Solid state physics. Lecture 9: Magnetism. Prof. Dr. U. Pietsch

Transcription

1 Solid state physics Lecture 9: Magnetism Prof. Dr. U. Pietsch

2 Diamagnetism and Paramagnetsim Materie in magnetic field m 0 0 H M H(1 H 0 0M m M magnetiszation magnetic susceptibility - magnetic permeability Within magnetic field, alignemt of permanet magnetic dipoles paramagnetim m 0 Diamagnetism All atoms with paired spins : S=0 m 0Ze² N r² 6m m 10 6 Langevin equation N atoms/unit volume <r²> = <x²> + <y²> + <z²> lectron distribution within the atom Paul Langevin

3 The Langevin theory of diamagnetism applies to materials containing atoms with closed shells. A field with intensity, applied to an electrons with charge e and mass m, gives rise to Larmor precession with frequency ω = e / m. The number of revolutions per unit time is ω / π, so the current for an atom with Z electrons is I Ze² ( 4m The magnetic moment of a current loop is equal to the current times the area of the loop. Suppose the field is aligned with the z axis. The average loop area can be given as, ² Where ² is the mean square distance of the electrons perpendicular to the z axis. The magnetic moment is Ze² ( ² 4m If the distribution of charge is spherically symmetric, we can suppose that the distribution of x,y,z coordinates are independent and identically distributed. Then where is the mean square distance of the electrons from the nucleus. Therefore,. r² <x²> = <y²> = <z²>= 1/3 <r²> ² x² y² / 3 r² If N is the number of atoms per unit volume, the diamagnetic susceptibility in SI units is m 0N 0Ze² N r² 6m

4 Paramagnetsim Paramagnetism m 0 Atoms with odd number of electrons S<>0 ree atoms/ions with partly filled inner shells Metals Magnetic moment g 1 J g J J ( J 1 S( S 1 L( L 1 J ( J 1 gyromagnetic ratio ohr magneton g Lande factor nergy niveau splitting in -field m g J Is sum of orbital momenta L J m J =J, J-1,J-,,-J J( J 1 Spin alignment quantized Spin never parallel

5 Summing up over all possible spin orientations Curie s law M N J J g mj ( g mj exp( kt J g mj exp( kt J M Ng m J J J M max = Ng J ( y y g kt J J (y rillion function J ( y J 1 coth( J J 1 y J 1 J coth( y J Leon rilluoin m or large (y Langevin or y <<1 : coth(y = 1/y + y/3 y³/45+. y( J 1 J ( y 3J M Ng J J N 0 0 ² ( 1 0 eff 3kT 3kT Curie - law C T 1/ T Measurement of (T eff =p

6

7 Paramagnetism of ions Calculation of eff from electron configuration of atoms, considering - Pauli principle - Hund s rule lectrons in partially filled shell first towards maximum S followed by maximazing L J = L S for shell below half filling J = L + S for shell above half filling xample Ce 3+, 1 f-electron, L=3; S= ½ J=L-S = 5/, experimental finding offen L=0 Orbial moment is quentched, caused by time average in non- cubic crystal field ( see later xample 3d element do show spin magnetism only!!!! no oribital magnetism

8 Solid state magnetism Considers: (1 interatomic interaction ; ( interaction of magnetic moments Metals: magnetism of electrons in conduction band Magntism of inner, partially filled shells : i.e. e 3d group, rear elements 4f shell Ion crystals: l=o spin magnetism, rear earth elements L is not quenched, 4f electrons are screed by 5p, 5d and 6s electrons against external crystal fields Paramagnetism of conducting electrons magnetic moment of single electron xpection of classic free electron gas m 0 M 0N kt e mc M N per electron L( n kt kt Curie behaviour N kt xperimentally one observes 1/100 of this value: only electrons close to can contribute to magnetism ratio T/T 0M 0N m kt

9 Density of states =0 : N( =N( = ½ N m V D D 3 / 1 ² ² ( ( M=0 >0 : N( - N( =( ½ N + ½ N - =( ½ N - ½ N = N M= N N =??? 3/ 1 3/ 1 ( 3 ( ( ( 3 ( ( A d D D A d D D 3/ ² ² m V A

10 << /kt /kt 100 A N 3 A N 3 3/ 3/ (1 3 / A 3/ 1/ A 3 3/ (1 3 /... M 3 ( T 0 Ne( 3 kt ² N e 3 kt Pauli s spin susceptibility of conducting electrons Additional effect due to induced energetic splitting of electron levels Landau diamagnetism tot para dia para 1 3 para N ² e kt or T>0 ² T M ( T M (0[1 1 ( ²... T M is nearly independent from T

11 Cooling down to 0.3K by demagnetization 1. Cooling down to 4.K by pumping He 4 and He 3. Switching on =1T magnetic field - in equilibrium N1 = N 3. Spin switch energy first released via He bath, 4. Slow reducing the field spin switch energy taken from phonons reduces temperature 3 Paramagnetic salt 4

12 Appears below Curie temperature T c erromagnetism At T > T c - paramagnetism T C T c Curie- Weisslaw Typical ferromagnets : e, Co, Ni group 3d Gd, Dy, - group 4f Origin of ferromagnetism is the internal interaction caused by exchange fields between next neighbors or by molecular fied.

13 Internal magnetic field Weiß supposed the existence of internal magnetic field y molecular field : M Spin at one particular atom is feeling magnetic moments of neighbored atoms, creating a mean field and small magnetisation M Using Curie law : Resulting in C m T 0 M m M ( 0 C ( 0M T M T C C C T C T C T Providing singularity at T=T c ; Curie- Weiss law is well confirmed experimentally More advanced calculations predict : C NJ( J 1 g 4/3 C T T ( c or e Tc= 1000K, = 5000; saturation M s = 1700 Gauss *M s = 10^7 Gauss = 10^3 T!!! T< Tc, Ms becomes T-dependent C 3k

14 Crystal field or example : en 6 complex

15 xplanation by rillion function T< T c, Ms becomes T-dependent, behavior explained by evaluation of rillouin function, kt determine crossing point between M x and M N tanh 0 M 0 kt M with x 0 kt Slope M(x increases for increasing T M(T/M(0=1 at T=0 and zero at T=T c. Origin of internal field is molecular field: an atom with unpaired spin is surrounded by other atoms in certain geometrical arrangement

16 xchange interaction Depending on next neighbor distances the energy separation between the bonding and antibonding orbital changes and results in paired or mainly unpaired arrangement, Phase transition between different magnetization can be induced by local distortions of the crystal field. xchange interaction is expressed by Heisenberg exchange energy U JSˆ i Sˆ j where J is exchange integral, ferromagnetic coupling: J>0, antiferromagnetic coupling: J<0, ² 0N J Z J cannot be explained by dipole-dipole interaction but interplay between kinetic energy and Coulomb energy xchange interaction, or metals like e, Co, Ni Slater criterion expresses J as function of r/r a, where r is next neighbor distance and r a the atomic radius. J>0 for r/r a > 1,5 and J<0 for r/r a < 1.5.

17 Kinds of magnetic exchange interaction Direct exchange interaction Indirect exchange interaction superexchange Oscillatry interaction via electrons in C, proposed by Rudermann, Kittel, Kasuya, Yoshida = RKKY interaction

18 Antiferromagnetism J<0 C ( T A magnetism vanishes above T N Neel temperature T N M 1 is the antiferromagnetic x-change interaction m curve shifted to negative Curie-Weiß T temperature. In range 0> T>T N the total magnetization is zero and A order increases xperimental evidence by spin resolved neutron scattering A