Marder M.P. Condensed Matter Physics

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Experimental Observation and Creation of Surfaces 83

Figure

4.7.

The Ewald construction for surfaces is shown, emphasizing that points in the

reciprocal lattice become rods, and that the condition for scattering is that these rods be

intersected by a spherical shell.

solid surface. Call the scattering surface the x-y plane. The condition for obtaining

a strong scattering peak therefore is given by Eq. (3.17) in the form

■

, R

, 0) = 2πΙ.

ι must be an

integer, but will vary depending (4.5)

on the choice of R.

The difference between scattering from a bulk solid and scattering from a solid

surface results from the fact that while the lattice vectors R in Eq. (4.5) lie in a

plane, the experiment takes place in a three-dimensional world, where radiation

is free to exit in a set of directions indexed by angles

and φ. Then Eq. (4.5) is

satisfied by any a of the form

and K

are components of a reciprocal lattice

q — [K

, Ky, q

). vector Kchosen so that

+ KyR

2-πΙ.

One (4.6)

can take q

to be anything at all, because the z

component of R is zero.

Constructing the Ewald sphere corresponding to Eq. (4.6), one sees that the

geometrical condition to be satisfied is that the surface of a sphere intersect a col-

lection of

rods,

as in Figure

4.7.

This condition is guaranteed to be satisfied at some

scattering angles for any choice of incoming wave vector

and for any orientation

of the sample, in contrast with the bulk case described in Section 3.3. There is no

need to rotate the sample, scan through incoming wave vectors, or use powdered

samples in order to obtain scattering peaks.

Despite LEED's crucial historical role in demonstrating the wave nature of the

electron, it is not easy to determine detailed features of surfaces based upon LEED

measurements. Because electrons interact very strongly with solids, multiple scat-

tering is impossible to avoid. Quantitative comparison of theory and experiment

requires one to make detailed guesses about surface structure and then carry out

lengthy quantum-mechanical calculations for the scattering of electrons from the

surfaces, based upon the guesses.

Chapter 4. Surfaces and Interfaces

4.3.2 Reflection High-Energy Electron Diffraction (RHEED)

In this technique, electrons of energy on the order of 100 keV are reflected off a

surface at a grazing angle. The wave vectors associated with such energies are

on the order of 200 Â

-1

and are much larger than the spacing between reciprocal

lattice vectors, leading to streaky scattering patterns where the Ewald sphere in-

tersects reciprocal lattice rods. Slight rotations of the sample are needed to obtain

strong signals in desired directions.

4.3.3 Molecular Beam Epitaxy (MBE)

With the ability to measure the properties of surfaces with precision has come the

ability to control their composition with comparable precision. The technique of

molecular beam epitaxy (MBE) allows building up a solid one atomic layer at a

time with precise control over the composition of each layer and the ability to alter

composition at will.

Figure

4.8.

Schematic view of

MBE.

Ballistic beams of

atoms

escaping from heated cham-

bers impinge upon a substrate and form atomically perfect layers, monitored by RHEED.

A caricature of a molecular beam epitaxy facility appears in Figure 4.8. An

atomically flat crystalline surface is placed within an ultra-high vacuum chamber,

meaning it is at a pressure of around 10~

torr. Various Knudsen cells, each

containing a different element to be added to the surface, are aimed at the substrate.

Atoms leave the Knudsen cell simply by being heated until they evaporate, and

then flying ballistically to the substrate, if the control shutter is open. To obtain

atomic control of

the

deposition process, reflection high-energy electron diffraction

is carried out in

situ.

The strength of the specularly reflected electron beam depends

upon the state of the surface, as shown in Figure 4.9. If a new surface layer is only

half-formed, the electron beam is scattered diffusely, while if

is nearly perfect and

smooth, the specular reflection grows stronger. The result is a periodic oscillation

of the strength of the electron beam, in which each period represents the deposition

of precisely one atomic layer.

Experimental Observation and Creation of Surfaces

10 15

Time t (s)

Figure 4.9. The intensity of RHEED scattering from a growing surface depends upon whe-

ther the surface is complete, or partially formed. These data show oscillations in RHEED

intensity during the growth of a (001) GaAs surface, monitoring the [210] reflection as

electrons reflect off the surface at an angle of 0.91°. [Source: Braun et al. (1998), p.

4937.]

To a great extent, layer growth in molecular beam epitaxy is dependent upon

the physics by which atoms attach to surfaces, still more of an empirical than a

theoretical science. Attempting to deposit silver on GaAs, for example, by this

technique produces blobs and islands rather than uniform atomically flat films.

Conversely, there are fortunate cases in which deposition layer by layer is possible.

One important example is the deposition of GaAlAs on GaAs, where the aluminum

may be substituted for the gallium at any desired concentration and where layers of

varied composition may be grown with exceptional precision, one upon the other.

The interfaces are not absolutely atomically perfect, but errors in which a gallium

or aluminum atom finds itself out of place are typically confined to the width of a

single atomic spacing.

4.3.4 Field Ion Microscopy (FIM)

A specimen is formed in the shape of a sharp tip. This tip is given a large positive

electrical potential, and because sharp tips create singular electrical fields, a field

on the order of 10

V/cm is generated near the surface. The tip is placed in a neutral

gas such as helium or neon at moderately low pressure, and the large fields cause

occasional ionization of the gas, thereby capturing an electron and repelling the ion.

The ions are captured on a screen. The capture process happens preferentially near

protruding atoms on the tip, so that the technique can actually image individual

atoms on the tip surface. Disadvantages include the fact that one must be able to

build the microscope tip out of the material one wishes to examine, and one can

only look at materials that can stand the high fields. This technique has largely

been supplanted by tunneling microscopy. It is reviewed by Tsong (1993).

Chapter 4. Surfaces and Interfaces

4.3.5 Scanning Tunneling Microscopy (STM)

The idea that quantum mechanics should permit particles to pass through barriers

occurred to J. R. Oppenheimer during a drive from the eastern United States to a

position as research fellow at Cal. Tech. in 1927. The symbols

-C

i r^ exp See Davis (1968), p. 23. The tunneling (4.7)

E formula in more familiar notation might

read φ ~ exp[—x

had just been scrawled on the windshield of his car when he ran off the road into

a county courthouse, a rather unsuccessful first attempt to put the formula into

practice.

Tunneling spectroscopy became a powerful tool for investigation of metals

and superconductors over the next four decades. As discussed by Wolf (1975), it

was typically performed between flat millimeter-scale samples with separations be-

tween the plates much larger than an angstrom. The macroscopic size of

the

sample

offset the exponentially small current that could flow between the two plates, and

the large size also produced an average over the thermal vibrations of the plates.

Gradual improvement of numerous branches of technology made it possible

by the 1980s to employ Eq. (4.7) for imaging of surfaces on the atomic scale, in

a device invented by Binnig and Rohrer (1987) and called the scanning tunneling

microscope or STM. The essential idea is to bring an atomically sharp metallic tip

near to a conducting surface. Because the current flowing from surface to tip varies

exponentially with the distance between them, it can be used as an exceptionally

sensitive indicator of surface height, and because the tip is very sharp, it is sensitive

to small-scale variations in the horizontal direction as well.

Detailed calculations of

the

precise rate at which tunneling takes place between

surface and tip are not needed to appreciate the device, but it is worth asking how

all the quantities appearing in Eq. (4.7) are to be defined in order to apply to the

case of the tunneling microscope. A schematic picture of the device appears in

Figure 4.10. First, consider the case in which the voltage difference V between tip

and sample is zero. The equilibrium chemical potentials of different conductors

are not in general the same, which means that in order for tip and sample to be at

the same voltage, a small transient current has had to flow between them soon after

they were connected together, resulting in a situation where the minimum energy

needed to raise an electron from tip or surface to the energy of the vacuum is

φ =

—

(n\ -\- /Ì2). Thus φ is the average of the work functions (4.8)

2 of the tip and surface; work functions are dis-

cussed further in Sections 19.2.1 and

23.6.1.

The tunneling current Eq. (4.7) follows from the Wentzel-Kramers-Brillouin

(WKB) approximation, which gives as an approximate solution of Schrödinger's

equation

^(x)~exp [(i/h) I dx'^2m(E-U(x'))

See Landau and Lifshitz (4 9)

(1977),

p. 164, or Schiff

(1968),

p. 268.

Experimental Observation and Creation of Surfaces

Figure 4.10. Schematic view of scanning tunneling microscope. Electrons tunnel through

the vacuum from a metal surface to a metal tip.

An electron wave function which begins on the left side of Figure 4.10 and travels

distance s through the vacuum will drop in amplitude by a factor of

exp

2ηιφ/Η

Because U

—

£ = φ in the forbidden region.

The calculation is performed in the limit of

vanishing voltage V, so one need not consider

the slight drop by amount V in the potential

U(x).

(4.10)

by the time it reaches the other

side.

The current must be proportional to the applied

voltage V, to the square of the wave function, to the initial density of electrons n\

ready to spring out of the sample, and to the final density of locations in the tip «f

to which they can travel. Thus one estimates the current J to be

JocninfV exp[—2sy 2m0/ft

]

exp

■\M[s/k]J[<f>/éV]

(4.11)

(4.12)

Achieving atomic-scale resolution depends upon careful experimental control

of a number of different elements. First, external noise and vibrations must be

eliminated, because motion of the tip by even 0.5 angstrom relative to the sam-

ple is fatal. The seeming impossibility of controlling vibrations is one explanation

for why the STM took so long to discover. In the first STM, vibration was con-

trolled by levitating the entire apparatus on permanent magnets over a bowl of

superconducting lead. Nested collections of mechanical springs were soon found

to be equally effective, but much cheaper. A schematic view of an STM from the

point of view of vibration isolation appears in Figure

4.11.

The central idea is that

large soft springs connect the outside world to the machine through progressively

88 Chapter 4. Surfaces and Interfaces

smaller and stiffer ones, terminating in the tip and sample housing that needs to be

as small and stiff as possible.

The next experimental challenge is to bring the tip reliably to within

angstrom

of the sample and then maintain it either at constant current or constant voltage

during scans across the surface. So that the sample can be removed and inserted

under the tip, there needs initially to be clearance of around

mm. This difficulty

also can be surmounted with appropriate use of springs, as indicated in Figure

4.11.

Gross vertical motions of the sample are controlled by a large weak spring,

which also helps ensure that external vibrations do not reach the sample. The

large spring is in series with smaller, much stiffer springs, so pushing up and down

upon it results in comparatively small motions of the sample. Fine motions of the

tip,

which must be controlled well within an angstrom but range over a micron,

are controlled by mounting the tip upon a number of piezoelectric elements. The

conceptually simplest geometry of the piezoelectric elements controls tip motion

with two horizontal elements (whose contraction moves the tip about in the plane)

and one vertical element (which controls its height). A widely used piezoelectric

material is PbZrTi (PZT), which contracts by around 4 parts in IO

per volt applied

across it, up to 1000 V. Therefore for 2-cm-long piezoelectric strips sub-angstrom

resolution is obtained by controlling voltages to within a percentage of a volt, but

at the same time excursions on the order of

μπ\ are obtainable. The piezoelectric

strips in such a geometry are too long and floppy to provide optimal control, and

numerous other designs are employed in practice.

Figure

4.11.

Schematic view of mechanical operation of

STM.

Finally, there remains the challenge of creating a tip capable of achieving

atomic resolution. Experimental groups have their own secret magical recipes,

and no single method dominates because no method has yet been found which is

particularly reproducible or reliable. The best tips have literally a single atom pro-

truding from their ends; they are hard to make and last only so long as mechanical

stresses during measurement do not make the atom jump about.

It would be difficult to operate a scanning tunneling microscope without mod-

ern electronics. On the one hand, the piezoelectric elements must be controlled

Experimental Observation and Creation of Surfaces 89

continuously and in precise sequence in order to scan the tip across the surface. In

addition, all the data for tip height as a function of

y location have to be recorded

and assembled later for interpretation. Computer control and data acquisition are

therefore almost indispensable.

Figure 4.12. Stereo pair showing the locations of atoms in the Si 7x 7 reconstruction of

the (111) surface. The first three layers of atoms are shown, with different colors assigned

to atoms in each layer. Top layer, ; middle layer, ; bottom layer, . The structure

occupies 7x7 unit cells, as may be seen by counting the number of cells separating the

centers of the large voids at each corner.

Figure 4.13. Atoms on the (111) surface of silicon form the pattern shown here, which

involves a 7x 7 atom surface unit cell. The image is produced by scanning tunneling

microscopy. [Source: Wolkow and Avouris (1988), p. 1050.]

The STM only became a standard tool in laboratories around the world after

proving itself by resolving a long-standing controversy that had resisted all other

types of surface analysis, the structure of the Si

( 111 )

surface. Although the crystal

surface of silicon used in the electronics industry is (100), the (111) surface is easier

to prepare for research purposes, because unlike (100) it can be formed by cleavage

and made atomically flat with relatively little effort. Prior to the STM, locations of

the atoms on the surface were not possible to determine, and the number of differ-

ent proposals was approximately equal to the number of experimental and theoret-

ical papers published on the subject. The STM resolved the question decisively;

90 Chapter 4. Surfaces and Interfaces

Figure 4.14. (A) Scanning electron micrograph of piezoresistive cantilever and tip used in

atomic force microscopy. [Courtesy of M. Tortonese, ThermoMicroscopes and Stanford

University.](B) Atomic-scale image of the surface of graphite. The image is a map of the

force between the microscope tip and the sample at a height of 12 pm, with forces ranging

from —30 pN (white) to 30 pN (black). Hexagons show crystalline lattice. However, the

experimental image has three-fold rather than six-fold symmetry because the AFM tip is

not symmetrical and interacts differently with the crystalline lattice in different directions.

pm over the sample, showing full symmetry of graphite lattice. [(B) and (C) courtesy of

U. Schwartz, Yale University. For further details, see Albers et al. (2009).]

the atoms are located in a 7x7 structure illustrated in Figure 4.12. Within a few

years,

images such as in Figure 4.13 became commonplace. Having shown it could

answer questions beyond the capability of any other instrument, very rapid growth

both in research use and commercial development began immediately thereafter.

Tempting though it may be to regard the white and black spots of an STM

image as atoms, one needs always to think back to Eq. (4.11) in viewing an STM

image. What one sees is a contrast between electron densities at various points

on the surface. Therefore, an atom on the surface which keeps a close grip on its

electrons will be invisible, and some atoms may come in and out of view as one

varies the bias voltage between tip and sample. In metals, the electron density is

frequently sufficiently uniform that an STM image is featureless; an insulator has

no free electrons to donate and is invisible, which leaves semiconductor surface

physics as the domain in which the method is most profitable.

Problems

4.3.6 Atomic Force Microscopy (AFM)

The atomic force microscope (AFM) was developed by Binnig et al. (1986) as a

modification of the STM, to permit investigation of insulators. A tip is lowered

toward a sample on a lever until it virtually touches, and either the force on the

lever is maintained constant as it scans over the sample, or else deflection of the

lever is measured by mounting a mirror on it and bouncing a laser beam off the

top.

The difficulties of controlling motion at the angstrom scale find the same

solutions as with STM. An image of the surface of graphite is shown in Figure

4.14. Interpretation of the image requires some caution. It accurately reflects the

force between the surface of graphite and the AFM tip, but because the tip is not

completely symmetrical, the image has three-fold rather than six-fold symmetry.

Bustamante and Keller (1995) show that the AFM technique can be modified to

function even in liquids at room temperature, although in such noisy environments

atomic-scale resolution is no longer possible.

4.3.7 High Resolution Electron Microscopy (HREM)

High resolution electron microscopy (HREM), described by Spence (1988), allows

the study of solids with atomic resolution, and it can only be used on extremely thin

sections of

samples.

An example of

the

technique has been displayed in Figure 4.5.

The image looks like a photograph of atoms, but it must be approached with cau-

tion. It actually reflects a complicated smearing out of the actual electron density,

and any attempt to extract quantitative density information requires an attempt to

deal with the smearing quantitatively.

Problems

Superlattice: Show that when two crystal surfaces meet in a commensurate

fashion, so that there is a superlattice along the interface, the areas of the

primitive cells for the two surface lattices must be rational multiples of one

another.

Interplanar spacing: Consider a bcc lattice, and find how e in Eq. (4.3)

depends upon /,

and k for the planes (110), (111), (114), and (137).

Bond breaking:

(a) Consider an fee lattice in which the energy of a surface is given by the num-

ber of broken nearest-neighbor bonds. With reference to Figure 4.4, find the

energy needed per area to cut along the (001) plane.

(b) Consider the diamond structure, shown in Figure 2.6. How many bonds per

area need to be cut to expose the (001) plane?

How many bonds per area need to be cut to expose a (111) plane? Which of

(001) and (111) should be expected to cleave most easily?

Chapter 4. Surfaces and Interfaces

Faceting:

(a) Consider a two-dimensional square crystal. Suppose the energy needed to

form a surface of Miller index (ij) is simply the number of nearest-neighbor

bonds broken in forming the surface. Show that the surface energy

per unit

length is

where

is the energy per bond and a is the lattice spacing.

(b) Draw a polar plot of Eq. (4.13); that is, draw a curve in polar coordinates

where r(9) is given by the free energy

appropriate for a surface cut at angle

Θ.

two-dimensional crystal whose surface free energy per

unit length is α(θ) for a surface at orientation

Θ,

or equivalently

a(y'),

where

y(x) is a function describing the height of the crystal in Cartesian coordinates,

and

dy/dx. Argue that the equilibrium shape of such a crystal is given by

minimizing the functional

f dxy/l+y^a^-X f

dxy(x),

(4.14)

where λ is a Lagrange multiplier.

(d) Dehne

f{y

) = \J^+y'My')·

(4.15)

Using the calculus of variations, show that

dx dy'

and that after obtaining a first integral and then multiplying by y", Eq. (4.16)

can be integrated to give

y/l+y

a(y)

\(y'x-y)

(4.17)

The arbitrary constants obtained during the integrations can be ignored. Why?

(e) Rewrite Eq. (4.17) in terms of the unit normal h to the surface described by

y{x).

Equation (4.17) describes a geometrical construction in which for every

normal direction n one draws a line perpendicular to « at a distance r(n) from

the origin such that the line hits the polar plot of a(/) from part

(b).

Then y(x)

is the envelope of all these lines. Use this construction to draw a sketch of the

equilibrium crystal resulting from the surface energy of Eq. (4.13). For more

information on this topic, see Rottman and Wortis (1984) and Tosi (1964)

92.