The magnetocaloric effect of ferromagnetic manganites: modeling and interpretation of properties with Landau and mean field theory
J.S. Amaral, M.S. Reis, V.S. Amaral
Departamento de Física da Universidade de Aveiro and CICECO
J.P. Araújo, T.M. Mendonça
Departamento de Física da Universidade do Porto and IFIMUP
Departamento de Química and CQ-VR
Departamento de Cerâmica e Vidro da Universidade de Aveiro
13th Workshop on magnetism and Intermetallics – Porto 2007
- Introduction to the magnetocaloric effect and its applications
- Ferromagnetic manganites and magnetocaloric properties
- Landau Theory of phase transitions in the study of the magnetocaloric effect
- Molecular mean-field theory and its application in magnetocaloric measurements
The Magnetocaloric Effect
- Discovered in pure iron in 1881 by Emil Warburg
- Applying a magnetic field reduces magnetic entropy of a magnetic material, increasing temperature in an adiabatic process
- Temperature decreases when magnetic field is removed
- Maxwell relation used to estimate the magnetocaloric effect from magnetization measurements
Development of magnetic cooling
- A magnetic cooling cycle can be obtained with isothermal and isofield processes
- First breakthrough application by chemist Nobel Laureate William F. Giauque and colleague D.P. MacDougall in 1933, reaching 0.25 K
- In 1997, the first near room temperature magnetic refrigerator was demonstrated by K. A. Gschneidner
- Magnetic cooling device development fuels an increasingly active field of research in material, device and fundamental physics
E. Brück et al.
Magnetic cooling prototypes
Toshiba Corp. 2003
- Devices are being developed in several research centers
- Field sources can be either permanent magnets or superconducting coils
- Magnetic material used is usually pure Gd or Gd based alloy
Astronautics corp 2001, USA
Magnetic material research
- Since magnetocaloric effect is proportional to ∂M/ ∂ T, material should be ferromagnetic, with Tc near operating (room) temperature.
- Material research brings several candidates to “optimal” magnetocaloric material, where several properties need to be considered.
- Manganite materials are promising.
- By chemical substitution, easily tunable Tc
- Magnetoelastic coupling, charge ordering effects, colossal magnetoresistance, all can contribute either positively or negatively to the magnetocaloric effect
- First or second-order magnetic phase transition
- Chemically stable (oxide), preparation method easily scaled to large quantities (ball milling, sol-gel,…)
- Cheaper than Gd based compounds
T is a trivalent rare-earth ion
D is a divalent dopant)
Manganite materials for magnetocaloric studies and applications
- La0.70Sr0.30MnO3 has a high Tc of ~ 90ºC
- La can be substituted by a rare earth ion, gradually decreasing TC and maintaining carrier density
- Properties of the substituted ion affect structural, electronic and magnetic properties of the manganite
Asamitsu et al.
- Study the magnetic properties of La0.70-xErxSr0.30MnO3 and La0.70-xEuxSr0.30MnO3
- Interpret how the magnetic Er ions and non-magnetic Eu ions change the magnetic and magnetocaloric properties of the LaSrMnO system?
- Manganite samples where synthesized by Sol-Gel and solid state methods
- Er and Eu content up to 21%
- Structural characterization of samples by X-Ray diffraction and Rietveld refinement
- Microstructural analysis by SEM and chemical analysis by EDS
- Magnetization measurements - TC at low applied field and isothermal MvsH measurements from 0 to 5 T applied field using a SQUID magnetometer
Results – Tc versus substitution
- Curie temperature decreases ~ linearly Eu substitution, but reaches a limit for Er substitution
- Secondary phase formation due to La/Er ionic size mismatch
What happens in the solubility limit?
- Above 6% of Er substitution, a secondary phase of ErMnO3 is formed, with drastic changes in microstructure
- Magnetocaloric properties should reflect this phenomenon
Secondary ErMnO3 phase
Magnetocaloric measurements near the Er solubility limit
- Maximum value of magnetic entropy change (in J.K-1.kg-1) lowers slightly due to formation of secondary phase
- The magnetic entropy curves widen, increasing the magnetic Relative Cooling power.
- This effect is interesting in an applications point of view, since Tc is ~15 K above ambient and the widening appears only below Tc.
- This effect can be optimized, and should depend strongly on the kinetics of Er ion difusion during preparation (sintering).
Magnetocaloric effect of Er and Eu series
Landau theory of phase transitions and the magnetocaloric effect
Gibbs free energy expansion:
Minimizing the free energy, we obtain the magnetic equation of state:
By representing isothermal magnetization data in an Arrott plot (H/M versus M2), polynomial fits give the values of A, B and C coefficients.
Landau theory of phase transitions and the magnetocaloric effect
With A,B and C coefficients determined, magnetic entropy change can be estimated by
And can be compared with results obtained by numerical integration of the Maxwell relation.
In the case of manganites, the B coefficient represents magnetoelastic couplings and electron spin condensation energy.
By changing the dependence of B with T it is possible to calculate and estimate the dependence of the magnetocaloric effect with such couplings (REF)
Mean-Field (molecular) theory application
We begin by considering the general mean-field law:
If the f function is monotonous (like the Brillouin function), then for corresponding values of M:
We can then plot plot a graph of H/T versus 1/T for regular M intervals
And linear fits for each M value will show the dependence of the exchange field with magnetization
Exchange field and scaling ‘f’ function
- For a 2nd order Phase transition, exchange field is ~λM
- We can then scale experimental data as a function of (H + Hexch)/T
- The f function is then obtained, and can be fitted with a odd-terms polynomial of arbitrary order.
Magnetic entropy estimation
The magnetic entropy change with applied magnetic field can be estimated by
Therefore the magnetic entropy variation between an applied field H1 and H2 is given by:
Which can be easily calculated numerically by using
- Manganite systems offer a rich field to study magnetocaloric effect and coupling influences
- Landau theory can be used to estimate the magnetocaloric effect and a deeper physical study of the system
- Mean field theory allows the estimation of magnetic entropy, including well below Tc, and also a deep theoretical analysis of the experimentally obtained ‘f’ function.
- FCT for financial support (POCI/FP/63438/2005 and POCI/CTM/61284/2004) and PhD. grant SFRH/BD/17961/2004
- M. Armanda Sá for assistance in magnetic measurements at IFIMUP, Porto