R?ssler U. Solid State Theory: An Introduction
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6.4 Spin Waves in Anti-Ferromagnets 177
H
spin
≃ J
a
n.n.ij
(
−S
2
+ S
a
†
1i
a
1i
+ a
†
2i
a
2i
+ a
1i
a
2j
+ a
†
1i
a
†
2j
)
. (6.55)
The first term ∼−S
2
on the right hand side gives the energy of the anti-
ferromagnetic configuration E
a
= −2J
a
νN S
2
. It will be shown later that this
configura tion does not represent the ground state. The remaining terms can
be formulated again in the Bloch representation for each sublattice and we
obtain
H
spin
≃ E
a
+2J
a
νS
k
&
b
†
1k
b
1k
+ b
†
2k
b
2k
+ γ
k
b
†
1k
b
†
2k
+ b
1k
b
2k
'
. (6.56)
The first terms under the sum, count the elementary excitations on each sub-
lattice, while the last term describes the coupling between the sublattices
which still has to be removed. This is achieved by the Bogoliubov
6
transfor-
mation
α
k
= u
k
b
1k
− v
k
b
†
2k
,α
†
k
= u
k
b
†
1k
− v
k
b
2k
β
k
= u
k
b
2k
− v
k
b
†
1k
,β
†
k
= u
k
b
†
2k
− v
k
b
1k
, (6.57)
where the coefficients u
k
and v
k
are real. Also the new operators obey the
boson commutation rules (Problem 6.6)
[α
k
,α
†
k
′
]=[β
k
,β
†
k
′
]=δ
kk
′
, [α
k
,β
k
′
]=[α
†
k
,β
†
k
′
]=[α
k
,β
†
k
′
]=0. (6.58)
which leads to the constraint u
2
k
−v
2
k
= 1. Note, that the b oson operators for
different sublattices commute with each other.
The Bo goliubov transformation is an important concept to exactly elim-
inate a bilinear coupling between two boson systems, of those r epresented
by the spin wave operators on the two sublattices. It can also be applied
to phonon–photon or exciton–photon coupling (leading to polaritons, see
Chap.
10) or to plasmon–phonon coupl ing (to yield the coupled plasmon–
phonon modes (see Sect.
4.6)). A further example is the magnon–phonon
coupling to be treated in Problem 6.7. An anal ogous transformation for bilin-
ear fermion coupling, the Bogoliubov–Valatin tran sformation is used in the
theory of superconductivity [
19].
This transformation is applied here in its inverted form according to which
b
1k
= u
k
α
k
+ v
k
β
†
k
,b
2k
= u
k
β
k
+ v
k
α
†
k
(6.59)
and corresp onding expressions for Hermitian adjoint operators, to find (Prob-
lem 6.8)
H
spin
= E
a
+2J
a
νS
k
&
− 1+
u
2
k
+ v
2
k
+2u
k
v
k
α
†
k
α
k
+ β
†
k
β
k
+1
+
2u
k
v
k
+ γ
k
(u
2
k
+ v
2
k
)
α
†
k
β
†
k
+ α
k
β
k
'
. (6.60)
6
Nikolai Nikolaevich Bogoliubov 1909–1992.
178 6 Spin Waves: Magnons
The last term on the rhs , which is bilinear in the α and β operators, vanishes
for
2u
k
v
k
+ γ
k
(u
2
k
+ v
2
k
)=0, (6.61)
which together with u
2
k
− v
2
k
=1leadsto
u
2
k
+ v
2
k
+2γ
k
u
k
v
k
=
8
1 − γ
2
k
(6.62)
and
u
2
k
=
1
2
1
#
1 − γ
2
k
+1
,v
2
k
=
1
2
1
#
1 − γ
2
k
− 1
. (6.63)
Thus, we arrive at the Hamiltonian for anti-ferromagnetic spin waves
H
spin
= −2J
a
νN S(S +1)+
k
¯hω
k
α
†
k
α
k
+ β
†
k
β
k
+1
. (6.64)
This Hamiltonian differs from that of the ferromagnetic magnons (
6.46)by
containing numb er operators of two elementary excitations (α
†
k
α
k
and β
†
k
β
k
)
with the same energy and in addition, a zero point contribution.
The magnon energy is ¯hω
k
=2J
a
νS(1 −γ
2
k
)
1/2
with γ
k
as defined before.
In contrast with the dispersion of the ferromagnetic magnons, we have now
for k ≪ π/a with 1 − γ
2
k
∼ k
2
a linear dependence of the magnon frequency
with k. An example for this linear dispersion measured by inelastic neutron
scattering in MnF
2
is shown in Fig.
6.8. Note that the dispersion does not
exactly follow the linear dependence ∼ k, but exhibits a small gap at k =0.
It is due to an anisotropy field, which removes the degeneracy of the two
different magnons and will b e the subject of Problem 6.9.
100
50
0
0 0.5 1.0
k
hω [°K]
<001> direction
<100> direction
k/k
max
Fig. 6.8. Magnon dispersion in MnF
2
measured by inelastic neutron scattering for
two different directions in k space after [
188]. Note the small gap at k = 0 due to a
small anisotropy field (see Problem 6.9)
6.5 Molecular Field Approximation 179
The energy of the ground state (the magnon vacuum, here denoted by
|Ψ
0
)
E
0
= Ψ
0
|H
spin
|Ψ
0
= −2J
a
νN S
2
− 2J
a
νS
k
1 −
8
1 − γ
2
k
!
(6.65)
is composed of the energy of the perfect anti-ferromagnetic configuration
(first term) and the zero-point contribution, which is negative because of
1 −
#
1 − γ
2
k
> 0 (second term). This indicates that the ground state deviates
from the exact anti-ferromagnetic order a s can b e seen more clearly by look ing
at the departure from the maximum value of the z component of the spin,
NS −Ψ
0
|
j
S
z
j
|Ψ
0
, in the sublattices (which would vanish for the perfect
spin alignment). We find for the sublattice 1 (and similar for the sublattice 2)
with
j
S
z
j
= NS −
k
b
†
1k
b
1k
= NS −
k
u
k
α
†
k
+ v
k
β
k
u
k
α
k
+ v
k
β
†
k
= NS −
k
u
2
k
α
†
k
α
k
+ v
2
k
β
k
β
†
k
+ u
k
v
k
α
†
k
β
†
k
+ α
k
β
k
(6.66)
and by making use of the commutation rules for the β operators after taking
the expectation value with |Ψ
0
NS −Ψ
0
|
j
S
z
j
|Ψ
0
=
k
v
2
k
=
1
2
k
(1 − γ
2
k
)
−1/2
− 1
=0. (6.67)
Thus, in the ground state, the magnon vacuum, the spins in the individual
sublattices are not perfectly aligned but slightly disordered.
6.5 Molecular Field Approximation
The theory of spin waves outlined in Sects.
6.3 and 6.4 has provided some
insight into the low-energy excitations, which means a departure from the
state with ferromagnetic or anti-ferromagnetic order. It is known, that mag-
netic order exists only below a critical temperature at which a phase transition
takes place. For ferromagnets, this critical temperature is the Curie
7
temper-
ature T
C
at which a transition to the paramagnetic phase takes place. The
transition temperature for anti-ferromagnets is the N´eel
8
temperature T
N
.
7
Pierre Curie 1859–1906, shared the Noble prize in physics 1903 with his wife
Marie Curie.
8
Louis N´eel 1904–2000, Noble prize in physics 1970.
180 6 Spin Waves: Magnons
The quantity to be studied here is the magnetization M (T )anditstem-
perature dependence up to and beyond T
C
. Assuming its orientation (which
can be defined by a weak external magnetic field) in z direction we may write
M(T )=(0, 0,M(T )) with
M(T )=gμ
B
j
S
z
j
, (6.68)
i.e., the magnetization is determined by the thermal expectation value of the z
components of the localized spin operators. For temperatures below the Curie
temperature T
C
, it describes the spontaneous magnetization which in terms
of ferromagnetic spin waves reads
M(T )=gμ
B
NS −
k
b
†
k
b
k
. (6.69)
The first term M (0) = gμ
B
NS is the saturation magnetization and the sec-
ond term accounts for the departure from M(0) by thermal excitation of
magnons which follows a temperature dependence given by T
3/2
(Problem
6.10). When approaching T
C
the magnetization vanishes, which in the lan-
guage of magnons would require b
†
k
b
k
to approach NS in contrast with the
condition b
†
k
b
k
≪NS for the spin–wave theory, or else, this theory is valid
only for T ≪ T
C
.
Let us look, therefore, again at the Heisenberg Hamiltonian with a n exter-
nal magnetic field (to fix the orientation of the spontaneous magnetization)
writtenintheform
H
spin
= −
j
&
J
n.n.i=j
S
i
+ gμ
B
H
ext
'
· S
j
. (6.70)
It suggests to interpret the content of the curly bracket as an effective magnetic
field acting on the spin S
j
. Th is interpretation requires that the spin operator
S
i
is replaced by its thermal expectation value S
i
.
To be more explicit, by splitting the spin operator S
i
= S
i
+ δS
i
into
its thermal expectation (or mean) value and deviations from it, called spin
fluctuations, the spin–spin interaction can be rewritten
S
i
· S
j
= S
i
·S
j
+ S
i
·δS
j
+ δS
i
·S
j
+ δS
i
· δS
j
. (6.71)
Neglecting the last term quadratic in the fluctuations, which are assumed to
be small, the Heisenberg Hamiltonian can be cast into the form
H
spin
= −
j
&
2J
n.n.i=j
S
i
+ gμ
B
H
ext
'
· S
j
+
n.n.i=j
JS
i
·S
j
,
(6.72)
6.5 Molecular Field Approximation 181
where the last term is a constant. The first term contains besides the Zeeman
term the molecular field, defined by
2J
n.n.i=j
S
i
= gμ
B
H
M
(6.73)
and the spin Hamiltonian becomes
H
spin
≃−gμ
B
j
{H
M
+ H
ext
}·S
j
+
n.n.i=j
J S
i
·S
j
. (6.74)
The molecular field H
M
(originally introduced by P. Weiss
9
) accounts for the
interaction of the spin S
j
with all the other spins S
i
,i = j replaced by S
i
,
but can be quantified, only when the thermal expectation values of all theses
spins are known. Note that this seemingly simple concept is quite general as
it can be applied to any system of interacting particles as e.g. in the Hartree–
Fock approximation introduced in Sect.
4.4. In this more general context, the
molecular field is a lso called mean field and the approximation denoted mean
field approximation.
According to the translational symmetry of the system, the individual
localized spins contribute e qu ally to the magnetization. Therefore, it can be
written
NM (T )=gμ
B
j
S
j
= gμ
B
NS
j
(6.75)
and the molecular field can be expressed as
H
M
=
2J
Ngμ
B
n.n.i=j
S
i
= λM ,λ=
νJ
g
2
μ
2
B
. (6.76)
It is determined by the magnetization M (T ) of the system and a constant
∼νJ ,theWeiss constant, which is stronger the larger the exchange integral
J and the numb er ν of nearest neighbors in the lattice.
Now, we focus on the temperature dependence of the magnetization. Let
the external magnetic field and the magnetization point in the z direction to
define an effective field H
eff
= H
ext
+ λM . Then the spin Hamiltonian takes
the form (up to a constant)
H
spin
= −gμ
B
i
S
z
i
H
eff
(6.77)
and the magnetization is to b e calculated as thermal exp ectation value
M(T )=gμ
B
Tr
1
Z
e
−βH
spin
S
z
i
with β =1/k
B
T. (6.78)
9
Pierre Ernest Weiss 1865–1940.
182 6 Spin Waves: Magnons
The trace can be evaluated with the eigenstates of (
6.77)whichare
*
i
|SM
S
i
,M
S
= −S, −S +1,...S− 1,S. (6.79)
The eigenvalues of S
z
i
are independent of the site
S
z
i
|SM
S
i
= M
S
|SM
S
i
(6.80)
and we can write
M(T )=
i
1
Z
+S
M
S
=−S
2μ
B
M
S
e
gµ
B
βM
S
H
eff
(6.81)
which gives
M(T )=gμ
B
SB
S
(gμ
B
βSH
eff
), (6.82)
with the Brillouin function (with y = gμ
B
βSH
eff
)
B
S
(y)=
2S +1
2S
coth
2S +1
2S
y
!
−
1
2S
coth
y
2S
. (6.83)
In the limit of low temperature s, y →∞, the Brillouin function B
S
(y) → 1
and we find
T → 0: M (T ) → M(0) = gμ
B
S (6.84)
i.e., the magnetization correctly approaches the satu ration value.
Actually, according to (
6.82) M is a function of the variable x = H
eff
/T
in which it shows the saturation behavior depicted in Fig.
6.9. Because of its
definition, the effective field is itself a function of the magnetization. Thus,
(
6.82) represents an implicit equation for M (T ).Itcanbesolvedbyconsider-
ing besides (
6.82) the second expression of M (x) obtained from the effective
field for the case of vanishing external field
M(T)
M(0)
1
x
T=T
c
T/T <1
c
Fig. 6.9. Saturation behavior of the spontaneous magnetization and graphical
solution for M(T ) with the phase transition for a ferromagnet (after [12])
6.5 Molecular Field Approximation 183
M(x)=
T
λ
x. (6.85)
For sufficiently low temperature, t he graph of (
6.85), which is a straight line
(see Fig.
6.9), cuts the saturation curve (6.82) always at a finite value of M (T ).
With increasing temperature, this crossing point moves to the left until at
the critical temperature, the graph of (
6.85) b ecomes the tangent of (6.82)
at M (T ) = 0. This signifies the transition from the ferromagnetic to the
paramagnetic phase.
The critical temperature can be o btained from the derivative of the magne-
tization with respect to x, which is found from the high temperature expansion
(y ≪ 1) of the Brillouin function
B
S
(y)=
S +1
S
y
3
−
(2S +1)
4
− 1
(2S)
4
y
3
45
+ ... (6.86)
according to which t h e leading terms of the magnetization in the absence of
the external magnetic field (H
ext
→ 0) are
M(T )=T
C
M(T )
T
− A
M(T )
3
T
3
, (6.87)
wherewehaveidentifiedtheCurietemperaturewith
T
C
=
JνS(S +1)
3k
B
(6.88)
and A is another constant determined by the system parameters. After divid-
ing by M(T )(
6.87) becomes a quadratic equation which gives the qualitative
relation
M(T → T
C
) ∼ (T
C
− T )
1/2
,T<T
C
. (6.89)
Thus, the mean field theor y allows us to describe the expected vanishing of
the magnetization, when approaching the critical temperature from below,
and it gives also the temperature dependence with the critical exponent 1/2.
This behavior is typical for a second order phase transition, which is charac-
terized here by the magnetization M(T )asorder parameter. The usual plot
of M(T )/M (0) versus T/T
C
(Fig.
6.10) is universal for a second order phase
transition and does not depend on the ferromagnetic material. Thus, the data
points for Ni and Fe fall onto the same curve, which is well described by the
Brillouin function with S =1/2. The validity o f the mean-field approach has
been confirmed also by ab-initio calculations with the dynamical mean-field
theory.[
189]
Above the critical temperature the magnetization in the presence of an
external mag netic field is
M(T>T
C
)=g
2
μ
2
B
NS(S +1)
1
3k
B
T
(H
ext
+ λM)
=
C
T
H
ext
+
T
C
T
M(T ) (6.90)
184 6 Spin Waves: Magnons
Ni
Fe
1.0
0.8
0.6
0.4
0.2
0.2 0.4 0.6 0.8 1.00
S=1/2
S=1
S=
∞
M(T) / M(0)
T/T
c
Fig. 6.10. Dep endence of the reduced saturation magnetization on the reduced
temperature. Symbols are experimental data for Ni and Fe, solid lines are calculated
from the Brillouin function with different values for the total spin S (after [30])
Table 6.1. Curie temperatures (T
C
in K) and saturation magnetization (M(0)
in Gauss) for some ferromagnets and N´eel temperatures (T
N
in K) for some anti-
ferromagnets
Ferromagnets T
C
M(0) Anti-Ferromagnets T
N
Fe 1043 1752 MnO 122
Co 1388 1446 FeO 198
Ni 627 510 CoO 291
EuO 77 1910 NiO 600
EuS 16.5 1184 MnF
2
67.34
MnAs 318 870 CoF
2
37.7
and it follows the Curie–Weiss law
M(T>T
C
)=
C
T − T
C
H
ext
,T>T
C
. (6.91)
Similar considerations lead to the phase transition between the anti-
ferromagnetic and paramagnetic phases characterized by the N´eel temperature
as critical temperature. In Table
6.1 some systems with magnetic order are
given together with their critical temperatures and (for the ferromagnetic
systems) their saturation magnetization at T = 0 K. As an example of a fer-
rimagnet we refer to Fe
3
O
4
(magnetite) with T
C
= 858 K and M(T =0)=
510 G. All data are taken from [
164]. For further reading a bout phase transi-
tions and critical exponents we refer to [
190–192].
6.6 Itinerant Electron Magnetism 185
6.6 Itinerant Electron Magnetism
While in the previous sections we have assumed a dominating exchange inter-
action and strongly localized electrons, we want to study now the opp osite
situation of Bloch or itinerant electrons with weak exchange interaction. The
appropriate Hamiltonian for this case is [166, 171]
H =
kσ
ǫ
k
c
†
kσ
c
kσ
+
1
2
U
kk
′
q
σσ
′
c
†
k+qσ
c
†
k
′
−qσ
′
c
k
′
σ
′
c
kσ
. (6.92)
It describes electrons in a band ǫ
k
with an interaction independent of k,which
can be understood as a screened Coulomb interaction e
2
/V ε
0
(k
2
+ k
2
FT
)with
neglect of the dependence on k, i.e. it corresponds to the r eplacements
v
k
=
e
2
ε
0
Vk
2
→
e
2
ε
0
V (k
2
+ k
2
FT
)
→
e
2
ε
0
Vk
2
FT
= U. (6.93)
We apply the Hartree–Fock approximation by considering only contributions
of the interaction with k
′
= k + q and σ = σ
′
, which allow us to write the
interaction as
H
int
= −
1
2
U
k
1
k
2
σ
c
†
k
1
σ
c
k
1
σ
c
†
k
2
σ
c
k
2
σ
= −
1
2
U
k
1
k
2
σ
n
k
1
σ
n
k
2
σ
. (6.94)
This can be formulated also with the numbers N
σ
=
k
n
kσ
of spin-up (σ =+)
and spin-down (σ = −) electrons as
H
int
= −
U
2
1
2
(N
+
− N
−
)
2
+
1
2
N
2
!
, (6.95)
where N = N
+
+ N
−
is the total number of electrons. The HF Hamiltonian,
known also as the Stoner
10
model then reads
H
HF
=
k,σ
ǫ
k
− σ
∆
2
!
c
†
kσ
c
kσ
+
U
4
(N
+
− N
−
)
2
− N
2
. (6.96)
The energy spectrum ε
kσ
= ε
k
− σ∆/2 is depicted in Fig.
6.11,wherefor
simplicity, a parabolic dispersion as for free electrons is assumed. The up and
down spin electrons d iffer in their energies by ∆ = U(N
+
− N
−
). The n um-
ber difference between both kin ds of electrons determines the magnetization
thus indicating the relation with magnetism. We recognize that the exchange
interaction leads to a similar result as the Zeeman term in the discussion of
the Pauli spin paramagnetism in Sect.
4.2. However, the k independent shift
of the spin-up and spin-down band applies to any dispersion relation ǫ
k
as
e.g. in Fig.
6.1.
10
Edmund Clifton Stoner 1889–1973
186 6 Spin Waves: Magnons
∆
k
F
+
+–
k
F
–
E
F
ε
kσ
Fig. 6.11. Schematic dispersion for spin-split energy bands
The Stoner model is the starting point for investigating the dependence
of the total energy on the degree of spin polarization ζ =(N
+
− N
−
)/N .
This will lead us to a refinement of the estimate given a t the beginning of
this Chapter with respect to the existence of ferromagnetic order in a system
of itinerant electrons. The grou nd state energy in HF approximation fo ll ows
from (
6.96)as
E
HF
0
=
k
f
k+
ǫ
k
−
U
2
(N
+
− N
−
)
+ f
k−
ǫ
k
+
U
2
(N
+
− N
−
)
!
+
U
4
(N
+
− N
−
)
2
− n
2
(6.97)
with the Fermi distribution function f
k±
. The sums over the band energies
are carried out by assuming the free electron dispersion ǫ
k
=¯h
2
k
2
/2m and
T =0togive
k
f
k±
ǫ
k
=
3
5
N
±
E
F
(N
±
). (6.98)
Using the corresponding results from Sect.
4.4 but with E
F
(N
±
)=2
2/3
E
F
(N),
because of single occupancy of the states in k spacewithspinalignedelectrons,
the total energy in HF approximation is
E
HF
0
(N
+
,N
−
)=
3
5
N
+
E
F
(N
+
)+N
−
E
F
(N
−
)
−
U
4
(N
+
− N
−
)
2
+ N
2
. (6.99)
We are interested in the dependence on the degree of spin polarization ζ =
(N
+
− N
−
)/N , and replace N
±
according to
N
±
=
1
2
N(1 ± ζ), (6.100)
which gives for the mean energy p er particle