An introduction to the dynamical scattering of electrons by crystals

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An introduction to the dynamical scattering ofelectrons by crystalsRichard M. Stern a & Howard Taub a ba Department of Physics, Polytechnic Institute of Brooklyn Brooklyn, N.Y.b Cavendish Laboratory, Cambridge, EnglandVersion of record first published: 27 Sep 2006.

To cite this article: Richard M. Stern & Howard Taub (1970): An introduction to the dynamical scattering of electrons bycrystals, C R C Critical Reviews in Solid State Sciences, 1:2, 221-302

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AN INTRODUCTION TO THE DYNAMICAL SCATTEmG OF ELECTRONS BY CRYSTALS*

Authors: Richard M. Stem Howard Tanb** Department of Physics Polytechnic Institute of Brooklyn Brooklyn, N.Y.

I. PREFACE

The establishment of Solid State Physics as an organized scientific discipline rests on the firm knowledge of the ordered periodicity of crystals, which is, in turn, derived from the evidence of X-ray diffraction. The development of the dynamical theory of X-ray diffraction by Ewald’ probably represents the most val- uable single contribution to our understanding of the details of the diffracted intensities in terms of the dynamical wavefield excited in the crystal. The fundamental concept which allows this theory to continually provide a basis for contemporary work of great sophistication is its treatment of the problem in terms of the excitation of the normal modes of the crystal.

An almost identical parallel exists in the case of electron dfiaction. Following the simultaneous verification of the DeBroglie? hypothesis by G.P. Thompson3 for high energy electrons in England and C.J. Davisson

and L.H. Germer‘ for low energy electrons in the United States, a theory was developed by H. A. Bethes which allowed the detennina- tion of the reflected intensities in terms of the excited normal modes of electron propagating within the cry~tal.”~ These latter are the Bloch waves which are the well known solutions of the Schrodinger equation for electrons traveling in a periodic potential and, of course, have the‘ir counterparts in the X-ray case as well for electromagnetic waves.

The subsequent history of the development of electron diffraction is very dissimilar to that of X-ray diffraction. In particular, it should be pointed out that there has never been a periodic structure uniquely determined by low energy electron dsraction techniques. The origins of this discrepancy are in part acci- dental and partially rest on the relatively large strength of the electron interaction compared with that of X-rays.

The investigations of Thompson inaugurated a continuing study of electron difEraction mea-

* Supported in part by USAFOSR Grant 1263-67 **Present address: Cavendish Laboratory, Cambridge, England

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surements at high energies in England, Japans-'' and Germany,'? which fall into two categories: transmission measurements and back-reflection measurements at glancing angles. It should be noted at this point that these two types of measurements are funda- mentally different inasmuch as the transmission measurements (Laue reflections) allow a direct determination of the propagating wave field within the crystal while the reflection measurements (of Bragg reflections) allow only an in- direct measurement of the wave field since they are associated with the forbidden gaps in the band structureY8 a point which will be discussed later in detail. The transmission measurements established a tradition which led to the eventual development of electron microscopy and the beautiful treatment of the theoretical problem by Whelan and Hirsch13 in England. The early reflection studies by Shinohara'l showed what appeared to be anomalies in the reflected intensity not anticipated by a kinematical or first order dynamical theory. On the other hand, the work of Davisson

and Germer inaugurated the study of (back) reflected intensities at low energies by that group and others, in particular H. E. Farns- worth and co-workers.'* Thus, there is a geo- graphical distinction, the primary tradition of low energy electron difhaction remaining in the U.S. The original Bethe article appeared in German, as did much subsequent work con- necting crystallography and electron W a c - tion, a fact which did not make these efforts part of the current literature of LEED scien- tists in the US. until the past few years. The importance of this literature was, of course, recognized by those working in electrical con- ductivity and, in particular, electronic band structure1s l6 but the connection with LEED has only recently been appreciated by LEED practitioners. The prime reason is apparently that most people working in LEED in the U.S. are interested in surfaces, and a tradition of associating LEED with two-dimensional crystallography has developed, assisted by the general association of the LEED problem with epitaxy, catalysis and the like.

This preoccupation with surfaces is derived from the characteristic large magnitude of the interaction of electrons with crystals. The phenomenological observation that the back reflected intensities are extremely sensitive to the state of the surface has contributed to the development of LEED as a surface tool. It is the magnitude of the interaction which makes the development of electron d~rac t ion theory and practice different from that of X-rays. Because of the importance of the state of the surface, the LEED experimental development has been guided by the need of ultra- high vacuum apparatus for maintaining clean surfaces. Under the general assumption that only the surface periodicity was important, little emphasis has been placed on developing three-dimensional techniques for low energy electron diffraction measurements. In particular, the post-accelerated display system com- monly available within the past decade has precluded three-dimensional approaches developed by X-ray diffraction; the existence of a variable wavelength in LEED has also led to the consideration of experiments at constant diffraction geometry and variable wavelength while the X-ray experiments are usually made at constant wavelength and variable geometry.''*

It is the purpose of this article to review the basic theory of the interaction of electrons with periodic structures and to defme the role played by the surface. A systematic review of electron diffraction and surface literature cannot be made at this time, particularly in the light of the continuing development of LEED theories.

Emphasis has been placed on the high energy electron diffraction observations and theories between 1927 and 1969 in an attempt to indicate the underlying similarity between dsraction at high energy and diffraction at low energy. We have restricted the low energy electron scattering discussion to a systematic survey of the measurements made in our laboratory on the (1 10) surface of tungsten. We have made no attempt to discuss the contemporary LEED literature in any systematic way. This somewhat biased approach does al-

* The relative importance of inelastic effects in LEED also places demands on electron spectroscopy in terms of both energy and momentum resolution which have yet to be extensively met, in a way comparable with X-ray techniques.

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low a presentation of the underlying principles in electron diffraction; the reader may judge for himself the success of this approach. It must be borne in mind that although we have emphasized the three-dimensional nature of the diffraction process even at low energies, LEED is primarily a surface tool. Its use as such, however, still awaits a sulliciently versatile theory to allow the interpretation of LEED data in terms of the details of surface arrange- ments of atoms.

II. INTRODUCTION

Measurements of the intensity of diffracted beams in low energy electron diftraction from single crystal surfaces are characterized by extensive fine structure, This fine structure must be considered to be anomalous if it is assumed that the primary source of the diffraction is the surface plane.l*-?O On the other hand, the existence of additional diffraction features due to the periodic absorption of impurities on the surface, and the existence of maxima in the intensity which violate strict three-dimensional crystal symmetry must be considered anomalous if electron diffraction in the range of 10-1000 eV is considered purely a bulk phenomenon. Although most LEED measurements have been concerned with the two- dimensional aspects of the problem, dynamical dfiaction theories which have been developed are, in principle, exact for the three-dimensional problem. 21-31 Those most applicable at low energies have emphasized the heretofore unexplained surface related effects such as fractional order Bragg reflections and surface resonance^.^^-^^ Of particular importance in determining the adequacy of any diffraction model is the choice of diffraction parameters. The standard spherical diffraction coordinates in three dimensions are E, total incident energy; d , angle of incidence with respect to the surface normal; and 9, angle of orientation of the plane of incidence defined by the incident beam and the surface normal, with respect to a fixed direction in the surface.

Measurements in LEED are usually taken as a function of wavelength (variable E) at constant B and 9 and give the impression of strong surface (i.e., two-dimensional) diffraction.

The utilization of techniques common to X-ray diffraction where the wavelength and, hence, E are maintained constant indicates strong three-dimensional diffraction. In particular, it can be shown that the inclusion of effects of strong multiple reflection (simultaneous diffraction) can successfully account for much of the geometrical origin of the fine structure observed in LEED meas~rements.~~

The understanding of the origin of the intensity variation with diffraction parameter, therefore, requires the development of a three- dimensional dynamical ditfraction theory or its equivalent. It is intended in the following article to show that the formalism developed by Bethe is sufficient to give a first order approximation to the general details of the dzraction in the energy range studied. These arguments are based on the understanding of the origin of the band structure in the solid and the relationship between the band structure and the diffracted intensities.

We consider, first, the problem of the propagation of an electron wave in an infinite region of space by examining the solution to the Schrodinger equation for the potential which can be used to describe that space. The locus of all these solutions as a function of energy and momentum (E, complex K) is defined as a dispersion hypersurface.l If each possible solution represents a normal mode of the problem, then the total solution is considered to be a linear combination of the normal modes; if we are dealing with the region of interface between two semi-infinite regions (a crystal surface, for example), then the solution is defined by the relative amplitude of each of the normal modes, which is, in turn, determined by the boundary conditions at the surface.

The introduction of an idealized surface (which leaves the semi-infinite crystal intact) does no more than introduce the boundary conditions. For any given diftraction experiment, one can determine the sequence of values for the dzraction parameters. A considerable simplification of the problem results if two of the dynamical variables are fixed and one is systematically varied. Each of the three possible such experiments results in a somewhat merent set of prediction^.^^ A number of particularly simple, symmetric cases will be discussed below. Particular use will be made

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of the dispersion surface at constant energy where eigenfunctions of real K can be described by simple geometry.

It should be noted that it is only the existence of the boundary surface which is required for the present discussion, the structure of the dfiacted intensities being related to a three- dimensional phenomenon. The chemical state of the real surface and the nature of the transi- tion region between the exterior and interior of the crystal are known to strongly d e c t the details of the diffracted intensities, making LEED a potentially important and powerful tool for the study of surfaces. However, these surface effects can only be discussed once the importance of the three-dimensional character of the difkaction is recognized. In particular, the band structure to which the dif€racted intensities are related may not be the characteristic band structure of the bulk solid but that of the region of the surface within a few extinction lengths. This is due to the existence of inelastic effects which limit the penetration of the primary wavefield. If these effects did not exist, then the theory predicts that in spite of the strong interaction between the electrons and crystal, the dynamical wavefield extends throughout the entire crystal.

The following treatment is for elastic scattering only. Inelastic processes are important, particularly in metals at low energies above the threshold for plasma excitation of the order of 10-15 eV30 37 43 and at high ene~gies.~'-~~ The effect of including absorption, although it can be accomplished phenomenologically, is not at present understood in sufficient detail to permit predictions of expected effects at all energies. Where possible in the following discussion, mention of inelastic effects will be made, but it should be pointed out that they are by no means complete.*

111. REFLECTION AND REFRACTION AT A BOUNDARY

a. The Optical Case Following the development of Bethe we

recognize that the behavior of the electron is

determined by the solution to the Schrodinger equation. * *

V2Y ( r )+ ( 8r2m0e /h2 ) [ E+V ( r ) 1 '+ ( r ) = 0 , ( 3-1

where mo, e, h have their usual meaning, E is the total energy of the electron, and eV is the potential energy of the electron, which is a function of co-ordinate r. The details of the solution to the problem, therefore, depend on the choice of V(r). Let us consider for a moment the case of a continuous, infinite medium, a region of constant V ( r ) .

In the vacuum, (i.e., V = 0) the solution to (3-1) is the plane wave

Y(r) = Yoexp 2ni(Kor),

where \k, is a constant and, for an accelerating potential E, K is the wave vector with

( 3-2 ) .. - I

We can now consider the optical case where the crystal is simply represented by a region of constant non-zero potential, V = V, (Fig- ure 1). In this case the solution to (3-1) is a plane wave

Y(r) = Y o e q 2ni(k*r), (3-4) .. ..-.

where the crystalline wave-vector kZ = K? - U,

V(r). Note that the units 2m,e where U(r) = - h2 of U are A-2 while V is in volts.

Our dynamical diffraction theory is now concerned with the behavior of external electrons injected into the crystal with positive total energy E > 0. It is appropriate at this point to indicate that in band theory, usually cancerned with electrons of negative total energy, the energy of a state with momentum k is defined by E, = k: + U and there appears to be a dispersion in E if one considers electrons of constant k; in electron diffraction theory the momentum of the state of energy

* The most common observation of inelastic effects are those of Kikuchi

** In the equations the symbol - appearing under a letter will signify a vector.

there is both an extensive theoretical and experimental literature on the subject.


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FIGURE 1

O I X - U o d 1 1 11

Model potential for the case of a semi-infinite crystal without diffraction effects. The crystal is represented by a region of constant non-zero potential. The crystal-vacuum interface is at the potential step.

El, is defined as k = (Ek - U)n, where there appears to be a dispersion in K at constant E.36

When the crystal surface is introduced into the problem (here the interface between the region of zero and constant non-zero potential), the conditions existing on the boundary as defined by a particular experiment (i.e., initial, E, 19, $) determine in a self consistent way which of the normal modes (Bloch waves in the periodic solid, plane waves in the optical case) allowed in the bulk are excited. The boundary conditions are:

a) conservation of energy, b) conservation of momentum parallel to

the surface, c) continuity of total wave function and

its normal derivatives at each surface (or at a l l surfaces for a finite slab),

d) finite electron densities everywhere. For a plane wave description to be valid

(a description which will be used for the general and special cases described later as well) condition (c) requires that we include solutions for both real and complex k. The solutions of complex k are the evanescent waves of the problem: they must be considered in both the crystal and the vacuum for the proper matching of the wave field at the surface. Con- dition (d) requires that the complex solutions have imaginary parts normal to the surface since E - U = k2 = (k, + iki)* = k: + k: + 2ik, * ki and E must be real: then k, is perpendicular to ki. This fact is common to all experiments at a single surface. Conditions (a) and (b) are used to determine the proper set of eigenfunctions excited in the crystal and (c) is

used to determine the relative amplitude of each of the eigenfunctions on the crystal and in the vacuum. The incident beam is assumed to have unit amplitude, and there can be no other waves but the incident wave traveling in the vacuum towards the crystal. Condition (d) prohibits the excitation in a real semi-inlinite crystal with absorption of solutions corresponding to a particle flow towards the entrance surface since if one considers the attenuation due to absorption, this would require an infinite source deep within the crystal. In the optical case being considered here,

the condition at the surface is given as

Here \k, $ are the wave functions in the two media and the derivative is taken normal to the surface. The conditions must be satisfied at all points on the infinite plane surface. The wave function in the vacuum is then made up of the sum of the incident and reflected waves:

Y(r)=A exp2ri K*r+B exp 2ri K"r , ( 3 - 6 ) - - - - - while the refracted wave is:

$(r) = C exp 2~i(k*r). (3-7 1 " - - The incident wave need not necessarily

come from the vacuum; consider the case where the incident wave originates within the semi-inlinite region of non-zero U. The incident wave is now

$(r) = A exp 2~i(k.r)

and the crystalline wavefunction is the superposition of the incident and reflected waves:

(3-8) - - -

6 ' (r)=A exp 2*i(k*r)+B exp 2ri(k'*r) ,(3.9)

the refracted wave in the vacuum is

- - - -

'Y(r) = C exp 2ri(K.r). ( 3-10 - - - July 1970 225

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The relative amplitudes of the reflected and refracted (transmitted) waves at the boundary are then given by the Fresnel reflection and transmission coe5cients obtained in the optical case. The eigenvectors can be determined with the help of the dispersion surfaces which can be shown to be generally useful for determining the excited wave functions even in rather more complicated cases; the description in the optical case will serve as a general guide, however.

b. The Dispersion Surface We define the dispersion hypersurface as the

locus of all the allowed eigenvectors as a function of E, K, and K,. For simplicity, without the loss of any generality, let us consider the case of real K only. In momentum space, at constant E, this surface is the spheroid of radius Ikl in the crystal (the Lorentz sphere) and of radius IKI in the vacuum (the Laue Sphere). If we consider a two-dimensional section of E vs. k, this is just the parabola

E = K2. We may also consider a particular plane of momentum space in which case the surface (E,K,K,) is the paraboloid of rotation (Figure 2).

Assuming that the semi-infinite plane is at constant potential, there are no fields parallel to the plane; thus the momentum parallel to the surface must be conserved: this is just Snell's law. The locus of all possible K vectors having the same value of momentum parallel to the surface and drawn to the same point (i.e., the origin of momentum space) is a cylindrical surface of radius lKl~ = lklr whose axis is the surface normal. Thus, if the direction of Kn in the crystal surface is defined by the incident wavevector, then all allowed wavevectors both in the vacuum and in the crystal lie on the intersection of this surface with the plane of difEraction, a straight line parallel to the surface normal, passing through the end of the incident K vector drawn to the origin (Figure 3) .

Figure 4b, c shows the superposition of the

FIGURE 2

1 K I

Section of the dispersion hypersurface (E, K,, KJ. At constant E the circle of radius K = Eln is the intersection of the sphere of propagation in the plane K, = const.


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FIGURE 3 Y

const.

Y A series of K vectors drawn to the ongin, terminat-

ing in the line vv', which is drawn parallel to the surface normal. This is the family of vectors having the same value of Ku, the component of K parallel to the surface. If the surface normal is drawn through the end of the vacuum K vector, then this construction allows a determination of all points on the dispersion surface which correspond to plane waves having the same component of K parallel to the surface, thus satisfying the boundary condition.

intersection of the Lorentz and Laue spheres with the plane of incidence, for the case of real K, constant V. The incident beam defines KO. The intersection of the surface normal w' with the dispersion surface determines the excited wave vectors; these intersections are known as the Tiepoints.' Within the crystal we will find that there are 2N such tiepoints, (both real and complex), where N is the order of the characteristic equation, here taken to be unity. Within the vacuum the two tiepoints correspond to the incident wave vector KO and the reflected wave vector Kor. Conservation of total energy and of momentum parallel to the surface requires that the angle of incidence equal the angle of reflection. In the crystal we find two wavevectors corresponding to the two tiepoints: k,, and k,,.. If we consider a semi-infinite crystal, the wave k,, does not

exist (Figure 4b). This means that the lower tiepoint is not excited; its excitation corresponds to the reflection of the transmitted wave from the lower surface. If the crystal is a plane parallel slab (Figure 4a), then all four waves would be excited (Figure 4c).

We can now see the meaning of the separate treatment of the wave incident from the vacuum or from the interior of the crystal. The presence of two boundaries requires the existence of both waves within the crystal. It should also be noted that for a range of values of 8, there exists no intersection of the surface normal with the Laue sphere, i.e., the vacuum dispersion surface. These geometries represent the range of diffraction angle for total internal reflection of the crystalline wave at the surface: The tiepoints are complex and the vacuum waves are evanescent.

The foregoing discussion has allowed us to determine the direction of the reflected and refracted plane waves in the case of a medium of constant potential. The dispersion surface is shown to directly determine the refraction correction and, also, in this case, to predict the region of total reflection. Although we have not included a discussion of the evanescent waves, it can be seen that as far as the determination of the reflected and transmitted intensities is concerned, they contribute nothing new since they carry no currentso and, hence, the inclusion of an arbitrary number of vacuum and crystalline evanescent waves (they must appear in pairs) results in the same values for the real wave amplitudes. There is one difference, however: in the case of total reflection. Here the number of real solutions is reduced; the matching at the boundary requires a non-zero amplitude in the vacuum. This wave is an exponentially decaying wave of real K parallel to the surface, purely imaginary K normal to the surface. We will see later that the disappearance of a number of real crystalline solutions also leads to reflection: if there are no real solutions then the reflection coefficient is unity; otherwise it may be quite small.

c. Inelastic EBects: Absorption

It is useful to consider the effect of inelastic scattering at this point. If we d e h e all inelastic mechanisms as those which remove electrons

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FIGURE 4a

-

I

I

CRYSTAL

OPTICAL M O D E L

(ONE BEAM CASE)

n = v

The model potential and wavevectors associated with a finite, parallel crystal slab, in the optical case.


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FIGURE 4b I v

I

The two-dimensional sections of the vacuum dispersion surface at constant energy (upper and lower circles) and the crystal dispersion surface (center circle). If only the upper surface is considered (semi-infinite case), then there is only a refracted wave in the crystal.

FIGURE 4c

d

- kll U

Superposition of the vacuum and crystal constant energy dispersion surfaces, showing the construction described in Figure 3. There are two plane waves excited in the vacuum and two excited in the crystal which sat is fy the boundary conditions of conservation of total E and KI .

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from the coherent beam, then one phenomenological way to introduce the resulting absorption is to include an imaginary part to the po-

tentia140 which leads to an imaginary part of k, i.e.:

v = v' + iVi; k = kr + iki; k2 = k: + ki + 2ikrki ( r ) o

k = (K2+V )+ + iVi/2(KZ+Vo 0 1% = k + i ki . 0

If this value for k is substituted in the crystalline plane wave, it leads to an exponentially decreasing plane wave amplitude where the amplitude absorption coefficient is given by:

( 3-13 !do = 2n Vo/K i (K2<<Vo).

(Note that if the real part of the potential is unchanged, the imaginary part of the potential results only in an exponential modulation of the wave.)

From the foregoing discussion it can be concluded that the fundamental electron problem below the diflraction limit is similar to that of plane wave optics where one includes the possibility of a complex dielectric constant. It is of interest to note that the index of refraction, as determined by the Snells construction of the dispersion surface, can be of the same order of magnitude for the electron case as fo'r the optical case. The index of refraction is dehed as the ratio of the vacuum to crystal wavelengths :

n = k/K = (l+Vo/E) 4 .

The dzerence between X-ray and electron scattering experiments now becomes clear. In the X-ray case the index of refraction is of the order of n = 1 - 1 X not very much different from unity, while in the electron case, in the range of energies from 20-20,000 eV the index ranges from (for V, = 20eV) 1.4 to 1.01. The implication of the magnitude

of the index of refraction will be discussed subsequently.

IV. THE PERIODIC POTENTIAL AND DIFFRACTION

a. Introduction The foregoing discussion has developed the

optical analog for electrons in order to introduce the existence of dispersion surfaces and boundary conditions and to point out that the problem in electron scattering from solids is really to determine the proper excitation of the normal modes. The optical problem is equivalent to a crystal without order-one need not consider the existence of atoms at all. The one-beam case also applies to a regular lattice provided the electron wavelength is sufficiently long so as to excite no reflections, i.e., for energies below the daact ion limit. For higher energies in the electron case, and as is usual in the X-ray case, one must consider in detail the possibility of diffraction. Here the dispersion surface becomes very useful since one is faced with the choice of either treating the response of the lo2? atoms of the crystal in terms of loz2 equations of motion or of examining the possible normal modes of the problem, the latter being a far simpler but not necessarily trivial task.

Let us now consider the case of a perfect, periodic crystal and determine the extension of the one electron model to this case. We will consider the lattice as determined only by


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the space group of the atoms; the treatment ignores the nature of the atoms themselve~.~~ In order to determine the effect of the periodicity, it is necessary to establish the conditions for dBraction. This can be done in a geometrical way with the aid of the concepts of the reciprocal lattice and the Ewald construction.

b. Reciprocal Lattice and Ewald Construction If a, b, c are the primitive translation vectors

of the crystal lattice, the primitive translation vectors of the reciprocal lattice (a*, b*, c*) are defined by the relation^^'-^^

where V = abxc is the volume of the unit cell in the crystal. The reciprocal lattice has a definite orientation relative to the crystal lattice which is determined by the above equations. One can easily show that the following relations are true:

Two properties make the reciprocal lattice of value in diffraction: (i) the vector r* (W) from the origin to the point (hk.?) of the crystal lattice (ii) the length of r* (hkt) is the reciprocal of the spacing of the planes (hkC) of the crystal lattice. To prove the above state- ments we first note that from the way in which Miller indices are defined,s3 the plane (hk4) intercepts the coordinate axes as shown in the diagram of Figure 5. It follows that the vector a/h - b/k lies in the (hkt) plane. Therefore, using Equation 4-2 one obtains

r*(hkll) (a/h-b/k) - - * * *

= (ha +kb +Rc ) - (a /h-b /k) = 0 - - - - - The same can be done for a second vector in the plane, say a/h - c / l . If r*(hk4) is perpendicular to two lines in the plane, it must

be normal to the plane; hence the first statement is proven. If n is a unit vector normal to the plane, then an/h is the interplanar spacing. Since n = r*/Ir*l we have

d(hka) = s'n/h=r*.a/hlr*(=l/lr*l, (4-4) - - - - - proving the second statement.

Ewald has introduced a simple geometrical interpretation of the Bragg conditions using the reciprocal lattice (Figure 6 ) . One proceeds by drawing a vector A 0 of length l / h in the direction of the incident wave and terminating at the origin of the reciprocal lattice; a sphere is then drawn of radius l / h centered at A.

FIGURE 5

Intersection of the plane (h, k , l ) with the axis of reciprocal space.

FIGURE 6

.

Reciprocal lattice and Ewald construction. When the reciprocal lattice point B lies on the sphere of radius l /k, then the Bragg conditions are fulfilled for the diffraction from the set of planes B.

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The possible directions of the diffracted rays for incident wavevector are determined by the intersections of the sphere with points of the reciprocal lattice. This may be proven by not- ing that OB is normal to one of the lattice planes (hkl) and therefore is of length l/d(hkC). From the geometry one has that

l/d(hkll-) = 2(sin8)/~

2 d(hkll) s i n 8 = (4-5)

which is just the Bragg condition. This means that every point in the reciprocal lattice corresponds to a possible reflection from the crystal although this reflection may be of higher order (i.e., d(hk4) = d(h’kZ’)/n for some other reciprocal lattice point (hlc‘d’); n is an integer).

Thus a reflection is excited (i.e., the Bragg- Laue conditions are satisfied) whenever a reciprocal lattice point lies on the Ewald sphere. The foIlowing theoretical approach is designed to allow a determination of the reflected intensities as the dynamical diffraction conditions

FIGURE 7 1400 V

110 1 L O O 1

Intersection of several spheres of reflection for different energies, at normal incidence, with a plane of the reciprocal lattice. The indexing of several reciprocal lattice points is shown, as is the crystal energy for each sphere. To scale for the (110) surface of tungsten. .

232 CRC Criticnl Reviews in Solid State Sciences

are varied and different reflections are excited. From Figure 7 it can be seen that at large vaues of k, many reflections can be excited simultaneously, and it is the eventual solution of this problem towards which we are

It should again be pointed out that we choose to describe the crystal in terms of a three-dimensionally periodic potential. If the crystal were composed of a single biperiodic plane, the Fourier transform (reciprocal lattice) would be a set of rods, normal to the plane, and there would be intersections with the sphere of reflection at all values of the diffraction parameters (Figure 8).17 We find no justification for such a model. Even at relatively low energies where the intensity appears

5 ~ 3

FIGURE 8

0 0.. 0 ...a. Intersection of several spheres of reflection

with a plane of a two-dimensional reciprocal lattice. The reciprocal lattice rods are drawn normal to the surface. In dynamical diffraction theory the range in energy over which a reflection is excited can be large at low energies. Thus, even if the reciprocal lattice is three-dimensional as in Figure 7, the non- vanishing of the reflections can be misinterpreted a~ indicating a two-dimensional lattice as in this figure.

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5 3 0 I- - aF, +-

9

9

a 0 0

‘ 1 a.

. . 0 cv

.

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to not go to zero between reciprocal lattice points, the relative strength of the interaction requires that a reciprocal Iattice point be excited far from the sphere of refle~tion.~? Thus it appears justified to attribute the low energy behavior of the dfiaction to the strength of the interaction rather than to any two-dimensional character of the crystal as seen by the electron beam. Figure 9 shows the similarity between an X-ray and an electron Laue illus- trating the three-dimensional nature of the reciprocal lattice in the energy range 75-150 volts.

V. DYNAMICAL THEORY OF BETHE

a. General Theory

1. Secular Equation and Dispersion Surface The primary objective of a dynamical theory

is to solve exactly or to an appropriate approximation the Schrodinger wave equation (Equa- tion 3-1). Kinematic theory, a zeroth-order approximation to dynamical theory, which assumes that the various defracted waves do not interact, is shown to be inadequate for describing electron diffraction intensities. Various formulations of dynamical diffraction employ- ing Green's functions, scattering matrices, vari- ational techniques and/or different expansions and approximations have been developed. A significant difference between these approaches is the efficiency of each in utilizing high-speed computers.

The dynamical theory as developed by Bethes and extended by several authorss8 is both the simplest to formulate and (perhaps for that reason) the least efficient to use in many-beam calculations.* Since the interpretation of experimental data will be restricted in this work to the two- and three-beam approximations, it is sufiicient to use, and appropriate to develop, Bethe's approach.

In a perfect crystal the potential V ( r ) can be represented by a three-dimensional Fourier series

where G = ha, + ka, +Ca, is a reciprocal lattice vector (where ai are basis vectors of the reciprocal lattice) and v, is the Fourier coefficient associated with that vector.** The crystal is assumed to be bounded by two parallel infinite net planes; the vacuum potential is defined to be zero. The solution of the Schrod- inger equation inside the crystal is a set of Bloch waves, each of which can be expanded in a plane wave series:

(5-3 1

where

k - = k + G (5-4) - g -0 -

* An n-beam theory is defined as a theory in which only the incident beam and the difIracted intensities associated with n-1 reciprocal lattice points are considered strong. For example, in the two-beam approximation d y one reciprocal lattice point (and the origin) is close to the Ewald sphere.

* * The scattering potential V ( r ) is negative for electrons. For convenience U(r), called the lattice potential and defined by U(r) = -2mV(r)/h', is often introduced." The coefficients v are actually the Fourier coefficients of U(r) and not of V ( r ) . However, it is common practice to refer to the v 3 as the Fourier coefficients of the potential and this will be done here. Also for convenience, subscripts and summation indices will be denoted by lower case letters which will represent the reciprocal lattice vector. As defined, the v, are positive quantities; if V ( r ) is given in volts, then the vg are expressed in A-'. (U(r)[A-'] = 6.65 x In real crystals vg contains a temperature dependence given by the Debye-Waller factor,= (for cubic crystals

of lattice constant a )

V(vo1ts)

and the other symbols have their usual meaning. For tungsten, this temperature factor has little effect on low- index reflections (h' + k' ft' < 10) but reduces high-index reflections (h* + k' +c' - 100) by about a factor of two at 1000°C.


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and Ubo ( r ) is that part of the Bloch wave which has the periodicity of the crystal:

Substituting (2) and ( 3 ) into (1) and letting E = h2kz/2m one obtains:

K$ + v - k2 V V ... -% -432

0 0

v K:+vo - k v ... g l g l g1-2

V K2+v -k2 . .. V

But k h + Q = k,, + H + Q = kh+q. If the summation indices are changed to h and g = h + q one obtains the equation

Removing the term corresponding to g - h = 0 from the summation, this equation, called the secular equation, can be rewritten as:

$0

g1 J,

J,

This identity is true for all r if, and only if, each term in the brackets vanishes for each g:

The system of linear homogeneous equations represented by the secular equation can also be written in matrix form:

= o (5-7 1

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Non-trival solutions for + exist if and only if the determinant of the coefficients vanishes:

K2+v -k2 0 0 0

V Q1

V g2

Kt+vo-k2 Q1

V g p 2 ‘

Kk+v -k2 .. . V gn-g, 0 ~C l.

This equation, which is of infinite order in the set of vectors k,, and Equation 5-4 define the totality of allowed wavevectors (the eigen- values) and Bloch waves (the eigenfunctions) in the crystal for a given incident wavevector KO. For IK,l = constant Equation 5-8 represents a surface in complex k-space called the (constant energy) dispersion surface;s6 at any point on this surface the ratio $,/$, is determined by Equation 5-6. A generalized dispersion hypersurface incorporating both the dispersion surface of dynamical diffraction theory (which is concerned with the dispersion in momentum at constant energy) and the band structure of band theory (which considers dispersion in energy at constant momentum) has been discussed by Stem, Perry and Boudreaux.SB

If Equation 5-6 is written in the form

\ 5-9 1

the amplitude of +g is seen to be large when KO + v, - k: is small or, equivalently, when the reciprocal iattice point is near the Ewald sphere. It is usually assumed that only reciprocal lattice points sufficiently close to the Ewald sphere contribute significantly to the diffracted intensity and, hence, the summations over all reciprocal lattice vectors are truncated to some finite set of reflections. Calculations of higher accuracy are obtained by including a greater number of reflections.

2. Boundary Conditions For a given primary electron energy, all

allowed wave vectors in the crystal are determined from the dispersion surface. The

= o ( 5-8

Q2

presence of a surface plane (which partitions space into ‘inside’ and ‘outside’ the crystal) and an extemalIy incident electron having a definite energy, momentum, and amplitude, introduces further restrictions on the crystal waves. The boundary conditions which must be satisfied are:

i. Conservation of total energy-In the diffraction approach the energy eigenvalue of each crystal wave is equal to the incident electron energy; conservation of energy is implicit in the use of the constant energy dispersion surface. In band theory, the Bloch waves are not in the same energy eigenstate as the incident electron and one must require as a separate condition that the expectation value of the energy for the total crystal wave equal the primary electron energy.

ii. Conservation of tangential component of momentum--This~condition is true for each plane wave component and is represented graphically by a plane of constraint in complex k-space drawn perpendicular to the crystal surface. The intersections of the plane with the dispersion surface, called tiepoints, select the allowed crystal wavevectors k; having the same tangential component of momentum as the incident wavevector and, using Equation 5-4, determine all other crystal wavevectors.

iii. Continuity of the total wavefunction and its normal derivative at the boundary-It follows that each plane wave component of the wave field and its normal derivative are continuous across the boundary. For a many-beam theory this subsidiary form is needed to provide enough boundary condition equations to solve for the unknown amplitudes.

Let + I n and $Out be the total wavefunctions


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inside and outside the crystal, respectively. Using the subscripts n and t to represent the normal and tangential components of a vector, where a is a tiepoint on the complex dispersion surface, and z is the coordinate in the normal direction, then qtin and $Out can be expanded in terms of plane waves by:

27riK * r 27r iK z out -gt -t gn e

Q

From (ii) the tangential momentum of each plane wave component is conserved and is the same for all tiepoints, belonging to the same g, that is

( 5-10 )

Continuity of the total wavefunction then requires that

2 a i K *r 2 a i K z -gt -t -gn

Q

For this to be true for all r t , the expression in brackets must vanish identically:

This is the standard form of the continuity condition (usually with z = 0) that appears in the literature. Continuity of the normal derivative for each plane wave component follows by changing +iUt to K,, +yt and +g +;, to k$ in every step of the above derivation.

iv. Finite electron densities everywhere- This requires that wavevectors have non-zero imaginary parts normal to the surface only. In a semi-infinite crystal with non-zero absorption, this implies that there are no waves propagating toward the entrance surface since this would require unbounded wave amplitudes deep within the crystal.

3. Evanescent Waves Wavefunctions having complex wavevectors

decay exponentially away from the boundary in the direction of either the inward normal, corresponding to primary extinction (dynarni- cal attenuation) in X-rays, or the outward normal corresponding to the vacuum evanescent wave in the optical case of total internal reflection. The fact that such waves have a physical reality and are not spurious solutions of the wave equation can be demonstrated op- tically by the phenomenon known as frustrated total reflection.Ro Exponentially attentuating waves are also used in describing all electron tunneling effects.

Evanescent waves have their greatest amplitude at the boundary and, consequently, have been used to describe surface states (Tamm states) various surface properties of solids,65 and diffraction phenomena resulting from an ordered surface layer.6s One may un- derstand the extreme sensitivity at low energies of back-reflected (Bragg) intensities to surface conditions in terms of the evanescent waves, In the two-beam approximation, the evanescent wave associated with a Bragg reflection G has an extinction length (the inverse of the dynamical attenuation coefficient) proportional to the wavelength and inversely proportional to the Fourier coefficient v,. For low primary

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energies the evanescent wave rapidly attenuates and, consequently, is strongly localized at the surface. The amplitude of the wave is then very sensitive to the absorption of foreign atoms on the surface or to reconstruction of the surface. As the primary energy is increased, the extinction length increases and the evanescent wave is more strongly af€ected by bulk properties: the relative importance of the surface decreases.

4. Absorption It is widely accepted that absorption is an

important factor in the determination of electron diffraction intensities. There exists no general treatment of the effects of inelastic scattering on diffraction, although various theories for fast electrons have been developed. The use of a complex potential to phenomenologically represent absorption in dynamical theory [SlateF and, independently, by Molieres7] has become the standard procedure for converting an elastic theory into an inelastic theory.

The most generally applied formulation due to Yoshioka*O derives the complex potential on a quantum-mechanical basis to incorporate inelastic scattering into Bethe's dynamical theory. The inelastic secular equation (see Equation 5-6) has the form:

where chg = cfr + ic& is, in general, a function of energy since it characterizes those inelastic processes which are excited. The coefficient cig, interpreted as due to virtual inelastic scattering processes (dynamical polarization), modifies the elastic potential of the crystal, while the imaginary part c&, which results in an imaginary component in the wavevectors, is associated with damping of the waves by inelastic scattering.

Complex atomic scattering factors have been employed in the dynamical theory of X-ray

diffraction to describe the Borrmann effe~t .~~-~O In high energy electron diffraction Yoshioka's approach has been applied to calculations and models involving interband and intraband transitions,71 plasmon excitation,2s 38 thermal s~at teMg,~?-~ ' weak beam effects,75 and various anomalous transmission effects.7s The complex Fourier components of the potential have been measured for a number of substances by studying extinction contours with the electron micro- s c ~ p e . ~ ~ - ~ ~ These measurements and calculations based on various models indicate that at high energies the imaginary potential is approximately proportional to the real potential. In a typical metal v:/v: - O.l*.'B This widely used approximation has been investigated for different temperatures and materials the choice of v:/v: = 0.1 is found to be a resson- able approximation for many, but not all, cases.

Absorption is incorporated into dynamical theories of the diffraction of electrons at low energies by introducing an imaginary part to the potential in the form of either complex Fourier componentss1 *? or complex scattering phase shifts.21 This approach, which is an extension of the high energy formulation, has not yet been theoretically justified at low energies and, possibly, more sophisticated theoretical techniques are required.82

Attempts have been made to explain the unexpectedly large widths of Bragg reflections observed experimentally by the introduction of an hag-potential into the dynar-iical calculations. There are two standard approximations: One of these assumes, as in the high energy case, that v:/v; = C (a constant) for all reflections G,81 where C is assumed to be an adjustable parameter usually varying between 0.001 and 0.1. The second approach assumes a constant imaginary potential (i.e., only v: is non-zero) of about 2V corresponding to the absorption expected from an isotropic plasma.82 This latter model precludes the possibility of anomalous absorption effects;

*For two-beam dynamical theory or where V' ( r ) a V ( r ) , one can introduce the notation

r i i v = v + c r v = c Q Q w ' g og

where vy is the Fourier coefficient of the real scattering potential.


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while it may be valid for energies near to the plasma threshold, the existence of the anisotropic secondary emission at 50 eV and above suggests that it is not completely valid for that energy range.

The secondary emission calculation discussed later uses neither of the above models but, being a semi-empirical formulation, introduces an adjustable parameter R,=v:/vA for each low-index reflection considered and one other parameter to relate vb to the average escape depth for secondary electrons.84-87

5. Dimensionless Notation The secular Equation 5-6 and the dispersion

Equation 5-8 have been written in terms of the linearly dependent set of wavevectors k,. It is convenient to introduce a new notation involving dimensionless quantities in which all wavevectors associated with a given tiepoint are expressed in terms of a single variable.

The secular Equation 5-6 can be rewritten as

where the Vg-h are the complex Fourier coefficients of the potential. If n is a unit vector in the direction of the outward normal and (,) represents the inner product, the external and internal wavevectors for the transmitted wavefield can be related using conservation of tangential momentum (Equation 5-10) :

The dimensionless quantity T, defined by

is called the Anpassung‘ 88 (adaptation) and measures the deviation of the transmitted crystal wavevector from the external incident wavevector. Adding G to both sides of Equa- tion 5-13 and using Equations 5-13 and 5-4:

where, analogous to Equation 5-4, K, is dehed by

The excitation error or ‘Anregungsfehler’ pg, defined by

is illustrated in Figure 10. Squaring Equation 5-15 and substituting from the above equation:

which may be rewritten in the form

FIGURE 10

Construction defining the “Anre- gungsfehler,” the relative error in excitation of a reciprocal lattice point. The observation of diffracted intensities at large excitation error indicates poor crystal resolving power, a kinematical effect for small crystals or a dynamical effect for strong interactions.

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where Then the secular Equation 5-6 can be rewritten as :

Introducing a dimensionless Fourier component pg defined by:

V

one can put Equation 5-12 into the h a 1 form:

1

-1 Og-,% = 0. h (5-20 )

All quantities in this equation are dimensionless. For X-ray dfiaction and high energy electron diffraction (-50 keV) 'ps - lo", therefore gg has appreciable amplitude only when ps and T are of the same order (see Equation 5-9). Consequently, one of the approximations made in X-ray and high energy electron diffraction is to neglect the quadratic terms p i and 2 in the secular Equation 5-20; at very low energies ps and T may be on the order of unity and the quadratic terms must be retained.s7

6 . Dynamical Potential The dynamical potential5 58 (also known as

the Bethe additional potential or Bethe's second approximation) is not formally a pseudopotential but its concept is also introduced to reduce the number of plane waves in the series expansion of the wavefunction. The wavefunction is expanded in terms of that small number of plane wave states corresponding to the strongly excited diffraction conditions. The contribution to the diffraction intensities due to reciprocal lattice points far from the Ewald sphere (i.e., the weak beams) is included through a first order transformation of the strong beam potential coefficients so that to first order the secular equation remains unchanged.

The procedure for obtaining the dynamical potentials from Bethe's dynamical theory is shown below for the general n-beam case:

p, q = strong reflections w = a weak reflection

Let

for each strong beam p, and

for each weak beam. be solved for q,,, and first equation:

. v

The latter equation can this substituted into the

JIW

(K2+v -laz)$ + 1' #qvp-q-l 1' v+%$;O w q#o Ko+vo-kw O O

p q # p , o

which can be written in the form

Redefining v 3 1

the transformed secular equation becomes

(K>V -k2)$ + 1' VpsJIq = 0 ( 5-22) pp q#p,o

which is identical in form to the original secular equation except that only the strong beams are considered and the Fourier coefficients are transformed in a manner defined by Equation 5-21. The similarity between this result and the pseudopotential approach is noted. The magnitudes of the dynamical potential coefficients are clearly dependent on the dynamical diffraction conditions.

Although dynamical potentials have not been widely used in low energy electron diffraction, they were originally introduced by Bethe to describe the electron diffraction observations of Davisson and G e ~ m e r . ~

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There exists a major discrepancy between the energy half-widths observed for Bragg reflections (10-30 eV) and those predicted by elastic dynamical theories. Absorption broadening can account for the observed widths but is accompanied by a drastic reduction in crystal reflectivity.43 91 In one model calc~lation,~~ absorption broadening consistent with experimental widths leads to reflectivities on the order of a few per cent or less. Since there is a scarcity of absolute reflectivity measurements, it is not yet known if this is a reasonable result over the entire range of dif€raction variables. On the other hand, the incorporation of the dynamical potential into the dynamical calculation will result in a better agreement between the observed and calculated half- widths without a loss in reflectivity. For example, Bethe found that the introduction of the dynamical potential for the case of the (662) reflection at 160 eV for nickel doubles the calculated half-width of this Bragg maxima.

At high energies, where two-beam dynamical theory is found to be applicable, the dynamical potential correction is less than 5% of the total scattering potential for some geom- e t r i e ~ , ~ ~ O4 but may be large where a ‘weak’ reflection begins to become a ‘strong’ reflection. Dynamical potentials have been used to explain the observed decrease in apparent inner potential for low-order reflections (see part 31, and are included in studies of the specular reflectiongs and the validity of Friedel’s Lawvs which states that the intensities of the reflections (hkl) and (hkj) are equal.

The effect of the dynamical potential correction on the energy half-widths of B r a g maxima for tungsten is presently being inve~tigated.~~

7. Phenomenological Potential Even without the explicit introduction of the

pseudopotential or possible corrections due to the dynamical potential, it is evident, experimentally, that the low energy Fourier coefficients must be smaller than the high energy coefficients. The inner potential, which is the coefficient associated with sin e/,i = 0, is almost 40 V in the high energy limit of .the Born

appro~imation.~* The inner potential, defined experimentally as the average shift in energy at which the Bragg maxima occur from those energies expected on the basis of the nearly- free-electron model, is about 20 V.ag

This discrepancy between theoretical high energy and experimental low energy inner potentials may have several possible origins. It may result from the variation of the potential seen by the incident electron due to variations with the primary energy of the many-body effects (screening, exchange, and correlation). Other elastic models exist which consider the crystal potential to be independent of energy, but still result in an energy dependence of the inner potential.

The most general treatment of the elastic dsraction problem, then, requires consideration of all the reciprocal lattice points, divided into three groups:8o those which lie on the sphere of reflection; those which do not exactly lie on the sphere but lie sufliciently close to it to influence the shape of the dispersion surface and, hence, the strength of the difiracted beams; and those which lie far from the sphere of reflection and can be ignored.

We will now consider two severe approximations: the situation where only one reflection lies on the sphere of reflection (the two-beam case) and the case where two simultaneous reflections are excited (the three-beam case). It must be pointed out that there appears to be experimental justification for treating certain selected measurements in these approximations: the angular dependence of the secondary electron emission in terms of a sequence of two-beam excitations, and the anomalous structure in the back reflected intensity in terms of a sequence of three-beam excitations. It must be emphasized that this is always a phenomenological treatment; it appears as if the two- and three-beam approximations are appropriate. In each case it is known that there are strong contributions due to the weak beams; their effects are incorporated in the phenomenological potentials used. In other words, the exact values of the potentials used in the two- or threebeam approximations are determined by the diffraction geometry and not the index of the reflections alone.

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B. Two-Beam Case

1. Dispersion Surface When only one strong diffraction condition

is considered in the crystal, the dispersion Equation 5-8 is:

-%

which leads to a fourth order polynomial equation in the adaptation coefficient S:

~4 + a ~ 3 + bT2 + C? + d = 0

where

In general, a, b, c, d and T may be complex quantities although, for the moment, they are assumed to be real.

A two-dimensional section in real k-space of a typical low energy two-beam dispersion surface is shown in Figure l l ( a ) at 20 eV for the (002) reflection (V,,, = 10 Volts). This is compared with a typical X-ray dispersion surface Figure l l ( b ) . In the low energy case the dispersion is so large that the reflection is excited over an angular range of loo while in the X-ray case (and for electron energies on the order of 50 keV) the reflection is excited over a few seconds of arc. Conse- quently, the X-ray dispersion surface is usually considered only in the immediate neighborhood of the Lorentz point Q, which corresponds to the exact excitation of the reflection.

For the electron case shown there are four real tiepoints determined by the constraint plane AA' and, therefore, eight propagating plane waves with wavevectors indicated in Fig- ure 12. The following sections are concerned

with the derivation of the formulas for the anomalous transmission of electrons in crystals (the Borrmann effect) based on the two-beam approximation and the application of this effect to our model for the anisotropic secondary emission.

2. Anomalous Transmission of Electrons At moderately high energies (-1 keV) all

the low-index reflections (G? 5 10) are in the forward direction corresponding to the Laue case of M r a c t i ~ n . ~ ~ In X-ray and high energy electron dBraction this geometry results in anomalous transmission.

The anomalous transmission of radiation through a single crystal was first observed for X-rays by Borrmannss and has since been investigated extensively by 0thers.~3 The analogous effect for electrons was first reported by Honjo and Mihamaloo who noticed an intensity difference between spots af a doublet that could be explained by including absorption in the dynamical equations. Kohra and WatanabelOl later observed the anomalously strong transmission of electrons through an M,O crystal set at the Bragg angle and made quantitative measurements of the anomalous absorption coefficients. Although channeling phenomena have recently been observed for protons, a particles and other ions, the relative virtues of a classical or quantum-mechanical treatment are the subject of much controversy.1o3 los

The two-beam dynamical theory of fast electrons in perfect crystals, which leads to the prediction of anomalous transmission, has been extensively studied and successfully applied for the description of extinction contours due to bending and thickness variations in otherwise perfect crystals and the fringes resulting from grain boundaries and stacking faults observed in electron micro~copy.'~ lo' This phenomenon has also been used to explain the observed variation with crystal orientation in the production of characteristic X-rays from thin

An analytic expression for the effect of diffraction on the secondary emission may be obtained using appropriate approximations which can later be removed with the aid of a computer. The formulation used is similar to that employed by Hashimoto, Howie and

films.103-107


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FIGURE l l a

l la and b-Two-Dimensional Sections in Real K-Space of Typical Two-Beam Dispersion Surfaces. A typical low energy two-beam dispersion surface at 20eV primary energy for the (002) reflection Vm = [lo eV, V. = 20 eV].

WhelanT6 although the notation is not identical. The secular equations (5-6) in the high

the introduction of a new variable

$0

2yg

energy approximation are: T ' : T - -

(2Y0T-40)$0-4g~e = 0 as defined in Figure 13, which shows the upper and lower sheets of the upper half of the dispersion surface, transforms the secular equations to:

-$ $ +(2Pg+2YgT-$o)$e = 0

For the case of a symmetric Laue reflection,*

-e 0

* Using as the definition a symmetric Laue reflection (G,n) = 0 one has:

K = K + G -g -0 -

At sufficiently high energies 1K.I - 1K.I so that (n,K,)/K,I - (n,Ko)/Kol . From Equation 5-18 this gives Y I - 7.

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The deviation parameter x is defined in terms of the excitation error:

The dispersion equation is then:

(5-27 P I 4 l 2

T ' * + + - + = O yg g

Use of the high energy approximation has reduced the dispersion equation from a quartic to a quadratic polynomial and is equivalent to ignoring the solutions associated with the lower section of the dispersion surface. The high energy electron dBraction dispersion surface now resembles the X-ray dispersion surface of Figure 11. The solutions for T' are:

..-% ( 5-29)

Substituting Equation 5-29 into Equation 5-28 :

(5-30 )

The - and + roots correspond to tiepoints on the lower and upper sheets, respectively, and are denoted by the superscripts (1) and (2). The Bloch waves corresponding to the tiepoints 1 and 2, and called the Type 1 and Type 2 waves, respectively, can be written as:

FIGURE l l b

2 )

I )

( b ) Typical X-ray dispersion surface. In the low

energy electron case the dispersion is so large that the reflection is excited over an angular range of 10' while in the X-ray case the reflection is excited over a few seconds of arc only. The intersection of the Laue spheres (the Laue point) is indicated by 1. The intersection of the Lorentz spheres (the Lorentz point) by Q. Because of the small deviation of the index of refraction from unity in the X-ray case, the dispersion surface asymptotes appear linear.

where

The quantities C(1) &d C(?) are defined by

where use has been made of the secular equations. Therefore, the general solution of the wave equation in the crystal for the high energy two-beam approximation is

The incident plane wave in the vacuum is represented by

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FIGURE 12a

Internal and External Wavevectors for Symmetric Laue Reflection G. Geometry for zero excitation error. The surface normal passes through the

Lorentz point, and there are four tie points excited: 1 and 2 in the upper half of the dispersion surface, and 3 and 4 in the lower half.

FIGURE 12c

I .. II

The relative intensity of the wavefield due to the superposition of the plane waves k,’,kf, and plane waves ki,k:, in the case of i ) nonzero excitation error, and ii) zero excitation error. This is just the Pendellosung, of length -! = KO Cos Wv;.

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Applying the boundary condition which requires that the wavefunction be continuous across the surface boundary plane for each Fourier component (Equation 5-11) leads to the conditions

lower section of the dispersion surface (two additional unknowns) have been neglected. Continuity of the normal derivatives, however, is equivalent to continuity of the tangential components of the wavevectors in the high energy approximation, since the magnitudes of

(1) (2) the wavevectors inside and outside the crystal are approximately equal. This last condition has already been satisfied by choosing the

$0 $0 proper tiepoints; hence the boundary condi-

$o +Go = JIo

(1) (1) + J2) (2) = 0 . (5-34)

tion on the normal derivatives may be neglected.5a Making use of Equation 5-32, the solutions

of 5-34 are:

and can be solved without introducing the additional boundary condition that the normal derivative of the wavefunction be continuous across the surface. If the latter condition were

( 5-35s 1 used, it would result in an overdetermined set of equations since waves associated with the

(2 ) P=-* -C

FIGURE 12b

The eight plane wavevectors in the crystal, and the four vacuum wavevectors excited by K, in the case of 12a. The quantities Jo , J:’, Jo , and I F , arc the normal components of the electron current leaving the Crystal, associated with the wavevectors Ki, Ki, KO, and Kg, respectively.

dn


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The amplitudes of the Type 1 and Type 2 wavefields are:

(1)

The significance of this expression is made clear by considering the case of x = 0. The intensities of the two wavefields are then I*(')(r)12 = ($,I2 s in2hG*r) ( 5-38] - - -

. . (539b) I $ (2 1 (r)I2 = [$,I2 cos2(n G - r ) .

- - - The planes of constant intensity, determined by setting G.r = constant, are parallel to the reflecting planes of G. I€ the $ ( I ) ( r ) are plotted vs. r (for r parallel to G), $(l) ( r ) is found to have nodes at the atomic planes while $(2)

( r ) has antinodes as are illustrated in the diagram of Figure 14. If the absorption in the crystal is dependent on r, the absorption coefficients associated with these two waves will be different. In particular, if the absorption is stronger at the atomic planes than away from

FIGURE 13

Geometrical construction for the adaptation 7 and the reduced adaptation T': the difference between the tiepoint and the Laue and the Lorentz spheres, respectively.

the atomic planes, the Type 2 wave will be strongly attenuated and the Type 1 wave will be weakly attenuated. The Type 1 wave results in the Borrmann effect (anomalous transmission) observed for X-rays,1o* electrons and neutrons.1oo In general, the instantaneous cur-

FIGURE 14

+ + + Wavefield associated with the type one and type

two tiepoints. The type one wave, nearer to the origin, consists of electrons which spend little time near to the atomic sites and, hence, see a small time averaged potential (the Sine waves). The type two waves, associated with the sheet of the dispersion surface farthest from the origin (or the upper edge of the band gap), have large amplitudes at the atomic sites, experience a large average potential, and, hence, have a larger k (the Cosine waves). Note from Figure 12b that the plane waves from each tiepoint form standing wave patterns parallel to the surface, and the wavefield propagates normally into the crystal.


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rent flow vector is a complicated function of position in the crystal but is always in the direction normal to the dispersion surface.36 I1O The direction of energy flow for x = 0 in the symmetric Laue case is into the crystal normal to the surface plane.

The magnitude of the intensity oscillation is a maximum for x = 0 and decreases for increasing 1x1 (see Equation 5-38). The absorption of the Type 1 wave increases and of the Type 2 wave decreases from their values at x = 0. It is convenient to consider the intensity of each wave as the spatial average in planes parallel .to the surface plane (for a symmetric reflection such planes are perpendicular to the reflecting planes). This averaging process eliminates the cos (2&. r ) term in Equation 5-38, leaving

Absorption is phenomenologically incorporated into the problem by the addition of

a complex part to the Fourier coefficients of the potential. Using the subscripts r and i to denote the real and imaginary parts of the potential, respectively, the complex dispersion equation is : *

-"-Qg I T I 2~ g1 + 2 p g- i4

Setting yo = ye and expanding the determinant lead to the equation:

i P -i+

112+1' u v

' B

whose solutions are

Or

*b

2yg g h+x2

For 'p: >> 'pi this reduces to:

T' = 4 X d F i )

( 5-41) + ++02+ i i

The real part of T' is identical to Equation 5-30; the imaginary part of T', which results from introducing an imaginary part to the potential, leads to an absorption factor in the wavefunction amplitude. As in the one-beam case (Equation 25) the average amplitude absorption coefficient in the direction normal to

Or, finally the surface** is

* Use is made of the fact that for a crystal having a center of symmetry (such as tungsten), q E = pP and ?Ji = P I r .

* * The phrase "in the direction normal to the surface" will be omitted for brevity.

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The quantity AP* is defined by:

i RK $i BV

(5-43 1 dug - = o g = g yg Kocose

Using Equations 5-3, 13, 41, 42, and 43 the absorption coefficient for each of the two types of wave fields associated with the reflection G can be written as:

This equation formally expresses the basic feature of anomalous transmission: If Apg = po and x = 0, then p:) - 0 and the Type 1 wave will propagate through the crystal with almost no attenuation. It is interesting to note that this equation has been derived using the complex potential but without explicitly considering the interference between +!’ and +:) (see Figure 12). The interference is inherent in the use of the dispersion equation since the relative intensities and phases of the two waves are determined by Equation 5-32 independently of the boundary conditions.

If the Type 1 and Type 2 waves propagate and are absorbed independently, the intensity of the total wavefield as a function of depth z into the crystal can be written as:

(5-45)

where 12) (x), the initial intensity of the Type i wave at z = 0, is assumed to be the intensity obtained in the elastic case (Equation 5-40) :

The typical behavior of the absorption coefficients and initial intensities for each type of wave and the initial net absorption defined by

(1 )I (1 ( 2 )I (2 ) Llg 0 a o X = Po - A U -.

- g m 1) (2 )

v = g 1: +I0

is shown as a function of x in Figure 15 (a-c) . The intensity and the absorption coefficient of the total wavefield deep within the crystal approach the values for the Type 1 wave.

In the preceding discussion the possibility of interference effects between waves of difler- ent dispersion sheets has been ignored. This interference results in oscillations in the transmitted and diffracted wave intensities with dis- tance into the crystal, the phenomenon known as Pendellosung. Associated with this effect is the observation of thickness fringes in wedge-shaped crystals both for X-ray and for high energy electron diffraction (Figure 12).

C. Effect of Diffraction on Secondary Emission

1. The Reciprocity Theorem The reciprocity theorem involves alternately

placing a given electron source at one of two points in space while measuring the resulting field intensity at the other point. In our case, the source of electrons at the external point is the incident beam;* hence, the reciprocity theorem requires that the internal source also have the full primary energy. The discussion is temporarily restricted to those secondary electrons which have lost none (or a negligible amount) of their initial energy and later extended to include all observed secondaries.

Electrons which originate from an internal source may be diffracted before leaving the crystal, resulting in a dfise, angular-dependent emission of quasi-elastic secondary electrons known as a Kikuchi pattern. Observa-

*The incident beam, whose electrons may be treated as plane waves, is equivalent (in the limit of large distances) to any (perhaps even isotropic) internal source.


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FIG

UR

E 1

5 W

avef

ield

Par

amet

ers a

s a

Func

tion

of E

xcita

tion

Err

or.

I

:h

oA

;:

,

, -8

-6

-4 -2

0 2

4 6

8 X-

I I

I I

1 I

I

-8

-6

-4

2

U 2

4 6

-__

-

X-

r

I I

1 I

----

- 1

11

11

11

11

11

11

11

24

68

-8 -6

-4 -2

6' 2

4 6

8 X-

X-

LI E

a) R

elat

ive

inte

nsity

of

the

type

one

and

the

typ

e tw

o w

aves

, II

Ir

b) T

he a

bsor

ptio

ns f

or t

he t

ype

one

and

type

tw

o w

aves

, p

lr p

2.

c)

The

net

in

itial

abs

orpt

ion

(at

z =

0)

defin

ed b

y Y

4

0

c

W

Pg

= (cClIO1 +

C*IO*)/(Iol

+ 102)

whe

re

the

initi

al

valu

es

of

the

inte

nsiti

es

are

used,

and

the

abso

rptio

n co

effic

ient

s ar

e ch

osen

for

th

e re

flect

ion

G.

d)

The

rel

ativ

e se

cond

ary

emis

sion

as

a fu

nctio

n of

exc

itatio

n er

ror

(P is

an

arbi

trar

y no

rmal

izat

ion

cons

tant

).

!2 CI

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h)

01 N

FIG

UR

E 1

6

Phot

ogra

ph o

f th

e di

ffra

ctio

n pa

ttern

obs

erve

d at

nor

mal

inc

iden

ce f

rom

tu

ngst

en

(1 10

) at

200

0 eV

com

pose

d pr

imar

ily o

f K

ikuc

hi p

atte

rns.

A

com

pute

r dr

awin

g of

the

pos

ition

s of

lo

w i

ndex

Kik

uchi

lin

es h

avin

g h'

+ k' +

C =

6

supe

rim

pose

d on

the

pho

togr

aph.

The

pos

ition

s of

th

e ed

ges

of t

he e

xces

s ba

nds are

pred

icte

d b

y th

e po

sitio

ns o

f th

e co

mpu

ter

draw

n lin

es.

The

mos

t pr

omin

ent

lines

hav

e lo

w i

ndic

es.

The

pol

ar h

alf

angl

e su

b-

tend

ed b

y th

e sc

reen

is

appr

oxim

atel

y 60

'.

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FIGURE 17

4

Geometry for Kikuchi line excitation. The locus of all K vectors which excite the reciprocal lattice point h(h,k,G,) is the cone C. The intersection of this cone with the screen is the Kikuchi line (there is a mirror reflection H(h,k,C,). Note that the circle defined by the locus of K is the locus of the center of Ewald spheres which excite the reflection H.

tions of these patterns are quite common in low energy electron diffraction. For tungsten, the Kikuchi pattern is composed of reversing bands, deficiency lines, and deficiency limiting circles. It is the dominant daraction feature above 600 eV and is shown (at 2000 eV, 11 0 orientation) in Figure 16. The angular positions of the lines which make up the pattern can be determined from Bragg’s Law: each line in the pattern corresponds to (but is not quite coincident with) the locus of all incident directions for which an electron will satisfy the Bragg condition for a reflection G = (h, k,C). The indices associated with the line are those of the excited reflection; the most prominent lines in the Kikuchi pattern have small indices. The pattern, which results from electrons emitted by the internal source, has an

orientation fixed with respect to the orientation of the crystal; its intensity depends on the degree of excitation of the internal source and therefore varies with the angle of incidence.

The locus of incident wavevectors KO which excite the reflection G is shown in Figure 17 to form a diffraction cone; the apex of the cone is at the origin of reciprocal space and its base plane is the perpendicular bisector of the reciprocal lattice vector G. Alternately, the locus of wavevectors KO can be defined by the intersection of this plane with the constant energy sphere centered at the origin. This geometry is identical to that which defines the Brillouin zone boundary for the reflection G: (BZ), (Figure 18), a concept useful in band theory.

The Ewald construction and reciprocal lat-

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tice representation of the diffraction geometry are not appropriate for studying the effects of specific reflections. The construction is useful for considering the excitation of reflections in the reciprocal lattice under conditions of varying primary energy but is inconvenient for describing arbitrary rotations of the crystal at a fixed energy. Brillouin zone diagrams (which can alternately be called Kikuchi diagrams since both have identical geometries) are a more suitable representation of the dfiraction geometry for determining which reflections are excited at various orientations of the incident beam in constant energy measurements. In particular, we will be concerned with the low- index reflections since these result in the prominent Kikuchi lines. Figure 19 shows Kikuchi/Brillouin zone diagrams (2000 eV, tungsten (1 10)) for the excitation of reflections having G2 = h2 + kZ +G2 2 6 (left hand diagram) and 6 < G2 2 12 (right-hand diagram) drawn with the aid of an IBM 360/50 computer and Calcomp 470 plotter. These diagrams are the orthographic projec- tions of the intersections of the ditlraction cones with a sphere centered at the crystal and are drawn to an included angle of 120° so that they correspond to the Kikuchi pattern as observed on the diffraction screen. This is shown in Figure 16; the positions of the edges of the low-index excess* bands in (a) are approximately given by the positions of the lines in the simulated Kikuchi pattern which uses the computer-generated diagram having G2 5 6.

2. Diffraction Effects It is known that Brag’s Law need not be

exactly fulfilled in order for a reflection to have non-zero intensity; an error in excitation of the reflection is allowed as is evidenced by the finite half-widths of Bragg maxima. The Brillouin zone diagrams are particularly convenient for constant energy measurements in that they display not only the orientations at

which a specific reflection may be excited but a:so indicate, for a given variation of the incident beam direction, the rate of change of the error in excitation.

The observed intensity variations of the- en- tire Kikuchi pattern can be associated with the correlation between the incident beam directions and the orientation of lines in the Bril- louin zone diagrams.*’ This can be understood on the basis of the reciprocity theorem from the following argument: If an electron from the internal source difIracts in such a way as to give rise to a relative minimum (or maximum) in intensity for a particular direction of observation in the Kikuchi pattern, for example, a deficiency Kikhchi line, then, using this orientation for the incident beam direction, diffraction of the incident electrons will result in a minimum (or maximum) field intensity at the internal source.** If the probability of exciting the source is proportional to the field intensity at the source, the quasi-elastic Kikuchi pattern should decrease (or increase) in intensity as is observed. The excitation of the internal source depends on the incident beam geometry as determined by the Brillouin zone diagrams. Once excited, the source can emit both elastic and inelastic secondary electrons; hence, the variation of the total secondary emission is determined by the intensity variation of the (quasi-elastic) Kikuchi pattern.

Several observations concerning the nature of internal sources can be made in support of this interpretation. If the internal sources were uniformly distributed withii the bulk crystal, there would be no Kikuchi pattern and no anisotropic secondary emission (except for the non-crystalline cosine dependence). The former phenomenon would wash out because the total diffraction pattern produced by ran- domly placed sources is completely isotropic, while the latter effect would not exist because there would be no mechanism for selective absorption in the crystal. An electron incident from an arbitrary direction would uniformly

* Relative to the background intensity. It should again be pointed out that the dark line in the pattern does not exactly coincide with the Brillouin zone boundary but is displaced away from the plane of rdection. The exact amount depends on the energy and the line being considered but is of the order of a few degrees.

** The Kikuchi pattern emanates radially from the crystal so that a point in the pattern can be equivalently described in terms of a direction. The external source (a plane wave) is also specified by a direction of propagation. The association of a direction with a point in space is necessary for the reciprocity theorem as stated here.


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FIGURE 18

a

a a 0

Brillouin Zone geometry. The Brillouin Zone (B.Zo) associated with the reflection G is the perpendicular bisector of the reciprocal lattice vector G. This is also the locus of the center of all Ewald spheres which excite the reflection G.

excite internal sources; this is not allowed even in the geometric descriptions of the angular dependent secondary emission.

The existence of selective absorption in the crystal is to be expected: Of the many inelastic processes that can occur in a bulk crystal (atomic, thermal, plasma) all have a greater probability of occurring at the atomic

site except excitation of the plasma which is thought to be approximately isotropic within the crystal.ll'* Excitations of the plasma will, therefore, not directly contribute to the observed variations in emission but may enter through the cascading of secondary electrons which follows the initial inelastic scattering.** It is an important feature of this model that

* Recent observations indicate that in metals plasma cxcitations may have a greater probability of occurring

** See, for example, Ichinokawa, T. and Kamigaki, Y.,Jap. J . A p p f . Phys., 8, 869, 1969, and the papers of E.G. between atoms.'"

McRae and P.J. Jennings, and J.O. Porteus in the Volume of Ref. 19. and the recent work of C.B. Duke.

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FIGURE 19 the anisotropic secondary emission is a conse- quence of the greater probability of absorption at the atomic sites and that the field intensity at the atomic sites is a function of the incident beam direction. This interpretation is consistent with the concept of anomalous absorption and with the two-beam dynamical model.

The apparent correlation between the orientations of the primary beam resulting in relative maxima in emission and low-index directions in the crystal can be readily understood from the Brillouin zone diagrams in Figure 20. Above 300 eV (for tungsten (1 10)) distinct poles appear in the Brillouin zone diagrams corresponding to directions in which no Brillouin zone boundary exists. The position of each pole is seen to be independent of the energy and corresponds to some low-index direction in the crystal.

For cubic crystals each pole coincides with a dsraction zone of the lattice, which may be defined as a direction in the crystal perpendicular to a set of non-parallel planes of reflection. The two Brillouin zone boundaries associated with a particular plane of reflection (h,k,d) lie on either side of this plane and have indices (h,k,l) and (h,k,l). Consequently, the zone direction has lines surrounding it but not passing through it (except possibly for higher order lines associated with other zones). All crystals on which measurements have been reported have low-index surface planes (since these are the most stable), that is, a pole at normal incidence. This accounts for the apparently universal observation that at normal incidence there is a relative maximum in emission.

3. Dynamical Theory and Secondary EmissionaCa7

The use of the reciprocity theorem predicts a variation in the secondary emission associated with each reflection because of the existence of Kikuchi lines which correspond to reflections from the dense crystalline planes. It

Computer drawn Brillouin Zone (Kikuchi line) diagrams for HZ < 6 and 6 < H2 < 12. The upper curve is a plot of the total secondary electron emission during a rocking curve, as a function of crystal rotation from the position of normal incidence.


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FIGURE 20a

Computer generated Kikuchi/Brillouin Zone diagrams for reflections having H? < 6, at primary energies of a) 50 eV, b ) 70 eV, c ) 100 eV, d ) 180 eV, e)270 eV, f ) 380 eV.

is expected that the dynamical theory leads to the same result but includes a real description of the physical processes involved.

It is possible to apply the results of the two- beam dynamical treatment to the analysis of the angular dependence of the total secondary electron emission by introducing a parameter related to the escape probability of secondary electrons. This escape depth L is an average quantity for all secondary electron energies and initial directions. Since the primary extinction length (p;') (mean free path for inelastic scattering in the primary beam) is determined by the average of all processes which degrade

the beam in energy or phase, the escape depth (L) is no less than one extinction length (unless the inelastic cross-section is very energy dependent) and, in general, is expected to be many extinction lengths in magnitude. It is, therefore, possible to introduce two parameters in the determination of the secondary emission variation associated with the excitation of a forward reflection at a given energy: the ratio of the extinction. length to the escape depth (p,,L), and the relative strength of the absorption associated with each reflection R = v!&. The results of a calculation of the relative secondary emission variation across

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a Laue reflectionss as a function of R is shown in Figure 21. The important point to notice is that there is a relatively small rise in the secondary emission associated with the increased absorption since at high absorption the secondaries are produced near to the surface but their escape probability is not seriously afTected: it is large to begin with. At low absorption, in the region of anomalous transmission, the primary beam is allowed to penetrate

deep into the crystal, and the secondary emission is greatly reduced. Thus, the characteristic secondary emission dependence on incident angle is composed of a series of minima associated with a sequence of two beams excitations: the electron analog of the Borrmann effect.

It should be noted that in this model, at energies where is of the order of unity, the large absorption affects the secondary emis-

FIGURE 20b

g) 500 eV, h) 750 eV, i ) 1000 eV, j) IS00 eV, k ) 2000 eV, 1 ) 10,000 eV. Note that above 300 eV distinct poles appear in the diagrams in which no low index Brillouin boundary exists. The position of each pole is independent of energy and corresponds to a low index direction in the crystal labeled in diagram (i) .


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FIGURE 21a The Effect of Dynamical Diffraction on the Secondary Electron Emission.

-10 J 4 -4 -2 2 4 6 8 m X-

-10 -8 4 -4 -2 0 s 4 a 8 10

X- The reduced diffraction contribution to the secondary emission for a single reflec-

tion be/PIo is shown as a function of excitation error for (upper curve) RB = 0.9 and varying values of 2 hL; (lower curve): 2 ML = 6, and varying values of Rg.

sion, and the characteristic curve is composed resulting values of v: are then used to predict of maxima and minima.* the shape of the secondary electron emission

The success of this two-beam calculation at another energy (500 eV) (Figure 23). The is shown in Figure 22 where a best fit between excellent qualitative agreement between the data and computer experiment is shown. These prediction and experiment shown in Figure 23

* At extremely low and high energy poL may be less than unity: The characteristics of the secondary emission are maxima associated with each reflection. This effect can be seen in the indexing of contrast in the scanning electron microscope. Note that the effect is observed down to 70 eV in Figure 26. The large electron-plasmon and electron-electron cross-sections lead to a relatively non-localized imaginary potential and, hence, reduced contrast.

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L3

OI

0

*I

V

FIG

UR

E 2

1b

3'

rh

PR

IMA

RY

E

LEC

TRO

N E

NE

RG

Y

c 2

00

0e

V

1500

eV

IOO

OeV

PR

IMA

RY

E

LEC

TRO

N E

NE

RG

Y

35

0e

V

20

5e

V

b---

30.0" 4

A

50

0e

V

b------

48.3

"+

Seco

ndar

y el

ectr

on e

mis

sion

nea

r to

nor

mal

inc

iden

ce f

or p

rim

ary

elec

tron-

ene

rgie

s be

twee

n 70

-200

0 eV

. T

he a

ngle

s in

dica

ted

cor-

re

spon

d to

the

cal

cula

ted

angu

lar

sepa

ratio

n be

twee

n th

e (0

02)

and

the

(002

) B

rillo

uin

zone

bou

ndar

ies.

Thi

s is

the

on

ly l

ow i

ndex

re

flect

ion

exci

ted

with

in t

he a

ngul

ar r

ange

of

the

rota

tion.

The

pos

ition

of

th

e zo

ne b

ound

arie

s (i.

e.,

the

kine

mat

ical

ori

enta

tion

for

exci

tatio

n of

the

(00

2) r

efle

ctio

n) i

s se

en t

o be

clo

ser

to n

orm

al in

cide

nce

than

the

pos

ition

of

the

min

ima

in e

mis

sion

. T

his

shif

t is

of

th

e or

der

of

mag

nitu

de p

redi

cted

fro

m t

he c

urve

s of

Fig

ure

21a.

As

the

ener

gy is

low

ered

, th

e re

lativ

e ef

fect

of

the

(002

) re

flec-

tio

n is

con

tinuo

usly

red

uced

, al

thou

gh t

here

is

still

a c

ontr

ibut

ion

at 7

0 eV

. Thi

s in

dica

tes

that

R,

is v

olta

ge s

ensi

tive:

the c

ross

sec

tion

for

loca

lized

abs

orpt

ion

is la

rger

at

high

er e

nerg

ies.

Bel

ow 5

00 e

V 8 =

00

is n

o lo

nger

th

e or

ient

atio

n fo

r a

rela

tive

max

imum

: th

ere

is a

dis

tinct

min

imum

in

the

emis

sion

. T

his

can

be u

nder

stoo

d by

not

ing

that

at

270

eV p

rim

ary

ener

gy f

our

Bri

lloui

n zo

ne

boun

dari

es i

nter

sect

at

the

cent

er o

f th

e zo

ne d

iagr

am.

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FIGURE 22

-60 -50 -40 -30 -20 -10

9

1 O i l 'ii iiii

c . . . . . . , . . ,

10 20 30 40 SO 60

ROTATION FROM NORMAL ( DEGREES)

E I

Computer curve fit of secondary electron emission rocking curve at 2000 eV. The experimental dependence of the net electron current leaving the crystal on the polar angle of incidence 0 (lower curve), and the corresponding computer calculation of the emission (upper curve) using the value 2 h L = 6 and the values for the other parameters as indicated, considering the separate contributions for all reflections having h'+ k'+ C 5 10. For each reflection, the orientation corresponding to zero excitation error is indicated at the top of the figure. The background emission (broken curve, top) has the form IB = [a / ( l + $$ #.L)J - b where the values a = 245, and b = 198 (arbitrary values) have been used. The quantity b corresponds to the incident beam current and must be sub- tracted from to determine crystal, which

the total secondary emission the net current leaving the is the quantity measured.

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FIGURE 23

E primary = 500 eV

c

c u)

C 3 .-

z e c p

2 0

J

v) > a 0 E 0 a LL I- z W a 3 0 I- W z

Y

a

C A LCU LATlO N R202=.25 R013 =. I3

I 1 1 1 1 1 1 1 1 1 I 1 1 I l l l l l l l -50 -30 -10 0 10 30 50

EXPERIMENT


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gives a good degree of confidence in the approach to the secondary electron emission problem.

(E:o)-E-~v (-ev,) . - 0

(-E-eVo % ( 0 ) (-ev,)

VI. BRAGG REFLECTED INTENSITIES

a.

QB

=O (6-2)

8. Tw0-B- Ca~e

We now turn our attention to the interpretation of the reflected intensities in electron diffraction. We have seen that the Bragg case is considerably different than the Laue geometry: the former corresponds to the situation where there are no eigenfunctions of real k (in the two-beam case) associated with the external energy E, while the latter case corresponds to the excitation of traveling crystalline waves. The existence of Bandgaps then leads to increased reflectivity; an absolute gap corresponds to total reflection while, in general, there are always directions for propagation, the gap is not absolute and the reflectivity

The two allowed solutions on the Brillouin zone boundary (E+, E-) then can be written in terms of the vacuum electron energy:

where now the magnitude of the energy gap is given by

E+ - E- = A E = 2 Iv,I . In a higher order approximation, Bethe has considered not only the two strong beams in- dexed 0 and H, but also the weak reflections, I . The plane wave amplitude at of these reflections is considered to be weak not only because Vl is small but also 'because the energy de- nominator (EY - E - V,) of Equation 6-2 associated with the reflection is large. In diffraction experiments where the total electron

energy is conserved (the elastic case), Equa- tion 6-4 defines the two unique energy states permitted for a particular value of the momentum.

Figure 24 is a representation of the intersection of the dispersion surface by the plane of incidence under conditions near to a B r a g reflection H(h, k, I ) . The wave vector for electrons in the crystal k, is related to KO by two conditions;

(6 -5)

and conservation of momentum parallel to the crystal surface:

I ,ffbl cos0 = Ik -0 I cos 8 ' , (6-6)

where V is now the accelerating potential of the electrons (in vacuum). This relationship expresses the reflection of a wave passing from

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one medium to another with a different index of refraction. The independent variable in the case shown in Figure 24 is the angle of incidence 8. Another parameter possible during a measurement where the energy is maintained constant is the azimuthal angle +. Each value of 8 defines a normal to the surface of the crystal which, in turn, specifies a point of intersection with i, the surface of dispersion (point

A) , ii) the circle (0, lkl) (point L), iii) the circle (0, IKI ) (point K) .

Four important cases are represented in Figure 24, each associated with one of the positions of the normal vI corresponding to: the case of transmission ( Y ~ ) ; the case of Bragg reflection ( Y ~ ) , and two cases in the neighborhood of the Brillouin Zone approach- ing total reflection (Y1, Y.).

FIGURE 24

\/-

4 I

Two-beam dispersion surface, showing the geometry for Bragg reflection. At low energies the gap appears at normal incidence; as the energy is increased the gap moves outward, and splits. For measurements at constant incident direction, variable energy, one must construct a sequence of such diagrams. Note that in such an experiment, the magnitude of K parallel to the surface varies. The range of energy for the surface normal to remain within the gap is vg/cosZ 8’, where vg is the g’th Fourier coefficient and 8’ is the polar angle of incidence. The range of angles over which the surface normal remains in the gap at constant energy (away from normal incidence) is v,/V sin e Cos e.

CRC Critical Reviews in Solid State Sciences 264

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A B r a g reflection occurs for values of the angle of incidence corresponding to intersection of the surface normal with the Brillouin zone boundary within the line segment

It is now appropriate to establish a simple kinematical relationship between the eleotron energy (in the vacuum), the angle of incidence in the vacuum, the inner potential of the crystal and the effective electron wavelength in the crystal, in terms of the apparent vacuum condition for Bragg reflections. If one considers the excitation of a point of the dispersion surface, it is convenient to introduce a vector g such that the magnitude of g is equal to the tangential component of the incident electron wave,

A, - A,.

Is1 = cos e. ( 6-7 1

The point L’, intersection of the sphere (0, (kJ), centered at the origin of reciprocal space of radius k, and the Brillouin zone boundary defines the center of the Ewald sphere passing through the reflection H for the symmetrical case. The following relation can be deduced from simple geometrical considerations

By combining 6-5, 6-6 and 6-8 the accelerating voltage at which the point L’ is excited can be written as

The change in kinematical energy between the symmetrical point L’ and the intersection A, (or A,) of the dispersion surface with the Brillouin zone boundary is related to the Fourier component V H of the potential:

In a similar manner we can introduce a new vector go such that

Here g,, is expressed in terms of the experimental parameters V and do in the vacuum),

( 6-12)

V and 0, at which point A, is excited are related through

( 6-13 )

The first term on the right side of Equation 6-13 represents the expected position of the Bragg reflection (MO) modified by a correction for the inner potential corresponding to the effect of refraction. This determines the condition for the excitation of the point L’. (There are no points on the dispersion surface excited by the normal v’). The observed inner potential v,, can be defined as

- v 0 = 2 . vO Sin 8 (6-14)

From this expression it can be seen that the experimental determination of the magnitude v,, can be made with less uncertainty the further the incident direction is from the crystal normal.

Equations 6-9, 6-1 3 allow an additional con- clusion to be made regarding the nature of the Bragg reflection. In an experiment where the incident direction is fixed at some value of 0 and the electron energy is continuously varied, Figure 24 can be continuously redrawn for each new value of the total energy, the relations 6-9, 6-13 always being satisfied. They can be rewritten for this case as:

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where now at constant 6 the dif€erence V, - V’ is equal to the half-width of the B r a g reflection H*.

b. Experimental Observations

If a single set of reflecting planes of index G is considered to give rise to a particular maximum in the back-reflected intensity, then that maximum can be identified as the Bragg reflection G associated with a single Fourier coefficient vg. A number of inconsistencies arise with respect to the parametric behavior of the total integrated intensity: The Bragg reflections are observed to occur in a nonsystematic way, at electron energies below those corresponding to the electron wavelengths anticipated from X-ray measurements; in addition, the range of excitation (in energy) is an order of magnitude larger than one would expect on the basis of reliable high energy electron scattering data.

It is convenient to discuss the energy shift of the center of the Bragg reflection in terms of an effective inner potential Vo**; the width of the reflection is given by 271, in the two- beam approximation. Both Vo and v, are observed to vary with the diflraction parameters. The reflected intensities are also observed to be strongly dependent on incident angle over a range where, on the two-beam theory, they should be either constant or only slowly varying. These inconsistencies are due to the inadequacy of the two-beam approximation in describing low energy electron difiaction phenomena.

It is instructive to compare the predictions of the two-beam theory with the experimental observations and to examine the effect that a single additional difEraction beam may have on the position (in energy or angle) and half- width (in energy or angle) of a given Bragg reflection.

Electron diffraction measurements have traditionally been made as a function of primary electron energy. Figure 25 shows a typical plot of the intensity versus energy for the

specularly reflected beam from a tungsten (1 10) surface.oo By careful arrangement of the incident beam direction, a series of Bragg reflections are observed which can be identifled with the set of Miller incides (nnO). If the crystal is rotated about its surface normal at a primary energy corresponding to one of the Bragg peaks for some arbitrary incident (polar) angle, a plot of the intensity versus azimuthal angle is obtained s 4 5 6 as is shown for the (550) reflection at 8 = 4 5 O in Figure 26. If the azimuthal angle and energy are maintained constant and the crystal rocked in a plane normal to the surface, a plot of the reflected Bragg intensity versus polar angle of incidence (a rocking curve)113 is obtained as is shown in Figures 27a and 27b.

It is convenient to discuss the experiment in terms of dispersion surfaces. Figure 28 shows a typical two-beam dispersion surface for the excitation of the center of the Bragg reflection. The appropriate diffraction conditions are obtained either at constant primary energy by choosing that incident beam direction corresponding to the center of the gap in the dispersion surface or at fixed incident orientation by varying the energy until the incident wave vector corresponds to the center of the energy gap. The dispersion surface is drawn at constant energy; the boundary condition requiring conservation of momentum parallel to the surface for each excited reflection is satisfied by drawing the crystal normal through the end of the vacuum wave vector KO. The intersections of the normal with the dispersion surface determine the set of allowed propagating waves in the crystal; if no intersections exist, no crystal waves with real wave vectors are allowed and there is total reflection; if there are intersections, then crystal waves propagate and the reflection coefficient falls to zero where the dispersion is small (i.e., the dispersion surface corresponds to the nearly- free electron Lorentz sphere) and the transmitted crystal waves have wavevectors near to KO.

* In an experiment where the energy is maintained constant and the incident direction (diffraction angle 0) is varied (a rocking curve) the domains relative to the two half-widths of the total reflection are unequal. The Bragg reflection is made unsymmetrical with respect to the value of e, corresponding to the symmetrical h u e point L.

** Related to the measured inner potential by Equation 6-14.


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On the basis of the two-beam dispersion surface, one would expect the following behavior for the specular reflection (nnO) :

(i) total reflection in intensity versus energy about the energy (V, - V,l f vg)/sin28, where V, = 7.5n2 for tungsten (1 10) ;

(ii) total reflection at all azimuthal angles as the crystal is rotated about the surface normal when the energy and polar angle correspond to the center of the gap as in Figure 24, since the surface normal remains in the gap throughout the entire rotation;

(iii) total reflection about some range of angles 2A €)lo 114 at an energy V,: (2A 8 = v./V

Examination of the experiments for these three cases i) an intensity vs. voltage diagram (pseudo rocking curve); ii) a rotation diagram; and iii) a rocking curve: Figures 25, 26, 27, respectively, show that the two- beam model is inadequate to describe the observations.

In the intensity vs. voltage curve, the position of the maxima is not shifted by a constant

sin e coS e l .

550 6C

I 7

FIGURE 25

I I I 1

.I .2 .3 .4 .5 .6 .7 ELECTRON ENERGY (kev)

Intensity of the specularly reflected beam, as a function of accelerating voltage, from the (110) surface of tungsten. The upper curve is a t about 3' from normal incidence, and at an orientation which is carefully chosen to exhibit large integer order Bragg reflections, and regular, additional half orders. The measured inner potential is indicated. The lower curve is made at 5' from normal incidence, a t an orientation where a dark line appears across the reflection at 7th, 8th and 9th order. By shifting the incident direction a fraction of a degree, the strong reduction of intensity can be made to occur for each order in sequence. The additional reflections and resonance like minima in the Bragg reflected intensities are observed in high energy electron diffraction as well and are known as diffraction anomalies of the first and second kind. The latter are associated with the existence of particular Kikuchi lines in the same orientation as the diffracted spot.

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N

01

aD

FIG

UR

E 2

6

3 8 &

45O

Wll

O

92O

ev

1 I

1

O0

90'

1 80°

R

OT

AT

ION

AN

GL

E

Typ

ical

rot

atio

n di

agra

ms.

The

int

ensi

ty o

f th

e sp

ecul

arly

ref

lect

ed B

ragg

bea

m i

s m

easu

red

duri

ng a

rot

atio

n of

th

e cr

ysta

l ab

out

its s

urfa

ce n

orm

al,

for

8 =

45'.

(The

geo

met

ry is

sho

wn

in t

he i

nset

: th

e az

imut

hal a

ngle

is m

easu

red

with

res

pect

to

the

orie

ntat

ion

of t

he [

lIO

] di

rect

ion

in t

he (

110)

sur

face

pla

ne).

Upp

er c

urve

at

920 eV (

880

Bra

gg

refl

ectio

n),

low

er c

urve

at 335

eV (

550

refl

ectio

n).

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FIGURE 27a

Typical Rocking Curves from Tungsten ( 110).

L I I I 1 I 1 I I 1 -20 -15 -10 -5 0 5 I0 I5 20

8 (degrees) The intensity of the specularly reflected beam is measured as a function of incident polar angle. The 880

reflection has zero excitation error at normal incidence. The range of excitation of .the reflection is 10" which corresponds to the measured energy half-width. (Primary electron energy V = 467 eV).

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amount from the X-ray position, indicating a variable inner potential.8 g8 115-130 (The width of the maxima is an order of magnitude greater than the expected values of 2v,: this apparent surface effect may be due to absorption broadening and cannot be explained by an elastic process described by constant energy dispersion surfaces.) In the rotation diagram, the intensity falls

rapidly with azimuthal angle when the crystal

is rotated from that position where the reflectivity shows a maximum in both energy and azimuthal orientation.

In the rocking curve, there is considerable fine structure, dependent on both energy and azimuth. If the rocking curves are not made at normal incidence, then the structure is a strong function of tilt angle as welP ?O 113

(Figure 27b). All of the above elastic effects can be quali-

/-

G I

FIGURE 28a,b

a b a) Two-beam dispersion surface showing Bragg geometry: the excitation error is zero and the surface normal

lies in the center of the forbidden gap. Ignoring the intersection with the upper and lower spherical dispersion surfaces, this geometry corresponds to total reflection. b) The geometry for total reflection, showing the shape of the gap in the dispersion surface. The annular gap is shown projected on the plane of the surface. Rotation of the crystal about its surface normal results in the surface normal describing a cylinder which lies in the center of the annular gap. During the rotation the reflection is total, and constant, in the two-beam case. The surface normal can be brought out of the gap either by varying the energy, or by varying the incident angle, until the normal intersects the real sheet of the dispersion surface.

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FIG

UR

E 2

8c

E'

I/

I/

0:

I

f

ii Sc

hem

atic

rep

rese

ntat

ion

of

the

wav

e fie

ld i

n th

e tw

o-be

am B

ragg

cas

e, i

) in

the

regi

on o

f to

tal

refle

ctio

n an

d ii

) ou

tsid

e th

e ga

p bu

t st

ill in

the

re-

gion

of

disp

ersi

on.

Not

e th

at t

he e

xtin

ctio

n le

ngth

in

the

Bra

gg c

ase

is c

lose

ly r

elat

ed t

o th

e Pe

ndel

losu

ng p

erio

d of

th

e h

ue

case

. If

the

cry

stal

sla

b is

suff

icie

ntly

thi

n, t

he g

ap e

vane

scen

t w

aves

will

mat

ch w

ith f

inite

. tr

ansm

itted

pla

ne w

aves

in

the

vacu

um (

or i

n th

e su

bstr

ate

crys

tal

in th

e ca

se o

f th

in

epita

xial

ly g

row

n fi

lms)

. N

otic

e th

e Pe

ndel

losu

ng i

n th

e B

ragg

cas

e at

the

edge

of,

but

outs

ide,

the

gap

.

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tatively predicted by including the effect of the excitation of a second reflection simultaneous with the first: the threedbeam case.

c. Three-Beam Case An exact treatment of the three-beam case

can introduce as many as 18 plane waves in the crystal, 6 real tie-points each corresponding to 3 plane waves. Such a formulation rapidly becomes untractable as more simultaneous reflections are included. *

Considerable qualitative information can be extracted from the dispersion equation without explicitly considering the matching conditions at the boundary. For example, the existence of an absolute gap in the band structure corresponds to total reflection of the incident beam (i.e., only the excitation of evanescent waves in the crystal). It is known from detailed intensity calculation^^^ that a gap which is not absolute corresponds to reduced or structured back-diffracted (Bragg) and forward -acted (Laue) intensities.121-120

Similar predictions on the behavior of the intensities can be made by considering the constant energy dispersion ~ u r f a c e . ~ * ~ - ~ ~ ~ Stern, Perry, and B o ~ d r e a u x ~ ~ have discussed the mixed Bragg-Laue case on this basis. In their description, they consider the secular Equation 5-7 when three strong reflections (B,L,O) are excited.

where the notation

6 = K2 + v - k2 (j=B,L,O,) j 0 0 j - - -

is used. Only one of the three components of the Bloch wave may be arbitrarily specified, say t j 0 . The remaining two components are related to I+,, by the secular equations:

+B

$0

- =

0 v-L 1 6

B-L I V B V

V V -B -L V B-L

I 6 o

V -B

L-B V JIL I “L - =

From Equation 6-17a, the condition for $tB to vanish is

(6-18)

Substituting this into Equation 6-16 leads to

Although the numerators in Equations 6-17a and 6-17b are zero for the same value of So, $L is not zero at this point because the denom- inator in Equation 6-17b also vanishes and the ratio $‘/$to is non-zero. In fact, from the secular equation $tB = 0 corresponds to

B E-L

V

V $L

$0

- = - - ( 6-20)

in terms of the wavevectors, Equations 6-18 and 6-1 9 can be written as

(6-21 1

(6-22 I

The intersection of the spherical surfaces defined by these equations lies in a plane whose equation is found by subtracting 6-21 from 6-20:

* Although not carried out in detail in Bethe,’ the existence of N beam cases (N > 2) is implicit in the original treatment (including the evanescent waves of complex k.)

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v v v v -L B -L IrB E-L V-B k2-k:=--- V '

Since kL = k, + L, this may be rewritten as:

B VL-B - - - V 2ko'L + L* = v

-L VSL v -B

When the quantity on the right-hand side of this equation is zero, which occurs for VB = vB-L (i.e., B = B - L) , the plane defined by this equation coincides with the Bd- louin zone boundary for the reflection L. In general, the plane is parallel to the zone boundary but displaced in k-space by an amount Sk given by

L-B V V V -L B - *

6k = - qg- VBL V-B (6-23 I

In an experiment which continuously varies the position of the tie-points, a decrease in the Bragg intensity is predicted for that m a c - tion geometry in which a tie-point satisfying Equations 6-23 and 6-24 (or 6-21) is excited. For the special case of vB = vB-L the remaining Bloch components are related by Equation 6-20:

JIL

$0

c a -1

which is precisely the condition found for anomalous penetration of the Type 1 wave for a symmetric reflection (at x = 0). However, the paper by Stem et aLS8 does not include any calculations of Sk for typical reflections or show any calculated mixed Eragg-Laue three-beam dispersion surfaces to determine if the appropriate geometric conditions for the proposed effect can be fulfilled.

We have performed these calculations for several sets of reflections:* a typical three- beam dispersion surface (drawn using an IBM360/50 computer and Calcomp 470 plotter) is shown in Figure 29 for the (660) and (002) reflections in the plane defined by these reflections and the origin.

d. Comparison Between Three-Beam and Two-Beam Cases

The interpretation of the behavior of the reflected intensities in the three-beam approximation is now quite different from that of N = 2. Several new features may be noted: the gap is no longer absolute in the region of the three-beam excitation, the width of the gap is increased in angle compared to the two- beam case, the center of the gap no longer coincides with that of the two-beam case, the number of real intersections with the dispersion surface in the region of the three-beam excitation now depends strongly on the choice of reflections and the magnitude of the parameters. Under the assumption that the electron reflectivity is large when the number of solutions of real k allowed by the boundary conditions is small (i.e., the normal lies in a gap), the difIracted Bragg intensity can be qualitatively described during any particular experiment when the shape of the dispersion surface is known in detail. It is convenient to demonstrate the effect on the diffracted intensities of varying the diffraction parameters by using the three-beam approximation. At high primary energies this approximation has been used to describe certain diffraction phenomena; at low energies the approximation is inadequate for accurately describing realistic dif€raction experiments since many strong beams are simultaneously excited. It is assumed here that the three-beam approximation is applicable at low energies and that each three-beam case is excited only over a specific range of incident beam orientations.

In the rotation measurement the incident wavevector is constrained to remain at the center of the two-beam gap. During the rotation there will be some azimuthal orientation at which point the cylindrical constraint surface intersects the three-beam dispersion surface and leaves the gap; this condition occurs sym- metrically on either side of the plane containing the three reciprocal lattice points (Figure 29b). For those orientations between and $? the constraint cylinder lies outside the gap

* To be published in detail elsewhere (R.M. Stern and H. Taub.)


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(i.e., intersects the dispersion surface) in the region of the three-beam excitation and the reflected intensity is expected to be diminished. However, the Bragg reflection has not been eliminated completely; the region of total reflection corresponding to the Bragg reflection for azimuthal angles between t,bl and t,b2 is simply shifted in energy as compared with the two-beam case. For the geometry shown in Figure 29a, b, it is apparent that region of total reflection shifts to lower energies (for the same polar angle) as the three-beam region is approached during a rotation. The magni-

FIGURE 29a

Computer generated two-dimensional section in real k-space of a three-beam dispersion surface. If the crystal normal is parallel to the reciprocal lattice vector OB, then B corresponds to a Bragg reflection, and L a Laue reflection: i.e., this is the mixed three- beam Bragg-Laue case. The plane contains the reflections and the origin; the diagram is drawn for a vacuum energy of 265 eV, the inner potential is 20 volts, B = (660), L = (002), V, - 2.2 V, VL = 15.7 V, VII-L = 2.1 V. The points A and A' represent the center of the two-beam gap (see Figure 28a) i.e., if L did not exist (VL = VB-L =O).

tude of the shift, which is the change in the effective inner potential, should be of the order of the energy half-width of the gap. The effective inner potential is observed to exhibit just this behavior for the (880) reflection from tungsten ( 110) as the azimuthal angle is vaned (Figure 30) near to the point of simultaneous excitation of (1 12) .22

The intensity variations in the rocking curie may be interpreted in the same way: for those geometries where an additional sheet of the dispersion surface is intersected, the excitation of the propagating waves will result in the re-

FIGURE 29b

Schematic representation of the gap in the three- beam case, projected on the Brillouin zone boundary of B. The center of the gap is at A", shifted considerably from the position of the two-beam gap center A'. During the rotation the surface normal yo

is constrained to rotate in a circle of K p . r . ~ ~ . ~ = K. Cos 8. In this case yo does not remain within the gap for all azimuthal angles, \k. but intersects a real sheet of the dispersion surface a t angles, between .kl and \k:. In this region the reflectivity is reduced from unity and propagating waves are allowed within the crystal (see Figure 28c). For the orientation A, the surface normal yo is at the center of the gap for the energy shown. For other values of .k the constraint circle no longer passes through the center of the gap at the same energy. In mract ion from a real crystal, the region where a particular three-beam approximation is expected to be valid will be small; this entire description will then be reduced to a valid description of only a few degrees in the rotation diagram, for example. Rotation away from a two-beam geometry (for example, the pole of one of the appropriate Brillouin zone diagrams, corresponding to the point A) is expected to 'be accompanied by a continuous shift in the position of the normal with respect to the center of the gap, at the energy of the diagram. It can be brought back to the center by shifting the energy.

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c

- 0 - m lp!

I-

( Sll Nn AUV8119trV) -A 1 IS N 31 NI

Ln z 0 t-

i -I LL W U

c3 c3

cr a a3

(3 Z 0

W cr W 3

J

(3

a

Z 0 CK

h

U v

z 0 I- 0 W -I L L W cr 0 In In


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duction of the reflected intensity as is observed energies and (polar) daraction angles indi- in Figure 27. cates that the sharp maxima are associated

Examination of the rotation diagrams indi- with geometries which are not sensitive to the cated that there are two major types of fea- diffraction conditions. For example, there are tures: sharp maxima and a large number of certain such maxima which appear at the same relative minima (Figure 29c, d, e) . Compari- azimuthal angle independent of electron wave- son of the rotation diagrams made at various length or polar difEraction angle. These max-

> I- v) z W I- z

-

-

I

FIGURE 29e

0 w

I:

A

i rJ lo = JI

Lower curve shows the rotation about the (660) reflection at 0 = 5 7 9 , showing the position of several Brillouin zone boundaries; upper curve shows the secondary electron emission during the rotation. The orientation of the dense planes of the lattice is shown by the short vertical lines. The orientation and indices of several Brillouin zone boundaries are also shown. (4 ) It should be noted that the minima are shifted from the Brillouin zone orientations.

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FIGURE 29f Primary Energy = 920 eV

+ (degrees) Minimum

11.02 (A)

42.9 (C) 54. 7 (D) 64-4 (E) 68. 2 (F) 78'6 (G)

JI (degrees) BZ

9 (101)

42. 2 (112)

4 7 . 3 (020)

56'8 (lii) 64.7 (112)

-2 .02

3 . 9 -0- 7

4 .4

2' 1

0. 3

-~

A & (Theory)

-4' 7 9

0- 539

-0. 316 4. 928

1- 427

-1' 871

Table of angles of observed minima *mim1mum~ orientation of associated Brillouin zone boundaries ( ~ B . z . ) , the observed deviation (A*), and the calculated deviation, expected from the discussion of section VIc (Equa- tion 6-23).

ima occur when the incident beam lies in one of the dense crystallographic planes of the surface (normal) zone. Maxima related to similar geometry for planes not of the surface zone can also be traced out as the daraction angle is vaned. This effect would seem to be associated with the phenomena of blocking or channeling, observed for the scattering of charged high and low energy particles from periodic 1atti~es.I~'

The minima appear in symmetric pairs on either side of maxima associated with the dense (low index) planes (Figure 29e). This symmetry is just that of the pair of Kikuchi lines (or Brillouin zone boundary intersections) associated with the dense planes of the lattice in terms of the reciprocity theorem and the total secondary emission. It is found, however, that the minima are, in fact, shifted from the Brillouin zone positions by an amount of the order of magnitude predicted by the three- beam theory in describing the position of the point on the dispersion surface corresponding to the excitation of Bloch waves having $B = 0 (Figure 29f). It is not clear whether or not this agreement is fortuitous (especially since in the case of the specular reflection there is

always a third Bragg reflection excited) but one might expect some sort of resonance minimum associated with each many beam excitation,*

Examination of the rocking curves can be done with the same situation in mind. The diffi- culty in this case is that, near to normal incidence, there are an extremely large number of low index reflections which can be excited simultaneously with the specular Bragg reflection. Thus, the experimental uncertainty in beam direction makes the interpretation of the rocking curves ditficults7 (see Figure 20).

(It should be recognized that the relationship between simultaneous reflection, dispersion, and the anomalous behaviour of the reflected intensities was recognized by Morse in his original article discussing the Davisson Germer experiments, Figure 3 1 ) ,

VII. SURFACE EFFECTS

A. Reflected Intensities: Energy Half-Width The theoretical problem of LEED has re-

ceived considerable attention, primarily because of the hope that the technique can allow

* The secondary emission is seen to behave in the same manner as the specularly reflected intensity, except that it is insensitive to the excitation of Bragg reflections (Figure 29d, e) .


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a description of the electronic and crystallo- measurements at high energies) and in terms graphic properties of crystal surfaces. In addi- of the electronic density of states (inelastic tion, it has become evident (based on a dis- measurements at low and high energies). Be- cussion similar to those above) that LEED cause of the anticipated strength cf the elec- measurements provide techniques for investi- tron-solid interaction, LEED has primarily gating the electronic structure of solids, both in been considered a tool for surface studies, al- terms of the unfilled band structure (elastic though it must now be realized that the pre-

1 2 3 4 5 6

FIGURE 30a

* Ew) AV,"

0.0" 920 0 2.6" 913 - 7

-3.9" 906 -14 3.9" 900 -20 5.3" 894 -26 6.6" 889 -31

-M" -10" 0" 10" 20" ,4

1 1 I

854 900 950 PRIMARY ENERGY (eV)

Position of the Bragg (880) reflection as a function of diffraction geometry. The total integrated intensity is plotted as a function of electron energy (and the position of the center of the peak, E, is tabulated) for several different azimuthal orientations shown in the insert. On the basis of three-beam arguments this behavior is qualitatively predicted by the discussion of Figure 29. The curves are all normalized to that at + = 0. (Note that if the effective inner potential shifts as the diffraction geometry is varied, the intensity at constant voltage shifts as well.) Note that the observed inner potential, Vo" = v,,/Cos2e' can shift by as much as 31 volts from its value of the 40 Volts at $ = 0. This observation of systematic variation of the effective inner potential is not common: usually small rotations result in large nonsystematic variations in V," since, for certain geometries, there may be many reflections which can be excited within small variations of the diffraction geometry.

July 1970 279

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3

h) 0

0

I 5 6

3

7

4

E i

nsid

e

I 2

60

2

26

5

3 2

70

4

275

5 2

80

6

205

7 290

V0

02

15

.6

v660

= 2.2

vooo

=20.0

A s

eque

nce

of t

hree

-bea

m d

ispe

rsio

n su

rfac

es a

t di

ffer

ent

ener

gies

illu

stra

ting

the

beha

vior

of

the

posi

tion

and

wid

th o

f th

e ga

p. I

t sh

ould

be

note

d th

at t

he t

hree

-bea

m g

ap i

s ap

prox

imat

ely

twic

e th

e w

idth

(in

bot

h an

gle

and

ener

gy)

of t

he t

wo-

beam

gap

.

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dominant effect observed in the variation of the diffracted intensities with variation in the magnitude of the diffraction parameters is the three-dimensional scattering from the solid. In the elastic case the interaction does not limit the penetration of the wave field into the solid. Surface effects, therefore, can only be identified after bulk effects have been recognized and cataloged. There are, however, two features which characterize LEED measurements of

back diffracted intensities and can be associated with the surface: additional reflections (corresponding to high order surface periodicity) and an unanticipated large range of the diffraction parameters over which the reflections are excited.

Following Bethe, the value of vg decreases with increasing value of h2 + k2 + 12; hence, at normal incidence, as the vacuum energy is increased, one would expect a series of regu-

FIGURE 31a

A

0

0

906 1 . 1 I I 1 I 1 1

I I I I 42 44

-ms I p 48 7

Prediction of “anomalous dispersion” in the three-beam case.8 The position of the regular reflection as a function of wavelength and incident angle is shown, for the case of a cubic crystal, using a Kronig potential (P - 3 n/2, -V, -3V). If the index of refraction is constant, the locus of the excitation co-ordinates should be a straight line for each of the several reflections shown. The anomalous dispersion regions are due to the simultaneous excitation of a second reflection.

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larly excited reflections (say the nno reflections from the 110 surface) of decreasing energy half-width. The usual observation in LEED has been just the contrary: the absolute half- width in energy increases monotonically in the range between 50-1000 eV, varying approximately as EW (Figure 32.) The relative half-width AE/E, which is a measure of the crystal resolving power, may either increase, decrease, or remain constant in this range depending on the exact energy dependence of the half-width. In back reflection the energy half-widths are much larger than the expected values of the Fourier Coefficients: at 1000 eV from the (110) surface of tungsten, the half-width of the reflection is 14 f 1 eV while V,,, is not expected to be larger than about one volt. On the other hand, the energy width of the lowest order reflection which can be identiiied is comparable to that expected from a reasonable value for the appropriate low order Fourier coefficient.

One possible origin for the increased half- width (or range of the excitation) is dynamical broadening due to many beam effects: the dynamical potentials as discussed previously. It

can be seen from the three-beam dispersion surface that the gap in the presence of two strong reflections is about twice that of the two-beam case. It is, of course, not clear how rapidly this dynamical potential converges with the inclusion of weak beams and additional strong beams: measurements at high energy and calculations at low energy show that the inclusion of very large numbers of beams lead to dynamical potentials which are increased by about a factor or two. It is, therefore, not expected that the many beam effects provide an explanation for the observed half- widths associated with the high order B r a g reflections.

It should be pointed out that it was just this problem, based on the nickle data of Davisson and Germer, which prompted the original articles of Bethe and othe~s. '~ lJs The range of the diffraction variables (associated with the deviation parameter X) for the excitation of a reflection is associated with a crystal resolving power and can be related to the size of the diffracting crystal (i.e., the number of reflecting planes) as well as the strength of the interaction (i.e., vg). The energy (or

FIGURE 31b


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FIG

UR

E 3

1c

Plot

of

the

Bri

lloui

n zo

ne b

ound

ary

inte

rsec

tions

with

the

zon

e (9

90

), f

or n

orm

al i

ncid

ence

on

the

(110

) su

rfac

e of

tu

ngst

en.

It can be

seen

th

at t

rave

ling

alon

g an

y zo

ne b

ound

ary

in a

ny d

irec

tion

(i.e.

, a

line

of

Sin

WX

= c

onst

) re

sults

in

the continuous e

xcita

tion

of o

ther

sim

ulta

neou

s re

flect

ions

. T

he n

umbe

r of

suc

h ex

cita

tions

is

equa

l to

the

num

ber

of z

one

lines

whi

ch i

nter

sect

at

a gi

ven

poin

t, an

d can

be q

uite

lar

ge.

ha

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w

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I OO

L

IOL

AE

(v

olts

) IV

FIG

UR

E 3

2

T x

x

0

T 5

.

I' '

0

'

2Va

(app

rox.

) /'

IOV

IO

OV

IOO

OV

E(vo

lts)

IO

KV

IOO

KV

Obs

erve

d en

ergy

wid

th A

E

as

a fu

nctio

n of

en

ergy

E,

for

the

spec

ular

refle

ctio

n fr

om t

ungs

ten

(110

).

The

ant

icip

ated

wid

th,

2Vg,

due

onl

y to

the

Four

ier

coef

fici

ent o

f th

e po

tent

ial

is al

so s

how

n (b

ased

on

the

high

ene

rgy,

Bor

n ap

prox

imat

ion)

. T

he h

alf-

wid

ths

cann

ot b

e du

e to

the

dyn

amic

al p

oten

tial

as d

eter

min

ed b

y th

e tw

o-be

am c

ase.

It

is n

eces

sary

to

find

anot

her

orig

in t

o t

he la

ck o

f cr

ysta

l re

solu

tion.

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equivalently angular) half-widths measured in electron diffraction have their origins in both of these physical processes. In particular, it has been pointed out that the angular half- widths for low order reflections measured at grazing incidence at high energies can have their origin in either of the two mechanisms which are indistinguishable.1° Either the magnitude of the Fourier coefficient v440 is about 4 eV or the penetration of the primary beam is of the order of 250 planes at 150 keV: both these numbers are reasonable and one has, in fact, no reason to prefer one mechanism over the other.

There are two similar arguments which allow the crystal resolution to be determined in the case where it is assumed that inelastic mechanisms attentuate the primary beam. SlateP3 has pointed out that the introduction of absorption by means of an imaginary part (constant part) of the potential V, = V: + iV; reduces the reflectivity of a band gap (since the gap is no longer absolute in the two-beam case and the envanescent gap waves are no different from the attenuated waves associated with the real solution to the dispersion surface on either side of the gap). The range of large reflectivity increases, and the Bragg reflection half-width is now increased by the ratio of Vb/v, so that the measured energy haif-width is approximately 2Vi. This approximation is only valid for small values of the ratio Vi/V:.V, measured for tungsten is of the order of 20 eV, but the implied value Vb = 12 eV from the high energy half-width limit seems excessively large. On the other hand, Kikuchi'O has pointed out that if one considers an exponential decay of the primary beam due to absorption, then the resolving power of the crystal can be given in terms of the extinction length. Figure 33 shows a plot of the amplitude absorption coefficient pa as a function of energy resolving power p. Figure 34 shows a plot of the (extinction length)-' (determined by the data of Figures 32 and 33) as a function of electron energy from tungsten (1 lo), and indicates that the average penetration below the kilovolt region is interpreted to be of the order 5-6 A. It should be pointed out that from the secondary electron emission measurements, the average absorption is diminished by about a

factor of 5-15 in the region of anomalous penetration:Ss-a6 thus the excitation of a forward reflection can be accompanied by a penetration of about 25-100 A. This is of importance in both elastic diffraction measurements (since there is a reduction in the Bragg reflected intensity at this point) and in inelastic measurements (Auger spectroscopy) since the wavefield samples the crystal far from the surface.

B. TemperahveEffeds It has recently been suggested that the half-

widths are due to phonon broadening,13" but this mechanism predicts a strong temperature dependence of the half-width which has not been There has been considerable interest in the temperature dependence of the difTracted intensities in LEED138 since it i s assumed that the surface phonon spectrum is dzerent from that of the Analysis based on kinematical models has indicated the characteristic temperature (i.e., that number which indicates the effective temperature dependence of the diffracted intensity I = I, e ~ p ( - M T / e ) ~ ~ appears to be lower at low energies than at high energies (Figure 35) . We would like to point out that there would appear to be little justification in identifying this quantity 0 with the Debye temperature of the solid. The effect of dynamic interactions, in particular the restriction on the number of active phonons in the surface which conserve both energy and momentum, and the effect of multiple scattering make the interpretation of extremely difficult.ls It should be noted that the temperature dependences of several different reflections (made at the same incident energy and direction) are not the same, indicating possible anisotropy in the effective phonon spectrum.*4o In addition, examination of the multiple scattering formulation indicates that when there is a strong contribution of the weak beams to the intensity, the sign of the temperature dependence is uncertain: under certain conditions diffracted intensities are observed to increase at higher temperatures, especially when the Bragg conditions for the strong beam are not exactly fullilled. These observations indicate strong multiple phonon effects as well. It must be concluded that temperature dependent scattering measurements must be

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FIG

UR

E 3

3 N

OI

m

-8

.7

.6

.5

a

.4

.3

.2

.I

0 0

0.1

0

0.20

0.30

0.40

0.5

0

P D

eter

min

atio

n of

cry

stal

res

olut

ion

due

to d

iffra

ctio

n lim

ited

line

broa

deni

ng. T

he a

bsor

ptio

n co

effic

ient

fi as a

fun

ctio

n of

nor

mal

ized

res

olvi

ng p

ower

p i

s sh

own:

V

, 4-

V,

AV -

,/Vn

V,

= 4

7.5

7.5

whe

re AV

is

the

full

ener

gy w

idth

at

half

max

imum

of

a nt

h or

der

spec

ular

B

ragg

pe

akM

' fro

m

the

100

tung

sten

su

r-

face

(M

O).

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FIG

UR

E 34

0.2

P

0.1

L( E

Y

\o

b-.

X

X T

xi

x

X

X

X T 1 X

X X

X E

X

1 I

I

500

1000

I5

00

2

0

E (v

olts

) 4

0

Cal

cula

ted

abso

rptio

n co

effic

ient

(p,

) as a

fun

ctio

n of

inc

iden

t el

ectr

on e

nerg

y fr

om F

igur

es 3

2, 3

3. T

he f

act

that

the

cry

stal

res

olut

ion

and the

pene

tra-

tio

n de

pth

are

inde

pend

ent

of i

ncid

ent

ener

gy l

eads

to

the

spec

ulat

ion

that

the

re m

ay b

e an

othe

r m

echa

nism

lim

iting

the

res

olut

ion.

T

he r

esol

utio

n at

hi

gh e

nerg

ies.

(i.e.

, SO

-lOO

KeV

) is

that

exp

ecte

d fr

om th

e dy

nam

ical

int

erac

tion

(i.e.

, th

e en

ergy

hal

f-w

idth

s ar

e of

the

mag

nitu

de o

f th

e V

is-

seve

ral e

lec-

3

tron

vol

ts) .

la'

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FIGURE 35

00 SPOT

W-G

X W + CARBON

0 W + C L E A N

0 w + o2

I I I I I I 100 200 300 400 500 600

BEAM VOLTAGE ( e V )

. Calculated effective Debye temperature as a function of incident electron energy from

tungsten (1 lo), for clean, oxygen covered, and carbon covered surfaces. The effective Debye temperature is only a measure of the temperature dependence of the diffracted intensity: it is that number which describes the temperature dependence when it is made exponential by the subtraction of an appropriate temperature independent background. The reduction of the value of this number at low energies has been taken to indicate both a reduced penetration of the electron beam, and an increase in the amplitude of the surface vibrations. Re- strictions on the number of effective phonons, and the possibility of multiple diffraction effects, together with the effect of lattice thermal expansion (both surface and bulk) make a physical interpretation of lattice dynamics from such measurements di5cult. Proper temperature dependence measurements require spectroscopy in both E and K. It should be noted that the relative aperture of the detector in these measurements is a function of accelerating voltage (at low voltage the diffracted intensity is integrated over a smaller fraction of the Brillouin zone than at high voltage for constant detector opening) leading to a continuous variation in the active phonon spectrum detected with varying energy.


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discussed with great care."' It should be pointed out that the addition

of the Debye Waller factor to all the Fourier coefficients of the potential not only reduces the relative amplitude of the strong beams, but also introduces a temperature dependence to the range of interaction defined in Equation 5-29, through the deviation parameter, x, which is proportional to 1/V, exp(-MT/B). Hence, all of the dynamical variables which depend on the deviation parameter are temperature dependent as well. *

C. Surface Reflections The most common surface related effect in

LEED observations are the additional reflections indicating surface periodicity ditferent from the bulk. The existence of these new diffraction azimuths is strongly dependent on the chemical state of the surface, and the literature contains an extensive catalog of various two-dimensional diffraction pattern^.^' Al- though, in principle, the main object of any LEED theory is to determine the crystallography of the surface configuration responsi- ble for these patterns, the present state of the art does not allow any differentiation to be made between the scattering for different possible atomic positions within the unit cell. In other words, LEED measurements of surface structures allow a unique determination only of the two-dimensional space group of the surface lattice. It is, of course, possible to propose specific models for a particular structure; but until the details of the multiple scattering are understood,''? there is no way of determining the validity of the particular model. It is the solution to this problem that most contemporary LEED theories address themselves.

We would like to present one argument which allows an understanding of the variation in intensity of some of the additional diffraction azimuths based on the dispersion surface model used in this paper.

The boundary condition which we have used to determine the diffracted wave vectors has been the conservation of momentum parallel

to the surface with respect to a surface reciprocal lattice vector. All diffraction vectors which conserve energy (i.e., k2 = const) but differ only by a reciprocal lattice vector are also equivalent. Consider now the problem of a semi-infinite triply periodic lattice with a single monolayer absorbed on it; the monolayer has a surface periodicity Merent from that of the buk. The detailed solution to the diffraction problem involves matching the wave-functions at both the layer-vacuum interface and the layer-crystal interface. If the layer is not monatomic, then it wiU have its own band structure and the matching problem is rather complicated but, in principle, can be done in the usual way. If, however, we consider a single layer, then to a first approximation its contribution to the scattering is only to introduce a new surface periidicity to the problem.

To examine this situation in detail consider the daracted waves scattered by a two-dimensional net, Figure 36a. Their directions can be determined by drawing all wave-vectors of the same magnitude as KO, but having projec- tions in the plane of the net which differ by a reciprocal lattice vector of the net. This can be seen in the onedimensional case of Figure 36b where the lattice spacing is a and the reciprocal lattice spacing is a*; each diffraction vector is defined by (kgll) = Koll + la* where 1 is an integer. (Notice that there are additional wave-vectors which can be drawn to reciprocal lattice lines outside the sphere of reflection: these correspond to evanescent waves since the wave-vectors must be complex in order that kZ = E.)

Each one of these diffracted waves can now be considered as a new incident wave-vector: they are the set of incident wave-vectors mod. a*, as seen in Figure 36c.

We may consider the effect of each of these new incident waves in turn by examining the original bulk dispersion surface. This is done merely by constructing the family of surface normals, defined by the incident wave-vector K and the SURFACE periodicity a* (Figure

*The Debye Waller factor is also to be associated with the maximum intensity of the diffracted beam, while measurements are always made of the total integrated intensity. Recent measurements indicate an aperture dependent energy half-width, and a temperature dependence of the half-width at large aperture as well. The half- widths of Figure 32 are extrapolated values at zero detector aperture.

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36d). To a first approximation, the reflected intensity associated with each of these new incident waves can be determined by our previ- ous criteria: if the associated normal passes through a gap in the dispersion surface, we

expect strong reflection; otherwise not. In addition, the behavior of the bulk wave field, in terms of surface resonances, and special points o 1 the dispersion surface will be the same as without the new surface net. This behavior is

FIGURE 36

Zeroth Order Dynamical Treatment of an Ordered Surface Monolayer

....... v' 1

lv3

I

4 v vo v' v 2 v3 a ) The monolayer can be considered to be the source of new diffracted beams shown schematically. b) The

directions of these beams can be determined from the Ewald construction, assuming only the surface periodicity of the monolayer. c) Each of these diffracted beams can be considered a new incident beam. They all differ only by an integer number of surface reciprocal lattice vectors. d) The reflectivity of the incident beam can be determined by the usual construction: here its surface normal v is shown passing through the middle of a three-beam gap where its reflectivity is large. The other surface normals can now be drawn, each displaced only by a reciprocal lattice vector of the surface monolayer. The additional reflected azimuths have large intensities only if the surface normal passes through the gap, which results in a large reflectivity. Thus there may be special points such as the tangent points of the gap, and the points where one of the plane wave amplitudes goes to zero, which lead to additional modulation of the intensities of the extra azimuths.

290 CRC Critical Reviews in Solid Strafe Sciences

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qualitatively verified in many observations of high order surface structure where the additional diffraction azimuths have large intensity only in the region near to the regular diffraction pattern. The dynamical solution of the problem, however, must include the angular variation of the forward scattered intensity from the first monolayer: this must be determined in a self consistent way in terms of the details of the multiple scattering within the surface net. (It is undoubtedly the multiple scattering within the surface net, and within the surface atoms themselves, which is the origin of the large magnitude of the surface scattering potential.)

VIII. SUMMARY AND CONCLUSIONS

This article has been written in an attempt: a) to introduce the concept of the disper-

sion surface in electron diffraction and to show how the normal modes of the three-dimensional wavefield are excited in a given experiment. The principle use of the dispersion surface is that it allows the qualitative determination of the relative magnitude of the reflection coefficient to be made graphically. In particular, it can be seen that the interpretation of experiments made at constant energy is more direct than that of experiments made at varying energy;

b) to demonstrate the importance of the three-dimensional aspect of the difhction problem even at low incident electron energies. It is only after the three-dimensional effects are cataloged that the surface effects can be recognized and understood;

c) to explain the relationship between the band structure and dynamical dispersion theory. It is of historical importance that a separate language has developed in these two research disciplines although, in principle, they are mathematically equivalent;

d) to emphasize the importance of certain approximations in understanding the observations made in LEED. In particular, it is shown that the consideration of normal modes excited when only one set of reflecting planes is important leads to a series of important re-

sults; the details of Pendellosung, extinction contours, and, especially, the description of the behavior of the secondary electron emission from single crystals as a function of incident beam geometry. When two sets of reflecting planes are simultaneously excited, it can be shown that considerable structure is expected to be introduced in the reflected intensity;

e) to emphasize the fact that the underlying principles of electron diffraction are the same for all energies. The primary difference between elastic low and high energy electron dsraction is the ability to make transmission measurements at high energies. Observations in back reflections are, in principle, identical for all energy ranges;

f ) to point out the importance of inelastic processes: it i s the fact that each inelastic cross-section is energy dependent in a characteristic way which determines the characteristics of electron diffraction measurements at various energies on various materials;

g) to indicate the importance of multiple diffraction effects: the utility of two- and three-beam approximations is only in their qualitative predictions. In general, the number of reflections which, in fact, must be considered simultaneously are probably never less than 10, and more reasonably 50, over the entire energy range.

Because the fundamental three-dimensional nature of the diffracting crystal is important, the possibility of comparison of the results of a given experiment with those of other experiments depends critically on precise knowledge of the magnitude of the diffraction parameters. It is seen that low index Laue reflections are very important in the generation of much of the fine structure which is observed in LEED measurements. This can be determined from the development of two- and three-beam theory in detail, as an example of some of the predictions of a straightforward theory. The development of the theory, based on that of Bethe, is appropriate at both high and low energies, and is independent of the electronic structure of the atoms of diffracting crystals: it is unique for each three-dimensional space group. There is no mention of the contribution of the actual crystal surface since the periodicity of the semi-infinite crystal is maintained

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to the crystal vacuum interface. Since the momentum of the electron wave-vector normal to the crystal is determined by the dispersion and is not conserved, it is not expected that relaxing the periodicity slightly near to the surface will affect the results signS~antly. '~~

A survey of the general problem shows that the strong reflections are those associated with forward scattering (Laue Reflections) having low index: In LEED measurements are made of back reflected intensity and one of the important geometrical situations is the mixed Brag-Laue three-beam case. A major difEi- culty in determining the details of the origin of LEED fine structure is the fact that many measurements are made at normal incidence or close to it: since crystal surfaces are usually low index directions, there are many simultaneous diffraction conditions satisfied in this region. It requires care and precision in select- ing specific geometries for LEED measurements which reduce the effects of multiple diffraction.

In the future, both real and computer experiments should be done at constant energy in order to allow a systematic exploration of the dispersion hypersurface, and to determine the effect of complex symmetries on LEED intensities. In order to separate out surface effects and, in particular, to determine the importance of particular inelastic processes, it will be necessary to perform measurements with precision comparable to that usually employed in X-ray diffraction, including the introduction of high resolution energy spec- trometers. It would appear that a new generation of experiments will be necessary to explore properly and systematically the origins of various diffraction effects and to distinguish between the utility of the many LEED theories now being developed.

Acknowledgments This article is the result of numerous discus-

sions with members of the diffraction group at the Polytechnic Institute of Brooklyn and contains much which has appeared in the various publications of the group:

We acknowledge helpful discussions with Prof. D.S. Boudreaux, Dr. L.S. Ching, Mr. S. Freedman, Mrs. A. Gervais, Mr. P. Goldstone,

Prof. J.B. Kreiger, Miss J.J. Perry, Prof. H. Wagenfeld, and Mr. S. Shnharoy. One of us (H.T.) wishes to express his appreciation ,to Prof. P.P. Ewald for critically reading the manuscript of his thesis, several sections of which are incorporated into this present work.

Note Added in Proof Since the original manuscript was prepared,

a considerable amount of new data has become available, and some extremely impressive theoretical efforts have been made in other labora- tories working on LEED problems. Although the intent of the article has been to provide an introduction to dynamic4 electron diff raction theory, it is felt that at least reference to the most significant recent developments should be made. The reader's attention is called to the papers presented at the Fourth LEED Theory Seminar, held at the N.B.S. May 1, 1970 (a limited number of programs are available from Prof. R. Wallace, Univ. of Calif., Irvine), and the preceding session on LEED at the APS meeting, the abstracts of which can be found in Bull. A.P.S., 15, 671 1970, In particular, the Quantum Field Theo- retical approach of C.B. Duke and the inelastic matching approach of R.O. Jones and J.A. Strozier appear to be able to predict with impressive accuracy the observations on aluminum and Be. The work of E.G. McCrae and that of K. Kambe convincingly treat the existence of scattering resonances which are re- sponsible for some of the low energy structure observed in the diffracted intensities. An extremely striking elastic band structure approach has been developed by V. Hoffstein and D.S. Boudreaux to explain the aluminum observations as well.

In the present article we have stressed the importance of the dispersion surface and described some rather simple cases, in particular those when only one or two reciprocal lattice points lie on the sphere of reflection. In measuring the intensity of the specularly reflected beam from cubic crystals, the geometry always requires that simultaneous reflections appear in pairs. A comparison of real cases should, therefore, always be made for the four-beam case. Some four-beam discussions appear in


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the literature,* and others will appear shortly. The presence of the fourth beam does not d e c t the overall arguments presented above, but it does alter the symmetry of the dispersion surfaces. In addition, the case of a surface resonant reflection (a reciprocal lattice point at the equator of the sphere of reflection) bears special treatment because of the experhentally verified importance of this geometry. Recent computer experiments in this laboratory indicate that the loss of reflectivity in the three- beam cases described are due primarily to the intersections of the surface normal with the real sheets of the dispersion surface, and that special points on the dispersion surface itself contribute only minor structure to the intensities. The importance of this fact should be noted with respect to the arguments presented in reference 36, and repeated here in Section VI c, where the stress has wrongly been placed on the special points of the dispersion surface.

X. APPENDIX

I. One-Beam Case in Dimensionless Notation The one-beam case, where the incident

plane wave is partially reflected and partially transmitted by the surface potential barrier* * has been discussed in Section 111. We include a description of this case, using dimensionless notation, so that the results can be compared with the general form of the higher order solutions. It is expected that the one-beam case is never valid in low energy electron diffraction from periodic crystals because the magnitude of the inner potential usually precludes the possibility of external electrons E > 0 having internal k values below the diffraction limit. The model may be valid for the case of transmission for amorphous thin films, however.

If q0 is assumed to be much larger than a l l other wave amplitudes, the secular Equation 5-6 reduces to

Solving for T one obtains

The T- solution is associated with crystal waves propagating in the direction of the entrance surface, In thin crystals at low energies such solutions must be dowed; in a semi-inlinite crystal with non-zero absorption these waves are forbidden by the boundary conditions.

At high energies Equation 5-24 reduces to

where pi and p i are the real and imaginary parts of the dimensionless Fourier coefficient ‘p.

The wavefunction 5-3 has the form

-2nKo 0: z/ 2cos 8 x (a travelling wave 1. = e

The amplitude absorption coefficient in the direction normal to the surface*** is defined by

(1-2 1

Using Equation 5-19 this can also be written as

i mr 0 uo = - Kocos 8 *

(1-3

The absorption coefficient is seen to be proportional to the complex part of the potential and inversely proportional to KO (i.e., the inelastic mean free path in the direction normal to the surface is proportional to EX) and inversely proportional to the cosine of the polar angle of incidence.

As a result of the conservation of charge, electrons, unlike X-rays, are not destroyed.

* Kohra, K., Moliere, K., Nakano, S., and Ariyama, M. J . Phys. Soc. lop. 17 Suppl. BII, 82, 1962.

** The reflection coefficient as a function of energy for various models of a simple barrier potential have been

*** The phrase, “in the direction normal to the surface,” is omitted throughout for the sake of brevity.

investigated in the literature.w

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Absorption processes are those which remove an electron from the elastic diffraction wavefield by either changing its energy or introducing a random change in phase.

II. Direction of Electron Current In the discussion of the excitation of Block

waves in terms of the surface normal and the dispersion surface, it has been pointed out that the tiepoints correspond to two types of waves: those having electron currents (Schrodinger Flux) directed into the crystal, and those having currents directed towards the entrance surface. The direction of the electron current associated ,with each tiepoint is determined by the direction of the group velocity associated with each Bloch wave. The fact that the direction of the group velocity is given by the normal to the dispersion surface is well

but the following proof (due to J.J. Perry3") is more concise than most. It must be emphasized that only those tie points are excited in thick crystals (with absorption) which correspond to net currents into the crystal. The discussion of Figure 12 a-d indicates that in most cases, even in thin crystals, the effect of tie points corresponding to currents in the direction of the entrance surface can be neglected, except for geometries where the forward difkacted waves suffer total internal reflection at the exit surface of the crystal.

Proof of the direction of electron current (Schrodinger Flux) :

therefore

(11-1 1

(11-2)

On the other hand, if one starts from the Schriidinger equation

and operates with V . k , one obtains

(h2/m) kG$G-VkE(k)$C = 0.

Multiplication by +G* and summation over G yields-

1 (h2/m)kG G

and

G 1 (h2/m)k,

Therefore, from Equation 11-2,

<v> = ( l / h )VkE(k) . (11-4)

However, in the units of this paper ( h Z / 2m = 1) it is necessary to rewrite Equation (11-1) as

whereby Equation 11-4 becomes

III. Temperature Dependence of Secondary Electron Emission Contrast

It has been pointed outsa that the magnitude of the structure in the angular dependence of the secondary electron emission is temperature dependent. This temperature dependence arises from the fact that the structure is proportional to the magnitude of the Fourier coefficient v, associated with each reflection G, which is in turn temperature dependent through a Debye- Waller factor. It should be noted that the average absorption is independent of temperature-it is only the anomalous absorption (both the increase and decrease) which is temperature dependent.

The secondary emission of electrons from a solid is a two-stage process: the incident electron must first be scattered out of the primary wavefield. This is followed by a complex cascading process'*s in which many secondary electrons of all energies (less than the primary


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energy) are created. Some of these secondaries escape from the crystal and contribute to the secondary emission; others may be sufficiently degraded in energy to where they cannot over- come the surface work function and are trapped within the crystal, becoming conduc- tion electrons. There have been numerous investigations of secondary electron emission and many physical models proposed concerned with mechanisms by which secondary electrons are produced and distributed both in energy and in momentum.14T These studies usually as- sume a non-crystalline solid. The resulting dependence on incident electron energy of the efficiency of creating secondary electrons (the secondary yield) is due to the variation of the inelastic cross-section for each energy loss mechanism (atomic, phonon, plasmon).

For non-crystalline (here taken to mean polycrystalline or amorphous) solids, the dependence of the yield 6 on the incident polar angle t9 can be described by148

The yield for single crystals shows strong fluctuations as the polar angle of incidence is varied which is not described by Equation 111- 1 .

The experiments described in this work and the discussion which follows are concerned solely with the additional effect that dynamical diffraction in perfect crystals superimposes onto the noncrystalline secondary emission. It is assumed that all phenomena not directly associated with the difEraction of the primary electrons are included in the non-crystalline secondary emission contribution (referred to as the non-crystalline background emission). The complex process by which secondary electrons escape from the solid (undoubtedly complicated by diffraction of the secondaries) is described in terms of a model containing only two parameters: P, the efficiency of creating and emitting secondary electrons and L, the average escape depth for the secondaries, determined by averaging overall energies and escape trajectories. The parameters are the counterparts of 6, and ‘a’ in the non-crystalline theory.

With the above simplifications consider next the effect of a single diffraction condition on the secondary emission for a symmetric Laue reflection in the two-beam approximation. The secondary emission process for a single crystal is imagined to proceed as follows: The primary electron beam is incident on the crystal such that the deviation from exact excitation of the reflection G is specified by the deviation parameter x. The two wavefields established in the crystal as a result of diffraction are characterized by the absorption coefficients given by Equation 5-44:

(111-2

The intensity of the total wavefield as a function of depth z is given by Equation 5-45:

The number of electrons scattered out of the primary elastic wavefield in a layer of thickness dz at a depth z below the surface is

These electrons now undergo a cascading process in which many secondary electrons are created. Each scattering results in an identical momentum distribution for the secondaries shifted by the depth into the crystal of the initial scattering event. The number of secondaries created as a result of the initial scattering in a layer dz at a depth z which escape from the crystal is then

where P and L have been previously defined. The total number of secondary electrons emitted from the semi-inkite crystal is found by integrating dn(x,z) over all z:

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(1) ( 2 ) I. = I. + I.

Substituting for p t ) from Equation 5-44 and for 1:) from Equation 5-46 in terms of x yields :

Ise(x) = PIo 1 (1+ -1 + R

2uoL } (111-4) 1 1 (1+ -1 - R 7 2POL g l+x

where Rg= !!% is a positive constant less

than one, which determines the strength of anomalous transmission. From Equations 5-42

P o

and 5-43

The other important quantity in Equation 5-48 is the factor 2p&. Although f i is a function of O (and consequently of x) it varies slowly in the angular range, AO, over which the reflection is strongly excited (Ad is less than 2O at 2000 eV and smaller at higher energies). The computations performed include the explicit dependence of po on d . Figure 21a shows IJP as a function of x for R, = 0.9 and varying values of 2p&, while in (b) 2p,L = 6 and R, is varied. The values of x at which L ( x ) is an extremum is found by setting dI,,/dx = 0:

R X extremum - (111-6)

The shape of these curves can be physically interpreted as follows:

If the incident beam direction is oriented, such x is very negative, i.e., far from exact excitation of the reflection G, the incident electrons experience some average value of absorption corresponding to the background emission. As the incident direction is changed so that x approaches zero pi increases and p: decreases. For small, negative x the net absorption is greater than the average value since the Type 2 wave has a larger amplitude than the Type 1 wave. However, the inelastic mean free path of the incident electrons has been assumed to be much smaller than the escape depth of the secondary electrons (ha = b = 4 ) so that the increase in absorption is not expected to significantly increase the secondary emission. When x becomes positive, the Type 1 wave prevails, the primary absorption decreases and secondaries are produced at increasingly larger distances from the surface. This decreases the escape probability, thereby decreasing the resulting secondary emission. Finally, for x + eo the absorption once again returns to the average value and the emission approaches the background value.

This description is similar to the phenomenological treatment of the production of characteristic X-rays in thin metal crystals by C.R. Hall.lo6 The major difIerence between the phenomena of electron emission and X-ray production is that X-rays, unlike secondary electrons, can easily escape from any depth below the surface to which the incident electrons might anomalously penetrate: In Hall's formulation the X-rays escape from the crystal unattenuated. In addition, he assumes that


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X-ray production can take place in either of two ways: the primary electron can produce X-rays while scattering out of the Bloch wave into a final state described by a plane wave. The plane wave may then continue to produce X-rays but is assumed to be unattenuated in the crystal. As a result, the total X-ray production (Equation 10, reference 106) diverges as the film thickness becomes infinite, while the maximum to minimum X-ray intensity ratio approaches one. In the present formulation the emission Equation 111-4 has a somewhat diBer- ent form from that describing X-ray production due to Hall, but in the limit as L+ co I., also approaches a constant value (Lim I.. = PIa). There is no anomalous variation in secondary electron emission with incident beam direction unless the escape depth is finite. Similarly, since X-rays are assumed to have

L-r -

an infinite escape depth, there is no anomalous X-ray production with incident beam direction unless the thickness of the sample is finite.

Measurements made of the variation of X-ray production in thin gold films140 and thin films of nickel and germanium106 are in qualitative agreement with Hall’s model.

It is now possible to introduce the explicit temperature dependence into Equation In-4 by recognizing that the temperature dependent quantity is the magnitude of the excitation errorlSo

Recognizing that the important quantity is the difference between the maximum and the minimum in the secondary electron emission associated with each reflection, one can write

M ~e~

l+e 6 1 2 RZ (1+ -1 - -

2uoL l + e g

1 2M

2M

R e g (1+ + - 2M

l+e g

(1+ 1 ) 2 - g 2uoL l+e g

R Z

2M

M

We are currently investigating the details of the temperature dependence of the secondary electron emission structure to ascertain if this is in fact the nature of the dependence observed.

We would also like to point out that this

model may well be valid over a wide range of energies, with only the necessity of taking into account the explicit energy dependence of the parameters R,, v,, and paL. The contrast observed in the scanning electron microscope, and, in particular, the “Coates Pat-

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tem~”151-153 can be explained on the basis of this model and should exhibit the same type of temperature behavior. (It is obvious that based on the reciprocity theorem, the Coates Patterns are, in fact, direct, not “pseudo” Kikuchi diagrams.)

Recently, P.B. H i r s ~ h ~ ~ ‘ has developed an elastic theory to explain the contrast effects at high energies in the scanning electron microscope, which does not explicitly treat the escape probability of the secondary electrons. It may be that at high energies, where

poL < 1, it is only necessary to treat the behavior of the elastic beam, although the discussion presented above is still expected to give similar results as for a purely elastic treatment. At high energies it has been noted that the scanning electron microscope contrast is, in fact, improved when only the elastic back- scattered current is detected.lS5 On the other hand, at low energies, we have observed that the contrast is preserved for all portions of the secondary electron spectrum, from the elastic to completely inelastic, over the range of incident energies studied (60-2000 eV) .

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An introduction to the dynamical scattering of electrons by crystals

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