chapter 6 magnetism at surfaces: ground statephys48w/hafta11danilo1/script11.pdf · chapter 6....

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 Fe3O4 (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 earli-

est 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 Ampere and Oersted in the early decades of the 19th century, demon-

strating that i) a current is able to influence a magnetic needle (Oersted) and ii) a

mechanical force exists between two wires injected with current (Ampere). Later,

Faraday completed our knowledge of magnetic field by discovering that time depen-

dent 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: Ampere

postulated that magnetism in atoms originates from the existence of a closed atomic-

sized current. Poisson and later Maxwell, instead, favored magnetic charges that

appear always coupled into dipoles as the source of the magnetic field. Distinguish-

ing 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 Rene Descartes, from his

“Principia Philosophiae”, 1644. This was one of the first drawings of a magnetic

field.

Jackson1 shows. Ultimately, Ampere’s hypothesis turned out to be the correct one,

as J.D. Jackson demonstrated. Following Ampere’s hypothesis, the magnetic field

produced by a current I circulating within a small loop C is given in terms of the

magnetic-moment vector ~m = e2~x × ~x = e

2m~L, which for a closed loop amounts to

~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 Ampere 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

the classical Hamilton function for an electron in an applied magnetic field ~B writes

H = 12m

(~p− e ~A)2 + eφ it follows that the canonical average of the atomic magnetic

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 deter-

mined 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 angu-

lar momentum and orbital angular momentum of all electrons taking into account

1J. D. Jackson, The Nature of Intrinsic Magnetic Dipole Moments, CERN 77 − 17, Theory

Division, 1 September 1977.


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 con-

ducted 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:

~µatom = −gLande · µB · ~J

where µB is the Bohr magneton and is recognized as the unit for magnetic moments,

gLande is a number (Lande factor, equaling 2 for pure spins and 1 for pure orbital

angular momentum) and ~J is the dimensionless total angular momentum of the

ground state electronic configuration arising from the sum of orbital (~L) and spin

(~S) angular momentum. Notice that as a result of the exchange interaction (Pauli

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 mo-

ments produces a finite magnetic moment at any temperature – the so-called Curie

paramagnetism. The Hamilton operator expressing the so-called Zeeman energy of

such atomic magnetic moments in an applied magnetic field ~B reads: The Zeeman

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)


m = J, J − 1, ...,−J (Zeeman splitting). Accordingly, the partition function reads

ZN =

∑mj

exp(−gLSJ · µB ·mj ·B

kB · T

)N (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)

〈µz〉 = −∂f(T,B)

∂B= µB tanh

[µB ·BkB · T

](6.3)

For small magnetic fields we obtain the Curie law

〈µ(P )z 〉 ≈

(gLSJ)2 · J(J + 1) · µ2B

3kB · T·B (6.4)

which contains the purely quantum mechanical quantity (gLSJ)2 · J(J + 1). The

paramagnetic susceptibility in this limit amounts to

χP ≈ µoC

T(6.5)

with the Curie constant

C =N

V

(gLSJ)2 · J(J + 1) · µ2B

3kB(6.6)

At room temperature χP ≈ 10−3. This means that applying a magnetic field of

1 Tesla to such a gas of free atoms produces a net magnetization vector ~M = NV〈~µ〉

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 impor-

tant phenomena occur. First, the breaking of the spherical symmetry intrinsic to

isolated atoms by the molecular or crystal field perturbs the orbital angular momen-

tum. The largest orbital angular momentum is obtained in a spherically symmetric

potential. In a cubic environment L is almost vanishing. Adding nearest neigh-

bors is equivalent to embed an atom in a crystal potential. The cubic symmetry

is thus progressively approached and the orbital angular momentum is accordingly


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 Gd3SO4·8H2O, Fe3+ in

NH4Fe(SO4)2·12H2O, and Cr3+ in KCr(SO)4·12H2O 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 energet-

ically 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


Figure 6.4: Ab-initio calculation of the magnetic anisotropy energy, and the mag-

netic 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 half-

integer 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 = (3π2N/V )1/3, N/V being the

electron density. In virtue of this filling the electron gas has a total kinetic energy

amounting to Ekin = N 35

h2k2F2m

. The non-magnetic ground state foresees that all


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 for-

mulating 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


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 sur-

vived the contact with the free electrons by means of the Stoner-Wohlfahrt mecha-

nism, which spin configuration will be energetically favored: the parallel (ferromag-

netic) spin configuration, foreseeing parallel alignment of two neighboring magnetic

moments or the antiparallel (antiferromagnetic) spin configuration, foreseeing an-

tiparallel alignment of two neighboring magnetic moments? The simplest model of


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− JEs ≈ 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

2M. B. Stearn, Physics Today, April 1978, p. 34.


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 fer-

romagnetism 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 pro-

vided within the atom-derived Heitler-London method. This alternative mechanism

is provided, according to Stearns, by the indirect exchange between localized d elec-

trons through RKKY coupling with the delocalized part of the d wave functions (or

with the s-like electrons).


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 spin-

split 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 interac-

tion 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 con-

sidered 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 re-

mains uniformly distributed. This produces a local spin polarization of the electron

gas surrounding the impurity

P =2 · 〈Sz〉h

=ρ+ − ρ−

ρ+ + ρ−≈ O(

κ cos 2kFx

kFx) (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.


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 multi-

layered 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).

chapter 6 magnetism at surfaces: ground statephys48w/hafta11danilo1/script11.pdf · chapter 6....

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