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Angular dependence of the leed spectrum of AI (001)
SURFACE
SCIENCE
17 (1969) 442445
LETTER
ANGULAR
DEPENDENCE
0 North-Holland
TO THE
Publishing
EDITOR
OF THE LEED SPECTRUM
Received
18 July
Co., Amsterdam
OF Al (001)
1969
The identification of LEED peaks in observed LEED spectra (intensity versus incident energy) from a dynamical theory of electron scattering is hampered by two basic difficulties. On the one hand the measured peaks are smooth, featureless and considerably broader than calculated peaks (for elastic scattering)l); on the other hand the appropriate crystal potential for the scattering calculation is uncertain. Only a few attempts at this identification have been madea). Because both the peak shapes and their positions are unreliable indicators, we have been led to study the variation of the LEED spectrum with angle of incidence both experimentally and theoretically to assist in the identification of peaks. We report here results for LEED spectra of Al(001) up to 80 V at angles of incidence 0 (with the normal) from 6” to 25”, over which striking changes occur in the spectrum, and find good qualitative correspondence between measured and calculated spectra. The calculated spectra were found by the propagation matrix method3) which is sufficiently fast to permit entire LEED spectra to be calculated as functions of the parameters defining the scattering problem. We have chosen a simple smooth two-parameter model of the periodic potential in Al, conveniently described as the uniform charge potential (UCP), with which to study the spectrum, rather than more complicated potentials used in various band calculations. The UCP has the advantage that the parameters can be adjusted to fit various features of the observed spectrum, whereas the proper potential to use near the surface of a solid and at energies well above the Fermi energy is itself in doubt. Our emphasis is on the more prominent features of the spectrum and their behavior with variation of angles of incidence; these features can be systematically studied with the two parameters of our model and are not sensitive to small details of the potential. In addition, the simplicity and precisely defined nature of the UCP gives a universal character to the results, providing a standard set of spectra with which to compare the effects of detailed variation of the potentia14). The UCP is the electrostatic potential in an infinite lattice with Z positive electronic charges on each FCC lattice site (cube side a) and Z negative electroniccharges uniformly spread over each primitive ce115). The values 442
ANGULAR
chosen
DEPENDENCE
OF THE
for 2 and a correspond
three conduction
electrons,
LEED
SPECTRUM
to reasonable
values
OF
443
Al (001)
for Al; 2=3
for the
and a = 7.6524 a.u. is the bulk lattice spacing
at
23 “C. Other, less important, parameters of the calculation are as follows: the potential rises abruptly to zero in vacuum at a plane halfway between planes of atoms parallel to (OOl), the azimuth angle 4 of the incident beam (with respect to cubic axes in the (001) face) is 45 ’ throughout, and the number of Fourier coefficients N in the expansion of the wave function (in planes parallel to the surface) is nine. Calculations with larger values of N, up to 25, give small variations in the detailed shapes of the lines (and sometimes additional small peaks corresponding to additional Fourier components of the potential), but preserve the main features of the spectra. We now identify the two strong peaks at 25.5 and 66.5 V in the measured spectrum at 6’= 6 ’ (fig. 1) with the two primary Bragg reflections6) at 18 and 62 V in the calculated’) spectrum (fig. 2). Note first that in both figures the upper primary peak moves distinctly toward higher energies as 8 increases, while the lower one moves only slightly that way, with a small retreat at 9 = 15” in both cases. In addition the lower peak at 0 = 15 ’ develops additional structure on the high energy side, while at 0= 20” the upper peak in both cases breaks up into smaller peaks but reappears at 13= 25 ‘. We further identify the central smaller experimental peak showing a distinct shoulder at f3= 6 o with the two prominent secondaries in the calculated
o
20
‘v”o0&0
80
100
Fig. 1. Measured spectra of 00 beam for electron incident on (001) face of Al at a series of 5 angles 0 with the normal, and at azimuth angle 4 = 45” with cubic axes in the face. The individual intensity curves have been normalized to constant incident intentisy at all energies.
444
P.M. MARCUS,
D. W. JEPSEN AND E
F. JONA
IVOLTS)
E (RYDBERGS)
Fig. 2. Calculated spectra of 00 beam for electron incident on (001) face of FCC lattice at the same angles as in fig. 1 (and at normal incidence), using uniform charge potential with 2 = 3, a = 7.6524 a.“., and N = 9. The reflected intensity relative to the incident intensity is plotted versus energy E in Rydbergs (lower scale and repeated on each axis) and in volts (upper scale - larger one-sided tics, repeated at top of t) == 25” curve).
curve at 32 and 38 V. Note that the secondary structure in figs. 1 and 2 splits apart, one part shifting strongly to higher energies, and the other shifting more moderately to lower energies. In addition, both secondaries may be identified as 20 secondaries (the upper one also has 11 and 22 character)“) and correspond to observed peaks in the 20 beam. We do not consider the observed below 20 V reliable, but we note that the calculation also shows several peaks below the primary Bragg peak, and that they separate more widely from the first Bragg peak at 0= 15’ and above, as do the peaks in the measured
spectra. P. M, MARCUS, D. W. JEPSEN and F. JONA*
I.B.M. Watson Research Center, Yorktown Heights, New York 10.598, U.S.A. * Now at Department of Materials Science, State University of New York at Stony Brook, Stony Brook, Long Island, New York, U.S.A.
ANGULAR
DEPENDENCE
OF THE
LEED
SPECTRUM
OF
Al (001)
445
References 1) The source of the experimental peak width has not as yet been established, but several theories are current. Effects of strong inelastic scattering of the incident electron in broadening and rounding the spectrum have been noted by several workers - recently by C. B. Duke, R. 0. Jones and J. A. Strozier, and by us at the Yale LEED Theory Seminar, March 17, 1969; V. Heine and J. B. Pendry, Phys. Rev. Letters 22 (1969)1003 have proposed a model in which the fluctuations in strain fields due to lattice phonons will broaden the peak; we have considered models in which small systematic variation of atom layer spacings near the surface could broaden the peaks. 2) K. Hirabayashi and Y. Takeishi, Surface Sci. 4 (1966) 150, and Kanji Hirabayashi, J. Phys. Sot. Japan 24 (1968) 846 compare theory and experiment for graphite; J. B. Pendry has reported results for Ni (Yale LEED Theory Seminar). 3) P. M. Marcus and D. W. Jepsen, Phys. Rev. Letters 20 (1968) 925. 4) Because the potential used comes primarily from the three conduction electrons, it is weak enough to be handled directly by the propagation matrix method; stronger potentials, such as one which includes core effects in Al, would require introduction of a pseudopotential. 5) The specification is completed by the average value of the potential (compared to zero in the vacuum at infinity) which is chosen by an arbitrary but natural assumption corresponding to the value for symmetrical finite specimens in the limit of macroscopic size [see, e.g., J. Callaway and M. L. Glasser, Phys. Rev. 112 (1958) 231 with the value (477/3)2/a for an FCC lattice. Note that the UCP gives a one-parameter family of reduced spectra, intensity versus &, with strength fixed by Zu. The UCP potential for a simple cubic lattice is used in ref. 3 where explicit expressions for the Fourier coefficients occur. [There are several misprints in the formula for V”(z) in ref. 1, p. 928, which should read V”(z) = (- 2Z/nu) exp(- 2nnlz]/a) {[exp(4nnIzl/a) + l]/[exp(2an) - 11 + 1). The corresponding Fourier coefficients of the FCC lattice are obtained as the superposition of four SC lattices.] 6) The identification as primary Bragg peaks is readily made by tracing them continuously in a series of spectra as Z increases from 0 to 3; this can also be done for the secondary peaks. 7) The claculated separation of the two peaks at Z = 3 (44 V) is thus slightly greater than the observed value (41 V). This is more significant than the absolute positions of the peaks, which depend on the average potential, chosen arbitrarily here5). Increasing Z to 5 can give the experimental separation, but generates many more and stronger secondaries than desired.
SCIENCE
17 (1969) 442445
LETTER
ANGULAR
DEPENDENCE
0 North-Holland
TO THE
Publishing
EDITOR
OF THE LEED SPECTRUM
Received
18 July
Co., Amsterdam
OF Al (001)
1969
The identification of LEED peaks in observed LEED spectra (intensity versus incident energy) from a dynamical theory of electron scattering is hampered by two basic difficulties. On the one hand the measured peaks are smooth, featureless and considerably broader than calculated peaks (for elastic scattering)l); on the other hand the appropriate crystal potential for the scattering calculation is uncertain. Only a few attempts at this identification have been madea). Because both the peak shapes and their positions are unreliable indicators, we have been led to study the variation of the LEED spectrum with angle of incidence both experimentally and theoretically to assist in the identification of peaks. We report here results for LEED spectra of Al(001) up to 80 V at angles of incidence 0 (with the normal) from 6” to 25”, over which striking changes occur in the spectrum, and find good qualitative correspondence between measured and calculated spectra. The calculated spectra were found by the propagation matrix method3) which is sufficiently fast to permit entire LEED spectra to be calculated as functions of the parameters defining the scattering problem. We have chosen a simple smooth two-parameter model of the periodic potential in Al, conveniently described as the uniform charge potential (UCP), with which to study the spectrum, rather than more complicated potentials used in various band calculations. The UCP has the advantage that the parameters can be adjusted to fit various features of the observed spectrum, whereas the proper potential to use near the surface of a solid and at energies well above the Fermi energy is itself in doubt. Our emphasis is on the more prominent features of the spectrum and their behavior with variation of angles of incidence; these features can be systematically studied with the two parameters of our model and are not sensitive to small details of the potential. In addition, the simplicity and precisely defined nature of the UCP gives a universal character to the results, providing a standard set of spectra with which to compare the effects of detailed variation of the potentia14). The UCP is the electrostatic potential in an infinite lattice with Z positive electronic charges on each FCC lattice site (cube side a) and Z negative electroniccharges uniformly spread over each primitive ce115). The values 442
ANGULAR
chosen
DEPENDENCE
OF THE
for 2 and a correspond
three conduction
electrons,
LEED
SPECTRUM
to reasonable
values
OF
443
Al (001)
for Al; 2=3
for the
and a = 7.6524 a.u. is the bulk lattice spacing
at
23 “C. Other, less important, parameters of the calculation are as follows: the potential rises abruptly to zero in vacuum at a plane halfway between planes of atoms parallel to (OOl), the azimuth angle 4 of the incident beam (with respect to cubic axes in the (001) face) is 45 ’ throughout, and the number of Fourier coefficients N in the expansion of the wave function (in planes parallel to the surface) is nine. Calculations with larger values of N, up to 25, give small variations in the detailed shapes of the lines (and sometimes additional small peaks corresponding to additional Fourier components of the potential), but preserve the main features of the spectra. We now identify the two strong peaks at 25.5 and 66.5 V in the measured spectrum at 6’= 6 ’ (fig. 1) with the two primary Bragg reflections6) at 18 and 62 V in the calculated’) spectrum (fig. 2). Note first that in both figures the upper primary peak moves distinctly toward higher energies as 8 increases, while the lower one moves only slightly that way, with a small retreat at 9 = 15” in both cases. In addition the lower peak at 0 = 15 ’ develops additional structure on the high energy side, while at 0= 20” the upper peak in both cases breaks up into smaller peaks but reappears at 13= 25 ‘. We further identify the central smaller experimental peak showing a distinct shoulder at f3= 6 o with the two prominent secondaries in the calculated
o
20
‘v”o0&0
80
100
Fig. 1. Measured spectra of 00 beam for electron incident on (001) face of Al at a series of 5 angles 0 with the normal, and at azimuth angle 4 = 45” with cubic axes in the face. The individual intensity curves have been normalized to constant incident intentisy at all energies.
444
P.M. MARCUS,
D. W. JEPSEN AND E
F. JONA
IVOLTS)
E (RYDBERGS)
Fig. 2. Calculated spectra of 00 beam for electron incident on (001) face of FCC lattice at the same angles as in fig. 1 (and at normal incidence), using uniform charge potential with 2 = 3, a = 7.6524 a.“., and N = 9. The reflected intensity relative to the incident intensity is plotted versus energy E in Rydbergs (lower scale and repeated on each axis) and in volts (upper scale - larger one-sided tics, repeated at top of t) == 25” curve).
curve at 32 and 38 V. Note that the secondary structure in figs. 1 and 2 splits apart, one part shifting strongly to higher energies, and the other shifting more moderately to lower energies. In addition, both secondaries may be identified as 20 secondaries (the upper one also has 11 and 22 character)“) and correspond to observed peaks in the 20 beam. We do not consider the observed below 20 V reliable, but we note that the calculation also shows several peaks below the primary Bragg peak, and that they separate more widely from the first Bragg peak at 0= 15’ and above, as do the peaks in the measured
spectra. P. M, MARCUS, D. W. JEPSEN and F. JONA*
I.B.M. Watson Research Center, Yorktown Heights, New York 10.598, U.S.A. * Now at Department of Materials Science, State University of New York at Stony Brook, Stony Brook, Long Island, New York, U.S.A.
ANGULAR
DEPENDENCE
OF THE
LEED
SPECTRUM
OF
Al (001)
445
References 1) The source of the experimental peak width has not as yet been established, but several theories are current. Effects of strong inelastic scattering of the incident electron in broadening and rounding the spectrum have been noted by several workers - recently by C. B. Duke, R. 0. Jones and J. A. Strozier, and by us at the Yale LEED Theory Seminar, March 17, 1969; V. Heine and J. B. Pendry, Phys. Rev. Letters 22 (1969)1003 have proposed a model in which the fluctuations in strain fields due to lattice phonons will broaden the peak; we have considered models in which small systematic variation of atom layer spacings near the surface could broaden the peaks. 2) K. Hirabayashi and Y. Takeishi, Surface Sci. 4 (1966) 150, and Kanji Hirabayashi, J. Phys. Sot. Japan 24 (1968) 846 compare theory and experiment for graphite; J. B. Pendry has reported results for Ni (Yale LEED Theory Seminar). 3) P. M. Marcus and D. W. Jepsen, Phys. Rev. Letters 20 (1968) 925. 4) Because the potential used comes primarily from the three conduction electrons, it is weak enough to be handled directly by the propagation matrix method; stronger potentials, such as one which includes core effects in Al, would require introduction of a pseudopotential. 5) The specification is completed by the average value of the potential (compared to zero in the vacuum at infinity) which is chosen by an arbitrary but natural assumption corresponding to the value for symmetrical finite specimens in the limit of macroscopic size [see, e.g., J. Callaway and M. L. Glasser, Phys. Rev. 112 (1958) 231 with the value (477/3)2/a for an FCC lattice. Note that the UCP gives a one-parameter family of reduced spectra, intensity versus &, with strength fixed by Zu. The UCP potential for a simple cubic lattice is used in ref. 3 where explicit expressions for the Fourier coefficients occur. [There are several misprints in the formula for V”(z) in ref. 1, p. 928, which should read V”(z) = (- 2Z/nu) exp(- 2nnlz]/a) {[exp(4nnIzl/a) + l]/[exp(2an) - 11 + 1). The corresponding Fourier coefficients of the FCC lattice are obtained as the superposition of four SC lattices.] 6) The identification as primary Bragg peaks is readily made by tracing them continuously in a series of spectra as Z increases from 0 to 3; this can also be done for the secondary peaks. 7) The claculated separation of the two peaks at Z = 3 (44 V) is thus slightly greater than the observed value (41 V). This is more significant than the absolute positions of the peaks, which depend on the average potential, chosen arbitrarily here5). Increasing Z to 5 can give the experimental separation, but generates many more and stronger secondaries than desired.