- Email: [email protected]
Low energy electron diffraction rocking curves
SURFACE
LOW
SCIENCE 17 (1969) 214-231 0 North-HolIand
ENERGY
ELECTRON
DIFFRACTION S. FRIEDMAN
Naval Applied Science Laboratory,
Publishing Co., Amsterdam
ROCKING
CURVES t
*
Brooklyn, New York 11251, U.S.A.
and R. M. STERN department of Physics, Polytechnic institute of Brooklyn, Brooklyn, New York 11201, U.S.A.
Received 21 March 1969 The variation of specularly reflected beam intensity with angle of incidence and incident beam energy from the (110) surface of a tungsten single crystal for incident electron energies in the 300-600 eV range and for angles of incidence within 15’ of the surface normal was found to have a structure that could only be explained in dynamical terms. Further analysis indicates that present dynamical theories are only partially successful in explaining the observed intensity variations. 1. Introduction
The variation of the diffracted beam intensity with diffraction angle, commonly known as a rocking curve, is employed in both X-ray diffraction and LEED to explore the reciprocal lattice of the crystal in the neighborhood of a particular reflection. Because of the ease of variation of electron wavelength in LEED, this exploration is also carried out by study of the intensity variation with wavelength, i.e., the pseudo-rocking curve. Both types of measurements have been made in this laboratory and have exhibited a structure that is not explicable in terms of simple kinematic theory for ranges of electron energy for which this approximation is usually deemed valid. A brief summary of earlier corollary results will precede the description and analysis of these observations. The earliest reports of dynamical effects in pseudo-rocking curvesr) were analyzed by means of a simplified dynamical theorya) which predicted the observed discontinuity in the angular dependence of the energy of the specular Bragg maximum. The angles at which these discontinuities occurred corresponded to those angles and energies which were required to excite simult Supported in part by USAF OSR Grant 1263-67. * Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physics at the Polytechnic Institute of Brooklyn. 214
LOW
ENERGY
ELECTRON
DIFFRACTION
ROCKING
CURVES
215
taneous reflections from planes perpendicular to the reflecting surface. Observed anomalous structure in the specular beam pseudo-rocking curves) was interpreted as being due to simultaneous reflections4). However, no quantitative treatment was attempted. More recently5), application of dynamical theory has predicted the enhancement in intensity of a specularly reflected beam upon the simultaneous occurrence of a diffracted beam in the plane of the crystal surface. On the other hand, a self consistent multiple scattering approachc) predicts both a maximum and minima in specular beam intensities upon excitation of reflections parallel to the surface*. These have been observed in studies of the (001) surface of LIFT). Finally, studies of rotation diagramss) and the angular dependence of single crystal secondary emissiong) in the same energy range as the rocking curve investigations lend support to the dynamical influence on reflected beam intensities and the indirect evidence of the existence of strong scattering into the crystal which must be considered in any dynamical treatment. While this survey is by no means complete, it does indicate that dynamical effects may be expected in both the rocking curve and pseudo-rocking curve. In addition, a study of the dynamical theory6,r0-12) to date indicates that no really complete explanation of these effects has as yet been formulated, although several theoretical treatments result in partial success.
2. Experimental
technique
All quantitative observations were made on the (0,O) (specularly reflected) beam using the (110) surface of a tungsten single crystal in a standard Varian LEED apparatus. Surface orientation and preparation are described in detail elsewherels). Surface cleaning in vacua and removal of surface carbon also followed earlier practice14). A Pritchard spectrophotometer was used to record relative diffracted beam intensity. Uncertainty in the incident beam direction due to stray magnetic fields was removed by defining normal incidence as that orientation which achieved either symmetry of Bragg maxima in symmetric azimuths or symmetry of angular variation of the total crystal current. The two angles used to specify the orientation of the surface normal are defined as rotation about an ax is parallel to the (001) direction in the plane of the crystal surface (angle of tilt) and rotation about an axis parallel to the (ITO) direction in the plane of the crystal surface (angle of rotation)**. * This prediction may be an artifact resulting from the assumption of a purely s-wave scattering process. ** The crystal orientation was such that the (001) axis in the plane of the crystal surface formed an angle of about 5” with the horizontal axis of rotation thereby inducing some degree of asymmetry in the observations.
216
S. FRIEDMAN
AND
R. M. STERN
Using the technique described above, rocking curves were obtained for a range of electrcrn energies which included the (880) specular reflection. The reslrftant traces are shown in fig. 1 for each angle of ‘tilt’ studied. Secondary
4.0.
4.5'
5.0’
5.9’
b.0’
490.6
485.5 4
80.0
475.0 473.0 471.5 469
0
468.0 467.0
466.0 465.0 464.0 463.0 461.0 459.0 4520 455.0 454.5 454.0 453 D 451.0 449.0 447.21 4450 44Q.Q 435.0 430.0
Fig. 1. Left han.d curve: Total secondary emission (crystal current to ground) as a function of incident angle for a series of energies within the (880) pseudo-rocking curve Bragg r&e&-m. Tilt angle = 4.5”. Right hand curves : Racking curves (total integrated intensity} for various energies bounded by the range of the (880) pseudo-rocking curve Bragg reflection; (a) tilt angIe I= 4”, (b) tilt angle =4.5”,(c) tilt angle = 5.0”, fd) tilt angle 12 5.5”, (e) tiit angle = 6.0”.
emission curves were also obtained at one angle of tilt and are also shown. immediate features worthy of note are the definite structure in all three characteristics. There is a significant variation of rocking curve structure with electron energy, which is true of the secondary emission curve to a lesser extent. Variation of tilt angle produces minor but discernible variations in the rocking curve structure.
LOW
ENERGY
ELECTRON
DIFFRACTION
4. Pseudo-rocking
ROCKING
CURVES
217
curve observations
These were obtained over a range of energies encompassing excitation of the (770), (880) and (990) reflections. Some preliminary scanning was needed to obtain orientations which yielded curves of any particular significance. With the energy approximately that required for exciting the particular Bragg maximum of interest, the crystal was rotated until a sharp decrease in the specularly reflected beam was visually observed. This region was then explored until the desired characteristic was obtained; namely, a well defined decrease in intensity to the background level at the center of the specular Bragg peak. Observation of the diffraction spot as the energy was varied through the Bragg peak revealed that the diminution of spot intensity was associated with an apparent dark band moving across the diffraction spot, a phenomenon which has recently been reported elsewhere for significantly lower energies?). Fig. 2 depicts the appearance of the spot at an energy corresponding to the center of the Bragg peak. This strong dip in the Bragg maximum was only observed for particular ‘tilt’ angles, in contrast to the rocking curve structure which occurred for every ‘tilt’ angle examined. However, minor dips in the Bragg peak were more generally observed. The resulting pseudo-rocking curves obtained under these conditions are
Fig. 2. Sketch of the specular diffraction spot showing the dark band spot as it appears when the energy is at the center of the anomalous curve Bragg peak.
that crosses the pseudo-rocking
218
S. FRIEDMAN
AND
R. M.STERN
L
41 ,L
A0 ENERG”
do , r”
6bO
1
Fig. 3c Fig. 3. Pseudo-rocking curves with large dips in intensity at the center of the Bragg peak: (a) anomalous (770) peak; (b) anomalous (880) peak; (c) anomalous (990) peak.
shown in fig. 3 for orientations which result in sharp intensity dips at the centers of the (770) (880), and (990) Bragg peaks. The relative position of these dips with respect to the Bragg peak was found to be strongly dependent on the incident beam direction. The dip disappears if the incident angle is changed by more than 2-3”. This is borne out by the series of pseudo-rocking curves reproduced in fig. 4. These curves were obtained for many incident beam orientations in the vicinity of that orientation yielding a pronounced
LOW
?
’
300
ENERGY
ELECTRON
CURVES
I
DIFFRACTION
I
400
500
ROCKING
219
I
600
ENERGY
(av)
Fig. 4. Pseudo-rocking curves for various orientations in the vicinity of that required for generating an anomalous (770) Bragg peak. The energy at which the anomalous dip occurs relative to the center of the Bragg peak is seen to be strongly dependent on orientation. da = rotation angle increment; db = tilt angle increment.
dip in the (770) Bragg peak. The occurrence of the strong, central dip was also found to be independent of the condition of the crystal surface. Fig. 5 shows two pseudo-rocking curves, before and after high temperature flashing to remove surface adsorbed gases. The strong, central dip is seen to be present in both curves. The difference in relative intensity of the fractional order peaks is also evident in these curves. 5. Discussion The envelopes
of the same specular
Bragg reflection
of both the rocking
220
s. FRIKD~ANANDR.M.~~ER~
and pseudo-rocking curves discussed above, have full widths at half maximum that are mutually consistent in terms of crystal resolving power (dE/E) from the relationship derived in Appendix A. The significant structure observed within these envelopes in either case must consequently be a result of dynamical (e.g. multiple scattering) effects *. Various approaches to approximate dynamical treatment of electron diffraction in the three beam case have resulted in the following predictions regarding the intensity variation of the specular beam: 1) decrease in the intensity upon excitation of a simultaneous reflection from planes perpendicular to the surface 2, ; 2) intensity enhancement or minimization upon excitation of simultaneous reflections in the surface planes-?); 3) wave amplitude extinction upon exciting a particular portion of the dispersion surface of a Laue reflectionll,l7). Although all of these treatments suggest dynamical reasons behind the observed effects, they fall short of affording a complete explanation, primarily because of the fact that many more simultaneous reflections occur during a rotation than there are significant dips or peaks in the rocking curve. Those simultaneous reflections that would be considered important from the various dynamical treatments are summarized in table 1 for a particular ‘tilt angle’ and for three energies; the energy required to achieve a Bragg maximum and energies signi~cantly above and below this value. The rotation angles at which these particular reflections occur are also tabulated therein, having been computed using the relationship derived in Appendix B. This is by no means a complete tabulation of reflections but only encompasses: - low index Laue (forward)
reflections, h2 + k2 + 1’ < 16;
- reflections
close to the specular
Bragg reflection,
(h - fQ2 -t (k - 8)” -t- i2 < 16; - approximate surface reflections, h + k =8. A comparison of the tabulated simultaneous reflections to the observed dips and peaks reveals that the primary dips at energies near 460 eV coincide with the occurrence of the (013) and (002) Laue reflections. However, it is significant that neither the (i2i)reflection, which is of lower index than the * Kinematic diffraction theory predicts that the relative intensity of a particular Bragg peak is independent of the existence of sinlultaneous reflectionsr5). The introduction of finite crystal size and absorptionr6) predicts finite kinematic widths of both rocking and pseudo-rocking curves but not the fine structure observed in the curves reported herein. In the dynamical theory, these widths are proportional to the dynamical potentials of Bethel*).
LOW
ENERGY
ELECTRON
DIFFRACTION
li/^;
FfEFOqE
ROCKING
CURVES
FLASH
I
I
I
500
600
700
ENERGY
CeV)
Fig. 5. Anomalous pseudo-rocking curves before and after high temperature ffash to remove surface adsorbed gases. Note that the anomaly in the (770) peak is present in both instances. Also note the increase in overall intensity as well as the increase in the relative intensity of the half order maxima.
(013) reflection, nor the (103) reflection which has the same index as the (013) reflection, coincide with any major dips, although they both appear in table 1. Hence, neither index nor orientation with respect to the surface appear to have any direct relationship to the occurrence of dips in the rocking curve intensity. Similar observations can be made with respect to simultaneous reflections close to the specular Bragg reflections and reflections approximately parallel to the surface. For energies higher than 460 eV, there
222
S. FRIEDMAN
AND
TABLE
Tabulation energies
of the rotation and tilt angles
b
k
i
2 2 3 3 0 1
i i i
0 0 I 1 2 3 3 5 5 5 6 6 6
6 I 7
7
i
0
i
z
8 8 8 8 8 8 9 9 9 9 105i 10 5 1
460 eV
480 eV
4”
5”
6”
4”
0.95 -
1.22 _
3.45 _
1.59
_
_
_ _
0.57 _
2.78 _
1.40 10.38 4.13 12.84 2.82 7.98 12.35 9.35 -
2.59 10.38 4.99 12.13 2.60 7.27 10.53 -
4.42 10.38 5.25 11.42 2.38 6.56 11.68 _
2.11 10.16 4.62 11.79 2.23 6.74 14.87 I I .20 _
3.63 IO.16 4.88 1 I .08 2.02 6.03 10.37 12.34 _
9.19 _
6.49
-
0.63 10.16 4.36 12.51 2.45 7.45 _ 10.03 _ _
9.86 8.50 _ _
9.55 7.86 _ _
9.20 7.14 _ _
~ 2.95 11.76 II.03 _
II.66 5.95 I I .43 10.42 _
i
11.06 8.35
3 2 3
11.06 7.73 _
11.02 7.00 _
11.52 _
11.68 _
11.79 _
10.33 _
10.36 10.45 _
10.25 10.76 _
_
_
_
_
3.88 3.92 -
4.62 4.63 _
5.27 5.20 _
7.84 9.61 1 I .93 _
-
-
-
5”
6”
4”
5”
2.20
0.05
6”
1
i 1 Z 2 2 3 2 3 3 1
i z 9 9 9 I 8 9 9 6 8 8 6 I I 1 8 8 5 6 6 I
1
angles required to excite various reflections for several electron (b). The reflections are limited to those defined in the text
440 eV
h
R. M. STERN
z 0 2 3 z
i
I 3
i 10.40 3 0 2 2
i
3 0
1
_
7 2 II.01 1 7 4 9.77 2 6 ;1 5.37 3 5 6 7.82 4 4 6 4.84 5 3 6 5.70 6 2 4 10.11 6 2 6 10.66 713 7 1 4 0.14 802 8 0 2 2.82 _.__~~_~
_ 9.42 11.33 4.64 8.17 4.92 5.52 10.58 10.18 0.80 0.01 ~ ~~
0.81 12.05 I I .58 1.11
12.07 2.13, 8.87 13.16 13.16 12.62 12.16 5.32 6.11 14.23 14.37 _ _
13.13 11.62 7.07 14.47 _
14.22 2.81
14.11 2.99 _
14.19 2.86
_ 8.44 10.06 10.47 _
8.96 10.43 9.48
_
_
_
7.71 12.98 3.88 8.55 5.01 5.36 11.03 9.72 1.69 2.38 -
12.92 7.50 6.71 6.37 3.51 4.31 11.33 8.98 1.62 _ 1.06
0.09 _
_ 1.38 9.96 4.27 11.48 1.89 6.24 12.90 1 I .83 2.72 _ _
2.89 9.96 4.53 10.76 I .67 5.54 8.60 12.96 I.10
13.97 14.16 7.79
13.70 13.66 6.51 _
13.40 13.11 4.98
15.03 _
15.02
0.30 _
0.83 _
14.99 15.06 1.44
1.52 _ _
1.48
9.96 4.01 12.19 2.11 6.95 _ 10.66 4.12 _ _
_
Il.35 4.66 10.95 13.46 3.52 9.20
10.95 14.57 11.46 13.82 2.98 9.48
_ _ - 1 .17,9.33
.44 8.94 6.00 6.72 3.59 4.13 11.78 8.52 2.52 _
9.84 10.48 5.26 7.08 3.68 3.98 12.22 8.08 3.38 _
14.64 5.50 7.94 5.05 2.29 3.04 12.44 1.46 3.21
13.23 6.86 1.25 5.39 2.37 2.87 12.89 7.01 4.08 _
3.38 _
5.44
4.12
6.16
1I
_ 2.15
_
1.38 _ 10.44 6.97, 13.19 11.92 14.12 2.54 9.65 1.30 11.76 11.72 8.30 6.52 5.75 2.46 2.72 13.31 6.58 4.90 _ 8.00
LOW
ENERGY
ELECTRON
DIFFRACTION
ROCKING
i
(a)
ROTATION
ANGLE
(DEGREES)
ROTATION
223
CURVES
(b)
ANGLE (DEGREES)
Fig. 6. Histograms depicting the number of reflections excited between any two angles of rotation up to 12”. In each case the tilt angle is 5”. Shaded bars above the histograms correspond to intensity minima half widths. (a) Electron energy = 460 eV. (b) Electron energy = 470 eV.
appears to be some correlation between the peaks that occur at some angle of rotation and the incidence of satisfying the exact condition for the (880) reflection. However, even this cannot be completely supported, since there is usually more than one non-central peak in the rocking curve at these energies. In addition, non-central peaks are seen to exist at energies below 460 eV which precludes their being due to excitation of the (880) reflection. Brief consideration was given to the possibility that the major dips were associated with angles at which a relatively large number of simultaneous reflections would be excited. All of the reflections excited during a rotation excursion were determined by using the relationships derived in Appendix B for two energies of interest. The histogram plots shown in fig. 6 depict the density of simultaneous reflections at any particular angle. The shaded areas above the histogram correspond to the half widths of the major dips at these energies. The density of reflections is seen to be relatively high, although there seems to be some correlation between intensity minima and density peaks. The corollary secondary emission curves have been treated in detail elsewhere9). The primary dips in this characteristic are believed to be due to the excitation of the (002) reflection. In general, these curves show much less major structure than the corresponding rocking curve and hence imply somewhat different governing mechanisms. The pseudo-rocking curves shown exhibit strong anomalous dips at the
224
S. FRIEDMAN
AND R. M.STERN
---_--I
Fig. 7a
Fig. 7b
LOW
ENERGY
ELECTRON
DIFFRACTION
ROCKING
b ROTATION
225
CURVES
8
IO
12
ANGLE (DEGREES)
Fig. 7c Fig. 7. Intersections of the zone boundaries of important reflections with the specular reflection zone boundary (zone edges). The latter is in the plane of the drawing. The origin of the incident wave vectors corresponding to the center of the anomalous pseudorocking curves peaks are plotted as + signs on the specular reflection zone boundary plane to show their relationship to the zone edges. (a) (770) zone boundary plane; (b) (880) zone boundary plane; (c) (990) zone boundary plane. Only one of every four reflections is designated by its appropriate index. The other three may be derived from the two fold symmetry. The angular scale shown in the lower right quadrant determines the orientation required to place the origin of the incident wave vector at a particular point on a zone edge. Zone edges are not shown in this quadrant.
centers of the (770), (880) and (990) Bragg peaks for particular angles of incidence. To interpret these in terms of simultaneous reflections one may construct a diagram depicting the intersection of the particular Bragg peak Brillouin zone boundary and the zone boundaries of other possible simultaneous reflections. The origin of the incident wave vector at which these anomalous dips occur may then be plotted on this same diagram and their relationship to any particular reflection may be seen. The Brillouin zone boundary intersections for the three specular reflections of interest are shown in fig. 7. In many instances, the zone boundary intersections for two or more reflections are seen to coincide, so that a unique association with a particular reflection may not be obtained for these reflections. The points at which the anomalous dips occur fall relatively close to one or more zone
226
S.FRIEDMAN
AND
R. M. STERN
edges, some of which correspond to two or more reflections. Experimental uncertainty precludes a definite association with a particular zone edge. For these two reasons, a unique association cannot be established from the results thusfar obtained. In addition, the absence of a strong dip in some of the pseudo-rocking curves shown in fig. 4 is observed under conditions that still excite the same simultaneous reflections, which indicates that the underlying explanation must involve more than the occurrence of these reflections. Further evidence of this may be obtained. from consideration of the threebeam dynamical treatmentIT) in which a dip in the specular Bragg peak is predicted if a Laue reflection is excited at a point on the corresponding dispersion surface such that the amplitude of the specularly reflected wave vanishes. As is shown in Appendix C, it is generally impossible to simultaneously satisfy the exact kinematic conditions for a specular Bragg and Laue reflection and at the same time satisfy the condition for vanishing Bragg amplitude. if one exactly satisfies the kinematic condition for specular Bragg reflection only, it is further shown that the incident wave vector must originate on a line defined by the intersection of the Brillouin zone boundary of the specular Bragg reflection and a plane parallel to the Brillouin zone boundary of the Laue reflection under consideration, the displacement of this plane being a function of the corresponding Fourier coefficients of the potential. Although this condition can be satisfied by the incident wave vector originating anywhere along this line, in fact, the phenomenon is observed in relatively few instances. Thus, as in the rocking curves, simultaneous reflections appear to be associated with the structure, but in a manner that involves more than the simultaneity of their occurrence. In the theoretical treatment of the structure of the secondary emission characteristicg), the point is made that reflections of low index (k2 +k2 + i2 < small number) also have a retatively high intensity, as is predicted by kinematic theory. Consequently, one may hypothesize that it is only reflections of the very lowest indices that are involved. However, on the basis of both the rocking and pseudo-rocking curves, this cannot be completely supported. For example, the rocking curve structure appears to be associated with the (013) reflection but not with the (103) reflection which has the same index nor with the (121) reflection which has a lower index. The number of reflections that are excited in the course of a relatively small rotation for the range of electron energies under investigation as may be seen from fig. 6, implies that for any particular angle of incidence, a great many reciprocal lattice vectors are close to the sphere of reflection. Hence, analysis in terms of simple 3 beam dynamical theory may not be exact enough to predict the observed structure. A more exact treatment would involve machine computation and some
LOW
ENERGY
ELECTRON
DIFFRACTION
ROCKING
CURVES
227
approximation of the crystal potential. This refinement has not as yet been fruitful in explaining the observed results. The previously mentioned selfconsistent multiple scattering treatment6), in which a ‘muon-tin’ potential function was assumed, predicted intensity minima in all reflected beams upon simultaneous excitation of a reflection parallel to the surface. The results presented herein indicate that many more reflections are excited than there are intensity minima, although the observed intensity minima could be associated with surface reflections within the present limits of experimental uncertainty. it has been pointed out’) that the occurrence of reflectivity minima upon excitation of any surface reflection is essentially a twodimensional phenomenon and that a more exact three dimensional treatment should result in a substantial restriction of the conditions under which intensity minima will occur. 6. Conclusions Rocking curves and pseudo-rocking curves of the LEED specularly reflected beam from a (110) tungsten surface exhibit structure which is not accountable for by any kinematic theory and while dynamical, is not simply accounted for in terms of current dynamical theory. Structure observed at lower energies where there are fewer simultaneous reflections to be accounted for and diffraction from thin films allowing direct observance of transmitted beams may lend additional insight into the mechanics governing the observed structure. Further refinement of present dynamical theory is aIso suggested.
Acknowledgment Thanks are due to Professor H. Wagenfeld and Mr. H. Taub for many helpful discussions and to Messrs. E. Esposito and P. Goldstone for obtaining all of the rocking curves and secondary emission curves presented herein. Appendix A Assume that the width of the specular Bragg peak of the pseudo rocking curve is associated with an effective finite extent of the reciprocal lattice point along the reciprocal lattice vector. To satisfy the Bragg condition at B, the incident wave vector k must satisfy
To satisfy the condition
at B- AB, where AB is along B and is determined
228
S.FRlEDMANAND
by the pseudo-rocking
R.M.STERN
curve width, k must change correspondingly
so that
2k’*(B - AB) + (B - AB)2 = 0, where k’ is the new wave vector required For a rocking
curve, the magnitude
to satisfy this condition.
of the wave vector is constant,
i.e.
lk’l = Iki, so that 2Ikl IBI cos6’where 0 is the rocking
2jkl (LIB) cos8+B2-2~L?~(dB)+LlB~=0
angle. For initially B=-2k
so that, ignoring
and
normal
incidence,
AB=-2Ak,
AB2,
-4k2cosO+4Jk(
AkcosQ+4k2
-8Jkl
Ak=O.
Solving for cos6, we obtain cos 0 = l-_2dkilk’ 1 - Ak/lkl
ZI-A;.
E = h2k2,
= ;AEIE.
Since Akjk
For small tI c0se=i-'e2
2
.
Hence 1-
&e2 =
1 - AE/2E
and
8 = J
(AE/E) .
This relationship enables computation of the width of the rocking Bragg peak, given the width of the pseudo-rocking curve peak.
curve
Appendix B The angles in table 1 are computed by assuming that the energy required to excite the (880) reflection at normal incidence is 458 eV. Assuming an inner potential of 20 eV, the radius of the Ewald sphere in reciprocal lattice units for a particular energy V is given by Irl = J[(V
+ 20)/478]442.
A simple cubic lattice was assumed. The fact that tungsten is body-centered cubic is accounted for by ignoring all those reflections of odd index (h + k + I= odd integer). The angle at which a particular reflection is satisfied is determined by satisfying the relationship 2r.B
+ B2 = 0.
LOW
The angles
ENERGY
ELECTRON
of tilt and rotation
DIFFRACTION
are referred
ROCKING
229
CURVES
to the (110) direction.
Conse-
quently rX = - /r-l cos $rt cos a (cos b + sin b), ry = - Irl cos &r cos a (cos b - sin b), rz = - IJ-/sin a, where b is the tilt angle and a is the rotation Since B, = h, B,, = k and Bz = 1,
angle.
h2 + k2 + l2 -2Irl[cos$ncosa{(cosb+sinb)h+(cosb-sinb)k)+Zsina]=O. The value of a required to excite a particular reflection at a particular of tilt is obtained by solving the above equation for a.
angle
Appendix C The Bethe dispersion
equations,
in the three-beam
approximationr5)
reduce
to k2 - K,2 k2!K;
vB
(
VI‘
vBv!L
UL -
$;
k2-K;
B
>
0
4L
=o (1)
wherein k is the incident wave vector in vacua (incorporating the refractive increase due to the crystal inner potential), K, the incident wave vector in the crystal, and K, and K, the diffracted wave vectors in the crystal corresponding the reciprocal lattice vectors B and L, while vg, vL and us-L denote the corresponding Fourier coefficients of the crystal periodic potential, in appropriate units. &,, 4B and 4L denote the amplitudes of the initially assumed corresponding plane waves in the crystal. By definition, &=&+B,
(2)
K,=K,+L.
(3)
Since the set of eqs. (1) are homogeneous, they have non-trivial solutions for only those values ofK, that result in the vanishing of the secular determinant. These values of K, generate the well known dispersion surface in reciprocal space. In addition, one may inquire as to what auxiliary conditions imposed on K, will result in a zero value for 4B/40. As is shown in detail elsewherer’), two additional conditions are required, namely k2 - K,2 = vsvL/vs_ L,
(4)
and 2&.
L
+
L2
=
_‘zB
_
!Lv!:L. (5)
VB-L
vB
230
S.FRIEDMAN
AND R. M. STERN
To apply these auxiliary equations to the pseudo-rocking curves observations, it is further assumed B is perpendicular to the surface (specular Bragg reflection) and that K,=k+Ak, where, to satisfy the required to the surface. Hence
boundary
(6) conditions,
Ak is also perpendicular
Ak=aB,
(7)
where the scalar c( is to be determined. Substituting and (5) and, neglecting Ak* Ak, we obtain
eqs. (6) and (7) into (4)
2cck. B = v~v&,_~, 2k.L+L2+2aB.L=
~??!t_v!!vB-~L~ vB-L
If it is now assumed that k satisfies Bragg reflection B, then
(8)
the exact kinematic
2k.B=Substituting eq. (10) into eq. (8), solving into eq. (9), we obtain
(9)
vB
condition
for the
B2. for CIand then substituting
(10) the result
From eq. (I 1) it is first seen that it is only possible to satisfy the kinematic conditions for Bragg reflection for the Laue rellection L for the special situation for which the right-hand side of eq. (11) vanishes. In general, eq. (11) defines a plane parallel to the Brillouin zone boundary of the lattice vector L which is the loci of the origin of all k vectors that will result in the vanishing of 4B/40. The intersection of this plane with the Brillouin zone boundary plane of the reciprocal lattice vector B as defined by eq. (10) thus defines a line which is the loci of the origin of all k vectors that will simultaneously satisfy the kinematic condition for the Bragg maximum for the specular reflection corresponding to B and the condition for ~~Jc#J~=O. These lines are parallel to the zone boundary intersections depicted in fig. 7 and are shifted by an amount determined by the right hand side of eq. (I 1).
References 1) 2) 3) 4)
C. J. Davisson and L. H. Germer, Proc. Natl. Acad. Sci. (US) 14 (1928) 624. P. M. Morse, Phys. Rev. 35 (1930) 1310. H. E. Farnsworth, Phys. Rev. 40 (1932) 684. W. E. Laschkarew, Trans. Faraday Sot. 31 (1935) 1081.
LOW
5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)
ENERGY
ELECTRON
DIFFRACTION
ROCKING
CURVES
S. Miyake, K. Kohra and M. Takagi, Acta Cryst. 7 (1954) 393. E. G. McRae, J. Chem. Phys. 45 (1966) 1258. E. G. McRae and C. W. Caldwell, Surface Sci. 7 (1967) 41. A. Gervais, R. M. Stern and M. Menes, Acta Cryst. A24 (1968) 191. R. M. Stern and H. Taub, Phys. Rev. Letters 20 (1968) 1340. D. S. Boudreaux and V. Heine, Surface Sci. 8 (1967) 426. H. Niehrs, Z. Physik 139 (1954) 88. K. Kohra, J. Phys. Sot. Japan 9 (1954) 690. R. Baudoing, R. M. Stern and H. Taub, Surface Sci. 11 (1968) 255. R. M. Stern, Appl. Phys. Letters 5 (1964) 218. R. D. Heidenreich, Phys Rev. 77 (1950) 271. J. J. Lander, Progr. Solid State Chem. 2 (1965) 26. R. M. Stern, J. Perry and D. Boudreaux, Rev. Mod. Phys. 41 (1969) 275. H. Bethe, Ann. Physik 87 (1928) 55.
231
LOW
SCIENCE 17 (1969) 214-231 0 North-HolIand
ENERGY
ELECTRON
DIFFRACTION S. FRIEDMAN
Naval Applied Science Laboratory,
Publishing Co., Amsterdam
ROCKING
CURVES t
*
Brooklyn, New York 11251, U.S.A.
and R. M. STERN department of Physics, Polytechnic institute of Brooklyn, Brooklyn, New York 11201, U.S.A.
Received 21 March 1969 The variation of specularly reflected beam intensity with angle of incidence and incident beam energy from the (110) surface of a tungsten single crystal for incident electron energies in the 300-600 eV range and for angles of incidence within 15’ of the surface normal was found to have a structure that could only be explained in dynamical terms. Further analysis indicates that present dynamical theories are only partially successful in explaining the observed intensity variations. 1. Introduction
The variation of the diffracted beam intensity with diffraction angle, commonly known as a rocking curve, is employed in both X-ray diffraction and LEED to explore the reciprocal lattice of the crystal in the neighborhood of a particular reflection. Because of the ease of variation of electron wavelength in LEED, this exploration is also carried out by study of the intensity variation with wavelength, i.e., the pseudo-rocking curve. Both types of measurements have been made in this laboratory and have exhibited a structure that is not explicable in terms of simple kinematic theory for ranges of electron energy for which this approximation is usually deemed valid. A brief summary of earlier corollary results will precede the description and analysis of these observations. The earliest reports of dynamical effects in pseudo-rocking curvesr) were analyzed by means of a simplified dynamical theorya) which predicted the observed discontinuity in the angular dependence of the energy of the specular Bragg maximum. The angles at which these discontinuities occurred corresponded to those angles and energies which were required to excite simult Supported in part by USAF OSR Grant 1263-67. * Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physics at the Polytechnic Institute of Brooklyn. 214
LOW
ENERGY
ELECTRON
DIFFRACTION
ROCKING
CURVES
215
taneous reflections from planes perpendicular to the reflecting surface. Observed anomalous structure in the specular beam pseudo-rocking curves) was interpreted as being due to simultaneous reflections4). However, no quantitative treatment was attempted. More recently5), application of dynamical theory has predicted the enhancement in intensity of a specularly reflected beam upon the simultaneous occurrence of a diffracted beam in the plane of the crystal surface. On the other hand, a self consistent multiple scattering approachc) predicts both a maximum and minima in specular beam intensities upon excitation of reflections parallel to the surface*. These have been observed in studies of the (001) surface of LIFT). Finally, studies of rotation diagramss) and the angular dependence of single crystal secondary emissiong) in the same energy range as the rocking curve investigations lend support to the dynamical influence on reflected beam intensities and the indirect evidence of the existence of strong scattering into the crystal which must be considered in any dynamical treatment. While this survey is by no means complete, it does indicate that dynamical effects may be expected in both the rocking curve and pseudo-rocking curve. In addition, a study of the dynamical theory6,r0-12) to date indicates that no really complete explanation of these effects has as yet been formulated, although several theoretical treatments result in partial success.
2. Experimental
technique
All quantitative observations were made on the (0,O) (specularly reflected) beam using the (110) surface of a tungsten single crystal in a standard Varian LEED apparatus. Surface orientation and preparation are described in detail elsewherels). Surface cleaning in vacua and removal of surface carbon also followed earlier practice14). A Pritchard spectrophotometer was used to record relative diffracted beam intensity. Uncertainty in the incident beam direction due to stray magnetic fields was removed by defining normal incidence as that orientation which achieved either symmetry of Bragg maxima in symmetric azimuths or symmetry of angular variation of the total crystal current. The two angles used to specify the orientation of the surface normal are defined as rotation about an ax is parallel to the (001) direction in the plane of the crystal surface (angle of tilt) and rotation about an axis parallel to the (ITO) direction in the plane of the crystal surface (angle of rotation)**. * This prediction may be an artifact resulting from the assumption of a purely s-wave scattering process. ** The crystal orientation was such that the (001) axis in the plane of the crystal surface formed an angle of about 5” with the horizontal axis of rotation thereby inducing some degree of asymmetry in the observations.
216
S. FRIEDMAN
AND
R. M. STERN
Using the technique described above, rocking curves were obtained for a range of electrcrn energies which included the (880) specular reflection. The reslrftant traces are shown in fig. 1 for each angle of ‘tilt’ studied. Secondary
4.0.
4.5'
5.0’
5.9’
b.0’
490.6
485.5 4
80.0
475.0 473.0 471.5 469
0
468.0 467.0
466.0 465.0 464.0 463.0 461.0 459.0 4520 455.0 454.5 454.0 453 D 451.0 449.0 447.21 4450 44Q.Q 435.0 430.0
Fig. 1. Left han.d curve: Total secondary emission (crystal current to ground) as a function of incident angle for a series of energies within the (880) pseudo-rocking curve Bragg r&e&-m. Tilt angle = 4.5”. Right hand curves : Racking curves (total integrated intensity} for various energies bounded by the range of the (880) pseudo-rocking curve Bragg reflection; (a) tilt angIe I= 4”, (b) tilt angle =4.5”,(c) tilt angle = 5.0”, fd) tilt angle 12 5.5”, (e) tiit angle = 6.0”.
emission curves were also obtained at one angle of tilt and are also shown. immediate features worthy of note are the definite structure in all three characteristics. There is a significant variation of rocking curve structure with electron energy, which is true of the secondary emission curve to a lesser extent. Variation of tilt angle produces minor but discernible variations in the rocking curve structure.
LOW
ENERGY
ELECTRON
DIFFRACTION
4. Pseudo-rocking
ROCKING
CURVES
217
curve observations
These were obtained over a range of energies encompassing excitation of the (770), (880) and (990) reflections. Some preliminary scanning was needed to obtain orientations which yielded curves of any particular significance. With the energy approximately that required for exciting the particular Bragg maximum of interest, the crystal was rotated until a sharp decrease in the specularly reflected beam was visually observed. This region was then explored until the desired characteristic was obtained; namely, a well defined decrease in intensity to the background level at the center of the specular Bragg peak. Observation of the diffraction spot as the energy was varied through the Bragg peak revealed that the diminution of spot intensity was associated with an apparent dark band moving across the diffraction spot, a phenomenon which has recently been reported elsewhere for significantly lower energies?). Fig. 2 depicts the appearance of the spot at an energy corresponding to the center of the Bragg peak. This strong dip in the Bragg maximum was only observed for particular ‘tilt’ angles, in contrast to the rocking curve structure which occurred for every ‘tilt’ angle examined. However, minor dips in the Bragg peak were more generally observed. The resulting pseudo-rocking curves obtained under these conditions are
Fig. 2. Sketch of the specular diffraction spot showing the dark band spot as it appears when the energy is at the center of the anomalous curve Bragg peak.
that crosses the pseudo-rocking
218
S. FRIEDMAN
AND
R. M.STERN
L
41 ,L
A0 ENERG”
do , r”
6bO
1
Fig. 3c Fig. 3. Pseudo-rocking curves with large dips in intensity at the center of the Bragg peak: (a) anomalous (770) peak; (b) anomalous (880) peak; (c) anomalous (990) peak.
shown in fig. 3 for orientations which result in sharp intensity dips at the centers of the (770) (880), and (990) Bragg peaks. The relative position of these dips with respect to the Bragg peak was found to be strongly dependent on the incident beam direction. The dip disappears if the incident angle is changed by more than 2-3”. This is borne out by the series of pseudo-rocking curves reproduced in fig. 4. These curves were obtained for many incident beam orientations in the vicinity of that orientation yielding a pronounced
LOW
?
’
300
ENERGY
ELECTRON
CURVES
I
DIFFRACTION
I
400
500
ROCKING
219
I
600
ENERGY
(av)
Fig. 4. Pseudo-rocking curves for various orientations in the vicinity of that required for generating an anomalous (770) Bragg peak. The energy at which the anomalous dip occurs relative to the center of the Bragg peak is seen to be strongly dependent on orientation. da = rotation angle increment; db = tilt angle increment.
dip in the (770) Bragg peak. The occurrence of the strong, central dip was also found to be independent of the condition of the crystal surface. Fig. 5 shows two pseudo-rocking curves, before and after high temperature flashing to remove surface adsorbed gases. The strong, central dip is seen to be present in both curves. The difference in relative intensity of the fractional order peaks is also evident in these curves. 5. Discussion The envelopes
of the same specular
Bragg reflection
of both the rocking
220
s. FRIKD~ANANDR.M.~~ER~
and pseudo-rocking curves discussed above, have full widths at half maximum that are mutually consistent in terms of crystal resolving power (dE/E) from the relationship derived in Appendix A. The significant structure observed within these envelopes in either case must consequently be a result of dynamical (e.g. multiple scattering) effects *. Various approaches to approximate dynamical treatment of electron diffraction in the three beam case have resulted in the following predictions regarding the intensity variation of the specular beam: 1) decrease in the intensity upon excitation of a simultaneous reflection from planes perpendicular to the surface 2, ; 2) intensity enhancement or minimization upon excitation of simultaneous reflections in the surface planes-?); 3) wave amplitude extinction upon exciting a particular portion of the dispersion surface of a Laue reflectionll,l7). Although all of these treatments suggest dynamical reasons behind the observed effects, they fall short of affording a complete explanation, primarily because of the fact that many more simultaneous reflections occur during a rotation than there are significant dips or peaks in the rocking curve. Those simultaneous reflections that would be considered important from the various dynamical treatments are summarized in table 1 for a particular ‘tilt angle’ and for three energies; the energy required to achieve a Bragg maximum and energies signi~cantly above and below this value. The rotation angles at which these particular reflections occur are also tabulated therein, having been computed using the relationship derived in Appendix B. This is by no means a complete tabulation of reflections but only encompasses: - low index Laue (forward)
reflections, h2 + k2 + 1’ < 16;
- reflections
close to the specular
Bragg reflection,
(h - fQ2 -t (k - 8)” -t- i2 < 16; - approximate surface reflections, h + k =8. A comparison of the tabulated simultaneous reflections to the observed dips and peaks reveals that the primary dips at energies near 460 eV coincide with the occurrence of the (013) and (002) Laue reflections. However, it is significant that neither the (i2i)reflection, which is of lower index than the * Kinematic diffraction theory predicts that the relative intensity of a particular Bragg peak is independent of the existence of sinlultaneous reflectionsr5). The introduction of finite crystal size and absorptionr6) predicts finite kinematic widths of both rocking and pseudo-rocking curves but not the fine structure observed in the curves reported herein. In the dynamical theory, these widths are proportional to the dynamical potentials of Bethel*).
LOW
ENERGY
ELECTRON
DIFFRACTION
li/^;
FfEFOqE
ROCKING
CURVES
FLASH
I
I
I
500
600
700
ENERGY
CeV)
Fig. 5. Anomalous pseudo-rocking curves before and after high temperature ffash to remove surface adsorbed gases. Note that the anomaly in the (770) peak is present in both instances. Also note the increase in overall intensity as well as the increase in the relative intensity of the half order maxima.
(013) reflection, nor the (103) reflection which has the same index as the (013) reflection, coincide with any major dips, although they both appear in table 1. Hence, neither index nor orientation with respect to the surface appear to have any direct relationship to the occurrence of dips in the rocking curve intensity. Similar observations can be made with respect to simultaneous reflections close to the specular Bragg reflections and reflections approximately parallel to the surface. For energies higher than 460 eV, there
222
S. FRIEDMAN
AND
TABLE
Tabulation energies
of the rotation and tilt angles
b
k
i
2 2 3 3 0 1
i i i
0 0 I 1 2 3 3 5 5 5 6 6 6
6 I 7
7
i
0
i
z
8 8 8 8 8 8 9 9 9 9 105i 10 5 1
460 eV
480 eV
4”
5”
6”
4”
0.95 -
1.22 _
3.45 _
1.59
_
_
_ _
0.57 _
2.78 _
1.40 10.38 4.13 12.84 2.82 7.98 12.35 9.35 -
2.59 10.38 4.99 12.13 2.60 7.27 10.53 -
4.42 10.38 5.25 11.42 2.38 6.56 11.68 _
2.11 10.16 4.62 11.79 2.23 6.74 14.87 I I .20 _
3.63 IO.16 4.88 1 I .08 2.02 6.03 10.37 12.34 _
9.19 _
6.49
-
0.63 10.16 4.36 12.51 2.45 7.45 _ 10.03 _ _
9.86 8.50 _ _
9.55 7.86 _ _
9.20 7.14 _ _
~ 2.95 11.76 II.03 _
II.66 5.95 I I .43 10.42 _
i
11.06 8.35
3 2 3
11.06 7.73 _
11.02 7.00 _
11.52 _
11.68 _
11.79 _
10.33 _
10.36 10.45 _
10.25 10.76 _
_
_
_
_
3.88 3.92 -
4.62 4.63 _
5.27 5.20 _
7.84 9.61 1 I .93 _
-
-
-
5”
6”
4”
5”
2.20
0.05
6”
1
i 1 Z 2 2 3 2 3 3 1
i z 9 9 9 I 8 9 9 6 8 8 6 I I 1 8 8 5 6 6 I
1
angles required to excite various reflections for several electron (b). The reflections are limited to those defined in the text
440 eV
h
R. M. STERN
z 0 2 3 z
i
I 3
i 10.40 3 0 2 2
i
3 0
1
_
7 2 II.01 1 7 4 9.77 2 6 ;1 5.37 3 5 6 7.82 4 4 6 4.84 5 3 6 5.70 6 2 4 10.11 6 2 6 10.66 713 7 1 4 0.14 802 8 0 2 2.82 _.__~~_~
_ 9.42 11.33 4.64 8.17 4.92 5.52 10.58 10.18 0.80 0.01 ~ ~~
0.81 12.05 I I .58 1.11
12.07 2.13, 8.87 13.16 13.16 12.62 12.16 5.32 6.11 14.23 14.37 _ _
13.13 11.62 7.07 14.47 _
14.22 2.81
14.11 2.99 _
14.19 2.86
_ 8.44 10.06 10.47 _
8.96 10.43 9.48
_
_
_
7.71 12.98 3.88 8.55 5.01 5.36 11.03 9.72 1.69 2.38 -
12.92 7.50 6.71 6.37 3.51 4.31 11.33 8.98 1.62 _ 1.06
0.09 _
_ 1.38 9.96 4.27 11.48 1.89 6.24 12.90 1 I .83 2.72 _ _
2.89 9.96 4.53 10.76 I .67 5.54 8.60 12.96 I.10
13.97 14.16 7.79
13.70 13.66 6.51 _
13.40 13.11 4.98
15.03 _
15.02
0.30 _
0.83 _
14.99 15.06 1.44
1.52 _ _
1.48
9.96 4.01 12.19 2.11 6.95 _ 10.66 4.12 _ _
_
Il.35 4.66 10.95 13.46 3.52 9.20
10.95 14.57 11.46 13.82 2.98 9.48
_ _ - 1 .17,9.33
.44 8.94 6.00 6.72 3.59 4.13 11.78 8.52 2.52 _
9.84 10.48 5.26 7.08 3.68 3.98 12.22 8.08 3.38 _
14.64 5.50 7.94 5.05 2.29 3.04 12.44 1.46 3.21
13.23 6.86 1.25 5.39 2.37 2.87 12.89 7.01 4.08 _
3.38 _
5.44
4.12
6.16
1I
_ 2.15
_
1.38 _ 10.44 6.97, 13.19 11.92 14.12 2.54 9.65 1.30 11.76 11.72 8.30 6.52 5.75 2.46 2.72 13.31 6.58 4.90 _ 8.00
LOW
ENERGY
ELECTRON
DIFFRACTION
ROCKING
i
(a)
ROTATION
ANGLE
(DEGREES)
ROTATION
223
CURVES
(b)
ANGLE (DEGREES)
Fig. 6. Histograms depicting the number of reflections excited between any two angles of rotation up to 12”. In each case the tilt angle is 5”. Shaded bars above the histograms correspond to intensity minima half widths. (a) Electron energy = 460 eV. (b) Electron energy = 470 eV.
appears to be some correlation between the peaks that occur at some angle of rotation and the incidence of satisfying the exact condition for the (880) reflection. However, even this cannot be completely supported, since there is usually more than one non-central peak in the rocking curve at these energies. In addition, non-central peaks are seen to exist at energies below 460 eV which precludes their being due to excitation of the (880) reflection. Brief consideration was given to the possibility that the major dips were associated with angles at which a relatively large number of simultaneous reflections would be excited. All of the reflections excited during a rotation excursion were determined by using the relationships derived in Appendix B for two energies of interest. The histogram plots shown in fig. 6 depict the density of simultaneous reflections at any particular angle. The shaded areas above the histogram correspond to the half widths of the major dips at these energies. The density of reflections is seen to be relatively high, although there seems to be some correlation between intensity minima and density peaks. The corollary secondary emission curves have been treated in detail elsewhere9). The primary dips in this characteristic are believed to be due to the excitation of the (002) reflection. In general, these curves show much less major structure than the corresponding rocking curve and hence imply somewhat different governing mechanisms. The pseudo-rocking curves shown exhibit strong anomalous dips at the
224
S. FRIEDMAN
AND R. M.STERN
---_--I
Fig. 7a
Fig. 7b
LOW
ENERGY
ELECTRON
DIFFRACTION
ROCKING
b ROTATION
225
CURVES
8
IO
12
ANGLE (DEGREES)
Fig. 7c Fig. 7. Intersections of the zone boundaries of important reflections with the specular reflection zone boundary (zone edges). The latter is in the plane of the drawing. The origin of the incident wave vectors corresponding to the center of the anomalous pseudorocking curves peaks are plotted as + signs on the specular reflection zone boundary plane to show their relationship to the zone edges. (a) (770) zone boundary plane; (b) (880) zone boundary plane; (c) (990) zone boundary plane. Only one of every four reflections is designated by its appropriate index. The other three may be derived from the two fold symmetry. The angular scale shown in the lower right quadrant determines the orientation required to place the origin of the incident wave vector at a particular point on a zone edge. Zone edges are not shown in this quadrant.
centers of the (770), (880) and (990) Bragg peaks for particular angles of incidence. To interpret these in terms of simultaneous reflections one may construct a diagram depicting the intersection of the particular Bragg peak Brillouin zone boundary and the zone boundaries of other possible simultaneous reflections. The origin of the incident wave vector at which these anomalous dips occur may then be plotted on this same diagram and their relationship to any particular reflection may be seen. The Brillouin zone boundary intersections for the three specular reflections of interest are shown in fig. 7. In many instances, the zone boundary intersections for two or more reflections are seen to coincide, so that a unique association with a particular reflection may not be obtained for these reflections. The points at which the anomalous dips occur fall relatively close to one or more zone
226
S.FRIEDMAN
AND
R. M. STERN
edges, some of which correspond to two or more reflections. Experimental uncertainty precludes a definite association with a particular zone edge. For these two reasons, a unique association cannot be established from the results thusfar obtained. In addition, the absence of a strong dip in some of the pseudo-rocking curves shown in fig. 4 is observed under conditions that still excite the same simultaneous reflections, which indicates that the underlying explanation must involve more than the occurrence of these reflections. Further evidence of this may be obtained. from consideration of the threebeam dynamical treatmentIT) in which a dip in the specular Bragg peak is predicted if a Laue reflection is excited at a point on the corresponding dispersion surface such that the amplitude of the specularly reflected wave vanishes. As is shown in Appendix C, it is generally impossible to simultaneously satisfy the exact kinematic conditions for a specular Bragg and Laue reflection and at the same time satisfy the condition for vanishing Bragg amplitude. if one exactly satisfies the kinematic condition for specular Bragg reflection only, it is further shown that the incident wave vector must originate on a line defined by the intersection of the Brillouin zone boundary of the specular Bragg reflection and a plane parallel to the Brillouin zone boundary of the Laue reflection under consideration, the displacement of this plane being a function of the corresponding Fourier coefficients of the potential. Although this condition can be satisfied by the incident wave vector originating anywhere along this line, in fact, the phenomenon is observed in relatively few instances. Thus, as in the rocking curves, simultaneous reflections appear to be associated with the structure, but in a manner that involves more than the simultaneity of their occurrence. In the theoretical treatment of the structure of the secondary emission characteristicg), the point is made that reflections of low index (k2 +k2 + i2 < small number) also have a retatively high intensity, as is predicted by kinematic theory. Consequently, one may hypothesize that it is only reflections of the very lowest indices that are involved. However, on the basis of both the rocking and pseudo-rocking curves, this cannot be completely supported. For example, the rocking curve structure appears to be associated with the (013) reflection but not with the (103) reflection which has the same index nor with the (121) reflection which has a lower index. The number of reflections that are excited in the course of a relatively small rotation for the range of electron energies under investigation as may be seen from fig. 6, implies that for any particular angle of incidence, a great many reciprocal lattice vectors are close to the sphere of reflection. Hence, analysis in terms of simple 3 beam dynamical theory may not be exact enough to predict the observed structure. A more exact treatment would involve machine computation and some
LOW
ENERGY
ELECTRON
DIFFRACTION
ROCKING
CURVES
227
approximation of the crystal potential. This refinement has not as yet been fruitful in explaining the observed results. The previously mentioned selfconsistent multiple scattering treatment6), in which a ‘muon-tin’ potential function was assumed, predicted intensity minima in all reflected beams upon simultaneous excitation of a reflection parallel to the surface. The results presented herein indicate that many more reflections are excited than there are intensity minima, although the observed intensity minima could be associated with surface reflections within the present limits of experimental uncertainty. it has been pointed out’) that the occurrence of reflectivity minima upon excitation of any surface reflection is essentially a twodimensional phenomenon and that a more exact three dimensional treatment should result in a substantial restriction of the conditions under which intensity minima will occur. 6. Conclusions Rocking curves and pseudo-rocking curves of the LEED specularly reflected beam from a (110) tungsten surface exhibit structure which is not accountable for by any kinematic theory and while dynamical, is not simply accounted for in terms of current dynamical theory. Structure observed at lower energies where there are fewer simultaneous reflections to be accounted for and diffraction from thin films allowing direct observance of transmitted beams may lend additional insight into the mechanics governing the observed structure. Further refinement of present dynamical theory is aIso suggested.
Acknowledgment Thanks are due to Professor H. Wagenfeld and Mr. H. Taub for many helpful discussions and to Messrs. E. Esposito and P. Goldstone for obtaining all of the rocking curves and secondary emission curves presented herein. Appendix A Assume that the width of the specular Bragg peak of the pseudo rocking curve is associated with an effective finite extent of the reciprocal lattice point along the reciprocal lattice vector. To satisfy the Bragg condition at B, the incident wave vector k must satisfy
To satisfy the condition
at B- AB, where AB is along B and is determined
228
S.FRlEDMANAND
by the pseudo-rocking
R.M.STERN
curve width, k must change correspondingly
so that
2k’*(B - AB) + (B - AB)2 = 0, where k’ is the new wave vector required For a rocking
curve, the magnitude
to satisfy this condition.
of the wave vector is constant,
i.e.
lk’l = Iki, so that 2Ikl IBI cos6’where 0 is the rocking
2jkl (LIB) cos8+B2-2~L?~(dB)+LlB~=0
angle. For initially B=-2k
so that, ignoring
and
normal
incidence,
AB=-2Ak,
AB2,
-4k2cosO+4Jk(
AkcosQ+4k2
-8Jkl
Ak=O.
Solving for cos6, we obtain cos 0 = l-_2dkilk’ 1 - Ak/lkl
ZI-A;.
E = h2k2,
= ;AEIE.
Since Akjk
For small tI c0se=i-'e2
2
.
Hence 1-
&e2 =
1 - AE/2E
and
8 = J
(AE/E) .
This relationship enables computation of the width of the rocking Bragg peak, given the width of the pseudo-rocking curve peak.
curve
Appendix B The angles in table 1 are computed by assuming that the energy required to excite the (880) reflection at normal incidence is 458 eV. Assuming an inner potential of 20 eV, the radius of the Ewald sphere in reciprocal lattice units for a particular energy V is given by Irl = J[(V
+ 20)/478]442.
A simple cubic lattice was assumed. The fact that tungsten is body-centered cubic is accounted for by ignoring all those reflections of odd index (h + k + I= odd integer). The angle at which a particular reflection is satisfied is determined by satisfying the relationship 2r.B
+ B2 = 0.
LOW
The angles
ENERGY
ELECTRON
of tilt and rotation
DIFFRACTION
are referred
ROCKING
229
CURVES
to the (110) direction.
Conse-
quently rX = - /r-l cos $rt cos a (cos b + sin b), ry = - Irl cos &r cos a (cos b - sin b), rz = - IJ-/sin a, where b is the tilt angle and a is the rotation Since B, = h, B,, = k and Bz = 1,
angle.
h2 + k2 + l2 -2Irl[cos$ncosa{(cosb+sinb)h+(cosb-sinb)k)+Zsina]=O. The value of a required to excite a particular reflection at a particular of tilt is obtained by solving the above equation for a.
angle
Appendix C The Bethe dispersion
equations,
in the three-beam
approximationr5)
reduce
to k2 - K,2 k2!K;
vB
(
VI‘
vBv!L
UL -
$;
k2-K;
B
>
0
4L
=o (1)
wherein k is the incident wave vector in vacua (incorporating the refractive increase due to the crystal inner potential), K, the incident wave vector in the crystal, and K, and K, the diffracted wave vectors in the crystal corresponding the reciprocal lattice vectors B and L, while vg, vL and us-L denote the corresponding Fourier coefficients of the crystal periodic potential, in appropriate units. &,, 4B and 4L denote the amplitudes of the initially assumed corresponding plane waves in the crystal. By definition, &=&+B,
(2)
K,=K,+L.
(3)
Since the set of eqs. (1) are homogeneous, they have non-trivial solutions for only those values ofK, that result in the vanishing of the secular determinant. These values of K, generate the well known dispersion surface in reciprocal space. In addition, one may inquire as to what auxiliary conditions imposed on K, will result in a zero value for 4B/40. As is shown in detail elsewherer’), two additional conditions are required, namely k2 - K,2 = vsvL/vs_ L,
(4)
and 2&.
L
+
L2
=
_‘zB
_
!Lv!:L. (5)
VB-L
vB
230
S.FRIEDMAN
AND R. M. STERN
To apply these auxiliary equations to the pseudo-rocking curves observations, it is further assumed B is perpendicular to the surface (specular Bragg reflection) and that K,=k+Ak, where, to satisfy the required to the surface. Hence
boundary
(6) conditions,
Ak is also perpendicular
Ak=aB,
(7)
where the scalar c( is to be determined. Substituting and (5) and, neglecting Ak* Ak, we obtain
eqs. (6) and (7) into (4)
2cck. B = v~v&,_~, 2k.L+L2+2aB.L=
~??!t_v!!vB-~L~ vB-L
If it is now assumed that k satisfies Bragg reflection B, then
(8)
the exact kinematic
2k.B=Substituting eq. (10) into eq. (8), solving into eq. (9), we obtain
(9)
vB
condition
for the
B2. for CIand then substituting
(10) the result
From eq. (I 1) it is first seen that it is only possible to satisfy the kinematic conditions for Bragg reflection for the Laue rellection L for the special situation for which the right-hand side of eq. (11) vanishes. In general, eq. (11) defines a plane parallel to the Brillouin zone boundary of the lattice vector L which is the loci of the origin of all k vectors that will result in the vanishing of 4B/40. The intersection of this plane with the Brillouin zone boundary plane of the reciprocal lattice vector B as defined by eq. (10) thus defines a line which is the loci of the origin of all k vectors that will simultaneously satisfy the kinematic condition for the Bragg maximum for the specular reflection corresponding to B and the condition for ~~Jc#J~=O. These lines are parallel to the zone boundary intersections depicted in fig. 7 and are shifted by an amount determined by the right hand side of eq. (I 1).
References 1) 2) 3) 4)
C. J. Davisson and L. H. Germer, Proc. Natl. Acad. Sci. (US) 14 (1928) 624. P. M. Morse, Phys. Rev. 35 (1930) 1310. H. E. Farnsworth, Phys. Rev. 40 (1932) 684. W. E. Laschkarew, Trans. Faraday Sot. 31 (1935) 1081.
LOW
5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)
ENERGY
ELECTRON
DIFFRACTION
ROCKING
CURVES
S. Miyake, K. Kohra and M. Takagi, Acta Cryst. 7 (1954) 393. E. G. McRae, J. Chem. Phys. 45 (1966) 1258. E. G. McRae and C. W. Caldwell, Surface Sci. 7 (1967) 41. A. Gervais, R. M. Stern and M. Menes, Acta Cryst. A24 (1968) 191. R. M. Stern and H. Taub, Phys. Rev. Letters 20 (1968) 1340. D. S. Boudreaux and V. Heine, Surface Sci. 8 (1967) 426. H. Niehrs, Z. Physik 139 (1954) 88. K. Kohra, J. Phys. Sot. Japan 9 (1954) 690. R. Baudoing, R. M. Stern and H. Taub, Surface Sci. 11 (1968) 255. R. M. Stern, Appl. Phys. Letters 5 (1964) 218. R. D. Heidenreich, Phys Rev. 77 (1950) 271. J. J. Lander, Progr. Solid State Chem. 2 (1965) 26. R. M. Stern, J. Perry and D. Boudreaux, Rev. Mod. Phys. 41 (1969) 275. H. Bethe, Ann. Physik 87 (1928) 55.
231