6.2 Electron emission processes	207

ations with this characteristic decay time t are caused by adatom diVusion in or out of the probe hole, and t ,r2/4D. So measuring the decay time yields the diVusion coeYcient. The subtlety can be increased further: using a slot in diVerent orientations allows one to explore diVusion anisotropy, since the measurement is dominated by diVusion parallel to the short axis of the slot of half-width x; now t,x2/2D. An example of O/W(110) is shown in ®gure 6.15; in this case the work function of the oxygen covered surface is greater than the clean metal, so the adsorbate reduces emission. Note that O-diVusion was found to be anisotropic in the ratio about 2:1, faster

Å

parallel to [110] (Tringides & Gomer, 1985).

Adsorption is useful for an electron source if the adsorbate increases emission. The stringent vacuum requirements of ®eld emission can be reduced somewhat if one both increases the operating temperature, and also uses an adsorbed layer which reduces the work function. This is thermal ®eld emission (TFE). Typical thin layers, which have long been used in TV and other sealed tube applications, are refractory (Ba, Sr) oxides, pasted onto, or indirectly heated by, the W ®lament; however, these coatings degrade badly if let up to atmosphere.

For high performance SEM applications, W(001) tips coated with Zr/O have been used as TFE sources; this cathode has f ,2.6±2.8 V (OrloV 1984, Swanson 1984). When operated at T,1800 K, the molecules adsorbed from the vacuum, or diVused from the support, do not remain on the surface long enough to cause the current to decay with time; the tips can also have a larger radius than for CFE at a given anode voltage V0, due to the lower f. In higher current applications, it is the angular current density (I/V ) which is more important than the brightness: the angular current density can be in excess of 1 mA/sterad. TFE is a successful compromise for many applications: a reasonable current which is stable in moderate (,1028 mbar) vacuum, an in®- nite lifetime barring accidents, requiring only routine preparation after bakeout: all of which is just as well, considering the initial cost!

Research into alternative CFE/TFE emitters also continues, and papers occasionally appear in the journals. An example detailing the TFE properties of LaB6 (Mogren & Reifenburger, 1991) emphasized that the emission process is rather more complex than the simplest Fowler±Nordheim treatment presented here, and needs to take into account the actual density of states in the material just above the Fermi level. This eVect is indicated by the diVerence between the full and dashed lines in ®gure 6.16; additionally, this paper showed that the current decay after ¯ashing may be due to the removal of emission from surface states. Note that the narrow energy distribution is a feature of CFE which gets lost to some extent in TFE. The highest performance analytical STEM and STEM-spectroscopy instruments use CFE primarily for this last reason. Once the vacuum in the source region has been improved to ,10210 mbar, the advantages of CFE can be realized in such applications.

6.2.4Secondary electron emission

When a sample is bombarded with charged particles, the strongest region of the electron energy spectrum is due to secondary electrons. We have discussed this extensively

208 6 Electronic structure and emission processes

Figure 6.15.(a) Current±current correlation functions for two orientations, diVusion
Å	Å

perpendicular and parallel to [110]; (b) diVusion coeYcients for O/W(110) parallel to [110] (from Tringides & Gomer 1985, reproduced with permission).

in chapter 3, for example in relation to ®gure 3.7 in section 3.3, and to the operation of the SEM. The secondary electron yield depends on many factors, and is generally higher for high atomic number targets, and at higher angles of incidence. There is a lot of information in this secondary electron `background', but, unlike Auger and other electron spectroscopies, it is not directly chemical or surface speci®c in general.

However, there are cases where the secondary electrons can be seen to convey more

6.2 Electron emission processes	209

Figure 6.16. Energy distributions during TFE from LaB6 as a function of temperature. Full lines are a calculation for a detailed ®eld emission model including the eVect of a peak in the density of states at energies just above EF, compared with the Fowler±Nordheim expression for f53.5 eV and F52.5 V/nm (Mogren & Reifenburger, 1991, reproduced with permission).

speci®c surface information. Under clean surface conditions, a change of surface reconstruction, or an adsorbed layer, will change the work function, the surface state occupation, and may also, in a semiconductor, change the extent of band bending in the surface region. A technique developed from this eVect is biased-secondary electron imaging (b-SEI), since biasing the sample negatively (anywhere from 210 to ,2500 V) causes the low energy electrons to escape the patch ®elds at the surface. This signal is much more sensitive for imaging than the corresponding Auger microscopy, as discussed in section 3.5. It has been shown that this technique is suYcient to detect subML deposits with good SNR, as illustrated in ®gure 3.21 for Ag/W(110). The case of Cs/W(110), studied earlier by Akhter & Venables (1981), showed that phase transitions could be observed and the activation energy for Cs surface diVusion measured, complementing original values by Taylor & Langmuir in the 1930s. The patch ®eld eVect is very strong in this case, extending for distances of more than 0.1 mm away from the Cs/W boundary at low bias ®elds.

Once again, the corresponding spectroscopy is useful in determining the origin of the contrast (Janssen et al. 1980, Futamoto et al. 1985, Harland et al. 1987). For 2 ML Ag deposited onto W(110) an increase in the secondary yield at the lowest electron energies E was observed, which is readily explained by a decrease in the work function. This form of surface microscopy has been exploited to measure diVusion of sub-ML

2106 Electronic structure and emission processes

and multilayer deposits over large distances (.10 mm) for both Ag/W(110) (Jones & Venables 1985) and Ag/Fe(110) (Persaud et al. 1994, Noro et al. 1996), as described in section 5.5.2. However, in the metal±semiconductor case of sub-ML Ag/Si(111) there is also a yield increase at higher E, a change in band-bending is involved, as discussed later in section 8.1. In the case of sub-ML Cs/Si(100), the detection sensitivity of b-SEI was pushed to below 1% of a ML, re¯ecting the large surface dipole moment caused by Cs adsorption at low coverage (Milne et al. 1994, 1995).

6.3Magnetism at surfaces and in thin ®lms

In this section, some concepts that are important in magnetism at and near surfaces are introduced, and examples of techniques sensitive to magnetic eVects are given. Magnetism has a long history, and it is not possible to cover this adequately in one section of one chapter. If you are going to study thin ®lm magnetism in detail, you will also need access to a modern textbook (e.g. Jiles 1991 or Craik 1995), plus review articles and chapters (e.g. Heinrich & Cochran 1993, Heinrich & Bland 1994). Here we ®rst consider aspects of symmetry and symmetry breaking in relation to phase transitions, and then introduce some surface techniques which are sensitive to magnetic eVects. Finally, we discuss some aspects which may become important in the next generation of thin ®lm devices based on magnetism.

6.3.1Symmetry, symmetry breaking and phase transitions

Arguments about symmetry, and symmetry breaking, ®gure strongly in the magnetic literature. Magnetism is often the prototype system in discussions of phase transitions. The Weiss theory of ferromagnetism published in 1908 (Ashcroft & Mermin, chapter 33, or Kittel, chapter 15) for example, is one of the earliest examples of a mean ®eld theory applied to a (second order) phase transition. Onsager's exact solution of the 2D Ising model in 1944 was couched in terms of a magnetic transition. In the book series edited by Domb, Phase Transitions and Critical Phenomena, many of the articles and theorems stem from magnetic interests. There is a discussion of Onsager's papers in volume 1 of this series by Temperly (1972).

One such theorem is that due to Mermin & Wagner, which shows that magnetic long range order (in the absence of anisotropy) is impossible for a Heisenberg spin system in 2D or 1D, whereas it is clearly possible in 3D systems. The argument goes as follows. In an ordered lattice of magnetic spins which can have any orientation, such as exists in a ferroor antiferro-magnet, the excitations are spin waves at low temperature. In these waves, the spins on neighboring lattice sites twist with respect to each other, giving rise to a magnetic energy v(k) proportional to k2; these quantized excitations are called magnons. Then we count the number of magnons, using Bose±Einstein statistics and obtain

nm5eg(k) Kn(k)Ldk,

(6.10)

6.3 Magnetism at surfaces and in thin ®lms	211

integrating from zero to in®nity, with the density of states g(k)5dn/dk5k/2p appropriate for v5Ak2. The number of magnons, nm, then is given by substituting for Kn(k)L, and in the high temperature limit, this gives

n	m	5e (k k T/hAk2)dk,edk/k,[ln (k)],	(6.11)
		B

which diverges at the lower limit.1 This means that theoretically we cannot have long range order in 2D, because long wavelength (low k) excitations are possible in these systems with negligible energy. The same Mermin±Wagner theorem applies to positional order in 2D, due to the divergence, also logarithmic, of long range positional correlations; thus the corresponding theorem has also been invoked in theoretical studies of adsorption, as discussed in chapter 4.

This theorem has been shown to be of interest in some situations, but usually the length scales are too long to be of practical interest, and what happens ®rst has to do with symmetry breaking. For instance, you can't make a free standing monolayer, or a truly 2D magnetic system. Once we have a monolayer or a magnetic system on a substrate, we have broken the symmetry. Logarithmic divergences are very easy to break; examples are the ®nite energies in the core of dislocations due to atomic structure, or the inductance of a ®nite, versus an in®nitesimal diameter, wire. The breakdown of the Mermin±Wagner theorem for such practical reasons is another case.

6.3.2Anisotropic interactions in 3D and `2D' magnets

In 3D magnetic systems we have many examples of symmetry breaking. The basic magnetic interaction is the exchange interaction related to the spins on a lattice as

E52 oJSi´Sj ,

(6.12)

where the summation is typically limited to neighboring sites only; this is referred to as the Heisenberg Hamiltonian between spins Si and Sj with exchange coupling constant J. In the presence of an external magnetic ®eld H, a unique axis is imposed (orientational symmetry breaking), because EH5 2 omi´H. The combination of these two terms for Si561/2 is the Ising model which Onsager solved exactly to ®nd the magnetization M as a function of T. The approach to the Curie temperature, above which the system is paramagnetic, goes like

M,(T2 Tc )1/8,

(6.13)

rather than the mean ®eld exponent of 1/2. These critical exponents are characteristic of the models as Tc is approached, and the dimensionality (two, three or higher dimensions), but are not dependent on the details of the interactions. This is the basis of interest in universality classes, within which the critical exponents are the same: impress your friends with `¼ as in the 2D XY model, we can see that ¼'! An introduction to these critical exponents is given by Stanley (1971), and the details for 2D systems are

1Note that we use kB for Boltzmann's constant in this section and the next, to distinguish it from the wavenumber k.

212 6 Electronic structure and emission processes

described by Schick (1981), Roelofs (1996) and others, as discussed in section 4.5.1. Magnetism has many other symmetry breaking interactions, and we can't realistically discuss them all here. But one very important case is the magneto-crystalline anisotropy (MCA) energy EK, which is due to the anisotropic charge distribution in the crystal ®eld, and orients the magnetic moments along speci®c crystalline axes. The form of this energy depends on the crystal symmetry, the most often encountered being uniaxial anisotropy, which, for example, makes the c-axis in h.c.p. cobalt the easy axis of magnetization; the leading term has the form Ksin2u. In a cubic crystal, such as b.c.c. Fe, we have cubic anisotropy, which is expressed in terms of the direction cosines

a1,a2,a3 to the three cube axes as

E	K	5K	(a	2a	21a	2a	21a	2a	2)1K	a	2a	2a 21. . .		(6.14)
		1	1	2	2	3	3	1	2	1	2	3
We can see that there won't be a second order term because a													21a	21a 251. The easy
												1	2	3
axis for Fe is along the K100L directions, and this corresponds to K1.0 and K1.2K2/9;
for Ni the easy axis is K111L (Craik 1995, section 1.11). A practical example is the use of

Fe±4% Si for transformer cores. Why? Not because Si does anything wonderful for the magnetization of Fe, but because it gives polycrystalline Fe a {100} texture, making it easy to magnetize in the plane of transformer laminations, leading to small energy losses when used with alternating currents.

There are several other anisotropic terms, which can be important in particular circumstances. A very important term is the demagnetizing energy, which is a macroscopic eVect caused by the shape of the sample, and derives from the magnetic self energy, ES. This self energy can be expressed as either the interaction of the demagnetization ®eld inside the sample with the magnetization, or equivalently, the integral of the energy density of the stray ®eld over all space. If, for example, the magnetization is perpendicular to a thin ®lm, there is a large energy due to the dipolar ®eld outside the ®lm; but if the magnetization is in the plane of the ®lm, this eVect is minimized. In real ®lms, this causes the formation of domains. These domains can be seen in transmission, even in quite small samples (not necessarily single crystals), by Lorentz microscopy (coherent Fresnel, Foucault, and diVerential phase contrast imaging), as described in several papers from Chapman's group in Glasgow (e.g. Chapman et al. 1994, McVitie et al. 1995, Johnston et al. 1996, Chapman & Kirk 1997). They can also be seen using electron holography as developed initially by the Möllenstedt school in Germany and Tonomura's group in Japan, and further developed and reviewed by Mankos et al. (1996). In uniaxial crystals such as h.c.p. cobalt, there will still be a small ®eld outside the ®lm, connecting two oppositely oriented domains. In cubic crystals, even this can be avoided, by the formation of small closure domains at the ends of the ®lm. The price paid for these domains is the energy of the interfaces between oppositely magnetized regions: these are known as Bloch or Néel walls, depending on the details of how the spins rotate from one domain to the other (Kittel 1976).

Another term relevant to thin ®lms is magnetoelastic anisotropy, or magnetostriction. In this eVect, the crystal parameters change because of the magnetism; this also implies that structure and symmetry changes will in¯uence the magnetism, as