Induced dipole contributions to the potential in the surface region of an ionic solid

Induced dipole contributions to the potential in the surface region of an ionic solid

Surface Science 62 (1977) 397-405 0 North-Holland Publishing Company INDUCED DIPOLE CONTRIBUTIONS TO THE POTENTIAL IN THE SURFACE REGION OF AN IONI...

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Surface Science 62 (1977) 397-405 0 North-Holland Publishing Company

INDUCED DIPOLE CONTRIBUTIONS

TO THE POTENTIAL IN THE SURFACE

REGION OF AN IONIC SOLID

C.A. MORRISON, R.P. LEAVITT, D.E. WORTMAN and N. KARAYIANIS Harry Diamond Laboratories, 2800 Powder Mill Road, Adelphi, MD 20783, USA Received 10 September 1976

The theory of the electric potential near the surface of an ionic solid is extended to include induced dipolar effects. Numerical results are given for the (100) and (110) faces of several solids having the NaCl structure. General matrices are given which allow the determination of the dipolar contribution for any ionic solid of the NaCl type. It is shown that the contribution is small for the two surface planes considered.

1. Introduction

The electrostatic potential of an atom in the surface layers of a solid is of interest in the analysis of data taken by X-ray photoelectron spectroscopy [ 11. Since the X-ray photoelectron spectroscopy technique probes the first few layers below the surface, the potential in this region is of primary importance. Generally, the resulting shift in the binding energy of the core electrons is the same for identical sites in the bulk material but is different for ions near the surface. Furthermore, the binding energy of electrons near the surface depends on the crystallographic plane of the surface of the solid. Most theoretical analyses of the potential near the surface of a solid consider only the point charge contributions to the shift in binding energy [2-41. In this paper, the theory is extended to include, in a self-consistent manner, the potential due to induced dipoles of the ions in the solid. For simple cubic structures such as NaCl, the point charge theory leads to a relationship between the potential of a surface ion and an ion in the bulk [2] that depends only on the crystal structure and the surface plane, but is independent of the ion charges (++) and the lattice dimension (a). However, when the contributions of the induced dipoles are considered, this relationship no longer holds. The reason for this is that in the point charge theory, the bulk potential and the surface,potential for a given crystallographic plane of the surface are both proportional to q/a. When the polarizabilities of the ions are introduced, their contribution to the potential depends on their proximity to the surface, vanishing deep in the interior of the solid. ‘Ihe total potential of the surface ions therefore is intricately dependent on the polarizability of the constitu397

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C.A. Morrison et al. /induced dipole contributions ?a potential

ent ions of the solid and is not simply related to the interior potential. Higher-order induced multipoles also contribute to the potential at a surf&e ion, and these can be accounted for in the same manner as the dipole contributions but with a great deal of added analytical complexity and therefore are not treated in this paper. In this paper, the theory previously developed for the crystalline electric fields in ionic solids [S] is applied to the evaluation of the potential near the surface of cubic ionic solids. In particular, the dipole contribution to the potential near the surface is evaluated for the surface planes {loo) and {110) in the NaCl structure.

2, Theory For the purposes of developing the techniques employed, we assume that the z axis of the coordinate system is normal to the surface of the semi-in~nite crystal. Since we discuss only cubic crystals, the electric field is then parallel to the z axis so that the z component of the electric field is denoted simply by E. The potential at a site in the nth layer is given by

where Qenns= Z!XI and Pm< is the z component the lattice position lmn’ a distance R m’ from dipole moments are determined by the position determined by the pola~zabilities. The induced given by

Pi = CYfEf

of the dipole moment of the ion at point n. The signs of the individual of the layer, but the magnitudes are dipole moment of a tiarticular ion is

(2)

where the electric field is evaluated at the site occupied by ion i, and pi is the polarizability of the ion. The electric field to be used in (2) can be written E=Eo+ED,

(3)

where E. is the point-charge contribution to the electric field and ED is the dipole contribution to the field. Then for the point charge term, we have

and since the terms from n’ = 0 to ~1’= 2n cancel, this term has to be evaluated only

C.A. Morrison et al. /Induced

dipole contributions

for n’ > 2n t 1. The dipole contribution

E&z)=

c &m n’>O

3PImn’(Zn’

-

zn)*

Rfrnnl

_-

to potential

can be written

Plmnl R?rnn*

(5)

l-

The dipole moment of a particular ion in a given layer is identical chose a particular site, say sodium, then we can write

E?(n) = $ G(n, n’)P& +

399

so that if we

c H(n, n’)Pgy , n’

with G and H given by choosing the origin at a sodium site and evaluating the sum over I and m in (5) for each of the particular ions. The result given in (6) can be written more compactly as E~=G+%H+i

>

(7)

where E and P are column vectors, and G and H are matrices. Inspection of ,(5) readily shows that G and H are symmetric and further depend only on Iz, - z,‘l. Similarly, if the origin in (5) is chosen as the chlorine type ion, then it can be shown that E$‘=G&‘tH.pa,

(8)

a result to be expected from the symmetry given in (4), we can further show that E$‘(n) = - EoNa

of the problem. By considering

.

the sum

(9)

Using eq. (2) and the above results, we can then write the following two equations: PNa=al(EotG~PNa+H~~‘),

~1=a2(-EotG~~‘tH~~),

(10)

where o1 and cr2 are the polarizabilities of Na and Cl types of ions, respectively, and This result then gives two coupled equations to solve to determine the dipole moments of each site which can then be used in (1) to obtain the potential at any point. From the second of equations (lo), we have

E. = Ey.

fl’ = a,B * (-E. with B-’

t Ha PNa) ,

= 1 - r_~aG.Th en using this result in the first of equations

~“=cqA*(l

-a2H*Bj*Eo,

(11) (lo), we obtain (12)

where A-’ = 1 - aIG - alcu2H. B * H. The polarization of the chlorine like site can be obtained by using (12) in (11). It should be noted that while G and H given in

400

C.A. Morrison et al. f induced

dipole contributions

to potentiaf

(7) are fundamental to the lattice, the A and B occurring in (I 1) and (12) depend also on the polarizability. Since each term of (1) involves the dipole moments of the constituent ions, each term can be written separately, or (13) n’t0

n’Z0

This term can be written also in matrix form as +;“=v9%

V-F’,

(14)

where $3 is the potential evaluated at the sodium-like cirlorine-like site, then it can be shown that d#=U.pc’t

site. If $Q is evaluated at a

v.pNa.

(15)

Thus, the dipole contribution to the potential at any point in the lattice can be found by using (14) and (15) along with the results given in (11) and (12).

3. Completion It is convenient for computation purposes to cast all of the results into drmensionless units. To do this computation, we assume that the charge is measured in units of q, the charge on the Na-like atoms, and the lengths are measured in units of the lattice constant, a (the nearest Na to Na distance). The potential is then in units

Table 1 The matrices G(n,n’) and H(n, n’) for the two surface planes (100) parenthesis are powers of 10 n’ - n

0 1 2 3 4 5 6 7 8 9

-2.55510 -4.04299 6.40157 -7.22669 8.55612 -1.00532 1.18266 -1.39100 1.63591 -1.92437

(1) (0) (-2) (-4) (-6) (-7) (-9) (-11) (-13) (-15)

numbers in

ill O> Plane

{loo} Plane .-.--.- __ G(n, n’)

and (110);

H(tz, n’)

G(n, n’)

-4.67181 (1) 6.6627 1 (0) -5.95758 (-2) 7.30901 (-4) -8.54075 (-6) 1.00561 (-7) -1.18260 (-9) 1.39102 (-11) -1.63609 (-13) 1.92468 (-1.5)

-1.61111 -1.03918 1.82863 -1.51109 1.60830 -1.69739 1.82762 -1.97491 2.13905 -2.31858

H(n, n’) (1) (1) (0) (-1) (-2) (-3) (-4) (-5) (-6) (-7)

-2.94580 (1) -9.84845 (-1) -1.15395 (0) 1.26264 (-1) -1.49727 (-2) 1.64990 (-3) --1.80704 (-4) 1.96602 (-5) -2.13520 (-6) 2.31692 _-.. (-7) _.._-._

C.A. Morrison et al. /Induced dipole contributions to potential

401

Table 2 Monopole electric field at the Na-lie site for the (100) and (110) surface planes (units of qfa2), EC’ = -ENa n

(100)

Plane

(110) Plane

Eg(n) 0 1

2 3 4 5 6 7 8 9

E&)

1.17863 -1.37467 1.61682 -1.90171 2.23678 -2.63090 3.09446 -3.63970 4.28100 -5.03531

2.02184 (0) -3.44253 (-1) 4.01986 (-2) -4.42104 (-3) 4.80783 (-4) -5.21702 (-5) 5.65859 (-6) -6.13702 (-7) 6.65579 (-8) -7.2183$ (-9)

(0) (-2) (-4) (-6) (-8) (-10) (-12) (-14) (-16) (-18)

of q/a, and the electric field is in terms of q/a’. Also, the polarizability is written in terms of cx/a3. The formulas convenient for the computation of G and Hare given in Appendix 1, and these have been used to calculate these matrices for surfaces in the (100) and (110) plane. Only the elements of G(n, n’) and H(n, n’) for values of n’ - n have been computed since each of these matrices is a function of n’ - n only and is symmetric. The results of these computations are shown in table 1. Similarly, the electric Geld, ED, at the Na-like site has been computed from eq. (4) for the two types of surface planes, and the results are given in table 2. Rapid summing formulas for the matrices U and V defined in eq. (15) also are given in the Appendix and have been used to calculate U and V for the two selected surface planes. The results Table 3 The matrices U and Y of eq. (14) for the two surface planes (100) and (110); only tile values for n’ > n are given since U(n’,n) = -U(n,n’) and V(n’,n) = - V(n,n’)

n’ - n

(100)

Plane

Un, n’) 4.95110 -7.13043 8.14646 -9.62648 1.13143 -1.33094 1.56542 -1.84125 2.16568

(110) Plane Wk n’)

VW, n’) (-1) (-3) (-5) (-7) (-8) (-10) (-12) (-14) (-16)

-6.97266 6.77792 -8.21196 9.61425 -1.13166 1.33090 -1.56543 1.84125 -2.16568

(-1) (-3) (-5) (-7) (-8) (-10) (-12) (-14) (-16)

1.33071 -1.85039 1.67817 -1.78776 1.90323 -2.05323 2.22113 -2.40663 2.60904

V(n) n’) (0) (-1) (-2) (-3) (-4) (-5) (-6) (-7) (-8)

-3.89144 1.33585 -1.47905 1.69961 -1.86541 2.03685 -2.21406 2.40357 -2.60772

(-1) (-1) (-2) (-3) (-4) (-5) (-6) (-7) (-8)

CA Morrison et al. /Induced dipole contributions to potential

402

Table.4 Lattice constants and polarizabifities used in the computations Compound

a C.4

ff+ (A3)

01- (A3)

NaCl BaO BaTe

5.64056 5.523 6.986

0.179 1.55 1.55

3.66 3.88 14.0

Lattice constants are from ref. [6], and poiarizabilities are from ref. (71.

Table 5 Shifts in the mono ale potential at the Na-like site for the (100) and {llO) planes; [email protected]= K d& [email protected]) and 6+ = --&#?a and $?) = - 3.495 129 in units of q/a

n __.-.-.0 1 2 3 4 5

&pa .(100) Plane

(110) Plane

1.32022 -1.54701 1.81957 -2.14017 2.51726 -2.96080

4.16278 -6.30730 6.44521 -7.07403 7.65417 -8.30370 ____

-.----. (-1) (-3) (-5) (-7) (-9) (- 11)

(-1) (-2) (-3) (-4) (-5) (-6) .-.__--.

Table 6 Dipole potential at the Na- and Cl-like sites for NaCl in units of q/a --~ n

(100) Plane

(110) Plane _._.. .

-

-~0

1 2 3 4 5

-1.08598 -1.27917 1.20649 -1.06758 9.32074 -8.18096

(-3) (-2) (-3) (-4) (-6) (-7)

8.39568 9.74565 -9.67274 8.56793 -7.48519 6.56842 ___

(-4) (-3) (-4) (-5) (-6) (-7) -.._ _.

_

-5.98635 -1.54391 9.98276 -3.77431 1.26815 -4.19430

(-3) (-2) (-3) (-3) (-3) (-4)

1.65138 4.15455 - 2.08642 7.43688 -2.435 15 8.13542

(-2) (-2) (--2) (-3) (-3) (-4)

of these calculations are shown in table 3. The matrices G, H, U, and V and the vector Eo are all fundamental to the type of lattice and the particular plane of the surface and are all that are needed to obtain the dipole contribution to the potential at any site. This contribution can be calculated by using the results of tables 1, 2, and 3 along with (1 l), (12), (14) and (1.5). This calculation has been done for only

C.A. Morrison et al. /Induced dipole contributions to potential

403

Table I Dipole potential at the Na- and Cl-like sites for BaO in units of q/a n

(100) Plane &?

0 1 2 3 4 5

-7.96275 -3.03609 7.81836 -1.93251 4.77992 -1.24244

(110) Plane @E

& (-3) (-2) (-3) (-3) (-4) (-4)

7.43352 2.81763 -7.32527 1.81004 -4.47721 1.16351

(-3) (-2) (-3) (-3) (-4) (-4)

-2.37816 -4.95058 3.00850 -1.34973 5.76051 -2.59230

& (-2) (-2) (-2) (-2) (-3) (-3)

3.44093 6.64106 -3.96460 1.79482 -7.68919 3.50565

(-2) (-2) (-2) (-2) (-3) (-3)

Table 8 Dipole potential at the Na- and Cl-like sites BaTe in units of q/a n

@P 0

1 2 3 4 5

(110) Planes

(100) Planes

-6.56253 -2.93121 6.56838 -1.42082 3.06896 -6.89439

@F (-3) (-2) (-3) (-3) (-4) (-5)

5.59626 2.47041 -5.65528 1.22263 -2.64114 5.93078

@Z

$3 (-3) (-2) (-3) (-3) (-4) (-5)

-2.67640 -3.90131 2.98764 -1.62768 8.63032 -4.86069

_

(-2) (-2) (-2) (-2) (-3) (-3)

5.63046 7.57525 -5.36650 2.92578 -1.55046 8.98536

(-2) (-2) (-2) (-2) (-2) (-3)

a few selected cases, but any NaCl type of lattice can be calculated by using the results of the tables. The results of these calculations are given in tables 4 to 8.

4. Conclusion The results given in tables 1 to 3 were used to calculate the dipole contribution to the potential for the three solids listed in table 4. No particular reason was used in the selection of these particular solids other than that there is a large variation of polarizability in the constituent ions. The results of these computations are shown in tables 6 to 8 and should be compared to the monopole shift given in table 5. In all cases, the monopole shift in the surface layer (n = 0) is a couple of orders of magnitude larger and becomes comparable at a distance of a/2. Surprisingly, the dipole contribution decays much less rapidly than the monopole shift as one goes deeper into the solid. Fortunately, the dipole contribution is small and, perhaps, can be neglected except for comparison to exceedingly accurate experimental results. The above theory can be extended to include crystal symmetries lower than

C.A. Morrison et al. /Induced

404

dipole contributions

to potential

cubic solids. Such an extension would become rather involved, and at present we have no plans to extend the theory.

Acknowledgement The authors thank Dr. Richard Kaufman of the Army Material Command Headquarters for suggesting that we extend our work in crystal field theory to encompass surface effects. We thank him also for his enthusiastic interest, support, and encouragement.

Appendix A number OS sums involved in the evaluation of the elements of the tensors G, H, U and I/in the text have extremely slow convergence in their direct form. These sums can be converted to rapidly converging sums by using techniques given in Whittaker and Watson [8]. For a general form of sum we have

c

[(1 + x)2a2+ (m t y)2b2t

l,m=--

z2]-3/2

=?!!-C’ cos 2Zrrx cos 2mny exp(-2n(zl ablzl l,m

[(Z/U)~ + (m/b)*] ‘12)

(A.1)

where tz( > 0. Also,

cg

(2n)2 -$)=r-

(-*X [[i)’

t (T)2]1’2

C

cos 2nlx cos 27rmy

e,:I_27rlzl

[(i)’

+ (F)‘]“‘),

G4.2)

where R = [(I + ~)~a* + (m + y)2b2 + z2] 1/2 and lz ( > 0. When z = 0, it is obvious that eq. (A.l) does not converge, and we have

z’

[12a2 + m2b2]-3/2

= 2,

+$

where Kr(z) is a Bessel function and c(z) the zeta function. monopole potential, the sum involved is

c’ km h

=,L..g

cos 2nlx

cos 2nmy exp(-2rrlzl

((3) +A

((2)

In the evaluation

64.3) of the

[(Z/a)? + (m/b)*] ‘/*) ,

ab IPI [(Z/CZ)~ + (m/b)*] 1/2 (A.4)

C.A. Morrison et al. /Induced dipole contributions to potential

405

where R is the same as in eq. (A.2) and equality is taken in the meaning that a similar negative term is to be used in conjunction with this result so that the total left side is convergent. In the evaluation of the potential at a point within a sheet (z = 0), it is convenient to arrange the sums involved so that the series are convergent. The result for a (100) plane is &

(-l)z+m

=f2_-g ---= !I! [8 5 5 (-1)” I=0 m=1 a l,m Q2 t f722)1/2 a

K0[2rr4

+ f)] - 2 In 21 ,

(A.9

and the corresponding term for the 1110) plane is

By using the above results, each of the quantities in the text can be cast into a rapidly converging form.

References [l] D.E. Parry, 3. Chem. Sot. Faraday II 71 (1975) 344. [2] J.D. Levine and P. Mark, Phys. Rev. 144 (1966) 751. [3] R.R. Sister, Sbrface Sci. 23 (1970) 403. (41 DE, Parry, Surface Sci. 49 (197.5) 433. [S] C.A. Morrison, Solid State Commun. 18 (1976) 153. [6] R.W.G. Wyckoff, Crystal Structures, Vol. 1 (Wiley, New York, 1968) p. 86. [ 71 C. Kittel, Introduction to Solid State Physics, 2nd ed. (Wiley, New York, 1956) p. 165. [8] E.T. Whittaker and G.N. Watson, A Course in Modern Analysis, 4th ed. ~Cambridge Univ. Press, Cambridge, 1927) p. 124.