Download Chapter 3 MAGNETISM OF THE ELECTRON

Chapter 3
MAGNETISM OF THE ELECTRON
The magnetic moments in solids are associated with electrons. The origin of the
magnetism is quantized angular momentum, with two distinct sources; orbital
motion and spin. Spin itself and spin-orbit coupling have a relativistic origin.
The description of magnetism in solids is fundamentally di¤erent depending on
whether the electrons are localized on the ion cores, or delocalized in energy
bands. In either case crystal structure and bonding modify the magnetism
profoundly. A starting point for discussion of magnetism in metals is the freeelectron model, which leads to temperature-independent Pauli paramagnetism
and Landau diamagnetism, in contrast to the Curie paramagnetism of localised
spins
3.1
Orbital and Spin Moments
Electrons are the main source of magnetic moments in solids. The electron is an
elementary particle with charge -e, where e = 1.602 10 19 C and mass m = 9.109
10 31 kg. It also possesses an intrinsic angular momentum known as spin of 21 ~, where
~ = 1.055 34 J s is Planck’s constant h divided by 2 . Magnetism is inherently
connected with the angular momentum of charged particles, so the quantum theory
of magnetism is closely related to the quantization of angular momentum. For the
electron there are two distinct sources of angular momentum; one is associated with
orbital motion, the other is due to its intrinsic spin (Fig 3.1)
The nucleons, proton and neutron, which have mass 1.67 10 27 kg and
charges of e and 0 respectively, both possess the same spin as the electron, but
nuclear spin gives rise to much smaller magnetic moments than electron spin because
of the much greater nucleon mass. For our purposes, nuclear magnetism can usually
be neglected.
Fig 3.1 Magnetic moments associated with (a) orbital and (b) spin angular momenta
of an electron.
Table 3.1 Properties of the electron
2
Magnetism of the electron
mass
charge
angular momentum
spin g-factor
magnetic moment
classical radius
me
e
1
~
2
g
m
re
9.109 10 31 kg
-1.6022 10 19 C
5.273 10 34 J s
2.0023
-9.274 10 24 A m2
2.818 10 15 m
3.1.1 Orbital moment
The orbital moment is most simply introduced in terms of the Bohr model of the atom,
where the electrons revolve around a nucleus of charge Ze in circular orbits under
the in‡uence of the Coulomb potential Ze2 =4 0 r (Fig 3.2). An electron circulating
in an orbit is equivalent to a current loop where the current direction is opposite
to the sense of circulation because of the negative electronic charge. If the speed of
the electron is v, the equivalent current is I = ev=2 r and the moment m=IA has
magnitude 12 rev. In terms of angular momentum, l = me r v, the moment is
m=
(e=2me )l
(3.1)
The factor of proportionality between magnetic moment and angular momentum in
m = l is known as the magnetomechanical (or gyromagnetic) ratio. For orbital
motion is -(e/2m); the minus sign means that m and l are oppositely directed,
since the electron charge is negative. The orbital angular momentum is quantized in
units of ~, so that
m=
(e=2me )ml ~ where ml = 0; 1; 2
(3.2)
Here ml is an orbital magnetic quantum number. The natural unit for electronic
magnetism is the Bohr magneton, de…ned as
B
= e~=2me
(3.3)
1 B = 9:274 10 24 A m2 : The quantized orbital magnetic moment is an integral multiple of the Bohr magneton.
The remarkable di¤erence between an electron and a classical charged particle is that the electron can continue circulating in its orbit inde…nitely as a sort of
perpetual motion or electronic supercurrent, whereas the classical particle must radiate energy on account of its continuous centripetal acceleration. Unquantized orbital
motion would soon cease as a result of radiation loss.
The relation (3.1) is often expressed in terms of a g-factor, which is de…ned as
the ratio of the magnitude of the magnetic moment in units of B to the magnitude
of the angular momentum in units of ~. Hence g is exactly 1 for orbital motion.
The relation between m and l can be generalized to non-circular orbits. From
(2:4); m = IA for a planar loop of any shape and area A. The angular momentum
of an electron moving with velocity v is r me v: Averaging r me v over the loop
Orbital and Spin Moments
3
H
gives (vme =s) r ds where s is the circumference of the loop. The integral is 2A
and I = ev=s, hence
e )l:
R m = (e=2m
R
1
3
[OR] m = 2 v r j m d r; j m (r ) = e v (r r)r; since l = r me v, it
follows that m = 12 er v = (e=2m)l .[CHECK]
The Bohr model provides us with natural units of length and energy in atomic
physics. If Z = 1, Newton’s second law for the centrally-accelerated electron ,
Ze2 =4 0 r2 = mv 2 =r, and angular momentum quantization mvr = n~ with n = 1
give r = a0 ; the Bohr radius, de…ned as
a0 = 4
0~
2
=me e2
(3.4)
value of a0 is 0.05292 nm. The corresponding binding energy of the electron is
ZR=n2 , where R0 = (m=2~2 )(e2 =4 0 )2 is known as the Rydberg. The value of
R0 is 2.180 10 18 J, which is equivalent to 13.606 eV.
Fig 3.2. An electron in a circular orbit of radius r. Angular momentum l and
magnetic moment m are oppositely directed.
3.1.2 Spin moment
The electron is known to possess an intrinsic magnetic moment, unrelated to any
orbital motion, which can only adopt one of two discreet orientations relative to a
magnetic …eld. By analogy with (3.1), we introduce the intrinsic angular momentum
called spin. The electron really is a point particle, with radius <10 20 m, so the picture of a spinning ball of charge in Fig 3.1 is ultimately misleading. The mysterious
built-in spin angular momentum simply emerges from relativistic quantum mechanics
(§3.5). The spin magnetic quantum number is ms = 21 , so that the angular momentum associated with the two distinct orientations is 12 ~. It turns out that the
magnetic moment associated with the electron spin is not a half. but almost exactly
one Bohr magneton. The magnetomechanical ratio is (e=m) and the g-factor is
2.
1
mz = (e=m)ms ~ with ms =
(3.5)
2
The spin angular momentum is therefore twice as e¤ective as the orbital angular
momentum at creating a magnetic moment. With higher-order corrections, g for the
electron’s intrinsic spin moment turns out to be 2.0023.
The reality of the fundamental link between magnetism and angular momentum, known as the Einstein-de Haas e¤ect, was demonstrated in an experiment carried out by Stuart in 1918. A ferromagnetic rod is suspended from a torsion …bre so
that it can turn about its own axis (Fig 3.4). A vertical magnetic …eld created by a
solenoid, is su¢ cient to overcome the demagnetizing …eld and saturate the magnetization of the ferromagnet. The current in the solenoid is then reversed, switching
the direction of magnetization of the rod. The impulse due to the reversal of the
electronic angular momentum of the electrons causes the rod to rotate, and from the
angle of rotation and the torsion constant of the …bre it is possible to deduce the angular momentum. In the case of iron, the speci…c magnetization, s = 217 A m2 kg 1 ,
4
Magnetism of the electron
corresponds to Ms = 1710 kA m 1 . The g-factor is found to be 2.09, which indicates
that the magnetization of iron is essentially due to electron spin. More surprising
is the magnitude of the ferromagnetic moment, which turns out to be only 2:2 B
per atom . The number of electrons per iron atom is equal to the atomic number,
Z = 26, yet the ferromagnetic moment of iron corresponds to the spin moment of
just over two of them. The other electrons form pairs with oppositely-aligned spins,
thereby contributing nothing to the resultant moment.
Fig 3.3 The Einstein-deHaas e¤ect, demonstrating the relation between angular
momentum and magnetic moment. An iron rod is suspended on a …bre. The …eld in the
solenoid is reversed, switching the direction of magnetization of the iron. The rod rotates.
3.1.3 Spin-orbit coupling
Generally, an atomic electron possesses both spin and orbital angular momentum.
They may be coupled by spin-orbit interaction to create a total electronic angular
momentum j. It is conventional to use lower-case letters l; s; j to denote the the
angular momenta of a single electron. Upper-case letters L; S; J are reserved for the
multi-electron atoms and ions discussed in the next chapter.
From the electron’s geocentric point of view, the nucleus revolves about it
with speed v. The motion is equivalent to a current loop In = Zev=2 r;which creates
a magnetic …eld 0 In =2r (2.6) at the centre. The spin-orbit interaction is due to
this magnetic …eld,B = 0 Zev=4 r2 ; acting on the intrinsic magnetic moment of the
electron. The interaction energy (2.63) Uso =
B B can be written approximately
in terms of the Bohr radius, since r = a0 =Z and me vr ~:
Uso
2 4
0 B Z =4
a30
(3.6)
The Z 4 variation means that the spin-orbit interaction, while weak for light elements
(the associated magnetic …eld is about ten tesla for hydrogen, Z = 1); becomes
much more important for heavy elements. The expression (3.6) is actually invalid
for the inner electrons of heavy elements because their velocities are close to that of
light. The correct version of the spin-orbit interaction, resulting from a relativistic
calculation, is given in §3.3.3.
3.1.4 Quantum mechanics of angular momentum.
The Bohr model is an oversimpli…cation of the quantum theory of angular momentum. In quantum mechanics, physical quantities are represented by operators —
momentum by p^ = i~r and energy by p^2 =2m = ~2 r2 :The angular momentum
operator is ^
l = r p;
^ with components
^
l = i~(y@=@z
z@=@y)ex + i~(z@=@x
x@=@z)ey + i~(x@=@y
y@=@x)ez
To calculate the magnetic moment m in B per molecular unit from the magnetic moment per
kilogram, multiply by the molecular weight M and divide by 5585:(m = M=1000NA B where
NA is Avogadro’s number). By chance, this factor for iron, which has M =55.85, is exactly 1/100
Orbital and Spin Moments
5
In terms of the spherical polar coordinates, the cartesian coordinates are x = sin cos ;
y = sin sin and z = cos . and the operators for the components of the angular
momentum become
^
lx = i~(sin @=@ + cot cos @=@ )
^
ly = i~( cos @=@ + cot sin @=@ )
^
lz =
i~(@=@ )
(3.7)
The square of the total angular momentum is
2
2
2
2
^
l =^
lx + ^
ly + ^
lz =
~2
@
1 @2
@2
+
cos
+
@
sin2 @ 2
@ 2
(3.8)
Spherical polar coordinates
An alternative way of representing angular momentum, which is particularly
useful when considering the magnetism of localised electrons, is with matrices. The
magnetic systems have a small number of magnetic basis states denoted by the magnetic quantum numbers mi , and they can be represented by
square hermitiany
matrices.
The spin angular momentum of the electron is represented by the 2 2 spin
operator s^. The three components of angular momentum are represented by the
operators s^x ; s^y ; s^z . Of these, s^z is conventionally chosen as the diagonal one and it
has eigenvalues ~=2 and ~=2, corresponding to ms = 21 . The two possible states
of the electron are known as the " or ’spin up’and # or ’spin down’states. Chemists
sometimes refer to them as the and spin states. The eigenvectors corresponding
1
0
1 0
to j"i and j#i are
and
, so that s^z takes the matix form
~=2.
0
1
0
1
0 1
A coordinate rotationz by = =2 yields s^x =
~=2 and a further rotation
1 0
0
i
~=2: The operator ^ obtained by multiplying s^ by
by = 2 yields s^y =
i 0
2=~; has components known as the Pauli spin matrices.
^=
0 1
1 0
;
0
i
i 0
;
0
i
i 0
(3.9)
The characteristic property of angular momentum in quantum mechanics is
that the operators representing the x, y, and z components satisfy the commutation
rule [^
sx ; s^y ] = i~^
sz ; and cyclic permutations thereof. The commutator in the square
bracket is s^x s^y s^y s^x :The operators have to be hermitian so that their eigenvalues are
y
A hermitian matrix [Aij ] is one for which Aij =Aji , where denotes the complex conjugate,
obtained by replacing i by -i.
z
A coordinate rotation is e¤ected by ...
6
Magnetism of the electron
real. s^x ; s^y and s^z satisfy this commutation rule, and all three of them have eigenvalues 12 ~. In quantum mechanics, only physical quantities whose operators commute
can be measured simultaneously. This is not true of the three spin components. It is
possible to measure simultaneously one component and the total spin, but it us not
possible to measure more than one of the components at the same time. The reason
1 0
s^z and s^2 commute is because both are diagonal; s^2 = s^2x + s^2y + s^2z =
3~2 =4.
0 1
The value of the square of the total angular momentum h^
s2 i = h"; #j s^2 j"; #i is 3~2 =4
for both eigenstatesy . [Footnote for Dirac notation][Footnote on commutators]
Pictorially
p(Fig 3.4), the electronic angular momentum can be represented by a
vector of length 3~=2 which precesses around Oz, and takes one of two orientations
along Oz. The component of spin angular momentum along Oz can only take the
values ms ~ where ms = 21 . The two states with ms = 12 have opposite magnetic
moments and a Zeeman spliting of the two energy levels develops in a magnetic …eld
B. The Zeeman Hamiltonian HZ = (e=m)s:B has eigenvalues g B ms B =
B B,
which are the energy levels in the applied …eld. The spin splitting of the states is
therefore 2 B B.
Other useful operators are the ladder operators
s^+ = s^x + i^
sy and s^ = s^x
i^
sy :
They do not correspond to measurable quantities, but they are helpful for manipulations as they raise or lower ms by unity, which in the case of s = 21 simply
means that they transform j"i to j#i and vice versa. The operators ^
s+ and ^
s are
0 1
0 0
represented by the matrices
~=2, and
~=2; respectively. Hence
0 0
1 0
s^+ j"i = 0; s^+ j#i =j"i; s^ j"i =j#i; s^+ j#i =j 0i: s^2 may be written in the form
1
s^2 = (^
s+ s^ + s^ s^+ ) + s^2z
2
.
p
To summarize, an electron with spin 12 has total angular momentum 3~=2.
There are two spin states for the electron, ms = 21 with a projection of the angular
momentum along a speci…ed direction Oz of ~=2. The states are degenerate in zero
…eld, but split in a magnetic …eld. They are denoted j 12 iand j 12 i; j"iand j#i, or
and
Fig 3.4. Vector model of electron spin. The total spin vector, of length X3~=2
precesses around the applied …eld direction with the Larmor frequency. It has two possible
projections ~/2 along Oz, corresponding to the j"iandj#i states. The Zeeman splitting
of the two magnetic energy levels in …eld is shown.
The matrix representation can be used for values of angular momentum other
than s= 12 . For example, if an electron is an orbital p-state with ` = 1, there are three
Magnetic Field E¤ects
7
possible eigenstates for ^
lz ;corresponding
0 1 0 to
1 ml =01: 0:1 1: The three states are rep1
0
0
resented by column vectors @ 0 A ; @ 1 A and @ 0 A ;and the three components of
0
0
1
p
0
1 0
0
i= 2
0p
0p
p
^
^
^
@
A
@
the angular momentum lx ; ly ,lare represented by the matrices
i= 2
0
i= 2 ~; 1= 2
p
0
0
i= 2
0
2
2
^
square of the total angular
0 momentum
1 l = `(` + 1)~ for all three states. It is rep1 0 0
+
resented by the matrix@ 0 1 0 A 2~2 ;while the raising and lowering operators ^
l
0 0 1
p
1
0
1
0
0 0
0
2 p0
p0
and ^
l are represented by @ 0 0
2 A ~ and @ 2 p0 0 A ~; respectively.
0
2 0
0 0
0
These ideas are readily generalized to any integral or half-integral quantum
number. The eigenvectors have 2`+1 elements, and the hermitian matrices have 2`+1
2
2
rows and columns. The diagonal matrices for ^
l and ^
lz have elements [^
l ]pq = `(` +
1)~2 p;q and [^
lz ]pq = (`+1 p)~ p;q ; where p;q is the Kronicker delta ( p;q = 1:for p = q;
p
^
^
(p)(2` + 1 p) p;p 1 , [
p;q = 0:for p 6= q). The operator l has elements [ l ]pq =
+
1
^
l ] is the re‡ection of [ ^
l ] in the diagonal, and ^
lx = 2 (^
l+ + ^
ly ), ^ly = 2i (^
l+ ^
ly ): The
magnetic moments m in units of Bohr magnetons associated with angular momentum
in units of ~ can be represented by similar matrices, with proportionality factors of
1 or 2 for obital or spin moments. The interaction of these moments with an applied
…eld B is represented by a term in the Hamiltonian -( B =~)(^
l + 2^
s):B. When B
^
is in the z-direction, this term is -( B =~)(lz + 2^
sz )B:[Is it necessary to show l,s as
operators; would vectors be clear enough]
3.2
Magnetic Field E¤ects
The e¤ects of a magnetic …eld on an electron with orbital or spin moment are to
modify its linear or angular motion, and to induce some magnetization. In this
section, we discuss the e¤ects of a magnetic …eld on the electron semi-classically.
3.2.1 Cyclotron frequency
If an electron travels with velocity v across a magnetic …eld B, the Lorentz force
ev B causes an acceleration perpendicular to its velocity, which produces a circular
motion. Newton’s second law gives F = mv 2 =r = evB so the cyclotron frequency
fc = v=2 r is proportional to the …eld;
fc =
eB
2 m
(3.10)
8
Magnetism of the electron
The angular frequency ! c = 2 fc :Any component of the electron velocity parallel
to the magnetic …eld is unin‡uenced by the Lorentz force, so the trajectory or the
electron is a helix along the …eld direction. Electrons which follow cyclotron orbits
radiate energy of frequency fc . The cyclotron frequency is 28 GHz T 1 : The wavelength = c=f of 28 GHz radiation is about 1 cm. The transverse motion of the
electron in the magnetic …eld is quantized, giving rise to a series of Landau levels
En = (n + 21 )~! c ;which are discussed in §3.3.
An example of the use of cyclotron radiation is the domestic microwave oven
where a ‘magnetron’generates 2.45 GHz microwaves from the motion of electrons in
a …eld of about 0.1 T generated by hard ferrite magnets. The 2.45 GHz radiation is
absorbed by water, and it is ideal for cooking. [Check]. On another scale, electrons
accelerated to a large multiple e of their rest energy and constrained to move in
curved paths by bending magnets in a synchrotron source emit linearly-polarised
white radiation in a narrow beam of width 1= e radians. Synchrotron radiation is a
valuable tool for probing the electronic structure of solids.
3.2.2 Larmor frequency
If an electron is already somehow constrained to move in an orbit, it has an associated
magnetic moment m = l where is the magnetomechanical ratio (3.1) The e¤ect of
the magnetic …eld is to exert a torque = m B on the current loop. Newton’s law
for angular momentum = dl=dt gives
dm
=
dt
B
m
(3.11)
Setting ! L = B, this shows that the orbital moment m precesses around the applied
…eld with the Larmor frequency fL = ! L =2 :
fL =
B
2
(3.12)
Note that the Larmor precession frequency for an orbital moment ( = e=2m) is
just half the cyclotron frequency, whereas it is equal to the cyclotron frequency for
a spin moment ( = e=m). The precession of the spin angular momentum around
Oz in Fig 3.5 occurs at the Larmor frequency.
Fig 3.6. An electron moving freely in a magnetic …eld follows a helical path. The
projection of the path in a plane normal to B is a circle where the electron circulates at
the cyclotron frequency eB /2 m:
Fig 3.7 Torque on a magnetic moment in a magnetic …eld induces a precession
around the …eld direction at the Larmor frequency.
3.2.3 Orbital diamagnetism
A semi-classical expression for the diamagnetic susceptibility of orbital electrons can
be deduced from the Larmor precession. There is some angular momentum, and
Magnetic Field E¤ects
9
therefore a magnetic moment is associated with the precession of the electron orbit
induced by the magnetic …eld. By Lenz’s law, the induced moment is expected to
oppose the applied …eld. The induced angular momentum is m! L h 2 i where h 2 i =
hx2 i + hy 2 i is the mean square radius of the electron’s orbit projected onto the plane
perpendicular to B. Since ! L = B, the induced moment is - 2 mhr2 iB, which gives
a susceptibility 0 M=B of
= n 0 e2 hr2 i=6me
(3.13)
where n is the number of electrons per cubic meter, and hr2 i = (3=2)h 2 i is the
average radius of the electron orbit.
The order of magnitude
of the orbital diamagnetism for an element with
p
28
3
2
n t 6 10 atoms m , hr i t 0:2 nm is 10 5 : It is a small e¤ect, present to
some extent for every element and molecule. Orbital diamagnetism is the dominant susceptibility when there are no partially-…lled shells, which produce a larger
paramagnetic contribution if there are unpaired electronic spins. Relatively large
diamagnetic susceptibilities are observed for materials such as graphite with delocalized electrons, where the induced currents can run around the carbon rings. The
diamagnetic susceptibility is then quite anisotropic, and it is greatest when the …eld
is applied perpendicular to the plane (Table 3.3).
Unfortunately there is an underlying problem with classical calculations of
the response of electrons to magnetic …elds. Since the magnetic force F = e(v
B) is perpendicular to the electron velocity, the magnetic …eld does no work on
a moving electron, and cannot therefore modify its energy. Hence W is zero in
(2.79), and it follows that there can be no change of magnetization. The idea was set
out in the Bohr-van Leuven theorem, a famous and disconcerting result of classical
statistical mechanics which states that at any …nite temperature and in all …nite
electric or magnetic …elds, the net magnetization of a collection of electrons in thermal
equilibrium vanishes identically. Every sort of magnetism is impossible for electrons
in classical physics! The semi-classical calculation of the orbital diamagnetism works
because we have assumed that there is a …xed magnetic moment associated with the
orbit.
It is clear from Fig 1.8 and 3.8 that the diamagnetic susceptibility of more than
half the elements in the periodic table is overwhelmed by a positive paramagnetic
contribution. We now grant the electron its intrinsic spin moment, and examine how
paramagnetic susceptibility arises in the two extreme models of magnetism, those of
localized and delocalized electrons.
Fig 3.8 Magnetic susceptibility of the elements.
3.2.4 Curie-law paramagnetism
Considering a single, localized electron in a magnetic …eld, the Zeeman splitting
of the j"i and j#i states is as shown in Fig 3.4. The splitting is 2 B B. As the
…eld increases, so does the Boltzmann population of the j"i state relative to the
j#i state. If we have n electrons per unit volume, the magnetization M is (n"
10
Magnetism of the electron
n# ) B where n";# depends on the Boltzmann populations. Hence we …nd n";# =
n exp( B B=kB T )=[exp( B B=kB T ) + exp( B B=kB T )]; and as a result,
M =n
B
tanh x
(3.14)
where x = B B=kB T: At room temperature, B B
kB T , so x is small and we can
make the approximation tanh x x: Hence we …nd the Curie-law expression for the
susceptibility = 0 M=B;
(3.15)
= n 0 2B =kB T
The Curie law is often written = C=T; where C is the Curie constant. A typical
value of n = 6 1028 m 3 for one unpaired electron per atom gives C = 0.5 K and a
susceptibility of 1.6 10 3 at room temperature, which diverges as T ! 0:
3.2.5 Pauli paramagnetism
Free electron model
The simplest delocalized-electron model is the free-electron model, where the electrons
are described as noninteracting waves con…ned in a box of dimension L that represents
the solid. The Hamiltonian is the sum of terms representing the kinetic and potential
energy
H = [(p2 =2m) + V (r)]
(3.16)
When V (r) is a constant, which can be set to zero, and p is replaced by
the operator i~r, Schrodinger’s equation is -(~2 =2me )r2 = E :Solutions are
the free electron waves
= L 3=2 exp(ik:r);where the L 3=2 factor is required for
normalization. The corresponding momentum and energy of the electron obtained
from the operators p^ = ihr and p^2 =2m are p = ~k and E = ~k 2 =2m, respectively.
The boundary conditions restrict the allowed values of k so that the components
ki (i = x; y; z) are 2 ni =L where ni is an integer. Since the indistinguishable
electrons obey Fermi-Dirac statistics, each quantum state represented by the integers
nx ; ny ; nz can accomodate at most two electrons, one ", the other #. The allowed
states form a simple cubic lattice in ‘k-space’, where the coordinates of a lattice point
are (kx ; ky ; kz ): Each state occupies a volume (2 =L)3 ; so the density of states in kspace is 2(L3 =8 3 ) where the 2 takes account of spin degeneracy. At zero temperature
the N = nL3 electrons in the box occupy all the lowest available energy states, up to
an energy known as the Fermi energy.
EF = (~2 =2m)(3 2 n)2=3
(3.17)
The surface separating occupied and unoccupied states is the Fermi surface, which
is a sphere in the free electron model since E k 2 : Again for a typical free electron
density (one per atom) of n: = 6:1028 m 3 , the Fermi energy is 9 10 19 J or 5.6
Magnetic Field E¤ects
11
eV. The corresponding Fermi temperature TF = EF =kB = 65; 000 K. The density of
states D(E) = dn=dE for either spin is
D";# (E) = (L3 =2 2 )(2m=~2 )3=2 E 1=2
(3.18)
k -space. Each point represents a possible state for the electrons in a free electron
gas contained in a box of side L, Each state can accomodate a "and a # electron.
From 3.10 and 3.11, the density of states at the Fermi level for a sample of
unit volume is
D";# (EF ) = 3n=2EF
(3.19)
The density of states of one of the bands is shown in Fig 3.9
The e¤ect of applied magnetic …eld B acting on the spin moment is to shift
the two subbands by
B B, as shown in Fig 3.10. Since B B=kB = 0:67 K for a …eld
of 1 T and TF
65; 000 K, the shifts are tiny compared to the Fermi energy. From
the …gure, it can be seen that magnetization (n" n# ) B is 2D";# (EF ) 2B B. Hence
the susceptibility
2
(3.20)
P = 2 0 B D";# (EF )
This result is rather general. As it depends only on the density of states at the
Fermi level, it is not restricted to the free-electron model, but for this model, using
(3.8) and setting EF = kB TF , the result can be written in the form
P
= 3n
2
0 B =kB TF
(3.21)
The Pauli susceptibility is temperature-independent to …rst order. Comparison with
Eq 3.9 shows that the Pauli susceptibilty is about two orders of magnitude smaller
than the Curie susceptibility at room temperature. It is of order 10 5 :
At …nite temperature, the occupancy of the states is given by the Fermi-Dirac
distribution function
f (E) = 1=fexp(E
)=kB T ] + 1g
(3.22)
where is the chemical potential, which is t EF at T t 0: The result is a small
correction to Eq 3.13, varying as T 2 :GIVE THE FORMULA.
Fig 3.9. Density of " or #states in the free-electron model. The states are occupied
with a probability given by the Fermi function (3.15), which is approximately a step
function.
Fig 3.10. Spin splitting of the" and #densities of states in a magnetic …eld. A net
moment results from the transfer of electrons at the Fermi level from the " to the # band.
3.2.6 Landau diamagnetism
The free electron model was used by Landau to calculate the susceptibility due to
orbital diamagnetism of the conduction electrons. The result is
L
=n
2
0 B =kB TF
(3.23)
12
Magnetism of the electron
which is eactly one third of the Pauli paramagnetism, but of opposite sign. It looks
as if the diamagnetism of the conduction electrons is just a correction to their paraamagnetism, and never the dominant contribution,. but this is not quite correct. The
real band structure of solids may be approximately taken into account, in the free
electron model, by renormalizing the electron mass; me is replaced by m : Then
(3.20) is replaced by L -(1/3)(me =m )2 P : For some semiconductors, and semimetals such as graphite or bismuth, m
0:01me ; despite the low electron densities, the
susceptibilty can be rather large. ( ? = 4 10 4 for graphite):
A summary of the susceptibility of the elements is provided in Fig 3.8, and
numerical values for some compounds are given in Table 3.3. Some of the elements
are naturally gaseous at room temperature, so the de…nition = M=H, where M is
the induced magnetic moment per unit volume is not always the most useful. Instead,
we plot the molar susceptibility mol = M=H where
A m2 kg 1 is the induced
magnetic moment per unit mass and M is the atomic weight which has units of m3
mol 1 :The units of mol are m3 mol 1 : A mole of most solid elements occupies roughly
10 cm3 = 10 5 m3 , so the molar susceptibility is roughly …ve orders of magnitude
less than the dimensionless susceptibility : Susceptibility units and conversions are
a rich source of confusion; they are summarized in Appendix 4. Since Avogadro’s
number NA = 6:022:1023 , the number density of atoms in solids is typically 6 1028
m 3 : An atom occupies a volume of (0.25 nm)3 :
Fig 3.11. Temperature dependence of the susceptibility of electrons, comparing the
Curie-law susceptibility for localized electrons with the Pauli and Landau contributions for
free electrons. The orbital diamagnetic susceptibility is also included.
Table 3.3. Mass susceptibilities of some paramagnetic and diamagnetic materials.
10
MgO
Al2 O3
TiO2
SrTiO3
ZnO
ZrO2
HfO2
SiO2
MgAl2 O4
H2 O
3.3
9
m3 kg
-3.1
-4.5
0.9
-6.2
-1.1
-1.4
-7.1
-9.0
1
10
C(diamond)
C(graphite)
Si
Ge
NaCl
GaAs
GaN
InSb
Perspex
9
m3 kg
-6.2
k -6.3
? -138.0
-1.8
-1.5
-6.4
-5.0
1
10
Cu
Ag
Au
Al
Ta
Zn
Pd
Pt
In
Bi
9
m3 kg
1.1
2.4
1.9
7.9
10.7
-2.2
67.0
12.2
-7.0
-17.0
1
Theory of the electronic magnetism
Maxwell’s equations relate magnetic and electric …elds to their sources. The other
fundamental relation of electrodynamics is the Lorentz expression for the force on a
moving particle with charge q,
Theory of the electronic magnetism
13
F = q(E + v
B)
(3.24)
The two terms are respectively the electric and magnetic forces. In a moving frame of
reference, the separation of the electric and magnetic components may di¤er, but the
force is always given by this equation. The.magnetic force gives the torque equation
(2.11).
When the charged particle is moving in a magnetic …eld, the momentum and
energy are each the sum of potential and kinetic terms.
p = pkin + qA;
H = (1=2m)p2kin + q'e
It is the canonical momentum p, including the vector potential A, which is represented by the operator i~r in quantum mechanics. The vector potential A represents the magnetic …eld B (B = r A) and the scalar potential 'e represents the
electric …eld E (E = r'e ): Hence the Hamiltonian ( 3.15) [(1=2m)(pkin )2 + q'e ]
becomes
H = [(1=2m)(p + eA)2 + V (r)
(3.25)
where V (r) = -e'e
3.3.1 Orbital moment
The orbital paramagnetism and diamagnetism of the electron can be derived from
(3.24). Setting V (r) = 0 and making use of the Coulomb gauge where A and p
commute, (p + eA)2 is expanded as p2 + e2 A2 + 2eA:p, so there are three terms in
the Hamiltonian.
H = [p2 =2m + V (r)] + (e=m)A:p+(e2 =2m)A2
(3.26)
H = H0 +H1 +H2
(3.27)
where H0 is the unperturbed Hamiltonian, H1 gives the paramagnetic response of
the orbital moment and H2 describes its small diamagnetic response. Consider a
uniform …eld B along Oz. The vector potential in component form can be taken
as A = (1=2)( yB; xB; 0), so B = r A = ez (@Ay =@x @Ax =@y) = ez B. More
generally
1
A= B
2
r
(3.28)
Now (e=m)A:p = (e=2m)B r:p = (e=2m)B:r p ) (e=2m)B:^
l since ^
l = r p.
^
The angular momentum operator ^
l is therefore r ( i~r). Its z-component, for
example, is -i~(x@=@y y@=@x); or in polar coordinates ^
lz = i~@=@ .
The second term in the Hamiltonian is therefore the Zeeman interaction for
the orbital moment
14
Magnetism of the electron
H1 =
(
B =~)Blz
(3.29)
The eigenvalues of ^
lz in a central potential (CHECK) are ml ~; where ml = `; ` +
1; :::::::`: The Hamiltonian (3.28) therefore gives an average orbital magnetic moment
P̀
ml B exp( ml B B=kB T ); and a susceptibility n < m > =H; where n is
< m >=
`
the number of atoms per cubic meter.
The third term in (3.25) is (e2 =8m)(B r)2 = (e2 =8m)B 2 (x2 + y 2 ) If the
electron orbital is spherically symmetric, hx2 i = hy 2 i = (1=3)hr2 i: so the energy
corresponding to H2 is E = (e2 B 2 =12m)hr2 i: This is the Gibbs free energy, because
the Hamiltonian depends on the applied …eld B = 0 H: Hence from (2.80) M =
@E=@B. In agreement with (3.7), the susceptibility 0 nM=B is
= n 0 e2 hr2 i=6m
(3.30)
3.3.2 Quantum oscillations
We now examine the diamagnetic response of the free-electron gas in more detail. The
Hamiltonian of the electrons in a magnetic …eld is (3.24). Choosing A = (0; xB; 0) to
represent the magnetic …eld, which is applied as usual in the z-direction, and setting
V (r) = 0 and me = m we have Schrodinger’s equation
1 2
[p + (py + xB)2 + p2z ] = E
2m x
(3.31)
where pi = i~@=@xi : It turns out that the y and z components of p commute with
H; so the solutions of this equation are plane waves in the y and z directions, with
wave function (x)eiky y eikz z : Substituting (x; y; z) back into Schrodinger’s equation,
we …nd
~2 d2
1
+ m ! c (x x0 )2 (x) = E 0 (x)
(3.32)
2
2m dx
2
where ! c = eB=m is the cyclotron frequency, x0 = ~ky =eB and E 0 = E
(~2 =2m)kz2 : Equation (3:31) is the equation of a one-dimensional harmonic oscillator,
with motion centred at x0 : The oscillations are at the cyclotron frequency for a particle
of mass m : The eigenvalues of the oscillator are (n + 21 )~! c ; E 0 = En = (n + 12 )~! c ;is
the energy associated with the motion in the xy plane, and the energy levels labelled
by the quantum number n re known as Landau levels. The motion in the z-direction
is unconstrained, so that
~2 kz2
1
E=
+ (n + )~! c
(3.33)
2m
2
The electron in the …eld, which classically follows the spiral trajectory illustrated
in Fig 3.6, represented in quantum mechanics by a plane wave along z and a onedimensional harmonic oscillator along x.
Theory of the electronic magnetism
15
The states of the free-electron gas form a closely-spaced lattice of points in
k-space with spacing 2 =L, where L is the dimension of the container. When the
magnetic …eld is applied, these states collapse onto a series of tubes, as shown in
Fig 3.12. Each tube corresponds to a single Landau level, with index n. As B
increases, the tubes expand and fewer of then are contained within the original Fermi
sphere. The degeneracy of the Landau levels can be calculated from the fact that
2
the area of k-space in the xy plane between one level and the next is (kn+1
kn2 ) =
(2m =~2 )[(n + 1 + 12 )~! c (n + 12 )~! c ] = 2m ! c =~. Each state occupies an area
(2 =L)2 : The degeneracy of the Landau level is therefore gn = 4m ! c =~(2 =L)2 =
m L2 ! c = ~:
Oscilliatory variations of the magnetization or onductivity with increasing …eld
arise because of the periodic emptying of the uppermost Landau level as the …eld
is increased. The Landau tubes expand with B, and an oscilliatory variation as
B 2 is observed. These oscillations of magnetic moment are known as the de Haas
van Alphen e¤ect, whereas the oscillations of the coonductivity are known as the
Shubnikov-de Haas e¤ect. From the periodicity of the oscillations, it is possible to
deduce the area of the Fermi surface normal to the direction of applied …eld. Hence
the Fermi surface can be mapped. [Include 1/B2 derivation]
Fig 3.12. The states of the free electron gas in a magnetic …eld coalesce onto a series
of tubes. Each tube represents a Landau level.
3.3.3 Spin moment
The time-dependent Schrödinger equation
(~2 =2m) 52
+V
= i~@ =@t
(3.34)
is not relativistically invariant because the derivatives @=@t and @=@x in the energy
and momentum operators do not appear to the same power. We need to use a 4-vector
X = (ct; x; y; z) with derivatives @=@X.
Dirac discovered the relativistic quantum mechanical theory of the electron,
which involves the Pauli spin operators ^ i ., and coupled equations for electrons and
positrons. The nonrelativistc limit of his theory, including the interaction with a
magnetic …eld represented by the vector potential A is written as a Hamiltonian
H = [(1=2m)(p+eA)2 +V (r)] p4 =8m3 c2 +(e=m)(r A):s+(1=2m2 c2 r)(dV =dr)l:s (1=4m2 c2 )(dV =dr)
(3.35)
* The …rst term is the non-relativistic Hamiltonian (3.24)
* The second term is a higher-order correction to the kinetic energy
* The third term is the interaction of the electron spin with the magnetic …eld.
Taken together with 3.28, the complete expression for the Zeeman interaction of the
electron is
16
Magnetism of the electron
HZ = (
^+ 2^
s)
(3.36)
B =~)B:(l
The factor 2 is not quite exact. The correct expression from quantum electrodynamics
is 2(1 + =2
:::::) t 2:0023, where = e2 =4 0 ~c t 1=137 is the …ne-structure
constant.
* The fourth term is the spin-orbit interaction, which for a central potential
V (r) = Ze2 =4 0 r with Ze as the nuclear charge becomes Ze2 0 l:s=8 m2 r3 since
2
0 0 = 1=c : The one-electron spin-orbit coupling is usually written as l:s. In an
atom h1=r3 i(0:1 nm)3 so the magnitude of the spin-orbit coupling is 2.5 K for
hydrogen (Z = 1), 60 K for 3d elements(Z t 25), and 160 K for actinides (Z t 65).
In a noncentral potential, the spin-orbit interaction is (^
s rV ):p^
* The …nal term just shifts the levels when l = 0.
3.3.4 Magnetism and relativity.
The classi…cation of interactions according to their relativistic character is based on
the kinetic energy
p
E = mc2 [1 + (v 2 =c2 )]
(3.37)
The order of magnitude of the velocity of electrons in solids is c where c is the
velocity of light and t 1=137: Expanding (3.18) in powers of shows the heirarchy
of interactions.
E = mc2 + (1=2)
2
mc2
(1=8)
4
mc2
(3.38)
Here mc2 = 511 keV; the second and third terms, which represent the order of
magnitude of electrostatic and magnetostatic energies are respectively 13.6 eV ( the
Rydberg, R0 ) and 0.18 meV. Magnetic dipolar interactions are therefore of order 2
K.
The Pauli susceptibility (3.13) can also be written in terms of the …ne-structure
constant as P = 2 kF a0 = ; where kF is the Fermi wavevector and a0 is the Bohr
radius.
3.4
Magnetism of electrons in solids.
Before examining the ways the electronic moments can pair o¤ in solids, …rst look
at the situation for free atoms, summarized in Table 3.2. The electronic moments
are completely paired for some of the elements with even atomic number Z such as
the alkaline earths or the noble gasses, but most elements retain a magnetic moment
in the atomic state. Except for hydrogen, their atomic moments are much less than
Z B . The largest values of 10 B are found for dysprosium and holmium. Electrons
in …lled shells have paired spins and no net orbital moment. Only unpaired spins in
un…lled shells, usually the outermost one, contribute to the atomic moment.
Magnetism of electrons in solids.
17
Table 3.2 Magnetism of free atoms. Atoms which are nonmagnetic are shown boxed,
in bold type. All of them have Z even. Radioactive elements with no stable isotope are
shown in italics.
Assembling the atoms together to form a solid is a traumatic process insofar
as the atomic moments are concerned. Magnetism in solids tends to be destroyed by
chemical interactions of the outermost electrons, which can occur in various ways:
electron transfer to form …lled shells in ionic compounds
covalent bond formation in semiconductors
band formation in metals.
Fig 3.5 Formation of d and s-bands in a metal. The broad s-bands have no moment.
The d-bands may have one if they are su¢ ciently narrow.
Atomic iron, for example, has an electronic con…guration (Ar)3d6 4s2 . Four of
the 3d electrons are unpaired, so the spin moment of the atom is 4 B . When the iron
atoms are brought together in a solid, as illustrated in Fig 3.4, the outer 4s orbitals
…rst overlap to form a broad 4s band, and then the smaller 3d orbitals follow suit to
form a band that is considerably narrower. This results in 4s ! 3d charge transfer,
producing an electronic con…guration in iron metal of approximately (Ar)3d7:4 4s0:6 .
The narrow 3d band has a tendency to split spontaneously to form a ferromagnetic
state, and this occurs below TC = 1044 K in normal Fe, which has the body-centred
cubic structure. The electrons occupy states with their magnetic moments either
parallel (") or antiparallel (#) to the ferromagnetic axis. A spin pair "# has no net
moment. The electrons are paired in all the inner shells, and in the 4s band. The spin
con…guration of the 3d band is approximately 3d"4:8 3d#2:6 so the number of unpaired
spins is 2.2. In Fig 3.4, the " and # electrons are shown as occupying di¤erent, spin
split subbands.
It must be emphasized that the nature of the chemical bonding, and therefore
the character of the magnetism depends critically on crystal stucture and composition. The very existence of a moment in face-centred cubic Fe depends on the
lattice parameter. Intermetallic conpounds such as YFe2 Si2 exist where there is no
spontaneous spin splitting of the d-band. Insulating ionic compounds containing the
Fe3+ ion have large unpaired spin moments of 5 B per iron, but covalent compounds
with low-spin F eII are nonmagnetic. A few examples to illustrate the scope of the
magnetism of this one element are given in Table 3.2
Table 3.2 Atomic moments of iron in di¤erent compounds, in units of Bohr magnetons
-F e2 O3
-F e
Y F e2
-F e
Y F e2 Si2
F eS 2
ferrimagnet ferromagnet ferromagnet antiferromagnet Pauli paramagnet diamagnet
5:0
2:2
1:45
unstable
0
0
18
Magnetism of the electron
3.4.1 Localized and delocalized electrons
It is a huge task to make a physical theory that will adequately account for the
behaviour of 3d electrons in narrow bands, but two limits are accessible. One is the
limit of localised electrons, where correlations of the electrons on the ion cores due
to their mutual Coulomb interaction are strong, but transfer of electrons from one
site to the next is negligible. The other is the limit of delocalised electrons where
the electrons are con…ned in the solid, but Coulomb correlations among them and
with the nuclear charges are relatively weak, of order 1 eV. Numerical methods for
calculating the electronic structure, outlined in Ch 5, then come into their own.
The localised model is best suited for 4f electrons in the rare earth series, R =
Ce .... Yb, which occupy orbitals belonging to an inner shell that barely participate
in the bonding. The outer electrons have 5s; 5p; 5d or 6s character. There are usually
two or three outermost 5d=6s valence electrons which either form a conduction band,
as in the metals, or else get transferred to an electronegative ligand such as oxygen in
the rare-earth oxides R2 O3 . There the rare earth becomes R3+ and the oxygen accepts
two electrons to …ll the 2p shell to become O2 , with a stable closed-shell con…guration
2p6 x . Either way, the 4f shell is su¢ ciently well buried inside the 5s and 5p shells
to be unconcerned with chemical bonding. It has an atom-like con…guration with a
well-de…ned integral number of electrons. There is a series of discreet energy levels for
this strongly-correlated set of electrons, discussed in the next Chapter. The electron
orbitals describe localised states, and the ions are properly described by Boltzmann
statistics. The localised model is also suitable for insulating ionic compounds of 3d
elements such as Fe2 O3 .
The delocalised model describes the magnetic electrons by wave-like extended
states which form energy bands. The number of electrons in the bands at the Fermi
level are not integral and the electrons are now described by Fermi-Dirac statistics.
The delocalised model applies to the magnetism of 3d metals and alloys, and to some
conducting compounds. It also applies to 5f (actinide) magnetism.
Neither localized nor delocalized moments disappear bove the Curie temperature; they just become disordered in the paramagnetic state, T > TC
Table 3.3 Summary of localised and delocalised magnetism.
x
The closed shell con…guration 2p6 is especially stable. Over 80% of the Earth’s crust (Fig 1.12)
is made up of atoms that are present there as 2p6 ions.
Magnetism of electrons in solids.
19
LOCALIZED MAGNETISM
Integral number of 3d or 4f electrons
Integral number of unpaired spins
Discreet energy levels
DELOCALIZED MAGNETISM
Nonintegral number of unpaired spins on the ion core;
per atom.
Spin-polarized energy bands with strong correlations.
N i2+ 3d8 m = 2
N i3d9:4 4s0:6 m = 0:6
B
exp( r=a0 )
Boltzmann statistics
4f metals and compounds
Some 3d compounds
B
exp( ik:r)
Fermi-Dirac statistics
3d metals
Some 3d compounds
1. Consider a hypothetical ferromagnet with an orbital moment of 1 B : The
atoms are dense-packed with rasius 0.1 nm. What is the magnetization? Show that
this is equal to the surface current density.