Phase transitions Experimental studies on magnetic materials Ekkes - PowerPoint PPT Presentation

Phase transitions Experimental studies on magnetic materials Ekkes Brück, Fundamental Aspects of Materials and Energy, TNW 27-08-11 Delft University of Technology Challenge the future

Outline • Basic magnetics (classical) •Origin of first order transition? •Magnetic materials •Gd 5 Ge 2 Si 2 magnetic Rare-Earth •MnFe(P,Si) magnetic transition metals •TbFe 4 Al 8 magnetic RE & TM 2 Magnetic phase transitions

Magnetization processes Magnetic (spin) moments no field no field no net moment no net moment µ - µ B B 0 - 0 Effect of temperature. Classical statistical physics: Effect of temperature. Classical statistical physics: probability of finding atomic dipole in state with energy E: probability of finding atomic dipole in state with energy E: 3 Magnetic phase transitions

Magnetization processes 7 M N µ 5 B 3 1 B/T (T/K) µ - µ B B 0 - 0 0 0 1 2 3 4 4 Magnetic phase transitions

Magnetization processes M S M M M S Hysteresis loop M R M R B C B B 0 B C 0 from unmagnetized from unmagnetized state state once magnetized once magnetized -M S -M S 5 Magnetic phase transitions

Magnetization processes Ferromagnetic domains www.ee.umd.edu/~rdgomez/permalloy.htm www.ee.umd.edu/~rdgomez/permalloy.htm 6 Magnetic phase transitions

Magnetization processes For example 2.5 MnFeP0.46As0.54 M ( µ B /f.u.) at 310 K 2.0 1.5 1.0 Materials with field induced first 0.5 order phase transition. B (T) 0.0 0 1 2 3 4 5 7 Magnetic phase transitions

Zeeman effect for state with total moment J Ground state J is 2J+1 times degenerated: J z =-J, -J+1, … J • • Splits in magnetic field into sublevels J z H = − μ ⋅ B = − µ ⋅ B P z 2 1 E = < H > = g µ J B 0 P Lande B z z J -1 ∆ E = g µ B -2 L B z 3 L ( L + 1 ) − S ( S + 1 ) = − g Lande 2 2 J ( J + 1 ) B Spectroscopic splitting factor g Landee depends on L, S, and J • • Splitting at B=1 Tesla in the order of meV Atom behaves as if it has effective moment: µ eff =-g L µ B J • 8 Magnetic phase transitions

Statistical physics description When a system, in contact with a heat bath at temperature T can be in a state with energy E , the probability for this is given by the Gibbs rule: where k is Boltzmann's constant. Z is called the partition sum, 9 Magnetic phase transitions

Z is needed to have the proper normalization The strength of statistical physics is that by calculating Z a lot of information about the system can be derived. The Helmholtz free energy is: while the Gibbs free energy is: 10 Magnetic phase transitions

Basic magnetocalorics Two energy reservoirs spins lattice  E 11 Magnetic phase transitions

Basic magnetocalorics spins lattice  E 12 Magnetic phase transitions

Magnetic cooling: Debye and Giauque 1926 → 61g Gd 2 (SO 4 ) 3 ·8H 2 O, ΔB=0.8T, 1.5K 0.25K Nobel prize 1949 13 Magnetic phase transitions

Thermodynamic relations: Differential of Gibbs free energy   ∂ ∂ ∂     G G G ( ) ( ) ( ) = − = − = −   S T , B , p   M T , B , p   V T , B , p , ,   ∂ ∂ ∂ T B p       B , p T , p T , B Entropy Magnetization Volume Differential of entropy ∂ ∂  ∂      S S S = + +       dS dT dB dp   ∂ ∂ ∂ T B p       B , p T , p T , B 14 Magnetic phase transitions

Identification of terms C ∂   S B , p = + − α   dS dT dB Vdp ∂ T  B  T , p ∂   T S Adiabatic process at = −   dT dB constant pressure ∂ C  B  B , p T , p ∂ ∂     S M Maxwell relations =     ∂ ∂  B   T  B ∂   M T B ∫ =   Δ S dB Magnetic entropy m ∂ T   B 0 15 Magnetic phase transitions

From definition of specific heat S 0 can be set to zero because it is not depending on field 16 Magnetic phase transitions

Experimental determination from magnetic measurements − M ( T , B ) M ( T , B ) ∑ ∆ = ∆ S ( T , B ) + + B i 1 i 1 i i − m T T i + i 1 i 17 Magnetic phase transitions

Outline •Basic magnetics (classical) • Origin of first order transition? •Magnetic materials •Gd 5 Ge 2 Si 2 magnetic Rare-Earth •MnFe(P,Si) magnetic transition metals •TbFe 4 Al 8 magnetic RE & TM 18 Magnetic phase transitions

Continuous phase transition In the absence of an external field, H=0, the system with exchange interaction J/k=1may spontaneously order. 1  1 1 1 1  2 = − F NJm - NHm+NT ( -m) ln ( -m)+( +m) ln ( +m)     2 2 2 2 2 T=0.2J/k T=0.25J/k T=0.3J/k 19 Magnetic phase transitions

First order phase transition If interactions with quartets play a role this may result in local minima in the free T=0.02 J/k energy. 1  1 1 1 1  T=0.0225 J/k 4 = − F NJm - NHm+NT ( -m) ln ( -m)+( +m) ln ( +m)     2 2 2 2 2 T=0.025 J/k 20 Magnetic phase transitions

Reason for first order transition Ansatz T C = T 0 [1 + ß(V - V 0 )/V ], (Bean and Rodbell, 1962). T C Curie temperature, T 0 Curie temperature if lattice incompressible. V 0 volume in absence of exchange interaction, β effect of volume change on Curie temperature. 21 Magnetic phase transitions

Gibbs energy 2     − V V 1 j 1   = −   σ − σ σ + − + + 2 0 G NkT B T ( S S ) pV     C 0 j l + 3 j 1 2 K V     0 N number of magnetic atoms per unit mass, k Boltzmann constant, σ 0 saturation magnetization, σ relative magnetization, K the compressibility, S j entropy of spin sublattice, S l entropy of lattice subsystem 22 Magnetic phase transitions

State equation 2 σ − η σ + σ T 2 0 . 867 B / NkT 0 0 = . 3 5 T σ + σ + σ − α β σ 2 0 . 867 0 . 606 2 T 0 0 α Thermal expansion Good agreement with exp. data Second order transition for η <1 23 Magnetic phase transitions

Gibbs energy as function of σ Local minimum near Tc ⇒ First order transition 24 Magnetic phase transitions

Outline •Basic magnetics (classical) •Origin of first order transition? • Magnetic materials •Gd 5 Ge 2 Si 2 magnetic Rare-Earth •MnFe(P,Si) magnetic transition metals •TbFe 4 Al 8 magnetic RE & TM 25 Magnetic phase transitions

Phase transition in iron 10 6T 0T ∆ T [K] 3T 3T C p [ J m / o l · K 0.8T ] 0 T [K] T [°C] Magnetic ordering T 26 Magnetic phase transitions

MCE in gadolinium Total entropy vs reduced temperature of gadolinium in low field (blue) and high field 9T (purple) (Gschneidner et al) 27 Magnetic phase transitions

Outline •Basic magnetics (classical) •Origin of first order transition? •Magnetic materials • Gd 5 Ge 2 Si 2 magnetic Rare-Earth •MnFe(P,Si) magnetic transition metals •TbFe 4 Al 8 magnetic RE & TM 28 Magnetic phase transitions

Giant magnetocaloric effect in Gd 5 Ge 2 Si 2 Magnetically dilute yet higher effect double transition? d ( K ΔT a ) Pecharsky & Gschneidner PRL 78 (1997) 4494 29 Magnetic phase transitions

Crystal growth Crystal growth D=4mm Sphere was cut by spark erosion from as grown rod Crystal was grown in a mirror furnace by means of traveling solvent floating zone method 30 Magnetic phase transitions

Unusual behavior Unusual behavior  Extraordinary magnetic behavior: first-order character of the paramagnetic-ferromagnetic transition. 3.0 Gd 5 Si 1.65 Ge 2.35 Crystal Sphere d=4mm 2.5 B=0.05T Stepwise heating mode 2.0 a M ( µ B /f.u.) b 1.5 c 40 1.0 0.5 30 0.0 600 0 50 100 150 200 250 300 350 400 T (K) 550 M ( µ B /f.u) a 20 500 b c 450 400 10 350 Gd 5 Ge 2.35 Si 1.65 crystal C (J/mol ·K) Sphere d=4mm 300 at 5K 250 0 200 0 1 2 3 4 5 B (T) 150 100 50 0 0 20 40 60 80 100 120 140 160 180 200 220 240 260 31 T (K) Magnetic phase transitions

Unusual behavior Unusual behavior  The high temperature paramagnetic monoclinic phase transforms to the low temperature ferromagnetic orthorhombic phase. The low temperature phase has a higher symmetry than the high temperature, which is the opposite of what is normally observed for other polymorphic systems. b b <T c : Pnma, No.62 >T c : P112 1 /a, No.14 32 Magnetic phase transitions Crystallographic data comes from W Choe PRL v84, n20, p4617, 2000

8 a-axis 7 6 5 -3 ) heating ∆ L/L (10 4 cooling 3 2 1 0 -1 0.5 b-axis thermal expasion of 0.0 Gd 5 Ge 2.4 Si 1.6 heating -3 ) -0.5 ∆ L/L (10 cooling -1.0 -1.5 -2.0 c-axis 0.5 0.0 heating -3 ) ∆ L/L (10 -0.5 cooling -1.0 -1.5 180 200 220 240 260 280 T (K) 33 Magnetic phase transitions