Download Lecture 21: Mean Field Theory of Ferromagnetism

Lecture 21:
Mean Field Theory of Ferromagnetism
Chapter IV. Magnetism and Phase Transitions
A.G. Petukhov, PHYS 743
April 16, 2014
Chapter IV. Magnetism and Phase Transitions Lecture 21: Mean Field Theory of Ferromagnetism
A.G. Petukhov,
April 16, PHYS
2014 743 1 / 14
Magnetic Interactions
The simple theory of paramagnetism considered in the previous lecture
assumes that the discrete sources of magnetic moment (localized
moments of partially filled shells in insulators containing transitional
metal ions, conduction electrons) do not interact with each other
The assumption about noninteracting magnetic moments must be
dropped to describe variety of phenomena caused by magnetic
interactions. For instance, some materials known as ferromagnets
have a nonvanishing magnetic moment , or ”spontaneous
magnetization” even in the absence of a magnetic field
One’s first expectation might be that the magnetic interactions
between discrete moments arise from their magnetic fields. In most of
the cases, however, the dominant source of magnetic interaction is
ordinary is the ordinary electrostatic electron-electron interaction.
Chapter IV. Magnetism and Phase Transitions Lecture 21: Mean Field Theory of Ferromagnetism
A.G. Petukhov,
April 16, PHYS
2014 743 2 / 14
Exchange Interaction
When we use the term ”magnetic interactions” we mean the
dependence of the energy of two or more magnetic moments on their
relative orientation. The main cause of ferromagnetism is the
so-called exchange interaction or exchange energy, which is due to: i)
Coulomb interaction between electrons; ii) Pauli exclusion principle
Because of the Pauli principle the energy of a two-electron system
(e.g. diatomic molecule) strongly depends on the relative orientation
of electron spins even though the Hamiltonian is spin-independent
and contains only Coulomb interactions.
Since the Coulomb force between electrons is several orders stronger
than the direct interaction arising from their magnetic fields the scale
of the exchange interaction is several orders of magnitude larger than
that of the magnetic dipole-dipole interaction
The exchange interaction is of purely quantum mechanical nature and
does not have a classical analogue.
Chapter IV. Magnetism and Phase Transitions Lecture 21: Mean Field Theory of Ferromagnetism
A.G. Petukhov,
April 16, PHYS
2014 743 3 / 14
Heisenberg Exchange Hamiltonian
Consider the eigenvalue problem for a diatomic molecule with two
electrons in a minimal 4 × 4 Hilbert space spanned by four
two-electron states such as |↑↓i, |↓↑i, |↑↑i, |↓↓i.
From quantum mechanics we know that the Hamiltonian of the
system commutes with the operator of the total spin Ŝ = ŝ1 + ŝ2 and
that the eigenstates of this Hamiltonian are also the eigenstates of
the operator S.
Due to this symmetry the Hamiltonian of the system has two
eigenvalues: Es , corresponding to a non-degenerate singlet state
(S = 0); and Et corresponding to a three-fold degenerate triplet state
(S = 1).
Consider the operator
Ŝ 2 = (ŝ1 + ŝ2 )2 = ŝ21 + ŝ22 + 2ŝ1 · ŝ2 =
Here we used that ŝ2i = s(s + 1) = 12 ( 12 + 1) =
3
+ 2ŝ1 · ŝ2
2
3
4
Chapter IV. Magnetism and Phase Transitions Lecture 21: Mean Field Theory of Ferromagnetism
A.G. Petukhov,
April 16, PHYS
2014 743 4 / 14
Heisenberg Exchange Hamiltonian
Therefore, the operator ŝ1 · ŝ2 has two eigenvalues:
S(S + 1)/2 − 43 = 41 ,
S=1
ŝ1 · ŝ2 =
S(S + 1)/2 − 34 = − 34 , S = 0
Finally, the original Hamiltonian can be mapped onto the so-called
Heisenberg spin Hamiltonian:
1
Hspin = (Es + 3Et ) − (Es − Et )ŝ1 · ŝ2
4
Now we can drop the constant term and rewrite this Hamiltonian in
the original form proposed by Heisenberg
Hspin = −2J ŝ1 · ŝ2 ,
J = (Es − Et )/2
We can generalize the two-particle (spin 1/2) Hamiltonian to the
arbitrary number of magnetic atoms with arbitrary spins and obtain
X
Hspin = −
Jij Si · Sj
(1)
ij
Chapter IV. Magnetism and Phase Transitions Lecture 21: Mean Field Theory of Ferromagnetism
A.G. Petukhov,
April 16, PHYS
2014 743 5 / 14
Mean Field Approximation
Even though the Heisenberg Hamiltonian (1) is approximate, it is still
enormously complex and cannot be treated exactly
The earliest attempt at a quantitative analysis of the ferromagnetic
transition was put forward by P. Weiss and is known as mean (or
molecular) field theory
Here we will apply the Weiss idea to the Heisenberg model (1)
First, we add a magnetic field H to the Hamiltonian:
X
X
Hspin = −
Jij Si · Sj + gµB H
Si
ij
i
Second, we separate all the terms containing spin Si :


X
Hi = −Si 
Jij Sj + gµB H 
(2)
j6=i
Chapter IV. Magnetism and Phase Transitions Lecture 21: Mean Field Theory of Ferromagnetism
A.G. Petukhov,
April 16, PHYS
2014 743 6 / 14
Mean Field Approximation
One can formally represent the Hamiltonian (2) as a Hamiltonian of
spin Si in the effective magnetic field, i.e.
Hi = −gµB Si · Heff ,
where
Heff = H +
1 X
Jij Sj
gµB
(3)
j6=i
Unfortunately, the “field” Heff is an operator, which depends on
configurations of all other spins in a very complicated way and the
problem is still intractable
The mean field approximation consists in replacing all Sj in Eq. (3)
with their thermodynamic expectation value, Sj → hSi
We note that the magnetization is related to hSi as
gµB X
gµB N
N
M=
hSi = gµB · n hSi , n =
(4)
hSi i =
V
V
V
i
Chapter IV. Magnetism and Phase Transitions Lecture 21: Mean Field Theory of Ferromagnetism
A.G. Petukhov,
April 16, PHYS
2014 743 7 / 14
Mean Field Approximation
The exchange interaction is determined by overlap of the wave
functions. As a result it is very short-range. Without any loss of
generality we assume that Jij = J if i, j are the nearest neighbors in
a crystal lattice and 0 otherwise
We choose H = (0, 0, H) along z-axis. Then M and Heff are both
parallel to z-axis as well.
It follows from Eqs (3) and (4) that:
Heff = H + λM,
where
zJ
,
n(gµB )2
and z is the number of the nearest neighbors in a crystal lattice.
Then we can calculate M using the Brillouin function:
gµB S(H + λM )
M = n · gµB S · BS
T
λ=
Chapter IV. Magnetism and Phase Transitions Lecture 21: Mean Field Theory of Ferromagnetism
(5)
A.G. Petukhov,
April 16, PHYS
2014 743 8 / 14
Critical Temperature
Eq. (5) is a transcendental equation for magnetization M
A solution of Eq. (5) exists at any temperature if H 6= 0
When H = 0 Eq. (5) describes spontaneous magnetization. Let us
introduce:
gµB S
(H + λM )
(6)
α=
T
and consider H → 0. Then we can rewrite Eq. (5) as
Tα
= BS (α)
(gµB S)2 nλ
(7)
Let us examine solutions of Eq. (7) at the onset of spontaneous
magnetization. This equation has a solution when the slope of the
l.h.s. is lower than that of the r.h.s, i.e. when the temperature T is
sufficiently small. The temperature corresponding to the matching
slopes of l.h.s and r.h.s. is called critical temperature Tc .
Chapter IV. Magnetism and Phase Transitions Lecture 21: Mean Field Theory of Ferromagnetism
A.G. Petukhov,
April 16, PHYS
2014 743 9 / 14
Critical Temperature
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
2
4
6
8
10
To find Tc we expand the Brillouin function at small α:
Tc · α
S+1
α
=
2
(gµB S) nλ
3S
(8)
From here we obtain:
1
Tc = S(S + 1)zJ
3
Chapter IV. Magnetism and Phase Transitions Lecture 21: Mean Field Theory of Ferromagnetism
A.G. Petukhov,
April 16, 2014
PHYS 74310 / 14
Magnetization
To find the magnetization M we have to solve Eq. (5) numerically.
The result is shown below
1.0
0.8
0.6
0.4
0.2
0.5
1.0
1.5
2.0
2.5
3.0
Near T = Tc magnetization behaves singularly, which is a signature of
a phase transition, i.e. emergence of the spontaneous magnetization
when T < Tc or, conversely, disappearance of the magnetization when
T > Tc .
Chapter IV. Magnetism and Phase Transitions Lecture 21: Mean Field Theory of Ferromagnetism
A.G. Petukhov,
April 16, 2014
PHYS 74311 / 14
Magnetic Susceptibility
Our final task is to find the susceptibility χ. We use:
χ=
∂M
∂H
and differentiate α (Eq. (6)) with respect to H:
∂α
gµB S gµB Sλ
=
+
χ
∂H
T
T
(9)
On the other hand:
χ=
Now we can eliminate
∂M
∂α
= ngµB · SBS0 (α)
∂H
∂H
∂α
∂H
χ(T ) =
(10)
from Eq. (9)-(10). This yields:
n(gµB S)2 BS0 (α)
T − n(gµB S)2 BS0 (α) · λ
Chapter IV. Magnetism and Phase Transitions Lecture 21: Mean Field Theory of Ferromagnetism
(11)
A.G. Petukhov,
April 16, 2014
PHYS 74312 / 14
Magnetic Susceptibility
Eq. (11) can be further simplified if we use Eq. (8)
χ(T ) =
n(gµB S)2 BS0 (α)
,
3S
T − S+1
BS0 (α) · Tc
(12)
where α is the solution of Eq. (7).
For T > Tc , α = 0 and BS0 (0) = (S + 1)/3S, which allows us to
simplify χ(T ) even further:
χ(T ) =
n(gµB )2 S(S + 1)/3
χ0
=
,
T − Tc
1 − Tc /T
(13)
where χ0 is the paramagnetic susceptibility. Eq. (13) is called the
Curie-Wiess law
When T < Tc one has to solve Eq. (7) numerically. The result is
shown in the Fig. below.
The divergence of the magnetic susceptibility is one of the main
signatures of the ferromagnetic phase transition
Chapter IV. Magnetism and Phase Transitions Lecture 21: Mean Field Theory of Ferromagnetism
A.G. Petukhov,
April 16, 2014
PHYS 74313 / 14
Magnetic Susceptibility
8
6
4
2
0.0
0.5
1.0
1.5
2.0
2.5
Chapter IV. Magnetism and Phase Transitions Lecture 21: Mean Field Theory of Ferromagnetism
3.0
A.G. Petukhov,
April 16, 2014
PHYS 74314 / 14