Venables J. Introduction to Surface and Thin Film Processes

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6.3 Magnetism at surfaces and in thin ﬁlms 217

Figure 6.19. (a) SEMPA spin polarization image, contrasted with (b) SEM intensity image of

an Fe–3% Si single crystal. The gray levels in (a) give the four diﬀerent magnetization

directions in the domains as marked by arrows (Hembree et al. 1987, reproduced with

permission).

rapidly and the sharper the tip, the faster the ﬁeld gradient decay. Measurements are

usually made in an a.c. detection scheme where the tip is vibrated at some resonance fre-

quency, and the departure from that resonance due to the tip’s interaction with the ﬁeld

gradient is detected with lock-in ampliﬁers. In this fashion, an image can be made at

about 50 nm resolution. But inevitably, the entire integrated ﬁeld gradient proﬁle from

the sample contributes to the image, so the reconstruction of the local sample magnet-

ization from such measurements may not be entirely straightforward (Rugar et al. 1990).

6.3.4 Theories and applications of surface magnetism

Magnetic interactions in 3d metals are dominated by the d-electrons and perturbed by

s-d hybridization. The 3d-electrons, responsible for the magnetism of Fe, Ni and Co,

218 6 Electronic structure and emission processes

Figure 6.20. Magnetic image of a 6 ML Co layer on W(110): (a, b) images taken with

P parallel and antiparallel to M in the two domain orientations; (c) diﬀerence image between

(a) and (b); (d) diﬀerence image between two images with P ⬜ M (from Bauer et al. 1996,

reproduced with permission).

form a relatively narrow band which overlaps with the wide 4s band. The question of

why these members of the 3d series are ferromagnetic, while others are antiferromag-

netic, and why the 4d and 5d series are not magnetic, is a typically subtle problem in

cohesive energy, in which several terms of diﬀering sign are closely balanced (Moruzzi

et al. 1978, Sutton 1994, Pettifor 1995). The magnetism of the parent atoms is a result

of Hund’s rule, which asserts that the ﬁrst ﬁve d-electrons are populated with parallel

spins, and the remaining ﬁve then ﬁll up the band with antiparallel alignment. This is

due to the reduced electron–electron Coulomb interaction between pairs with parallel

spins, because the exchange-correlation hole which accompanies each electron (see

Appendix J) keeps these electrons further apart on average. The rare earth elements are

an important class of magnetic materials based on 4f-electrons, but are not discussed

here.

When these atoms are assembled into solids, several eﬀects occur which we should

not try to oversimplify. The d-band is very important for cohesion, and the simplest

model is that due to Friedel (1969), which predicts a parabolic dependence of the bond

energy as the number of d-electrons N

is increased across the series. This model leads

to the contribution of d-d bonding to the pair-bond energy, E

兰

(E2

) (5/W)dE52 (W/20)N

(102 N

), (6.16)

where

is the unperturbed atomic d-level energy and W is the d-band width in the

solid. This parabolic behavior with N

is quite closely obeyed by the 4d and 5d series,

leading to surface energies displaying similar trends (Skriver & Rosengaard 1992). In

terms of the second moment of the energy distribution

, the overlap integrals

between d-orbitals of strength

, the band width are related by

W5(12z)

1/2

|, (6.17)

with z nearest neighbors; this can be derived for a rectangular d-band, where the second

moment

/12 (Sutton 1994). However, when magnetic eﬀects are considered, the

shape of the d-band is also very important, and ferromagnetism only results when both

the d-d nearest neighbor overlap is strong and the density of states near the Fermi

energy is large. These conditions are fulﬁlled towards the end of the 3d series, aided by

the two-peaked character of the density of states, sketched in ﬁgure 6.21(a); this energy

distribution has a large fourth moment

, which is also implicated in the discussion

of why Fe has the b.c.c. structure, points which can be explored further via project 6.4.

When detailed band structure calculations are done including magnetic interactions,

we have to account separately for the majority spin-up (

↑) and minority spin-down

(

↓) densities. By analogy to LDA, there is a corresponding local spin density (LSD)

approximation. This is illustrated in ﬁgures 6.21(b, c) and 6.22 by the calculations for

b.c.c. Fe by Papaconstantopoulos (1986); the up and down spins bands are shifted by

almost 2 eV. Above the ferro-paramagnetic transition at T5770°C these spins lose

long range order, but short range order is still present.

These spin density methods have been pursued intensively by Freeman & co-workers

(Weinert et al. 1982, Freeman et al. 1985), particularly in the version known as the

FPLAPW (full potential, linearized APW). Several features of thin ﬁlm magnetism

have been studied by this method as described by Wu et al. (1995). Comparisons of

6.3 Magnetism at surfaces and in thin ﬁlms 219

Figure 6.21. (a) Schematic distribution of s-d band overlap with the d-band ha

ving a double-peaked density of states; (b) the calculated spin-up and

5[100], after Papaconstantopoulos (1986, reproduced with permission). The symmetry points at

labelled 1 are s-like; the d-like states are 12 (e

53z

, x

) and 25 (t

5yz, zx, xy).

(a)

s-band

d-band

Spin-up

Spin-down

Figure 6.22. (a) Majority and (b) minority spin density of states, as calculated for b.c.c. Fe by

Papaconstantopoulos (1986, reproduced with permission); the vertical dashed line corresponds

to the Fermi energy. Note that both the 3d-bands have large fourth moments, with the t

band

having a large DOS at E

, and that the s- and p-bands are much broader than the d-bands.

(a)

(b)

magnetism in the bulk 3d transition series with freestanding monolayers, with monolay-

ers on non-magnetic substrates, and with isolated atoms or ions have been made. The

general feature is that reduced dimensionality goes part way to restoring the individual

magnetic moment per atom to the atomic value. This is illustrated in ﬁgure 6.23 by com-

parison with isolated ions in paramagnetic salts, on the assumption that the solids have

1 s-electron, whereas the ions have only d-electrons. In the bulk, the magnetic moment

per atom is reduced from the atomic value, in part from the itinerant character of d-elec-

trons, in part from the quenching of orbital angular momentum in a crystal (Kittel 1976,

Jiles 1991). These reductions are less marked at the surface and in monolayers.

Perhaps the most dramatic eﬀect is that these changes may be suﬃcient to change

the sign of the coupling between layers from ferromagnetic (F) to antiferromagnetic

(AF) or vice versa. Some of these eﬀects have been seen over the last few years in mag-

netic multilayers, in which thin magnetic layers are separated by non-magnetic spacers.

The coupling between the layers can be either F or AF, and can be changed, both by

the thickness of the spacer layers, and by the application of a magnetic ﬁeld.

The coupling between magnetic layers separated by noble or transition metals, as in

Fe–Cr, Fe–Ag or –Au, or Co–Cu superlattices, have all the magnetic interactions we

have discussed, plus a coupling due to the conduction electrons in the non-magnetic

spacers. The phenomenon is best thought of as a quantum size eﬀect with magnetic

222 6 Electronic structure and emission processes

Figure 6.23. Calculated magnetic moments of 3d transition metals in their bulk, surfaces and

monolayers, in comparison with isolated ions in paramagnetic salts. Open (ﬁlled) circles

denote ferromagnetic (antiferromagnetic) ground states (after Wu et al. 1995, replotted with

permission, and Jiles 1991). This plot assumes that the solid states contain 1 s-electron/atom.

complications (Stiles 1993, 1996). In eﬀect, there are Friedel oscillations at each metal

interface, and in the case of magnetic materials these are spin-dependent; there are

standing waves in the spacer layer, and reﬂection and transmission amplitudes at the

interfaces. The period is given by (2k

)

of the spacer layer in the direction perpendic-

ular to the layers, but in noble and transition metals there can be more than one value

of k

due to the topology of the Fermi surface.

The competition between these length scales and the ML period produces complex

magnetic patterns in superlattice ‘wedges’ which have been seen by SEMPA, as shown

in ﬁgure 6.24 (Unguris et al. 1991, 1994, Pierce et al.1994). These studies show that

observing the ﬁner periods is dependent on the quality of the interfaces, i.e. on crystal

growth processes. The use of wedged samples is a clever way of studying several

diﬀerent thicknesses in the same experiment, by using a microscope to pinpoint the

place where the multilayer is being sampled. It can even be done in 2D to probe two

thickness variables at once (Inomata et al. 1996); this is clearly very advantageous as a

6.3 Magnetism at surfaces and in thin ﬁlms 223

Figure 6.24. SEMPA magnetization images of the magnetic coupling of Fe layers in an

Fe/Cr/Fe sandwich grown on a wedge shaped Cr layer whose thickness increases from left to

right, on a single crystal Fe(001) substrate. There are two domains in the substrate with

magnetization to the left and right, giving the sharp horizontal demarcation in both panels of

size ⬃300 mm square viewed obliquely. In the lower panel, the rough Cr layer was grown at

room temperature giving long period reversals between antiferro- and ferro-magnetic

coupling, whereas in the upper panel, layer by layer growth at T5300 °C reveals additional

short, ⬃2 ML, period reversals (after Unguris et al. 1991, reproduced with permission).

Monolayers

110 20 30

0123 4

Thickness, (nm)

means of homing in on particular thicknesses which have desired properties. These

observations are not only very pretty science, but they hold out the prospect of device

applications, such as high density non-volatile memories, and sensitive read/write

devices. In particular, the giant magneto-resistance (GMR) and related multilayer

eﬀects are being actively researched, as described in section 8.3.

A ﬁeld like surface and thin ﬁlm magnetism, which builds on a long history of elec-

tric and magnetic properties, surface physics and growth processes, can be especially

diﬃcult for anyone trying to get started. In this situation, a reasonable initial strategy

is to skip all the preliminary work, and go straight to the latest (international) confer-

ence proceedings. One conference (from which some examples are taken) was the

Second International Symposium on Metallic Multilayers (MML’95) (Booth 1996); it

is especially useful to read the invited papers, since these have more perspective, and

typically survey several years of work.

Further reading for chapter 6

Ashcroft, N.W. & N.D. Mermin (1976) Solid State Physics (Saunders College) chap-

ters 8–11, 14, 15 and 33.

Craik, D. (1995) Magnetism: Principles and Applications (John Wiley).

Desjonquères, M.C. & D. Spanjaard (1996) Concepts in Surface Physics (Springer)

chapter 5.

Jiles, D. (1991) Magnetism and Magnetic Materials (Chapman and Hall).

Kittel, C. (1976) Introduction to Solid State Physics (6th Edn, John Wiley) chapters 14

and 15.

Pettifor, D.G. (1995) Bonding and Structure of Molecules and Solids (Oxford University

Press) chapters 2, 5, 7.

Stanley, H.E. (1971) Introduction to Phase Transitions and Critical Phenomena (Oxford

University Press).

Sutton, A.P. (1994) Electronic Structure of Materials (Oxford University Press) chap-

ters 7, 8 and 9.

Sutton, A.P. & R.W. Balluﬃ (1995) Interfaces in Crystalline Materials (Oxford

University Press

) chapter 3.

Woodruﬀ, D.P. & T.A. Delchar (1986, 1994) Modern Techniques of Surface Science

(Cambridge University Press) chapters 6 and 7.

Zangwill, A. (1988) Physics at Surfaces (Cambridge University Press) chapter 4.

Problems and projects for chapter 6

Problem 6.1. Why is (2k

)

the characteristic length for electrons at

surfaces?

A whole series of problems can be devised to get a feel for the size and relevance of this

characteristic length; the following questions require access to a standard solid state

textbook (e.g. Ashcroft & Mermin 1976, or Kittel 1976).

224 6 Electronic structure and emission processes

(a) For b.c.c Li, ﬁnd the value of the lattice parameter a, and hence of the nearest

neighbor distance. Assuming one electron per atom, evaluate the radius of the

Wigner-Seitz sphere r

, the magnitude of (2k

)

, and hence the periodicity Dz of

Friedel oscillations. Indicate these lengths (a and Dz) in relation to ﬁgure 6.6, and

identify the corresponding periods in the electron density oscillations.

(b) Consider the discussion based on ﬁgure 6.5, and use this to derive equation (6.4),

which is correct asymptotically for z well inside jellium. Note that we cannot use

(6.4) near z50. Use the positions of maxima and minima in Lang & Kohn’s cal-

culation for r

55 to estimate the value of the phase factor

, and plot this predic-

tion of n(z) for comparison with ﬁgure 6.2(a).

of superconductivity, noting how electrons of opposite momenta with energies

within "

of E

form the BCS ground state, via coupling by phonons of wave-

vector 2k

(d) Similarly note how scattering of electrons across the Fermi surface by phonons of

wavevector 2k

is thought to contribute to structural transitions (charge density or

static distortion waves) at metal surfaces, for example in the case of Mo(001).

(e) Estimate the wavelength of the ripples seen in the quantum corral of ﬁgure 6.4, and

relate this length to the Fermi surface of copper, incorporating the cylindrical

geometry. Consult the paper by Petersson et al. (1998) to see how such measures

can be made quantitative.

Problem 6.2. Derivation of the Richardson–Dushman equation from

free electron theory

Consider the Fermi–Dirac distribution in the form f(E )5(11exp(E2

)/k

where the chemical potential of the electron gas,

is the same as the Fermi energy E

(a) Use this form, together with the density of states shown in ﬁgure 6.5 and the

concept of the perpendicular energy, E

5("k

)

/2m, to derive the free electron form

of the Richardson–Dushman equation (6.7) when barrier transmission occurs for

all E

(b) By performing a 1D calculation, matching electron waves at the surface at z50,

investigate how the reﬂection coeﬃcient for electrons incident on the surface

changes as a function of E

when E

. Use this result to show how energy-depen-

dent barrier transmission aﬀects the formula derived in part (a).

Project 6.3. Barrier transmission, the Fowler–Nordheim equation and

models of STM operation

The Fowler–Nordheim equation can be derived from free electron theory in a similar

manner to problem 6.2, except that we now need the probability of transmission

through the ﬁnite barrier as a function of E

. This is typically given by the

Wentzel–Kramers–Brillouin (WKB) approximation, where the transmission coeﬃcient

Problems and projects for chapter 6 225

T⬃exp[2(2/")兰(V(z)2E

)

1/2

dz]. In this formula, the potential V(z) is as in ﬁgure 6.11,

and the limits of the z-integration are set by the perpendicular energy E

(a) Show that the result (6.8) arises in the limit of a triangular barrier at zero temper-

ature, and that the energy distribution will be broadened at elevated temperature

as in ﬁgure 6.12.

(b) Show that the same set of arguments applied to the operation of the STM, predict

qualitatively the observed exponential dependence of tip-sample voltage on tip

separation at constant tunneling current, in the limit of small voltages.

the particle current in terms of gradients of the (1D) wavefunction, and show using

ﬁrst order perturbation theory that the tunneling current I is given by

I5(e/") 兺

[ f(E

)2 f(E

)] |M

d(E

1V2E

where the matrix element M

is given by

5("/2m)

兰

dS·(

*),

where dS is the element of surface area lying in the barrier region. In these formulae,

f(E) is the Fermi function and d(E) the Dirac d-function.

Project 6.4 Interactions between d-electrons, magnetism and

structures in transition metals

Starting from the relevant sections of Sutton (1994) and/or Pettifor (1995), and other

literature cited in section 6.3.4, investigate models of b.c.c. and f.c.c. Fe, and neighbor-

ing elements in which you are interested.

(a) Describe how magnetism stabilizes the b.c.c. structure at low temperature, and ﬁnd

out about f.c.c. Fe at high temperatures, and the magnetism of neighboring ele-

ments in the 3d series.

(b) What features of the 4d and 5d series make the elements in corresponding columns

of the periodic table non-magnetic?

can create angular dependent interactions, and thus help to stabilize crystal struc-

tures which are not close-packed.

(d) Show that the angular dependent interactions can be formulated in terms of higher

moments (

with i.2) of the valence electron density of states, and that the b.c.c.

stability is largely attributable to

, while the h.c.p.–f.c.c. diﬀerence is aﬀected by

. From this viewpoint, investigate the contribution such eﬀects can make to the

surface energy of transition metals.

226 6 Electronic structure and emission processes