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## Thermodynamic Approach of the Magnetocaloric Effect

A general introduction to the thermodynamics of the magnetocaloric effect is given in this section. I will start with describing some aspects of adiabatic cooling and explain how the MCE can be utilized in magnetic cooling by adiabatic demagnetization as well as presenting some basic notions and theories related to magnetocaloric phenomenon.

### Thermodynamics of Adiabatic Cooling

The principle behind the magnetic refrigeration is the adiabatic demagnetization which is based on the fact that the total entropy, S, of a system is a thermodynamic state function that can be controlled by the the use of external thermodynamic state variables such as temperature (T), pressure (p), and the magnetic field (H) : S = S„T, p, H” (1.1).

The change in entropy, dS, can be expressed as follows : dS„T, p, H” = „ @S ”p,H dT + „ @S ”T,H dp + „ @S ”p,T dH (1.2).

In general, pressure variations of magnetic refrigerators are too small to play a signif-icant role in the entropy change and can be therefore overlooked. Under isobaric conditions, dS becomes : dS„T” = „ @S ”H dT + „ @S ”T dH (1.3).

Isothermal entropy change ( S) can be induced by keeping the temperature constant and applying a magnetic field ( from H0 to H1) and can be expressed as follows : 4SH0!H1 = S„T, H1” S„T, H0” (1.4).

#### Employment of Maxwell relation

The magnetocaloric effect in terms of 4Tad or 4SM can be determined either directly or indirectly. Direct measurements are ultimately desirable, but can be very challenging, hence it is common practice to use indirect and quasidirect methods to obtain values of isothermal 4SM and adiabatic 4Tad from the thermodynamic analysis of experimental data that record the temperature or the magnetic-field dependence of the magnetization mag-nitude using the well-known Maxwell relation (MR) obtained by integrating equation 1.13. Idealy, the magnetization should be obtained in thermodynamic equilibrium as a single valued function of magnetic field and temperature which may be achieved near SOPT. Whereas, however, there has been some controversy in the literature regarding the applicability of MR near FOPT [36, 37, 33] with significant hysteresis since the ma-terial is assumed to be in equilibrium in order to use MR. The validity of this equation depends on the details of the experimental procedure since it is acceptable to use data obtained after thermal excursion away from the hysteretic regime in strongly hysteretic transitions[25, 26, 27, 28, 29] preventing the occur of the so-called colossal magnetocaloric effect manifesiting in a spurious peak in 4SM calculated by indirect approach [34, 35].

A. Giguère et al [36], have proposed a magnetic analogue of the Clausius-Clapeyron equation which is equivalent to the Maxwell method for FOPT materials allowing the use of non equilibium data without performing thermal excursion away from the hysteretic regime. One typically measures the temperature dependence of magnetization at fixed ap-plied magnetic field, in order to determine the magnetic field dependence of equilibrium transition temperature T0. 4SM can be obtained using the following equation : dHc 4SM = 4 M (1.18).

Where DM is the difference of magnetization between the two phases, while „dHdTc ”1 is the change rate of phase transition equilibium temperature with magnetic field.

**Quantum versus Jahn teller orbitals physics**

At present, the main controversy about vanadate is the description of low energy physics.. In 2001, Khaliullin et al. proposed a quasi-1D spin-orbital model [56] which is at the origin of the discussion. It was pointed out that in order to stabilize the stronger than expected ferromagnetic super-exchange along the c-axis in the C-type magnetic phase, a strong t2g quantum orbital fluctuations must exist ina half-lled system of yz and zx orbitals in RVO3 systems. In this case, JT distortion as well as the C-type magnetic phase should not occur. However, the Jahn-Teller effect would still stabilize the G-type magnetic phase ( with C-type OO) at low temperatures. Khaliliullin et al have predicted that orbitals will expand along the c-axis to give a so-called orbital orbital Peierls state when orbital fluctuations are dominant and there is no JT distortion at all. Evidence for an orbital Peierls state [67, 68], where orbitals are dimerised along the c- axis, was obtained by magnetic neutron scattering by Ulrich et al. in YVO3 [69]. This is consistent with Pb11 symmetry observed earlier by Tsvetkov et [72]. In the C-type magnetic phase, the width of the magnon band along the FM c-axis was larger than that in the AFM ab plane. This violates the classical Goodenough-Kanamori rules described in section 1.3.2.3 stating that the ferromagnetic super-exchange interactions are usually weaker than antiferromagnetic interactions. These observations were analyzed using the quasi-1D spin-orbital model, which successfully explains most of the anomalies of the Ts<T<Too phase of YVO3. Most of the subsequent theoretical studies on yvo3 and lavo3 using this model confirmed its validity [71, 70].

**Magnetic moments and exchange interactions**

Table 4.4 summarizes the electronic configurations, the value of spins and moments (gJB J) of magnetic ions in the case of different R. The ligand field (O2) of manganese induces a lifting of degeneracy of their 3d orbitals. Figure 1.13 reminds low-field energy levels of d orbitals in a squar pyramidal environment and in an octaedral environment. The filling of these orbitals is done with respect to the Hund rule (see section 1.3.2.1). It should be noted that theMn4+ and Mn3+ ions have a low anisotropy with an easy magnetization plan in the plane (a,b) (easy axis of the Mn3 follows the apical direction of the pyramid) [30]. Estimating the degeneracy and filling of these orbitals requires further calculations based onspecific crystallographic structure since the rare earth environment with 8 first neighbors and the nature of 4f orbitals are quit more complex. As for rare earth, they have a strong anisotropy because of their spin orbit coupling. This anisotropy varies according to the compound and we will see in chapter 4 how this anisotropy will impact the magnetism and magnetocaloric properties in HoMn2O5.

**The Born-Oppenheimer approximation**

In 1927, the two physicists Max Born and Robert Oppenheimer developed an approach to quantum theory describing molecules, called the Born-Oppenheimer approximation. The purpose of this approximation is to simplify and solve the Schrodinger equation. In fact, electrons and atomic nuclei have very different masses and the magnitude of the electromagnetic forces acting on them are same. As a result, the electronic motion (106m/s) and nuclear motion (103m/s) can be separated leading to a molecular wave function depending on the electorn positions ( nuclear positons are fixed). The assumption of instananeous equilibrium for every nuclear configuration implies that the electron wave function is a solution of equation 1.1.

**Density functional theory**

Despite the fact that the resolution of the Schrödinger equation is simplified and reduced to the behavior of electrons, it is still very complex to solve because of the electron-electron interactions and therefore the intervention of other complementary approximations will be necessary. The Hartree-Fock (HF) approximation is one of the most important ways to deal with that problem, this method is based on an important element which is Pauli exclusion principle, this principle means that two fermions cannot occupy the same quantum state leading us to consider the anti-symmetry of the wave function that will be described by the Slater determinant. This method is, however, quite complex to solve for solids due to the presence of the exchange potentiel taking into account the anti-symmetry of the wave function. The resolution of Schrodiner equation was beyond human reach until discovery of Hohenberg [10], Kohn and sham [11] that leads to a remarkable reduction in difficulty giving birth to the density functional theory (DFT). In principle, the only input required for DFT calculations is the most basic informations related to the system: atomic species and atomic positions.

**The theorems of Hohenberg and Kohn**

The Hohenberg and Kohn [10] approach is linked to any system consisting of electrons moving under the influence of external potential. It aims to make the DFT an exact theory for many-body systems and is based on two theorems that state:

First theorem The total energy of a particle system is expressed as a unique functional of total density (r) [xxx]. In other words, there is unique link between the external potentiel and electronic density. Therefore, the hamiltonian operator is determined only by electronic density and from that operator the properties of the system can be calculated. Second theorem A universal functional for the energy in terms of the density !r can be defined for any Vext . The exact ground state energy of the system is the global minimu that corresponds to the exact ground state density. The ground state energy H »¼ = EVext »¼ is of the form : EVext »¼ = h jT + Vj i + h jVext j i (2.5).

**Table of contents :**

**1 Literature review **

1.1 Introduction – what is all about

1.1.1 Magnetic refrigeration

1.1.2 Thermodynamic Approach of the Magnetocaloric Effect

1.1.2.1 Thermodynamics of Adiabatic Cooling

1.1.2.2 Magnetocaloric Effect and Adiabatic Demagnetization

1.1.2.3 Entropy and its Dependence on the Magnetic Field

1.1.3 Classification and phenomenology of magnetocaloric materials

1.1.3.1 Magnetic transition order

1.1.3.2 Employment of Maxwell relation

1.1.4 Selection Criteria

1.2 Multiferroic materials

1.2.1 Why multiferroics are interesting ?

1.2.2 Classifying Multiferroics

1.3 Type-I multiferroic : Vanadate RVO3

1.3.1 Multiferroic structure

1.3.2 Magnetic configuration

1.3.2.1 Hund rules

1.3.2.2 Super-exchange interactions

1.3.2.3 Goodenough-Kanamory rules

1.3.3 Phase diagram in rare-earth vanadates

1.3.4 Quantum versus Jahn teller orbitals physics

1.4 Type-II multiferroic: Orthorhombic RMn2O5

1.4.1 General structure

1.4.2 Magnetic moments and exchange interactions

1.4.2.1 Magnetic moment

1.4.3 Exchange interactions

1.4.4 Dzyaloshinskii-Moriya interaction

1.5 Outline of the thesis

**2 Experimental and theoretical techniques **

2.1 Sample elaboration and characterization

2.1.1 Sample elaboration

2.1.2 Magnetic characterization

2.2 Ab initio calculations

2.2.1 Introduction to ab initio calculations

2.2.2 Quantum many-body systems

2.2.3 The Born-Oppenheimer approximation

2.2.4 Density functional theory

2.2.4.1 The theorems of Hohenberg and Kohn

2.2.4.2 The Kohn-Sham equations

2.2.4.3 The exchange-correlation functional

2.2.5 Solving the equations

2.2.5.1 The pseudopotentiel basis set

2.2.5.2 The LAPW basis set

2.3 Monte Carlo simulation

2.3.1 Theoretical models

2.3.2 Basic of Monte Carlo simulation

2.3.3 Metropolis algorithm

**3 Magnetocaloric effect in oxyde thin films **

3.1 Introduction

3.2 Research progress in rare earth perovskite oxide magnetocaloric materials : From micro to nanoscale

3.2.1 ABO3 perovskites

3.3 PrVO3: An inhomogeneous antiferromagnetic material with random field .

3.3.1 Bulk and film magnetic properties

3.4 Strain-induced giant magnetocaloric effect in epitaxial PrVO3 thin films

3.4.1 Outline of the experiments

3.4.2 Magnetic characterization

3.4.2.1 PrVO3 (100nm) deposited on 001-oriented SrTiO3 (STOº substrate

3.4.2.2 PrVO3 (41.7nm) deposited on 001-oriented ¹La, Srº¹Al,TaºO3 (LSAT) substrate

3.4.3 Magnetocaloric properties

3.5 Theoretical calculations

3.5.1 Calculations details

3.5.2 Electronic and structural properties

3.5.3 Magnetic properties

**4 Magnetocaloric effect in HoMn2O5 single crystals **

4.1 Introduction

4.2 Why RMn2O5

4.3 Common features

4.3.1 Magnetic structure

4.3.2 Magnetocaloric effect in RMn2O5 at low temperature regime

4.4 Study of HoMn2O5 single crystals

4.4.1 Computational details

4.4.2 Electronic properties

4.4.3 Magnetic properties

4.4.3.1 XMCD at the Ho M4,5edge

4.4.3.2 XMCD at the Mn L2,3 edge

4.4.4 Magnetocaloric properties

4.4.5 Summary

**5 General conclusions and future challenges **

**Résumé en Francais**