Marder M.P. Condensed Matter Physics

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References 73

6. Mott-Bethe relation: Demonstrate Eq. (3.47). Use the fact that the electron

density satisfies Poisson's equation, V

V = Απηβ, where V is the electrostatic

potential.

Powder pattern intensities : Consider an array of identical scatterers from

which is made a powdered crystal suitable for the Debye scattering method.

The intensity of a measured line appearing at Bragg angle

\f(e)\%M

. *f (3.70)

sin

2Θ

Here M

is the multiplicity of the relevant &-space point. Sketch the derivation

of Eq. 3.70 by showing that the ^-dependent part is the product of two types

of terms:

• Form factors / and F such as those in Eqs. (3.47) and (3.31).

• Geometrical factors resulting from averaging over crystallite directions.

Formally, the geometrical factors are obtained by observing that the Bragg

reflection due to a reciprocal lattice point K is

Ô(q

-K

)ô(q

-K

)ô(q

-K

). (3.71)

In a powder diffraction experiment, the probability that a reciprocal lattice

vector of magnitude K should point along angles Θ and Φ is sin(6)/47r. Av-

eraging over Θ and Φ, and writing the resulting single delta function as a

function of the Bragg angle

will give the desired result. Use the mathemat-

ical result that 5(f(x)

—

y) = Σ,· 5(x

—

Xi)/\f(xì)\, where x,· are the roots of

y-f(x).

8. Zinc powder pattern: Consider the hep metal zinc in polycrystalline or pow-

dered form, and consider conducting an X-ray scattering experiment in which

radiation of 1 Â wavelength is scattered off the sample. Write a computer

program calculating the scattering intensity as a function of Bragg angle that

should be observed, treating zinc atoms as point scatterers. Use experimental

numbers for the lattice constants of

zinc,

and use Eq. (3.70) for the intensities

of the lines.

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Surfaces and Interfaces

4.1 Introduction

Surface physics might seem to be unimportant because the number of atoms sitting

at a surface relative to those lying within the bulk is negligible for macroscopic bod-

ies.

However, this judgment is incorrect. Surfaces provide the doorways through

which material inevitably passes when entering or leaving a solid. The state of

a material's surface determines both its strength and its resistance to chemical at-

tack. The manufacture of integrated circuits consists in the deposition of complex

sequences of surface layers to build up the desired pathways for electrons. Partly

for this last reason there is a continual rapid development of experimental meth-

ods for describing and characterizing surfaces. Characterization has in fact outrun

the immediate needs of technology, and it allows even surfaces without crystalline

order to be pictured in atomic detail.

4.2 Geometry of Interfaces

The simplest deviation from perfect crystalline behavior occurs when a crystal

comes to an end, abutting either another crystal or empty space. The first case

is a grain boundary, and the second one is a surface. As discussed by Wolf (1992),

these two types of interface can be described in a common way from the geometri-

cal point of view.

One can initially simplify the problem of interfaces by assuming that when a

perfect crystal is sliced along a plane, none of the remaining atoms moves from its

original location. This assumption is not generally true, but is often a reasonable

first approximation. The way in which it fails has to do with the fact that the crys-

tal deforms as it approaches a boundary. The deformation may take a mild form,

such as a slight increase or decrease of the volume per atom. It may involve the

formation of an entirely new crystal structure right at the interface, called surface

reconstruction. Or it may involve the formation of an entirely new phase; for ex-

ample, a thin liquid layer on an otherwise solid body. The region of deformation

relative to the perfect crystal is called the selvage.

Ignoring surface reconstructions and other deformations, a plane interface be-

tween two crystals may be described by ten variables, describing the ways that two

perfect planar interfaces can be brought together. Three of the variables describe

the precise microscopic three-dimensional positioning of one crystal with respect

to another, as shown in Figure 4.1(A): exactly how far the two crystals are from

one another, and how far each has been slid along the other in the two in-plane

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

78 Chapter 4. Surfaces and Interfaces

directions. The seven remaining variables have macroscopic significance. Each of

the two crystals has been sliced along a plane. The conventional way to describe

planes is with Miller indices, using three numbers, but of course to describe a plane

in three dimensions needs only two numbers, the angles along which its unit nor-

mal points. However, one needs to describe the precise point along the unit normal

where the cut is made. So specifying the two crystal planes takes a total of six

numbers. A tenth and final variable

is needed to specify the relative orientations

of the two crystals as they rotate about in the plane where they meet, shown in

Figure 4.1(B).

Figure

4.1.

(A) Three microscopic variables describe the precise positioning of

two

ideal

interfaces with respect to one another. (B) Angle specifying the relative rotations of two

crystal interfaces.

For a free surface, only two variables need to be described; those giving the

plane along which it cuts the crystal.

4.2.1 Coherent and Commensurate Interfaces

When two crystal surfaces meet, it is valuable to know if they will mesh well with

one another. This question is particularly interesting if one is building a layered

structure consisting of alternating crystal types and wants to know if they will bind

together. If two crystal surfaces are placed together and the atoms are in perfect

registry with one another, then one has a coherent interface, and the process of

growing such an interface is called epitaxy. There is a related concept, that of a

commensurate interface which is slightly more general. Two interfaces are com-

mensurate if there is some larger two-dimensional lattice on which their atoms

coincide, as shown in Figure 4.2. The general condition for two lattices to be

commensurate is slightly complicated because one must include the possibility of

rotating them at some peculiar angle, as in Figure 4.2. Let a\,

a-i

and b\, &2 be

two sets of primitive vectors for the atoms on the two interfaces. Then the two are

commensurate if there is an infinite set of integers n\,ri2 and m\,

m-i,

together with

Geometry of Interfaces 79

an angle

such that

- , - / cos

sin ö\ ,

r τ Λ

n\a

=\ . ,, „ \ [m\b\ + m

). (4.1)

—

sin #cos 0 /

The set of points on which the two lattices agree is called the superlattice. One

can always define b\ and b

so that the rotation angle in Eq. (4.1) is zero, although

it is included for generality.

Figure 4.2. Two square lattices whose lattice constants differ by VS/2 are commensurate

but incoherent, because not all the atoms lie on top of

one

another.

4.2.2 Stacking Period and Interplanar Spacing

Another interesting geometrical quantity is the stacking

period,

which is defined by

counting the number of lattice planes one encounters, while heading directly away

from the interface, before hitting a new lattice plane that has all its atoms sitting

directly above those on the interface, along the normal. There are some fairly

general results for crystals of cubic symmetry. If the Miller index of a surface is

(ijk),

then the stacking period in a cubic crystal is

P = ö(i

+ k

), (4-2)

where δ equals 1 or 2, depending on the particular lattice being considered and

whether /, j, and k are odd or even, as shown in Figure 4.3.

Probably more important is the interplanar spacing

d = ea/yji

+ k

, (4.3)

where e equals 1/2 or 1 depending upon circumstances. In fee crystals, for ex-

ample, e is 1 if i, j, and k are all odd, but otherwise it equals 1/2. Because the

Chapter 4. Surfaces and Interfaces

fee

( 100)

surface, P = 2 fee

( 111 )

surface, P = 3

Figure 4.3. The stacking periods of the (100) and (111) planes for the fee lattice. In the

first

case,

δ of

Eq.

(4.2) is 2 and e of Eq. (4.3) is 1/2; in the second case, δ =

and e = 1.

product of the interplanar spacing and the area per atom on the interface must

be constant, interfaces with large interplanar spacings have the highest density of

points on the surface. This situation is energetically favorable if two crystals are

brought together, because it gives them opportunities to bond that resemble those

of the unbroken crystal. For this reason, interfaces with small Miller indices, (110),

(100),

and (112), occur frequently in practical situations.

Yet another concept, not so easy to define precisely, is the number of nearest-

neighbor bonds one must snap in order to build a given interface. This idea is

connected approximately with surface energy, although the energy needed to form

a surface cannot usually be encompassed completely by so simple a calculation as

the snapping of nearest-neighbor bonds. The process of snapping bonds is rela-

tively easy to visualize in simple cases, such as the (100) surface of an fee lattice,

shown in Figure 4.4.

There is a large vocabulary that has developed to describe the geometries of

Figure 4.4. Diagram showing the bonds broken in cutting along the (001) plane in an

fee crystal. For planes at oblique angles, determination of the analogous quantity is more

difficult, particularly because it may be advantageous for the surface of separation to rise

and fall slightly on the atomic scale to cut the smallest number of bonds.

Geometry of Interfaces 81

interfaces. If a single crystal is sliced along some plane, and the top half turned

upside-down and replaced, the result is called a twin boundary, shown in Figure

4.5.

The grain boundary in this case is a mirror plane; the high degree of symmetry

makes such boundaries energetically favorable and quite common. If instead one

rotates the top half of the crystal through some angle around a normal to the inter-

face,

one produces a twist boundary. If a single crystal has a wedge cut out of it and

is then glued back together, the result is a tilt boundary; if the wedge is small, the

result may be called a low-angle grain boundary. Finally, if the two halves of the

crystal are brought back together without any tilting or rotation, but out of registry,

the interface is a stacking fault.

1 nm

Ni· Al O

(110) X

C?.

o /^

/Tuo

(001) r 1

(001)

(A) (B)

Figure 4.5. (A) (111) twin boundary in NiAl visualized with atomic resolution in a high-

resolution electron micrograph. (B) Sketch clarifying locations of

atoms.

The mirror sym-

metry between right- and left-hand sides virtually eliminates defects along the interface.

[Source: Nadarzinski and Ernst (1996), p. 651.]

4.2.3 Other Topics in Surface Structure

Because surfaces are open to the environment, they easily form thin layers of new

structures, generically referred to as adsorption. If the attraction of the foreign

material is weak, it is called physisorption. De Gennes (1985) reviews the phe-

nomenon of wetting in which a thin fluid layer covers a solid. Gomer (1975) re-

views chemisorption, where a chemical species forms a strong chemically bonded

layer. All these topics are discussed in more detail by Zangwill (1988).

Chapter 4. Surfaces and Interfaces

4.3 Experimental Observation and Creation of Surfaces

Many of the most dramatic recent advances in experimental physics have con-

cerned studies of surfaces. There is an unpleasant tradition of giving each new

technique an acronym, which tends to acquire new letters as the technique is re-

fined.

4.3.1 Low-Energy Electron Diffraction (LEED)

Figure 4.6. (A) Sketch of a LEED experiment. Electrons accelerate from a region with

voltage V toward a sample surface. Reflected electrons pass through two grids whose volt-

age is arranged to allow past only those electrons that lost minimal energy during contact

with the sample, and finally are attracted to a screen where they are imaged. (B) Low-

energy electron diffraction from a diamond (111) surface with a single hydrogen terminat-

ing each dangling carbon bond. [Source: Cheng et al. (1997), p. 3714.]

Low-energy electron diffraction (LEED) is the method used by Davisson and

Germer (1927) to establish the wave nature of the electron. The technique uses the

fact that electrons of energy less than 10

eV have huge scattering cross sections in

many materials, due to the intense interaction of electrons with any charged object,

and is reviewed by Webb and Lagally (1973).

In the LEED configuration, electrons are directed toward a sample, shown in

Figure 4.6. They collide with the surface, penetrating at most a few atomic layers,

and some are scattered backwards. These reflected electrons are filtered, and only

those suffering minimal energy loss in the scattering process are retained. Effec-

tively, these electrons diffract only off the first layer or two of atoms with which

they collide. The wavelength of electrons is given by

λ = 12.2 [energy/eV]"'/

À (4.4)

so for typical energies of a few hundred electron volts, wavelengths are on the order

of angstroms.

The basic equation for scattering from a surface is still given by Eq. (3.11), but

the sum over / is taken only over a two-dimensional collection of atoms lying at the