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Chapter 6
Magnetism at Surfaces: Ground
State
6.1
Introduction
The experimental facts about magnetism in solids are as old as the history of
mankind: some materials have the property of producing a sizable magnetic field
that attracts or repels other materials. One of the first references to the magnetic
properties of what we know now to be magnetite Fe3 O4 (lodestone) is by the Greek
philosopher Thales of Miletus and dates back to the 6th century BC. The name
“magnet” may come from the lodestones found in Magnesia. In China, the earliest literary reference to magnetism lies in a 4th century BC book called Book of
the Devil Valley Master: “The lodestone makes iron come or it attracts it”. The
lodestone-based compass was used for navigation in medieval China by the 12th
century.
The main observation about the origin of the magnetic field dates back to the
experiments of Ampère and Oersted in the early decades of the 19th century, demonstrating that i) a current is able to influence a magnetic needle (Oersted) and ii) a
mechanical force exists between two wires injected with current (Ampère). Later,
Faraday completed our knowledge of magnetic field by discovering that time dependent magnetic fields can produce a magnetic current. J.C. Maxwell gave a complete
description of electromagnetic fields that is still very precise (Maxwell equations).
The origin of magnetism in matter remained debated for a long time: Ampère
postulated that magnetism in atoms originates from the existence of a closed atomicsized current. Poisson and later Maxwell, instead, favored magnetic charges that
appear always coupled into dipoles as the source of the magnetic field. Distinguishing between the two hypotheses is a subtle problem, as a paper in 1977 by J. D.
Prof. Dr. Danilo Pescia ([email protected]), Laboratory of Solid State Physics, ETHZ,
Zurich, Switzerland, has generously contributed this and the next Chapter on Magnetism at
Surfaces.
135
CHAPTER 6. MAGNETISM AT SURFACES: GROUND STATE
136
Figure 6.1: Drawing of the magnetic field of the Earth by René Descartes, from his
“Principia Philosophiae”, 1644. This was one of the first drawings of a magnetic
field.
Jackson1 shows. Ultimately, Ampère’s hypothesis turned out to be the correct one,
as J.D. Jackson demonstrated. Following Ampère’s hypothesis, the magnetic field
produced by a current I circulating within a small loop C is given in terms of the
e ~
magnetic-moment vector m
~ = 2e ~x × ~ẋ = 2m
L, which for a closed loop amounts to
. R
m
~ = I S ~nds, where S is the surface within the loop C and ~n is the vector normal
to S. The origin of “atom-sized currents” resides within the quantum mechanics.
In fact, while Maxwell equations and the postulate of Ampère are very exact in
describing the origin of magnetic fields, the Bohr-van Leuwen theorem forbids the
existence of such currents in classical physics. This theorem states that “given that
~ writes
the classical Hamilton function for an electron in an applied magnetic field B
1
~ 2 + eφ it follows that the canonical average of the atomic magnetic
H = 2m (~p − eA)
moment ≺ m
~ vanishes exactly and at any finite temperature.
It was Landau (1930) who explicitly produced an average magnetic moment for
a quantum mechanical free electron in a uniform magnetic field and thus determined the “birth” of magnetism in matter. Landau solved the problem of a free
electron moving in a uniform magnetic field using quantum mechanics to describe
the motion, i.e., breaking away from Lorentz force and thus from Bohr theorem.
However, the most widespread magnetic moment in matter (Fe, Co, Ni, Gd, etc.) is
the one originating within an atom (bound electrons). It can be understood within
the Dirac theory of the electron as resulting from the addition of the spin angular momentum and orbital angular momentum of all electrons taking into account
1
J. D. Jackson, The Nature of Intrinsic Magnetic Dipole Moments, CERN 77 − 17, Theory
Division, 1 September 1977.
CHAPTER 6. MAGNETISM AT SURFACES: GROUND STATE
137
Figure 6.2: Pierre Pelerin de Maricourt (French), Petrus Peregrinus de Maricourt
(Latin) or Peter Peregrinus of Maricourt was a 13th century French scholar who conducted experiments on magnetism and wrote the first extended treatise describing
the properties of magnets. His work is particularly noted for containing the earliest
detailed discussion of freely pivoting compass needles, a fundamental component of
the dry compass soon to appear in medieval navigation.
Pauli principle and Hund’s rules:
µ
~ atom = −gLande · µB · J~
where µB is the Bohr magneton and is recognized as the unit for magnetic moments,
gLande is a number (Landé factor, equaling 2 for pure spins and 1 for pure orbital
angular momentum) and J~ is the dimensionless total angular momentum of the
~ and spin
ground state electronic configuration arising from the sum of orbital (L)
~ angular momentum. Notice that as a result of the exchange interaction (Pauli
(S)
principle and Hund’s rules) the excited states of the atom, carrying a different
magnetic moment than that of the ground state, are separated from the ground
state by the intraatomic exchange energy JHund , which amounts to 1 − 5 eV.
The partition function for an ensemble of such non-interacting magnetic moments produces a finite magnetic moment at any temperature – the so-called Curie
paramagnetism. The Hamilton operator expressing the so-called Zeeman energy of
~ reads: The Zeeman
such atomic magnetic moments in an applied magnetic field B
Hamiltonian produces a splitting of the 2J + 1 degenerate atomic ground state level
carrying the quantum numbers n, J, En,J into
En,J (B) = En,J + m · µB · B · gLande
(6.1)
CHAPTER 6. MAGNETISM AT SURFACES: GROUND STATE
138
m = J, J − 1, ..., −J (Zeeman splitting). Accordingly, the partition function reads
ZN =

X
−gLSJ

exp
mj

N
· µB · mj · B 
kB · T
(6.2)
and the mean magnetic moment per atom as minus the derivative according to the
variable B of the Helmholtz Free Energy per atom amounts to (here is the result
given for the particularly simple case of spin 1/2)
∂f (T, B)
µB · B
hµz i = −
= µB tanh
∂B
kB · T
(6.3)
For small magnetic fields we obtain the Curie law
)
hµ(P
z i ≈
(gLSJ )2 · J(J + 1) · µ2B
·B
3kB · T
(6.4)
which contains the purely quantum mechanical quantity (gLSJ )2 · J(J + 1). The
paramagnetic susceptibility in this limit amounts to
χP ≈ µo
C
T
(6.5)
with the Curie constant
C=
N (gLSJ )2 · J(J + 1) · µ2B
V
3kB
(6.6)
At room temperature χP ≈ 10−3 . This means that applying a magnetic field of
~ = N h~µi
1 Tesla to such a gas of free atoms produces a net magnetization vector M
V
along the direction of the applied magnetic field which in turn enhances the field
with a strength of ≈ µo · M ≈ 10−3 Tesla. We conclude that quantum mechanics
was successful in producing magnetism in atoms, which, however, is too weak to
explain the usual magnetic phenomena observed in everyday life.
6.2
The Magnetic Ground State of Solids and
Surfaces
When two atoms, carrying a net total angular momentum J – and accordingly a
net magnetic moment – are brought together to form a bound state three important phenomena occur. First, the breaking of the spherical symmetry intrinsic to
isolated atoms by the molecular or crystal field perturbs the orbital angular momentum. The largest orbital angular momentum is obtained in a spherically symmetric
potential. In a cubic environment L is almost vanishing. Adding nearest neighbors is equivalent to embed an atom in a crystal potential. The cubic symmetry
is thus progressively approached and the orbital angular momentum is accordingly
CHAPTER 6. MAGNETISM AT SURFACES: GROUND STATE
139
Figure 6.3: Dependence of the mean magnetic moment per atom in units µB on
the ratio H/T for three paramagnetic ions, namely Gd3+ in Gd3 SO4 ·8H2 O, Fe3+ in
NH4 Fe(SO4 )2 ·12H2 O, and Cr3+ in KCr(SO)4 ·12H2 O are in S states with S = 7/2
for Gd3+ , and S = 5/2 for Fe3+ . The orbital moment of Cr3+ is quenched; it has
an effective spin of S = 3/2. From W. E. Henry, Phys. Rev. 88, 559 (1952).
suppressed (in Fig. 6.4, the suppression of L produces a progressive reduction of
the magnetic moment). Surface and thin films are intermediate situations between
isolated atom and full bulk symmetry and thus allow for some sizable amount of
orbital angular momentum. Notice that the presence of orbital angular momentum
is tied to the strength of an interaction of relativistic origin, the spin-orbit coupling.
This interaction is responsible for the existence of an anisotropy energy, because
of which the total energy of the spin configuration depends on the direction of the
spin with respect to the lattice. Accordingly, a direction along which it is energetically more favorable for the spin to point along is established through spin-orbit
coupling.
Second, having been left with a net total magnetic moment per atom, arising
mostly from the spin component, the question is now: how much of the magnetic
moment – typically originating from the localized d or f shells of the atoms –
is left when a localized orbital is embedded into a “sea” of free electrons which
CHAPTER 6. MAGNETISM AT SURFACES: GROUND STATE
140
Figure 6.4: Ab-initio calculation of the magnetic anisotropy energy, and the magnetic orbital moment per Co atom on Pt(111). From P. Gambardella et al., C. R.
Physique 6, 75 (2005).
Figure 6.5: Experimental results: magnetic anisotropy energy as a function of the
number of Co atoms in Co clusters on Pt(111). From P. Gambardella et al, C. R.
Physique 6, 75 (2005).
typically appears during the formation of the (metallic) solid? In metals with
quenched angular momentum the magnetic moments per atom should be a halfinteger multiple of 2µB , i.e., their magneton number should be an integer. This
would lead, for example, to atomic magnetic moments of 2, 3, and 4 µB , respectively
for Ni (total spin S = 1), Co (S = 3/2), and Fe (S = 2). Instead, the experimentally
measured values in bulk amount to 0.616 (Ni), 1.715 (Co), and 2.216 (Fe) µB . The
Stoner-Wohlfahrt (SW) Slater model of magnetism provides th e correct framework
to explain the size of magnetic moments in solids and the of non-integer magneton
numbers. The SW model in its simplest version considers free electrons where
energy levels are filled up to the Fermi radius kF = (3π 2 N/V )1/3 , N/V being the
electron density. In virtue of this filling the electron gas has a total kinetic energy
h̄2 k2
amounting to Ekin = N 53 2mF . The non-magnetic ground state foresees that all
CHAPTER 6. MAGNETISM AT SURFACES: GROUND STATE
141
states up to EF are filled with two electrons carrying opposite spins. Introducing an
exchange interaction shifts the energy levels of minority electrons to higher energies,
while the energy levels of majority spin electrons are shifted downwards. This
produces an energy gain that actually favors the relative shift of energy bands and,
ultimately, the formation of a magnetic moment (which in this model, results from
the imbalance between the number of majority spin electrons and the number of
minority spin electrons). On the other side, the radius of the Fermi sphere must
be increased to host all the electrons, as the original double occupancy of each
level (Pauli principle) is no longer possible. This produces an increase of the total
kinetic energy that goes against the formation of a spin imbalance. Therefore, the
formation of the magnetic moment is the result of a delicate energy balance and is
subject to strong restrictions.
Figure 6.6: (Left) Local DOS of Mn in Ag according to LSDA computations by R.
Podloucky et al. Phys. Rev. B 33, 5777 (1980). The spin splitting is about 3 eV
(experiment: 4 eV). (Right) Computed and measured values of 3d impurity atoms
in Ag, Cu, and Al. The highest moment appears in the middle of the 3d row.
A more quantitative approach to the problem within the SW model allows formulating a Stoner criterion for the formation of a magnetic moment: provided the
density of states at the Fermi level of the non-magnetic electronic structure is large
enough, it is energetically favorable for the spin gas to have a spin imbalance, i.e.,
to form a net magnetic moment per atom. Typically, the broadening of the atomic
states to energy bands lower the resulting spin moment with respect to the atomic
value, as the broadening favor the population of minority spin as well as majority
ones. Here is where dimensionality – respectively the number of nearest neighbors
– plays a role. At surfaces and thin films, where some neighbors are missing, the
broadening of atomic levels is less severe than in the bulk. This allows at surfaces
CHAPTER 6. MAGNETISM AT SURFACES: GROUND STATE
142
and thin films the recovery of some magnetic moment with respect to bulk systems.
The value of the magnetic moment per atom at surface and thin films is closer to
the atomic one.
Figure 6.7: Paramagnetic FeRh: dispersion curves for high-symmetry directions.
The broken line is the Fermi level. The abscissa is in arbitrary units. From C.
Koenig, J. Phys. F: Met. Phys. 12, 1123 (1982).
Figure 6.8: FeRh: dispersion curves for the two spin directions; (a) spin +, (b)
spin -. The broken line is the Fermi level. From C. Koenig, J. Phys. F: Met. Phys.
12, 1123 (1982).
Third, the question remains: provided some magnetic moment per atom has survived the contact with the free electrons by means of the Stoner-Wohlfahrt mechanism, which spin configuration will be energetically favored: the parallel (ferromagnetic) spin configuration, foreseeing parallel alignment of two neighboring magnetic
moments or the antiparallel (antiferromagnetic) spin configuration, foreseeing antiparallel alignment of two neighboring magnetic moments? The simplest model of
CHAPTER 6. MAGNETISM AT SURFACES: GROUND STATE
143
chemical bond – the H2 molecule – gives a straightforward answer to this question.
In H2 one can explicitly compute the energy of both states (here denoted with t for
triplet, S = 1 and s for singlet, S = 0) using the Heitler-London method:
Et ≈ 2E1s + Q − J
Es ≈ 2E1s + Q + J
Q (Coulomb integral ) contains all Coulomb energies (electrons with the nuclei,
Coulomb interaction of the two electrons and Coulomb repulsion of the two nuclei. J
instead represents the interatomic exchange energy, arising form the requirement of
symmetrizing the wave functions according to the purely quantum mechanical Pauli
principle. Q is positive, but J – in contrast to the atomic Hund’s-rule exchange
JHund – is negative. Thus, the chemical bond favors the singlet ground state. We
point out that the strength of the effective exchange energy J is one to two orders
of magnitude smaller than the one-site (intraatomic) exchange interaction (which
amounts to about 3−5 eV). This means that rotating one spin in the presence of the
other ones needs much less energy than suppressing the magnetic moment. It is the
interatomic exchange interaction which is relevant for determining the temperature
scale at which collective ferromagnetic order vanishes (in the next chapter, the so
called Curie Temperature). Notice that this result, obtained by a simple calculation
of the H2 molecule, is robust and there exists a very strong theorem by Lieb and
Mattis that states that in a linear arrangement of atoms the non-magnetic state, i.e.,
the state with lowest total spin, is the actual ground state. One needs to go higher
than one dimension to escape this theorem, because in higher dimensions electrons
states with different symmetry – atomic orbitals with different quantum numbers
– can hybridize: it is this degeneracy between orbital wave functions with different
symmetry that provides a route to escape the strong Pauli principle which favors
antiparallel alignment between spins of electrons occupying the same orbital. The
situation is exactly the same as in atoms: only if the electronic states participating
to the formation of the magnetic moment have different quantum numbers, the
exchange interaction can act to lower the energy of the triplet state. The situation
can be therefore summarized as follows. The chemical bond between same orbitals
centered at different atoms favors the antiparallel ground state, in virtue of the
Pauli principle. The crystal potential, however, can act to mix different symmetries
and different orbitals into the wave functions forming the valence bands in solids.
As in atoms, different symmetries might favor energetically the parallel coupling,
thus producing ferromagnetic alignment between neighboring atoms. However, it
depends on the crystal potential and on the orbitals involved whether a total spin
in the ground state is formed or not.
The long sought explanation for the origin of ferromagnetic coupling in metals
like Fe was provided after years of research.2 The first condition for ferromagnetism
2
M. B. Stearn, Physics Today, April 1978, p. 34.
CHAPTER 6. MAGNETISM AT SURFACES: GROUND STATE
144
Figure 6.9: Band structure of Fe along the ∆ direction. The minority-spin bands
(broken lines) are shifted toward higher energies with respect to majority-spin bands
(continuous lines). The symmetry label of the bands is also shown. In contrast to
the Stoner model, the shift is not exactly rigid but depends slightly on ~k. On the
top part of the figure, the ~k-points selected by the used photon energy are indicated.
is that of having some localized magnetic moments, which are provided in Fe by
the localized d electrons. The last ones keep part of their atomic magnetic moment
produced by the Hund’s-rule intraatomic exchange. The second condition for ferromagnetism it that all these moments line up parallel to each other, in apparent
contrast to Pauli principle and to the negative interatomic exchange intervening
during formation of the chemical bond. This second condition requires a novel
mechanism for exchange other than direct exchange between the d electrons provided within the atom-derived Heitler-London method. This alternative mechanism
is provided, according to Stearns, by the indirect exchange between localized d electrons through RKKY coupling with the delocalized part of the d wave functions (or
with the s-like electrons).
CHAPTER 6. MAGNETISM AT SURFACES: GROUND STATE
145
Figure 6.10: On the left is an energy-resolved photoemission spectrum taken at
normal emission from Fe(100). In the middle, electrons are analyzed for their spin
polarization. On the right, the photoemission intensity for the two spin channels is
plotted separately, by suitably compounding the total intensity and the polarization
data. The two peaks are identified as due to electrons originating from the spinsplit bands close to the Γ point. From E. Kisker et al., Phys. Rev. Lett. 52, 2285
(1984).
6.2.1
RKKY Oscillations
The presence of a more or less localized magnetic moment – arising from d-wave
functions – creates a potential sink with the strength of the s − d exchange interaction at the location of the magnetic moment for majority (spin up) s electrons, by
virtue of the atomic Hund’s rules. The minority spin-down electrons can be considered as non-affected by the impurity. A local perturbation in one spin channel
produces an oscillating density in the affected spin channel, while the other spin remains uniformly distributed. This produces a local spin polarization of the electron
gas surrounding the impurity
P=
2 · hSz i
ρ+ − ρ−
κ cos 2kF x
= +
≈
O(
)
h̄
ρ + ρ−
kF x
(6.7)
that propagates far away from the perturbing magnetic moment. At some location
x within the spin polarized s-electron gas a spin imbalance appears. This spin
imbalance acts as an effective exchange field for d-wave functions and tends to align
a d-derived magnetic moment at that location parallel to itself: A magnetic moment
at the location x would lower his energy by aligning along the direction of P. In
this way, the exchange interaction can propagate, oscillating between positive and
negative depending on the position x and can couple spins which are quite distant
from each other.
CHAPTER 6. MAGNETISM AT SURFACES: GROUND STATE
146
Figure 6.11: a) The left-hand side shows a typical hysteresis curve (M versus
magnetic field H) recorded for exchange-coupled Co films. At the shift field H = Hj
the magnetizations of the individual films are aligned to the direction specified by
the external magnetic field. The critical field Hj is measured as a function of the
Cu spacer thickness τ by scanning a focused laser beam over a wedge-like multilayered structure, shown schematically on the right-hand side. b) Hj versus τ
for a room-temperature grown wedge-like multi-layered structure. A finite shift
field means antiferromagnetic coupling in the ground state. A vanishing shift field
means ferromagnetic coupling. The thickness of the Co films are 13.2 and 15.8 ML,
respectively. Inset, the Fourier transform, the two peaks corresponding to the two
periodicities 2.4 and 5.4 ML. The long period dominates. c) same as b) but with
the Cu wedge and the final Co film deposited and measured at 160 K. The short
period now dominates, as seen by the Fourier transform).