- Email: [email protected]
Formation energies of vacancies at a (100) sodium chloride surface
J. Phys. Chem. So&&
Pergamon
Press 1965. Vol. 26, pp. 1863-1867.
FORMATION
Printed in Great Britain.
ENERGIES OF VACANCIES
A (100) SODIUM CHLORIDE J. M. BLAKBLY Department
and CHE-YU
of Materials Science and Engineering,
AT
SURFACE Lt
Cornell University,
Ithaca, New York
(Receiwed 16 February 1965; in revisedform 4 May 1965)
Abstract-The formation energies of cation and anion vacancies at 0°K at a (100) surface pIane of sodium chloride have been calculated using the method of Mott and Littleton. The value for a cation, anion vacancy pair is found to be 2.12 eV, slightly greater than the corresponding bulk value of 1.90 eV. The calculated value for the surface vacancy pair energy represents a first order approximation. The assumptions involved are discussed in some detail.
INTRODUCXION
THERE are a number of areas in surface physics where a knowledge of the atomic defect structure of the surface is of importance. It is probable that point and other imperfections are active sites for many surface reactions including nucleation processes and they may be intimately involved in considerations of surface ionic conduction and diffusion. Both experimental info~ation and theoretical predictions on the defect structure of surfaces are extremely scarce. In the present paper, the method first applied by MOTT and LITTLETON in their calculation of defect energies in the interior of ionic crystals, is used in the evaluation of the increase in internal energy at 0°K associated with the formation of cation and anion vacancies at a (100) surface of a sodium chloride crystah A number of simplifying assumptions are used in the present analysis. These are noted throughout the calculation and are enumerated in the discussion. The results to be presented here thus represent only a first order appro~mation to the solution of the problem. However the calculation serves to indicate the difficulties involved and to suggest future modifications. CALCULATION OF SURFACE VACANCY FORMATION
ENERGIES
The following calculation will refer to the work required to remove a surface cation, the anion case being entirely analogous. We shall assume 1863
throughout that in the absence of defects there are no lattice distortions nor electronic dipoles in the surface layers of the type considered by VERWEY(~) and by BENSON et aZ.(Q Thus we are considering the energy change on forming the defect at an idealized surface rather than at a real surface where there is some electronic and ionic polarization. Consider the cation situated at site (i) in Fig. 1 on the surface of a sea-ignite sodium chloride crystal. The binding energy of this ion is given by the summation over all ions of the crystal of the pair interaction energy &j, i.e. 2 Ccr $ We shall assume
j#i
(1)
drl = &*++c; (2) where the superscripts c and r denote the couIomb and short range repulsive interactions respectiveIy. #f* = t ;
5%= A expt-hddl (4) The work W+ to remove the cation from position (i) in the surface to infinity is given as the sum of three contributions to the energy of the crystal, i.e. w, = (increase in Coulomb interaction energy) + (polarization energy) + (increase in repulsive energy). = w,+w,+w, (5)
J. M. BLAKELY
1864
We shall consider each of these energy contributions separately and outline the methods used in their evaluation. a. TOP VIEW +-+-+-+
and CHE-YU
LI
Polarization energy
On removing the positive ion from the surface layer a net electrostatic field is set up in the crystal due to the existence of an effective negative charge at the vacant site resulting in induced dipole moments at the ions. If v is the potential at the site of the missing ion due to the dipoles induced on all other ions of the crystal then W, = ;eV
-+-+-+-
The evaluation of the polarization energy thus reduces to a calculation of the potential V at the vacant site. Assuming the crystal to be a continuum of dielectric constant K, the nolarization P due to
+-+-+-+ -+_@++-
+
(7)
-
+
-
+
-
+ (8)
-+-+-+where r is the vector from the vacant site. Following the procedure of MOTT and LITTLETON(~) we shall use equation (8) to evaluate the induced dipoles ~11 and pa associated with the positive and negative ions respectively except for the anions which are nearest neighbors of the vacant site. The electronic dipole moments and displacements of the nearest neighbors in the surface plane will be denoted by pp and fpao respectively (Fig. 2). The corresponding quantities for the nearest neighbor in the second crystal layer are cLpand &co.
+-++-+-+ b.SIDE
VIEW
-+_~l+li_++
+-+-+-+ _
+
_
+
_
+
_
psT
+-+-+-+
Q--L Coulomb energy
The change in the Coulomb interaction energy of the crystal on removing the positive ion is simply given by*
-_-y__.
a
= e~(l.679) a0
---
I <
30
I
1____ &pO
(6)
We find * Details of the evaluation of the various lattice summations used in this calculation may be found in Ref. 5.
ao
0
CLQ
+ w, = 2 & I
lq____
-
CL1 = -*-*-*
P
q----4
+
FORMATION
ENERGIES
OF VACANCIES
AT
with a corresponding expression for ~2. a1 and a2 are the electronic polarizabilities of the cations and anions respectively and 6 the displacement polarizability.
6 = e2/p wherep is the force constant due to overlap forces. In general p will be a function of position in the crystal. We shall consider only repulsive interactions between first nearest neighbors so that we distinguish only the force constants ps for the outermost layer and PB the value in the rest of the crystal. Thus =
(11)
Ps =
(12)
PB
Due to this difference in displacement polarizabilities of surface and bulk ions we must distinguish CL:, pi and pt, t~f where the superscripts S and B refer to surface and bulk respectively. The potential at the vacant site is then given by
A (100) SODIUM
CHLORIDE
SURFACE
1865
The method used to determine tp, tq and m,, mq was essentially that already described by MOTT and LITTLETON and consists of the solution of a set of simultaneous equations representing the balance between electrostatic and repulsive forces at the sites of the nearest neighbor anions. This procedure involves considerable algebra and several new series summations.(s) The values used in the numerical evaluations were@) til = 0.41 X 1O-24 ems, a2 = 2.97 x lo-24 cm3, K = 5.62. We obtain [p = O-057, tq = 0.082, mP = 5.55 x 10-2, mq = 7.30 x 10-s and a value of the polarization energy W, = qO.592)
(15)
a0
Repulsive energy The contribution of the repulsive forces to the work of extraction of the ion may be computed by considering the work that must be done against these forces to replace the ion and restore the surrounding ions to their original positions. Considering only nearest neighbor repulsive forces Wr =
4[4a05pd’r(m + [htqrb’r(ao
+ ao&) - @Ia0 + ao&.J] + a&d
- fl(ao
+ aOt31
(16)
where +’ is the first derivative of 4. Using the values of the nearest neighbor displacements given above,
+
4m,e
ao(l +
5~)~
+
mcle
ao(l+ 5#
w, = “2(0.125) (13)
where we have set pp = -mPeao, pq = - m,ea,-,. The subscripts on the summations indicate that, excluding the nearest neighbors of the vacancy, they are to be taken over surface or bulk ions respectively. The last four terms in the above equation represent the contribution from the five nearest neighbors. By evaluating the series of equation (13) we obtain V = “~1~5067_4~+2~4140A~+0~5201A~ UOL 4&J ____ ~ 5Q +1*83884:+ (1+[@)2+ (1+fq)2
4mp -
+ (1 +&Js+
~ 32 (1 +fa)2
1
a0
(17)
The corresponding values of the energy contributions involved in the formation of an anion vacancy are listed in Table 1. In evaluating the energy of formation of a vacancy pair at the surface we have followed FRENKEL(~) in imagining the ions removed from the surface to form new regions of the semi-infinite crystal, i.e. energy to form a vacancy pair = W+ + W_ - WL (18) where WL is the binding energy per ion pair of the crystal.
(14)
DISCUSSION In Table 1 the energy values given for the bulk of the crystal are from the work of Mott and
1866
J.
M.
BLAKELY
and
CHE-YU
LI
Table 1
__I---
Polarization energy
Coulomb binding energy
Cation
Anion
Cation
Anion
Surface
8.65
3.05
2.82
O-65
0.62
Bulk (from MOTT and kl"IU3TON~'~2~
9‘01
3.56
3.01
0.77
0.72
Overlap energy
Work to remove ion W+
W_
Energy to form vacancy pair*
4.96
5-21
2.12
4.68
5.27
t-90
~_______ * Energy to form a vacancy pair = (WC f W_- WL). Following Frenkel we imagine ions removed from such positions to be reconstituted to form new regions of crystal. (WL = 8.047 eV.)
Littleton since these are obtained by methods similar to those used in the present work. More recent modifications(s) are only slightly different and show no improvement in the agreement with experimental results. It should be noted that the final value for the vacancy pair formation energy at a (100) sodium chloride surface plane is slightly greater than that in the interior of the crystal. This result is somewhat surprising and some comments on the nature of the assumptions involved in the present analysis may be worthwhile. In the treatment of the polarization energy it has been implicitly assumed that the dielectric tensor for the surface region may be approximated by a simple constant equal in magnitude to the bulk dielectric constant. In principle this assumption can be overcome by extending the treatment to higher order approximations where we would consider the induced dipoles up to the nth nearest neighbors to be unknown. For sufficiently large n the error would become insignificant. We have not attempted to include the effect of normal polarization or displacement of the surface ions. The electronic and displacement polarization at a (100) sodium chloride surface have been discussed by VJSRWEY@) and by BENSON et aZ.@)In principle the technique of Benson et al. could be applied directly to compare the energy of a (100) surface containing a vacant site with that of the complete surface and hence derive the formation energy of the defect. The calculation for the case of the missing ion is much more complex due to the loss of symmetry. This effect will be considered in future work.
The constants assumed in the Born-Mayer repulsive potentials are those derived for interaction within the interior of the crystal. Any modification of these quantities for the surface region would require detailed experimental information on a uniquely surface property such as surface free energy. Such info~ation is not available. At finite temperatures the properties of the free surface will be modified by the existence of the space charge discussed by FRENKEL(~) and LEHOVEC.@) There is also the possibility that vacancies may exist at the surface at 0°K to decrease the strain energy associated with the boundary and hence to minimize the total surfacefree energy. In conclusion, the calculations shown here represent the first attempt to evaluate the energy associated with a vacancy pair at an idealized (100) sodium chloride surface. This info~ation is of paramount importance for the understanding of surface structure. Only a first order approximation has been obtained but it is hoped that future work will treat some of the problems pointed out in the present analysis. Acknowledgements-This work was supported by the United States Atomic Energy Commission on contract A(30-1)3228 and by the Advanced Research Projects Agency through the Materials Science Center at Cornell University. The authors are grateful to Mrs. WILMA BFXNA~EI for assistance in carrying out the series summations and to Professor H. S. SACK for a critical review of the manuscript.
FORMATION
ENERGIES
OF VACANCIES
AT
A [loO) SODIUM
CHLORIDE
SURFACE
1867
REFERENCES 1. MOTF N. F. and LITTLETON M. J., Ilrans. Faraday Sor. 34,485 (1938). 2. FUMI F. G. and TOSI M. P., Disc. Faraday Sot. 23, 92 (1957). 3. VBR~IX E. J. MT., Rec. Truer. Chim. 65, 521 (1946). 4. BTIWSONG. C., FREEMAN P. I. and DEMPSEV E., So& Surfaces, Advances in Chem. Series, No. 33, Ameu. Chem. Sot. (1961) p. 26.
5. BLAKELY J. M. and Lx CHE-Yu, Materials Science Center Repoti No. 308. Cornell University (1965). 6. KITTEL C., Introduction to Solid State Physics (2nd edition), p, 165, Wiley, New York (1956). 7. FRENKEL J., Kinetic Theory of Liquids, Oxford University Press, London (1946). 8. LEHOYECK., f. Cirern. P&s. Z&l123 (1953).
Pergamon
Press 1965. Vol. 26, pp. 1863-1867.
FORMATION
Printed in Great Britain.
ENERGIES OF VACANCIES
A (100) SODIUM CHLORIDE J. M. BLAKBLY Department
and CHE-YU
of Materials Science and Engineering,
AT
SURFACE Lt
Cornell University,
Ithaca, New York
(Receiwed 16 February 1965; in revisedform 4 May 1965)
Abstract-The formation energies of cation and anion vacancies at 0°K at a (100) surface pIane of sodium chloride have been calculated using the method of Mott and Littleton. The value for a cation, anion vacancy pair is found to be 2.12 eV, slightly greater than the corresponding bulk value of 1.90 eV. The calculated value for the surface vacancy pair energy represents a first order approximation. The assumptions involved are discussed in some detail.
INTRODUCXION
THERE are a number of areas in surface physics where a knowledge of the atomic defect structure of the surface is of importance. It is probable that point and other imperfections are active sites for many surface reactions including nucleation processes and they may be intimately involved in considerations of surface ionic conduction and diffusion. Both experimental info~ation and theoretical predictions on the defect structure of surfaces are extremely scarce. In the present paper, the method first applied by MOTT and LITTLETON in their calculation of defect energies in the interior of ionic crystals, is used in the evaluation of the increase in internal energy at 0°K associated with the formation of cation and anion vacancies at a (100) surface of a sodium chloride crystah A number of simplifying assumptions are used in the present analysis. These are noted throughout the calculation and are enumerated in the discussion. The results to be presented here thus represent only a first order appro~mation to the solution of the problem. However the calculation serves to indicate the difficulties involved and to suggest future modifications. CALCULATION OF SURFACE VACANCY FORMATION
ENERGIES
The following calculation will refer to the work required to remove a surface cation, the anion case being entirely analogous. We shall assume 1863
throughout that in the absence of defects there are no lattice distortions nor electronic dipoles in the surface layers of the type considered by VERWEY(~) and by BENSON et aZ.(Q Thus we are considering the energy change on forming the defect at an idealized surface rather than at a real surface where there is some electronic and ionic polarization. Consider the cation situated at site (i) in Fig. 1 on the surface of a sea-ignite sodium chloride crystal. The binding energy of this ion is given by the summation over all ions of the crystal of the pair interaction energy &j, i.e. 2 Ccr $ We shall assume
j#i
(1)
drl = &*++c; (2) where the superscripts c and r denote the couIomb and short range repulsive interactions respectiveIy. #f* = t ;
5%= A expt-hddl (4) The work W+ to remove the cation from position (i) in the surface to infinity is given as the sum of three contributions to the energy of the crystal, i.e. w, = (increase in Coulomb interaction energy) + (polarization energy) + (increase in repulsive energy). = w,+w,+w, (5)
J. M. BLAKELY
1864
We shall consider each of these energy contributions separately and outline the methods used in their evaluation. a. TOP VIEW +-+-+-+
and CHE-YU
LI
Polarization energy
On removing the positive ion from the surface layer a net electrostatic field is set up in the crystal due to the existence of an effective negative charge at the vacant site resulting in induced dipole moments at the ions. If v is the potential at the site of the missing ion due to the dipoles induced on all other ions of the crystal then W, = ;eV
-+-+-+-
The evaluation of the polarization energy thus reduces to a calculation of the potential V at the vacant site. Assuming the crystal to be a continuum of dielectric constant K, the nolarization P due to
+-+-+-+ -+_@++-
+
(7)
-
+
-
+
-
+ (8)
-+-+-+where r is the vector from the vacant site. Following the procedure of MOTT and LITTLETON(~) we shall use equation (8) to evaluate the induced dipoles ~11 and pa associated with the positive and negative ions respectively except for the anions which are nearest neighbors of the vacant site. The electronic dipole moments and displacements of the nearest neighbors in the surface plane will be denoted by pp and fpao respectively (Fig. 2). The corresponding quantities for the nearest neighbor in the second crystal layer are cLpand &co.
+-++-+-+ b.SIDE
VIEW
-+_~l+li_++
+-+-+-+ _
+
_
+
_
+
_
psT
+-+-+-+
Q--L Coulomb energy
The change in the Coulomb interaction energy of the crystal on removing the positive ion is simply given by*
-_-y__.
a
= e~(l.679) a0
---
I <
30
I
1____ &pO
(6)
We find * Details of the evaluation of the various lattice summations used in this calculation may be found in Ref. 5.
ao
0
CLQ
+ w, = 2 & I
lq____
-
CL1 = -*-*-*
P
q----4
+
FORMATION
ENERGIES
OF VACANCIES
AT
with a corresponding expression for ~2. a1 and a2 are the electronic polarizabilities of the cations and anions respectively and 6 the displacement polarizability.
6 = e2/p wherep is the force constant due to overlap forces. In general p will be a function of position in the crystal. We shall consider only repulsive interactions between first nearest neighbors so that we distinguish only the force constants ps for the outermost layer and PB the value in the rest of the crystal. Thus =
(11)
Ps =
(12)
PB
Due to this difference in displacement polarizabilities of surface and bulk ions we must distinguish CL:, pi and pt, t~f where the superscripts S and B refer to surface and bulk respectively. The potential at the vacant site is then given by
A (100) SODIUM
CHLORIDE
SURFACE
1865
The method used to determine tp, tq and m,, mq was essentially that already described by MOTT and LITTLETON and consists of the solution of a set of simultaneous equations representing the balance between electrostatic and repulsive forces at the sites of the nearest neighbor anions. This procedure involves considerable algebra and several new series summations.(s) The values used in the numerical evaluations were@) til = 0.41 X 1O-24 ems, a2 = 2.97 x lo-24 cm3, K = 5.62. We obtain [p = O-057, tq = 0.082, mP = 5.55 x 10-2, mq = 7.30 x 10-s and a value of the polarization energy W, = qO.592)
(15)
a0
Repulsive energy The contribution of the repulsive forces to the work of extraction of the ion may be computed by considering the work that must be done against these forces to replace the ion and restore the surrounding ions to their original positions. Considering only nearest neighbor repulsive forces Wr =
4[4a05pd’r(m + [htqrb’r(ao
+ ao&) - @Ia0 + ao&.J] + a&d
- fl(ao
+ aOt31
(16)
where +’ is the first derivative of 4. Using the values of the nearest neighbor displacements given above,
+
4m,e
ao(l +
5~)~
+
mcle
ao(l+ 5#
w, = “2(0.125) (13)
where we have set pp = -mPeao, pq = - m,ea,-,. The subscripts on the summations indicate that, excluding the nearest neighbors of the vacancy, they are to be taken over surface or bulk ions respectively. The last four terms in the above equation represent the contribution from the five nearest neighbors. By evaluating the series of equation (13) we obtain V = “~1~5067_4~+2~4140A~+0~5201A~ UOL 4&J ____ ~ 5Q +1*83884:+ (1+[@)2+ (1+fq)2
4mp -
+ (1 +&Js+
~ 32 (1 +fa)2
1
a0
(17)
The corresponding values of the energy contributions involved in the formation of an anion vacancy are listed in Table 1. In evaluating the energy of formation of a vacancy pair at the surface we have followed FRENKEL(~) in imagining the ions removed from the surface to form new regions of the semi-infinite crystal, i.e. energy to form a vacancy pair = W+ + W_ - WL (18) where WL is the binding energy per ion pair of the crystal.
(14)
DISCUSSION In Table 1 the energy values given for the bulk of the crystal are from the work of Mott and
1866
J.
M.
BLAKELY
and
CHE-YU
LI
Table 1
__I---
Polarization energy
Coulomb binding energy
Cation
Anion
Cation
Anion
Surface
8.65
3.05
2.82
O-65
0.62
Bulk (from MOTT and kl"IU3TON~'~2~
9‘01
3.56
3.01
0.77
0.72
Overlap energy
Work to remove ion W+
W_
Energy to form vacancy pair*
4.96
5-21
2.12
4.68
5.27
t-90
~_______ * Energy to form a vacancy pair = (WC f W_- WL). Following Frenkel we imagine ions removed from such positions to be reconstituted to form new regions of crystal. (WL = 8.047 eV.)
Littleton since these are obtained by methods similar to those used in the present work. More recent modifications(s) are only slightly different and show no improvement in the agreement with experimental results. It should be noted that the final value for the vacancy pair formation energy at a (100) sodium chloride surface plane is slightly greater than that in the interior of the crystal. This result is somewhat surprising and some comments on the nature of the assumptions involved in the present analysis may be worthwhile. In the treatment of the polarization energy it has been implicitly assumed that the dielectric tensor for the surface region may be approximated by a simple constant equal in magnitude to the bulk dielectric constant. In principle this assumption can be overcome by extending the treatment to higher order approximations where we would consider the induced dipoles up to the nth nearest neighbors to be unknown. For sufficiently large n the error would become insignificant. We have not attempted to include the effect of normal polarization or displacement of the surface ions. The electronic and displacement polarization at a (100) sodium chloride surface have been discussed by VJSRWEY@) and by BENSON et aZ.@)In principle the technique of Benson et al. could be applied directly to compare the energy of a (100) surface containing a vacant site with that of the complete surface and hence derive the formation energy of the defect. The calculation for the case of the missing ion is much more complex due to the loss of symmetry. This effect will be considered in future work.
The constants assumed in the Born-Mayer repulsive potentials are those derived for interaction within the interior of the crystal. Any modification of these quantities for the surface region would require detailed experimental information on a uniquely surface property such as surface free energy. Such info~ation is not available. At finite temperatures the properties of the free surface will be modified by the existence of the space charge discussed by FRENKEL(~) and LEHOVEC.@) There is also the possibility that vacancies may exist at the surface at 0°K to decrease the strain energy associated with the boundary and hence to minimize the total surfacefree energy. In conclusion, the calculations shown here represent the first attempt to evaluate the energy associated with a vacancy pair at an idealized (100) sodium chloride surface. This info~ation is of paramount importance for the understanding of surface structure. Only a first order approximation has been obtained but it is hoped that future work will treat some of the problems pointed out in the present analysis. Acknowledgements-This work was supported by the United States Atomic Energy Commission on contract A(30-1)3228 and by the Advanced Research Projects Agency through the Materials Science Center at Cornell University. The authors are grateful to Mrs. WILMA BFXNA~EI for assistance in carrying out the series summations and to Professor H. S. SACK for a critical review of the manuscript.
FORMATION
ENERGIES
OF VACANCIES
AT
A [loO) SODIUM
CHLORIDE
SURFACE
1867
REFERENCES 1. MOTF N. F. and LITTLETON M. J., Ilrans. Faraday Sor. 34,485 (1938). 2. FUMI F. G. and TOSI M. P., Disc. Faraday Sot. 23, 92 (1957). 3. VBR~IX E. J. MT., Rec. Truer. Chim. 65, 521 (1946). 4. BTIWSONG. C., FREEMAN P. I. and DEMPSEV E., So& Surfaces, Advances in Chem. Series, No. 33, Ameu. Chem. Sot. (1961) p. 26.
5. BLAKELY J. M. and Lx CHE-Yu, Materials Science Center Repoti No. 308. Cornell University (1965). 6. KITTEL C., Introduction to Solid State Physics (2nd edition), p, 165, Wiley, New York (1956). 7. FRENKEL J., Kinetic Theory of Liquids, Oxford University Press, London (1946). 8. LEHOYECK., f. Cirern. P&s. Z&l123 (1953).