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Surface Madelung potentials in electron spectroscopy
SURFACE SCIENCE23 (1970)403-408 0 North-Holland PublishingCo.
LETTERS TO THE EDITOR
SURFACE
MADELUNG
ELECTRON
POTENTIALS
IN
SPECTROSCOPY
Received 3 August 1970 Electron spectroscopic chemical shifts in crystalline solids may contain a structure-dependent part which is a function of the electrostatic potential generated by the charged species which surround the atom or ion of interest. Hence, it becomes important to obtain at least an estimate of this selfpotential in order to correlate chemical shifts with crystal structure, and initially one would prefer to choose a simple model for the system. The most obvious first choice in this effort is the idealized point-charge model of an ionic crystal. However, most of the standard formulations which are discussed in the literature consider the crystal to be of infinite extent, and since electron spectroscopy typically samples only a very thin surface layer11 s), it is not clear that they are applicable. It would, therefore, be more instructive to consider a semi-infinite lattice, and for that reason, we have applied a rigid-lattice model developed by Ewalds). We have found, contrary to past speculations, that the local (Madelung) potential at a site of given charge is neither uniformly raised now lowered due to the presence of a nearby surface. Furthermore, the magnitude of the potential cannot be estimated by any simple “rule of thumb” [e.g., that a surface ion should have a self-potential which is roughly one-half as large as its infinite-lattice value4)]. Ewalds) has shown that the self-potential in a semi-infinite lattice may be split into two parts. The first, called $inf, is the potential calculated for an infinite lattice, and the second, called $epi, is the so-called epipotential which contains the influence of the surface alone. The two combine additively to give the net potential $ as:
b+= $inf
+
*epi
*
(1)
The first term has been the subject of several formulations which have appeared in the literature, but the best of the available methods for calculating ~inf appears to be the one of Ewald5) as formulated by Tosis) and as later reduced to a computer program by Van Go01 and Piken’). The second term has, to the best of our knowledge, never been calculateds); that is the primary object of the present paper. 403
404
RICHARD
R. SLATER
ti epi is given (in esu) by the Fourier series $epi(x)
=
{lb31‘laHI}-’ ’ ,,,i$_
lexp(2rria,.x
Sh
+ in IM
i
- 2rclanl zj(b,l)*
-Cl - exp(2~ldM
+ 2~i~~rlH/lb31)1-1, }
(2)
where if a,, a2, a3 are the lattice translation vectors, x =xlul +_~,a, +~,a, is the position vector in direct space; if bl, b,, b, are the reciprocal lattice vectors, qn =h,b, +h,b, +h,b, (h,, h,, h3 irh?gerS) is a running vector in reciprocal space; if /It, /IZ are the projections of b, and b, onto the cut-off plane, an =h& +hz/Iz is a running vector in reciprocal space; where n is the inward unit normal to the cut-off plane; z is the distance along u3, expressed in units of a,; C;ll,h2 indicates summation over all h, and hz, except for the (0,O) term; S(q) = C q,d-’ exp(- 2niq*x,) s is the structure factor, summed over all ions in the basis; A is the volume of a unit cell, and qs is the charge carried by the sth ion in the basis. The cut-off plane is taken to be parallel to the plane determined by a, and a,. The preceding formula does not contain the (0,O) term because such a term would be constant over all space, and hence would neither allow $+ to go to zero deep within the interior of the crystal, nor would it allow for a finite work function (the removal of a test charge from the surface to infinity would involve an indefinitely large amount of work). In any case, the omission is not serious since electron spectroscopy should reveal only difserences in the local electrostatic field. In order to reduce the previous equation to a more easily managed form, it is rationalized and all of the imaginary exponents are replaced by their corresponding cosine terms [this is possible because each term (h,, h2) of the series has a corresponding conjugate term (-h,, -A,) located somewhere within the expansion, and the imaginary parts cancel]. Once this simplification is made, it is a rather straight-forward task to write a computer program to compute $,pi(X), and the particular program we have formulated allows one to use any compound whose unit-cell structure is known. It passes the qualitative test that the epipotential obtained at a particular value of x shall, for a given crystal, be (1) independent of the unit-cell description, (2) independent of the unit-cell origin, and (3) dependent only upon the location of the physical surface. Computationally, we find that the “infinite” summation never has to exceed the range defined by (h,, h,) =( - 10, - 10) and (h,, h,) =(lO, IO), due to the rapidity with which the series converges. For
positions lying deeper into the crystal than several atomic layers, this range may be reduced even further, since the series converges even more rapidly. We next present some of our results for corundum (cr-Al,O, in both the British and Alcoa systems of nomenclature) whose lattice symmetry is rhombohedral and for which t+Qinf has been previously reported’). It is a logical first choice since it is an ionic compound for which LEED experiments have shown that the surface structure can be of the type that one would expect from the rigid lattice approximation “). In calculating the potential tr/ at each ion, we have used the Van Gool-Piken program to obtain +inf and our OWII program to obtain 1J/,pi,assuming cut-off along only a few of the many possible planes. Since cut-off along the (l,l,l) plane (rhombohedral description) requires that a new unit cell be chosenlo), we have used the hexagonal description and have taken the cut-off plane to lie perpendicular to the c-axis. The results are summarized in tables 1 through 3, and fig. 1 demonstrates the rapid decay of ~epi with penetration into the crystal.
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.16 0.20 DISTANCE
Fig. 1.
Inner @potential
ALONG
along the hexagonal
C-AXIS
(3, &, c] direction in corundum.
There are two points which the preceding calculation exemplifies: (1) In agreement with the comments of Ewalds), we find that ~epi decays very rapidly with penetration into the crystal. It decays so rapidly in fact that, in all of the structures examined to date, it is found to be negligible below the first several layers of atoms. (2) In a surface unit-cell, all ion positions can become electrostatically non-equivalent. Because of (2), one might expect to see a very complicated electron spectrum provided that electrostatic nonequivalence implies a corresponding chemical shift. If, however, the electrons
406
RICHARDKSLATER TABLET
Bulk parameters for a-A1~03 in two descriptions The coordinates are those of ref. 14. a-Al203 rhombohedral
description
a = 5.128 A; a = 55” 17’ Symmetry: R~c-Dad6 Aluminum parameter: UN = 0.3520 (4c position) Oxygen parameter: UO = 0.5560 (6e position) Unit cell origin at J position lrnr at an aluminum ion&: - 36.592 V (IBMat an oxygen iona: 26.392 V a-Al203 hexagonal description _. a=b=4.759& c=12.991 8, a=B=90”; y=120° Symmetry: R~c-D& Aluminum parameter: VA1= 0.3520 (12~ position) Oxygen parameter: UO = 0.3060 (18e position) Unit cell origin at 3 position vinr at an aluminum ion&: - 36.585 V vi,,* at an oxygen ions: 26.397 V
&From Van Gool-Piken
computer program.
TABLE
2
Inner epipotential in a surface rhombohedral unit-cell of corundum for two different cleavage planes; the values for a (100) cleavage plane are obtainable by symmetry considerations, the coordinates are those of ref. 14 Rhombohedral coordinates a b c 0.1480 0.3520 0.6480 0.8520 0.5560 0.4440 0.9440 0.0560 :
0.1480 0.3520 0.6480 0.8520 0.9440 0.0560 : 0.5560 0.4440
Charge
3 3 3 3
0.1480 0.3520 0.6480 0.8620 : 0.5560 0.4440 0.9440 0.0560
Cleaved along (001)8
-2 -2 -2 -2 -2 -2
&Atom surface intersects c-axis at 0.0560 units. b Atom surface intersects b-axis at 0.0560 units.
Cleaved along (010)b
VePiw)
vepi WI
4.219 0.821 - 0.431 0.090 - 2.579 -0.100 0.795 0.586 - 0.060 - 6.273
4.219 0.821 -0.431 0.090 0.795 0.586 - 0.060 - 6.273 - 2.579 -0.100
407
SURFACEh4ADELUNGPOTENTIALSINELECTl7ONSPECTROSCOPY TABLE 3
Inner epipotential in a surface hexagonal unit-cell of corundum, whose cleavage plane passes through the c-axis at 0.0187 units; ions with a vepr whose absolute value is smaller than 1O-s V are not listed; the coordinates are those of ref. 14 Hexagonal coordinates (I
b
0 0
0 0
5 ;3 0.3060 0
0.6940 0.0273
3 0.6393 0.3607 3 0.9727
:
3 3 3 0 0.3060 0.6940 3 0.3607 0.9727
f 0.0273 0.6393
Charge
VlePi 09
c
0.3520 0.1480 0.0187 0.3147 0.5187 0.1853 0.4813 t : 0.4167 0.4167 0.4167 0.0833 0.0833 0.0833
3 3 3 3 3 3 3 -2 -2 -2 -2 -2 -2 -2 -2 -2
0.009
0.472 5.440 0.027 0.001 - 0.726 - 0.002 0.096 - 0.040 -0.015 0.004 - 0.001 - 0.001 - 0.871 - 1.186 1.526
rise to the spectrum come from a layer which is, say, 10 unit-cells thick (50-100 A in the case of CX-A1,03),then this model predicts that the calculation of ~inf will be adequate and that it will not be a prime necessity to consider cut-off effect@). Since the epipotential is a measure of the electrostatic “range” of the surface, its rapid disappearance appears to be in qualitative agreement with the calculations of Benson et al.is~is), which predict that the surface distortion in an alkali halide crystal should not extend for more than three or four atomic layers into the crystal. We have not presented the calculations which have been completed for other systems due to the fact that they would occupy far too much space. Instead, it is suggested that those who are interested in other crystal systems contact the author for a copy of his program. giving
RICHARD
Gulf Research & Development Co., P.O. Drawer 2038, Pittsburgh, Pennsylvania 15230, U.S.A.
R.SLATER
408
RICHARD
R. SLATER
References 1) Kai Siegbahn et al., ESCA, Atomic, Molecular and Solid State Structure Studied by Means of Electron Spectroscopy (Almquist & Wiksellsboktryckeri, A. B., Uppsala, Sweden, 1967). 2) P. W. Palmberg and T. N. Rhodin, J. Appl. Phys. 39 (1968) 2425. 3) P. P. Ewald and H. Juretschke, in: Structure and Properties of Solid Surfaces, Eds. R. Gomer and C. S. Smith (University of Chicago Press, 1953). 4) C. S. Fadley et al., J. Chem. Phys 48 (1968) 3779. 5) P. P. Ewald, Ann. Physik 64 (1921) 253. 6) M. P. Tosi, in: Solid State Physics, Vol. 16, Eds. F. Seitz and D. Turnbull (Academic Press, New York, 1964). 7) W. Van Go01 and A. G. Piken, J. Mat. Sci. 4 (1969) 95; J. Mat. Sci. 4 (1969) 105. 8) P. P. Ewald, private communication. 9) C. C. Chang, J. Appl. Phys. 39 (1968) 5570. 10) The new lattice-cell (which in this case can be a unit-cell) must be chosen in such a way that contiguous cells can form the desired atom surface. In general, it will be different from a unit-cell. 11) In this conclusion we agree with the speculations of W. L. Jolly et al., J. Am. Chem. Sot. 92 (1970) 1863 to the effect that there exist systems for which cut-off effects are not visible to electron spectroscopy. Furthermore, recent work of our own (to be published) shows that bulk structure in the alumina system can account for the observed chemical shifts. 12) G. C. Benson, P. I. Freeman and E. Dempsey, J. Chem. Phys. 39 (1963) 302. 13) G. C. Benson and T. A. Claxton, J. Chem. Phys. 48 (1968) 1356. 14) International Tables for X-ray Crystallography (Kynoch Press, Birmingham, England, 1952).
LETTERS TO THE EDITOR
SURFACE
MADELUNG
ELECTRON
POTENTIALS
IN
SPECTROSCOPY
Received 3 August 1970 Electron spectroscopic chemical shifts in crystalline solids may contain a structure-dependent part which is a function of the electrostatic potential generated by the charged species which surround the atom or ion of interest. Hence, it becomes important to obtain at least an estimate of this selfpotential in order to correlate chemical shifts with crystal structure, and initially one would prefer to choose a simple model for the system. The most obvious first choice in this effort is the idealized point-charge model of an ionic crystal. However, most of the standard formulations which are discussed in the literature consider the crystal to be of infinite extent, and since electron spectroscopy typically samples only a very thin surface layer11 s), it is not clear that they are applicable. It would, therefore, be more instructive to consider a semi-infinite lattice, and for that reason, we have applied a rigid-lattice model developed by Ewalds). We have found, contrary to past speculations, that the local (Madelung) potential at a site of given charge is neither uniformly raised now lowered due to the presence of a nearby surface. Furthermore, the magnitude of the potential cannot be estimated by any simple “rule of thumb” [e.g., that a surface ion should have a self-potential which is roughly one-half as large as its infinite-lattice value4)]. Ewalds) has shown that the self-potential in a semi-infinite lattice may be split into two parts. The first, called $inf, is the potential calculated for an infinite lattice, and the second, called $epi, is the so-called epipotential which contains the influence of the surface alone. The two combine additively to give the net potential $ as:
b+= $inf
+
*epi
*
(1)
The first term has been the subject of several formulations which have appeared in the literature, but the best of the available methods for calculating ~inf appears to be the one of Ewald5) as formulated by Tosis) and as later reduced to a computer program by Van Go01 and Piken’). The second term has, to the best of our knowledge, never been calculateds); that is the primary object of the present paper. 403
404
RICHARD
R. SLATER
ti epi is given (in esu) by the Fourier series $epi(x)
=
{lb31‘laHI}-’ ’ ,,,i$_
lexp(2rria,.x
Sh
+ in IM
i
- 2rclanl zj(b,l)*
-Cl - exp(2~ldM
+ 2~i~~rlH/lb31)1-1, }
(2)
where if a,, a2, a3 are the lattice translation vectors, x =xlul +_~,a, +~,a, is the position vector in direct space; if bl, b,, b, are the reciprocal lattice vectors, qn =h,b, +h,b, +h,b, (h,, h,, h3 irh?gerS) is a running vector in reciprocal space; if /It, /IZ are the projections of b, and b, onto the cut-off plane, an =h& +hz/Iz is a running vector in reciprocal space; where n is the inward unit normal to the cut-off plane; z is the distance along u3, expressed in units of a,; C;ll,h2 indicates summation over all h, and hz, except for the (0,O) term; S(q) = C q,d-’ exp(- 2niq*x,) s is the structure factor, summed over all ions in the basis; A is the volume of a unit cell, and qs is the charge carried by the sth ion in the basis. The cut-off plane is taken to be parallel to the plane determined by a, and a,. The preceding formula does not contain the (0,O) term because such a term would be constant over all space, and hence would neither allow $+ to go to zero deep within the interior of the crystal, nor would it allow for a finite work function (the removal of a test charge from the surface to infinity would involve an indefinitely large amount of work). In any case, the omission is not serious since electron spectroscopy should reveal only difserences in the local electrostatic field. In order to reduce the previous equation to a more easily managed form, it is rationalized and all of the imaginary exponents are replaced by their corresponding cosine terms [this is possible because each term (h,, h2) of the series has a corresponding conjugate term (-h,, -A,) located somewhere within the expansion, and the imaginary parts cancel]. Once this simplification is made, it is a rather straight-forward task to write a computer program to compute $,pi(X), and the particular program we have formulated allows one to use any compound whose unit-cell structure is known. It passes the qualitative test that the epipotential obtained at a particular value of x shall, for a given crystal, be (1) independent of the unit-cell description, (2) independent of the unit-cell origin, and (3) dependent only upon the location of the physical surface. Computationally, we find that the “infinite” summation never has to exceed the range defined by (h,, h,) =( - 10, - 10) and (h,, h,) =(lO, IO), due to the rapidity with which the series converges. For
positions lying deeper into the crystal than several atomic layers, this range may be reduced even further, since the series converges even more rapidly. We next present some of our results for corundum (cr-Al,O, in both the British and Alcoa systems of nomenclature) whose lattice symmetry is rhombohedral and for which t+Qinf has been previously reported’). It is a logical first choice since it is an ionic compound for which LEED experiments have shown that the surface structure can be of the type that one would expect from the rigid lattice approximation “). In calculating the potential tr/ at each ion, we have used the Van Gool-Piken program to obtain +inf and our OWII program to obtain 1J/,pi,assuming cut-off along only a few of the many possible planes. Since cut-off along the (l,l,l) plane (rhombohedral description) requires that a new unit cell be chosenlo), we have used the hexagonal description and have taken the cut-off plane to lie perpendicular to the c-axis. The results are summarized in tables 1 through 3, and fig. 1 demonstrates the rapid decay of ~epi with penetration into the crystal.
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.16 0.20 DISTANCE
Fig. 1.
Inner @potential
ALONG
along the hexagonal
C-AXIS
(3, &, c] direction in corundum.
There are two points which the preceding calculation exemplifies: (1) In agreement with the comments of Ewalds), we find that ~epi decays very rapidly with penetration into the crystal. It decays so rapidly in fact that, in all of the structures examined to date, it is found to be negligible below the first several layers of atoms. (2) In a surface unit-cell, all ion positions can become electrostatically non-equivalent. Because of (2), one might expect to see a very complicated electron spectrum provided that electrostatic nonequivalence implies a corresponding chemical shift. If, however, the electrons
406
RICHARDKSLATER TABLET
Bulk parameters for a-A1~03 in two descriptions The coordinates are those of ref. 14. a-Al203 rhombohedral
description
a = 5.128 A; a = 55” 17’ Symmetry: R~c-Dad6 Aluminum parameter: UN = 0.3520 (4c position) Oxygen parameter: UO = 0.5560 (6e position) Unit cell origin at J position lrnr at an aluminum ion&: - 36.592 V (IBMat an oxygen iona: 26.392 V a-Al203 hexagonal description _. a=b=4.759& c=12.991 8, a=B=90”; y=120° Symmetry: R~c-D& Aluminum parameter: VA1= 0.3520 (12~ position) Oxygen parameter: UO = 0.3060 (18e position) Unit cell origin at 3 position vinr at an aluminum ion&: - 36.585 V vi,,* at an oxygen ions: 26.397 V
&From Van Gool-Piken
computer program.
TABLE
2
Inner epipotential in a surface rhombohedral unit-cell of corundum for two different cleavage planes; the values for a (100) cleavage plane are obtainable by symmetry considerations, the coordinates are those of ref. 14 Rhombohedral coordinates a b c 0.1480 0.3520 0.6480 0.8520 0.5560 0.4440 0.9440 0.0560 :
0.1480 0.3520 0.6480 0.8520 0.9440 0.0560 : 0.5560 0.4440
Charge
3 3 3 3
0.1480 0.3520 0.6480 0.8620 : 0.5560 0.4440 0.9440 0.0560
Cleaved along (001)8
-2 -2 -2 -2 -2 -2
&Atom surface intersects c-axis at 0.0560 units. b Atom surface intersects b-axis at 0.0560 units.
Cleaved along (010)b
VePiw)
vepi WI
4.219 0.821 - 0.431 0.090 - 2.579 -0.100 0.795 0.586 - 0.060 - 6.273
4.219 0.821 -0.431 0.090 0.795 0.586 - 0.060 - 6.273 - 2.579 -0.100
407
SURFACEh4ADELUNGPOTENTIALSINELECTl7ONSPECTROSCOPY TABLE 3
Inner epipotential in a surface hexagonal unit-cell of corundum, whose cleavage plane passes through the c-axis at 0.0187 units; ions with a vepr whose absolute value is smaller than 1O-s V are not listed; the coordinates are those of ref. 14 Hexagonal coordinates (I
b
0 0
0 0
5 ;3 0.3060 0
0.6940 0.0273
3 0.6393 0.3607 3 0.9727
:
3 3 3 0 0.3060 0.6940 3 0.3607 0.9727
f 0.0273 0.6393
Charge
VlePi 09
c
0.3520 0.1480 0.0187 0.3147 0.5187 0.1853 0.4813 t : 0.4167 0.4167 0.4167 0.0833 0.0833 0.0833
3 3 3 3 3 3 3 -2 -2 -2 -2 -2 -2 -2 -2 -2
0.009
0.472 5.440 0.027 0.001 - 0.726 - 0.002 0.096 - 0.040 -0.015 0.004 - 0.001 - 0.001 - 0.871 - 1.186 1.526
rise to the spectrum come from a layer which is, say, 10 unit-cells thick (50-100 A in the case of CX-A1,03),then this model predicts that the calculation of ~inf will be adequate and that it will not be a prime necessity to consider cut-off effect@). Since the epipotential is a measure of the electrostatic “range” of the surface, its rapid disappearance appears to be in qualitative agreement with the calculations of Benson et al.is~is), which predict that the surface distortion in an alkali halide crystal should not extend for more than three or four atomic layers into the crystal. We have not presented the calculations which have been completed for other systems due to the fact that they would occupy far too much space. Instead, it is suggested that those who are interested in other crystal systems contact the author for a copy of his program. giving
RICHARD
Gulf Research & Development Co., P.O. Drawer 2038, Pittsburgh, Pennsylvania 15230, U.S.A.
R.SLATER
408
RICHARD
R. SLATER
References 1) Kai Siegbahn et al., ESCA, Atomic, Molecular and Solid State Structure Studied by Means of Electron Spectroscopy (Almquist & Wiksellsboktryckeri, A. B., Uppsala, Sweden, 1967). 2) P. W. Palmberg and T. N. Rhodin, J. Appl. Phys. 39 (1968) 2425. 3) P. P. Ewald and H. Juretschke, in: Structure and Properties of Solid Surfaces, Eds. R. Gomer and C. S. Smith (University of Chicago Press, 1953). 4) C. S. Fadley et al., J. Chem. Phys 48 (1968) 3779. 5) P. P. Ewald, Ann. Physik 64 (1921) 253. 6) M. P. Tosi, in: Solid State Physics, Vol. 16, Eds. F. Seitz and D. Turnbull (Academic Press, New York, 1964). 7) W. Van Go01 and A. G. Piken, J. Mat. Sci. 4 (1969) 95; J. Mat. Sci. 4 (1969) 105. 8) P. P. Ewald, private communication. 9) C. C. Chang, J. Appl. Phys. 39 (1968) 5570. 10) The new lattice-cell (which in this case can be a unit-cell) must be chosen in such a way that contiguous cells can form the desired atom surface. In general, it will be different from a unit-cell. 11) In this conclusion we agree with the speculations of W. L. Jolly et al., J. Am. Chem. Sot. 92 (1970) 1863 to the effect that there exist systems for which cut-off effects are not visible to electron spectroscopy. Furthermore, recent work of our own (to be published) shows that bulk structure in the alumina system can account for the observed chemical shifts. 12) G. C. Benson, P. I. Freeman and E. Dempsey, J. Chem. Phys. 39 (1963) 302. 13) G. C. Benson and T. A. Claxton, J. Chem. Phys. 48 (1968) 1356. 14) International Tables for X-ray Crystallography (Kynoch Press, Birmingham, England, 1952).