Novel complex materials under pressure : Novos materiais complexos sob pressão

Instituto de Física Gleb Wataghin

Mario Moda Piva

Novel complex materials under pressure

Novos materiais complexos sob pressão

CAMPINAS

2019

Novel complex materials under pressure

Novos materiais complexos sob pressão

Thesis

presented

to

the

Gleb

Wataghin Institute of Physics of

the University of Campinas in partial

fulllment of the requirements for the

degree of Doctor of Sciences.

Tese apresentada ao Instituto de Física

Gleb Wataghin da Universidade

Es-tadual de Campinas como parte dos

req-uisitos exigidos para a obtenção do

tí-tulo de Doutor em Ciências.

Advisor/Orientador: Prof. Dr. Pascoal José Giglio Pagliuso

Este exemplar corresponde à versão nal da Tese de Doutorado defendida pelo aluno Mario Moda Piva, orientado pelo Prof. Dr. Pascoal José Giglio Pagliuso

CAMPINAS

2019

Biblioteca do Instituto de Física Gleb Wataghin Lucimeire de Oliveira Silva da Rocha - CRB 8/9174

Piva, Mario Moda,

P688n PivNovel complex materials under pressure / Mario Moda Piva. – Campinas,

SP : [s.n.], 2019.

PivOrientador: Pascoal José Giglio Pagliuso.

PivTese (doutorado) – Universidade Estadual de Campinas, Instituto de Física

Gleb Wataghin.

Piv1. Magnetismo. 2. Supercondutividade. 3. Experimentos de altas pressões.

I. Pagliuso, Pascoal José Giglio, 1971-. II. Universidade Estadual de Campinas. Instituto de Física Gleb Wataghin. III. Título.

Informações para Biblioteca Digital

Título em outro idioma: Novos materiais complexos sob pressão Palavras-chave em inglês:

Magnetism Superconductivity

High pressure experiments Área de concentração: Física Titulação: Doutor em Ciências Banca examinadora:

Pascoal José Giglio Pagliuso [Orientador] Eduardo Miranda

Narcizo Marques de Souza Neto Raimundo Lora Serrano

Marcos de Abreu Avila Data de defesa: 06-12-2019

Programa de Pós-Graduação: Física

Identificação e informações acadêmicas do(a) aluno(a)

- ORCID do autor: https://orcid.org/0000-0003-4744-3397 - Currículo Lattes do autor: http://lattes.cnpq.br/6633919599667720

MEMBROS DA COMISSÃO JULGADORA DA TESE DE DOUTORADO DE MARIO MODA PIVA – RA 118069 APRESENTADA E APROVADA AO INSTITUTO DE FÍSICA

“GLEB WATAGHIN”, DA UNIVERSIDADE ESTADUAL DE CAMPINAS,

EM 06 / 12 / 2019.

COMISSÃO JULGADORA:

- Prof. Dr. Pascoal José Giglio Pagliuso – Orientador – DEQ/IFGW/UNICAMP

- Prof. Dr. Eduardo Miranda – DFMC/IFGW/UNICAMP

- Prof. Dr. Narcizo Marques de Souza Neto – CNPEM

- Prof. Dr. Raimundo Lora Serrano – IF/UFU

- Prof. Dr. Marcos de Abreu Avila – CCNH/UFABC

OBS.: Ata da defesa com as respectivas assinaturas dos membros encontra-se no

SIGA/Sistema de Fluxo de Dissertação/Tese e na Secretaria do Programa da

Unidade.

CAMPINAS

2019

Acknowledgments

I would like to thank my parents, Maria Elvira and Marcos, for always being by my side. Your constant support was always fundamental in my life.

I would like to sincerely thank my wife, Tamaira, for your companionship, always helping me and supporting me in all my achievements since 2012. I am very grateful to your understanding and patience in the periods that I was far away. You are crucial in my life, and therefore, crucial to the elaboration and conclusion of my PhD.

I would like to thank my advisor, Pascoal, for his commitment and determi-nation in my academic and professional education. Your ethics, honesty and excellence are examples that I will carry for all my life. Finally, I also appreciate the comprehension and patience in times of excessive anxiety of my part.

I would like to thank my friends Lucas, Fernando, Samara and Samuel. Your friendships are invaluable and I always have a good time when we meet.

I am thankful to my capixaba friends, Laiza, Rodolfo and Gabriel for our fun conversations and events that make life happier. Undoubtedly, your friendships are the best discoveries of my PhD.

I would like to also thank my lab colleagues Cris, Rettori, Granado, Davi, Lina, Ricardo, Camilo, Denise, Jean, Kevin, Paulo and Rogério, for our friendship and for the advices provided throughout my PhD.

I am also thankful for the support of the lab technicians Zairo, Milton, Ailton, Renato and Gustavo.

I would like to thank Ricardo Reis, Ajeesh and Michael for the help, kindness and hospitality during my stay in Dresden.

I would like to thank my LANL colleagues Soonbeom, Marein, Tomo, Eric, Filip, Satya, Roman, Sean, Priscila and Joe for the useful discussions. I would like to give special thanks to Priscila, Sean and Joe for your friendship, kindness and hospitality.

2017/10581-1. We note that the opinions, hypotheses and conclusions or recommendations expressed in this material are responsibility of the author and do not necessarily reect the opinion of FAPESP.

This study was nanced in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

Resumo

Vários compostos com estados fundamentais inéditos, como supercondutivi-dade não-convencional, estruturas magnéticas complexas e estados não-líquidos de Fermi cristalizam em estruturas com camadas. Além disso, esses estados fundamentais po-dem ser controlados por substituição química, campos magnéticos e aplicação de pressão hidrostática. Em particular, a aplicação de pressão é um parâmetro de controle ideal, já que nenhuma desordem química é introduzida no sistema. Portanto, a aplicação de pressão externa é uma rota desejável para induzir estados fundamentais não-usuais. Nessa tese, nós relatamos a resposta sob pressão das propriedades dos materiais

antiferromag-néticos com camadas: CeY Bi2 (Y = Cu e Au), CeAu2Bi, CaMn2Bi2 e Sr(Fe1−xCox)2As2.

Monocristais desses compostos foram sintetizados pela técnica de uxo metálico. A fase sintetizada foi caracterizada por experimentos de difração de raios-x, EDX, susceptibili-dade magnética, calor especíco e resistivisusceptibili-dade elétrica. Medidas de resistivisusceptibili-dade elétrica e calorimetria AC sob pressão foram realizadas para estudar a evolução das propriedades

em função da pressão. O ordenamento antiferromagnético no CeCuBi2 foi suprimido

com a aplicação de pressão e um estado de resistência nula (ZRS) foi encontrado em

altas pressões. Para o CeAuBi2 a aplicação de pressão externa favoreceu o

antiferro-magnetismo (AF) e um ZRS foi encontrado novamente com temperaturas de transição

similares ao CeCuBi2 e em pressões similares. Além disso, CeAu2Bi e CaMn2Bi2 também

mostraram um aumento da temperatura de Nèel (TN) em função do aumento da pressão.

A aplicação de pressão também aumentou a energia de ativação do CaMn2Bi2. Por m,

Sr(Fe1−xCox)2As2 apresentou uma supressão da fase do tipo onda de densidade de spin

(SDW) junto com o aparecimento de um estado supercondutor com o aumento da pressão e/ou concentração de Co. Em todos os casos, os experimentos dependentes da pressão foram aplicados com sucesso para extrair parâmetros relevantes e forneceram novas per-cepções aos fenômenos físicos fundamentais dos sistemas estudados. Essas ideias serão discutidas, especicamente, para cada família de compostos ao longo da tese e resumidas no capítulo 5, Conclusões e Perspectivas.

Abstract

Many compounds with novel ground states, such as unconventional supercon-ductivity, complex magnetic structures and non-Fermi-liquid states crystallize in layered structures. Moreover, these ground states can be tuned by chemical substitution, mag-netic elds and applied hydrostatic pressure. In particular, applied pressure is an ideal tuning parameter, as there is no chemical disorder introduced in the system. Therefore, the application of external pressure is a desirable route to induce unusual ground states. In this thesis, we report the pressure dependence of the properties of the antiferromagnetic

layered materials: CeY Bi2 (Y = Cu and Au), CeAu2Bi, CaMn2Bi2 and Sr(Fe1−xCox)2As2.

Single crystals of these compounds were synthesized by the metallic-ux technique. The synthesized phase was characterized by x-ray diraction, EDX, magnetic susceptibility, specic heat and electrical resistivity measurements at ambient pressure. Electrical re-sistivity and AC calorimetry experiments under pressure were performed to study the evolution of the properties as a function of pressure. The antiferromagnetic order in

CeCuBi2 was suppressed with the application of external pressure and a zero resistance

state (ZRS) was found at high pressures. For CeAuBi2the application of external pressure

favored the antiferromagnetism (AF) and a ZRS was again found with similar transition

temperatures to CeCuBi2 and at similar pressures. Moreover, CeAu2Bi and CaMn2Bi2

have also shown an enhancement of the Nèel temperature (TN) as a function of

increas-ing pressure. In addition, applied pressure enhanced the activation energy of CaMn2Bi2.

Finally, Sr(Fe1−xCox)2As2 presented a suppression of the spin-density-wave (SDW) phase

along with the appearance of a superconducting state with increasing pressure and/or Co concentration. In all cases, the pressure dependent experiments were successfully ap-plied to extract relevant parameters and provided new insights to the underlying physical phenomena of the studied systems. These insights will be discussed, specically, for each family of compounds throughout the thesis and summarized in chapter 5, Conclusions and Perspectives.

List of Figures

1.1 Superconducting transition temperatures as a function of the discovery year, extracted from [1]. Note the similar crystalline structures for dierent

families of superconductors. . . 17

1.2 (a) Crystalline structure for selected cuprate superconductors, adapted from [6]. (b) Temperature-composition phase diagram for La1−xSrxCuO4, extracted from [7]. (c) Crystalline structures of CeIn3, CeMIn5and Ce2MIn8, extracted from [8]. (d) Temperature-pressure phase diagram for CeRhIn5 extracted from [4]. (e) Crystalline structure of selected iron based super-conductors, extracted from [9]. (f) Temperature-pressure phase diagram for SrFe2As2, extracted from [5]. . . 19

1.3 Crystalline structures of CeY Bi2(Y = Cu or Au) (a), CeAu2Bi (b), CaMn2Bi2 (c) and SrFe2As2 (d). . . 20

2.1 (a) Aluminum electronic bands as a function of compression. (b) Radial wavefunction Rnl as a function of r with bonding (dotted curves) and an-tibonding (dashed curves) for three dierent Wigner-Seitz radii S. Both gures were extracted from [20]. . . 24

2.2 Ground states of ions with partially lled d and f shells, extracted from [21]. 31 2.3 RKKY exchange parameter (JRKKY) as a function of r. . . 41

2.4 Schematic picture of a ferromagnetic transition. . . 42

2.5 Magnetization as a function of x at three dierent temperatures. . . 44

2.6 Schematic picture of an antiferromagnetic transition. . . 45

2.7 Magnetic susceptibility of an antiferromagnetic system with applied mag-netic elds in dierent directions. . . 47

2.9 Magnetic susceptibility as a function of temperature for: (a) Pauli param-agnetism and diamparam-agnetism; (b) Curie paramparam-agnetism; (c) Ferromparam-agnetism and (d) Antiferromagnetism. Adapted from [23]. . . 49 2.10 (a) Commensurate and (b) Incommensurate spin-density waves. Extracted

from [24]. . . 51 2.11 Fundamental properties of superconductors: (a) innite conductivity and

(b) the Meissner eect, extracted from [25, 26], respectively. . . 53

2.12 Microscopic magnetic eld (h0_{) as a function of appled magnetic eld for}

(a) type I and (b) type II superconductors. . . 57 2.13 Schematic explanation of the attractive potential between two electrons. . . 60 2.14 Interaction potential for a (a) spin triplet and (b) spin singlet wavefunctions

between two electrons, extracted from [30]. . . 65

2.15 Radial charge distribution for the Ce ion in the conguration 4f15d16s2

adapted from [32]. . . 66 2.16 (a) CEF splitting of the p orbitals for a L = 1 ion in an uniaxial positive

CEF environment, adapted from [23]. (b) CEF scheme for a Fe2+ _{ion. . . 67}

2.17 CEF splitting for a Mn2+ ion (a) and for a Ce3+ ion (b), adapted from [24]. 69

2.18 (a) Interaction of magnetic impurities and the conduction electrons at high (a) and low (b) temperatures. (c) Eletrical resistivity of gold with iron impurities, extracted from [35]. . . 72 2.19 (a) Kondo lattice. (b) Electrical resistivity for CeCu6, adapted from [36]. . 74 2.20 Doniach phase diagram, extracted from [37]. . . 75 3.1 Schematic representation of the quartz tube (a) and the thermal treatment

(b) used in the metallic ux technique. . . 77 3.2 (a) Schematic representation of a single crystal with six contact leads for

electrical and Hall resistivity measurements. (b) Electrical resistivity puck for a PPMS system, adapted from [42]. . . 80 3.3 Schematic representation of a MPMS SQUID magnetometer, adapted from

[43]. . . 82 3.4 (a) Exploded view of a PPMS specic heat puck. (b) Side view of a PPMS

3.5 (a) Schematic view of the components of a commercial piston cylinder cell (Pcell 30) from Almax EasyLab, adapted from the cell manual. . . 84 3.6 Schematic representations of a mounted cell from Almax Easy Lab (a),

extracted from the cell application note, and C&T Factory (b), extracted from [45]. Pictures of the pressure cells sold by Almax Easy Lab (c) and C&T Factory (d). . . 85 3.7 (a) Schematic representations of a Bridgman cell. (b) Picture of the parts

of a Bridgman cell, extracted from [37]. . . 86 3.8 Picture of an electrical feedthrough for piston cylinder type cells. . . 87 3.9 Assembled bottom anvils for Bridgman cells with solid (a) and (b) liquid

pressure transmitting media. . . 87

3.10 (a) ω|TAC|as a function of the modulation frequency, adapted from [47]. (b)

Schematic representation of the sample assembly used in AC calorimetry experiments under pressure. . . 90

4.1 Crystalline structure shared by CeCuBi2 and LaCuBi2. . . 93

4.2 (a) Electrical resistivity and (b) temperature dependence of the specic

heat versus temperature for CeCuBi2 and LaCuBi2. The insets show the

low temperature regions. . . 93 4.3 Normalized electrical resistivity as a function of temperature for LaCuBi2

(a) and CeCuBi2 (b) for several pressures. Insets show a low temperature

zoom of ρxx(T )/ρxx(20 K). . . 94

4.4 (a) Temperature-pressure phase diagram for RCuBi2 (R = La, Ce). Run

#1 was carried out using a piston cylinder type pressure cell and run #2 using a Bridgman cell. AFM and ZRS denote antiferromagnetic and zero

resistance state, respectively. (b)Hc2 as a function of pressure. Open

sym-bols are for H parallel to the ab-plane and solid symsym-bols are for H parallel to c-axis. The dashed lines are a guide to the eyes. . . 95

4.5 (a) Field dependence of ρxx(H)/ρxx(4 T) for CeCuBi2, with H parallel to

the c-axis. (b) Hf lip-pressure phase diagram. Open symbols are used for measurements carried out at 1.5 K, solid ones for experiments at 2.0 K. . . 96 4.6 (a)Crystalline structure of CeAuBi2. (b) Specic heat as a function of

4.7 Magnetic susceptibility of CeAuBi2 as a function of temperature for elds parallel (a) and perpendicular (b) to the c-axis. (c) Magnetization as a function of applied magnetic eld. The solid red lines are ts using a CEF mean eld model. . . 98 4.8 (a) Magnetization as a function of applied magnetic eld at several

tem-peratures. Open symbols represent decreasing magnetic eld. (c)

Field-temperature phase diagram of the metamagnetic transitions of CeAuBi2. . 100

4.9 (a) AC calorimetry as a function of temperature at several pressures. (b) Electrical resistivity as a function of temperature at several pressures. (c) Temperature-pressure phase diagram for CeAuBi2. . . 101 4.10 Magnetoresistance and Hall resistivity as a function of applied magnetic

eld for several pressures at three dierent temperatures. The solid orange lines are two-band model ts. . . 102 4.11 Electrical resistivity (a) and AC calorimetry (b) as a function of

tempera-ture at 25 kbar for CeAuBi2. . . 103

4.12 (a) Hexagonal structure of CeAu2Bi. (b) c-axis view of the crystalline

structure of CeAu2Bi displaying the triangular lattice formed by Ce3+ ions. 104

4.13 (a) χ(T ) at µ0H = 0.1 T. The top inset shows a zoomed-in view of χ(T )

in the low-temperature range. The bottom insets shows (χ − χ0)−1 as a

function of temperature. (b) M as a function of applied magnetic eld. The solid red lines are ts using a CEF mean eld model. . . 105

4.14 (a) Specic heat measurements for CeAu2Bi and LaAu2Bi. The insets show

cp/T as a function of T2. The solid line is an extrapolation of the t used to

extract the Sommerfeld coecient. (b) Magnetic specic heat and entropy of CeAu2Bi. The solid red line is a t using a CEF mean eld model. . . . 107 4.15 (a) ρ(T ) for CeAu2Bi and LaAu2Bi. (b) ρ(T ) of CeAu2Bi for dierent

applied pressures. The insets show the low-temperature range of the re-sistivity for several applied pressures and the temperature-pressure phase diagram, respectively. . . 108

4.16 (a) Specic heat behavior as a function of temperature of CeAu2Bi for

several applied magnetic elds. (b) CeAu2Bi electrical resistivity for several applied magnetic elds. . . 109

4.17 (a) Comparison between the crystalline structure of CaMn2Bi2and BaFe2As2.

(b) c-axis view of the crystalline structure of CaMn2Bi2 and BaFe2As2. . . 109

4.18 (a) Magnetic susceptibility as a function of temperature. (b) Specic heat

as a function of temperature. The inset shows cp/T as a function of T2_{, at}

low temperatures. . . 110 4.19 (a) Electrical resistivity at several pressures. The inset shows the

antifer-romagnetic transition. (b) Natural logarithm of ρxx as a function of 1/T .

The red lines are the extrapolation of linear ts. . . 111 4.20 Pressure dependence of the antiferromagnetic transition temperature (a)

and the activation energy (b). Solid lines are linear ts. . . 112 4.21 Temperature dependence of the activation energy (a) and carrier mobility

(b) of CaMn2Bi2 at several pressures. Solid lines are linear ts. . . 113

4.22 Hall resistivity as a function of applied magnetic eld for several pressures at 200 K, 100 K and 10 K, the red lines are two-band model ts. The inset shows the c exponents of allometric ts. . . 114 4.23 X-ray powder diraction for selected samples of Sr(Fe1−xCox)2As2. . . 115 4.24 (a) Dependence of the lattice parameter c as a function of Co concentration

(x). The dashed gray line is a guide to the eyes. (b) Specic heat measure-ments as a function of temperature for selected samples of Sr(Fe1−xCox)2As2.116 4.25 (a) Magnetic susceptibility as a function of temperature for selected samples

of Sr(Fe1−xCox)2As2. (b) Normalized zero-eld cooling and eld cooling

curves for Sr(Fe1−xCox)2As2. . . 117 4.26 Electrical resistivity as a function of temperature (a) and

temperature-composition phase diagram for Sr(Fe1−xCox)2As2. . . 118

4.27 (a) AC calorimetry as a function of temperature for SrFe2As2 at several

pressures. (b) Temperature-pressure phase diagram obtained with the AC calorimetry data. . . 119 4.28 (a) Electrical resistivity as a function of temperature for x = 0 (a) and

x = 0.03(b) at several pressures. Temperature-pressure phase diagram for

4.29 (a) Electrical resistivity as a function of temperature for x = 0.12 (a) and

x = 0.16(b) at several pressures. Temperature-pressure phase diagram for

x = 0.12 (c) and x = 0.16 (d). Solid lines for TS and TSDW are linear ts. . 121

4.30 (a) Electrical resistivity as a function of temperature for x = 0.23 (a) and

x = 0.32(b) at several pressures. Temperature-pressure phase diagram for

x = 0.23 (c) and x = 0.32 (d). Solid lines for TS and TSDW are linear ts. . 123

4.31 (a) Derivative of the electrical resistivity as a function of temperature for

x = 0 at several pressures, obtained using the Bridgman cell. The

in-set shows a zoomed-in view of the low temperature range. Temperature-pressure phase diagram for x = 0. . . 124

4.32 Temperature-pressure phase diagram for SrFe2As2 considering that 0.12 Co

Contents

1 Introduction 17

2 Theoretical Aspects 22

2.1 Eects of Pressure in Solids . . . 22

2.2 Magnetism . . . 24

2.2.1 Diamagnetism and Paramagnetism . . . 28

2.3 Magnetic Ordering . . . 37 2.3.1 Magnetic Interactions . . . 37 2.3.2 Ferromagnetism . . . 42 2.3.3 Antiferromagnetism . . . 45 2.3.4 Spin-Density Waves . . . 49 2.4 Superconductivity . . . 52 2.4.1 Conventional Superconductivity . . . 53 2.4.2 Unconventional Superconductivity . . . 63 2.5 Rare-Earth Ions . . . 65

2.6 Crystal Field Eects . . . 66

2.7 Kondo Eect . . . 71

3 Experimental Details 76 3.1 Single crystal synthesis . . . 76

3.2 Synthesized phase structural and elemental characterization . . . 79

3.3 Transport measurements at ambient pressure . . . 79

3.4 Magnetic susceptibility and magnetization measurements . . . 81

3.5 Specic heat measurements . . . 82

3.6.1 Electrical resistivity experiments under pressure . . . 86

3.6.2 AC calorimetry under pressure . . . 89

4 Results and Discussion 92 4.1 Ce-based compounds . . . 92

4.1.1 CeCuBi2 and LaCuBi2 . . . 92

4.1.2 CeAuBi2 . . . 97

4.1.3 CeAu2Bi . . . 104

4.2 CaMn2Bi2 . . . 109

4.3 Sr(Fe1−xCox)2As2 . . . 115

5 Conclusions and Perspectives 127

Bibliography 131

Chapter 1

Introduction

One of the main goals of condensed matter research is to achieve unique phases that will enable new technologies and enhance our knowledge. Among several physical phenomena, superconductivity has unique potential to technological applications, due to the possibility of conducting electricity without losses, and it still possesses open ques-tions, such as "What is the microscopic theory for unconventional superconductivity?". Nevertheless, superconductors are already applied on our daily lives in magnetic

reso-Figure 1.1: Superconducting transition temperatures as a function of the discovery

year, extracted from [1]. Note the similar crystalline structures for dierent families of superconductors.

nance imaging or in some prototypes of levitating trains. However, one major issue is the low superconducting transition temperatures (Tc) present in current superconductors. Therefore, the search for Tc's of the order of room temperature has driven the attention of the condensed matter community in the last decades. One path to reach higher Tc's is

the synthesis of new materials. Figure 1.1 presents the Tcas a function of the discovery's

year of new superconducting materials. At ambient pressure, the highest Tc is 135 K,

which is much lower than room temperature. Another way to increase Tc is to favor the

superconducting state by using external parameters, such as chemical doping, applied pressure or strain. The application of external pressure is a powerful tuning parameter as no chemical disorder is introduced in the system. As we can see in Figure 1.1 the

application of external pressure raised Tc to 165 K for HgBaCaCuO. Moreover, Tc's as

high as 203 K and 260 K for H3S and LaH10, respectively, at pressures close to 1500 kbar have been recently reported [2, 3].

Remarkably many superconducting compounds present layered crystalline struc-tures, as clearly evidenced in the right side of Figure 1.1. In particular all cuprates present Cu-O layers in their crystalline structure as better displayed in Figure 1.2 (a). Moreover, they share a temperature-composition phase diagram, in which an antiferromagnetic insu-lator is driven into a superconducting metal by chemical doping, as presented in Figure 1.2 (b). Similar layered crystalline structures are observed for the well-known heavy-fermion

superconductors CeIn3, CeMIn5 (M = Co, Rh, Ir) and Ce2MIn8 (M = Co, Rh, Pd) [4],

as presented in Figure 1.2(c). Also, a similar phase diagram occurs in CeRhIn5, for in-stance, by the application of external pressure the antiferromagnetism is suppressed and a superconducting state takes place, as can be seen in Figure 1.2(c). Finally, the iron based superconductors also share layered crystalline structure, presented in Figure 1.2(e), with Fe-As or Fe-Se layers. Again, the suppression of an antiferromagnetic phase by chemi-cal doping or pressure leads to a superconducting state, as displayed in Figure 1.2(f) for

SrFe2As2 [5].

In this regard, the large similarity between the crystalline structures of many novel materials may be a strong indication that layered crystalline structures are prone to the appearance of unique ground states. Therefore, aiming to improve our knowledge on complex phases, such as magnetism and superconductivity, we performed electrical resistivity and/or AC calorimetry experiments under applied pressure in two dierent

Figure 1.2: (a) Crystalline structure for selected cuprate superconductors, adapted from [6]. (b) Temperature-composition phase diagram for La1−xSrxCuO4, extracted from

[7]. (c) Crystalline structures of CeIn3, CeMIn5 and Ce2MIn8, extracted from [8]. (d)

Temperature-pressure phase diagram for CeRhIn5 extracted from [4]. (e) Crystalline

structure of selected iron based superconductors, extracted from [9]. (f) Temperature-pressure phase diagram for SrFe2As2, extracted from [5].

types of layered materials: Ce-based compounds and X-P n (X = Fe or Mn, P n = As or Bi) materials.

Ce-based compounds present the interplay between Ruderman - Kittel - Ka-suya - Yoshida (RKKY) magnetic interaction, crystalline electrical eld (CEF) and Fermi

surface (FS) eects, which arise from the strong hybridization between Ce3+ _{4f electrons}

and conduction electrons (ce). As discussed before, the application of external pressure tunes this hybridization and may lead the system into novel phases. On this thesis we

focus on CeY Bi2 (Y = Cu or Au) and CeAu2Bi. CeCuBi2 and CeAuBi2 crystallize in

the tetragonal P 4/nmm crystalline structure with layers of CeBi-Y -CeBi-Bi, as presented in Figure 1.3(a). Both compounds order antiferromagnetically at low temperatures with transition temperatures of 16 K, in the Cu case, and 19 K for the Au compound. More-over, experiments under pressure have shown a decrease of the antiferromagnetic

transi-tion temperature (TN) as a function of applied pressure for both compounds, as previously

Figure 1.3: Crystalline structures of CeY Bi2(Y = Cu or Au) (a), CeAu2Bi (b), CaMn2Bi2

reported [10, 11]. CeAu2Bi is a new compound discovered during the elaboration of this thesis, that crystallizes in the hexagonal P 63/mmc structure with Ce planes separated by AuBi layers, as shown in Figure 1.3(b).

Finally we also studied the layered structurally related CaMn2Bi2and SrFe2As2

compounds. Both materials present X-P n (X = Fe or Mn, P n = As or Bi) layers with the same site symmetry for Fe and Mn. However, the Mn-Bi layers form a puckered honeycomb lattice instead of the square net formed by the Fe-As layers. In particular,

CaMn2Bi2 displays an antiferromagnetic order at 150 K with magnetic moments lying

in the honeycomb plane [12]. At ambient pressure, CaMn2Bi2 has been reported to be

a narrow-gap semiconductor with extremely large magnetoresistance [12, 13]. At high temperatures, electrical resistivity measurements display metallic behavior [12]. At low

temperatures, CaMn2Bi2 displays an activated behavior with ∆ ∼ 20 K and a nonlinear

Hall resistivity. Band-structure calculations suggest that one of the 3d5 Mn electrons

strongly hybridizes with the Bi p bands giving rise to a hybridization gap. Therefore applied pressure presents an ideal tuning parameter to test this framework. The last

studied compound SrFe2As2 is a member of the iron based superconductors. Although

this class of superconductors has been extensively studied, there are just a few reports combining the application of external pressure with chemical substitution [14, 15, 16]. Moreover those experiments were performed in polycrystalline samples, which present a higher amount of disorder than single crystals [14, 15] or for just one concentration of Co

substitution in single crystals of SrFe2As2 [16]. Therefore we performed a combined study

of Co substitution and applied pressure in Sr(Fe1−xCox)2As2 single crystals, synthesized

by the In-ux technique [17]. At ambient pressure, SrFe2As2 presents a tetragonal to

or-thorhombic structural transition closely connected to a spin density wave phase transition at around 203 K. The application of external pressure and/or Co substitution suppress both transitions inducing a superconducting state. Furthermore the combined study al-lows to estimate the equivalence between applied and chemical pressure.

Chapter 2

Theoretical Aspects

In this chapter the theoretical details used in this thesis will be discussed. A brief explanation of the most important physical properties of the studied compounds will be made. The eects of pressure in solids, magnetism, superconductivity, rare-earth compounds and crystal eld eects will be discussed to improve the reader understanding of our results.

2.1 Eects of Pressure in Solids

The application of external pressure on a material has the main thermodynamic eect of decreasing its volume. The relation between the applied pressure (P ) and the volume (V ) reduction is given by the bulk modulus (K). At a constant temperature T ,

K can be dened as:

K = −V Å ∂P

∂V ã

T

. (2.1)

The compression of the solid has many consequences due to the changes in its electronic conguration, as we will further discuss. Let us rst consider the simple example of a electron in a 1D box of size L. The electron wavefunction can be written as:

Ψ =… 2 Lsin πnx L  , (2.2)

in which n is the quantum number. The energy of the electron levels is dened by: En= ~ 2_π2 2m n L 2 , (2.3)

where m is the electron mass and ~ is the reduced Planck's constant. If we consider changes in the volume of this system, for instance going from L to yL with y < 1, the energy dierence between the levels of the uncompressed system and the compressed one will be: ∆En= ~ 2_π2_n2 2mL2 Å 1 −1 y ã2 . (2.4)

Therefore ∆En < 0 and the energy of the electron levels are increased as a function of

applied pressure. Even though this is an over simplied example, it can be related to real materials in an illustrative level. In solids the electrons are distributed in bands and the application of external pressure aects dierently each band. However, it pushes the electron bands closer to the Fermi level. This is caused by the increase of the overlap between adjacent electron orbitals that also leads to the delocalization of these orbitals and increases the hybridization with the conduction electrons. As we can see in Figure 2.1 (a),

the 3s and the 3p bands of Al rise in energy, crossing the Fermi level (EF) at V ≈ 0.06V0,

in which V0 is the volume at ambient pressure. This results in an electron transfer, from

the 3s band to the 3d band, leading to electronic transitions.

The other eect of applied external pressure is the broadening of the bands caused by the decrease of the Wigner-Seitz radius (the radius of a sphere whose volume is the average volume of an atom in the condensed phase). As displayed in Figure 2.1 (b), the

reduction of the Wigner-Seitz radius from S1 to S2changes the bonding (dotted) and

anti-bonding (dashed) relation of the radial wavefunction Rn,l(r), creating new stable regions.

If enough pressure is applied on the material, the core levels are broaden into bands, due to the increased overlap between the inner and external orbitals, what is known as pressure ionization. Furthermore, the compression caused by the external pressure may induce structural transitions, as dierent crystalline structures could be energetically favored in a compressed environment. For instance, elemental bismuth displays several structural transitions as a function of pressure [18]. In extreme cases, external pressure may even

Figure 2.1: (a) Aluminum electronic bands as a function of compression. (b) Radial

wavefunction Rnl as a function of r with bonding (dotted curves) and antibonding (dashed

curves) for three dierent Wigner-Seitz radii S. Both gures were extracted from [20]. induce the amorphization of the crystalline structure [19].

The eects presented so far change the electronic band conguration and in

consequence, the density of states at the Fermi level η(EF). Usually, they tend to enhance

the metallic behavior of the materials as they decrease the magnitude of the gaps between the bands. However in some materials, the application of external pressure moves bands with the same symmetry close to the Fermi level. These bands cannot cross, and therefore

a hybridization gap is open at EF leading to semiconducting behavior. This eect is

present in the diamond phase of the group IV elements [20] and maybe in CaMn2Bi2. In summary the application of external pressure can change the electronic conguration of a solid, which is usually the most important component to the physical properties of interest in a given material. In this thesis, we will focus on the eects of pressure in magnetism and superconductivity.

2.2 Magnetism

Macroscopic systems present a great number of atoms that interact with each other. These interactions can lead the material to unusual ground states, that could not be achieved if the atoms were isolated. These phenomena are called collective phenomena and magnetism is an example of it. Usually the magnetic properties of a system come from the electrons of its atoms and their interactions with each other. The electrons present

a magnetic moment, which combines their spin (S) and angular orbital momentum (L). In a classical scenario, the presence of a current (I) in a nite loop of area |dA| generates a magnetic moment dµ with the form:

dµ = IdA, (2.5)

the vector dA has the direction normal to the loop determined by the current. Now we can calculate the magnetic moment of the atoms. For simplicity, we consider Bohr's

hydrogen, in which an electron of charge −e and mass meperforms a circular orbit around

a proton. The orbital period is τ = 2πr/v, in which r is the circle radius and v = |~v|. Therefore the current I is I = −e/τ. Moreover, the angular momentum of the electron

(L = R × P ) is mevr and is equal to ~ in the ground state. Then the magnetic moment

becomes:

µ = πr2I = − e~

2me

= −µB, (2.6)

µB is known as the Bohr magneton (9.274 × 10−24 A.m2) and it is a convenient unit

to measure atomic magnetic moments. Furthermore, −e/2me is dened as the electron

gyromagnetic ratio γe.

The magnetic moments of real atoms will depend on the electron spin and on its electronic state. The rst is an intrinsic magnetic moment associated with the electron. The latter is orbital angular momentum and it is associated with the orbital movement of the electron around the nucleus. The energy of a magnetic moment µ in an applied magnetic eld H can be written as

E = −µ · H. (2.7)

Therefore, for an electron in a magnetic eld parallel to the z axis, it becomes:

where Sz is the component of the spin angular momentum along the eld direction.

Moreover, g0 is called the electron g-factor, which is approximately 2, and it depends on

the relative contribution of spin and orbital angular momenta.

Furthermore, the electrons in a real atom also present orbital angular

momen-tum. If we consider an electron in a position ri and momentum pi, we can dene the total

angular momentum as:

~L = X

i

ri× pi, (2.9)

in which we consider L in units of ~ and the sum is over all electrons in the atom. The Hamiltonian of the free atom can be written as:

H0 = X i Å p2 i 2m + Vi ã , (2.10)

where we summed the electrons kinetic energy (T = p2

i/2m) and potential energy (Vi).

Applying a magnetic eld H will add the interaction energy of the eld with each electron

spin (si):

E = g0µBH

X i

si. (2.11)

Furthermore, in the presence of a magnetic eld we should consider the canon-ical moment of the electron:

p = mv + e

cA(r), (2.12)

where A is the vector potential. If we consider a convenient vector potential A = H×r

2 ,

T = 1 2m X i h p_i+ e cA(ri) i2 = 1 2m X i Å p_i+ e c H × ri 2 ã2 = T0 + µBL · H + e2 8mc2H 2X i x2_i + y2_i
. (2.13)

Finally, the Hamiltonian of the atom in the presence of an applied magnetic eld is:

H = H0+ µB(L + g0S) · H + e2 8mc2H 2X i x2_i + y_i2
. (2.14)

The energy shifts produced by the eld-dependent terms in 2.14 are small. Therefore, we can apply second-order perturbation theory to calculate the changes in the energy levels (∆En) caused by the application of magnetic eld:

∆En = µBH ·hn|L+g0S|ni+ X n0_6=n |hn|µBH · (L + g0S)|n0_i|2 En− En0 + e 2 8mc2H 2_hn|X i (x2_i+y_i2)|ni. (2.15) Equation 2.15 is the basis for the calculation of magnetic susceptibilities of isolated atoms, ions or molecules. The magnetic susceptibility (χ) is a measure of the evolution of the magnetization of a material as a function of applied magnetic eld. It can be dened as:

χ = ∂M

∂H. (2.16)

The magnetization (M) of a compound is dened as the magnetic moment per unit volume. At T = 0 K and considering a system of volume V , the magnetization can be described as:

M (H) = −1

V

∂E0(H)

∂H , (2.17)

nite temperature T , if the system is at thermal equilibrium, we need to consider the magnetization as the thermal equilibrium average of the magnetization of each excited

state of energy En(H): M (H, T ) = − P nMn(H)e −En/kBT P ne−En/kBT , Mn(H) = − 1 V ∂En(H) ∂H , (2.18)

in which kB is the Boltzmann constant. Considering the Helmholtz free energy (F ) for a

system with N independent distinguishable particles:

F = −kBT ln(Z), (2.19)

where Z is the single particle partition function Z = Pne

−En/kBT. The magnetization at

a nite temperature becomes:

M (H, T ) = −1

V ∂F

∂H. (2.20)

The magnetic susceptibility at temperatures T > 0 K can be written as:

χ = −1

V

∂2_F

∂H2. (2.21)

Next we are going to apply equation 2.15 to calculate the magnetic suscep-tibility of the simplest cases in magnetism: diamagnetism (χ < 0) and paramagnetism (χ > 0).

2.2.1 Diamagnetism and Paramagnetism

Although diamagnetism is present in all materials, it is often a weak negative response to applied magnetic elds. In a diamagnetic material, the application of an external eld induces magnetic moments opposed to the applied magnetic eld. We consider a solid composed of atoms with all electronic shells lled. These atoms present

zero spin and orbital angular momentum in their ground state (|0i), due to its sperical symmetry.

J |0i = L|0i = S|0i = 0, (2.22)

in which J is the total angular momentum (J = L + S). Substituting 2.22 in 2.15 results in: ∆E0 = e 2 8mc2H 2_h0|X i (x2_i + y2_i)|0i = e 2 12mc2H 2_hr2_{i, (2.23)}

where hr2_{i is the mean square atomic radius (}1

Zih0|P r

2

i|0i). Assuming that the

popula-tion of the atoms in an excited state is negligible, the magnetic susceptibility of a solid composed by N atoms is:

χd= −N Z V ∂2∆E0 ∂H2 = − e2 6mc2 N Z V hr 2_{i, (2.24)}

which is known as the Larmor diamagnetic susceptibility. Equation 2.24 describes accu-rately the magnetic response of ionic crystals as the alkali halides and of the solid noble gases.

Solids composed by ions with partially lled electronic shells present a dierent response to applied magnetic elds. Let us consider an ion with just one partially lled shell and the other ones are lled or empty. The electronic states can be described by the eigenvalues of L, S and J, because these operators often commute with the Hamiltonian. There are 2(2l+1) one-electron levels, therefore we consider n as the number of electrons in the shell, with 0 < n < 2(2l+1). In the absence of electron-electron Coulomb interactions and the spin-orbit interaction, the ground state of the ion would be degenerate. However both interactions break this degeneracy, changing the ground state of the ion. This ground state can be predicted by the empirical Hund's rules.

1. Hund's First Rule

This decreases the Coulomb energy, due to the Pauli exclusion principle, which prevents electrons with parallel spins being in the same place, reducing their Coulomb repulsion.

2. Hund's Second Rule

The lowest lying levels in energy are the ones with the largest total orbital angular momentum L, that is consistent with Hund's rst rule and the exclusion principle. This also reduces the Coulomb repulsion between the electrons.

3. Hund's Third Rule

This rule determines the the value of the total angular momentum J. Consid-ering the rst two rules, we found L and S that minimizes the Coulomb repulsion and, therefore, the energy of the state. However, this still leaves (2L + 1)(2S + 1) degenerated states. This degeneracy is broken by the spin-orbit coupling, that can be described in the Hamiltonian for these states as λ(L · S), in which λ is a proportionality constant. The spin-orbit coupling favors maximum J (L parallel to S), if λ is negative. For positive λ, a minimum J is favored ( L antiparallel to S). Shells less than half lled present a positive

λ and J = |L − S|. For shells lled more than the half, λ is negative and J = L + S.

The ground state is written as(2S+1)_L

J and L is described with a letter from

the spectroscopic code:

L = 0, 1, 2, 3, 4, 5, 6 X = S, P, D, F, G, H, I

In gure 2.2, we show the ground states of the most interesting ions in mag-netism, the ones with partially lled d or f shells.

Let us consider an insulating solid with N/V ions with partially lled shells per unit volume. The ground state (|0i) is nondegenerate if the shell is one electron short to be half lled (J = 0). In this case, the rst term in 2.15 vanishes, but the second term is not zero. Therefore, the energy shift of the ground state is:

Figure 2.2: Ground states of ions with partially lled d and f shells, extracted from [21]. ∆E0 = e2 8mc2H 2_h0|X i (x2_i + y_i2)|0i −X n |h0|µBH · (L + g0S)|ni|2 En− E0 . (2.25)

The magnetic susceptibility becomes:

χV = −N V " e2 8mc2H 2_h0|X i (x2_i + y_i2)|0i − 2µ2_BX n |h0|(Lz+ g0Sz)|ni|2 En− E0 # , (2.26)

in which the rst term is the Larmor diamagnetism. The second term, however, is positive

(since En> E0) and favors a parallel alignment of the magnetic moment with the applied

magnetic eld. This behavior is known as paramagnetism and the paramagnetic response of ions one electron short to be half lled is called Van Vleck paramagnetism. Both Larmor diamagnetism and Van Vleck paramagnetism are temperature independent. Moreover, if

we assume that the excited states population are negligible, the magnetic susceptibility of the solid will be a balance between these two susceptibilities.

If the shell does not present J = 0 the rst term in 2.15 will not be zero. The contribution to the energy shift of this term is much bigger than the other two terms, therefore we can neglect them. The ground state is (2J + 1)-fold degenerate and we need

to diagonalize hJLSJz|(L + g0S)|J LSJz0i.

The Wigner-Eckart theorem lets us write the matrix elements of any vector op-erator with a given value of J as matrix elements of J itself multiplied by a proportionality constant (g). Therefore, we have:

hJLSJz|(L + g0S)|J LSJ_z0i = ghJLSJz|J |JLSJ_z0i. (2.27)

By using the completeness relation we can write:

hJLSJz|(L + g0S) · J |J LSJz0i = ghJLSJz|J2|JLSJz0i. (2.28)

Considering g0 = 2 and using the fact that L · J = 1₂(L2 + J2 − S2_{) and S · J =}

1 2(S 2 + J2− L2), we get: g = 3 2+ 1 2 ï S(S + 1) − L(L + 1) J (J + 1) ò . (2.29)

This proportionality constant is known as the Landé g-factor and along with the

Wigner-Eckart theorem enable us to write (L + g0S) = gJ. Therefore, we can consider the

magnetic moment of the ions as µ = −gµBJ. We should note that this assumption

is only valid considering that the dominant contribution to the free energy is from the

(2J + 1)states in the ground state. In other words, the splitting between the ground state

and the rst excited state is much bigger than kBT. However, this prevents us to calculate

the susceptibility by equating the free energy to the ground state energy, because as the

eld goes to zero, the splitting between the states becomes of the order of kBT.

Therefore, if we consider a solid with N ions with J 6= 0 in a volume V with only the lowest (2J + 1) thermally excited, our partition function becomes:

Z = J X Jz=−J e−βgµBHJz_, in which β = 1 kBT , (2.30)

which is a geometrical progression with initial term e−βgµBHJ multiplied by e−βgµBH.

Therefore, we can simplify the summation to:

Z = sinh(2J + 1) βgµBH 2  sinhβgµBH 2  . (2.31)

The magnetization can be calculated by:

M (H, T ) = −N V ∂F ∂H = − N βV ∂ln(Z) ∂H , (2.32) yielding: M (H, T ) = N V gµBJ BJ(βgµBJ H). (2.33)

BJ(βgµBJ H) is the Brillouin function dened by:

BJ(x) = 2J + 1 2J coth Å 2J + 1 2J x ã − 1 2Jcoth Å 1 2Jx ã , x = βgµBJ H. (2.34)

As T → 0 at a constant applied eld, the magnetization reaches its maximum

value (saturation value) Ms = (N/V )gµBJ and each ion is aligned by the eld. This

condition is only satised when kBT  gµBH. gµBH/kB is of the order of 1 K in a eld

of 104 _{gauss, which is satised only at low temperatures and high elds. Therefore, the}

magnetization is usually in the opposite limit gµBH  kBT. Considering the expansion

of the Brillouin function for small-x:

BJ(x) ≈

J + 1

3J x + O(x

3_{), (2.35)}

M (H, T ) = N

V (gµB)

2J (J + 1)

3kBT

H. (2.36)

The magnetic susceptibility obtained from this magnetization is:

χC =

N

V (gµB)

2J (J + 1)

3kBT , (2.37)

which is known as the Curie's law. Although this law is consistent in a huge range of

temperature and elds, it is only valid when gµBH  kBT. Moreover, this paramagnetic

contribution is much bigger than the Van Vleck paramagnetism and the Larmor diamag-netism. It is also the dominant term in the energy shift of the ground states when a magnetic eld is applied in a solid with ions with partially lled shells and J 6= 0.

Curie's law is often written as:

χC = N 3V µ2 Bp2 kBT , in which p = g [J(J + 1)] 1/2 , (2.38)

in which p is called the eective moment of the ion. This law accurately describes the magnetic response of rare-earth ions in insulating crystals. The fact that the 4f shells are deep inside the atoms enable us to treat them as isolated ions in a good approximation. However, this is not the case for the transition metals. The partially lled d-shells are the outermost electronic shells and are more prone to the eect of crystalline electrical elds. These elds present the symmetry of the crystalline site of the ions and should be introduced as a perturbation on the (2S + 1)(2L + 1) states determined by the rst two Hund's rules. Therefore, Hund's third rule needs to be modied considering both spin-orbit and crystal eld eects. For light atoms, as the ions from the iron group, the crystal eld eect is much stronger than the spin-orbit coupling. Thus the spin-orbit coupling can be ignored in the construction of Hund's third rule in a rst approximation. The crystal elds aect only the orbital angular momentum, and if they are suciently asymmetric, they can create a ground state in which the mean value of the components of L are zero. This is called the quenching of the orbital angular momentum. For heavier atoms, the spin-orbit coupling increases and may become of the order of the crystal eld eects. In these cases, both eects need to be accounted in Hund's third rule and group

theory needs to be applied to calculate the energy shifts in the electron levels.

The previous discussion was based considering an insulating crystal composed by ions with partially lled or lled shells. Now we need to calculate the magnetic response of the conduction electrons in metals. This is a complex problem, due to the itinerant behavior of the conduction electrons and to the response of the electron orbital motion to the magnetic eld. However, this problem can be solved within the independent electron approximation and neglecting the orbital response of the electrons.

In this scenario the magnetization will be:

M = −µB(n+− n−), (2.39)

considering g0 = 2 and n = N/V with the subscript denoting spins parallel (+) or

antiparallel (-) to the applied magnetic eld. Because we neglected the orbital motion of the electrons, the application of a magnetic eld only shifts the energy of the electronic

levels (E) by ±µBH. The density of levels becomes:

η±(E ) = 1

2η(E ∓ µBH). (2.40)

The total number of electrons per unit volume for each spin is:

n± =

Z

d(E )η(E )f (E ), (2.41)

in which f(E) is the Fermi distribution function: f(E) = (e(β(E−µ))_{+ 1)}−1 _{for a chemical}

potential µ. Because in metals we are on the degenerate limit of the Fermi distribution

function and µBH is of the order of 10−4EF, we can expand the density of levels to:

η±(E ) = 1 2η(E ) ∓ 1 2µBHη 0 ±(E ), (2.42)

and the number of electrons for each spin is:

n± = 1 2 Z η(E )f (E )d(E ) ∓ 1 2µBH Z d(E )η0(E )f (E ). (2.43)

The magnetization then becomes: M = µ2_BH Z d(E )η0(E )f (E ) = µ2_BH Z d(E )η(E ) Å −∂f (E ) ∂E ã . (2.44)

Considering −∂f/∂E = δ(E − EF), which is valid approximation even at room

tempera-tures, due to the minute value of the T 6= 0 corrections for temperatures much smaller

than the Fermi temperature (TF ∼ 104), the magnetization can be written as:

M = µ2_BHη(EF), (2.45)

leading to the following magnetic susceptibility:

χp = µ2_Bη(EF). (2.46)

This susceptibility is known as the Pauli paramagnetic susceptibility, which is temperature independent and of the order of the diamagnetic susceptibilities. The small magnitude of the Pauli susceptibility compared to the susceptibility of the magnetic ions can be attributed to the fact that the exclusion principle is much stronger than the thermal disorder in suppressing the spins alignments with the eld.

To calculate the Pauli magnetic susceptibility we neglected the orbital motion of the conduction electrons. In magnetic eld, the orbital motion of the electrons couples with the eld and generates a magnetization antiparallel to the applied eld. This dia-magnetic response of the conduction electrons is known as Landau diamagnetism and it is related to the Pauli paramagnetic susceptibility through:

χL= −

1

3χp. (2.47)

In real materials, the magnetic susceptibility is the addition of all contributions presented before. Therefore, magnetic susceptibility measurements result in the total magnetic susceptibility of the bulk magnetic moment induced by a eld. In this regard,

it is hard to experimentally separate each contribution in the nal result of susceptibility. Until now, our discussion on magnetism did not consider the interactions be-tween the sources of magnetic moments. In the next section we are going to review the eects of these interactions on the magnetic properties of solids.

2.3 Magnetic Ordering

In the previous section we neglected any kind of interactions between the sources of the magnetic moments in the material. As we will see, these interactions can create a system with a net magnetization without the application of a magnetic eld. These compounds are called ferromagnets. In other systems, the sources organize them-selves in a particular conguration that there is no net magnetization in the absence of the eld, however they are not randomly oriented. These materials are called an-tiferromagnets. In this regard, in this section we are going to introduce the magnetic interactions present in solids. After that, the ferromagnetism and antiferromagnetism will be discussed.

2.3.1 Magnetic Interactions

The rst interaction that might be important in the magnetic properties of a

solid is the dipolar magnetic interaction. If we consider two magnetic dipoles µ1 and µ2

separated by a distance r, the energy of the magnetic dipolar interaction will be:

E = 1

r3 [µ1· µ2− 3(µ1· r)(µ2· r)] . (2.48)

To estimate the order of magnitude of this interaction, we can consider µ ≈ 1 µB and

r ≈ 1Å, to obtain E ≈ 10−23 J ≈ 10−4 eV, which is equivalent to 1 K. Many compounds

present high ordering temperatures, some order at 1000 K. Therefore, the magnetic dipolar eect is not important in most materials, but it must considered in compounds that order in the mK range.

magnetic ordering of most compounds. Therefore, stronger interactions must play an important role in these systems. These interactions are called exchange interactions and are electrostatic, arising from the Pauli exclusion principle.

To illustrate the appearance of the exchange interaction in solids, we are going to consider a two-electrons problem and then expand this result to real materials.

Considering the spatial positions of the electrons as r1 and r2, their electronic

states will be φa(r1)and φb(r2), respectively. Therefore the spatial part of the joint wave

function is φa(r1)φb(r2). We also need to consider the spin part of their wave functions,

which can be linear combination of the following possibilities: | ↑↑i, | ↑↓i, | ↓↑i and | ↓↓i. The arrows denote the z component of the rst and second electrons, respectively. This basis can be separated in two groups: the singlet state (S = 0) and the triplet states

(S = 1). The singlet state presents the wave function of the form _√1

2(| ↑↓i − | ↓↑i), which

is antisymmetric with the exchange of the particles. The triplet states can be: | ↑↑i,

| ↓↓i and _√1

2(| ↑↓i + | ↓↑i). We can see that the triplet states are symmetric with the

exchange of the electrons. Because they are fermions, their total wave function must be antisymmetric to obey Pauli exclusion principle. Therefore, the overall singlet state wave function must have a symmetric spatial part and for the triplet state an antisymmetric spatial part is required. In this regard, we can write two wave functions considering both the spin and spatial parts.

ΨS = 1 √ 2[φa(r1)φb(r2) + φa(r2)φb(r1)]χS, ΨT = 1 √ 2[φa(r1)φb(r2) − φa(r2)φb(r1)]χT. (2.49)

Where χ denotes the normalized spin part of the function. The energies of the two states are: ES = Z Ψ∗_SHΨSdr1dr2, ET = Z Ψ∗_THΨTdr1dr2. (2.50)

ES− ET = 2 Z

φ∗_a(r1)φ∗_b(r2)Hφa(r2)φb(r1)dr1dr2. (2.51)

The spin Hamiltonian for a two-electron system can be constructed considering

the individual spin operators Si. They satisfy the relation S2

i = 12(

1

2+ 1) =

3

4. Therefore,

the total spin operator can be written as:

S2 = (S1+ S2)2 =

3

2 + 2S1· S2. (2.52)

This results in S1· S2 = −3₄ for the singlet state and S1· S2 = 1₄ for the triplet state.

In this regard, we can consider the spin Hamiltonian as an eective Hamiltonian of the form:

Hspin ₌ 1

4(ES + 3ET) − (ES− ET)S1· S2. (2.53)

The rst term is just a constant, which can be omitted redening the zero energy. The second term, however, depends on the relative orientation of the two spins. By dening

the exchange parameter Jex as:

Jex = ES − ET

2 =

Z

φ∗_a(r1)φ∗b(r2)Hφa(r2)φb(r1)dr1dr2, (2.54)

the Hamiltonian becomes:

Hspin = −2JexS1· S2. (2.55)

For positive Jex_{, parallel spins (triplet state S = 1) are favored and for negative J}ex

an antiparallel arrange (singlet state S = 0) is favored. It is important to note that 2.55 depends only on the relative orientation of the two spins and not on the position of

the spins with respect to r1 and r2. This will be important just if rotational symmetry

breaking terms, as spin-orbit coupling or magnetic dipolar interaction, are present in the Hamiltonian and results in anisotropic magnetizations.

Hspin = −XJ_ijexSi · Sj, (2.56)

in which Jex

ij are the exchange parameters between the ithand jthspins. This Hamiltonian

is known as the Heisenberg Hamiltonian.

The simplest exchange interaction present in solids is the direct exchange. It occurs when the exchange interaction is between two neighboring magnetic atoms, with-out the need for an intermediary atom. Due to the small overlap between the neighboring magnetic orbitals this interaction is not the most important one for the magnetic proper-ties of a compound. Within the rare earths the 4f electrons are very localized and close to the nucleus. Therefore they present a minimum overlap. Even in the 3d transition metals, as Fe, Co and Ni, the overlap is still insucient.

In this regard, it is common to nd other exchange interactions between the magnetic atoms. In fact, there are several exchange interactions in solids, that are listed below:

1 Superexchange:

It is present in ionic crystals and the interaction between two non-neighboring magnetic atoms is mediated by a non-magnetic atom in between them.

2 Double Exchange:

It occurs due to the hopping of electrons between mixed valence ions, creating a ferromagnetic interaction.

3 Anisotropic Exchange:

It is also called Dzyaloshinsky-Moriya interaction. It is based on the coupling between the ground state of a magnetic ion with an excited state of another magnetic ion, produced by the spin-orbit coupling.

4 Itinerant Exchange:

It is the exchange interactions of the conduction electrons between themselves. 5 RKKY Interaction:

It occurs in metals, when the localized magnetic moments polarize the con-duction electrons around them. These polarized concon-duction electrons couple to the other localized magnetic moments that are a distance r away. This indirect exchange interac-tion is known as the Ruderman-Kittel-Kasuya-Yosida (RKKY) interacinterac-tion. The exchange parameter of the RKKY interaction can be written as [22]:

JRKKY = − 9π 8 (η(EF)Jf s)2 EFr3 ï 2kFcos(2kFr) − sin(2kFr) r ò , (2.57)

in which EF is the Fermi energy, kF is the radius of the Fermi surface, Jf s is the exchange

interaction between the localized moments and the conduction electrons and η(EF)is the

density of the conduction electrons states. Due to its oscillatory behavior, JRKKY can be

either positive or negative, favoring parallel or antiparallel spins, respectively. Figure 2.3 presents the oscillatory response of the RKKY exchange parameter as a function of r. The long range behavior of the RKKY interaction is responsible for the magnetic ordering in several rare earth based compounds.

Figure 2.3: RKKY exchange parameter (JRKKY) as a function of r.

As briey discussed before, the presence of magnetic interactions can lead the material to magnetic ordering. In the next sections we are going to review the ferromag-netic and antiferromagferromag-netic orders.

2.3.2 Ferromagnetism

As previously discussed the magnetic interactions between two localized mag-netic moments can create an ordered state in the solid. This ordered state is reached by decreasing the temperature, as thermal disorder favors a random orientation of the spins

at high temperatures and prevents the ordered state. Below a critical temperature (TC)

the localized moments become parallel aligned and add up to a net magnetization even in the absence of applied magnetic elds, as presented in Figure 2.4. This behavior is called ferromagnetism and the compound is a ferromagnet.

Figure 2.4: Schematic picture of a ferromagnetic transition. In an applied magnetic eld, the Hamiltonian for the ferromagnet is:

H = −X

ij

J_ijexSi· Sj + gµBX

j

Sj · H, (2.58)

in which Jex _{> 0 to ensure the ferromagnetic alignment and we considered an ion with}

no orbital angular momentum for simplicity. We can rewrite this Hamiltonian to:

H = −X i Si· X j J_ijexSj+ gµBH ! . (2.59)

Then, an eective molecular eld is dened at ith _{site as:}

Hmf = H + 1 gµB X j J_ijexSj. (2.60)

This eective eld is a mean eld approximation of the external magnetic eld environ-ment in which the magnetic ions are placed. This method, known as the Weiss model, does not consider magnetic uctuations or other short range interactions that may be present close to TC. However, due to its simplicity, it is always a good starting point to analyze the magnetic properties of a new compound. The Hamiltonian then becomes:

H = −gµB

X i

Si· Hmf, (2.61)

which is the same for a paramagnet in a magnetic eld Hmf. In a ferromagnet, on the

other hand the mean value of every spin is the same, therefore, we can represent then as a function of the total magnetization:

hSii = V N M gµB . (2.62)

By substituting each spin by its mean value in 2.60, the eective eld becomes:

Hmf = H + λM , where λ = V N Jex 0 (gµB)2 and J ex 0 = X j Jij. (2.63)

We found the magnetization for a paramagnet in 2.33. Now we need to solve it replacing

H by Hmf. The magnetization in the mean eld approximation becomes:

M (H, T ) = M0

Å H_mf

T ã

= MsBS(y). (2.64)

in which M0 is the magnetization without magnetic interactions and y = gµBS(H+λM )

kBT . We

can solve this equation graphically considering x = λM

T at H = 0. We have to plot the

following pair of equations:

M (T ) = M0(x),

M (T ) = T

and the solutions will be the intersections between the two curves. At high temperatures

(T > TC) the intersection only occurs at M = 0, as displayed in Figure 2.5, and therefore

the system is in the paramagnetic state. At low temperatures (T < TC), there are two

intersections: one at M = 0 and another one with M > 0, the latter being more stable.

In this regard, the compound will be ferromagnetically ordered. We dene TC as the

temperature in which the slopes of both equations are the same. In this regard, we have:

∂ ∂x Å TC λ x ã = ∂M0(x) ∂x , TC = N V λ(µBp)2 3kB = S(S + 1) 3kB J₀ex. (2.66)

Figure 2.5: Magnetization as a function of x at three dierent temperatures. If a small eld is applied, a small magnetization is induced even at T > TC. The magnetic susceptibility can be calculated by:

χ = ∂M ∂Hmf ∂Hmf ∂H , χ = χC 1 − TC T
= N 3V kB (µBp)2 T − Tc, (2.67)

in which χC is the susceptibility without magnetic interactions. This expression is known as the Curie-Weiss law. As discussed before this is a good starting point to study the mag-netic properties of a new compound. However, this is a simple equation, and corrections need to be done for more accurate results.

2.3.3 Antiferromagnetism

If the magnetic ions of the compound align themselves antiparallel to each other below a particular temperature, the net magnetization will be zero and this behavior is called antiferromagnetism as displayed in Figure 2.6.

Figure 2.6: Schematic picture of an antiferromagnetic transition.

The antiferromagnetism is favored by a negative exchange interaction Jex_{< 0}

between the magnetic moments. Usually, we can consider the antiferrmagnetic lattice as the sum of an up sublattice (+) with a down sublattice (−). The molecular eld then

becomes Hmf = H − |λ|M±, in which λ is negative now. The magnetization can be

written as: M±(H, T ) = MsBS Å gµBS(H − |λ|M∓) kBT ã . (2.68)

We can assume for simplicity that the two sublattices dier only on the directions of the

M = MsBSÅ gµBS|λ|M

kBT

ã

, (2.69)

for H ≈ 0. This equation is almost the same as 2.64 and therefore the solution is very similar. The molecular eld will present the same behavior, disappearing for temperatures

above the transition temperature, which is known as the Néel temperature (TN) in this

case and dened by:

TN = N V λ(µBp)2 3kB . (2.70)

The magnetization of both sublattices behave as two separate ferromagnetic

magnetizations with opposite directions. Below TN the net magnetization M++ M− will

be zero. We can dene the staggered magnetization as the dierence M+− M−, which is

non-zero for T < TN.

For T > TN the application of a small magnetic eld can be discussed as in the

ferromagnetic case. The magnetic susceptibility for the antiferromagnetic case is then:

χ = χ0 1 + TN T
= N 3V kB (µBp)2 T + TN . (2.71)

In this regard, we can summarize the magnetic susceptibility at high temper-atures (in the paramagnetic state) by:

χ = N

3V kB

(µBp)2

T − θCW

, (2.72)

in which θCW is the Weiss temperature. For θCW = 0 the compound is paramagnetic. If

θCW > 0 the system is ferromagnetic and we expect θCW = TC. Finally, if θCW < 0 the

material is antiferromagnetic and θCW = −TN. This equation plus a constant term was

used in this thesis to estimate the eective moment of the magnetic ions present in our compounds.

Applying a magnetic eld in an antiferromagnetic compound at temperatures

for the resultant magnetic susceptibility. The energy saved by one sublattice to align its moments with the eld is compensated by the cost in energy to change the direction of the moments on the other sublattice, considering that the magnetization is the same in each sublattice, but with opposite direction.

Let us neglect thermal agitation eects by considering T = 0. The magneti-zations of both sublattices will be Ms, and therefore the application of a small magnetic

eld will have no eect because |M±|are both saturated. In this case the net

magnetiza-tion of the material is zero, thus χ|| is also zero. However if the magnetic eld is applied

perpendicular to the magnetization, it induces a tilt on both sublattices and induces a

magnetization on the perpendicular plane. Therefore, χ⊥ is not zero.

By increasing the temperature, thermal uctuations start to develop and aect dierently the system for elds applied parallel or perpendicular to the magnetization. For elds parallel to the magnetization the thermal uctuations have a huge eect, because the applied eld enhances the magnetization of one sublattice and reduces it for the other. For elds applied perpendicular, however, the thermal agitation has almost no eect, because the magnetic eld reduces the magnetization of both sublattices in the

Figure 2.7: Magnetic susceptibility of an antiferromagnetic system with applied magnetic elds in dierent directions.

same way. Therefore, we have that χ|| → χ⊥ as T → TN, as presented in Figure 2.7. If the applied magnetic eld is not small, it will eventually dominate over the magnetization of the sublattices and will be the dominant term in the molecular eld. In this regard, the applied magnetic eld will force all magnetic moments to be aligned parallel to each other. The route that this is reached depends on the direction of the applied magnetic eld.

If the magnetic eld is applied perpendicular to the magnetization, it slowly rotates the magnetic moments, until they are aligned parallel to the eld.

If the eld is applied parallel to one sublattice magnetization, the magnetic moments suddenly align themselves parallel to the applied eld at a critical eld. This is known as a spin-ip transition and is displayed in Figure 2.8. This transition occurs in compounds that the magnetization presents a preferred direction due to the exchange in-teractions between its magnetic moments. This preferred orientation of the magnetization in some particular direction is called anisotropy.

If the system present a small anisotropy, the spin-ip transition can happen in two steps, what is called spin-op transition. At a particular applied magnetic eld,

the magnetic moments suddenly organize themselves tilted by an angle to the applied magnetic eld. Further increasing the magnetic eld reduces this angle and then the magnetic moments become aligned parallel to the eld. This behavior is also shown in Figure 2.8. The representations of both spin-ip and spin-op transitions were performed considering T = 0 K. For higher temperatures the thermal uctuations make the curves less steep, but the overall behavior is still the same.

Figure 2.9 presents a summary of the magnetic susceptibilities as a function of temperatures studied so far.

Figure 2.9: Magnetic susceptibility as a function of temperature for: (a) Pauli para-magnetism and diapara-magnetism; (b) Curie parapara-magnetism; (c) Ferropara-magnetism and (d) Antiferromagnetism. Adapted from [23].

2.3.4 Spin-Density Waves

The presence of Pauli paramagnetism in metals may create a magnetic ordering in the conduction electrons. In the absence of a magnetic eld, the kinetic energy needed to change the electrons from the spin down band to the spin up band is: