The Journal of
Physical Chemistry
Copyright 1993 by the American Chemical Society
(Q
VOLUME 97, NUMBER 4, JANUARY 28,1993
LETTERS Local Softness, Scanning Tunneling Microscopy, and Surface Reactivity Marcel0 Calvh,'J Amaldo DaI Pino, Jr.,* Jing Wang, and John D. Joannopoulos Department of Physics, Massachusetts Institute of Technology. Cambridge, Massachusetts 02139 Received: August 28, 1992; In Final Form: November 6. 1992
Within the finite temperature extension of density functional theory, it is shown that local softness measures the ability of a chemical system to gain or donate charge as the external potential changes. This feature allows us to establish a formal connection between local softness and scanning tunneling microscopy (STM) images. We show that, under appropriate conditions, STM images can be used to measure local softness for surfaces. Finally, a potential application of those images as reactivity criteria within the context of the hard and soft acids and bases (HSAB)principle is discussed.
Hardness and softness were originally empirical concepts.1-2 Recent developments in density functional theory (DFT)have helped to place them on a more rigorous foundation.3 Indeed, an absolute scale for hardness was established,e6 and local softness and hardness were defined as extensionsof the global q~antities.~J If a local interpretaticn of the hard and soft acids and bases (HSAB) principle is invoked, these new local properties are potentially useful as chemical reactivity indexes.+" In this work the physical significance of local softness, s(r), is analyzed in the context of the finite temperature extension of DFTI2in order to establish a relationshipof s(r) with scanning tunneling microscopy (STM)l3.l4images. This relationship shows that it is possible to obtain experimental local softness for surfaces, and consequently weinfer the potential useofSTM imagesaslocal reactivitycriteria within the context of the HSAB principle. The relation of local softness with chemical reactivity arises from twodifferent sources. First, Parr and Yang have postulated that the preferred sites for attacking groups in a molecule are described by the maximum values of the Fukui function,15fir). Bothflr) and local softness contain equivalent local information becausefir) can be mapped onto s(r) through a scale factor (the Permanent address: Departamento de Quhnica, Universidad Aut6noma Metropolitana-Iztapalapa,A.P. 55-536, Mtxico, D.F. 09340, MCxico. Sup ported by Consejo Nacional de Ciencia y Tecnologfa and Secretarfa de Educacidn Plblica (Mtxico). Permanent address: Departamento de Fisica do lnstituto Tecnologico de Aeronautica, CTA, Sao JosedosCampos, Sao Paulo, Brazil 12225. Supported by theConselhoNacionalde DesenvolvimentoCientificoe Tccnologico (CNPqBrazil). +
global softness, S):7
s(r) = Sflr) (1) The second way in which s(r) may be used as a reactivity criterion is through the local interpretation9J0of the HSAB principle:'V2J6 "hard regions of a system prefer to interact with hard reactants whereas soft regions will prefer soft species". Both uses of s(r) are helpful to determine the initial attacking points during a reaction process.+'lJ5 In the finite temperature extension of DFT,12the grand potential is a functional of the density:
Q [ m l = Jiu(r) - CcJp(r) dr + F[p(r)l
(2)
where p is the chemical potential, v(r) is the external potential, and F[p(r)]is a universal functional of the density, p(r), that incorporatesthe kinetic energy, the electron4ectron interaction, and an entropy term. In this representation of DIT,the natural variables are temperature, T ,volume, V,and chemical potential. This formalism permits a straightforward investigation of variations of the number of electrons, N, induced by changes in the external potential. From the grand potential, a, one can obtain a particularly useful Maxwell relation:
Therefore, local softness, s(r), measures the ability of the system to donate or accept electrons at a particular point r as the external
0022-3654/5S/2091-01S3~04.~0~0Q 1993 American Chemical Society
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184 The Journal of Physical Chemistry, Vol. 97, No. 4, 1993
potential is changed at that position. This interpretation is consistent with the workof Yang and Parr, in which local softness is related to local fluctuations in the charge den~ity.~ In STM experiments,both modesof operation (constant current or height)13J4 measure the local ability of the system to donate or accept electronsdependingon the bias voltageand tip properties (change in the external potential). For small bias voltage, the states involved with the charge transfer are only those close to the Fermi level. This fulfills the condition of constant chemical potential expressed in eq 3 and allows us to derive a relation between local softness and STM images. Yang and Parr7have proved, for a particular system (a metal at 0 K), that the local softness is the local density of states at the Fermi level, ps(r,Er):
Their derivation, however, can be immediately extended to semiconductor systems where there is an energy gap and the occupation numbers are restricted to the values zero or one. On the other hand, a standard approximation to produce theoretical STM images, in the limit of small bias voltage and low temperature, id7
where u is the conductance in ohm-' for distances in au and energies in eV, R is the radius of curvature of the tip, and k is the minimum inverse decay length for the wave functions in vacuum. To obtain eq 5 , Tersoff and Hamman17considered the limit of point probe. It represents the ideal case of a nonintrusive measurement with the maximum possible of resolution. Accordingly, from eqs 4 and 5 u
0:
s(r)
(6)
This equation links local softness with a directly measurable quantity and opens the possibility of obtaining qualitarioe experimental softness (or Fukui functions) for surfaces. Since the derivative of the energy as a function of the number of electrons is discontinuous,I8one may define different types of local softness: s(r)- which is related to the density of the lowest unoccupied states (LUMO density) and arises when electronic charge is transferred to the system; and s(r)+ which is related to the density of the highest occupied states (HOMO density) and arises when charge is transferred out of the system. Thus, when STM experiments are performed by donating or extracting electrons (depending on the bias voltage), their images are related to s(r)- and s(r)+,respectively. For metals, in the limit of small bias voltage, s(r)- and s(r)+ will be very similar. For semiconductors, however, s(r)- and s(r)+ will generally differ greatly. The application of STM images to study surface reactions with atomic resolution is well-kn~wn.'~It has been shown that the reactivity of different surface sites and even of defects can be explained in terms of pS(r,Ef).20J Usually, the analysis is performed by comparing images before and after the reaction t a k a place. Combining the results of our work with the local interpretation of the HSAB principle, a new potential use of STM images as reactivity maps emerges. In this picture, hard regions of the clean surface will interact with hard species whereas soft areas will react with soft reactants. The interesting feature of thisapproachisitscapacity todistinguishthereactivityofdifferent sites on a surface based exclusively on the electronic structure (or the STM images) of the clean surface and that of the adsorbant. It is important to note that s(r)is a reactivity criterion applicable in the initial stages of a reaction (at large distances). Thus, for the scope of the present work, STM images will have the same restriction, and they can only be applied for obtaining the initial attacking sites. To find these sites is important for reaction mechanism elucidation purposes; in addition, for those reactions
in which kinetically controlled and thermodynamically controlled products coincide, STM images will predict the stable product. The Tersoff and Hamman approximation17(eq 5 ) is concordant with the limit of low temperatures implied in eq 4. It also satisfies the low bias voltage limit required for interpreting s(r) in terms of eq 3. However, there is a better procedure to isolate ps(r,Ef) from STM experiments.22 In this method the quantity which is equal to p,(r,Ef) is the ratio of differential conductivity to total conductivity. Hence, anaccurate p r d u r e toestimates(r) would be to obtain (dI/dV)(I/V)-I for small bias voltages and low temperatures. In order to test the qualitative validity of eq 6 and to illustrate the use of STM images as reactivity indexes in the context of the HSAB principle, let us consider an example that involves the Si(ll1) 7 X 7 surface. This complex surface reconstruction presents several nonequivalent sites,23 whose reactivities are different.19 Hammers, Tromp, and D e m ~ t h 2 ~ have obtained the constant current STM image of this surface in a range of bias voltages. When a small negative bias is applied, the image is dominated by the first layer of atoms (adatoms). (See Figure 1 of ref 24.) It consists of 12 3-fold coordinated atoms; six of them belong to the faulted half and the other six to the unfaulted half of the surface unit cell. The image clearly distinguishes between these two regions: the faulted half presents higher values of current than the unfaulted one. According to our discussion of eq 6, the STM image obtained through this process is related to s(r)+. This implies that (for small negative bias) the faulted region is softer than the unfaulted one. These features of s(r)+ seem to be strongly related with the observed reactivity of the Si( 11 1) surface with Ag or Pd. These metals exhibit a strong preference to condense in the faulted r e g i ~ n ~ ~ ? ~ ~ and, according to the local HSAB principle, behave as soft acceptors. On the other hand, the STM image24reveals s(r)when a small positive voltage is applied. In this case, the image does nor exhibit asymmetry between the two regions. The second layer of atoms (rest atoms) correspond to dark (hard) areas independently of the applied bias. Experimentally, the rest atoms of unfaulted and faulted regions react much faster with NH, than ad atom^,^^ indicating that this reactant behaves as a hard species. Notice that the classification of hard (NH3) and soft (metals), obtained from the reactive patterns, is consistent with the standard chemical scale of hardness.28 This example indicates the potential use of STM images obtained at small bius as a measure of local softness. Presently, we are working on the ab initio determination of the softness of surfaces and its application to surface interactions.
Acknowledgment. This work was supported in part by US. AFOSR Grant AFOSR-90-0276A and by ONR Grant No. "14-90-5-1370. We thank Prof. R. G. Parr for his encouraging interest in this work. References and Notes (1) Pearson, R. G. J . Am. Chem. Soc. 1%3,85, 3533-3539. (2) Pearson, R.G. HardandSoft Acidsand Bases; Dowdcn, Hutchinson, and Ross: Stroudsburg, 1973. (3) Parr, R. G.; Yang, W. Derrsity Functional Theory of Atoms and Molecules; Oxford University Press: New York. 1989; pp 87-104. (4) Robles, J.; Bartolotti, L. J. 1.Am. Chem. Soc. 1984,106,3723-3724. (5) Pearson, R. G. Inorg. Chem. 1988, 27,734-740. (6) Pearson, R.G. J . Org. Chem. 1989,54, 1423-1430. (7) Yang. W.; Parr, R.G. Prof. Natl. Acad. Sci. U S A . 1985,82,67236726. (8) Berkowitz, M.;Gosh,S. K.; Parr, R. G. J . Am. Chem. Soc. 1985,107, 68 1 1-68 14. (9) Lee,Ch.; Yang, W.; Parr, R. G. J . Mol. Struct. ( T H E O C H E M ) 1988, 163, 305-313. (10) Yang, W.; Mortier, W. J. J . Am. Chem. Soc. 1986,108,5708-571 1 . (,I 1) Mendez. F.;GalvBn,M. In Density Functional Methods in Chemisrry; Springer-Verlag: New York, 1991; UP 387-400. . (15) Mermcn, N . D. Phys. Rev. ik5,137, A1441-A1443. (13) Hansma, P. K.; Tersoff, J. J. Appl. Phys. 1987, 61, Rl-R23. (14) Wu, X. L.;Lieber. Ch. M. Progr. Inorg. Chem. 1991,39,431-511. (15) Parr, R. G.; Yang, W. J . Am. Chem. Soc. 1984, 106.4049-4050.
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