...... ............... unesp TÃT UNIVERSIDADE ESTADUAL PAULISTA PROGRAMA DE , - POS-GRADUAÇAO , EM FISICA ÁREA DE FÍSICA APLIC-ADA INSTITUTO DE GEOCI NCIAS E Cl RIO CLARO The Grüneisen parameter applied to critical phenomena and experimental investigations of correlated phenomena in molecular conductors Lucas Cesar Gomes Squillante 2023 - SP UNIVERSIDADE ESTADUAL PAULISTA “Júlio de Mesquita Filho” Instituto de Geociências e Ciências Exatas Câmpus de Rio Claro Lucas Cesar Gomes Squillante The Grüneisen parameter applied to critical phenomena and experimental investigations of correlated phenomena in molecular conductors Tese de Doutorado apresentada ao Instituto de Geociências e Ciências Exatas do Câmpus de Rio Claro, da Universidade Estadual Paulista “Júlio de Mesquita Filho”, como parte dos requisitos para obtenção do título de Doutor em Física Aplicada. Orientador: Prof. Dr. Valdeci Pereira Mariano de Souza Rio Claro - SP 2023 S773g Squillante, Lucas Cesar Gomes The Grüneisen parameter applied to critical phenomena and experimental investigations of correlated phenomena in molecular conductors / Lucas Cesar Gomes Squillante. -- Rio Claro, 2023 208 p. : il., tabs., fotos Tese (doutorado) - Universidade Estadual Paulista (Unesp), Instituto de Geociências e Ciências Exatas, Rio Claro Orientador: Valdeci Pereira Mariano de Souza 1. Critical phenomena. 2. Grüneisen parameter. 3. strongly correlated electron systems. I. Título. Sistema de geração automática de fichas catalográficas da Unesp. Biblioteca do Instituto de Geociências e Ciências Exatas, Rio Claro. Dados fornecidos pelo autor(a). Essa ficha não pode ser modificada. UNIVERSIDADE ESTADUAL PAULISTA “Júlio de Mesquita Filho” Instituto de Geociências e Ciências Exatas Câmpus de Rio Claro Lucas Cesar Gomes Squillante The Grüneisen parameter applied to critical phenomena and experimental investigations of correlated phenomena in molecular conductors Tese de Doutorado apresentada ao Instituto de Geociências e Ciências Exatas do Câmpus de Rio Claro, da Universidade Estadual Paulista “Júlio de Mesquita Filho”, como parte dos requisitos para obtenção do título de Doutor em Física Aplicada. Comissão Examinadora Prof. Dr. Valdeci Pereira Mariano de Souza (orientador) IGCE / UNESP/Rio Claro (SP) Prof. Dr. Aniekan Magnus Ukpong Universidade de KwaZulu Natal / Pietermaritzburg, África do Sul Prof. Dr. Ricardo Paupitz Barbosa dos Santos IGCE / UNESP/Rio Claro (SP) Prof. Dr. Ricardo Egydio de Carvalho IGCE / UNESP/Rio Claro (SP) Prof. Dr. Rafael Sá de Freitas IFUSP / USP/São Paulo (SP) Conceito: Aprovado. Rio Claro (SP), 16 de Junho de 2023. Acknowledgements I would like to express my profound gratitude to Prof. Dr. Mariano de Souza for the immense support during my entire scientific career and for proposing the research topics investigated in this Thesis, as well as for all his advice and guidance throughout the entire progress of this work. He supported me beyond the academic activities always encouraging me to reach the best version of myself and to flourish my humane personal side in the academic activities and in life. He always saw an inner potential in me that even I didn’t know it existed. In a world full of hollowness, superficiality, and lacking true academic mentors, I feel honored and proud to have him as my advisor. For those who know me since I started my undergraduate studies, I once was a non-fully committed student. Today I feel totally different, committed with the academic path and looking forward to become a professor of Physics in a Brazilian university in a near future. In the Talmud it is written that “Whoever saves one life, saves the world entirely” and I feel academically saved! My profound gratitude for you, Prof. Dr. Mariano de Souza! I cannot express my love and gratitude for my fiancé Mariana Alvarinho Lorizola. She always encouraged me to pursue my dreams and to work hard to achieve my goals. She was also very patient, kind, and lovely during the entire period of the developing and writing of this Thesis. She is the best life partner I would ever imagine having in my life. All the love and affection that came from our relation are somehow spread throughout all chapters of this Thesis in a lovely fashion. Arriving home everyday to find you with a bright smile and our baby cat Frida asking to be petted make me feel loved and my heart warm. I love you, thank you for believing in me! At this point, I recall Robin Willians quote in the role of John Keating in the classical movie “Dead Poets Society” (1989): “...We don’t read and write poetry because it’s cute. We read and write poetry because we are members of the human race. And the human race is filled with passion. And medicine, law, business, engineering, these are noble pursuits and necessary to sustain life. But poetry, beauty, romance, love, these are what we stay alive for. To quote from Whitman, "O me! O life!... of the questions of these recurring; of the endless trains of the faithless... of cities filled with the foolish; what good amid these, O me, O life?" Answer: that you are here - that life exists, and identity; that the powerful play goes on and you may contribute a verse. That v vi the powerful play goes on and you may contribute a verse. What will your verse be? ...” Thank you for contributing a verse with me in this beautiful journey, my love! I also would like to thank you my family, my mother Lindalva Gomes and my sister Isis Gomes Marques, for always supporting me in the best way they could, even from a distance. I wish this Thesis will spark love and affection in their lives! If my mother hadn’t registered me in English classes when I was a kid, I wouldn’t be writing this Thesis in English. Thank you, you are very special to me and I love you both! I acknowledge all members of the Solid State Physics group, with special thanks to my Ph.D. colleague Isys Mello for all her support throughout the years. Also, I thank everyone from the Physics department at Unesp who supported us during the Ph.D., with special thanks to Leandro Xavier Moreno and Elizabete Pereira das Neves. I also thanks Maria from Klips Papelaria & Copiadora who printed all the posters I have presented in conferences for always caring too much about the exceptional printing quality of the posters, contributing to the best poster awards during the Ph.D. I thank my great friends Renan Vieira Barreto and Vinicius Cavassano Zampier for the time we have lived together. Our home was always filled with nice discussions, celebrations, and a lot of hard work. I remember that we were always working in our manuscripts and projects and it really helped me during the first years of my Ph.D. to live with other people committed with their studies. Special thanks to Renan who really teached me everything I know about training exercises and helped me to get a better shape during the pandemics. Renan always told me what I needed to hear instead of what I wanted to hear, which was very important to me. Special thanks to one special friend André Kyoshi Fujii Ferrazo who always kindly supported me and encouraged me to do great things. Thank you for all endless philosophical talks, walks at night, and emotional support as well. I profoundly acknowledge Carlos Eduardo Ortolani Prado de Moura and Cristina Rosa Campos for the amazing support during this Ph.D. period and for making, in particular, this time of my life much lighter and smoother. Thank you both for the enlightenments and emotional support. I also acknowledge all professors who did not gave me an opportunity and did not believe in my potential. If it weren’t by them, I wouldn’t be in this lovely academic path I am today. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. Finally, I thank everyone who directly or indirectly supported me in this Thesis. Abstract Critical phenomena are of great interest to the scientific community and can be widely extended to various fields of research, such as classical and quantum phase transitions, caloric effects, and even Biology in brain functionality, for instance. As broadly discussed in the literature, the smoking-gun physical quantity to experimentally access both finite-temperature and quantum critical points is the so-called Grüineisen ratio. In this Ph.D. Thesis, a systematic review is performed on the derivation and generalization of the Grüneisen parameter followed by its unprecedented applications to several distinct scenarios, such as magnetic model systems, zero-field quantum phase transitions, the maximization of caloric effects close to any critical-end point based on entropy arguments, the here-proposed adiabatic magnetization of a paramagnetic salt, as well as for Cosmology in the frame of the universe expansion. Since this Ph.D. Thesis is a symbiosis between theoretical and experimental results, an experimental investigation of correlated phenomena was carried out for molecular conductors of the (TMTTF)2X family, where TMTTF is the base molecule tetramethyltetrathiafulvalene and X a monovalent counter- anion such as PF6, SbF6, or AsF6. Such strongly correlated electron systems are considered suitable ones for the exploration of Mott insulating phase, charge-ordering, spin-Peierls, and superconductivity. In particular, the investigation of a possible multiferroic character in these salts was performed via quasi-static (low-frequency) dielectric constant ε′ measurements as a function of temperature where a maximum in ε′ as a function of temperature was observed at the corresponding charge- ordering temperature for both hydrogenated and 97.5% deuterated (TMTTF)2SbF6 salts. Furthermore, Raman measurements were performed on the 97.5% deuterated (TMTTF)2PF6, showing a possible magneto-optical effect on the ν4(ag) vibrational mode of the TMTTF molecule. Yet, fluorescence measurements demonstrated that the fully-hydrogenated (TMTTF)2AsF6 presents an expressive fluorescence background, which is roughly five orders of magnitude lower than that for the 97.5% deuterated variant of (TMTTF)2PF6. Keywords: critical phenomena; Grüneisen parameter; strongly correlated elec- tron systems. Resumo Fenômenos críticos são de grande interesse para a comunidade científica e podem ser estendidos para várias áreas de pesquisa, como transições de fase clássicas e quânticas, efeitos calóricos e até mesmo na Biologia no contexto da funcionalidade cerebral, por exemplo. É discutido amplamente na literatura que a grandeza física crucial para se acessar experimentalmente tanto pontos críticos a temperatura finita quanto pontos críticos quânticos é a chamada razão Grüneisen. Nesta tese de doutorado, uma revisão sistemática é realizada sobre a derivação e generalização do parâmetro de Grüneisen seguido pelas suas aplicações inéditas para diversos cenários físicos distintos, tais como sistemas modelo para o magnetismo, transições de fase quântica a campo magnético zero, a maximização de efeitos calóricos próximos de qualquer ponto crítico baseado em argumentos de entropia, para a magnetização adiabática de um sal paramagneto proposta nesta tese, bem como para a Cosmologia no contexto da expansão do universo. Uma vez que esta tese de doutorado é uma simbiose entre resultados experimentais e teóricos, uma investigação experimental de efeitos correlacionados em condutores moleculares também foi realizada para os condutores moleculares da família (TMTTF)2X, onde TMTTF é a molécula base tetrametiltetratiafulvaleno e X um contra-ânion monovalente, tais como PF6, SbF6 ou AsF6. Tais sistemas eletrônicos fortemente correlacionados são considerados apropriados para a exploração de isolantes de Mott, fase de carga ordenada, spin-Peierls e até mesmo supercondutividade. Em particular, uma investigação do possível caráter multiferroico destes sais foi realizada utilizando medidas de constante dielétrica quase estática (baixa frequência) ε′ em função da temperatura onde um máximo de ε′ foi observado na temperatura de ordenamento de carga correspondente para a variante totalmente hidrogenada e 97,5% deuterada do sal (TMTTF)2SbF6. Além disso, experimentos Raman foram realizados na variante 97,5% deuterada do sistema (TMTTF)2PF6, onde um possível efeito magneto-óptico no modo vibracional ν4(ag) da molécula TMTTF foi observado. Ainda, experimentos adicionais mostraram que o sistema totalmente hidrogenado (TMTTF)2AsF6 apresenta uma expressiva fluorescência, a qual é cerca de cinco ordens de magnitude maior do que a da variante 97,5% deuterada do sistema (TMTTF)2PF6. Palavras-chave: fenômenos críticos; parâmetro de Grüneisen; sistemas eletrônicos fortemente correlacionados. Contents 1 Introduction 1 2 Theoretical investigations employing the Grüneisen parameter 5 2.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Detailed deduction of the effective Grüneisen parameter . . . 7 2.2.2 The Brillouin-like paramagnet . . . . . . . . . . . . . . . . . 15 2.2.3 The Friedman equations . . . . . . . . . . . . . . . . . . . . 20 2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.1 The magnetic Grüneisen parameter for the Brillouin paramagnet 27 2.3.2 Genuine zero-field quantum phase transitions? . . . . . . . . 28 2.3.3 Grüneisen parameter and the canonical definition of temperature 33 2.3.4 Adiabatic magnetization in a paramagnetic insulating system 36 2.3.5 Elastocaloric-induced effect adiabatic magnetization . . . . . 38 2.3.6 Grüneisen parameter and second-order phase transitions . . 48 2.3.7 Generalization of the Grüneisen parameter . . . . . . . . . . 52 2.3.8 Maximization of caloric effects . . . . . . . . . . . . . . . . . 56 2.3.9 Electric Grüneisen parameter and quantum paraelectricity . 59 2.3.10 Grüneisen parameter and the expansion of the universe . . . 62 3 Experimental investigations on the possible multiferroic character of the Fabre salts 74 3.1 Strongly correlated electronic phenomena in the (TMTTF)2X salts 74 3.2 Experimental aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.2.1 Principle of work of the Teslatron-PT cryostat . . . . . . . . 83 3.2.2 Cleaning up of the zeolite trap . . . . . . . . . . . . . . . . . 90 3.2.3 Pump and purge of the 4He circulation lines . . . . . . . . . 93 3.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.3.1 Dielectric constant . . . . . . . . . . . . . . . . . . . . . . . 94 3.3.2 Raman spectra and fluorescence background of the TMTTF salts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4 Summary and conclusions 101 ix Contents x 5 Perspectives and outlook 103 Bibliography 104 6 Appendices 117 6.1 Absence of a perfect alignment between µ⃗ and B⃗ . . . . . . . . . . . 117 6.2 The impossibility of connecting adiabatic deformations and the concept of negative temperatures . . . . . . . . . . . . . . . . . . . 119 6.3 Power down during experiments . . . . . . . . . . . . . . . . . . . . 122 6.4 Technical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.5 Joint reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.6 Tutorships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.7 Participation in scientific events . . . . . . . . . . . . . . . . . . . . 126 6.8 Prizes and awards . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.9 Graduate student representative . . . . . . . . . . . . . . . . . . . . 128 6.10 Published manuscripts as a co-author . . . . . . . . . . . . . . . . . 129 6.11 Published manuscripts as first author . . . . . . . . . . . . . . . . . 129 6.12 Fapesp agency divulgation articles . . . . . . . . . . . . . . . . . . . 192 “The graveyard is the richest place on Earth, because it is there that you will find all the hopes and dreams that were never fulfilled, the books that were never written, the songs that were never sung, the inventions that were never shared, the cures that were never discovered, all because someone was too afraid to take that first step, keep with the problem, or determined to carry out their dream.” — Les Brown (1945 -) 1 Introduction Thermodynamics plays an unquestionable crucial role in Physics covering mathe- matical descriptions from heat engines [1, 2] to black holes [3], being considered as one of the most relevant fields in Physics in the last centuries [4]. Upon browsing classical Thermodynamics textbooks [1, 2], it is easily noticed that most of its applications are based on industrial machines and this is not a mere coincidence. Indeed, Thermodynamics was the driving intellectual force behind the industrial revolution and all its applications that emerged from this process in the beginning of the nineteenth century [4]. Only in the last decades, Thermodynamics started to play an increasingly key-role in other fields, such as Biology. A few examples lies, for instance, in the description of cell compartmentalization [5] or brain criticality in neural networks [6]. Afterwards, Thermodynamics was employed in modern technology associated with eco-friendly cooling, the so-called caloric effects [7], which refers to the temperature decrease in response to an adiabatic removal of a tuning parameter, which can be pressure or magnetic field, for example. In this context, the smoking-gun to explore critical phenomena and caloric effects is the so-called Grüneisen ratio Γ [8, 9], which is defined as the ration between thermal expansivity and the heat capacity at constant pressure. Also, such physical quantity is the appropriate experimental tool to probe quantum critical points [10]. Hence, it becomes evident that Γ is a key quantity in Thermodynamics. This Ph.D. Thesis 1 1. Introduction 2 is based in two main chapters: a theoretical and an experimental one. At this point, it is worth mentioning that all theoretical topics covered in this Thesis, including the elastocaloric effect, adiabatic magnetization, among others, were possible to be investigated only because of the intense experimental activities and the robust experimental background of the Solid State Physics Laboratory at Unesp - Rio Claro, SP. Thus, in the frame of this Ph.D. Thesis, a theoretical background is reviewed followed by the obtained results regarding modern and cutting-edge applications of Γ. In the last decades, molecular conductors have been recognized as an appropriate playground for the exploration of correlated phenomena in low-dimensions [11, 12]. A class of systems of particular interest is the (TMTTF)2X, where TMTTF is the base molecule tetramethyltetrathiafulvalene and X a monovalent centrosymmetric counter-anion, such as PF6, AsF6, and SbF6. This electronic system enables the investigation of contemporary aspects in the field of correlated phenomena, like the metal-Mott insulator transition [12] and the charge-ordered phase [11]. It is possible to change the physical properties of such systems either by employing chemical substitution in the base molecules or by applying external hydrostatic pressure [13, 14, 15]. The samples of such systems are single-crystals, which enables the investigation of correlated phenomena in a clean environment since no doping is made, i.e., absence of disorder via doping, being only the replacement of the counter-anions X in the electrochemical synthesis performed. In 2018, the so-called Mott-Hubbard ferroelectric phase was investigated in the Fabre salts [16]. A proper comparison between the physical properties of the fully-hydrogenated and the 97.5% deuterated variants of the system (TMTTF)2PF6 was performed via systematic quasi-static dielectric constant measurements. It was theoretically predicted that the Mott-Hubbard ferroelectric phase would present magnetoelectric effects [17]. Hence, the investigations regarding possible multiferroic/magnetoelectric effects in these systems is a hot topic nowadays, which is the main focus of the experimental part of this Ph.D. Thesis. After this brief introduction, it is worth mentioning that this Ph.D. Thesis is divided in the following way: 1. Introduction 3 • Chapter 2: this chapter is focused on the obtained theoretical results. In the beginning, a theoretical background on the detailed derivation of the effective Grüneisen parameter and its relation with the Grüneisen ratio is presented. After, a brief recapitulation of the fundamental aspects of the Brillouin-like paramagnet is performed, followed by the theoretical results obtained in this Ph.D. regarding zero-field quantum criticality, the relation between the canonical definition of temperature and the magnetic Grüneisen parameter, and caloric effects. Also, a discussion about the Friedmann equations and the Einstein field equations are also presented, as well as the application of the Grüneisen parameter to Cosmology (submitted manuscript). In this context, the following manuscripts were published with the obtained results discussed in this chapter: Physical Review B 100, 054446 (2019), Scientific Reports 10, 7981 (2020), Scientific Reports 11, 9431 (2021), Materials Research Bulletin 142, 111413 (2021). • Chapter 3: this chapter is dedicated to the experimental results obtained in this Ph.D. Thesis regarding the molecular conductors of the (TMTTF)2X salts. A brief introduction, motivation, and state-of-the art are presented. Before discussing the results, a compilation of the technical aspects regarding the operation and use of the Teslatron-PT cryostat is included. The experimental results are then presented followed by its discussion. • Chapter 4: here, the summary and conclusions of this Thesis are provided. • Chapter 5: perspectives and outlook are briefly discussed. • Some appendices are included in the end of the Thesis regarding the non- perfect alinement between the magnetic moment and the external magnetic field, the impossibility of connecting adiabatic deformations and the concept of negative temperatures, power down during experiments, technical aspects like the fixing of a SMB connector, participation in joint reports during the Ph.D. degree, tutorships, attendance to scientific events, prizes and awards, 1. Introduction 4 participation as the graduate student representative, and the full texts of the manuscripts that were published in the frame of this Thesis. “Take a good rest, small bird,” he said. “Then go in and take your chance like any man or bird or fish.” — Ernest Hemingway, The Old Man and the Sea. 2 Theoretical investigations employing the Grüneisen parameter Contents 2.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Theoretical background . . . . . . . . . . . . . . . . . . . 7 2.2.1 Detailed deduction of the effective Grüneisen parameter 7 2.2.2 The Brillouin-like paramagnet . . . . . . . . . . . . . . . 15 2.2.3 The Friedman equations . . . . . . . . . . . . . . . . . . 20 2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.1 The magnetic Grüneisen parameter for the Brillouin paramagnet . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.2 Genuine zero-field quantum phase transitions? . . . . . 28 2.3.3 Grüneisen parameter and the canonical definition of tem- perature . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.4 Adiabatic magnetization in a paramagnetic insulating system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.5 Elastocaloric-induced effect adiabatic magnetization . . 38 2.3.6 Grüneisen parameter and second-order phase transitions 48 2.3.7 Generalization of the Grüneisen parameter . . . . . . . 52 2.3.8 Maximization of caloric effects . . . . . . . . . . . . . . 56 2.3.9 Electric Grüneisen parameter and quantum paraelectricity 59 2.3.10 Grüneisen parameter and the expansion of the universe 62 In this chapter, several distinct physical scenarios that were explored using the various definitions of the Grüneisen parameter, namely effective, magnetic, 5 2. Theoretical investigations employing the Grüneisen parameter 6 electric, elastic, and the Grüneisen ratio, are presented, which essentially covers magnetism at ultra low-temperatures, finite-temperature critical points, quantum critical points, caloric effects, and the dark energy problem in modern Cosmology. The obtained results for each topic of research are presented, as well as the published manuscripts that originated from these investigations. Key equations are boxed throughout the text. 2.1 State of the art During the last four decades, there have been an enormous interest to explore the behaviour of the thermodynamical quantities in the critical regime, i.e., close to a finite-temperature critical point (CP). Particular examples include brain criticality in the frame of neural networks [6] and the onset of unconventional superconductivity [18, 19]. Besides finite-temperature critical points, investigations on quantum critical points have been growing continuously since they can be related to superconductivity [20, 21] or even to quantum Griffiths phases [22], just to mention a few key examples. In this context, the so-called Grüneisen ratio Γ [9, 10], which is given by the ratio between thermal expansivity and the specific heat, plays a fundamental role in exploring critical phenomena, since it quantifies the entropy variation in terms of temperature and the tuning parameter, which can be pressure p or external magnetic field B, for instance. It is well known that upon approaching a finite-temperature CP, Γ is enhanced close to the critical parameters [23]. Over the years, Γ has been employed not only to explore finite-temperature CP [23], but also magnetic-field induced quantum CP [21] and zero-field quantum CP [24, 25, 26]. When the tuning parameter is B, the magnetic Grüneisen parameter Γmag is the proper quantity to probe, for instance, quantum criticality [10]. Also, Γmag quantifies the so-called magnetocaloric effect [8], which is the temperature decrease due to an adiabatic removal of B [1]. Hence, this chapter focus on the obtained theoretical results regarding critical phenomena employing the various definitions of Γ. 2. Theoretical investigations employing the Grüneisen parameter 7 2.2 Theoretical background Before starting the discussions of this chapter followed by the obtained results, a theoretical background based on the discussion on Ref. [27] regarding the effective and magnetic Grüneisen parameters, the Grüneisen ratio, and the well-known Brillouin-like paramagnet are provided. 2.2.1 Detailed deduction of the effective Grüneisen param- eter First, it is considered a system where an infinitesimal pressure and temperature variation, namely dp and dT , respectively, can take place as a result of both an infinitesimal entropy variation dS and an infinitesimal strain variation dε, so that the coupled equations for the differentials can be written as [27]: dp = ( ∂p ∂ε ) S dε+ ( ∂p ∂S ) ε dS, (2.1) dT = ( ∂T ∂ε ) S dε+ ( ∂T ∂S ) ε dS. (2.2) Also, an isotropic compression is considered, so that ε = ∆v/v0 = (v − v0)/v0, being v the system’s volume and v0 its initial volume. Dividing both sides of Eq. 2.1 by dε, it reads: dp dε = ( ∂p ∂ε ) S + ( ∂p ∂S ) ε dS dε , (2.3) dp dε = ( ∂p ∂ε ) S + ( ∂p ∂S ) ε dS dε . (2.4) Assuming an isothermal case, i.e., dT = 0, Eq. 2.2 becomes: 0 = ( ∂T ∂ε ) S dε+ ( ∂T ∂S ) ε dS, (2.5) − ( ∂T ∂S ) ε dS = ( ∂T ∂ε ) S dε, (2.6) 2. Theoretical investigations employing the Grüneisen parameter 8 −dS = ( ∂T ∂ε ) S dε( ∂T ∂S ) ε , (2.7) dS dε = − ( ∂T ∂ε ) S( ∂T ∂S ) ε . (2.8) Replacing Eq. 2.8 in Eq. 2.4, it reads: dp dε = ( ∂p ∂ε ) S + ( ∂p ∂S ) ε − ( ∂T ∂ε ) S( ∂T ∂S ) ε  , (2.9) dp dε = ( ∂p ∂ε ) S − ( ∂p ∂S ) ε ( ∂S ∂T ) ε ( ∂T ∂ε ) S . (2.10) Considering the generalized differential relation for the state functions given by [28]:( ∂x ∂z ) g = ( ∂x ∂y ) g ( ∂y ∂z ) g . (2.11) Assuming x = p, y = S, z = T , and g = ε in Eq. 2.11, it reads:( ∂p ∂T ) ε = ( ∂p ∂S ) ε ( ∂S ∂T ) ε . (2.12) Replacing Eq. 2.12 into Eq. 2.10: dp dε = ( ∂p ∂ε ) S − ( ∂p ∂T ) ε ( ∂T ∂ε ) S . (2.13) Assuming that dp = (∂p/∂ε)Tdε, Eq.2.13 becomes:( ∂p ∂ε ) T = ( ∂p ∂ε ) S − ( ∂p ∂T ) ε ( ∂T ∂ε ) S , (2.14) ( ∂p ∂ε ) T = ( ∂p ∂ε ) S − ( ∂p ∂T ) ε ( ∂T ∂ε ) S . (2.15) The term (∂p/∂ε)T in Eq. 2.15 can be rewritten based on the definition of ε = ∆v/v0 = (v − v0)/v0, so that:( ∂p ∂ε ) T = [ ∂p ∂(∆v/v0) ] T = [ ∂p 1 v0 ∂(∆v) ] T = v0 ( ∂p ∂∆v ) T = v0 [ ∂p ∂(v − v0) ] T . (2.16) Considering the case of a compression, v is always lower than v0 so that ∂[v − v0] = −∂v and Eq. 2.16 reads:( ∂p ∂ε ) T = v0 ( ∂p −∂v ) T = −v0 ( ∂p ∂v ) T . (2.17) 2. Theoretical investigations employing the Grüneisen parameter 9 The right side of Eq. 2.17 is the definition of the isothermal bulk modulus BT [28], so that: ( ∂p ∂ε ) T = −v0 ( ∂p ∂v ) T = BT . (2.18) The same analysis can be made in terms of (∂p/∂ε)S in Eq. 2.15, so that: ( ∂p ∂ε ) S = BS, (2.19) where BS is the adiabatic bulk modulus [28]. The term (∂T/∂ε)S in Eq. 2.15 can also be rewritten in terms of ε = ∆v/v0 = (v − v0)/v0, so that: ( ∂T ∂ε ) S = [ ∂T ∂(∆v/v0) ] S = [ ∂T 1 v0 ∂(∆v) ] S = v0 [ ∂T ∂(∆v) ] S = v0 [ ∂T ∂(v − v0) ] S , (2.20) so that: ( ∂T ∂ε ) S = v0 ( ∂T −∂v ) S = −v0 ( ∂T ∂v ) S . (2.21) Now, replacing Eqs. 2.18, 2.19, and 2.21 into Eq. 2.15, it reads: BT = BS − ( ∂p ∂T ) ε [ −v0 ( ∂T ∂v ) S ] , (2.22) (BT − BS) = ( ∂p ∂T ) ε ( ∂T ∂v ) S v0. (2.23) Since the condition of a constant v strictly implies in a constant ε as well: ( ∂p ∂T ) ε = ( ∂p ∂T ) v . (2.24) The term (∂p/∂T )v can be rewritten in terms of the well-known relation [28]: ( ∂v ∂p ) T ( ∂p ∂T ) v ( ∂T ∂v ) p = −1, (2.25) so that: ( ∂p ∂T ) v = − 1( ∂v ∂p ) T ( ∂T ∂v ) p . (2.26) 2. Theoretical investigations employing the Grüneisen parameter 10 The term (∂T/∂v)p is related to the definition of the thermal expansion coefficient at constant pressure αp by (∂v/∂T )p = αpv0 [28], so that (∂T/∂v)p = 1/(αpv0) and Eq. 2.26 reads: ( ∂p ∂T ) v = − 1( ∂v ∂p ) T 1 αpv0 = − αpv0( ∂v ∂p ) T . (2.27) Following the definition of BT , the term in Eq. 2.27 (∂v/∂p)T = −v0/BT and thus Eq. 2.27 becomes: ( ∂p ∂T ) v = −αp��v0 −��v0 BT = αpBT . (2.28) Now, the term (∂T/∂v)S in Eq. 2.23 can be rewritten employing the relation of Eq. 2.11 considering x = T , y = p, z = v, and g = S, so that:( ∂T ∂v ) S = ( ∂T ∂p ) S ( ∂p ∂v ) S . (2.29) The term (∂T/∂p)S can be rewritten following the relation [28]:( ∂T ∂p ) S ( ∂S ∂T ) p ( ∂p ∂S ) T = −1, (2.30) ( ∂T ∂p ) S = − 1( ∂S ∂T ) p ( ∂p ∂S ) T . (2.31) Plugging Eq. 2.31 into Eq. 2.29: ( ∂T ∂v ) S = − 1( ∂S ∂T ) p ( ∂p ∂S ) T ( ∂p ∂v ) S = − ( ∂S ∂p ) T( ∂S ∂T ) p ( ∂p ∂v ) S . (2.32) Given the fact that αp = (1/v0)(∂v/∂T )p = (1/v0)(−∂S/∂p)T and the heat capacity at constant pressure cp = T (∂S/∂T )p [28], Eq. 2.32 becomes:( ∂T ∂v ) S = αpv0T cp ( ∂p ∂v ) S . (2.33) Given the definition of BS = −v0(∂p/∂v)S [28], the term in Eq. 2.33 (∂p/∂v)S = −BS/v0, so that Eq. 2.33 reads:( ∂T ∂v ) S = αp��v0T cp ( −BS ��v0 ) = −αpTBS cp . (2.34) 2. Theoretical investigations employing the Grüneisen parameter 11 Finally, inserting Eqs. 2.28 and 2.34 in Eq. 2.23, it reads: (BT − BS) = αpBT ( −αpBST cp ) v0 = −αpBTT ( αpBSv0 cp ) . (2.35) Multiplying both sides of Eq. 2.35 by −1, it reads: (BS − BT ) = αpBTT ( αpBSv0 cp ) , (2.36) BS = BT + αpBTT ( αpBSv0 cp ) = BT [ 1 + αpT ( αpBSv0 cp )] , (2.37) BS BT = 1 + αpT ( αpBSv0 cp ) . (2.38) Employing the relation for the partial derivatives in Eq. 2.11 assuming x = p, y = T , z = S, and g = v, it reads:( ∂p ∂S ) v = ( ∂p ∂T ) v ( ∂T ∂S ) v . (2.39) Employing the Maxwell relation (∂p/∂S)v = −(∂T/∂v)S [28] and the term (∂T/∂v)S computed in Eq. 2.34, so that, (∂p/∂S)v = αpBST/cp. The latter can be replaced in Eq. 2.39 together with Eq. 2.28 and the definition of the heat capacity at constant volume cv = T (∂S/∂T )v [28], i.e., (∂T/∂S)v = T/cv, so that Eq. 2.39 now reads: ��αp BS��T cp = ��αp BT ��T cv , (2.40) BS cp = BT cv , (2.41) which can be rewritten simply as: BS BT = cp cv . (2.42) Replacing Eq. 2.42 into Eq. 2.38, it reads: cp cv = 1 + αpT ( αpv0 cp BS ) = 1 + αpT ( αpv0 �� cp �� cp BT cv ) = 1 + αpT Γeff︷ ︸︸ ︷( αpv0BT cv ) . (2.43) The dimensionless physical quantity between parenthesis in Eq. 2.43 is the effective Grüneisen parameter [29, 9]: Γeff = αpv0BT cv . (2.44) 2. Theoretical investigations employing the Grüneisen parameter 12 Note that Γeff is also related to the specific heat ratio, namely Γeff = (cp/cv) − 1 [30]. It is worth mentioning that Γeff can be rewritten in terms of the internal energy U . Employing the definition of Γeff in Eq. 2.44 and, considering αpBT = (∂p/∂T )v [27] and cv = (∂U/∂T )v, we have: Γeff = v0 ( ∂p ∂T ) v( ∂U ∂T ) v = v0 ( ∂p ∂T ) v ( ∂U ∂T )−1 v . (2.45) Employing the relation from Eq. 2.11, one can write: ( ∂p ∂U ) v = ( ∂p ∂T ) v ( ∂T ∂U ) v = ( ∂p ∂T ) v ( ∂U ∂T ) v −1 . (2.46) Replacing Eq. 2.46 into Eq. 2.45, it reads [9, 31]: Γeff = v0 ( ∂p ∂U ) v . (2.47) Strictly speaking, Γeff must be computed upon taking into account all the entropy contributions to the system, namely phononic Sph and electronic Sel, so that Stot = Sph + Sel. Rewriting Γeff upon considering that (∂S/∂ ln v)T = v0αpBT and (∂S/∂ lnT )v = cv, Γeff reads [31]: Γeff = ( ∂S ∂ ln v ) T( ∂S ∂ ln T ) v . (2.48) Replacing the expression for Stot in Eq. 2.48, we have: Γeff = [ ∂(Sph+Sel) ∂ ln v ] T[ ∂(Sph+Sel) ∂ ln T ] v , (2.49) Γeff = ( ∂Sph ∂ ln v ) T + ( ∂Sel ∂ ln v ) T( ∂Sph ∂ ln T ) v + ( ∂Sel ∂ ln T ) v . (2.50) Employing Eq. 2.48 into Eq. 2.50, it reads: Γeff = Γph effc ph v + Γel effc el v cph v + cel v , (2.51) 2. Theoretical investigations employing the Grüneisen parameter 13 where the phononic and electronic contributions are taken into account both in Γeff and in cv. Therefore, Eq. 2.51 can be written in its final form [31]: Γeff = ∑ i Γi effc i v∑ i ci v . (2.52) Equation 2.52 shows that although S is an additive quantity in a system, Γeff is not. After almost 100 years that Eduard Grüneisen (1877-1949) proposed the relation shown in Eq. 2.44, it was proposed in the literature that a singular portion of Γeff can be employed to explore the vicinities of a pressure-induced quantum critical point [8]. Equation 2.44 can be rewritten as [8]: Γeff = αp cp cp cv v0BT = Γcp cv v0BT , (2.53) where Γ is the so-called Grüneisen ratio [8]. Since αp is more singular than cp close to a pressure-induced quantum critical point [8], Γ is the sensitive quantity that presents a divergent-like behaviour in the vicinity of the quantum critical point. The reason αp is more pronounced than cp close to a QCP, or a finite-T critical end point, is that usually T is fixed and the tuning parameter is varied, making the S variation more pronounced upon varying the tuning parameter. Another way of writing Γ is employing the definition of αp and the molar heat capacity at constant pressure cp = T/N(∂S/∂T )p [28], where N is the number of particles in the system, so that: Γ = αp cp = 1 v0 ( ∂v ∂T ) p  1 T N ( ∂S ∂T ) p  = 1 vmT ( ∂v ∂T ) p( ∂S ∂T ) p , (2.54) where vm = v/N is the molar volume. Employing the Maxwell relation (∂v/∂T )p = −(∂S/∂p)T [28], Γ reads [8]: Γ = − 1 vmT ( ∂S ∂p ) T( ∂S ∂T ) p . (2.55) Using the relation [28]: ( ∂S ∂T ) p ( ∂T ∂p ) S ( ∂p ∂S ) T = −1, (2.56) 2. Theoretical investigations employing the Grüneisen parameter 14 ( ∂T ∂p ) S = − 1( ∂S ∂T ) p ( ∂p ∂S ) T = − ( ∂S ∂p ) T( ∂S ∂T ) p . (2.57) Replacing Eq. 2.57 in Eq. 2.55, Γ can be written as: Γ = 1 vmT ( ∂T ∂p ) S . (2.58) Equation 2.58 shows that Γ quantifies the temperature change upon adiabatically varying p, i.e., the so-called barocaloric effect. A couple of years after the proposal that Γ is the singular contribution of Γeff in the vicinities of quantum critical points [8], it was also proposed by Prof. Mariano and collaborators that Γ can be also employed to explore finite-T critical points as well [23]. The authors of Ref. [8] also proposed that Γ can be rewritten for a magnetic-field-induced quantum critical point, so that [8]: Γmag = − ( ∂M ∂T ) B cB = − 1 T ( ∂S ∂B ) T( ∂S ∂T ) B , (2.59) where M is the modulus of the magnetization, cB the heat capacity at constant magnetic field, Γmag the magnetic Grüneisen parameter, and B the modulus of the external magnetic field. Employing the relation [28]: ( ∂S ∂T ) B ( ∂T ∂B ) S ( ∂B ∂S ) T = −1, (2.60) − ( ∂T ∂B ) S = 1( ∂S ∂T ) B ( ∂B ∂S ) T = ( ∂S ∂B ) T( ∂S ∂T ) B . (2.61) Replacing Eq. 2.61 in Eq. 2.59, it reads: Γmag = − 1 T [ − ( ∂T ∂B ) S ] , (2.62) Γmag = 1 T ( ∂T ∂B ) S . (2.63) 2. Theoretical investigations employing the Grüneisen parameter 15 Equation 2.63 shows that Γmag quantifies the so-called magnetocaloric effect [1], i.e., the temperature decrease due to an adiabatic removal of the external magnetic field at a starting temperature T , which is accounted by the pre-factor 1/T . Hence, it becomes evident that Γ and Γmag are key-parameters in probing both classical and quantum criticality, as well as in the quantification of caloric effects. 2.2.2 The Brillouin-like paramagnet Before discussing the results, for the sake of completeness, the fundamental physical aspects of the Brillouin-type paramagnet are briefly recalled, since those are connected with some of the obtained theoretical results obtained. Just to mention, the modulus of many vectorial physical quantities discussed in this Ph.D. Thesis, such as the external magnetic field, magnetization, and related quantities, are considered. The well-known from textbooks Brillouin-type paramagnet is a non- interacting model that describes the behaviour of a paramagnetic insulating system [32]. The spins are localized and the system has only two eigenstates, i.e., there is a probability that the magnetic moment is either aligned in a parallel or an anti-parallel configuration with respect to B, being hereafter defined as Ppar and Panti−par, respectively. Such probabilities are given by [1]: Ppar(T,B) = eµBB/kBT 2 cosh (µBB/kBT ) , (2.64) Panti−par(T,B) = e−µBB/kBT 2 cosh (µBB/kBT ) , (2.65) where kB is Boltzmann constant (1.38×10−23) J/K and µB the Bohr magneton (9.27×10−24 J/T). Recalling that the entropy S can be computed employing [1]: S = −kB ∑ j P (ψj) ln (P (ψj)), (2.66) where ψj represents the jth eigenstates. Since for the case of an insulating paramagnet there is only the probabilities Ppar and Panti−par, it is then written: S = −kB[Ppar ln (Ppar) + Panti−par ln (Panti−par)]. (2.67) 2. Theoretical investigations employing the Grüneisen parameter 16 Inserting Eqs. 2.64 and 2.65 into Eq. 2.67, it is straightforward to write [1]: S(T,B) = −BµB T tanh ( BµB kBT ) + kBln [ 2 cosh ( BµB kBT )] , (2.68) Upon analysing the high-temperature limit for Eq. 2.68 at a fixed B, it is obtained S(T → ∞, B = constant) = kB ln (2), which indicates that the two eigenstates are equally probable. In the opposite side, at a fixed B, when T → 0 ⇒ S → 0, as expected by the third law of Thermodynamics [1]. The latter indicates that the ground-state of a Brillouin-type paramagnet is a ferromagnetic phase. This can be analysed in terms of the two spin populations N1 and N2, which refer, respectively, to the number of spins aligned in parallel or anti-parallel with respect to B. The expressions for N1 and N2 with respect to the total spin population NT = (N1 +N2) are well-known in textbooks [33], being given by: N1 NT = exp(µBB/kBT ) exp(µBB/kBT ) + exp(−µBB/kBT ) (2.69) and N2 NT = exp(−µBB/kBT ) exp(µBB/kBT ) + exp(−µBB/kBT ) . (2.70) Figure 2.1 depicts that, on one hand, when T → 0 K, all magnetic moments are in a parallel alignment with respect to B. On the other hand, at high temperatures, i.e., T → ∞, the populations N1/NT and N2/NT are equally probable. Also, the heat capacity at constant magnetic field cB can be computed employing the expression cB = T (∂S/∂T )B and Eq. 2.68, so that: cB (T,B) = T ( ∂S ∂T ) B = B2µB 2 kBT 2 sech2 ( µBB kBT ) . (2.71) The maximum in cB shown in Fig. 2.2 for various fixed values of B is associated with the famous Schottky anomaly [34, 35]. Such an anomaly is a fingerprint of two-level systems and it shows that, upon lowering T , there is a particular condition in which entropy is dramatically varied as a consequence of the ordering of the system when T → 0 K. Upon approaching such a condition, it becomes energetically more favorable for the system to order in a long-range fashion and thus the population N1/NT is favored, which in turn dramatically affects the available eigenstates of 2. Theoretical investigations employing the Grüneisen parameter 17 Figure 2.1: Spin populations N1/NT (up) and N2/NT (down) represented by the red and blue color data, respectively. Note that when T → 0 K, N1/NT → 1 and N2/NT → 0, which means that the ground-state of the Brillouin-type paramagnet is a ferromagnetic phase. In the regime of high temperatures, i.e., T → ∞, both N1/NT and N2/NT go to 0.5, which means that both configurations, namely spin up and down configurations are equally probable [S = kB ln (2)]. The arrows illustrate the spin populations in two distinct temperatures. the system and, as a consequence, the entropy. As well-known from textbooks, such a maximum in cB takes place when (kBT )/(µBB) ≃ 0.834 [34, 35, 36], i.e., when the thermal energy matches exactly the value of the energy between the two levels ∆. Hence, as depicted in Fig. 2.2, upon increasing the fixed value of B, the Schottky anomaly will take place at higher temperatures. This happens because when a higher B is applied, ∆ is increased so that the condition for the Schottky anomaly to take place is thus reached at higher temperatures. In other words, given that (kBT )/(µBB) ≃ 0.834 [34, 35], if the magnetic energy increases then the thermal energy has to be increased as well in order to satisfy such a condition, which in turn makes the Schottky anomaly to be reached at higher temperatures upon increasing B. At very high temperatures, all curves shown in Fig. 2.2 converge to the same value of cB, since S does not vary much with increasing T , namely S → kB ln (2). Regarding S for the Brillouin-type paramagnet, for B = 0 T ⇒ S = kB ln (2) which is a non-temperature dependent S and thus it violates the 2. Theoretical investigations employing the Grüneisen parameter 18 0 1 2 3 4 0 1 2 3 4 5 6 c B (x 10 -24 J/ K) T ( K ) B = 0 . 5 T B = 1 T B = 2 T Figure 2.2: Heat capacity at constant magnetic field cB as a function of T for B = 0.5 T (blue line), B = 1 T (red line), and B = 2 T (orange color line). The maximum on cB is associated with the Schottky anomaly for two-level systems [34, 35]. More details in the main text. third law of Thermodynamics, i.e., S → 0 as T → 0 [1]. Hence, to solve this issue, the mutual interactions between neighboring magnetic moments must be considered [37, 1]. At this point, the magnetic energy E associated with the dipolar interactions between adjacent spins is recalled, namely [32]: E = µ0 4π| r⃗ |3 [ µ⃗1 · µ⃗2 − 3 | r⃗ |2 (µ⃗1 · r⃗)(µ⃗2 · r⃗) ] , (2.72) where µ0 is the vacuum permeability (4π×10−7) T·m/A, r⃗ the distance between neighbouring spins, and µ⃗1 and µ⃗2 two arbitrary neighbouring magnetic moments. Considering no applied B, it can be written |E| = µBBloc, where Bloc is the modulus of the effective local magnetic field that emerges into the system as a result of the dipolar interactions between neighboring moments. Then, Eq. 2.72 becomes: µBBloc = µ0 4π| r⃗ |3 [ µ⃗1 · µ⃗2 − 3 | r⃗ |2 (µ⃗1 · r⃗)(µ⃗2 · r⃗) ] . (2.73) Assuming |µ⃗1| = |µ⃗2| = µB = (9.27 × 10−24) J/T and a typical distance between spins given by | r⃗ | = (5Å = 5 × 10−10) m, it is straightforward to notice that the 2. Theoretical investigations employing the Grüneisen parameter 19 second term on the right side of the brackets does not significantly contribute to E, so that Eq. 2.73 can be approximated to: Bloc ≃ µ0µ 4π| r⃗ |3 ≃ (4π × 10−7)(9.27 × 10−24) 4π(5 × 10−10)3 ≃ 0.01T. (2.74) Equation 2.74 enables us to conclude that Bloc = 0.01 T is intrinsically present in real paramagnets, since interactions between magnetic moments are always present. Therefore, since both B and Bloc are acting on the system, it is natural to assume that the paramagnetic system is under the influence of a resultant magnetic field, being its modulus Br, based on the angle θ between B⃗ and B⃗loc, which, employing the cosine rule, reads: Br 2 = B2 +Bloc 2 − 2BBloc cos (θ). (2.75) For the sake of simplicity, it is assumed that θ ≈ 90◦, so that: Br 2 = B2 +Bloc 2 − ≈ 0︷ ︸︸ ︷ 2BBloc cos (θ) . (2.76) Hence, Br takes the form: Br ≃ √ B2 +Bloc 2. (2.77) Now, replacing B for Br in Eq. 2.68, it reads: S(T,B) = − µB √ B2 +Bloc 2 T tanhµB  √ B2 +Bloc 2 kBT + kBln 2 cosh µB √ B2 +Bloc 2 kBT  . (2.78) Note that, even when B = 0 T, Bloc is still non-zero and thus S assumes a temperature-dependent mathematical form, which is now in perfect agreement with the third law of Thermodynamics [37]. Thus, although the Brillouin-type paramagnet model is non-interacting, the effective dipolar interactions between adjacent spins must be considered. It is worth mentioning that the modulus of the magnetic energy Emag associated with Bloc is given by Emag = µBBloc = (9.27 × 10−24)(0.01) = (9.27 × 10−26) J. Comparing such a result with the thermal 2. Theoretical investigations employing the Grüneisen parameter 20 energy T ≃ (9.27 × 10−26)/kB ≃ (9.27 × 10−26)/(1.38 × 10−23) ≃ 6.7 mK. Such a result indicates that the relevance of the mutual interactions takes place on temperatures T ≤ 6.7 mK. In Section 2.3, the obtained results regarding the Brillouin-type paramagnet and an investigation of the role played by Bloc are discussed in a broader context. 2.2.3 The Friedman equations Before discussing the results obtained in the frame of this Thesis regarding the application of both Γ and Γeff to Cosmology, a Newtonian-based derivation of the Friedman equations [38] without the cosmological constant Λ is properly given based on discussions presented in Ref. [39]. Note that Friedman has derived his equations in 1922 using Einstein field equations to describe a spatially homogeneous and isotropic expanding universe, so that a full derivation of Friedman equations including all general relativistic effects is not covered here. The latter is properly reported in Ref. [40]. Friedman has published his equations 7 years before Hubble reported what is known today as the Hubble law [41], showing that galaxies are moving away from each other and the further away they are, the faster they move away. Hubble proposed a linear relationship between the radial velocity vR of galaxies and the distance to a particular galaxy d, so that H = vRd [41], where H is the Hubble parameter broadly discussed in Cosmology. The Hubble law is considered one of the most important relations in Cosmology [39]. Just to mention, the first experimental evidence that the universe is expanding was reported in 1912 by V. M. Slipher [42], while the observational evidence that the expansion of the universe was accelerated was only reported in 1998, cf. Ref. [43]. Back to the Newtonian-based derivation of the Friedman equations, first it is considered that the universe is a homogeneous sphere of matter with a mass Ms that does not vary in time. Such a sphere can be either contracting or expanding isotropically over time, so that its radius Rs(t) is time-dependent. Considering a test mass m at the surface of the sphere, the modulus of the gravitational force F experienced by the 2. Theoretical investigations employing the Grüneisen parameter 21 test mass considering Newton’s law of gravity is given by [44]: F = −GMsm Rs 2 , (2.79) where G is the universal gravitational constant. Considering Newton’s second law, F can be written in terms of the gravitational acceleration, which is given by: F = ma = m d2Rs dt2 , (2.80) which reads: ��m d2Rs dt2 = −GMs��m Rs 2 , (2.81) d2Rs dt2 = −GMs Rs 2 . (2.82) Multiplying both sides of Eq. 2.82 by dRs/dt, it reads: dRs dt d dt ( dRs dt ) = −GMs Rs 2 dRs dt . (2.83) Integrating both sides:∫ ( dRs dt ) d dt ( dRs dt ) = ∫ −GMs Rs 2 ( dRs dt ) = GMs ∫ − 1 Rs 2 ( dRs dt ) . (2.84) Upon analysing the term (dRs/dt)d/dt(dRs/dt), one can notice employing the chain rule that such a term represents the time derivative of the function 1/2(dRs/dt)2, i.e., d/dt[1/2(dRs/dt)2] = (dRs/dt)d/dt(dRs/dt). Hence, Eq. 2.84 is rewritten as: ∫ d dt 1 2 ( dRs dt )2  = GMs ∫ − 1 Rs 2 ( dRs dt ) . (2.85) Employing basic Calculus, the integral of the derivative of a function yields in the function itself, so that: 1 2 ( dRs dt )2 + c1 = GMs ∫ − 1 Rs 2 ( dRs dt ) . (2.86) Now, the same analysis is employed for the term −1/Rs 2(dRs/dt), i.e., employing the chain rule one can notice that d/dt[1/Rs] = −1/Rs 2(dRs/dt), so that: 1 2 ( dRs dt )2 + c1 = GMs ∫ d dt ( 1 Rs ) , (2.87) 2. Theoretical investigations employing the Grüneisen parameter 22 1 2 ( dRs dt )2 + c1 = GMs Rs + c2, (2.88) 1 2 ( dRs dt )2 = GMs Rs + c3, (2.89) where c1 and c2 are integration constants and c3 = (c2 − c1). Given that Rs varies in time, the volume of the sphere Vs will vary as well and, as a consequence, so the matter density ρm, so that: ρm = Ms Vs ⇒ Vs = Ms ρm . (2.90) Since a spherical shape is considered, Eq. 2.90 reads: Vs = 4 3πRs 3 = Ms ρm , (2.91) Ms = 4 3πRs 3ρm. (2.92) The temporal variation of Rs can be inferred as a consequence of the change in the metric of space, as well as in so-called scale factor a(t), so that: Rs = a(t)rs, (2.93) where rs is the comoving radius of the sphere. Essentially, the comoving radius is a definition of measuring distances in the universe that takes into account the fact that the universe is expanding over time. It helps to keep track of how far apart objects are from each other, even as the universe expands. By making use of Eq. 2.93, the term dRs/dt = ȧ(t)rs. At this point, for the sake of simplicity, a(t) is given solely by a. Then, together with Eq. 2.92, Eq. 2.89 reads: 1 2(ȧrs)2 = G Rs 4 3πRs 3ρm + c3, (2.94) 1 2 ȧ 2rs 2 = 4πG 3 Rs 2ρm + c3. (2.95) Employing Eq. 2.93 into Eq. 2.95: 1 2 ȧ 2rs 2 = 4πG 3 (ars)2ρm + c3, (2.96) 2. Theoretical investigations employing the Grüneisen parameter 23 1 2 ȧ 2rs 2 = 4πG 3 a2rs 2ρm + c3. (2.97) Dividing both sides of Eq. 2.97 by rs 2a2/2, it yields: 1 2 ȧ 2rs 2 rs 2a2 2 = 4πG 3 a2rs 2ρm + c3 rs 2a2 2 , (2.98) ȧ2 a2 = 8πG 3 ρm + 2c3 rs 2 1 a2 , (2.99) ( ȧ a )2 = 8πG 3 ρm + 2c3 rs 2 1 a2 . (2.100) Equation 2.100 is the first Friedman equation in a Newtonian form. It connects the temporal behaviour of the scale factor with ρm(t). However, the correct form of the first Friedman equation must take into account relativistic effects since it is derived from Einstein field equations in the frame of general relativity. Hence, the correct form of the first Friedman equation reads [38]: ( ȧ a )2 = 8πG 3c2 ρ− κc2 rs 2 1 a2 , (2.101) where ρ is the energy density, κ is the space curvature constant, and c is the speed of light. Note that the relation between ρ and ρm is ρm = ρ/c2 [39]. For the derivation of the second Friedman equation, the first-law of Thermodynamics is recalled: dS = dU + pdVS, (2.102) where dU is the infinitesimal internal energy variation. Since the expansion of the universe is adiabatic dS = 0, so that: dU + pdVS = 0. (2.103) Assuming that VS(t) and U(t), Eq. 2.103 is given by: U̇ + pV̇S = 0. (2.104) 2. Theoretical investigations employing the Grüneisen parameter 24 Recalling that Rs(t) = ars, VS(t) reads: VS(t) = 4 3πRs(t)3 = 4 3πa 3rs 3, (2.105) thus: V̇S = 4 3πrs 3(3a2ȧ) = VS(t) ( 3 ȧ a ) . (2.106) Now, U(t) can be rewritten as: U(t) = VS(t)ρ, (2.107) then: U̇ = VS ρ̇+ V̇Sρ = VS ( ρ̇+ 3 ȧ a ρ ) . (2.108) Replacing Eqs. 2.106 and 2.108 into Eq. 2.104: VS ( ρ̇+ 3 ȧ a ρ+ 3 ȧ a p ) = 0, (2.109) ρ̇+ 3 ȧ a ρ+ 3 ȧ a p = 0, (2.110) ρ̇+ 3ȧ a (ρ+ p) = 0. (2.111) Equation 2.111 is the so-called fluid equation [39], connecting the energy density and pressure with the temporal evolution of the scale factor. Now, rewriting the first Friedman equation as: ȧ2 = 8πG 3c2 ρa 2 − κc2 rs 2 . (2.112) Taking the time derivative in both sides of Eq. 2.112, it reads: 2ȧä = 8πG 3c2 (ρ̇a2 + 2ρaȧ). (2.113) Dividing both sides by 2ȧa: ä a = 4πG 3c2 ( ρ̇ a ȧ + 2ρ ) . (2.114) Rewriting Eq. 2.111 as: ρ̇ = −3ȧ a (ρ+ p). (2.115) 2. Theoretical investigations employing the Grüneisen parameter 25 Replacing Eq. 2.115 into Eq. 2.114: ä a = 4πG 3c2 {[ −3 � � �̇a a (ρ+ p) ] � �� a ȧ + 2ρ } , (2.116) ä a = 4πG 3c2 (−3ρ− 3p+ 2ρ) = 4πG 3c2 (−ρ− 3p), (2.117) ä a = −4πG 3c2 (ρ+ 3p) . (2.118) Equation 2.118 is the second Friedman equation, also known as the acceleration equation [39]. As broadly discussed in the literature, the universe is usually described by the picture of a perfect fluid, i.e., an isotropic fluid with no viscosity [45]. The equation of state (EOS) of a perfect fluid is p = ωρ [45, 39], where ω is the so-called EOS parameter. After the Big Bang, various eras of the universe took place. First, the radiation-dominated era and, later on, the matter-dominated era was achieved. During both radiation- and matter-dominated eras, the universe expanded in a decelerated way [45, 39]. Currently, the universe is expanding in an accelerated way [43] because of the so-called dark-energy (DE). Essentially, DE is the vacuum energy and it is taken into account both in Einstein field equations and Friedman equations when the cosmological constant Λ is incorporated in such equations. The term “dark” refers to the fact that the origin of such energy is still unknown during the writing of this Thesis. Just to mention, our universe is composed nowadays of about 69% of dark energy, 7% of dark-matter, and 4% of regular matter, i.e., stars and planets, for instance [39]. The relationship between p and ρ was different for each era, given by distinct values of ω. For the radiation-dominated era ω = 1/3, for the matter-dominated era ω → 0, and for the DE-dominated era ω = −1. It is worth mentioning that ω is merely discussed in the literature as a numerical value, which is discussed in more details in the results section of this Thesis. Regarding the accelerated expansion, the most accepted explanation for such an acceleration is that DE is a fluid with positive ρ, but with a negative pressure [45], namely p = −ρ. For the expansion to be accelerated, the condition ä > 0 must be fulfilled in Eq. 2.118, being the opposite also true for a decelerated expansion ä < 0. Just to mention, since a represents the metric of space, ȧ has units 2. Theoretical investigations employing the Grüneisen parameter 26 of velocity while ä of acceleration [39]. By assuming that the total energy density ρtot of the universe is given by ρtot = (ρradiation + ρmatter + ρDE) and that p is an additive physical quantity, i.e., the total pressure ptot = (pradiation + pmatter + pDE), Eq. 2.118 can be straightforwardly rewritten: ä a = −4πG 3c2 (ρradiation + ρmatter + ρDE + 3pradiation + 3pmatter + 3pDE). (2.119) Considering the perfect fluid EOS and the value of ω for the various eras: ä a = ( ρradiation + ρmatter + ρDE + 31 3ρradiation + �������:0 3 · 0.ρmatter + 3(−1)ρDE ) , (2.120) ä a = −4πG 3c2 (2ρradiation + ρmatter − 2ρDE). (2.121) Note that as the universe expands, a is increased. Also, based on Friedman equations, it is established in the literature that ρradiation ∝ 1/a4 and ρmatter ∝ 1/a3 [46]. Also, ρDE is considered to be time-independent, which is still a topic under debate in the literature. Thus, the expansion of the universe and the increase in the scale factor makes the energy density of radiation and matter to be decreased and, in the DE-dominated era, such contributions can be considered negligible in comparison with ρDE, so that Eq. 2.121 reads: ä a ≃ −4πG 3c2 (−2ρDE), (2.122) ä a ≃ 8πG 3c2 ρDE. (2.123) The minus sign of −2ρDE, which comes from the consideration of a negative pressure associated with DE, makes ä/a to be positive in the DE-dominated assuming that G is always greater than zero. This implies that DE drives the accelerated expansion of the universe. This is one of the key results emerging from the Friedman equations in modern Cosmology [39]. Based on these discussions regarding the perfect fluid picture, Friedman equations, and the accelerated expansion of the universe, we have unprecedentedly connected well-established concepts of condensed matter physics, namely Γ and Γeff , to Cosmology, which is properly discussed in Section 2.3.10. 2. Theoretical investigations employing the Grüneisen parameter 27 2.3 Results 2.3.1 The magnetic Grüneisen parameter for the Brillouin paramagnet By making use of the partition function Z = 2 cosh ( µBB kBT ) of the Brillouin-type paramagnet [1], the free energy F can be computed as: F (T,B) = −kBT ln (Z) = −kBT ln [ 2 cosh ( µBB kBT )] . (2.124) Hence, employing M = −(∂F/∂B)T [28], it reads: M(T,B) = − ( ∂F ∂B ) T = − 2µB kBT sinh ( µBB kBT ) . (2.125) Employing Eq. 2.125, the behaviour of M as a function of T and B can be analysed. It turns out that in the regime of both T → 0 K and B → 0 T, M presents a step-like behaviour, cf. Fig.2.3. As a step further, the analytical calculation of Γmag for the Brillouin-type Figure 2.3: Magnetization M as a function of T and B for the Brillouin-type paramagnet. Note that in the regime of both T and B close to zero, a step-like behaviour of M is observed. Figure extracted from Ref. [47]. paramagnet, namely [47, 8]: Γmag = − ( ∂M ∂T ) B cB = − 1 T (∂S/∂B)T (∂S/∂T )B = 1 T ( ∂T ∂B ) S , (2.126) 2. Theoretical investigations employing the Grüneisen parameter 28 which quantifies the so-called magnetocaloric effect, i.e., the temperature change upon the adiabatic removal of B. For the Brillouin-type paramagnet, Γmag was already reported in the literature by us and is temperature-independent, namely Γmag = 1/B [47, 48], showing a divergent-like behaviour and sign-change when B → 0 T. Such behavior of Γmag motivated a deeper investigation of the Brillouin-type paramagnet in the frame of zero-field quantum phase transitions and the role played by the effective mutual interactions between neighboring magnetic moments in such a model. Note that, as previously discussed, the Brillouin-type paramagnet does not take into consideration the interaction between neighbouring magnetic moments and, in order to not violate the third law of Thermodynamics, the magnetic dipolar interactions must be considered. Some of the obtained theoretical results discussed in this Section were published in (Appendix 6.11): • Gabriel O. Gomes, Lucas Squillante, A.C. Seridonio, Andreas Ney, Roberto E. Lagos, and Mariano de Souza, Magnetic Grüneisen parameter for model systems, Physical Review B 100, 054446 (2019). 2.3.2 Genuine zero-field quantum phase transitions? In this Section, the focus lies on magnetic-field driven quantum phase transitions (QPT), more specifically zero-field quantum phase transitions [25], i.e., a quantum phase transition that takes place when both T → 0 K and B → 0 T [10]. There are several real system reported in the literature as candidates to undergo a zero-field QPT, such as YbCo2Ge4 [24], β-YbAlB4 [25], and Au-Al-Yb [26]. In this context, Γmag plays a key role in probing a genuine magnetic-field-induced QPT, being the following criteria needed to be fulfilled for such [10]: i) Γmag must diverge at the critical magnetic field modulus Bc, i.e., the value of B in which the QPT occurs; ii) concomitant with the last condition, Γmag must change sign upon crossing Bc; iii) a typical scaling of the type T/(B −Bc)ϵ must be observed, where ϵ is a scaling parameter. 2. Theoretical investigations employing the Grüneisen parameter 29 a)T B a) 0 magnetic ordered phase quantum-critical point T B magnetic ordered phase Bloc ~ B² + B ² quantum paramagnet loc B Bloc b) Br mutu al int era cti on s θ Figure 2.4: Schematic representation of T versus B for a magnetic-field-induced quantum phase transition. In panel a), the blue bullet represents a quantum critical point at finite B where a quantum phase transition, for instance, from a magnetic ordered to a quantum disordered phase takes place. The white dotted lines represent the suppression of an energy scale upon approaching the quantum critical point [8], followed by the corresponding emergence of another energy scale associated with the new phase. In panel b), the hypothetical zero-field quantum critical point is depicted (yellow bullet). The yellow gradient represents the increasing role played by the mutual interactions upon decreasing temperature. It is assumed that Br ≃ √ B2 + Bloc 2, so that the angle θ between B and Bloc is assumed to be θ ≃ 90o. Figure extracted from Ref. [49]. When such three conditions are fulfilled, a genuine magnetic-field-driven QPT is probed [10], cf. Fig. 2.4. The divergence and sign-change of Γmag was analysed in the frame of this Thesis when considering the intrinsic presence of Bloc in the Brillouin-type paramagnet and also in the case of the real-system β-YbAlB4 [25], where a divergence of Γmag was reported for both B and T → 0 [25]. At this point, Bloc is estimated for β-YbAlB4 by making use of Eq. 2.74 in an analogous way as for the Brillouin-type paramagnet. By making use of the effective magnetic moment of the Yb atoms µ ≃ 1.94µB and a typical distance between Yb atoms r ≃ 3.5 Å [50] ⇒ Bloc ≃ 0.04 T for β-YbAlB4. At this point, an analysis of the impact of considering the intrinsic presence of Bloc in both the Brillouin-type paramagnet and β-YbAlB4 is carried out in terms of the behaviour of Γmag at ultra-low temperatures for vanishing B. It is assumed that Br acting on the system is a sum between B and Bloc so that Br ≃ √ B2 +Bloc 2, cf. Eq. 2.77. Hence, B is replaced by Br in the expression for 2. Theoretical investigations employing the Grüneisen parameter 30 entropy of the Brillouin-type paramagnetic (Eq. 2.78) and Γmag is computed, so that: Γmag = B B2 +Bloc 2 . (2.127) Note that for vanishing B ⇒ Γmag → 0 and its divergent-like behaviour is suppressed when Bloc is taken into account. As a consequence, Γmag is only enhanced for vanishing B but does not diverge upon crossing B = 0 T. For the real-system β- YbAlB4, Γmag can be computed via its proposed quantum critical free energy FQC , given by [25]: FQC(T,B) = − 1 (kBT̃ )1/2 [(gµBB)2 + (kBT )2]3/4 , (2.128) where kBT̃ ≈ 6.6 eV ≈ (1.06×10−18) J, T̃ refers to a characteristic temperature and g = 1.94 is the Landé factor [51]. Note that the proposed FQC refers to the valence-fluctuating character of 4f electrons in Yb atoms in β-YbAlB4 [25]. Using Eq. 2.128, it is straightforward to calculate the quantum critical entropy, SQC = −(∂FQC/∂T )B, namely: SQC(T,B) = 3k2 BT 2(kBT̃ )1/2(B2g2µB 2 + kB 2T 2)1/4 . (2.129) An analysis of the behaviour of the entropy for both the Brillouin-type paramagnet and β-YbAlB4 considering a finite Bloc shows that the entropy accumulation at ultra-low temperatures, usually associated with the proximity of the quantum critical point, is suppressed since the presence of Bloc brings order to the system, as can be seen in the upper and lower panels of Fig. 2.5. Replacing B by Br in Eq. 2.129 and computing Γmag by Eq. 2.126, it reads: Γmag = Bg2µB 2 kB 2T 2 + 2g2µB 2(B2 +B2 loc) . (2.130) The behaviour of Γmag for the Brillouin-type paramagnet and for β-YbAlB4 is depicted in Fig. 2.6, showing that a divergent-like behaviour is suppressed when Bloc is taken into account for both cases, being only an enhancement of Γmag observed instead of a divergent-like behaviour. For the case of β-YbAlB4 without considering Bloc ≃ 0.04 T, cf. orange color solid line in Fig. 2.6, Γmag does not actually diverges 2. Theoretical investigations employing the Grüneisen parameter 31 0 2 4 6 8 1 0 - 0 . 0 4 - 0 . 0 2 0 . 0 0 0 . 0 2 0 . 0 4 2 3 4 5 k B l n ( 2 ) S Q C ( x10 -27 ) (J /K) B l o c = 0 T B l o c = 0 . 0 1 T S ( x10 -24 ) (J /K) B r i l l o u i n P a r a m a g n e t T = 5 m K B l o c = 0 T B l o c = 0 . 0 4 T B ( T ) β - Y b A l B 4 T = 5 m K Figure 2.5: Entropy S versus B at a fixed T = 5 mK for the Brillouin-type paramagnet (upper panel) considering Bloc = 0 T (blue stars) and Bloc = 0.01 T (red circles) and for β-YbAlB4 for Bloc = 0 T (green stars) and Bloc =0.04 T (orange color circles). Figure extracted from Ref. [49]. for B = 0 T since a fixed T = 5 mK is considered. In the case of the Brillouin-type paramagnet, Γmag is non-temperature dependent. However, for β-YbAlB4, Γmag depends on temperature. Upon analysing the temperature dependence of Γmag for β-YbAlB4, one can notice that Γmag presents a divergent-like behaviour followed by a sign-change when T → 0 K and vanishing B, cf. upper panel of Fig. 2.7. Hence, upon considering finite Bloc, Γmag is only enhanced and changes-sign at low-T and vanishing field but its divergent-like behaviour is suppressed, as can be seen in the lower panel of Fig. 2.7. Hence, it is shown that when finite Bloc is considered in 2. Theoretical investigations employing the Grüneisen parameter 32 - 3 0 0 - 2 0 0 - 1 0 0 0 1 0 0 2 0 0 3 0 0 - 0 . 0 5 0 - 0 . 0 2 5 0 . 0 0 0 0 . 0 2 5 0 . 0 5 0 - 9 0 - 6 0 - 3 0 0 3 0 6 0 9 0 B l o c = 0 . 0 1 T B l o c = 0 T Γ ma g(T -1 ) B r i l l o u i n P a r a m a g n e t B l o c = 0 . 0 4 T B l o c = 0 T Γ ma g(T -1 ) B ( T ) β - Y b A l B 4 T = 5 m K Figure 2.6: Magnetic Grüneisen parameter Γmag as a function of B for the Brillouin-type paramagnet (upper panel) considering Bloc = 0 T (blue line) and Bloc = 0.01 T (red line) and for β-YbAlB4 at T = 5 mK for Bloc = 0 T (orange color line) and Bloc = 0.04 T (green line). Note that the divergence of Γmag is suppressed at B = 0 T when finite Bloc is considered. Figure extracted from Ref. [49]. real paramagnets one of the key conditions to probe a genuine zero-field quantum phase transition is not fulfilled. Such an analysis in terms of Γmag highlights the need of considering the intrinsic mutual interactions in real insulating paramagnets in the frame of zero-field quantum criticality. Next, a connection between Γmag and the canonical definition of temperature is performed. 2. Theoretical investigations employing the Grüneisen parameter 33 T(K) β-YbAlB (B = 0 T) β-YbAlB (B = 0.04 T)4 4 loc loc -1 -1 Figure 2.7: Magnetic Grüneisen parameter Γmag versus T versus B for β-YbAlB4 considering Bloc = 0 T (upper panel) and Bloc = 0.04 T (lower panel). Figure extracted from Ref. [49]. 2.3.3 Grüneisen parameter and the canonical definition of temperature As well-known from textbooks, the concept of temperature T is universally de- fined as [34, 1]: 1 T = ( ∂S ∂E ) B , (2.131) where E is the average magnetic energy. The upper panel of Fig. 2.8 depicts the regimes where the slope of (∂S/∂E)B determines the temperature. For vanishing temperatures, (∂S/∂E)B → ∞, while for (∂S/∂E)B → 0 indicates that T → ∞ 2. Theoretical investigations employing the Grüneisen parameter 34 [1]. Note that for a positive (negative) slope of (∂S/∂E)B, positive (negative) temperatures can be inferred, being positive and negative temperatures defined when a parallel or an anti-parallel alignment between B⃗ and µ⃗ takes place [34, 1], respectively. Indeed, a perfect alignment between µ⃗ and B⃗ is prevented by Quantum Mechanics, cf. Appendix 6.1. The concept of negative absolute temperatures can be a little bit indigestible at first, mainly because negative temperatures are hotter than positive ones [1]. Interestingly enough, negative temperatures were verified experimentally in the last century in the celebrated experiment of Purcell and Pound, where they used the nuclear spins of Li-7 in a LiF single-crystal that presents a long relaxation time of the order of 5 minutes under B ≃ 0.6376 T [52]. Essentially, the experiment consisted in applying B⃗ on the sample, so that B⃗ and the nuclear magnetic moments are in a parallel configuration (positive temperature). Then, B⃗ was reversed in a time-scale of micro-seconds so that an anti-parallel configuration between B⃗ and the nuclear magnetic moments is achieved, making the system to be at a negative temperature. In the regime where E → 0 J, the value S = kB ln (2) is nicely restored meaning that at very high temperatures, there is an equal probability for the spins to be aligned in a parallel or an anti-parallel configuration in respect to B⃗. Recalling that the average magnetic energy for the Brillouin-type paramagnet is given by [1]: E(T,B) = −NµBB tanh ( µBB kBT ) , (2.132) it is possible to rewrite E as a function of the average magnetic moment ⟨µ⟩ = NµB tanh (µBB/kBT ) along B⃗, as follows: E = −⟨µ⟩B. (2.133) From the expression for ⟨µ⟩ given by [1]: ⟨µ⟩ = NµB tanh ( µBB kBT ) , (2.134) it is straightforward to write B as a function of ⟨µ⟩: B(T, ⟨µ⟩) = kBT µB arctanh ( ⟨µ⟩ NµB ) . (2.135) 2. Theoretical investigations employing the Grüneisen parameter 35 - 1 5 - 1 2 - 9 - 6 - 3 0 3 6 9 1 2 1 50 2 4 6 8 1 0 - 0 . 0 2 0 . 0 0 0 . 0 2 0 3 6 9 0>     ∂ ∂ TB S 0<     ∂ ∂ TB S 0=     ∂ ∂ BE S 0<     ∂ ∂ BE S B = 1 T B = 2 T ( B l o c = 0 T ) S ( x1 0-24 ) (J /K) E ( J ) 0>     ∂ ∂ BE S B µ B µ B ( T ) S ( x10 -24 ) (J /K) T = 5 m K T = 1 m K B l o c = 0 . 0 1 T T = 5 m K T = 1 m K B l o c = 0 T k B l n ( 2 ) k B l n ( 2 ) Figure 2.8: Upper panel: Magnetic entropy S as a function of the average magnetic energy E at a fixed B = 1 (purple line) and 2 T (orange color line). The slope of (∂S/∂E)B indicates positive, infinite, or negative temperatures. The definition of positive and negative temperatures are associated with a parallel or an anti-parallel configuration, respectively, between B⃗ (red arrow) and the magnetic moment µ⃗ (blue arrow). The background colors indicate that negative temperatures are hotter than positive ones. Lower panel: Magnetic entropy S as a function of B for Bloc = 0 T at T = 5 mK (blue stars) and T = 1 mK (white circles). For Bloc = 0.01 T the red and green circles represent T = 5 and 1 mK, respectively. The background colors depict the insensitivity of (∂S/∂B)T regarding positive or negative temperatures. Figure extracted from Ref. [49]. Equation 2.135 indicates that at a certain temperature T , B is associated with a spin configuration that corresponds to a specific average magnetic moment ⟨µ⟩. 2. Theoretical investigations employing the Grüneisen parameter 36 Recalling that the magnetocaloric effect can be quantified by Γmag [8]: Γmag = 1 T ( ∂T ∂B ) S = 1 T ( ∂B ∂T ) S −1 , (2.136) the temperature derivative of B in Eq. 2.135 can be computed and replaced it into Eq. 2.136, resulting: Γmag = µB TkBarctanh ( ⟨µ⟩ NµB ) . (2.137) Since Γmag = 1/B for the Brillouin-type paramagnet [49], T as a function of B and ⟨µ⟩ is given by: T (B, ⟨µ⟩) = BµB kBarctanh ( ⟨µ⟩ NµB ) . (2.138) Equation 2.138 connects in an unprecedented way the Purcell and Pound experiment [53] and Γmag itself. When B⃗ and ⟨µ⟩ are ∥, positive temperatures are inferred. However, when B⃗ and ⟨µ⟩ are anti-∥, then negative temperatures can be associated. Remarkably, the definition of temperature is encoded in Γmag and vice-versa. 2.3.4 Adiabatic magnetization in a paramagnetic insulating system The so-called adiabatic demagnetization is well-known from textbooks [1]. In short, it refers to the cooling of a paramagnetic system upon adiabatically demagnetizing it. The basic concept behind this powerful cooling technique lies in the adiabaticity of the process, i.e., when B is adiabatically removed, the thermal energy must be reduced to maintain the entropy constant and, as a consequence, the system cools down. But what if an opposite situation sets in? What if the temperature is varied adiabatically instead of B? Although counter-intuitive, it is indeed possible to vary the temperature adiabatically employing the application of stress [54], which is discussed in more details in the next Section. Regarding a paramagnetic insulating system without the influence of any B, Bloc still remains finite due to the intrinsic mutual interactions between neighboring spins, cf. previously discussed. Upon decreasing temperature until the magnetic energy associated with Bloc is 2. Theoretical investigations employing the Grüneisen parameter 37 comparable to the thermal energy, usually below ≈ 6.7 mK, the mutual interactions become more and more relevant. In this temperature regime, the process of adiabatic magnetization can be performed, cf. shown step-by-step in Fig. 2.9. 0 1 0 2 0 3 0 4 0 5 0 0 2 4 6 8 1 0 2 . 0 0 2 . 0 5 2 . 1 0 2 . 1 5 2 . 2 00 . 0 6 0 . 0 8 0 . 1 0 0 . 1 2 0 . 1 4 B r = B l o c ( B = 0 T ) B r = B l o c + ∆ B l o c ( T ) S ( 10 -24 J/ Km ol) T ( m K ) k B l n ( 2 ) T 2T 1 CB S ( 10 -24 J/ Km ol) T ( m K ) a d i a b a t i c m a g . A T = 4 . 3 5 m K Figure 2.9: Magnetic entropy S as a function of temperature T for the Brillouin-type paramagnet considering Br = Bloc (cyan solid line) and Br = (Bloc + ∆Bloc) (red dashed line). The inset shows a zoom in the low-T regime describing the proposed steps for performing the adiabatic magnetization. First, the system should be cooled to a temperature where the magnetic energy associated with Bloc is comparable to or higher than the thermal one, which is represented by going from point A to B. Then, at an initial temperature T1, the temperature is adiabatically increased from T1 to T2 (path B to C) and, to keep the entropy constant, the mutual interactions must rearrange itself to increase the magnetic energy associated with Bloc to increase it by an increment ∆Bloc, giving rise to the proposed adiabatic magnetization. The cycle can be restarted upon increasing the temperature non-adiabatically and then cooling down the system again to point A. Figure extracted from Ref. [49]. As can be seen in Fig. 2.9, the crucial point in performing the adiabatic magnetization refers to an adiabatic temperature increase that makes the neighboring spins rearrange themselves to increase Bloc in order to compensate the adiabatic increase of the thermal energy. Upon increasing the temperature adiabatically from an initial temperature T1 to a final temperature T2, the ratio between magnetic and thermal energies can be written as [49]: µBBloc kBT1 = µB √ Bloc 2 + ∆Bloc 2 kBT2 , (2.139) 2. Theoretical investigations employing the Grüneisen parameter 38 where Br after the adiabatic temperature increase is given by Br = √ Bloc 2 + ∆Bloc 2. From Eq. 2.139, the local magnetic field increment ∆Bloc is given by [49]: ∆Bloc = Bloc √√√√T2 2 T1 2 − 1. (2.140) Upon assuming T1 = 2 mK, T2 = 2.1 mK, and Bloc = 0.01 T: ∆Bloc = 0.01 √ 2.12 22 − 1 ≃ 0.0032T ≃ 3.2mT. (2.141) Amazingly, such adiabatic magnetization can be carried out by solely manipulating the mutual interaction upon increasing the temperature adiabatically, i.e., without applying B to the paramagnetic system. As a step further, two experimental setups to perform and detect the adiabatic magnetization were proposed by us in the literature, which is discussed in the next Section. The obtained theoretical results discussed in Sections 2.3.2, 2.3.3, and 2.3.4 were published in (Appendix 6.11): • Lucas Squillante, Isys F. Mello, Gabriel O. Gomes, A.C. Seridonio, R.E. Lagos-Monaco, H. Eugene Stanley, Mariano de Souza, Unveiling the physics of the mutual interactions in paramagnets, Scientific Reports 10, 7981 (2020). Also, the publication was reported by the Fapesp Agency in the following: • “Theoretical study shows that matter tends to be ordered at low temperatures” - Appendix 6.12. 2.3.5 Elastocaloric-induced effect adiabatic magnetization As previously mentioned, adiabatic temperature changes can be seem, at first glance, as counter-intuitive. However, this is totally possible due to an adiabatic compression [1]. At this point, the first law of Thermodynamics is recalled, which reads [28]: dQ = dU + dW, (2.142) where dQ is the infinitesimal heat variation and dW is the infinitesimal work performed on the system regarding its compression. As a consequence of an adiabatic compression, the temperature will change and so the internal energy, so that [28]: dU = ( ∂U ∂T ) v dT = cvdT. (2.143) 2. Theoretical investigations employing the Grüneisen parameter 39 The infinitesimal work performed on the system due to compression, is then given by [28]: dW = −pdv. (2.144) The pressure under adiabatic conditions can be expressed in terms of the variation of U in response to a change of v, so that p = −(∂U/∂v)S [28]. Thus, Eq. 2.144 reads: dW = −pdv = ( ∂U ∂v ) S dv = ( ∂U ∂S ) v ( ∂S ∂v ) T dv. (2.145) Note that the term (∂U/∂S)v = T , following the canonical temperature definition [1]. Employing the Maxwell relation (∂S/∂v)T = (∂p/∂T )v [28], Eq. 2.145 now reads: dW = T ( ∂p ∂T ) v dv. (2.146) Upon putting Eqs. 2.143 and 2.146 into Eq. 2.142, it becomes: dQ = cvdT + T ( ∂p ∂T ) v dv. (2.147) Since dQ = TdS [1, 28] and that in an adiabatic process dS = 0, hence dQ = 0 as well, so that: TdS = cvdT + T ( ∂p ∂T ) v dv (2.148) 0 = cvdT + T ( ∂p ∂T ) v dv (2.149) cvdT = −T ( ∂p ∂T ) v dv. (2.150) Employing the Thermodynamic relation [28]:( ∂p ∂T ) v ( ∂T ∂v ) p ( ∂v ∂p ) T = −1, (2.151) it can be rewritten as: ( ∂p ∂T ) v = − 1( ∂T ∂v ) p ( ∂v ∂p ) T . (2.152) Note that the isothermal compressibility is given by κT = −1/v(∂v/∂p)T [1], so that the term in Eq. 2.152 ( ∂v ∂p ) T = −κTv. Also, following the definition of the 2. Theoretical investigations employing the Grüneisen parameter 40 volumetric thermal expansion β = 1/v(∂v/∂T )p, the term in Eq. 2.152 ( ∂T ∂v ) p = 1/βv. Employing these substitutions in Eq. 2.152, it reads:( ∂p ∂T ) v = − 1 1 βv [−κTv] = β κT . (2.153) Then, replacing Eq. 2.153 into Eq. 2.150: cvdT = −Tβ κT dv. (2.154) Equation 2.154 can then be rewritten as: dT = − Tβ cvκT dv . (2.155) Equation 2.155 is key in analysing the temperature variation in response to an adiabatic compression of a system. Considering that T , β, cv, and κT are positive quantities, the sign of the adiabatic temperature variation is ruled solely by dv. On one hand, if dv < 0, the temperature of the system is adiabatically increased (dT > 0) in response to the decrease in its volume due to an adiabatic compression. On the other hand, if dv > 0, the temperature of the system is adiabatically decreased (dT < 0) as a consequence of an increase in the volume of the system due to an adiabatic decompression. The very same analysis can be performed now in terms of p instead of v. Considering that the entropy of the system can be varied due to p or T , so that: dS(T, p) = ( ∂S ∂T ) p dT + ( ∂S ∂p ) T dp. (2.156) Considering the definition of the heat capacity at constant pressure cp = T (∂S/∂T )p [1], the term in Eq. 2.156 (∂S/∂T )p = cp/T and the Maxwell relation (∂S/∂p)T = −(∂v/∂T )p [28], Eq. 2.156 can now be rewritten as: dS(T, p) = cp T dT − ( ∂v ∂T ) p dp. (2.157) Since an adiabatic process is considered, i.e., dS(T, p) = 0: cp T dT = ( ∂v ∂T ) p dp. (2.158) 2. Theoretical investigations employing the Grüneisen parameter 41 Recalling that β = 1/v(∂v/∂T )p, then (∂v/∂T )p = βv so that: dT = βvT cp dp. (2.159) Equation 2.159 is similar to Eq. 2.155, since for an adiabatic pressurization the final p is higher than the initial one, i.e., dp > 0, the volume of the system is decreased and the temperature of the system is adiabatically increased (dT > 0), while for an adiabatic decompression the final p is lower than the initial one (dp < 0), the volume of the system is increased and thus the temperature is adiabatically decreased. The effect of varying the temperature upon an adiabatic pressure variation is called the barocaloric effect, which is quantified by the singular portion of Γeff , the so-called Grüneisen ratio Γ = β/cp = 1/Tv(∂T/∂p)S [8, 9]. Indeed, Γ is naturally encoded in Eq. 2.159, which can be rewritten as:( ∂T ∂p ) S = βvT cp , (2.160) so that: 1 Tv ( ∂T ∂p ) S = β cp = Γ. (2.161) Experimentally, an adiabatic temperature change can be carried out due to the application of stress σ (Fig. 2.10), which refers to the compression or decompression of a particular direction. All the 9 stress components can be represented by a 3×3 matrix, given by [56]: σij = σ11 σ12 σ13 σ21 σ22 σ23 σ31 σ32 σ33  , (2.162) where σij represents all the stress components in Voigt’s abbreviated notation [56, 57], being the index i the normal plane on which the stress component is acting and j the direction of the applied stress. Hence, analogously to Eq. 2.156, it can now be written as: dS(T, σ) = ( ∂S ∂T ) σ dT + ( ∂S ∂σij ) T,σ dσij. (2.163) 2. Theoretical investigations employing the Grüneisen parameter 42 z x y ΔLx Ϭx,-x Ϭz,-z Ϭy,-y Ϭzy Ϭzx Ϭxz Ϭxy Ϭyx Ϭyz Figure 2.10: Schematic representation of both the normal (σx,−x, σy,−y, and σz,−z) and shear (σxy, σxz, σyx, σyz, σzx, and σzy) stress components in a solid. The dashed lines indicate the compression of the solid only in the x direction by the application of a uniaxial compressive stress component σx,−x, which in turn decreases the length L of the solid by a factor ∆Lx. Figure extracted from Ref. [55]. Note that the second term on the right side of Eq. 2.163, represents the entropy variation due to the application of only one stress component while the others are kept constant, being also T kept constant. In our analysis it is considered that only one of the normal components is varied, i.e., a uniaxial application of stress (Fig. 2.10), being this the reason why such an entropy variation is given at a fixed T and σ. An analogous decomposition of σ can be made in terms of the thermal expansion, so that αij = (∂S/∂σij)T,σ [54]. Note that, usually, stress has units of pressure [Pa], however, in this particular case, the notation of σij is given in terms of stress components per unit of volume, so that σij = [J] [56]. Also, considering that the heat capacity at constant stress cσ/T = (∂S/∂T )σ, Eq. 2.163 can be rewritten as: dS = cσ T dT + αijdσij = 0 (2.164) cσ T dT = −αijdσij, (2.165) dT = − T cσ αijdσij. (2.166) 2. Theoretical investigations employing the Grüneisen parameter 43 It becomes clear that in Eq. 2.166, dT is the infinitesimal adiabatic temperature change, from a starting temperature T , due to the application of an adiabatic stress component σij. Conventionally, σij < 0 means that an uniaxial compression is carried out, while σij > 0 is associated with an uniaxial decompression [56]. Hence, considering that cσ, T , and αij are positive, for dσij < 0 it implies in dT > 0, while for dσij > 0, dT < 0. In other words, upon adiabatically compressing (decompressing) the system along a particular axis, its length is reduced (increased) and, as a consequence, its temperature is adiabatically increased (reduced) to keep the system’s entropy constant. This is the so-called elastocaloric-effect, which is similar to the barocaloric effect, previously discussed. Analogously to the fact that Γ (Eq. 2.161) quantifies the barocaloric effect, Eq. 2.166 can be rewritten as [55]: ( ∂T ∂σij ) S = Tαij cσ (2.167) 1 T ( ∂T ∂σij ) S = αij cσ = Γec . (2.168) Equation 2.168 accounts for the definition of the elastic Grüneisen parameter Γec, which quantifies the elastocaloric effect and was proposed in the literature in the frame of this Ph.D. Thesis, cf. Ref. [55]. Just to mention, after the proposal of Γec it was employed by researchers of the Max Planck Institute - Dresden to investigate the phase diagram of the unconventional superconductor Sr2RuO4 [58]. Now, the proposed adiabatic magnetization of a paramagnetic insulating system due to an adiabatic temperature increase in terms of σij and the magnetization M of the system are analysed, so that it is possible to write: dS(T,M, σij) = ( ∂S ∂T ) M,σ dT + ( ∂S ∂M ) T,σ dM + ( ∂S ∂σij ) T,M dσij = 0, (2.169) so that: − ( ∂S ∂M ) T,σ dM = ( ∂S ∂T ) M,σ dT + ( ∂S ∂σij ) T,M dσij. (2.170) 2. Theoretical investigations employing the Grüneisen parameter 44 Employing the well-known Maxwell relation (∂S/∂M)T = −(∂B/∂T )S [28], it is possible to write that:( ∂B ∂T ) S dM = ( ∂S ∂T ) M,σ dT + ( ∂S ∂σij ) T,M dσij. (2.171) Equation 2.171 is key in analysing the proposed adiabatic magnetization upon an adiabatic uniaxial compression. Since no B is applied to the system, it becomes evident that B in Eq. 2.171 can be recognized as Bloc. Essentially, Eq. 2.171 tell us that, in response to an adiabatic temperature variation due to the application of uniaxial stress in the system, there should be an increase in the magnetization of the system by a factor dM to keep the entropy constant, which in turn is originated by a variation of Bloc by a factor ∆Bloc due to the temperature increase of the system, cf. previously discussed. At this point, it is tempting to make an analogy between positive and negative temperatures in the celebrated Purcell and Pound’s experiment [53] in terms of either an adiabatic expansion or contraction under application of stress. However, this is not in agreement with the well-established conditions for attaining negative temperatures [59], as discussed in Appendix 6.2. Just to mention, the proposed adiabatic magnetization resembles the celebrated Pomeranchuk effect [60], which predicts that 3He might solidify upon heating. In the case of the adiabatic magnetization, the system is magnetized upon adiabatically heating it. Next, two proposed experimental setups are discussed to carry out the adiabatic magnetization. In the first case, the paramagnetic insulating specimen is inserted into an adiabatic chamber inside a coil with two pistons holding it, cf. Fig. 2.11. As depicted in Fig. 2.11, the sample is compressed adiabatically by the pistons decreasing its volume by a factor ∆v. Hence, since it is an adiabatic compression, its temperature rises and the sample is adiabatically magnetized. The modulus of B⃗r is increased from Br = Bloc to Br = √ B2 loc + ∆Bloc 2. The change in Br in time generates a magnetic flux variation in time, which in turn gives rise to an electromotive force ξ = −(dΦ/dt) [37] opposing to such a change in flux Φ, so that an electric current flows through the coil and it can be detected by the ampere meter. Based on the electrical current measured, ∆Bloc can be determined. At 2. Theoretical investigations employing the Grüneisen parameter 45 Entropy = S1 Magnetization = M1 + ΔM Paramagnetic sample Pressure = p1 + Δp Adiabatic chamber v - Δv T2 = 2.1 mK Entropy = S1 Magnetization = M1 Br = Bloc Paramagnetic sample Pressure = p1 Adiabatic chamber A v T1 = 2 mK B Br = √Bloc+ ΔBloc 2 2 C i (A) A Bapplied pressure (stress) Δt quasi-staticBr = Bloc 0 Br = √Bloc+ ΔBloc 2 2 i (A) 0 i (A) 0 Magnetic energy minimization modulus modulus Figure 2.11: Schematic representation of the experimental setup for attaining adiabatic magnetization. In panel A, the paramagnetic insulating sample is inserted into an adiabatic chamber inside a coil connected to an ampere meter and it is supported by two lateral pistons. Essentially, an adiabatic chamber is a thermally isolated compartment designed to prevent heat exchange with the outside environment. The sample is at an initial temperature T1 = 2 mK, pressure p1, entropy S1, magnetization M1, and Br = Bloc. In panel B, the two pistons compress adiabatically the sample so its volume is reduced by a factor of ∆v. Hence, the final temperature is T2 = 2.1 mK, so that the entropy remains constant but the magnetization is increased in order to minimize the magnetic energy of the system so that Br = √ B2 loc + ∆Bloc 2. In C, it is depicted that, upon going from A to B in a time interval ∆t, the increase in Br generates a magnetic flux variation in time, which in turn leads to an electromotive force in the coil generating thus an electric current that is measured by the ampere meter. Based on the measured electrical current, ∆Bloc can be determined. Figure extracted from Ref. [55]. this point, it is plausible to ask “How to apply uniaxial stress to a sample?”, and more importantly, “How to actually attain the adiabatic character that is crucial for the experimental realization of the elastocaloric effect?”. The answer to both of these questions lies in the piezoelectric device proposed in Ref. [54], cf. depicted 2. Theoretical investigations employing the Grüneisen parameter 46 b) a) sample ∆L ∆Bloc nano-SQUID tip sensor PZT stacks PZT stacks V = 0 V ≠ 0 f nano-SQUID tip sensor movable sample plate fixed sample plate epoxy sample movable sample plate sample fixed sample plate Figure 2.12: Schematic representation of the proposed experimental setup to carry out and detect the elastocaloric-induced adiabatic magnetization due to the mutual interactions. a) The sample is attached with epoxy to a movable plate, which is connected to a piezoelectric device [54, 61, 62]. b) When A.C. voltage V is applied on the piezoelectric stacks, the sample is continuously uniaxially compressed and decompressed by a factor ∆L. Due to the adiabatic compression, Bloc is increased by a factor ∆Bloc, which is detected by a nano-SQUID tip sensor [63]. Figure extracted from Ref. [55]. in Fig. 2.12. The proposed experimental setup for attaining adiabatic temperature changes is depicted in Fig. 2.12 [54, 55]. The paramagnetic insulating sample is attached with epoxy to a movable plate, which is connected to a piezoelectric (PZT) device. As soon as A.C. voltage with a chosen frequency f is applied to the PZT, the movable plate starts to oscillate back and forwards with the same frequency f of the A.C. voltage, continuously compressing and decompressing the sample. 2. Theoretical investigations employing the Grüneisen parameter 47 Upon tuning f to achieve such an oscillation in a time scale higher than the one associated with thermal relaxation, the adiabatic character is attained [54]. In other words, the compression/decompression of the sample occurs faster than it can exchange heat with its surroundings. In this tiny time window of adiabaticity of about 30 ms [54], the temperature of the paramagnetic sample is continuously increased/decreased, following Eq. 2.166. Hence, considering the case of its adiabatic compression, its temperature increases and the sample is adiabatically magnetized, cf. previously discussed. To detect such ∆Bloc, we proposed that a nano-SQUID on a tip [63] can be employed, which is broadly used to detect magnetic fields of a few mT [63]. Just for the sake of completeness, the magnetic flux Φ through a SQUID with surface area S ′ is given by [37]: Φ = ∫ S′ B⃗ · dS⃗ ′ = n ( h 2e ) , (2.172) where n is an integer number, h is Planck’s constant, dS⃗ ′ a normal vector in respect to the SQUID’s junction inner area S ′, and e the fundamental electron charge. The factor (h/2e) is the quantum of Φ and it dictates the SQUID sensitivity to small values of Φ. Although the nano-SQUID on a tip [63] has its resolution compromised by its small S ′, its spatial resolution is outstanding making it very sensitive to small magnetic moments [63], which in turn can detect ∆Bloc ≃ 3.2 mT (Eq. 2.141). It is worth mentioning that the parameters involving the detection of ∆Bloc employing the nano-SQUID on a tip [63] shall depend on the physical properties of the investigated system, the temperature in which the experiment is carried out, and, of course, the characteristics of the nano-SQUID on a tip itself. Based on the proposed adiabatic magnetization due to an adiabatic temperature increase of a paramagnetic insulating specimen, it is natural to extend the concept of an adiabatic temperature increase to investigate other interacting systems [55], such as Bose-Einstein condensates in magnetic insulators [64] and spin-ice systems [65]. In such cases, we propose that the many-body interactions between magnetic moments are rearranged as a whole when the temperature is adiabatically increased similarly to the one proposed for the paramagnetic insulating system. Hence, this 2. Theoretical investigations employing the Grüneisen parameter 48 technique for adiabatically increasing temperature can be employed to explore the many-body character of the magnetic interactions in other systems. For instance, in the case of the Bose-Einstein condensate in an insulator, the Hamiltonian of the system incorporates a Zeeman term −gµBBzS z, cf. Eq. 1 of Ref. [64], that governs the density of triplons in the system, being Bz the modulus of the applied external magnetic field in the z direction, and Sz the spin projection in the z direction. Considering that the resulting magnetic field Br in the system is solely given by the effective local magnetic field generated by the mutual interactions between adjacent magnetic moments, an adiabatic temperature increase makes the magnetic moments to rearrange to compensate for the adiabatic thermal energy increase. As a consequence, the Zeeman term of the Hamiltonian is varied and so is the density of triplons. This is only one particular example regarding how the adiabatic temperature increase can be employed to investigate the many-body character of the magnetic interactions in various systems [55]. 2.3.6 Grüneisen parameter and second-order phase tran- sitions As discussed in the last Sections, Γ quantifies caloric effects and also the entropy change upon tuning the system close to the critical point. In a step further, we have proposed that Γ can also quantify the critical temperature Tc shift in the vicinity of a second-order phase transition upon varying a tuning parameter, which can be pressure, electric or magnetic field, for instance, in an analogous way as the canonical Ehrenfest relation, which is given by [66]: dTc dp = Tv ( ∆α ∆cp ) , (2.173) where ∆α and ∆cp refer to the jumps in the thermal expansion and the heat capacity at constant pressure due to a second-order phase transition. Equation 2.173 quantifies the shift in Tc upon applying pressure in terms of ∆α and ∆cp. For the case of an adiabatic stress application, the entropy variation reads [54]: dS = −ccr σ T ( dTc dϵ ) T dϵ+ cσ T dT = 0, (2.174) 2. Theoretical investigations employing the Grüneisen parameter 49 where ccr σ is the critical contribution to the heat capacity at constant stress, ϵ = (L − L0)/L0 refers to the strain, which is connected to σ through the Young modulus YM = σ/ϵ [56], L and L0 refer to the final and initial sample lengths, respectively. The term of Eq. 2.174 cσ T dT accounts for the entropy variation due to the elastocaloric effect, while − ccr σ T ( dTc dϵ ) T refers to shifting Tc and thus compensating the contribution to S from the elastocaloric effect in order to keep the global entropy constant. Upon adiabatically compressing the sample, for instance, its temperature is increased contributing to the increase of the system’s entropy and thus the term cσ T dT contributes to the enhancement of S. To keep the entropy constant, the term on Eq. 2.174 − ccr σ T ( dTc dϵ ) T must be negative, i.e., (dTc/dϵ)T must be positive. In other words, upon adiabatically compressing the sample, its Tc is increased. The opposite is also true, when the sample is adiabatically decompressed, its temperature decreases and the term in Eq. 2.174 cσ T dT contributes to decrease S. Hence, in order to keep S constant, the term in Eq. 2.174 − ccr σ T ( dTc dϵ ) T must be positive, contributing thus to compensate the decrease in S, so that the total S variation is zero. For such a term to be positive, (dTc/dϵ)T must be negative, i.e., its Tc is decreased. In summary, upon adiabatically compressing (decompressing) the system, its Tc is increased (decreased). An experimental verification of this analysis is reported in Ref. [54] for the iron-based superconductor Ba(Fe0.979Co0.021)2As2, cf. Fig. 2.13. Rewriting Eq. 2.174: ccr σ T ( dTc dϵ ) T dϵ = cσ T dT (2.175) ( dT dϵ ) S = ccr σ cσ ( dTc dϵ ) T , (2.176) which connects the elastocaloric effect, now written in terms of ϵ instead of σ, with the Tc shift. Employing the definition of the proposed Γec (Eq. 2.168) and multiplying both sides of Eq. 2.176 by 1/T , it now reads: 1 T ( dT dϵ ) S = 1 T ccr σ cσ ( dTc dϵ ) T = Γcr ec , (2.177) 2. Theoretical investigations employing the Grüneisen parameter 50 Figure 2.13: Peak to peak amplitude of the elastocaloric temperature oscillation T EC pp versus temperature T in which the elascoaloric effect was performed for Ba(Fe0.979Co0.021)2As2 [54]. The uniaxial strain in the x direction εdisp xx is also indicated. It is worth mentioning that εdisp xx > 0 means that the specimen length is higher than the original one (decompressed), while εdisp xx < 0 implies that its length was reduced from its original size (compressed). Following discussions in the main text, note that when the sample is compressed εdisp xx < 0, both the antiferromagnetic transition temperature TN and the temperature in which a nematic/structural phase transition TS takes place are increased, while for εdisp xx > 0, TN and TS decreases. Figure extracted from Ref. [54]. w