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A LEED intensity analysis of the W{001}(√2 × √2)R45° surface phase
Surface Science 104 (1981) 405-418 North-Holland Publishing Company
405
A LEED INTENSITY ANALYSIS
OF THE W {001}(42
X 42)R45”
SURFACE
PHASE
J.A. WALKER Department
of Physics, University of York, Heslington,
York YOI 5DD, UK
and
M.K. DEBE * and David The Donnan Laboratories, Received
A. KING The University, Liverpool L69 3BX, UK
30 July 1980;accepted
for publication
27 October
1980
A LEED intensity analysis of the low temperature (42 X J2)R4S0 structure of W (001) is reported, which makes use of extensive data from a surface with a dominance of one rotational domain over the other. Three different models are tested, each involving small lateral displacements of top layer tungsten atoms, and an R-factor analysis is used to determine optimum agreement between experiment and theory. Best agreement is found with the Debe-King model, which has p2mg space group symmetry and involves lateral displacements (closer to 0.16 than 0.32 A) of W atoms in the [ 1101 directions to form a zig-zag row structure. It is suggested that the high temperature (1 X 1) structure also involves a uniform lateral displacement of surface atoms from the bulk positions.
1. Introduction The W{OOl} surface has been the subject of several LEED studies. Above 370 K the clean surface exhibits the expected (1 X 1) periodicity, and LEED intensity analyses have shown that the surface is contracted, although the exact amount varies from study to study. The most extensive determination [l] yields a value of 8% contracted from the bulk, while other values of 6% [2], 11% [3], and 5% [4] have been reported. Ion channelling studies on the W (001) surface indicate that the surface layer is contracted by at most 6% [S]. If the crystal is cooled below 300 K a (d2 X d2)R45’ surface structure is formed and evidence has been presented to demonstrate that this low temperature phase is a clean surface phenomenon [6-81, and hence the observed LEED pattern must be due to a reconstruction of the clean surface. Debe and King [6,7] have proposed a model for the reconstructed surface (based on the observation of lack of four-fold symmetry in the measured spectra, at exactly normal incidence), involving atomic shifts parallel to the plane of the surface in the (ilO) direction. * Present
address:
0039-6028/8
Systems
Research
l/0000-OOOO/$
Laboratory,
3M Center,
St. Paul, Minnesota
02.50 0 1981 North-Holland
Publishing
55133,
Company
USA.
406
J.A. Walker et al. / LEED intensity analysis of W (001 )(J2 x J2)R45”
The same LEED pattern and similar 1(v) spectra are observed when hydrogen is adsorbed on the W(OO1) surface at room temperature (although the symmetry is different) [7], and it has been suggested that hydrogen stabilizes either the low temperature reconstruction [9] or a structure similar to it [lo]. It has been suggested [6,8] that the displacive phase transition is periodic lattice distortion - charge density wave driven, involving a similar mechanism to that proposed for the transition metal dichalcogenide layer materials [ 111, and this has been the subject of recent theoretical investigation [12-141. For the analogous transition on the MO {OOl} surface Inglesfield [ 141 found that there is only a small Kohn anomaly in response to atomic displacements with the wavevector coupling the surface states at the Fermi energy, and hence concluded that the transition is not due to the temperature dependence of the Fermi distribution as in charge density wave systems. The transition is attributed to the instability of the (1 X 1) surface for arbitrary atomic displacements, with the surface states favouring the displacement which couples them together. The surface atoms then move in an anharmonic potential, and it is the temperature dependence of the anharmonic phonons which determines the transition. Recently, the coupling of surface electronic states through the phase transition has been observed by angle-resolved photoemission [ 191. LEED calculations based on the Debe-King model have been published by Barker et al. [9] in comparison with the experimental data of Felter et al. [8]. The comparison was made on the basis of 5 experimental beams and it was concluded that the shifts were in the range 0.15-0.3 A, with a first interlayer spacing of 1.48-l .58 A, the latter value being the bulk spacing. This paper presents a more complete analysis based on the data described by Debe and King [7], using an R-factor analysis [ 151 to compare theory and experiment, and gives consideration to other structures involving small lateral displacements.
2. Experimental
evidence for p2mg symmetry
A complete description of the experimental data has been given elsewhere [6,7] and so will only be briefly summarized here. Normal incidence LEEDI(V) spectra from 24 half-order and 12 integral order beams collected at a temperature of -190 K clearly showed a 2mm point group symmetry:If the surface structure contained equal numbers of all its possible equivalent domains, the diffraction pattern including intensities must show the same rotational symmetry as the substrate. Therefore the observed 2mm symmetry implies a preference for one rotational orientation over another of domains each having 2mm symmetry, with mirror planes in the [ 1 lo] directions. It was also noted that the (S/2, *h/2) spectra and the (+-h/2, G/2) spectra were indentical in relative intensity but differed in absolute intensity by a factor of roughly 2 for the [l/2, l/2] and [3/2,3/2] beam sets. This can be explained by the systematic absences of the (*h/2, *h/2) beams from one
J.A. Walker et al. / LEED intensity analysis of W (OOI)(,/2 X J2)R45”
401
domain so that a predominance of one domain orientation causes the (&h/2, +h/2) beams to be uniformly weaker than the (kh/2, Th/2) beams. The only two-dimensional space group satisfying all the experimental observations is p2mg. If the surface reconstruction is restricted to a single layer then the only structure possible is the one proposed by Debe and King and shown as model A in fig. 1. The top layer atoms move along the diagonals of the substrate mesh by an amount SA, to form zig-zag rows parallel to the (110) direction. This gives a reconstructed surface which has both glide line and mirror plane symmetry [7], and preferential orientation of such (42 X d2)R4.5’ domains completely satisfy the experimental observations.
< lOO>
.l
MODEL
A
MODEL
B
MODEL
C
MODH
D
Fig. 1. Structural single-domain models for the W (001) clean surface; the crossover points on the square lattice represent the position of the 2nd layer W atoms, and the arrow heads the position of top layer atoms. Models A to C: possible structures for the low temperature (J2 X J2)R45” phase. Model D: possible structure for the high temperature (1 X 1) phase.
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412
J.A. Walker et al. / LEED intensity analysis of W (001)(~2
3. Theoretical
X J2)R45”
calculations
The calculated spectra were computed using the CAVLEED suite of LEED programs [ 161, which uses a direct summation method to handle the intralayer scattering and either RFS or layer doubling for the layer summation. In this case they were used with up to 9 phase shifts and 57 beams to ensure convergence up to about 220 eV. The phase shifts used were non-relativistic and were those by Van Hove and Tong [2]. The crystal potential was of the muffin-tin form with the surface-vacuum interface represented by a non-reflecting barrier. The value of the real part of the inner potential was determined from the R-factor analysis, which is an acceptable procedure since at normal incidence there is no refraction of the incident beam at the surface barrier, and the incidence angle needs no correction. Values of 12-14 eV were obtained depending on the structural model. Absorption was represented by the usual optical potential, with a value of Vei = -4 eV. Separate Debye temperatures were used to describe the bulk and surface atom vibrations, with values of OErk = 450 K and f3s)urface= 3 18 K [ 11. LEED spectra were calculated for the Debe-King model (model A in fig. 1) using two values of the parameter SA, viz. SA = 0.32 A and SA = 0.16 A. Assuming a hard ball model, an atom displaced along the (IlO) direction by an amount SA = 0.32 A will touch the two surface atoms moving towards it, and this value is therefore likely to represent a maximum value for the displacement, The surfacebulk layer spacing, d,, was allowed to vary between 1.42 and 1.67 A, corresponding to relaxations of -10% and +6% respectively from the bulk value of 1.58 A. The calculated spectra were averaged over the different possible domains, assuming a preferential orientation as discussed in the previous section. In order to fully investigate the sensitivity of the calculated spectra to the structural parameters. calculations were also performed for two further structural models, shown as models B and C in fig. 1. In model B only half of the atoms in the surface layer move along the diagonals of the substrate mesh (by an amount Sn), while in model C alternating rows of atoms move a distance SC in the (100) and (010) directions respectively. It can be seen that for SA = $u = 42Sc, all three structural models have the same reconstructed unit cell in the surface layer, but differ in the registry of the surface layer with respect to the unreconstructed substrate. Model B was worth examining in the light of recent ion scattering data [ 171. which showed that only half the surface atoms m.ove through the phase transition, although King and Thomas [lo] have suggested that this result may be anticipated from the inhibitive effect of steps and defects on the formation of the low temperature phase. 4. Results Fig, 2 shows normal incidence LEED spectra for seven beams calculated for model A with values of SA = 0.16 and 0.32 A. For each beam and each value of
J.A. Walker et al. f LEED intensity analysis of W (001 I(,/2 X J2)R45”
413
SA, the spectra are shown for two values of d,, namely d, = 1.49 A and d, = 1.52 A corresponding to relaxations of the surface layer of -6% and -4% respectively. In general most features in the experimental spectra are reproduced in the theory, although agreement on relative peak heights is sometimes poor. Typical examples of this are the peak near 100 eV in the (l/2, .5/2) beam which is very weak in the experimental data but is the strongest feature in the calculated spectra, and the peaks at 1.50 and 160 eV in the (S/2,3/2) beam which are similarly weak in the experimental spectra but strong in the theory. The spectra can be seen to be sensitive to the choice of SA and from a visual evaluation a value of SA = 0.16 a appears to produce better agreement between theory and experiment. For instance the experimental data below 130 eV in the (3/2,3/2) beam are reproduced much better with SA = 0.16 A as are the peaks at 208 and 212 eV in the (l/2,5/2) beam. In order to obtain a quantitative estimate of the agreement between theory and
R
0.41
0.2 1.42
1.50
Fig. 3. The reliability factor R plotted as a function layer atoms, d,, for the various trial structures.
I.58
of lattice
1.66
spacing
dz (ii)
between
top and second
414
J.A. Walker et al. / LEED intensity analysis of W (001 )(J2 X ~,/2)R45’
f
m” 2 I,
a? x(D wx
J.A. Walker er al. / LEED intensity analysis of W {OOI)(,,fZ X J2)R45”
415
416
J.A. Walker et al. / LEED intensity analysis of W (001 I(,+/2 X J2)R45”
experiment an R-factor analysis was carried out and the results are shown in fig. 3. It can be seen that Rmin = 0.27 for SA = 0.16 A, and this value occurs at d, = 1.49 A (6% contraction). For S A = 0.32 A, Rmin = 0.29 at d, = 1.47 A (7.3% contraction). Fig. 4 shows normal incidence LEED spectra for 4 beams calculated for Models A, B, and C with SA = 0.16 A, Sn = 0.32 A and SC = 0.12 A. All spectra are shown with the surface-bulk layer spacing d, = 1.49 A. For these values of the parameters the reconstructed unit cell is the same in each case but with a different registry with respect to the unreconstructed bulk. The differences between any two spectra in terms of peak positions and relative peak heights can be seen to be comparable to the differences shown in fig. 2 for the same structure model, but with different values for the parameter SA. An R-factor analysis was carried out for models B and C and the results are shown in fig. 3. It can be seen that for model B with Sn = 0.32 A, Rmin = 0.3 1 at d, = 1.49 A (6% contraction), and for model C with with SC = 0.12 A, R,i, = 0.31 at d, = 1.47 A (7.3% contraction). The results of the R-factor analysis therefore favour model A, which is the p2mg structure, as concluded from the symmetry in the experimental data. Neither model B nor C have the p2mg symmetry and when a comparison was made between experimental beams which were labelled as equivalent and a separate comparison between the corresponding theoretical beams calculated using models B and C, the differences between the theoretical spectra were noticeably greater than those in the experimental beam set, thus again giving us confidence that the correct symmetry had been assigned to the reconstructed surface.
5. Discussion LEED calculations have been performed for various structural models and an R-factor analysis has been applied to determine the best agreement between theory and experiment. Of the structural models tested the best agreement was obtained for the model originally proposed by Debe and King; we now find the atoms in the surface layer move along the diagonals of the surface mesh by 0.16 A, with a surface-bulk displacement of 1.49 A, corresponding to a 6% contraction. LEED I- I’ spectra calculated for the 3 different top layer to substrate registries represented by models A, B and C show relatively subtle differences, as might be expected, since the spectra are less sensitive to different lateral displacements of surface atoms than to different vertical displacements. For example, Barker et al. [9] found that a model with alternative vertical displacements of surface atoms gave very poor agreement with the experimental spectra. This is a further argument against model B, in which only every other surface atom is shifted from the bulk lattice position, since hard sphere considerations would lead to the conclusion that the shifted atoms, necessarily rolling up over atoms in the second layer, should be vertically shifted as well, giving alternate lateral and vertical shifts. This point was
J.A. Walker et al. f LEED intensity analysis of W (OOl)(J2
X J2)R4S”
417
not included in the model, but it can be anticipated that even poorer agreement with the experimental spectra would be obtained. Recently, Melmed et al. [18] have concluded from field ion microscopy results that every other atom is vertically displaced on W {OOl}, in apparent contradiction with the present analysis. However, several points need to be noted concerning their conclusions. What they observe is that during field ion stripping of a W{OOl} surface, alternate surface atoms are field desorbed, firstly producing a p(2 X 2) structure and then a c(2 X 2) structure. From this it is concluded that alternate surface atoms are vertically displaced prior to the stripping. We note the following: (i) The enthalpy change between (1 X 1) and (d2 Xd2)R45’ surface phases is only -1 kcal/mol (0.045 eV/atom) [7]; under the applied field required for field evaporation (-lo8 V cm-‘) it would be surprising if the (42 X d2)R45’ structure remained the most stable phase. (ii) It has been shown that both steps [7] and impurities [20] inhibit the formation of the low temperature phase. Apart from the influence of steps around the small (100) pole on a field ion tip [7], the creation of the first vacancy on the tip by field evaporation must have a profound effect on the surface structure in the immediate vicinity of this defect; whatever the field-stabilized structure prior to the formation of this first vacancy, it must be anticipated that the surface structure will be dramatically altered by it. It is therefore unlikely that the structure formed by successive field evaporation of surface tungsten atoms will bear any resemblance to the zero-field defect-free structure. The formation of the zig-zag chains characteristic of the top layers of models A, B and C is contingent on the energy gained from the near-neighbour proximity of surface atoms in this structure [ 191, which is clearly removed by vacancy formation. (iii) A p(2 X 2) structure is never observed by LEED during cooling of the W{OOl} surface between 1000 and 150 K, confirming that the zero-field, defectfree surface structure bears no relationship to the FIM results. (iv) The FIM results indicate formation of the stripped, ordered c(2 X 2) structure even at temperatures above 400 K, where LEED indicates only a (1 X 1) structure. Despite the conclusion, that the FIM results bear no relationship to the zero-field, defect-free stable surface structure, the unusual observation of ordered structures during field stripping is, we believe, further evidence for the inherent instability of atoms in the W (001) surface, From an analysis of the variation of the intensity of integral order LEED beams with temperature, Debe and King [7] concluded that the phase transition observed on cooling the W{OOl} surface below 370 K was an order-order transition. This conclusion is somewhat controversial, since the observation of a (1 X 1) structure alone at high temperatures is also consistent with a disordered surface layer. Thus, according to Inglesfield [14] the energy gained from electron-phonon coupling is relatively small, and he concludes that the W{OOl} surface atoms are unstable to arbitrary lateral displacements, with the surface layer therefore disordered above 370 K; only at lower temperatures does the small additional energy gained from coupling surface atoms into near-neighbour positions overcome the entropy
J.A. Walker et al. / LEED intensity analysis of W 1001 ](J2 X J2)R4S0
418
of the disordered phase. The analysis based on the variation of integral order LEED beam intensity with temperature [7] would, however, appear to be conclusively in favour of an order-order transition, particularly as the integral order spectra are unchanged through the phase transition. This last fact, coupled with the present analysis, leads to a major difficulty in the order-order phase transition model. The surface-bulk interlayer spacing is virtually the same (1.48 to 1.49 A) (refs. [l] and [7], and present work), for both the (1 X 1) and the (42 X 42)R4.5’ structures, although on hard sphere considerations the former would be expected to be -0.3 A less than the latter. This difficulty could be resolved if the phase stable above 370 K consisted of every top layer atom in a given domain being displaced in the same direction, giving a (1 X 1) top layer with every surface atom shifted by -0.2 A in the Ci 10) direction. An example of such a structure is given in fig. 1D. This model would also be in agreement with Inglesfield’s considerations [ 141. At some higher temperature, an order-disorder transition might be anticipated; the DebyeWaller plots of Debe and King [7] are linear over the range 370 to 1000 K, indicating that this transition must, if it exists, occur at a temperature above 1000 K.
Acknowledgements We acknowledge the Science Research Council for a grant to purchase the equipment and for Fellowships to J.A.W. and M.K.D. References [l] [2] [3] (41 [S] [6] [7] [8] [9] [lo] [ 111 [12] (131 [14] [15] [ 161 [ 171 (181
F.S. Marsh, M.K. Debe and D.A. King, J. Phys. Cl3 (1980) 2799; see also, M.K. Debe, D.A. King and F.S. Marsh, Surface Sci. 68 (1977) 437. M.A. Van Hove and S.Y. Tong, Surface Sci. 54 (1976) 91. B.W. Lee, A. Ignatiev, S.Y. Tong and M.A. Van Hove, J. Vacuum Sci. Technol. 14 (1977) 291. J. Kirschner and R. Feder, Surface Sci. 79 (1979) 176. L.C. Feldman, R.L. Kauffman, P.J. Silverman, R.A. Zuhr and J.H. Barrett, Phys. Rev. Letters 39 (1977) 38. M.K. Debe and D.A. King, J. Phys. Cl0 (1977) L333; Phys. Rev. Letters 39 (1977) 708. M.K. Debe and D.A. King, Surface Sci. 81 (1979) 193. T.E. Felter, R.A. Barker and P.J. Estrup, Phys. Rev. Letters 38 (1977) 1138. R.A. Barker, P.J. Estrup, F. Jona and P.M. Marcus, Solid State Commun. 25 (1978) 375. D.A. King and G. Thomas, Surface Sci. 92 (1980) 201. J.A. Wilson, F.J. DeSolvo and S. Mahajan, Phys. Rev. Letters 32 (1974) 882.
E. Tosatti, Solid State Commun. 25 (1978) 637. J.E. Inglesfield, J. Phys. Cl1 (1978) L69. J.E. Inglesfield, J. Phys. Cl2 (1979) 149. E. Zanazzi and F. Jona, Surface Sci. 62 (1977) 61. C.G. Kinniburgh and D. Titterington, Computer Phys. Commun., submitted. I. Stensgaard, L.C. Feldman and P.J. Silverman, Phys. Rev. Letters 42 (1979) 247. A.J. Melmed, R.T. Tung, W.R. Graham and G.D.W. Smith, Phys. Rev. Letters 43 (1979) 1521. [19] J.C. Campuzano, D.A. King, C. Somerton and J.E. Inglesfield, Phys. Rev. Letters, 45 (1980) 1649.
405
A LEED INTENSITY ANALYSIS
OF THE W {001}(42
X 42)R45”
SURFACE
PHASE
J.A. WALKER Department
of Physics, University of York, Heslington,
York YOI 5DD, UK
and
M.K. DEBE * and David The Donnan Laboratories, Received
A. KING The University, Liverpool L69 3BX, UK
30 July 1980;accepted
for publication
27 October
1980
A LEED intensity analysis of the low temperature (42 X J2)R4S0 structure of W (001) is reported, which makes use of extensive data from a surface with a dominance of one rotational domain over the other. Three different models are tested, each involving small lateral displacements of top layer tungsten atoms, and an R-factor analysis is used to determine optimum agreement between experiment and theory. Best agreement is found with the Debe-King model, which has p2mg space group symmetry and involves lateral displacements (closer to 0.16 than 0.32 A) of W atoms in the [ 1101 directions to form a zig-zag row structure. It is suggested that the high temperature (1 X 1) structure also involves a uniform lateral displacement of surface atoms from the bulk positions.
1. Introduction The W{OOl} surface has been the subject of several LEED studies. Above 370 K the clean surface exhibits the expected (1 X 1) periodicity, and LEED intensity analyses have shown that the surface is contracted, although the exact amount varies from study to study. The most extensive determination [l] yields a value of 8% contracted from the bulk, while other values of 6% [2], 11% [3], and 5% [4] have been reported. Ion channelling studies on the W (001) surface indicate that the surface layer is contracted by at most 6% [S]. If the crystal is cooled below 300 K a (d2 X d2)R45’ surface structure is formed and evidence has been presented to demonstrate that this low temperature phase is a clean surface phenomenon [6-81, and hence the observed LEED pattern must be due to a reconstruction of the clean surface. Debe and King [6,7] have proposed a model for the reconstructed surface (based on the observation of lack of four-fold symmetry in the measured spectra, at exactly normal incidence), involving atomic shifts parallel to the plane of the surface in the (ilO) direction. * Present
address:
0039-6028/8
Systems
Research
l/0000-OOOO/$
Laboratory,
3M Center,
St. Paul, Minnesota
02.50 0 1981 North-Holland
Publishing
55133,
Company
USA.
406
J.A. Walker et al. / LEED intensity analysis of W (001 )(J2 x J2)R45”
The same LEED pattern and similar 1(v) spectra are observed when hydrogen is adsorbed on the W(OO1) surface at room temperature (although the symmetry is different) [7], and it has been suggested that hydrogen stabilizes either the low temperature reconstruction [9] or a structure similar to it [lo]. It has been suggested [6,8] that the displacive phase transition is periodic lattice distortion - charge density wave driven, involving a similar mechanism to that proposed for the transition metal dichalcogenide layer materials [ 111, and this has been the subject of recent theoretical investigation [12-141. For the analogous transition on the MO {OOl} surface Inglesfield [ 141 found that there is only a small Kohn anomaly in response to atomic displacements with the wavevector coupling the surface states at the Fermi energy, and hence concluded that the transition is not due to the temperature dependence of the Fermi distribution as in charge density wave systems. The transition is attributed to the instability of the (1 X 1) surface for arbitrary atomic displacements, with the surface states favouring the displacement which couples them together. The surface atoms then move in an anharmonic potential, and it is the temperature dependence of the anharmonic phonons which determines the transition. Recently, the coupling of surface electronic states through the phase transition has been observed by angle-resolved photoemission [ 191. LEED calculations based on the Debe-King model have been published by Barker et al. [9] in comparison with the experimental data of Felter et al. [8]. The comparison was made on the basis of 5 experimental beams and it was concluded that the shifts were in the range 0.15-0.3 A, with a first interlayer spacing of 1.48-l .58 A, the latter value being the bulk spacing. This paper presents a more complete analysis based on the data described by Debe and King [7], using an R-factor analysis [ 151 to compare theory and experiment, and gives consideration to other structures involving small lateral displacements.
2. Experimental
evidence for p2mg symmetry
A complete description of the experimental data has been given elsewhere [6,7] and so will only be briefly summarized here. Normal incidence LEEDI(V) spectra from 24 half-order and 12 integral order beams collected at a temperature of -190 K clearly showed a 2mm point group symmetry:If the surface structure contained equal numbers of all its possible equivalent domains, the diffraction pattern including intensities must show the same rotational symmetry as the substrate. Therefore the observed 2mm symmetry implies a preference for one rotational orientation over another of domains each having 2mm symmetry, with mirror planes in the [ 1 lo] directions. It was also noted that the (S/2, *h/2) spectra and the (+-h/2, G/2) spectra were indentical in relative intensity but differed in absolute intensity by a factor of roughly 2 for the [l/2, l/2] and [3/2,3/2] beam sets. This can be explained by the systematic absences of the (*h/2, *h/2) beams from one
J.A. Walker et al. / LEED intensity analysis of W (OOI)(,/2 X J2)R45”
401
domain so that a predominance of one domain orientation causes the (&h/2, +h/2) beams to be uniformly weaker than the (kh/2, Th/2) beams. The only two-dimensional space group satisfying all the experimental observations is p2mg. If the surface reconstruction is restricted to a single layer then the only structure possible is the one proposed by Debe and King and shown as model A in fig. 1. The top layer atoms move along the diagonals of the substrate mesh by an amount SA, to form zig-zag rows parallel to the (110) direction. This gives a reconstructed surface which has both glide line and mirror plane symmetry [7], and preferential orientation of such (42 X d2)R4.5’ domains completely satisfy the experimental observations.
< lOO>
.l
MODEL
A
MODEL
B
MODEL
C
MODH
D
Fig. 1. Structural single-domain models for the W (001) clean surface; the crossover points on the square lattice represent the position of the 2nd layer W atoms, and the arrow heads the position of top layer atoms. Models A to C: possible structures for the low temperature (J2 X J2)R45” phase. Model D: possible structure for the high temperature (1 X 1) phase.
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J.A. Walker et al. / LEED intensity analysis of W (001 )(J2 X J2jR45”
412
J.A. Walker et al. / LEED intensity analysis of W (001)(~2
3. Theoretical
X J2)R45”
calculations
The calculated spectra were computed using the CAVLEED suite of LEED programs [ 161, which uses a direct summation method to handle the intralayer scattering and either RFS or layer doubling for the layer summation. In this case they were used with up to 9 phase shifts and 57 beams to ensure convergence up to about 220 eV. The phase shifts used were non-relativistic and were those by Van Hove and Tong [2]. The crystal potential was of the muffin-tin form with the surface-vacuum interface represented by a non-reflecting barrier. The value of the real part of the inner potential was determined from the R-factor analysis, which is an acceptable procedure since at normal incidence there is no refraction of the incident beam at the surface barrier, and the incidence angle needs no correction. Values of 12-14 eV were obtained depending on the structural model. Absorption was represented by the usual optical potential, with a value of Vei = -4 eV. Separate Debye temperatures were used to describe the bulk and surface atom vibrations, with values of OErk = 450 K and f3s)urface= 3 18 K [ 11. LEED spectra were calculated for the Debe-King model (model A in fig. 1) using two values of the parameter SA, viz. SA = 0.32 A and SA = 0.16 A. Assuming a hard ball model, an atom displaced along the (IlO) direction by an amount SA = 0.32 A will touch the two surface atoms moving towards it, and this value is therefore likely to represent a maximum value for the displacement, The surfacebulk layer spacing, d,, was allowed to vary between 1.42 and 1.67 A, corresponding to relaxations of -10% and +6% respectively from the bulk value of 1.58 A. The calculated spectra were averaged over the different possible domains, assuming a preferential orientation as discussed in the previous section. In order to fully investigate the sensitivity of the calculated spectra to the structural parameters. calculations were also performed for two further structural models, shown as models B and C in fig. 1. In model B only half of the atoms in the surface layer move along the diagonals of the substrate mesh (by an amount Sn), while in model C alternating rows of atoms move a distance SC in the (100) and (010) directions respectively. It can be seen that for SA = $u = 42Sc, all three structural models have the same reconstructed unit cell in the surface layer, but differ in the registry of the surface layer with respect to the unreconstructed substrate. Model B was worth examining in the light of recent ion scattering data [ 171. which showed that only half the surface atoms m.ove through the phase transition, although King and Thomas [lo] have suggested that this result may be anticipated from the inhibitive effect of steps and defects on the formation of the low temperature phase. 4. Results Fig, 2 shows normal incidence LEED spectra for seven beams calculated for model A with values of SA = 0.16 and 0.32 A. For each beam and each value of
J.A. Walker et al. f LEED intensity analysis of W (001 I(,/2 X J2)R45”
413
SA, the spectra are shown for two values of d,, namely d, = 1.49 A and d, = 1.52 A corresponding to relaxations of the surface layer of -6% and -4% respectively. In general most features in the experimental spectra are reproduced in the theory, although agreement on relative peak heights is sometimes poor. Typical examples of this are the peak near 100 eV in the (l/2, .5/2) beam which is very weak in the experimental data but is the strongest feature in the calculated spectra, and the peaks at 1.50 and 160 eV in the (S/2,3/2) beam which are similarly weak in the experimental spectra but strong in the theory. The spectra can be seen to be sensitive to the choice of SA and from a visual evaluation a value of SA = 0.16 a appears to produce better agreement between theory and experiment. For instance the experimental data below 130 eV in the (3/2,3/2) beam are reproduced much better with SA = 0.16 A as are the peaks at 208 and 212 eV in the (l/2,5/2) beam. In order to obtain a quantitative estimate of the agreement between theory and
R
0.41
0.2 1.42
1.50
Fig. 3. The reliability factor R plotted as a function layer atoms, d,, for the various trial structures.
I.58
of lattice
1.66
spacing
dz (ii)
between
top and second
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J.A. Walker et al. / LEED intensity analysis of W (001 )(J2 X ~,/2)R45’
f
m” 2 I,
a? x(D wx
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J.A. Walker et al. / LEED intensity analysis of W (001 I(,+/2 X J2)R45”
experiment an R-factor analysis was carried out and the results are shown in fig. 3. It can be seen that Rmin = 0.27 for SA = 0.16 A, and this value occurs at d, = 1.49 A (6% contraction). For S A = 0.32 A, Rmin = 0.29 at d, = 1.47 A (7.3% contraction). Fig. 4 shows normal incidence LEED spectra for 4 beams calculated for Models A, B, and C with SA = 0.16 A, Sn = 0.32 A and SC = 0.12 A. All spectra are shown with the surface-bulk layer spacing d, = 1.49 A. For these values of the parameters the reconstructed unit cell is the same in each case but with a different registry with respect to the unreconstructed bulk. The differences between any two spectra in terms of peak positions and relative peak heights can be seen to be comparable to the differences shown in fig. 2 for the same structure model, but with different values for the parameter SA. An R-factor analysis was carried out for models B and C and the results are shown in fig. 3. It can be seen that for model B with Sn = 0.32 A, Rmin = 0.3 1 at d, = 1.49 A (6% contraction), and for model C with with SC = 0.12 A, R,i, = 0.31 at d, = 1.47 A (7.3% contraction). The results of the R-factor analysis therefore favour model A, which is the p2mg structure, as concluded from the symmetry in the experimental data. Neither model B nor C have the p2mg symmetry and when a comparison was made between experimental beams which were labelled as equivalent and a separate comparison between the corresponding theoretical beams calculated using models B and C, the differences between the theoretical spectra were noticeably greater than those in the experimental beam set, thus again giving us confidence that the correct symmetry had been assigned to the reconstructed surface.
5. Discussion LEED calculations have been performed for various structural models and an R-factor analysis has been applied to determine the best agreement between theory and experiment. Of the structural models tested the best agreement was obtained for the model originally proposed by Debe and King; we now find the atoms in the surface layer move along the diagonals of the surface mesh by 0.16 A, with a surface-bulk displacement of 1.49 A, corresponding to a 6% contraction. LEED I- I’ spectra calculated for the 3 different top layer to substrate registries represented by models A, B and C show relatively subtle differences, as might be expected, since the spectra are less sensitive to different lateral displacements of surface atoms than to different vertical displacements. For example, Barker et al. [9] found that a model with alternative vertical displacements of surface atoms gave very poor agreement with the experimental spectra. This is a further argument against model B, in which only every other surface atom is shifted from the bulk lattice position, since hard sphere considerations would lead to the conclusion that the shifted atoms, necessarily rolling up over atoms in the second layer, should be vertically shifted as well, giving alternate lateral and vertical shifts. This point was
J.A. Walker et al. f LEED intensity analysis of W (OOl)(J2
X J2)R4S”
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not included in the model, but it can be anticipated that even poorer agreement with the experimental spectra would be obtained. Recently, Melmed et al. [18] have concluded from field ion microscopy results that every other atom is vertically displaced on W {OOl}, in apparent contradiction with the present analysis. However, several points need to be noted concerning their conclusions. What they observe is that during field ion stripping of a W{OOl} surface, alternate surface atoms are field desorbed, firstly producing a p(2 X 2) structure and then a c(2 X 2) structure. From this it is concluded that alternate surface atoms are vertically displaced prior to the stripping. We note the following: (i) The enthalpy change between (1 X 1) and (d2 Xd2)R45’ surface phases is only -1 kcal/mol (0.045 eV/atom) [7]; under the applied field required for field evaporation (-lo8 V cm-‘) it would be surprising if the (42 X d2)R45’ structure remained the most stable phase. (ii) It has been shown that both steps [7] and impurities [20] inhibit the formation of the low temperature phase. Apart from the influence of steps around the small (100) pole on a field ion tip [7], the creation of the first vacancy on the tip by field evaporation must have a profound effect on the surface structure in the immediate vicinity of this defect; whatever the field-stabilized structure prior to the formation of this first vacancy, it must be anticipated that the surface structure will be dramatically altered by it. It is therefore unlikely that the structure formed by successive field evaporation of surface tungsten atoms will bear any resemblance to the zero-field defect-free structure. The formation of the zig-zag chains characteristic of the top layers of models A, B and C is contingent on the energy gained from the near-neighbour proximity of surface atoms in this structure [ 191, which is clearly removed by vacancy formation. (iii) A p(2 X 2) structure is never observed by LEED during cooling of the W{OOl} surface between 1000 and 150 K, confirming that the zero-field, defectfree surface structure bears no relationship to the FIM results. (iv) The FIM results indicate formation of the stripped, ordered c(2 X 2) structure even at temperatures above 400 K, where LEED indicates only a (1 X 1) structure. Despite the conclusion, that the FIM results bear no relationship to the zero-field, defect-free stable surface structure, the unusual observation of ordered structures during field stripping is, we believe, further evidence for the inherent instability of atoms in the W (001) surface, From an analysis of the variation of the intensity of integral order LEED beams with temperature, Debe and King [7] concluded that the phase transition observed on cooling the W{OOl} surface below 370 K was an order-order transition. This conclusion is somewhat controversial, since the observation of a (1 X 1) structure alone at high temperatures is also consistent with a disordered surface layer. Thus, according to Inglesfield [14] the energy gained from electron-phonon coupling is relatively small, and he concludes that the W{OOl} surface atoms are unstable to arbitrary lateral displacements, with the surface layer therefore disordered above 370 K; only at lower temperatures does the small additional energy gained from coupling surface atoms into near-neighbour positions overcome the entropy
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of the disordered phase. The analysis based on the variation of integral order LEED beam intensity with temperature [7] would, however, appear to be conclusively in favour of an order-order transition, particularly as the integral order spectra are unchanged through the phase transition. This last fact, coupled with the present analysis, leads to a major difficulty in the order-order phase transition model. The surface-bulk interlayer spacing is virtually the same (1.48 to 1.49 A) (refs. [l] and [7], and present work), for both the (1 X 1) and the (42 X 42)R4.5’ structures, although on hard sphere considerations the former would be expected to be -0.3 A less than the latter. This difficulty could be resolved if the phase stable above 370 K consisted of every top layer atom in a given domain being displaced in the same direction, giving a (1 X 1) top layer with every surface atom shifted by -0.2 A in the Ci 10) direction. An example of such a structure is given in fig. 1D. This model would also be in agreement with Inglesfield’s considerations [ 141. At some higher temperature, an order-disorder transition might be anticipated; the DebyeWaller plots of Debe and King [7] are linear over the range 370 to 1000 K, indicating that this transition must, if it exists, occur at a temperature above 1000 K.
Acknowledgements We acknowledge the Science Research Council for a grant to purchase the equipment and for Fellowships to J.A.W. and M.K.D. References [l] [2] [3] (41 [S] [6] [7] [8] [9] [lo] [ 111 [12] (131 [14] [15] [ 161 [ 171 (181
F.S. Marsh, M.K. Debe and D.A. King, J. Phys. Cl3 (1980) 2799; see also, M.K. Debe, D.A. King and F.S. Marsh, Surface Sci. 68 (1977) 437. M.A. Van Hove and S.Y. Tong, Surface Sci. 54 (1976) 91. B.W. Lee, A. Ignatiev, S.Y. Tong and M.A. Van Hove, J. Vacuum Sci. Technol. 14 (1977) 291. J. Kirschner and R. Feder, Surface Sci. 79 (1979) 176. L.C. Feldman, R.L. Kauffman, P.J. Silverman, R.A. Zuhr and J.H. Barrett, Phys. Rev. Letters 39 (1977) 38. M.K. Debe and D.A. King, J. Phys. Cl0 (1977) L333; Phys. Rev. Letters 39 (1977) 708. M.K. Debe and D.A. King, Surface Sci. 81 (1979) 193. T.E. Felter, R.A. Barker and P.J. Estrup, Phys. Rev. Letters 38 (1977) 1138. R.A. Barker, P.J. Estrup, F. Jona and P.M. Marcus, Solid State Commun. 25 (1978) 375. D.A. King and G. Thomas, Surface Sci. 92 (1980) 201. J.A. Wilson, F.J. DeSolvo and S. Mahajan, Phys. Rev. Letters 32 (1974) 882.
E. Tosatti, Solid State Commun. 25 (1978) 637. J.E. Inglesfield, J. Phys. Cl1 (1978) L69. J.E. Inglesfield, J. Phys. Cl2 (1979) 149. E. Zanazzi and F. Jona, Surface Sci. 62 (1977) 61. C.G. Kinniburgh and D. Titterington, Computer Phys. Commun., submitted. I. Stensgaard, L.C. Feldman and P.J. Silverman, Phys. Rev. Letters 42 (1979) 247. A.J. Melmed, R.T. Tung, W.R. Graham and G.D.W. Smith, Phys. Rev. Letters 43 (1979) 1521. [19] J.C. Campuzano, D.A. King, C. Somerton and J.E. Inglesfield, Phys. Rev. Letters, 45 (1980) 1649.