R?ssler U. Solid State Theory: An Introduction

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6.2 The Heisenberg Hamiltonian 167

exchange term. The direct process requires R

= R

′

= R and R

= R

′

= R

′

and can be expressed as



′



σσ

′

†

Rσ

†

′

Rσ

. (6.12)

When applied to the ferromagnetic ground state, it removes two electrons at

sites R and R

′

with spins σ and σ

′

, resp ectively, and puts them back where

they are taken from. It is clear that this requires R = R

′

, because at each

lattice site, there is only one electron that can be removed, and the operator

part can be transformed to yield



†

Rσ



′

†

′

=1. (6.13)

Thus, we ﬁnd that H

is the energy of the electrostatic interaction between the

charge densities located around R and R

′

, i.e. of the electronic conﬁguration

associated with the spin conﬁguration of Fig.

6.4.

The exchange term is obtained with R

= R

′

= R and R

= R

′

= R

′

and reads



′



σσ

′

†

Rσ

†

′

Rσ

′

. (6.14)

The operator part can be transformed (again with R = R

′



σσ

′

†

Rσ

†

′

Rσ

′

= − c

†

R↑

†

′

↑

′

↑

− c

†

R↓

†

′

↓

′

↓

− c

†

R↑

R↓

†

′

↓

′

↑

− c

†

R↓

R↑

†

′

↑

′

↓

. (6.15)

The diﬀerent combinations of creation and annihilation operators at a given

site have the following meaning with respect to the site R (here we drop the

site index):

†

↑

†

↓

:countthe↑, ↓ electrons

†

↑

− c

†

↓

: counts the diﬀerence between ↑ and ↓ electrons

†

↑

↓

†

↓

↑

: cause spin ﬂips.

Taking into account the commutation relation

†

↑

↓

†

↓

↑

]=c

†

↑

− c

†

↓

(6.16)

we recognize that the operators c

†

↑

↓

†

↓

↑

, and c

†

↑

−c

†

↓

at each site R fulﬁll

the commutation rules of the su(2) algebra known from the angular momenta

(Problem 6.4):

]=iε

ijk

, i,j,k = x, y, z cycl. perm. (6.17)

168 6 Spin Waves: Magnons

with the Levi–Civita symbol ε

ijk

, and we may identify

= S

+iS

= c

†

↑

↓

−

= S

− iS

= c

†

↓

↑

†

↑

− c

†

↓

). (6.18)

Thus, it is possible to replace the annihilation and creation o perators by spin

vector operators S

=(S

) at each lattice site. Thi s is achieved by

adding and subtracting (c

†

R↑

†

′

↓

′

↓

+ c

†

R↓

†

′

↑

′

↑

)/2ontherhs of

(

6.15) to obtain for the operator part of the exchange term



σσ

′

†

Rσ

†

′

Rσ

′

= −



−

′

+ S

−

′



− 2S

′

−



σσ

′

†

Rσ

†

′

. (6.19)

The ﬁrst two terms on the rhs canbewrittenintheform

−

′

+ S

−

′

+2S

′

=2S

· S

′

(6.20)

while the last term can be combined with the direct term, and we ﬁnd for the

interaction part, H

int

of the Hamiltonian, the form

int

= −



′

R=R

′

· S

′

(6.21)

which is the Heisenberg

Hamiltonian with the exchange integral

′



′

∗

(r − R)φ(r − R

′

)φ

∗

′

− R

′

)φ(r

′

− R)

4πε

|r − r

′

. (6.22)

It can be shown that J

′

> 0. We add the Zeeman term and replace the

site index R by a number index i (which can be understood as replacing the

3-dimensional Bravais lattice with a linear chain but is meant also as short

notation for R

) to write the spin or Heisenberg Hamiltonian in the form

spin

= −



i,j

i=j

· S

− gμ

ext



. (6.23)

This Hamiltonian is the canonical starting point in the theory of magnetism

[

183]. In spite of its simplicity, it bears fundamental properties, one of which

is related to symmetry. Without external magnetic ﬁeld, (

6.23) is spheri-

cally symmetric, however, its ground state, the ferromagnetic conﬁguration,

has only axial symmetry. This situation is known as spontane ous symmetry

bre a king: Due to interaction between the spins and in order to gain energy,

Werner Heisenberg 1901–1976, Nobel prize in physics 1932.

6.2 The Heisenberg Hamiltonian 169

the system prefers a conﬁguration with lower symmetry than that of the

Hamiltonian. The further task of this chapter will be to describe elementary

excitations out of this ground state.

There exist several model Hamiltonians related to the Heisenberg

Hamiltonian, which have been under investigation in the context of magnetic

properties [

171]:

1. the anisotropic Heisenberg model

spin

= −



i,j

i=j



+ S



=

, (6.24)

2. the Ising model (with J

=0)

Ising

= −



i,j

i=j

, (6.25)

3. and the XY model (

=0)

= −



i,j

i=j

+ S

). (6.26)

The Ising and the XY models are used in statistical mechanics because of

their simplicity compared to the Heisenberg model, while still representing

reasonable approximations for real spin systems. As it turns out, however, the

low-dimensional spin models (except the two-dimensional Ising model) fall

short of describing the ferromagnetic phase transition (see Sect.

6.5). Instead

the XY model produces a transition between two disordered phases of diﬀer-

ent topology. This Kosterlitz–Thouless tr ansition is caused by the appearance

of spin vortices (see [

171]).

In all these mo dels, the exchange integrals app ear as parameters. The

microscopic mechanism, although it is always the exchange interaction, can

be quite diﬀerent depending on the lattice structure of the solid. The eas-

iest case is that of the transition metals, where the exchange takes place

directly between the d orbitals on nearest neighb or lattice sites of a close-

packed structure like bcc and fcc. Therefore, it is called direct exchange to

distinguish it from indirect or super-exchange in compounds of elements with

incomplete d shells, as e.g. the ferromagnetic insulators MnAs, EuO, and

EuS. In these solids forming lattices with basis, the orbitals which carry a

magnetic moment, are separated by the non-magnetic ions and the exchange

coupling has to be mediated by the orbitals of the intervening ions. A similar

situation is found in metallic ferromagnetic compounds, where the exchange

170 6 Spin Waves: Magnons

coupling b etween distant magnetic ions is mediated by the free electrons.

This is the Rudermann–Kittel exchange, which exists together with the direct

exchange interaction and is frequently the dominating mechanism. Depending

on the mechanism and the lattice conﬁguration, the exchange constant can

be positive (as anticipated so far), which leads to the ferromagnetic order,

but also negative. In the latter case, neighboring spins align anti-parallel and

exhibit the anti-ferromagnetic ordering. More complex crystal structures with

sublattices accomodating diﬀerent spin carrying orbitals, i.e. the spin operator

S depends on the sublattice, lead to the ferrimagnetic or anti-ferrimagnetic

order.

6.3 Spin Waves in Ferromagnets

In this Section, we want to describe low-energy excitations out of the ferro-

magnetic ground state. Let us assume for the moment the classical picture

of spins in Fig.

6.4 as localized magnetic dipoles. They are coupled from site

to site by the scalar product of their resp ective spin vectors, weighted by the

exchange integral. If o ne of the spins is tilted against the preferential direc-

tion thus raising the energy, the neighboring spins tend to follow this tilt.

If this spin is released from its tilted orientation the whole system will start

to perform a collective motion just as the linear chain of masses connected

by springs in Chap.

3: The masses now become the magnetic moments of the

spins and the role of the springs is taken by the exchange coupling. This col-

lective excitation of the localized interacting spins are the spin waves or, in

quantized form, the magnons.

In order to quantify this consideration, we start from the Heisenberg model

in tight-binding approximation, for which the exchange coupling is taken into

account o nly between nearest neighbors (n.n.i, j), thus the Hamiltonian (

6.23)

simpliﬁes to

spin

= −J



n.n.i,j

· S

− gμ

ext



. (6.27)

and contains only one exchange integral. The nature of the spin, being an

angular momentum, can be exploited by making use of the corresponding

representation with

|SM 

= {S(S +1)− M(M ± 1)}

1/2

|SM ± 1

, (6.28)

where |SM 

is an eigenstate of the spin operators S

and S

for the site i.Itis

convenient here to write the Hamiltonian with raising and lowering operators

(see (

6.18))

spin

= −J



n.n.i,j



−

+ S

−

)+S



− gμ

ext



. (6.29)

6.3 Spin Waves in Ferromagnets 171

Let us ﬁrst calculate the expectation value of H

spin

without external magnetic

ﬁeld (H

ext

= 0) in the ferromagnetic ground state, formulated as |Ψ

 =

|SS

= Ψ

spin

|Ψ



= −J



n.n.i,j

SS|S

|SS

SS|S

|SS

= −JνS

N, (6.30)

where ν is the number of nearest neighbors and N the number of sites in

the chain. Note, that the ferromagnetic ground state in angular momentum

representation is composed of angular momentum eigenstates with maximum

z component. Application of the raising operator to |SS gives zero, while

application of the z components of the spin operators to the ground state

yields



|Ψ

 =



|SS

= NS|Ψ

. (6.31)

As we see from (

6.30), the ground state energy is negative if the exchange

integral J, whose sign depends on the conﬁguration of nearest neighbors and

the atomic orbital, is positive. The stability of the ground state increases with

J, with the number of nearest neighbors ν, and with the total spin S.The

latter changes with the number of electrons in an incomplete shell and takes

a maximum value for half-ﬁlling of the shell according to Hund’s rule

Before studying the collective excitations of the spin lattice it is quite

instructive to have a look at the dynamics of the individual spin operator S

It is straightforward to derive the equation of motion with the Heisenberg

Hamiltonian (

6.23) (Problem 6.5)

¯h

spin

, S

]=−

¯h

× S

) , (6.32)

where H



+ gμ

ext

is an eﬀective magnetic ﬁeld acting on the

spin S

at the same site. The equation of motion (

6.32) is that of an angu-

lar momentum, whose dynamics is determined by a torque, or of a magnetic

dipole moving in a magnetic ﬁeld. This ubiquitous equation of motion, origi-

nally formulated in the context of magnetic resonance phenomena, is known

as Bloch equation [

184].

In order to solve (

6.32)wechooseasbeforeH

ext

=(0, 0,H

ext

), which

gives for the transverse spin components the equations of motion

¯h

= −



n.n.i



− S



+ gμ

ext

¯h

= −



n.n.i



− S



− gμ

ext

. (6.33)

Friedrich Hund 1896–1997.

172 6 Spin Waves: Magnons

Close to the ferromagnetic ground state, i.e. at low temperature, we may use

the replacement S

→S

≃S (while S

, S

≪S) to simplify these

equations:

¯h

= −S



n.n.i



− S



+ gμ

ext

¯h

= −S



n.n.i



− S



− gμ

ext

. (6.34)

They are combined with S

= S

± iS

in the equation

¯h

= ∓i





n.n.i

− S

)+gμ

ext



. (6.35)

The coupled motion of neighboring spins can b e decoupled by exploiting the

periodicity of the chain (o r solid) with the Bloch representation

√



−ik·R

. (6.36)

With these normal coordinates (see Chap.

3, where the same concept was

applied in lattice dynamics) we arrive at

¯h

⎛

⎝



n.n.i,j



1 − e

−ik·(R

−R

)



+ gμ

ext

⎞

⎠

. (6.37)

The exchange integrals J

to the ν nearest neighbors with d = R

− R

are

all the same in a cubic lattice and we may write instead the expression in

brackets as

¯hω

=2JνS(1 − γ

)+gμ

ext

, with γ



ik·d

. (6.38)

Thus, we come to the important result, that the whole spin conﬁguration of the

lattice (but also the individual localized spin) performs an oscillatory motion

with the frequency ω

, if small deviations from the ferromagnetic ground state

are consid ered. This collective mode, the spin wave characterized by the wave

vector k (depicted in Fig.

6.5), corresponds to a coherent precession of the

Fig. 6.5. Schematic of a ferromagnetic spin wave on a linear chain

6.3 Spin Waves in Ferromagnets 173

individual spins around the direction of the ferromagnetic orientation. It is

completely analogous to the lattice modes in Chap.

3. Its quantized form is

called magnon as it is r elated to magnetic properties of the solid.

For methodical reasons, it is worth presenting as a n alternative to the

Bloch equation, which essentially is a classical co ncept, a formulation based

on the occupation numb er representation and adapted to the quantized form

of elementary excitations. In the previous formulation of excitations o ut of

the ferromagnetic ground state, the emphasis was on the transverse compo-

nents of the local spins to quantify the precession of the spin vectors around

the z direction. The alternative is to quantify the deviation of S

from its

maximum value S. This is achieved with a transformation that replaces the

ladder operators S

by boson operators a

†

with [a

†

]=δ

according to

√



1 −

†



1/2

−

√

2Sa

†



1 −

†



1/2

(6.39)

without changing the commutation relations among the three angular momen-

tum operators S

−

.ItistheHolstein

–Primakoﬀ

transformation [185].

The z component o f the spin operator follows (with f(a

†

)={1 − a

†

2S}

1/2

)from

−

]=2S



f(a

†

f(a

†

) − a

†



=2(S − a

†

) (6.40)

such that

= S − a

†

(6.41)

describes the deviation from the maximum value S by the number opera-

tor a

†

which will turn out to be that of quantized excitations out of the

ferromagnetic ground state.

Let us assume again excitations of low energy which can be quantiﬁed by

a

†

≪S and allow us to expand the operator function f(a

†

). Then

√

1 −

†

+ ...

≃

√

2Sa

, and S

≃

√

2Sa

†

, (6.42)

and we have a one-to-one correspondence between the ladder ope rators S

andthebosonoperatorsa

and a

†

.Thefactorf(a

†

) is required here only to

obtain (

6.41). Under the condition of low-energy excitations the Heisenberg

Hamiltonian can now be written

spin

≃−J



n.n.i,j



†

+ a

†



+ S

− S



†

+ a

†



≃−E

+ JS



n.n.i,j

†

+ a

†

− a

†

− a

†

(6.43)

Theodor D. Holstein, 1915–1985.

Henry Primakoﬀ, 1914–1983.

174 6 Spin Waves: Magnons

where higher order terms in the occupation number operator (corresponding to

magnon–magnon coupling analogous to lattice dynamics beyond the harmonic

approximation) have been omitted and E

is the ground state e n ergy (see

(

6.30)). The last two terms represent a coupling between nearest neighbor

lattice sites which can be removed by transforming to normal coordinates (or

switching from the Wannier to the Bloch representation) with

√



−ik·R

†

√



ik·R

†

. (6.44)

The individual contributions to (

6.43) can be expressed in the new boson

operators for the collective excitation as



n.n.ij

†

= ν



′



−i(k−k

′

)·R



 

′

†

′

= ν



†



n.n.ij

†



′



n.n.ij

ik·R

−ik

′

·R

†

′



′



i(k−k

′

)·R



ik·d

†

′

= ν



†

(6.45)

and the Hamiltonian takes the approximate form

spin

≃ E

+2JνS



(1 − γ

†

= E



¯hω

†

. (6.46)

It is the Hamiltonian for spin waves in ferromagnets. The meaning of the boson

operators b

†

and b

can be read from the relation



= NS −



†

, (6.47)

where the ﬁrst term NS gives the maximum value of the z comp onents of all

spin operators in the ferromagnetic ground state and the second term counts

the number of quantized collective excitations, the magnons.

Because we have assumed ex change interaction only between nearest

neighbors, the magnon dispersion ω

, depending on the conﬁguration of the

magnetic ions, can be expressed in terms of cosine functions as is typical

for the tight-binding approximation (see Sect.

5.4). The width of the magnon

band, determined by the number of nearest neighbors and the exchange con-

stant, is in the range of a few to some tens meV, which is about the same as

for phonons. For |k|≪2π/a we may expand

1 − γ

≃ 1 − (1 −

2ν



|k ·d|

) ∼ k

(6.48)

6.4 Spin Waves in Anti-Ferromagnets 175

and ﬁnd a quadratic dependence on the wave vector around the minimum at

k = 0. This quadra tic dispersion has been measured by inelas t ic neutron scat-

tering,e.g.inthe3d transition metals [

186, 187], but will be shown only when

we discuss itinerant magnetism in Sect.

6.6. For a more recent introduction

into the concepts of neutron scattering in solids with examples of magnetic

excitations see [

96, 97].

Similar to phonons, magnons also can be thermally excited and contribute

to the speciﬁc heat. This contribution can be calculated from the thermal

expectation value of (

6.46)

E(T )=E



¯hω

b

†



= E

(2π)



¯hω

β¯hω

− 1

k. (6.49)

As we are interested in the low-temperatur e behavior of the speciﬁc heat,

which is determined by the quantum nature of the excitations, the dispersion

can be approximated as ¯hω

= αk

, where for a simple cubic lattice with

lattice constant a one has α =2JSa

. With this isotropic quadratic dispersion

the integral can be evaluated in polar coordinates and reads after substituting

x = βαk

E(T )=E

2π

5/2



max

3/2

− 1

dx. (6.50)

We may compare here with E(T )of(

3.64), obtained for the acoustic phonons

in the Debye model. For low-temperatures the upper limit of the integral can

be shifted to inﬁnity and we obtain one of the Bose integrals (see Appendix),

which can be expressed here as Γ(5/2)ζ(5/2; 1). Finally, we ﬁnd for the

magnon contribution to the thermal energy

E(T )=E

0.45

3/2

T )

5/2

(6.51)

and for the speciﬁc heat

(T )=

dE(T )

V =const.

=0.113 k

3/2

. (6.52)

This T

3/2

dependence is characteristic of the magnon contribution to the

speciﬁc heat and makes it d i stinct fr om the contributions of the acoustic

phonons (∼T

) or free electrons (∼T ).

6.4 Spin Waves in Anti-Ferromagnets

In the last section, we have assumed a ferromagnetic ground state, which

required a positive exchange integral b etween nearest neighbors. An anti-

ferromagnetic order, with the spins on neighboring lattice sites b eing oriented

176 6 Spin Waves: Magnons

Fig. 6.6. Linear chain with anti-ferromagnetic order, the Wigner–Seitz cell contains

two ions with opposite spin

Mn lattice 1

Mn lattice 2

Fig. 6.7. Structure of the anti-ferromagnetic insulator MnF

anti-parallel to each other, would mean an enlarged unit cell with a basis

consisting of (at least) two atoms with a negative exchange integral between

nearest neighbors, as depicted in Fig.

6.6. As a real three- dimensional system

we show in Fig.

6.7, the crystal structure of the anti-ferromagnetic insulator

MnF

. The conﬁguration of the ﬂuorine ions around the Mn ions on the

corners of the cube is diﬀerent from that around the Mn ion in the center,

thus the structure is that of two interpenetrating simple cubic lattices each

with a basis of one Mn and two F ions.

The Hamiltonian for this system (without external magnetic ﬁeld) can be

written

spin

= J



n.n.i,j

· S

= J



n.n.ij



−

+ S

−



+ S

. (6.53)

Here S

, S

are the spin vector operators on the two sublattices and the

negative sign of the exchange integral is absorb ed in the positive constant

. In order to derive a spin wave Hamiltonian similar to (

6.46)weemploy

the Holstein–Primakoﬀ transformation, introduced in the previous section, to

replace in a ﬁrst step, the spin op erators on each sublattice by boson op erators

≃

√

2Sa

−

≃

√

2Sa

†

=+S − a

†

(sublattice 1)

≃

√

2Sa

−

≃

√

2Sa

†

= −S + a

†

(sublattice 2)

(6.54)

and ﬁnd the approximate Heisenberg Hamiltonian for low-energy excitations

out of the anti-ferromagnetic state, i.e. for a

†

, a

†

≪S,