LEED surface crystallography, R-factors and the structure of the (110) surfaces of III–V semiconductors

Surface Science 177 (1986) L915-L924 North-Holland, Amsterdam

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S U R F A C E S C I E N C E LETTERS

LEED SURFACE CRYSTALLOGRAPHY, R - F A C T O R S AND T H E S T R U C T U R E OF T H E (110) SURFACES OF III-V SEMICONDUCTORS P.G. COWELL, M. P R U T T O N and S.P. T E A R Department of Pt~vsics, University of York, Heslington, York YOI 5DD, UK Received 20 May 1986; accepted for publication 17 July 1986

Investigation of lnSb(ll0) is presented which is used as the basis for a study of the way in which R-factors are used in LEED methods. These are general comments on the nature of R-factors and the methodology of LEED appropriate to m a n y of the complex surfaces such as the zincblende (110) surface now being studied.

Recently an interesting exchange of letters has occurred in the scientific press concerning the surface structure of G a A s ( l l 0 ) as determined by LEED analysis [1-3]. The technological importance of this compound makes the analysis of its surfaces of great interest and it claims the title of the first semiconductor with a surface structure determined by LEED [4]. The nature of the slight controversy indicated in the recent correspondence may seem slightly esoteric to those outside the LEED community as the conclusions of both parties on the structure of G a A s ( l l 0 ) are the same. In fact they represent refinements of the surface structure originally published in 1976 [4]. New interest was added to the problem of the structure of G a A s ( l l 0 ) with the publication of a new model structure from isochromat spectroscopy [5,6] and angularly resolved photoemission [7]. The L E E D analysis was reconsidered [1] and a new local minimum in R-factor was found near the new surface structure model. The global minimum in the parameter space investigated was still near the old 1976 model, i.e. with the G a - A s bonds in the surface making an angle of 29 ° to the surface. The new structure (with the G a - A s bonds at 7 ° to the surface) was therefore rejected. More data will be presented in this letter to confirm the results obtained for G a A s ( l l 0 ) by analogy with I n S b ( l l 0 ) which is expected to be physically similar to GaAs [9]. The general similarities between the I I I - V semiconductors surface structures have been discussed by Kahn [8]. This letter will also concern itself with some discussion and criticism of R-factors and the methodologies used in LEED analyses. It is our belief that part of the confusion which has arisen recently in the L E E D community over 0039-6028/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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the use of R-factors may stem from a lack of appreciation of the true meaning of an R-factor and a certain carelessness of language when discussing L E E D results. This might lead to the mis-interpretation of the structural models which result from a L E E D analysis.

What is an R-factor? There has been much discussion in the recent past of the use of R-factors in L E E D analyses [14,15]. When evaluating a potential R-factor the criterion nearly always used in choosing or constructing a function has been that it should be most sensitive to the positions of the peaks in an I ( V ) curve [12,16,17]. In other words it should act in the same way as the visual comparisons used before R-factors"arrived in the area of LEED analysis. This is stated explicitly as the reason for the quite complex form of the Zanazzi and Jona R-factor, but is this a legitimate criterion to use? It has been repeatedly demonstrated that there is no simple way of relating peak positions to crystal structures using dynamical multiple scattering theory. It might seem unreasonable if the improvement of the technique of L E E D analysis by the use of R-factors were to produce radically different structural results and this would tend to justify the use of this criterion. On the other hand, if by using R-factors it is hoped to improve the accuracy and credibility of structures determined using LEED, they should rest on more substantial or reliable foundations than the method they replace. It is by reason of their sound mathematical origin that Philip and Rundgren [14] have proposed that metrics should be used as R-factors. However, the definition of a metric includes many functions inappropriate for use as an R-factor. One simple example is the discrete metric [23] defined as D ( a , a) = 0, D(a, b) = 1 for all a 4: b. The problem of choice remains. If an R-factor is to be a metric which is it to be and why? An idea of the action and meaning of R-factors may be derived from considering the theory of convergence. A sequence ( x n } of real numbers converges to a limit x if, for every value of ~, all but a finite number of members of the sequence lie between x - ~ and x + c. This definition of convergence is equivalent to saying that, for the sequence { x~ } with a limit x, the distance between the point x~ and the point x tends to zero as n tends to infinity. The function I x , - x l is a metric which tests for uniform convergence. This argument may easily he extended to sequences of functions but then the concept of the convergence of functions is far more complicated. In theory any metric may be used to test for convergence but the kind of convergence will be different for different metrics. When applied, in the situation of LEED analysis, the conceptual complication is enormous. An R-factor metric tests for some quite esoteric kind of convergence between large sets of complicated functions. However, the use of a metric for the comparison should guarantee some measure of convergence (provided that it can be made to tend to zero as the parameters are varied).

P. (7,. Cowell et aL / (11 O) surfaces of I I l - V semiconductors

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In statistics, distance measures like metrics are used in areas such as multidimensional scaling [18]. A dissimilarity function (which may not be a metric but shares some of its properties) is used to order large amounts of data in such a way that conclusions may be drawn. A dissimilarity function is a non-negative, symmetric function that gives a zero result only when the things being compared are exactly similar. It need not obey the triangle inequality as all metrics must. If the analogy is drawn between R-factors and dissimilarity functions it is apparent that an R-factor may legitimately be used that does not satisfy the rules for being a metric. This allows great scope in the choice of R-factor and it is still not clear on what basis that choice should be made. Probably the safest way to regard R-factors is in terms of a demarcation criterion as described by Popper [19]. That is, a criterion which is able to distinguish between the currently accepted "scientific truth" (a "well defined" surface structure) and its potential replacement. The number of potential structures is so vast in an indirect surface structure method like L E E D that no one could reasonably be expected to search through more than a small subset of the possible parameter space. Therefore all the results should correctly be regarded as provisional. As Popper wrote: " T h e old scientific ideal of episteme - of absolutely certain, demonstrable knowledge - has proved to be an idol ... every scientific statement must remain tentative for ever" [20]. An R-factor analysis is one method of refining knowledge in the small world of LEED. Of course, a potential structure established by L E E D or any other method is more convincing if it is the same as a tentative structure determined by another, physically different, technique. A n I n S b ( l l O ) t h e o r y - t h e o r y comparison. The results we present for this surface are not a surface structure determination. They are part of the preliminary work that it was decided was necessary before attempting a thorough L E E D analysis of InSb(110). The work was also done so that it may serve as a model case for the investigation of techniques for searching complex parameter spaces efficiently. Fig. 2 shows the results of an R-factor comparison between a set of theoretical models in which only the top layer has been distorted from bulk positions (see fig. 1) and data calculated using the published surface structure of I n S b ( l l 0 ) [10,11]. The decision to perform theory-theory comparisons only was made to simplify the problem in several ways. In these comparisons there is a correct solution. The minimum in R-factor should correctly occur at the bond angle and relaxation chosen for the "experiment". Also the noise and other undesirable effects from experimental data were not introduced into the comparisons. All the work done up to the present indicates that the published model [10,11] is substantially correct and subsequent improvements will consist of refinements of this model. It is described in table 1. Fig. 2 clearly shows a minimum at the expected position where the rotation

P.G. Cowell et al. / (110) surfaces of I l l - V semiconductors

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Sb

In -

T:

Fig. 1. A view of the zincblende InSb(ll0) surface from the ( i l 0 ) direction. The left hand side shows a bulk termination and the fight hand side is a reconstructed surface which shows the way in which relaxation, ~ and surface bond rotation, w are defined.

_.-u

J"

/

\

/ I ~ 1'.

..--

" ~ "4

Fig. 2. Pendry R-factor plotted against rotation and relaxation of the top layer. The comparison is between the calculated published model and models with rotations and relaxations of the top layer only. Twelve I ( V ) curves were used in the comparison: (1, 0, ( - 1, 0), (0, 1), (1, 1), ( - 1, 1), (2, 0), ( - 2 , 0), (2, 1), ( - 2 , 1), (0, 2), (1, 2) and ( - 1, 2). The lines marked a, b and c are explained in the text and in fig. 4. The smallest R-factor in the parameter range was 0.528 at the point marked X. The subsidiary m i n i m u m is marked Y.

P.G. Cowell et al. / (110) surfaces of l l l - V semiconductors

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Table 1 S u m m a r y of the published model for InSb(ll0) a) Layer 1

Layer 2

Rumple (displacement between In and Sb ions normal to surface) as a percentage of the bulk inter-layer spacing

34%

-8%

Rotation of InSb bond out of surface plane

28.8 °

-6.4 °

Displacement parallel to the surface

Such as to conserve I n - S b bond length

No shifts (bond relaxation)

- 9%

0%

Relaxation of rotated layer as a percentage of the inter-layer spacing

a) The data is taken from refs. [1031] where it is called the "Best fit" model. The relaxation is defined as the displacement perpendicular to the surface of the mean ion position in a layer compared to the bulk positions.

and relaxation of the top layer correspond to the rotation and relaxation of the published model, i.e. the I n - S b bond rotated about 30 ° and the top layer relaxed about - 10% (inwards). There is, however, a clearly defined subsidiary minimum near zero rotation and 5% relaxation. The R-factor used for fig. 2 is that of Pendry [12]. Fig. 3 is a diagram of the Euclidean norm [23] values that result from the same comparisons. This is a more rational version of the X-ray

o f

-

o6

-20

Fig. 3. A contour plot of Euclidean norm against rotation and relaxation. The comparison is equivalent to that shown in fig. 2, The lowest value for this R-factor was 0,364 at the point marked X. The subsidiary m i n i m u m is marked Y.

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R-factor used by m a n y LEED groups [13]. The X-ray R-factor in most of its various forms is an asymmetric function, R ( a , b) does not equal R ( b , a). The Euclidean norm is symmetric in form. It is the integral of the root mean square difference between two curves both normalised to unit integral over the range of comparison. It is a metric [14]. It shows the effect of a minimum in the expected position with an unexpected subsidiary minimum near zero rotation, 5% relaxation. Other R-factors have been used for the same set of comparisons and have all yielded similar results. It is therefore reasonable to conclude that the occurrence of two comparably deep minima is caused by the data. In other words the I ( V ) curves calculated both for 30 ° rotation, - 10% relaxation and for near zero rotation, 5% relaxation resemble the "experimental" curves (in this case the curves calculated for the published model). In all the situations examined, there has been a slight preference (as indicated by the R-factor) for a model with a large surface bond rotation, but the alternative surface structure from the subsidiary m i n i m u m should be carefully considered as it is only slightly less probable. It would be wrong to dismiss a possible result by blaming the R-factor or deliberately constructing a new R-factor which forces one or other of the models to be chosen. Similar results have quite commonly been found in the investigations of I I I - V compound surfaces [1,21,22]. In these investigations the range of parameter space explored and the search techniques used have not always been such as would make it possible to decide unequivocally on the surface structure on the basis of the LEED analyses alone. However, the evidence from other surface structure analysis techniques is, on balance, in favour of structural models with a high (about 30 °) bond rotation in the top layer. With such a complicated shape to the parameter space to be searched an obvious pitfall to avoid is that of finding the wrong minimum or at least failing to notice the presence of the significant subsidiary minimum. This defect can be shown, with hindsight, to have been quite frequent in the past but now that more powerful computers have become available for making L E E D calculations much more rapidly it should be less common. Indeed the history of the G a A s ( l l 0 ) surface serves as an example: the original structure determinations [4] failed to find the potential surface structure with a bond rotation of about 7 ° until the model was proposed by workers in other fields of surface science. The dangers of inadequate search techniques such as searching only along lines of constant bond rotation or constant relaxation are demonstrated in fig. 4. Fig. 4 shows three cuts through fig. 2. The lines are marked on fig. 2 as a - a , b - b , c-c. Fig. 4a shows a cut at constant relaxation and shows the R-factor for rotations of the surface bond about a bulk termination, i.e. relaxation = 0%. A search through the simple 2D parameter space only along this line would lead one to expect the global minimum to be around zero rotation. If the limits of the search were then to be narrowed to probe only the area around this

P.G. Cowell et al. / (110) surfaces of I I I - V semiconductors

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L921

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0.950-

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0.875-

-20 j

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Rotation / *

Fig. 4a. A section through fig. 2 along line a-a. This is a line of constant relaxation (relaxation = 0%). The correct value of the rotation should be 30 ° but the lowest R-factor corresponds to 0 °. This section is equivalent to many used in the analysis of GaAs [1,2] and other surfaces.

suspected solution, an incorrect structural m o d e l w o u l d result. Fig. 4a does show a s u b s i d i a r y m i n i m u m at the correct r o t a t i o n value b u t fig. 4b shows o n l y one m i n i m u m . This figure shows a cut at c o n s t a n t rotation. T h e minim u m is at the correct r e l a x a t i o n value as the cut is taken through the global m i n i m u m in fig. 2, i.e. the r o t a t i o n is 30 ° . T h e r e is no sign of the s u b s i d i a r y m i m i m u m which is k n o w n to b e c o m p a r a b l e to the g l o b a l m i n i m u m . It was b y t a k i n g sections t h r o u g h p a r a m e t e r space similar to cut b that caused the early investigators o f G a A s ( l l 0 ) to fail to observe a s u b s i d i a r y m i n i m u m . Fig. 4c shows the cut t h r o u g h b o t h m i n i m a in fig. 2. This is the o n l y line t h r o u g h the p a r a m e t e r space o f r o t a t i o n a n d r e l a x a t i o n that faithfully shows the relative d e p t h s of the m i n i m a . W e m u s t therefore c o n c l u d e that it w o u l d b e very easy, using the search techniques a p p l i e d to G a A s ( l l 0 ) in the past, to find a w r o n g surface structure were they to be a p p l i e d to I n S b ( l l 0 ) o r m a n y o t h e r equally c o m p l e x surfaces.

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0.7-

-15

~

-10

i

-5

i

i

Relaxation

51

l 01

I

15

%

Pig. 4b, Section b-b through fig. 2. A section at constant rotation through the global m i n i m u m in the parameter space (rotation = 30 °). It was by searching along lines such as this that the early investigations of G a A s failed to find any subsidiary minima.

This discussion of the problems that must be faced in the practice of LEED is not at all unique to this area. Indeed there are many surface structure techniques where a comprehensive search of a parameter space is needed (e.g. angularly resolved X-ray emission, ion scattering and surface EXAFS). This is c o m m o n in principle to all indirect methods of analysis. Only the computational time for modelling each point in parameter space varies from one technique to another. The methodology involved in L E E D analysis makes two main assumptions. First that, if the experimental I(V) curves are exactly reproduced by the calculated curves, the structure used in the calculation corresponds exactly to the structure of the crystal. This ideal situation never occurs because of errors from both theoretical and experimental data. It is therefore assumed that, as the calculated curves become more and more similar to the experimental curves, the theoretical model becomes closer to the real crystal structure. As the situation when an R-factor of exactly zero has never occurred there seems

P.G. Cowell et al. / (110) surfaces of I I I - V semiconductors

L923

I

°'9°1 C 0.85/

o

g

0.65H

0.60"

-15

T

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-s'

0.55" 51

10 I

lS I

Relaxation 1% Fig. 4c. Section c - c through fig. 2. This section passes through both minima in the parameter space. There does not seem to be any a priori method of choosing this line. It m a y only be found after both minima have been located. In a more complex R-factor space with more than two minima there may not be a single line through all the minima.

little point in discussing the first assumption. It will serve until it can be disproved. The second assumption has much more significance particularly in discussions concerning the legitimate use of R-factors. The results presented here and elsewhere [1,21] which show the existence of two possible surface structures having comparable R-factors lead to important questions concerning these assumptions. Using the second assumption mentioned above would lead one to regard both potential structures as equally close to the real structure. This is not true in the case of InSb(ll0) that has been presented here. Although two minima were found in the set of R-factors we can be certain that only one is correct. The correct solution has been forced by the theoretical model chosen for the "experiment". The unexpected minimum is independent of the R-factor used for the comparisons. Those used include the Pendry R-factor [12], the Euclidean norm [23], the Lp metric [23] and the weak integrated distance [14]. It can safely be

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a s s u m e d to b e a p r o p e r t y of t h e I ( V ) c u r v e s a n d t h e r e f o r e of the crystal s u r f a c e r a t h e r t h a n the R - f a c t o r . It m a y b e t h a t t h e results p r e s e n t e d h e r e are a s y m p t o m o f the p o s s i b i l i t y t h a t L E E D m a y n o t b e a b l e in s o m e cases to d i s t i n g u i s h w i t h a r e a s o n a b l e d e g r e e of c e r t a i n t y b e t w e e n s o m e s u r f a c e s t r u c tures. T h i s is n o t a d i s a s t r o u s b l o w to t h e c r e d i b i l i t y o f L E E D . It i n d i c a t e s t h a t t h e l i m i t i t a t i o n s of the m e t h o d s h o u l d b e t a k e n seriously. T h e i n c r e a s i n g a c c u r a c y a v a i l a b l e in L E E D a n a l y s i s b e c a u s e o f the i n t r o d u c t i o n o f R - f a c t o r s s h o u l d n o t o n l y a l l o w m o r e a c c u r a t e s t r u c t u r e s to be f o u n d b u t s h o u l d also b e u s e d as a m a g n i f y i n g glass to s h o w u p d e f i c i e n c i e s in the e x p e r i m e n t a l , theoretical and methodological practice of LEED. T h e w o r k d e s c r i b e d is s u p p o r t e d b y the S c i e n c e a n d E n g i n e e r i n g R e s e a r c h C o u n c i l . D i s c u s s i o n s w i t h V.E. d e C a r v a l h o w h o is c o n d u c t i n g a n e w i n v e s t i g a t i o n o f the s u r f a c e s t r u c t u r e o f I n S b ( l l 0 ) h a v e b e e n i m p o r t a n t in the d e v e l o p m e n t of the i d e a s p r e s e n t e d here.

References [1] [2] [3] [4] [5] [6] [7] [8] [9} [10]

C.B. Duke, S.L. Richardson, A. Paton and A. Kahn, Surface Sci. 127 (1983) L135. M.W. Puga, G. Xu and S.Y. Tong, Surface Sci. 164 (1985) L789. C.B. Duke and A. Paton, Surface Sci. 164 (1985) L797. A.R. Lubinsky, C.B. Duke, B.W. Lee and P. Mark, Phys. Rev. Letters 36 (1976) 1058. V. Dose, H.-J. Grossman and D. Staub, Phys. Rev. Letters 47 (1981) 608. V. Dose, H.-J. Grossman and D. Staub, Surface Sci. 117 (1982) 387. D.J. Chadi, J. Vacuum Sci. Technol. 15 (1978) 631, 1244. A. Kahn, Surface Sci. Rept. 3 (1983) 193. C.B. Duke, A. Paton and A. Kahn, J. Vacuum Sci. Technol. A1 (1983) 672. C.B. Duke, R.J. Meyer, A. Paton, LL. Yeh, J.C. Tsang, A. Kahn and P. Mark, J. Vacuum Sci. Technol. 17 (1980) 501. [11] R.J. Meyer, C.B. Duke, A. Paton, J.L. Yeh, J.C Tsang, A. Kahn and P. Mark, Phys. Rev. B21 (1980) 119. [12] J.B. Pendry, J. Phys. B13 (1980) 937. [13] E.g.C.B. Duke, A. Paton, W.K. Ford, A. Kahn and J. Carelli, Phys. Rev. B24 (1981) 62; and cf. E. Zanazzi and F.P. Jona, Surface Sci. 62 (1977) 61. [14] J. Philip and J. Rundgren, in: Determination of Surface Structures by LEED, Eds. P.M. Marcus and F. Jona (Plenum, New York, 1980) p. 409. [15] J. Rundgren, Metrics on LEED Spectra. Application to Surface Segregation of PtxNi ~_ Alloys, given at International Seminar on Surface Structure Determination by LEED and other methods, Erlangen, 1985. [16] E. Zanazzi and F. Jona, Surface Sci. 62 (1977) 61. [17] R.J. Meyer, C.B. Duke, A. Paton, J.C. Tsang, J.L. Yeh, A. Kahn and P. Mark, Phys. Rev. B22 (1980) 6171. [18] E.g.C. Chatfield and A.J, Collins, Introduction to Multivariate Analysis (Chapman and Hall, London/New York) ch. 10. [19] K.R. Popper, Logik der Forschung (Vienna, 1934) [Eng. Trans. Logic of Scientific Discovery (Hutchinson, London/New York, 1957)]. [20] K.R. Popper, Logic of Scientific Discovery (Hutchinson, London/New York, 1957) p. 280. [21] C.B. Duke, A. Paton and A. Kahn, J. Vacuum Sci. Technol. A2 (1984) 515. [22] C.B. Duke, A. Paton, A. Kahn and D.-W. Tu, J. Vacuum Sci. Technol. B2 (1984) 366. [23] A.C. Sobrero and W.H. Weinberg, in: Determination of Surface Structures by LEED, Eds. P.M. Marcus and F. Jona (Plenum, New York, 1980) p. 437.