02 weiss heisenberg stoner
TRANSCRIPT
-
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
-
8/12/2019 02 Weiss Heisenberg Stoner
2/33
Types of magnetic structures
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
5/33
Towards (uantum theory
Langevin)s and especially Weiss theories descri'e a wide range of magneticphenomena relatively well
paramagnetism, ferromagnetism, antiferromagnetism /8
-
8/12/2019 02 Weiss Heisenberg Stoner
6/33
+acroscopic definitions
Thermodynamics2 free energy
+agneti3ation
*uscepti'ility
:orce in inhomogeneous magnetic field
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
14/33
"rillouin function
:rom free energy we get magneti3ation
where "rillouin function is defined 'y
4t low temperatures BJ6
4t high temperatures
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
16/33
&le2 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
-
8/12/2019 02 Weiss Heisenberg Stoner
17/33
&le2 ?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
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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+
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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.
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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 $$ e&ists 9 it is a ground state E0;
$int2
-
8/12/2019 02 Weiss Heisenberg Stoner
31/33
Low-T e&citations of ferromagnet
Lowering spin proection 'y one at one site is not an eigenstate of $$2
"ecause , i.e., translational invariance, wecan construct linear com'inations
which are eigenstatesof $eisen'erg $amiltonian2
One can show that
-
8/12/2019 02 Weiss Heisenberg Stoner
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
-
8/12/2019 02 Weiss Heisenberg Stoner
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