02 weiss heisenberg stoner

8/12/2019 02 Weiss Heisenberg Stoner

1/33

Outline Para-, ferro-, antiferro-, ferrimagnets, ...

Classical theory

Langevine theory of paramagnetism, Curie law

Weiss molecular field theory, TC, Curie-Weiss law

Quantum theory of dia- and paramagnetism

Larmor, van lec!, "rillouin function, Pauli

#nteraction of moments

$eitler-London model of $%and e&change interaction $eisen'erg model hamiltonian

mean field appro&imation, Curie-Weiss law, TC

(uantum $eisne'erg model, magnons, "loch)s law

*toner model


2/33

Types of magnetic structures


3/33

Classical theory of paramagnetism

+odel of Langevin magnetic moment as a vector in magnetic field

nergy of moment min field H/0,0,$12

+ean value of magneti3ation on atom

where Langevin function is defined as

*uscepti'ility /for large T1

Curie law /inverse proportionality to T1


4/33

Weiss molecular field theory

4ttempt to e&plain magnetic order

#ntroduction of an internal field caused 'y neigh'oring atomic moments

$uge fields 56000T7 8o physical interpretation 'y Weiss

#n 3ero e&ternal magnetic field, using Langevin theory

we get 9 graphical solution2

:or T;Tc there is only a solution +0

Critical temperature

*uscepti'ility 9 Curie-Weiss law


5/33

Towards (uantum theory

Langevin)s and especially Weiss theories descri'e a wide range of magneticphenomena relatively well

paramagnetism, ferromagnetism, antiferromagnetism /8


6/33

+acroscopic definitions

Thermodynamics2 free energy

+agneti3ation

*uscepti'ility

:orce in inhomogeneous magnetic field


7/33

lectrons in magnetic field

change of canonical momentum operator

the !inetic energy operator 'ecomes

add interaction of spins with magnetic field

summary of new terms in $amiltonian


8/33

Pertur'ation theory

the new terms are small at fields which we can produce in la'oratories,therefore we treat it as a pertur'ation

suscepti'ility is %ndorder in H

changes in energy levels up to %ndorder in H2

the linear term is of the order of 56 me in fields 560 T, the other terms are

?-@ orders of magnitude smaller

individual terms lead to a dia- or paramagnetic 'ehavior of the system inmagnetic field


9/33

Larmor diamagnetism

insulators, closed shells in ground state /LSJ01, i.e., only last termcontri'utes

ground-state energy change due to mag. :ield2

if all e&cited states are high in energy, then

This is Larmor diamagnetic susceptibility/negative1, typically 560-A


10/33

$und)s rules

/61 +a&imi3e the total spin S.4.!.a. Bthe 'us seat rule.+a&imi3ing spin reduces screening andallows electrons to get closer to cores.

/%1 While fulfilling /61, ma&imi3e thetotal or'ital momentum L.Classically2 or'iting in the same direction reduces pro'a'ilitythat electrons meet, i.e., reduces repulsion.

/?1 :or less than half-filled shells D EL-*E and for morethan half-filled shells D L F *.This is a conse(uence of spin-or'it coupling


11/33

an-lec! paramagnetism

if D0, 'ut not L or *, the first term vanishes

we get two non-3ero terms in suscepti'ility

first term is Larmor diamagnetism

second term is an-lec! paramagnetism /positive1

if the first e&cited state is close in energy to the ground state, morecomplicated formulas apply


12/33

Paramagnetism

:or atoms with unfilled shells with non3ero *, L and D, the first term is non3eroand dominates

We have /%DF61 degenerate state in 3ero field

#n non3ero field we need to diagonali3e /%DF61&/%DF61 matri& with elements

Wigner-c!art theorem states

with Land< factor


13/33

Paramagnetism

Than!s to Wigner-c!art theorem we see that the matri& is actually diagonal

we can interpret /within the lowest D-multiplet1

as a magnetic moment of the ion

To get suscepti'ility we need to consider all these /%DF61 states split 'ymagnetic field

:ree energy


14/33

"rillouin function

:rom free energy we get magneti3ation

where "rillouin function is defined 'y

4t low temperatures BJ6

4t high temperatures


15/33

Curie)s law

4t high temperatures we get for suscepti'ility

i.e., suscepti'ility is inverse-proportionalto temperature Curie law

This paramagnetic suscepti'ility is at room temperature of the order of 60 -%-60-?and thus dominates the diamagnetic contri'ution

Comparing to the Curie-law derived in the classical case, we can define aneffective moment G effective "ohr magneton num'erp2


16/33

&ample2 rare-earth paramagnets

Hood agreement theory-e&periment,e&cept for *m I u

*m I u have low-lying e&citedstates, which we neglected


17/33

&ample2 ?d transition metals

:or ?d transition metals Curie)slaw wor!s if we assume L0

Quenching of or'ital momentum

due to crystal field splitting+odification of the third $und)s

rule


18/33

+agnetism of conduction electrons

Jelocali3ed conduction electrons

+agnetic field shifts energy levels 'y

*ince we can e&pand density of states

and o'tain for num'er of occupied states

i.e., the magneti3ation is


19/33

Pauli paramagnetism

:rom magneti3ation

we get a suscepti'ility

which is independent of temperature

This contri'ution is called Pauli paramagnetism

#t is of order 60-K, i.e., compara'le to Larmor diamagnetic contri'ution

8ote2 conduction electrons also have Landau diamagnetic contri'ution tosusc., see 4I+


20/33

*ources /%%.@. I %[email protected]

+ain source2

4shcroft I +ermin2 Solid State Physics

Chapter ?62 Jiamagnetism and Paramagnetism /%%[email protected]

Chapter ?%2 lectron #nteractions and +agnetic *tructure Chapter ??2 +agnetic Ordering

*ee also2

+ohn2 Magnetism in the Solid State

Chapter >2 $eisen'erg $amiltonian

*ection 6K.62 $eitler-London Theory for the &change :ield


21/33

#nteraction 'etween moments

We developed a (uantum theory of magnetic suscepti'ility originating fromvarious terms /Larmor, van lec!, non3ero-D paramagnetism, Pauli1

*o far we included no interaction 'etween moments 9 no mechanism for aspontaneous magneti3ation in 3ero e&ternal magnetic field

Possi'le sources of such interaction2

Jipolar too wea!, 50.6me

*pin-or'ital stronger in heavy elements, up to 56e for actinides

lectro-static F Pauli principle the strongest, 56e and more

$eitler-London model of hydrogen molecule as a starting point for

constructing model $amiltonians for interactions of locali3ed moments


22/33

$ydrogen molecule /summary1

Two electron system, four possi'le spin arrangements

We can classify them asspin singlet /*01 and

spin triplet /*612

The ground state is singlet

#f we neglect all the higher e&cited states and restrict ourselves to singlet I

triplet, we can reproduce the energy levels 'y the following model$amiltonian e&pressed in spin-space only2


23/33

$eitler-London model of $%

% hydrogen atoms, % electrons, assumption that there is always one electronclose to every proton 9 symmetric and antisymmetric wavefunctions

the complete $amiltonian can 'e written as

energies of the spatially symm.Gantisymm. states are

where


24/33

$eitler-London model of $%

Let)s add spins 9 possi'le spin configurations for two electrons2

*0, +*0 spin singlet antisymmetric in spins

*6, +*-6,0,6 spin triplet symmetric

Total %-electron wave-function is always antisymmetric 9 lower lying state

that is symmetric in or'ital space must have antisymmetric spin part, i.e.,spin singlet I higher lying state, which is antisymmetric in or'ital space willhave 'e a spin triplet

sing relations

the same energy levels can 'e directly o'tained 'y

we introduced Be&change constant 9 generali3ation gives $eisen'erg

model hamiltonian


25/33

$eisen'erg model

Henerali3ation of the situation with hydrogen molecule2

&tractingJijis not a trivial pro'lem and to some e&tent it is still not

completely solved *olving the (uantum model itself without appro&imations is computationally

unsolva'le e&cept for smallest systems

+echanismsGsources ofJij/Olle)s lecture ne&t wee!12

Jirect e&change *uper-e&change

#ndirect e&change /MNN1

#tinerant e&change


26/33

+agnetic structures

D;0 for nearest neigh'ors 9 ferromagnet

D0 for nearest neigh'ors 9 antiferromagnet

non-negligi'le D)s for more distant neigh'ors or in geometries leading tomagnetic frustrations 9 more complicated magnetic structures, e.g., spin

spirals, non-collinear structures, etc.


27/33

+ean-field theory Weiss field

rewrite the $eisen'erg model, including field2

this loo!s li!e a set of spins in effectivefield

which does not depend on idue to periodicity

yet, the effective /Weiss1 field is an operator 9 the mean-field theoryreplaces it with its thermodynamic mean value


28/33

Critical temperature

Ta!ing mean-field appro&imation and 3ero e&ternal field we o'tain e(uation

When M/T1 goes to 3ero, and we get

the mean-field appro&imation of the magnetic transition temperature

Wea! points2 over-estimation of TC, wrong low-temperature 'ehavior, alsoaround T

C


29/33

+ore advanced methods

Mandom-phase appro&imation

+onte-Carlo simulations e&act answers within the classical $eisen'erg

model, though demanding calculations

sing numerical methods we can also get M/T1 within all three methods

Joing +onte-Carlo $eisen'erg model (uantum-mechanically is still anactive field of research


30/33

Hround state of ferromagnet

4t T0, all moments aligned parallel, i.e., total moment 8* 9 E8*,8*;

Mewrite $eisen'erg $amiltonian using raising and lowering operators2

4pplying the $$ on the E8*,8*; gives

i.e., it is an eigenstate of the $$

8o lower energy-state of ferromagnetic $$ e&ists 9 it is a ground state E0;

$int2


31/33

Low-T e&citations of ferromagnet

Lowering spin proection 'y one at one site is not an eigenstate of $$2

"ecause , i.e., translational invariance, wecan construct linear com'inations

which are eigenstatesof $eisen'erg $amiltonian2

One can show that


32/33

Low temperature magneti3ation

*uperposition of magnons /li!e for phonons1 is only an appro&imation here,'ut B(uite ON for low-lying e&cited states

The magneti3ation is reduced 'y one per magnon, i.e.

4t low temperatures only the lowestenergy e&citations happen, and for these

i.e.,

"loch)s ?G% law

+ermin-Wagner theorem 9 no magneti3ation in %J or 6J


33/33

Hround state of antiferromagnet

#ntuitively2 arrangement of alternating upGdown moments

#t is notan eigenstate of $eisen'erg $amiltonian7

4ssume *6G% chain and apply

#n classical case /spins as vectors1 this is a ground state with lowest energy

:or nearest-neigh'or /881 interaction the following 'ounds are valid2

which actually coincide in the classical case, where

i i

02 weiss heisenberg stoner

Documents