2006 Electronic structure and emission processes

are reliable experimental data. One important point in comparing experiment and theory is often only mentioned in passing: the electronic structure model typically refers not only to zero temperature, but indeed to a solid without zero-point vibrations. The experiment, on the other hand, is done at high temperature to avoid kinetic limitations, and exhibits a substantial temperature dependence due to entropic eVects, as shown in ®gure 1.8, and discussed in sections 1.1 and 1.2. Thus the point of comparison is often quite diYcult to establish.

Theorists can now calculate not only the surface energies of the diVerent low index {hkl} faces, but also the step energies in particular directions on these surfaces, enabling vicinal surfaces, the shape of 2D nuclei, and unstable facets to be explored, complementing the results for atomic diVusion on surfaces described in section 5.4 (Jacobsen et al. 1996, Ruggerone et al. 1997, Vitos et al. 1999). Current research is exploring the reliability of such models, and their usefulness in interpreting crystal growth experiments, which are strongly in¯uenced by kinetics. An example is the study of ¯uctuations in the shape of ML-deep pits on the Cu(111) surface by dynamic STM observations (Schlösser et al. 1999), leading to a value of the step energy of

0.22eV/atom along the close-packed directions; this value is close to that calculated by EMT models. These results and others have lead to the realization that entropic eVects associated with steps on metals are important, even at room temperature and below (Frenken & Stoltze 1999).

6.2Electron emission processes

Electron emission processes are central to many eVects at surfaces and interfaces, and to many techniques for examining the near-surface region. Most obviously we have emission from the solid into the vacuum, the electron overcoming the work function barrier in the process. This happens both in thermal emission, as described in section 6.2.1 below, and in photoemission and Auger electron spectroscopy, described in section 3.3. In a high electric ®eld, the barrier height can be substantially reduced, resulting in cold or thermally assisted ®eld emission, as discussed here in sections 6.2.2 and 6.2.3. Finally an incoming beam can result in secondary electron emission, as described in section 6.2.4, and hot electrons can penetrate internal barriers by ballistic emission, as described in connection with the microscopy of semiconductors in section

8.1.3.All of these eVects are connected with electron sources for various types of microscopy. Consequently, one can think of this section as providing a complement for those sections which deal with (electron) microscope techniques.

6.2.1Thermionic emission

The Richardson±Dushman equation, dating from 1923, describes the current density emitted by a heated ®lament, as

J(T)5AT 2exp(2 f/kT),

(6.7)

6.2 Electron emission processes	201

so that a plot of log(J/T2) versus 1/T yields a straight line whose negative slope gives the work function f. This value of f is referred to as the `Richardson' work function, since there is an intrinsic temperature dependence of the work function, whose value df/dT is of order 1024 to 1023 eV/K, with both positive and negative signs (Hölzl & Schulte 1979, table 4.1). When data is taken over a limited range of T, this temperature dependence will not show up on such a plot, but will modify the pre-exponential constant. This constant, A, can be measured in principle, but is complicated in practice by the need to know the emitting area independently, since what is usually measured is the emission current I rather than the current density, J.

The form of this equation can be derived readily from the free electron model, by considering the Fermi function, and integrating over all those electrons, moving towards the surface, whose `perpendicular energy' is enough to overcome the work function. In this calculation, ignoring re¯ection at the surface by low energy electrons, the value of A is 4pmk2e/h35120 A/cm2/K2. Where absolute values of current densities have been measured, values of this order of magnitude have been found. This derivation is quite suitable as an exercise (problem 6.2) but is also available explicitly in the literature (Modinos, 1984).

Thermionic emitters in the form of pointed wires or rods are used as electron sources in many electron optical devices such as oscilloscopes, TV and terminal displays, and both scanning and transmission varieties of electron microscopes. A good thermionic emitter has to have a combination of a low work function and a high operating temperature. However, as can be seen from tabulations such as table 6.2, higher melting point metals typically have higher work function. Thus the search is on for metals with a moderate work function which are suYciently strong, or creep-resistant, near to their sublimation temperature, which in many cases is a long way below the melting temperature. Note that an additional possibility is to take a high melting point material and to coat or impregnate it with a thin low work function layer. This is done for high current applications (TV and computer terminals) in sealed vacuum systems as described by Tuck (1983). For specialists, updates on current practice can be found in conference proceedings published in Applied Surface Science 111 (1997) and 146 (1999).

The standard material for comparison is a polycrystalline tungsten `hairpin' ®lament with f around 4.5 V, made of drawn wire a few tenths of a millimeter in diameter, bent, and situated in a triode structure, using a gate electrode called a Wehnelt. The competition is between the brightness of the source and its lifetime, which decreases markedly as the operating temperature is increased. For example, standard W-®laments used as electron microscope sources may have a lifetime of around 15 h when operated at 2800 K, but this extends to maybe 50 h when the operating temperature is dropped to 2700 K (OrloV, 1984).

The brightness, B, is typically the parameter which matters most in electron optical instruments, the current density per unit solid angle (J/V ); B is conserved if the energy of the beam is constant and geometrical optics applies. Tungsten ®laments have an eVective source diameter around 50 mm, an emission current around 50 mA, resulting in B,53 104 A/cm2/sterad at 100 kV electron energy; the brightness scales linearly with energy.

202 6 Electronic structure and emission processes

A material which has replaced tungsten ®laments very successfully for high brightness applications is LaB6, lanthanum hexaboride, which has f around 2.5 V, grown as small single crystal rods in [001] orientation with a square pointed end made of natural facets. When operated at around 1700 K, the lifetime is around 500 hours, and the brightness around 33 106 A/cm2/sterad at 100 kV, which is a major improvement, despite the increased cost and vacuum requirement. This increase in B mostly comes from a decrease in the emission diameter to around 5 mm; the actual current emitted is typically lower than the tungsten hairpin.

In instruments such as analytical SEM, TEM and STEM, we need to force as much current into a small spot as possible, in order to extract a high spatial resolution signal which has a suYcient signal to noise ratio (SNR). This means that there has been an intensive search for materials with better performance as thermionic emitters than LaB6. It is clear that the desired material must be very stable at high temperature, and moreover must have a stable surface. Borides, carbides and nitrides are natural candidates, which have strong (largely ionic) bonds and can be, or can be made, adequately conducting.

Futamoto et al. (1980, 1983) investigated mixed rare earth borides (LaxM12xB6), where several metals M were tried out. They found that these additions made the emission go down rather than up, but that after some use, they improved somewhat, but never exceeded the performance of pure LaB6. Using a microprobe AES apparatus, they investigated the surface composition of the tips, and found that the other metallic elements evaporate faster, leaving a surface layer, a few nm thick enriched with La; emission properties thus remained remarkably similar across the series. Swanson et al. (1981) changed the surface plane away from (001), measuring the lifetimes for a given emission current: no luck, (001) was the best!

Electron microscopy conferences typically have a few papers on carbides and nitrides; some of these have promising properties, but they have not proved to be stable enough to be used routinely. Thus LaB6(001) stays! The competition has come from ®eld emission as described below.

6.2.2Cold ®eld emission

A high electric ®eld near the emitter lowers the work function barrier; the barrier height can be suYciently reduced to increase emission substantially, as drawn for a ®eld F54 V/nm in ®gure 6.11. When the ®eld is this strong, the width of the barrier is of order 1nm, and electrons can escape even at low (room) temperature by tunneling. This is (cold) ®eld emission.

The ®eld F plays a similar role to the temperature in thermionic emission, and the governing equation is that by Fowler & Nordheim, derived in 1928 from free electron theory. The current density J, in the simplest case without the image force correction, is given by

J5AF2 exp (2 Bf3/2/F),

(6.8)

where the constants are

A56.23 106(m/f)1/2/(m1f) A/cm2 and B56.833107 V21/2/cm,

(6.9)

6.2 Electron emission processes	203

	10
	8	f
(eV)		EF
	6
Energy	4
	2
		metal	vacuum
	0	0	1	z (nm)	2

Figure 6.11. Electron energy diagram for ®eld emission as a function of distance z drawn for a ®eld F54 V/nm.

m being the Fermi level with respect to the bottom of the valence band, i.e. m5mÅ as expressed in section 6.1.1, with F measured in V/cm (Gomer 1961, 1994, Modinos 1984). Free electron theory is also able to calculate the electron energy distribution, as shown in ®gure 6.12(a), as a product of the Fermi±Dirac distribution for perpendicular energies, and the barrier transmission function, as discussed in problem 6.3. At low F, the distribution is sharp, but the intensity is weak, and vice versa.

Experimentally, cold ®eld emission requires a sharp tip, radius r, and UHV conditions. A voltage V0 is applied to a ®rst anode with a small hole in it, but most of the ®eld in generated very close to the tip, giving F5V0/kr, with k dependent on the tip shape, but typically k,5. With V053 kV and r5100 nm, a ®eld F53000/(531027)5 0.631010 V/m, or 6 V/nm is obtained. Field emission tips are usually operated with V0 from 1 to 5 kV, and radii around 100 nm. The linear dependence of F on the voltage V0 means that a Fowler±Nordheim plot of log(I/V02) versus 1/V0 gives a straight line, and is a good check on the ®eld emission mechanism.

A single crystal W wire emitter is used, in a low work function orientation. Both (310) and (111) orientations have been widely used in high performance SEM, TEM and STEM instruments; the ultimate single atom tip on W(111) has been demonstrated, and checked by comparison with FIM (Fink 1988). Improving the performance of CFE in an analytical STEM instrument used for electron energy loss spectroscopy (EELS) is described by Batson et al. (1992), with the measured ®eld emission spectrum shown in ®gure 6.12(b), which includes the Fermi tail at room temperature. Here the technical limits are at full stretch. The authors want to study the composition and electronic structure of materials, such as strained Ge/Si quantum wells, using the Si 2p energy loss edge at 100 eV, with nm spatial resolution (Batson & Morar 1993). They need a high current in order to get enough SNR in the spectrum, but if more current is drawn from the tip, both the energy and spatial resolution degrade. The trick is to achieve a modest improvement in energy resolution by deconvolution, using the Fermi tail of the emission (broadened by the spectrometer resolution as in ®gure 6.12(b)) as a sharp feature which enables the deconvolution to succeed.

204 6 Electronic structure and emission processes

Figure 6.12. Field emission energy distributions from tungsten tips (a) as a function of applied ®eld F, where the shaded area indicates those electrons having the energy component normal to the surface Ex.m ; the total emission increases strongly with F, the curves are scaled for easier comparison (after Gomer 1961); (b) energy distribution of a ®eld emission source with half-width 0.42 eV measured with an EELS spectrometer operating at 120 keV with 0.2 eV energy resolution, showing the Fermi tail due to emission at room temperature (after Batson et al. 1992, both diagrams reproduced with permission). Note that the energy scale is inverted right to left in these ®gures, corresponding to normal practice in these ®elds.

At the other end of the commercial spectrum, there have been widespread developments in ®eld emission for use in ¯at panel displays for TV and computer screens. This has now been demonstrated as prototype in industrial laboratories, so the real issues become manufacturability, reliability and of course cost relative to competitive schemes (Slusarczuk 1997). The systems have to work at relatively low voltage, which makes light output from the phosphors also an issue. Two very similar schemes are in competition: the ®rst is based on the Spindt cathode, a lithographically etched assembly based on micrometer-sized structures containing arrays of ®eld emission tips, as shown in ®gure 6.13(a); this technology is reviewed by Brodie & Spindt (1992). Other speci®c thin ®lm materials are in contention as the source, most notably diamond-like carbon (DLC) ®lms with speci®c nanometer scale structures, using the setup shown in ®gure 6.13(b). This is a very competitive area; recent progress is reviewed and possible mechanisms are discussed by Robertson & Milne

6.2 Electron emission processes	205

(a)	Glass screen		+1kV
	e-	Phosphor pixels
1 mm	Spacer		Vacuum
		Spindt tip
			Row electrode
0.5 m			Dielectric
			Cathode

Glass backplate

(b)	Glass screen		+1kV
	e-	Phosphor pixel
1 mm	Spacer
		Carbon film	Gate
0.5 m			Dielectric
			Source

Glass backplate

Figure 6.13. Field emission display geometries using (a) Spindt cathodes and (b) DLC ®lms (after Robertson 1997, reproduced with permission).

(1997, 1998) and Robertson (1997). One of the main issues is how to synthesize nanometer-scale structures reliably over large areas; we return to this topic in section 8.4.2.

Field emission requires a very good vacuum, and often, even in UHV, emission is not due to the clean surface. A typical ®eld emitter tip needs to be `¯ashed' to clean it, usually by passing a current through a loop on which it is mounted. After ¯ashing the emission current is high, but rather unstable; the current decays with time, and becomes more stable as it does so. This is due to contamination of the tip, either from the vacuum, or more often from diVusion of adsorbed surface species to the tip. Thus the nature of real ®eld emission tips during use, and indeed of real STM tips, is somewhat shrouded in mystery.

206 6 Electronic structure and emission processes

2.8		(110)				(112)		(111)			Na		K	Cs
2.4
2.0
						(a)							(b)
Current density , j
(arbitrary units)

Figure 6.14. Current jumps associated with the arrival of (a) individual tungsten atoms on W(hkl) planes as indicated; (b) the average eVect of alkali atoms on these planes of a W ®eld emission tip (after Todd & Rhodin 1974).

6.2.3Adsorption and diffusion: FES, FEM and thermal ®eld emitters

These adsorbate and diVusion eVects can be turned on their head, and put to good use scienti®cally, in ®eld emission spectroscopy (FES) and microscopy (FEM). This ®eld was pioneered by Gomer, whose 1961 book contains many of the important features of the methods: his 1990 review article should be consulted for details of methods and results on diVusion using the current ¯uctuation method. Three types of example are given here.

FEM images the tip itself, with a plate anode which may be coated with phosphor to detect the intensity of emission from diVerent crystal planes; in more recent experiments a channel plate would be used as an intermediate ampli®er. The main features are caused by the variation of emission with crystallographic orientation. It is on this basis that faces such as W(310) were subsequently chosen as ®eld emission tips for electron optical instruments.

In situ deposition of individual metal atoms on the tip has been shown to cause jumps in the emitted current, as illustrated in ®gure 6.14. Todd & Rhodin (1974) showed that they could distinguish 1, 2 and 3 W-atoms arriving on individual W (hkl) faces, and their subsequent desorption when the ®eld remained on. Then they investigated the response to adsorption of diVerent alkali adatoms (Na, K, Cs), which all increase emission markedly via lowering f.

A sophisticated technique was developed to measure diVusion coeYcients due to diVusion of these adatoms. A probe hole, or slot, is cut out of the screen, and the current through this hole is measured as a function of time. If no adatoms move in or

out of the `hole area', then the current stays constant; on the contrary, if they do, it changes. This can be expressed as a current±current correlation function KdI(0)´dI(t)L,

d 5 2 Å

which in normalized form is shown to decay with the delay time t ( I I I, the deviation from the average).

Rigorous results can be derived from the ¯uctuation-dissipation theorem, to show that the decay time for such correlations scales as the radius r of the probe hole squared, divided by the diVusion coeYcient. This makes sense qualitatively: the ¯uctu-