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Temperature dependence of low energy electron diffraction from aluminum
SURFACE
SCIENCE 32 (1972) 139-148 0 North-Holland
TEMPERATURE ELECTRON
DEPENDENCE DIFFRACTION
Publishing Co.
OF LOW ENERGY FROM
ALUMINUM
DENNIS T. QUINTO* Department
of Engineering and Applied Science, Yale University, New Haven, Connecticut 06520, U.S.A.
B. W. HOLLAND School of Physics,
University
of Warwick,
Coventry,
England
and W. D. ROBERTSON Department
of Engineering and Applied Science, Yale University, New Haven, Connecticut 06520, U.S.A.
Received 20 December 1971; revised manuscript received 2 March 1972 The temperature dependence of diffracted intensity from the clean (100) surface of aluminum has been measured in the range of 298 to 598°K. Effective Debye temperatures have been extracted from the data for five prominent intensity peaks of integral and nonintegral order. It appears that all the effective Debye temperatures, which range from 189 to 33l”K, are represented best by an increasing, almost linear, function of energy from 25 to 130 eV. The increase with energy is interpreted in terms of a larger mean square displacement of surface ions relative to the bulk with the result that there is a decreasing contribution from the surface with increasing energy and a corresponding rise in effective Debye temperature. More significantly, it is shown that the temperature dependence of intensity may be used to identify the operative scattering processes in terms of the order of back scattering contributing to the temperature dependence of intensity profiles. Thus, the temperature dependence of intensity maxima from aluminum indicates that they can be described by a calculation to first order in back scattering, as indeed they have been. On the other hand, the more complicated temperature dependence of diffracted intensity from copper, for example, can be identified in advance with higher orders of multiple scattering that will require calculations to higher order for its description.
1. Introduction Physical modelsl,z) and computational proceduress) used to describe the scattering of low energy (20 to 200 eV) electrons from crystal surfaces at finite temperatures are approaching a predictive capability corresponding to the range of uncertainty in available empirical data which are used to evaluate the models. In this context, the experimentalist is obliged to obtain data over a wide range of variables and with sufficient precision to provide a more searching test of physical models and realistic computational methods. * Present address: Inland Steel Company, Research Department, 46312, U.S.A. 139
East Chicago, Indiana
140
D. T. QUINTO,
B. W. HOLLAND
AND W. D. ROBERTSON
Aluminum is frequently used as a test of models and computational procedures but, until recently, the only comprehensive set of measured intensity profiles for aluminum was the set obtained by Jona4) at room temperature. However, at the time these data were collected, Auger spectroscopy was not available as a technique to evaluate the chemical state of the surface and, accordingly, an element of uncertainty remains to be resolved. The range of empirical data from a clean (100) surface of aluminum has since been extended by Quinto5) to include azimuthal and incident angular dependence of elastically scattered intensity, at angular intervals of 2 O.In addition to the angular dependence of scattering, the temperature dependence of intensity maxima was measured in the range of 300 to 600 “K. All of these data were obtained from surfaces continuously monitored by Auger spectroscopy to insure freedom from the refractory oxide produced by electropolishing and oxygen subsequently adsorbed from the ambient atmospheres). In this paper we report the measurement of the temperature dependence of the intensity of five prominent diffraction peaks and the extraction of effective Debye temperatures from the data. Beyond the obivous importance of the temperature dependence of intensity for purposes of comparing computed and experimental profiles, it can be anticipated that the effects of temperature on the different peaks of a profile may be used as a diagnostic device to identify the scattering mechanism responsible for particular peaks. The availability of such a device could assume a very significant role in identifying the origin of peaks by considerably simplifying the problem of extracting information concerning the structure of the free surface of a crystal from low energy diffraction profiles. We show that the interpretation of our results, together with a comparison with Reid’sr3) data on copper and in the light of calculations by Tong and Rhodin14) and Pendrys), leads to such a diagnostic method. 2. Experimental
procedure
2.1. CRYSTAL PREPARATION A 12 mm by 2 mm thick sample was spark-cut from a spectrographic grade aluminum single crystal. Parallel (100) faces were oriented to ++” by the combination of X-ray Laue patterns and the spark-cutter goniometer. The (100) faces were electropolished in 90 ml perchloric acid, 600 ml ethanol, 180 ml acetic acid, 32 g NaOH, 45 g sodium acetate. The refractory layer of aluminum oxide produced by electropolishing was minimized by chemical polishing in warm 0.1 N NaOH and 50 ~01% HN03 immediately before placing the crystal inside the LEED apparatus. A Varian 3-grid LEED system was used for all intensity measurements. Ion bombardment, Auger spectroscopy and mass spectrometry were used for
TEMPERATURE
DEPENDENCE
OF LEED
FROM
Al
141
cleaning the sample surface, and for monitoring the chemical state of the surface and the ambient during the time required to collect data. Final cleaning, to remove the refractory residue of anodic oxide, was accomplished by an accumulated total of 45 hr A-ion bombardment and heating to 570°C for 24 hr. As reported previously@), this procedure reduces the oxygen Auger peak below detection sensitivity (5% of initial p-p amplitude of oxygen) and maximizes the L,,VV aluminum peak at 67 eV. The importance of this preparation procedure is emphasized here because specular beam intensities before this treatment are only 25% of those measured after removal of oxygen. Subsequent maintenance of an oxygen free surface requires only routine bombardment and annealing to eliminate oxygen adsorbed from the ambient atmosphere of the apparatus operating at IO-” Torr. 2.2,
INTENSITY
MEASUREMENT
The intensity data shown in fig. 1 are isothermal profiles obtained at different (constant) temperatures. The aluminum crystal was heated by thermal conduction from a pyrolytic graphite resistor to which the crystal was attached. Temperature was measured by a ~hromel-alumel thermocouple embedded in the back face of the aluminum crystal. Due to the magnetic field produced by the dc heating current, the field-free angle of incidence (8 = 5 “) is changed in the low energy range of 2&30 eV and the effect is observed as motion of the specular beam. A first order correction for this change in angle was made by adjustment of a trimming magnet at each temperature (current). The observed change in angle of incidence is negligible at primary beam energies greater than 40 eV. Specular beam intensities were measured with a spot photometer (Photo Research Corporation). The acceptance angle of the photometer is 0.25”, which subtends a 1.3 mm diameter area on the fluorescent screen relative to a diffracted spot diameter less than 1 mm and, accordingly, the recorded data are integrated intensities. A step-voltage power supply (FIuke) provided for discrete measurements of intensity at 2 V intervals, after waiting for a suhicient time (2-5 set) at each voltage increment for the relatively slow photometer to reach equilibrium. The voltage is corrected for a 2 V contact potential difference between the bariated nickel cathode and the aluminum crystal {work functions of 2 and 4.2 eV, respectively). Measured intensities are normalized to the incident beam current, following the procedure described by Jona4). The intensity data in fig. 1 are plotted on a log scale to show clearly the fine-structure which is usually obscured by conventional linear plotting, particularly at higher energies.
142
D. T. QKJINTO, B. W. HOLLAND
AND
3. Experimental
W. D. ROBERTSON
results
The log of intensity of the specular beam from the (100) surface of aluminum at an angle of incidence, 0 = 5 ‘, and azimuthal angle, C$= 5’ [angular definitions as in Jona4)], normalized to the measured incident beam current, is presented in fig. 1 as a function of incident energy, 20 to 160 eV, for six different temperatures from 298 “K to 598 “K. I’
I
I
!
I I
n=2f37v)
I
I
I
n=3(63v) AI (100) e= 5’ += 5” n=4(148v
6> -
(26~I I’ 20
1
, (46~)
Il 40
(69v) I 60
ENERGY
i
(97Mt , 80
100
(EXTERNAL),
120
140
I 3
eV
Fig. 1. The logarithm of normalized intensity of the specular (ZOO)beam from the (100) surface of clean aluminum at 0 = 5” (angle of incidence) and 4 = 5” (azimuthal angle relative to the non-primative unit cella) as a function of (external) energy and temperature, 298 to 598°K. The positions of Bragg peaks (zero inner potential) are indicated by arrows for n = 2, 3 and 4. The dashed vertical lines indicate the energies at which the intensity is evaluated, at each temperature.
TEMPERATURE
DEPENDENCE
OF LEED
FROM
Al
143
The three strong peaks at external beam energies of 26, 69 and 134 eV correspond to “Bragg” peaks with n =2, 3 and 4 in the single scattering and the apparent inner potentials associated with these approximation*, peaks are 11, 14 and 14 eV (see fig. 1). The strong secondary peaks at 46 and 97 eV are labeled 23 and 34, respectively, for identification purposes only since these labels do not have a simple meaning in the framework of a multiple scattering theory. Before considering the exponential decrease of intensity with temperature, some general features of the profiles in fig. 1 should be noted particularly. (1) As the temperature rises, a new peak appears between 20 and 23 eV, and increases in intensity relative to its neighbor (n = 2) at 26 eV. Since the observation is close to the limit of experimental precision at low energies, it might be inferred that this unusual behavior is due to uncontrolled changes in the angle of incidence; but intensity profiles from the same crystal at the same azimuth, taken at 2’ intervals from the experimental lower limit of 5 ’ to 21’ at room temperature, do not exhibit the effect. Thus we conclude that it is uniquely associated with temperature and, furthermore, the behavior seems to be similar to that observed by Reid13) on Cu(100). (2) The integral order peaks exhibit changes in shape (appearance and disappearance of shoulders) and small changes in energy with increasing temperature. These effects are not included in the treatment of the data which is limited to intensity evaluated at the vertical dashed lines in fig. 1. (3) The secondary peaks in general follow the behavior of their low energy primary neighbours, except the IZ= 2: peak at 46 eV which becomes asymmetric and shifts to lower energies by about 5 eV in the temperature range of 298 to 598 “K. 3.1. EFFECTIVE DEBYE TEMPERATURES The maximum peak intensities for all prominent peaks (excepting the lowenergy peak in the range of 20-23 eV, for which the available data are insufficient) are presented in fig. 2 as a function of temperature, from which the Debye-Waller exponent and the effective Debye temperature are extracted in accord with the following expressions: I =
IoemzM,
* More precisely, the single-scattering “Bragg” condition is reflected in the intensity profiles as an envelope for multiple scattering peaks7). However, from a weakly scattering metal, such as aluminum, the prominent peaks appear at energies corresponding to the Bragg condition, within the limits of present resolution of 5 2 eV, and are usually designated as “Bragg” peaks.
144
D. T.
QUINTO,
B. W. HOLLAND
AND
W. D. ROBERTSON
where IO is the rigid lattice intensity, m is the ionic (atomic) mass, 0 is the angle of incidence, and i is the electron wavelength inside the crystal with an inner potential of 16.7 eV (V,=4 +I$) to conform with theoretical computations L 2*g). Effective Debye temperatures, obtained from the slopes of the lines, are shown in fig. 2 for each of the prominent maxima in the intensity profiles. These effective Debye temperatures are shown as a function of (external) beam energy in fig. 3. The extreme limits of available bulk Debye temperaturesl7) are shown as a band which must be approached asymptotically by the effective Debye temperature at some higher value of energy. I
I
t 300
I
I
Al(l00)
I
I
I :I&.-
I
I
I
I
I
I
350
400
450
500
550
600
TEMPERATURE
(OK)
Fig. 2. The logarithm of intensity for each prominent peak in fig. 1, as a function of temperature, from which the Debye-Waller factor (I= ZOe-ZM) and the corresponding effective Debye temperature are obtained.
Similarly, the effective BD presumably approaches asymptotically a lower limit at some lower energy associated with the surface layer. The data are represented by a linear function as an approximation to what is, in principle, the middle segment of a very flat sigmoid function. 3.2. DISCUSSION A
feature
of the results
which
is not
unexpected,
being
in accord
with
TEMPERATURE
DEPENDENCE
OF LEED FROM
145
Al
, 150
0
I 25
I 50
I 75
I 100
I 125
ENERGY (EXTERNAL),
I 150
I 175
200
eV
Fig. 3. The effective Debye temperature, obtained from the data in fig. 2, is shown as a function of energy. The range of bulk Debye temperatures, obtained from various sources, is shown in the upper band of the figure.
theoretical predictions 1,219) and other experimental works~10~11), is the rise in effective Debye temperature with increasing energy. Surface ions have considerably larger (perhaps, 2 x) mean square displacementsi2) than bulk ions and, therefore, the contribution of surface ions relative to bulk will fall off with increasing energy, resulting in a rise in the effective Debye temperature, approaching the bulk 8, in the limit of some high energy. The effect does not depend entirely on penetration, and it will appear even if the penetration is constant, because the vibrationally renormalized scattering cross section for the surface ions falls much faster with increasing energy than for the bulk ions. Indeed, Laramores) has modelled this aspect of our results with fair success by choosing different Debye temperatures for the surface and bulk ions, with the surface Debye temperature being chosen as 180°K; that is, less than the lowest measured effective On of 189 “K. The most striking feature of the experimental data is the almost linear (see above) dependence of the effective Debye temperature on energy, for all five (integral and non-integral) intensity maxima. This is in sharp contrast to the results of Reidr”9r3) for Cu(lOO), who finds that the effective Debye temperature is a violently fluctuating function of energy, and similar results have been observed by other workers, for example Tabor et al.il), and are to be expected on the basis of the theory of Duke and Laramorel). The characteristic intensity profiles from aluminum495) cannot be interpreted as im-
146
plying of the peaks. tion16)
D. T. QUINTO,
B. W. HOLLAND
AND
W. D. ROBERTSON
that only single-scattering contributions need be considered because existence of prominent features between the integral order “Bragg” Moreover, Tong and Rhodinl”) have shown that the Double-Diffracapproximation of the Inelastic Collision Modelt) gives a reasonably
good description of the room temperature data. In this respect also aluminum is unusual, and we shall see that these two anomalies are related. Duke and Laramorel) showed that, given certain assumptions [see also Hollandrs)], the rigid lattice theory in the momentum representation is modified for vibrating lattices simply by multiplying each ion-core t-matrix element by the corresponding Debye--Wailer factor. This factor is of the form :
where tiK=A(k-k’) is the momentum transfer (the incoming wave vector is k and the outgoing is k’) and (U,U,), is the thermal average of a product of ion-core displacement components. For single scattering, k is the incident and k’ the reflected external wave vector. In multiple scatterings an integration is performed over the wave-vectors of the Debye-Waller factor in calculating the scattered amplitude. Notice that forward scattering (i.e., k’= k) does not contribute to temperature dependence. Hence any contribution arising from a sequence of multiple forward scatterings followed by a single reversal will show the temperature dependence associated with single scattering. More generally, if the total back-scattered amplitude arises predominantly from scattering paths in which the direction is reversed only once, whether in a single scattering event or by a sequence of smaller angle scatterings, then the temperature dependence of such maxima will have effective Debye temperatures directly related to the Debye temperatures of the surface and bulk layers and a series of such peaks will show an effective Debye temperature varying smoothly with energy and converging on the bulk Debye temperature for high enough energies. Evidently this idea explains our results on aluminum. Neglecting multiple back-scattering within planes, the DoubleDiffraction picture allows only one complete reversal in all contributions to the total scattered amplitude, and we have noted that this approximation gives an adequate description of the diffracted intensity at room temperaturer4). Our interpretation can be substantiated by re-examining the behaviour of the anomalous doublet found by Reidra) on Cu(100) at about 30 to 40 eV. He found that the lower energy peak behaved in a kinematical manner, having an effective Debye temperature about equal to the value obtained from X-ray studies. The higher energy peak had a much lower Debye temperature which was accounted for by HollandIs) in terms of multiple scat-
TEMPERATURE
tering
theory,
DEPENDENCE
using a semi-empirical
OF LEED
isotropic
FROM
AI
Debye-Wailer
147
factor.
Now,
according to our hypothesis above, the lower energy peak should arise principally from scattering paths in which only a single reversal occurs, while the higher energy peak should involve higher orders of back-scattering. Work of Pendrys) bears closely on this question. Pendry made an important advance in the multiple scattering theory of LEED by showing that a perturbation scheme based on the number of backscatterings between layers is both fast and accurate. In this Renormalised Forward Scattering scheme all forward scattering is treated exactly. Pendry performed calculations for Cu(lOO), and though he used a different angle of incidence and azimuth from Reid, one can easily identify Reid’s doublet in the theoretical curves. The significant point is that Pendry’s first order calculation in back-scattering gives a close fit for the lower energy peak, but not for the anomalous higher energy member of the doublet. If we can neglect multiple back-scattering in the plane, this result explains why the temperature dependence of the lower member of the doublet can be interpreted in terms of single scattering. To get a reasonable description of the anomalous peak, Pendry had to go to the next approximation in his scheme, which involves triple back-scattering. Hence the low effective Debye temperature of this peak. It appears that our interpretation of the temperature dependence of scattering from aluminum is not only consistent with Tong and Rhodin’s secondorder perturbation (double-diffraction) calculations, but also provides an explanation for the difference between copper and aluminum, consistent with Pendry’s theory. This interpretation leads naturally to the suggestion that temperature dependence can be used as a diagnostic tool, to classify peaks according to Pendry’s scheme. A monotonic increasing effective Debye temperature for a sequence of diffraction peaks implies that they will be well described by a calculation to first order in back scattering. Diffraction profiles exhibiting more complicated energy dependence of effective Debye temperatures, involving increasing and decreasing values in a sequence of peaks, will in general require a calculation in the Renormalised Forward Scattering Scheme going beyond the first pass. Thus, before time consuming calculation of diffraction profiles, it is possible to specify from the observed temperature dependence the peaks that involve only a single reversal in the scattering paths. After a single pass (in which the layer scattering amplitudes are calculated exactly) we can say which peaks have important contributions from multiple back-scattering in the plane; namely those peaks that are well described in this approximation but show the temperature dependence corresponding to multiple back-scattering. The remaining peaks involve multiple back-scattering between planes, and they will require calculations to higher order for their description.
148
D.T.QUINTO,
B. W.HOLLAND
AND
W.D.ROBERTSON
Acknowledgement This, and associated investigations, are performed under Contract SAR/ N00014-67-0097-0003 with the Office of Naval Research. One of us, D.T.Q., is indebted to the International Nickel Company for a fellowship.
References 1) C. B. Duke and G. E. Laramore, Phys. Rev. B 2 (1970) 4765, 4783. 2) C. B. Duke, G. E. Laramore, B. W. Holland and A. M. Gibbons, Surface Sci. 27 (1971) 523. 3) J. B. Pendry, Phys. Rev. Letters 27 (1971) 856. 4) F. Jona, IBM J. Res. Develop. 14 (1970) 444. 5) T. Quinto, Dissertation, Yale University, Dec. 1971, to be published. 6) T. Quint0 and W. D. Robertson, Surface Sci. 27 (1971) 645. 7) C. W. Tucker and C. B. Duke, Surface Sci. 24 (1971) 31. 8) E. R. Jones, J. T. McKinney and M. B. Webb, Phys. Rev. 151 (1966) 476; 160 (1967) 523. 9) G. E. Laramore, private communication, submitted to Phys. Rev., Comments and Addenda, Feb. 1972. 10) R. J. Reid, Surface Sci. 29 (1972) 603, 623. 11) D. Tabor, J. M. Wilson and T. J. Bastow, Surface Sci. 26 (1971) 471. 12) R. F. Wallis, B. C. Clark and Robert Herman, in: The Structure and Chemisfry of Solid Surfaces, Ed. G. A. Somorjai (Wiley, New York, 1969) p. 17-1. 13) R. J. Reid, Phys. Status Solidi (a) 4 (1971) K 211. 14) S. Y. Tong and T. N. Rhodin, Phys. Rev. Letters 26 (1971) 711. 15) B. W. Holland, Surface Sci. 28 (1971) 258. 16) C. B. Duke, J. R. Anderson and C. W. Tucker, Jr., Surface Sci. 19 (1970) 117. 17) M. Blackman, in: Hundbuch der Physik, Vol. 7 (I), Ed. S. Fltigge (Springer, Berlin, 1955) p. 325.
SCIENCE 32 (1972) 139-148 0 North-Holland
TEMPERATURE ELECTRON
DEPENDENCE DIFFRACTION
Publishing Co.
OF LOW ENERGY FROM
ALUMINUM
DENNIS T. QUINTO* Department
of Engineering and Applied Science, Yale University, New Haven, Connecticut 06520, U.S.A.
B. W. HOLLAND School of Physics,
University
of Warwick,
Coventry,
England
and W. D. ROBERTSON Department
of Engineering and Applied Science, Yale University, New Haven, Connecticut 06520, U.S.A.
Received 20 December 1971; revised manuscript received 2 March 1972 The temperature dependence of diffracted intensity from the clean (100) surface of aluminum has been measured in the range of 298 to 598°K. Effective Debye temperatures have been extracted from the data for five prominent intensity peaks of integral and nonintegral order. It appears that all the effective Debye temperatures, which range from 189 to 33l”K, are represented best by an increasing, almost linear, function of energy from 25 to 130 eV. The increase with energy is interpreted in terms of a larger mean square displacement of surface ions relative to the bulk with the result that there is a decreasing contribution from the surface with increasing energy and a corresponding rise in effective Debye temperature. More significantly, it is shown that the temperature dependence of intensity may be used to identify the operative scattering processes in terms of the order of back scattering contributing to the temperature dependence of intensity profiles. Thus, the temperature dependence of intensity maxima from aluminum indicates that they can be described by a calculation to first order in back scattering, as indeed they have been. On the other hand, the more complicated temperature dependence of diffracted intensity from copper, for example, can be identified in advance with higher orders of multiple scattering that will require calculations to higher order for its description.
1. Introduction Physical modelsl,z) and computational proceduress) used to describe the scattering of low energy (20 to 200 eV) electrons from crystal surfaces at finite temperatures are approaching a predictive capability corresponding to the range of uncertainty in available empirical data which are used to evaluate the models. In this context, the experimentalist is obliged to obtain data over a wide range of variables and with sufficient precision to provide a more searching test of physical models and realistic computational methods. * Present address: Inland Steel Company, Research Department, 46312, U.S.A. 139
East Chicago, Indiana
140
D. T. QUINTO,
B. W. HOLLAND
AND W. D. ROBERTSON
Aluminum is frequently used as a test of models and computational procedures but, until recently, the only comprehensive set of measured intensity profiles for aluminum was the set obtained by Jona4) at room temperature. However, at the time these data were collected, Auger spectroscopy was not available as a technique to evaluate the chemical state of the surface and, accordingly, an element of uncertainty remains to be resolved. The range of empirical data from a clean (100) surface of aluminum has since been extended by Quinto5) to include azimuthal and incident angular dependence of elastically scattered intensity, at angular intervals of 2 O.In addition to the angular dependence of scattering, the temperature dependence of intensity maxima was measured in the range of 300 to 600 “K. All of these data were obtained from surfaces continuously monitored by Auger spectroscopy to insure freedom from the refractory oxide produced by electropolishing and oxygen subsequently adsorbed from the ambient atmospheres). In this paper we report the measurement of the temperature dependence of the intensity of five prominent diffraction peaks and the extraction of effective Debye temperatures from the data. Beyond the obivous importance of the temperature dependence of intensity for purposes of comparing computed and experimental profiles, it can be anticipated that the effects of temperature on the different peaks of a profile may be used as a diagnostic device to identify the scattering mechanism responsible for particular peaks. The availability of such a device could assume a very significant role in identifying the origin of peaks by considerably simplifying the problem of extracting information concerning the structure of the free surface of a crystal from low energy diffraction profiles. We show that the interpretation of our results, together with a comparison with Reid’sr3) data on copper and in the light of calculations by Tong and Rhodin14) and Pendrys), leads to such a diagnostic method. 2. Experimental
procedure
2.1. CRYSTAL PREPARATION A 12 mm by 2 mm thick sample was spark-cut from a spectrographic grade aluminum single crystal. Parallel (100) faces were oriented to ++” by the combination of X-ray Laue patterns and the spark-cutter goniometer. The (100) faces were electropolished in 90 ml perchloric acid, 600 ml ethanol, 180 ml acetic acid, 32 g NaOH, 45 g sodium acetate. The refractory layer of aluminum oxide produced by electropolishing was minimized by chemical polishing in warm 0.1 N NaOH and 50 ~01% HN03 immediately before placing the crystal inside the LEED apparatus. A Varian 3-grid LEED system was used for all intensity measurements. Ion bombardment, Auger spectroscopy and mass spectrometry were used for
TEMPERATURE
DEPENDENCE
OF LEED
FROM
Al
141
cleaning the sample surface, and for monitoring the chemical state of the surface and the ambient during the time required to collect data. Final cleaning, to remove the refractory residue of anodic oxide, was accomplished by an accumulated total of 45 hr A-ion bombardment and heating to 570°C for 24 hr. As reported previously@), this procedure reduces the oxygen Auger peak below detection sensitivity (5% of initial p-p amplitude of oxygen) and maximizes the L,,VV aluminum peak at 67 eV. The importance of this preparation procedure is emphasized here because specular beam intensities before this treatment are only 25% of those measured after removal of oxygen. Subsequent maintenance of an oxygen free surface requires only routine bombardment and annealing to eliminate oxygen adsorbed from the ambient atmosphere of the apparatus operating at IO-” Torr. 2.2,
INTENSITY
MEASUREMENT
The intensity data shown in fig. 1 are isothermal profiles obtained at different (constant) temperatures. The aluminum crystal was heated by thermal conduction from a pyrolytic graphite resistor to which the crystal was attached. Temperature was measured by a ~hromel-alumel thermocouple embedded in the back face of the aluminum crystal. Due to the magnetic field produced by the dc heating current, the field-free angle of incidence (8 = 5 “) is changed in the low energy range of 2&30 eV and the effect is observed as motion of the specular beam. A first order correction for this change in angle was made by adjustment of a trimming magnet at each temperature (current). The observed change in angle of incidence is negligible at primary beam energies greater than 40 eV. Specular beam intensities were measured with a spot photometer (Photo Research Corporation). The acceptance angle of the photometer is 0.25”, which subtends a 1.3 mm diameter area on the fluorescent screen relative to a diffracted spot diameter less than 1 mm and, accordingly, the recorded data are integrated intensities. A step-voltage power supply (FIuke) provided for discrete measurements of intensity at 2 V intervals, after waiting for a suhicient time (2-5 set) at each voltage increment for the relatively slow photometer to reach equilibrium. The voltage is corrected for a 2 V contact potential difference between the bariated nickel cathode and the aluminum crystal {work functions of 2 and 4.2 eV, respectively). Measured intensities are normalized to the incident beam current, following the procedure described by Jona4). The intensity data in fig. 1 are plotted on a log scale to show clearly the fine-structure which is usually obscured by conventional linear plotting, particularly at higher energies.
142
D. T. QKJINTO, B. W. HOLLAND
AND
3. Experimental
W. D. ROBERTSON
results
The log of intensity of the specular beam from the (100) surface of aluminum at an angle of incidence, 0 = 5 ‘, and azimuthal angle, C$= 5’ [angular definitions as in Jona4)], normalized to the measured incident beam current, is presented in fig. 1 as a function of incident energy, 20 to 160 eV, for six different temperatures from 298 “K to 598 “K. I’
I
I
!
I I
n=2f37v)
I
I
I
n=3(63v) AI (100) e= 5’ += 5” n=4(148v
6> -
(26~I I’ 20
1
, (46~)
Il 40
(69v) I 60
ENERGY
i
(97Mt , 80
100
(EXTERNAL),
120
140
I 3
eV
Fig. 1. The logarithm of normalized intensity of the specular (ZOO)beam from the (100) surface of clean aluminum at 0 = 5” (angle of incidence) and 4 = 5” (azimuthal angle relative to the non-primative unit cella) as a function of (external) energy and temperature, 298 to 598°K. The positions of Bragg peaks (zero inner potential) are indicated by arrows for n = 2, 3 and 4. The dashed vertical lines indicate the energies at which the intensity is evaluated, at each temperature.
TEMPERATURE
DEPENDENCE
OF LEED
FROM
Al
143
The three strong peaks at external beam energies of 26, 69 and 134 eV correspond to “Bragg” peaks with n =2, 3 and 4 in the single scattering and the apparent inner potentials associated with these approximation*, peaks are 11, 14 and 14 eV (see fig. 1). The strong secondary peaks at 46 and 97 eV are labeled 23 and 34, respectively, for identification purposes only since these labels do not have a simple meaning in the framework of a multiple scattering theory. Before considering the exponential decrease of intensity with temperature, some general features of the profiles in fig. 1 should be noted particularly. (1) As the temperature rises, a new peak appears between 20 and 23 eV, and increases in intensity relative to its neighbor (n = 2) at 26 eV. Since the observation is close to the limit of experimental precision at low energies, it might be inferred that this unusual behavior is due to uncontrolled changes in the angle of incidence; but intensity profiles from the same crystal at the same azimuth, taken at 2’ intervals from the experimental lower limit of 5 ’ to 21’ at room temperature, do not exhibit the effect. Thus we conclude that it is uniquely associated with temperature and, furthermore, the behavior seems to be similar to that observed by Reid13) on Cu(100). (2) The integral order peaks exhibit changes in shape (appearance and disappearance of shoulders) and small changes in energy with increasing temperature. These effects are not included in the treatment of the data which is limited to intensity evaluated at the vertical dashed lines in fig. 1. (3) The secondary peaks in general follow the behavior of their low energy primary neighbours, except the IZ= 2: peak at 46 eV which becomes asymmetric and shifts to lower energies by about 5 eV in the temperature range of 298 to 598 “K. 3.1. EFFECTIVE DEBYE TEMPERATURES The maximum peak intensities for all prominent peaks (excepting the lowenergy peak in the range of 20-23 eV, for which the available data are insufficient) are presented in fig. 2 as a function of temperature, from which the Debye-Waller exponent and the effective Debye temperature are extracted in accord with the following expressions: I =
IoemzM,
* More precisely, the single-scattering “Bragg” condition is reflected in the intensity profiles as an envelope for multiple scattering peaks7). However, from a weakly scattering metal, such as aluminum, the prominent peaks appear at energies corresponding to the Bragg condition, within the limits of present resolution of 5 2 eV, and are usually designated as “Bragg” peaks.
144
D. T.
QUINTO,
B. W. HOLLAND
AND
W. D. ROBERTSON
where IO is the rigid lattice intensity, m is the ionic (atomic) mass, 0 is the angle of incidence, and i is the electron wavelength inside the crystal with an inner potential of 16.7 eV (V,=4 +I$) to conform with theoretical computations L 2*g). Effective Debye temperatures, obtained from the slopes of the lines, are shown in fig. 2 for each of the prominent maxima in the intensity profiles. These effective Debye temperatures are shown as a function of (external) beam energy in fig. 3. The extreme limits of available bulk Debye temperaturesl7) are shown as a band which must be approached asymptotically by the effective Debye temperature at some higher value of energy. I
I
t 300
I
I
Al(l00)
I
I
I :I&.-
I
I
I
I
I
I
350
400
450
500
550
600
TEMPERATURE
(OK)
Fig. 2. The logarithm of intensity for each prominent peak in fig. 1, as a function of temperature, from which the Debye-Waller factor (I= ZOe-ZM) and the corresponding effective Debye temperature are obtained.
Similarly, the effective BD presumably approaches asymptotically a lower limit at some lower energy associated with the surface layer. The data are represented by a linear function as an approximation to what is, in principle, the middle segment of a very flat sigmoid function. 3.2. DISCUSSION A
feature
of the results
which
is not
unexpected,
being
in accord
with
TEMPERATURE
DEPENDENCE
OF LEED FROM
145
Al
, 150
0
I 25
I 50
I 75
I 100
I 125
ENERGY (EXTERNAL),
I 150
I 175
200
eV
Fig. 3. The effective Debye temperature, obtained from the data in fig. 2, is shown as a function of energy. The range of bulk Debye temperatures, obtained from various sources, is shown in the upper band of the figure.
theoretical predictions 1,219) and other experimental works~10~11), is the rise in effective Debye temperature with increasing energy. Surface ions have considerably larger (perhaps, 2 x) mean square displacementsi2) than bulk ions and, therefore, the contribution of surface ions relative to bulk will fall off with increasing energy, resulting in a rise in the effective Debye temperature, approaching the bulk 8, in the limit of some high energy. The effect does not depend entirely on penetration, and it will appear even if the penetration is constant, because the vibrationally renormalized scattering cross section for the surface ions falls much faster with increasing energy than for the bulk ions. Indeed, Laramores) has modelled this aspect of our results with fair success by choosing different Debye temperatures for the surface and bulk ions, with the surface Debye temperature being chosen as 180°K; that is, less than the lowest measured effective On of 189 “K. The most striking feature of the experimental data is the almost linear (see above) dependence of the effective Debye temperature on energy, for all five (integral and non-integral) intensity maxima. This is in sharp contrast to the results of Reidr”9r3) for Cu(lOO), who finds that the effective Debye temperature is a violently fluctuating function of energy, and similar results have been observed by other workers, for example Tabor et al.il), and are to be expected on the basis of the theory of Duke and Laramorel). The characteristic intensity profiles from aluminum495) cannot be interpreted as im-
146
plying of the peaks. tion16)
D. T. QUINTO,
B. W. HOLLAND
AND
W. D. ROBERTSON
that only single-scattering contributions need be considered because existence of prominent features between the integral order “Bragg” Moreover, Tong and Rhodinl”) have shown that the Double-Diffracapproximation of the Inelastic Collision Modelt) gives a reasonably
good description of the room temperature data. In this respect also aluminum is unusual, and we shall see that these two anomalies are related. Duke and Laramorel) showed that, given certain assumptions [see also Hollandrs)], the rigid lattice theory in the momentum representation is modified for vibrating lattices simply by multiplying each ion-core t-matrix element by the corresponding Debye--Wailer factor. This factor is of the form :
where tiK=A(k-k’) is the momentum transfer (the incoming wave vector is k and the outgoing is k’) and (U,U,), is the thermal average of a product of ion-core displacement components. For single scattering, k is the incident and k’ the reflected external wave vector. In multiple scatterings an integration is performed over the wave-vectors of the Debye-Waller factor in calculating the scattered amplitude. Notice that forward scattering (i.e., k’= k) does not contribute to temperature dependence. Hence any contribution arising from a sequence of multiple forward scatterings followed by a single reversal will show the temperature dependence associated with single scattering. More generally, if the total back-scattered amplitude arises predominantly from scattering paths in which the direction is reversed only once, whether in a single scattering event or by a sequence of smaller angle scatterings, then the temperature dependence of such maxima will have effective Debye temperatures directly related to the Debye temperatures of the surface and bulk layers and a series of such peaks will show an effective Debye temperature varying smoothly with energy and converging on the bulk Debye temperature for high enough energies. Evidently this idea explains our results on aluminum. Neglecting multiple back-scattering within planes, the DoubleDiffraction picture allows only one complete reversal in all contributions to the total scattered amplitude, and we have noted that this approximation gives an adequate description of the diffracted intensity at room temperaturer4). Our interpretation can be substantiated by re-examining the behaviour of the anomalous doublet found by Reidra) on Cu(100) at about 30 to 40 eV. He found that the lower energy peak behaved in a kinematical manner, having an effective Debye temperature about equal to the value obtained from X-ray studies. The higher energy peak had a much lower Debye temperature which was accounted for by HollandIs) in terms of multiple scat-
TEMPERATURE
tering
theory,
DEPENDENCE
using a semi-empirical
OF LEED
isotropic
FROM
AI
Debye-Wailer
147
factor.
Now,
according to our hypothesis above, the lower energy peak should arise principally from scattering paths in which only a single reversal occurs, while the higher energy peak should involve higher orders of back-scattering. Work of Pendrys) bears closely on this question. Pendry made an important advance in the multiple scattering theory of LEED by showing that a perturbation scheme based on the number of backscatterings between layers is both fast and accurate. In this Renormalised Forward Scattering scheme all forward scattering is treated exactly. Pendry performed calculations for Cu(lOO), and though he used a different angle of incidence and azimuth from Reid, one can easily identify Reid’s doublet in the theoretical curves. The significant point is that Pendry’s first order calculation in back-scattering gives a close fit for the lower energy peak, but not for the anomalous higher energy member of the doublet. If we can neglect multiple back-scattering in the plane, this result explains why the temperature dependence of the lower member of the doublet can be interpreted in terms of single scattering. To get a reasonable description of the anomalous peak, Pendry had to go to the next approximation in his scheme, which involves triple back-scattering. Hence the low effective Debye temperature of this peak. It appears that our interpretation of the temperature dependence of scattering from aluminum is not only consistent with Tong and Rhodin’s secondorder perturbation (double-diffraction) calculations, but also provides an explanation for the difference between copper and aluminum, consistent with Pendry’s theory. This interpretation leads naturally to the suggestion that temperature dependence can be used as a diagnostic tool, to classify peaks according to Pendry’s scheme. A monotonic increasing effective Debye temperature for a sequence of diffraction peaks implies that they will be well described by a calculation to first order in back scattering. Diffraction profiles exhibiting more complicated energy dependence of effective Debye temperatures, involving increasing and decreasing values in a sequence of peaks, will in general require a calculation in the Renormalised Forward Scattering Scheme going beyond the first pass. Thus, before time consuming calculation of diffraction profiles, it is possible to specify from the observed temperature dependence the peaks that involve only a single reversal in the scattering paths. After a single pass (in which the layer scattering amplitudes are calculated exactly) we can say which peaks have important contributions from multiple back-scattering in the plane; namely those peaks that are well described in this approximation but show the temperature dependence corresponding to multiple back-scattering. The remaining peaks involve multiple back-scattering between planes, and they will require calculations to higher order for their description.
148
D.T.QUINTO,
B. W.HOLLAND
AND
W.D.ROBERTSON
Acknowledgement This, and associated investigations, are performed under Contract SAR/ N00014-67-0097-0003 with the Office of Naval Research. One of us, D.T.Q., is indebted to the International Nickel Company for a fellowship.
References 1) C. B. Duke and G. E. Laramore, Phys. Rev. B 2 (1970) 4765, 4783. 2) C. B. Duke, G. E. Laramore, B. W. Holland and A. M. Gibbons, Surface Sci. 27 (1971) 523. 3) J. B. Pendry, Phys. Rev. Letters 27 (1971) 856. 4) F. Jona, IBM J. Res. Develop. 14 (1970) 444. 5) T. Quinto, Dissertation, Yale University, Dec. 1971, to be published. 6) T. Quint0 and W. D. Robertson, Surface Sci. 27 (1971) 645. 7) C. W. Tucker and C. B. Duke, Surface Sci. 24 (1971) 31. 8) E. R. Jones, J. T. McKinney and M. B. Webb, Phys. Rev. 151 (1966) 476; 160 (1967) 523. 9) G. E. Laramore, private communication, submitted to Phys. Rev., Comments and Addenda, Feb. 1972. 10) R. J. Reid, Surface Sci. 29 (1972) 603, 623. 11) D. Tabor, J. M. Wilson and T. J. Bastow, Surface Sci. 26 (1971) 471. 12) R. F. Wallis, B. C. Clark and Robert Herman, in: The Structure and Chemisfry of Solid Surfaces, Ed. G. A. Somorjai (Wiley, New York, 1969) p. 17-1. 13) R. J. Reid, Phys. Status Solidi (a) 4 (1971) K 211. 14) S. Y. Tong and T. N. Rhodin, Phys. Rev. Letters 26 (1971) 711. 15) B. W. Holland, Surface Sci. 28 (1971) 258. 16) C. B. Duke, J. R. Anderson and C. W. Tucker, Jr., Surface Sci. 19 (1970) 117. 17) M. Blackman, in: Hundbuch der Physik, Vol. 7 (I), Ed. S. Fltigge (Springer, Berlin, 1955) p. 325.