala, April 2002 Roland Mathieu

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