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SURFACE ENERGY AND SURFACE POTENTIALS OF TRANSITION METALS
J. Waber, E. Kennard, Yu-Ping Tsui
To cite this version:
J. Waber, E. Kennard, Yu-Ping Tsui. SURFACE ENERGY AND SURFACE POTENTIALS OF TRANSITION METALS. Journal de Physique Colloques, 1972, 33 (C3), pp.C3-103-C3-117.
�10.1051/jphyscol:1972315�. �jpa-00215049�
JOURNAL DE PHYSIQUE Colloque C3, supplkment au no 5-6, Tome 33, Mai-Juin 1972, page C3-103
SURFACE ENERGY AND SURFACE POTENTIALS OF TRANSITION METALS
J. T. WABER, E. KENNARD and YU-PING TSUI Northwestern University Evanston, Illinois
There are several factors which must be included in any satisfactory treatment of the interaction of the surface of a metallic substrate with an atom or ion approaching it. First the potential gradient, which has a two dimensional repeat lattice parallel to the free surface, and which arises near the free surface from the truncation of the lattice, raises the degeneracy of the orbitals of the approaching atom. Rao and Waber [I] have recently studied this using an ion- lattice model. Second, an ion near the free surfaces induces an electronic redistribution in the metal and the ion is subject thereby to a potential which approa- ches the classical image potantial at large distances.
Newns [2] has derived such a potential based on a specific alternating distribution of charges along a line in the metal perpendicular to the surface. It depended on the dielectric response of the metal. Rao and Waber [3] have used an ion-lattice model to obtain a potential which also converges to the classical image potential at large distances but which also exhi- bits the two dimensional granularity of the potential experienced by a charge when it is only one or two interatomic distances from the surfaces. Justification for such a model will be taken up in a later section.
These two aspects are primarily ones external to the metal and might well lend themselves to molecular orbital or to tight-binding methods. A third aspect of the general problem is the behavior of the electrons in the vicinity of the free surface. It will be this topic which is the major one of this paper.
When a volume of metal is constrained by two or more surfaces, its total energy is increased relative to that possessed by the small number of atoms in an infinite block. A large portion of this energy arises from an increase in the kinetic energy of the electrons.
Charge oscillations similar to Friedel oscillations occur near the surface and due to barrier penetration a double layer forms. Another contribution to the surface energy arises from the electrostatic energy.
However, Lang and Kohn [4], [5] show that the local variation in the exchange and correlation of the electrons in this region is very significant.
The latter authors show that the c( jellium >) model is quite inadequate to obtain correct surface energies
of metals with electronic densities higher than that of aluminum. We will discuss the generation of realistic crystal potentials which could be used in the self- consistent approach which they have employed.
We will first discuss the repopulation of momentum states which arises from intruding free surfaces into a bulk metal. We will retrace the argument with the
tc jellium )) model since the argument can readily be adapted to use on the density of states curves of transit, on metals which had been carried out by KKR or APW methods. The argument can also be used to estimate the effect of the anisotropy on various (hkl) planes. Both of these topics will be discussed from the standpoint of the calculational problems encoun- tered which primarily are similar to those encountered in energy band calculations of the perfect solid.
We are not presently at the stage where we can calculate the overlap of the crystal orbitals with those of the adsorbable atom or ion. We will content our- selves here with obtaining some of the features of the crystal orbitals of a bounded solid.
Model of a finite metallic solid. - Truncation of a solid by a plane at x = L and separation of it into two parts so that no atoms exist for x > L, causes a local breakdown in Bloch's theorem. That is, the wave function
$(r + na) = r - n a > L .
This causes certain momentum states to be unoccupied and in order to conserve the number of electrons per atom, it is necessary to increase EF. Let us look at this problem.
1. Infinite barriers. - Initially we will deal with a potential well which we will assume to have infinite potential walls in the x direction at 0 and L,, in the y direction at 0 and L, and in the z-direction at 0 and L,. This is a c( jellium )> model in the sense the positive charges are assumed to be smeared out so that the potential inside the well is uniformily - V,.
A typical solution is
Research supported by a grant from the National Aeronau- $,,,,,, = ALmn sin - sin -
tics and Space Administration. ( ) ( sin ( ) . (11
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972315
C3-104 J. T. WABER, E. KENNA RD AND YU-PING TSUI This wave function automatically satisfies some of
the boundary conditions. When x = Lx or y = L, or z = L,, the function vanishes for all integer values of 1, m and n as required by the infinite potential barrier. The coefficient A,, can be determined by box normalization. Note that for state I = 0, m = 0, n = 0 the wave function vanishes for all values of x, y and z values within the box. Therefore this state is
not occupied since it cannot be normalized.
Let us further consider L, and L, to be very large while Lx is only a small distance, namely one which corresponds to a few interatomic distances in a real solid. For the present we will consider the two sur- faces bounding a thin sheet of material.
The energy of the (Imn)th occupied state is
A value of EF exists such that the energy of a set of I, m and n values is less than EF and the energy of any other member of the { I, m, n ) set is greater than EF.
If E(1mn) were a continuous function of its parameters rather than discrete one, EF would be a ellipsoid Fermi surface. Let us call the maximum allowed values of the three integers I,, m, and n, ; the maximum value along either the x, y or z axes, respectively. Then N, the number of states allowed, ranging from - I,,, to
+ I,,, etc. is
N(I,, m,, n,) = (2 1, + 1) (2 m, + 1) (2nm + 1) G (3) where G is the fractional volume of an ellipsoid inscribed in a parallelepiped. For a change from I, to I,,,, the change in the number of states is
In both eq. (3) and (4), the spin degeneracy has not been included. The change in energy AE accompa- nying the increase of I, by unity is
The density of states AN/AE can be estimated in the following way
The volume of the minimum bounding ellipsoidal volume is
Some error will arise when the difference between discrete intervals is large. Then certain cells may be inside and some outside the enveloping ellipsoid.
An illustration of this feature is presented in figure 1.
Spherio Fermi
d a l SUI
Excluded States E i
FIG. 1. - A portion of phase space on the k z k , plane showing occupied cells which are by full circles (@), the excluded states Ei indicated by hollow circles (o), the traces of the ellipsoid Fermi envelope indicated by the Fermi Energy EF and the repopulated states EZ indicated by filled squares (W). The new Fermi level
is E;.
We can estimate the number of cells with energy less than EF by noting that the volume of a unit cell in phase or k space is (7c3/Lx L, Z,). Hence the number of cells is
Ignoring surface conditions for the moment, then the number of electronic states not counting spin degeneracy which can be accommodated in this paral- lelepiped is found by assuming only positive values of Kx, K, and K,.
Thus we can write a mean Fermi wave vector k, as follows
which is related to the expression that is usually written for a cubic block of material.
Positive and negative integers can be assumed for the wave functions which can be combined into complex linear combinations which differin form from eq. (1)
I lmn > = B,,, exp
Now let us apply the assumption that both Ly and L, 9 L,. In fact, it will be convenient later to assume that they are sufficiently large that cyclic boundary conditions can be accurately applied. Then the spectrum of discrete states is so dense, that we can then replace (mn/L,) by a continuous variable k,, and can define k, similarly. Then we can write
I lk, k, ; x y z > =
= 2 A sin (lK, x ) exp(iky y) exp(ik, z ) (12)
SURFACE ENERGY AND SURFACE POTENTIALS OF TRANSITION METALS C3-105 and this wave function satisfies the boundary condi-
tion of an infinite potential barrier at L,, where k, = .n/Lx. Sugiyama [6] showed that there is a region near the boundary where electrons are excluded and as a consequence Freidel oscillations occur in the interior.
2. Finite barrier. - We will replace the infinite barrier with a more realistic one such as
If we assume in addition that we have nearly free electrons with effective mass m* (times the free elec- tronic mass me) we can define an auxiliary quantity p such that
Here we will use the atomic units h = me = e = 1 and hence the energy will be in Rydbergs.
Because of the finite step, barrier penetration into Region 11 occurs and the charge density for I x I > L, becomes exponentially decaying of I x ( is the energy E(1mn) is less than I - Vo I.
In Region I, we have
I K,kykz;xyz > =
= 2 A, cos (IK, x) exp ( i(ky Y + k, 2) ) . (15) The exponential decay factor in Region I1 is defined as
7 = JZ (16)
and we can write the appropriate wave function as
* exp ( i(ky y + k, z) ) . (17) The rationale for introducing a new length in the X direction is that a some distance d beyond L,, the charge density will fall approximately to zero, so that effectively a new infinite barrier could be erected at + 1 and the total charge balance be conserved.
Region I is thus expanded and the maximum k in the k, direction will approximately be
Note that 6 is a phase shift which is related to d and the e-folding distance of the tail of the wave function.
The general behavior of metallic electrons near finite surface barriers was studied first by Huaug and Wyllie [7].
If the interior charge density p(xyz) is unaltered far from the barrier, the new potential step is deter- mined by p2 and one finds that v which is the ratio of p2/E, is greater than 1. Stratton [8] and Hun- tington [9] have also treated this problem in consi- derable detail. The charge density p(x) matched at the boundary of two regions is given by
1
-$ J" (,ti - k2) [sin' k x + k2,
P cos kx -- 2 sin (2 kx)] dk (Region I)
P(X) = (19)
\ (Region 11)
This expression also assumed that Kx can be replaced by a continuous variable k.
Analytic estimation of surface energy. - Above we defined surface energy as the increase in total energy when a piece of metal is cut and a free surface is formed. Brager and Schuchowitzky [lo] pointed out in 1928 that localizing the electrons to only a part of space had to increase their momentum as a consequence of the uncertainly principle. They sug- gested that if an infinite surface barrier were present at x = + Lx, that the wave function I kx k,, k, ; xyz >
could not propagate into Region I1 because the momen- tum hk, would be zero. Hence the coefficient of the functions with a component k, = 0 would be zero.
This would also be applicable if the artificial infinite barrier were at 1. This corresponds to localization of the electrons in space and deoccupation of certain momentum states.
The individual cells in phase space are shown in figure 1. The dimensions are unequal to reflect the difference in the finite size of L, and L,. Only the section on the plane k, = 0 is shown. The occupied cells are indicated by one full dot for the two spins.
The curve marked EF is the trace of the Fermi enve- lope on the plane k, = 0.
The states deoccupied because they have a zero momentum component along the k, axis are indicated by empty circles. Call Ei the energy of the states so excluded and p the number of them. New states above E, must be occupied to conserve the number of elec- trons/atom. We will indicate the newly occupied states by filled squares and indicate their energies as E,.
The new Fermi energy we will designate E:. The net increase in kinetic energy
Alternatively we may write
C3-106 J. T. WABER, E. KENNARD A N D YU-PING TSUI where E, is energy of the lowest state. The quantity a
would be determined by the mean value theorem and would be between 112 for a linear array of Ei states and 315 for a cubic array. Identify the mean Ei with k,.
If Ly and L, were sufficiently large and equal, the cells in phase space would approximately be needle- shaped. This feature of a finite dimension in the x direction merits further study. The E(k) would be continuous in two directions but discrete in the K, direction.
Assuming a spheroidal Fermi envelope as in figure 2, the excluded states for barriers at f 1 are
They occupied states above
FIG. 2. - Illustration of a slice Ak units wide of the spheroidal Fermi envelope which corresponds to sheet of metal either x or rZ units thick but for which the dimensions ^y and 2 are large enough that cyclic boundary conditions apply. For these k , and
kz are determined by a Iattice dimension.
confined to a spheroidal slice Ak units wide each side of the plane k, = 0. One can write the energy of any state as
If ky and k, can be regarded as continuous, then the radius of the slice or (( band >> would be k,. The area of the outer circular area of the slice would be n(ki - k2) and such a factor occurs in eq. 19.
In an octant of the phase space, the number of states available in the unperturbed sheet of metal is
assuming that I, K, is k. And if the average energy of the states is (315 k&/2 m*). Then the kinetic energy of the unperturbed solid is
If k , and k, are not too different, T, depends on the fifth power of k,.
The number of states p will be given by
Since the volume of the slice of excluded states is approximately (2 n Ak) k:.
Assuming that each state in this slice has the ave- rage increase in energy of 611 1 k&, then the influence of the two parallel surfaces on the kinetic energy of the electrons is
Thus the surface term for 2 parallel planes is approxi- mately the fraction (30 Aklll k,) of the unperturbed total kinetic energy T, and it depends roughly on k&.
Looking at the corrections due to the six surfaces, one term will contain an increase A, if we assume that the new k:, due to promotion into previously unoc- cupied states is (1 + A) kM = yk,. We can calcu- late A, by counting the number of states. Then
where V = (L; L,). Thus we can obtain
and because Aklk, is small, q > 1.
Slightly different numbers will occur if the solid is a parallelepiped instead of the thin sheet we have assumed so far. The surface area of such a solid would be (4 Ly L, + 2 L;). Alternatively we can find p from the number of states added to a thin shell outside the spheroid EF. Thus
Then the total energy E can be obtained from p by multiplying by the energy of each state which is equal to k& A2!8 m* and
where the second term arises from three intersecting excluded areas (from the slices) which occur in the thin shell. After dividing E by the rough surface area 6 L2 one gets
SURFACE ENERGY AND SURFACE POTENTIALS O F TRANSITION METALS C3-107
When barrier penetration is taken into account then there is a negative contribution as Stratton [8] showed.
where v was defined above before eq. (19).
Estimation of surface energy from density of state curves. - Let us turn our attention not to a spheroidal Fermi surface but to a more complex one determined by band calculation. In principle, one can obtain much the same results by excluding a slice 2 Ak wide from the true Fermi surface. The QUAD scheme developed by Mueller et al. [Ill was adapted to counting how many states randomly gene- rated states Kr fell in the excluded region. The aniso- tropy of the surface energy due to different (hkl) barrier planes could be found by excluding a properly rotated slice. For this study we used three free sur- face planes and the pertinent exclusion criteria are :
Plane Criteria
-
If the K: component of any random state (K:, Ki, K,') met the first criterion, the state was not occupied but the electrons were added to a previously unoccupied state above EF. The other two criteria are quite similar.
At the time of writing this paper, the method has only been checked with the parabolic band of a nearly free electron metal. The energy of a state is given by an equation which is equivalent to eq. (2). We have assumed a body centered cubic solid with four (sp) energy bands and assumed that there are only six electrons per atom. A statistical analysis indicates that an acceptable value of Ak is proportional to N,-"~ where N , is the number of states generated randomly. In the QUAD program, a quantity MESH is the number of division one takes along the r - X
direction in the Brillouin Zone. This sets the number of cubicles studied in the Brillouin Zone. Two values namely 4 and 8 were used in this study. The informa- tion found by excluding the states found in one slice 2 k wide for MESH = 4 are summarized in Table I.
With this small network, it is not possible to obtain the ideal parabolic density of states curve. The deri- ved curve is shown in figure 3. It exhibits several spikes even though we generated 2 000 points in each cubicle.
Most of these deviations are eliminated where MESH is doubled as we will presently see.
One may schematically represent the energy of the excluded states Ei and the new states E, as shaded areas in figure 4. The shift in the Fermi level at 0 OK is also indicated. It was assumed that the set of six ( 100 )
Pertinent information about the states excluded, the shift in the Fermi Level arzd the irzcrease in kinetic energy MESH = 4.
No of states excluded (En - Ei) New E$
Old EF Old T New T*
AT
Units (100)
- - (110)
- (1 11) -
7 114 6 930 4 510
mRy 11.22 9.29 6.44
RY 0.324 0 0.323 9 0.323 2
0.321 8 - -
1.155 93 - -
RY 1.162 79 1.164 27 1.16094
mRy 6.8 8.3 5 .O
ENERGY I N RYDBERGS 100
FIG. 3. - The density of states curve derived assuming that the energy of the states was proportional + k.k. Four bands were assumed and the parameter MESH was set at 4 in the QUAD
program.
F O R ~ N P I N I T E S U R F A C E B A R R I E R
FIG. 4a. - Curve A is for Density of States Curve for a infinite metal assuming nearly free electrons. The excluded states on the walls of the XY, YZ and XZ planes in the Fermi sphere are indicated by curve B. The net effect is to increase the Fermi
level at o OK to E:.
C3-108 J. T. WABER, E. KENNARD AND YU-PING TSUI F O R F I N I T E BARRIER
'
""
TFIG. 4b. -The dotted curve C is the N(E) curve obtained by approximating barrier penetration by relocating the effective infinite barrier at I and thus increasing the constant of the para- bola. The net densitv of states curve D (crosshatched) is obtained by subtracting B frdm C . The net effec; of the increased number
of occupied states is that E$ - EF is smaller.
planes formed boundaries of the solid. Then three intersecting belts of excluded states lie parallel to the k, k,, k, k, and k, k, walls of the space in figure 2.
The plots showing the distribution or the density of excluded the several thousand states Ei is shown for the (100) surface in figure 5, for the (110) surface
0.000
- 2 0 0 8.200 16.600 25.000 33.VOO L(I.800 50.200
ENERGY I N RYDBERGS 100
FIG. 5. Density of Excluded States for (100) boundary. Cubi- clevolume determined by dividing r- Xdistance in theBrillouin
Zone by MESH and cubing it.
in figure 6 and for the (1 11) surface in figure 7. There is some suggestion of structure in the curves of exclud- ed states rather than the featureless shaded area in figure 4. That is, there is a small increase in the signal- to-noise ratio peak observed in figure 5 near 33.4 Ryd- bergs (despite the indicated scale on the abscicca, the actual energy is one hundred times smaller namely
ENERGY I N RYDBERGS 100
! 4h
,200 8.200 16.600 25.000 33.UOO '11.800 50.200
FIG. 6. - Density of Excluded States Ei for the (110) Barrier.
0.000
-.ZOO 8.200 16.600 25.000 33.Y00 L(1.800 50.200
ENERGY I N RYDBERGS 1 0 0
FIG. 7. - Density of Excluded States for the (1 11) Barrier.
334 milli-Ry or mRy). For (1 10) surface, a broad level region is suggested above from 280 to 420 mRy.
For the (111) surface, a two peak structure is sug- gested.
Turning to the part of the study in which MESH = 8 was used, a nearly perfect parabola is seen in figure 8.
The parabola does not continue much above 330 mRy because only a limited number of bands, 4, were used and additional energy bands (in the reduced zone scheme) must occur for the parabola to continue.
The numerical results are presented in Table 11. The density curves of the excluded states Ei are presented for the (100) surface in figure 9. A small peak is also found near 330 mRy as it was in figure 5. The broad peak observed in figure 6 for the (1 10) surface is more
SURFACE ENERGY AND SURFACE POTENTIALS OF TRANSITION METALS C3-109
0.000 L---
- -
. - . + - . - - , +.-.- - - t- k-.ZOO 8.20f 16.600 25.000 33.400 41.800 SC.?nO
ENEKGY I N RYDBERGS 100
FIG. 8. - The Density of States Curve N ( E ) for a BCC metal.
Similar to figure 3 but with MESH set equal to 8.
0.000
-.200 B.200 16.600 25.DU0 33.'100 91.800 50.2001
ENERGY I N RYDBERGS 100
FIG. 9. -Density of Excluded States for (100) Barrier. The cubicles are one-eighth as large as in figure 5.
Similar information for MESH = 8
No of states excluded (Ef - Ei) New E:
Old EF Old T New T*
A T
Units (100)
- - (110)
- (1 11)
-
8 640 7 248 5 102
mRy 10.48 7.43 5.65
RY 0.324 83 0.324 54 0.324 06
0.322 91 - -
1.154 62 - -
RY 1.161 17 1.161 70 1.159 24
mRy 6.5 7.7 4.6
nearly resolved in figure 11 into two peaks located in the energy range which was mentioned above. The
two peaks suggested in figure 7 for the (1 11) surface have been merged into one broad peak in the compa- nion figure, namely figure 11. The location of the peak is approximately the same.
0.000
-.ZOO 8.200 16.600 25.000 33.1100 '11.800 50.200
ENERGY I N RYnBERGS 100
FIG. 10. - Density of Excluded States for (110) Barrier. Similar to figure 6 but with MESH set equal to 8.
FIG. 11. - Companion to figure 7 for the (1 11) Barrier with MESH twice as large.
Anisotropy. - One can obtain a reasonable measure of the anisotropy of the surface energy a(hk1) by tabulating the ratio o(hkl)/o(100). The ratio found in this study are presented in Table 111. These ratios are compared with 1121 the earlier calculations by Mme Cyrot-Lackman who used a random walk method to estimate certain overlap integrals which occur in the tight binding method and from these
C3-110 J. T. WABER, E. KENNA LRD AND YU-PING TSUI
Summary of the anisotropy found in estimating gurface energy a(hk1) from density of Jtates curves compared with o (100).
Calculated MESH (100) (110) (111)
- - - -
Present 4 1 .OO 1.20 0.72
Calculation 8 1 .OO 1.10 0.70
Cyrot
Calculation 1.00 1.04 0.67
Experimental 1.00 1.06 0.87
to estimated the band width, cohesive energy, sur- face energy by calculating various moments
of the density of states curve N(E). The experimental results on the surface energy of tungsten. The agree- ment is better than might have been expected since this metal is not nearly a free electron metal. Improve- ments are expected when the actual N(E) curve deriv- ed from band structure calculations are used.
The jellium model and its disadvantages. - It is appropriate to note that earlier calculations with the G jellium )) model have had variable success.
The surface energies which were found by Brager and Schuckowitzky [lo] and by Stratton [8] and Huntington [9] are presented in table IV. One sees
Surface energy o due to the change in the kinetic energy of the electrons
Metal - lithium sodium potassium copper silver gold zinc cadmium mercury
Surface energy (ergs/cm2)
Ref. Ref. Ref.
[lo] [71 PI E ~ P
that the success of these authors is reasonably good for low density metals such as the alkali metals. The results are the poorest for the noble metals such as Ag and Au.
After a lapse of two decades, Bennett and Duke [13], 1141, Smith [15] and Lang 1161 reformu- lated the problem and made self consistent calculation of the local charge density p(r) near interfaces. Here p(r) was a rapidly changing function and Friedel oscillations were observed. Agreement was obtained
with experiments on the work function and surface energy. This agreement is good as long as the den- sity p(r) is low, but it rapidly deteriorates as p(r) approaches the electron density of moderately dense metals such as aluminum. Even negative surface energies were found.
Lattice models. - In the present treatment; we have been aware of such limitations of using a (( jel- lium )) model. We have used it to (a) rationalize the exclusion of certain states and to (b) test whether anisotropy in the surface energy of the (hkl) plane could be found. The question of whether the success is fortuitous can only be assessed later when actual band structure calculations have been used in a simi- lar manner.
It is our conviction that a model similar to that of Lang and Kohn [4], [5] which does include ion- cores imbedded in a locally varying electron gas must be used to obtain realistic answers. To this end we have devoted considerable effort to obtain realistic potentials of the crystal, both in the bulk and near the surface.
Lang and Kohn [4] point that the ground state energy of a many-electron system in an external potential V(r) may be written in terms of the charge density p(r) as
In this expression the second term is the Coulomb interaction between electron, T[p] is the kinetic energy of the electron gas and E,,[pl is the appropriate exchange and correlation contributions to the energy. The last two quantities are functionals of the local charge density p(r). There are many things to commend them on in their analysis. One limitation of their method is of calculating the effective potential. They repre- sent the ion-cores by appropriate pseudopotentials.
This band structure technique is very successful for dealing with the bands of metals derived from s and p states of the free atoms. However, it has a tendency to break down for transition metals where the d-bands are no longer well separated from the bands with substantially s-p character.
Either the APW or the KKR methods lead to greater success in treating the conduction electrons of transition metals. Both are usually characterized by the use of the c< muffin-tin potential B, that is by a constant potential in the region outside the ion-cores.
It is worthwhile to point out that this is not a real limitation since Koelling and his colleagues [18] have successfully used various modifications and have calculated the local warping of the <( muffin-tin )) and included the latter in their band structure calcu- lations of the bulk d-transition and f-transition metals
Bennett and Duke [13] pointed out that the local
SURFACE ENERGY AND SURFACE POTENTIALS O F TRANSITION METALS C3-111
exchange and correlation contributions to the inter- there is free space in the + Y region. The X Y plane facial potential were very important in making any thus represents a (010) surface. Consider now one self-consistent calculations. This finding was in accord unit cell and a new set of coordinate axes x, y, z with the earlier calculations of Bardeen [I91 and with origin at the center of the unit cell, on the Y Cutler and Loucks [20]. The recent papers of Lang axis. Let the distance between the two origins be and Kohn [4], [5] stress this point and suggest that equal to D and consider first a unit cell for which it may be the dominant factor in calculating the D > R + 1 units. This unit cell may be considered surface energy. to be unaffected by the surface in that for all points The forthcoming paper of Kennard, Perrot and within the unit cell the sphere of radius R falls comple- Waber [21] is based on Perrot's self consistent APW tely within the lattice side of the surface and all band structure calculations [22] in which the exchange atoms contributing to the potential are present. Let energy is calculated with crystal orbitals. It is hoped us consider some particular point [ax, by, cz] within that good agreement will be achieved. the unit cell, together with its sphere of radius R and move the origin of the coordinate system x , y, z Lattice potentials. - Let us turn Our attention along the + Y direction, i. e. towards the surface. At to the generation of the potentials for semi-infinite some point, the sphere will touch the surface and as D lattices (*). The local effect of defects and adsorbed becomes more positive, some atom sites within the atoms will be indicated. sphere will be outside the surface and thus will be The input data is the self-consistent field densities empty. ~h~ potential at the point [ax, by, c z ~ will obtained using a relativistic Dirac-Slater calcula- thus be increased. some point the sphere will be tion [23]. t he atomic potentials were obtained by completely empty, i. e. this is a point in free space.
numerical integration of Poisson's equation using a hi^ is the way in which the summation is carried Simpson integration routine. As the summation proce- out by the computer. T - , ~ X, Y, co-or~inates are dure is essentially the same for potential and charge arranged so that the X Y plane contains the origin of density, the description given below supplies equally a unit cell. At the start D is set equal to R + I . A
to either. point xyz together with its set of vectors [I, m , n ] and
Consider a face-centered cubic lattice with a cell distances r,m, are read. ~h~ distance r,m, is
side of 2 units and with the origin at the center of the by the lattice parameter and the potential contribu- unit cell. The positions of the atoms on this lattice tion at that point v,,, is calculated by interpolation are given by the position vectors my, nzl where of the values of atomic potential. The charge density ( x , y, z) are unit vectors along the cube axis and p,,, is also calculated. This procedure is repeated for the integers I, m, n satisfy the conditions : all [Imn] values and the potential and charge density 1 + m + n = (2 N + 1) N a n integer. contribution are separately summed giving V O ( x y z )
the Coulomb potential at that point and the total The potential summation was carried out over those charge density p;y, at that point. A table of values sites located within a fixed sphere of radius R from of ,, ., r, V;,, ppm, is maintained. The exchange the point under consideration. That is, for each point potential contribution is then evaluated using Slater,s [ax, by, czI in the unit the atoms located within original free electron approximation ,241 and added a distance of R units must be found and the potential to v ~ ( x y z ) . Here
contributions from these atoms summed.
The unit cell of the direct lattice was divided into Vx,(r) = - 6 - p(r) 8 000 points, i. e. 20 points/cell side, and the atoms
within a distance of 5 a0 from each point were listed
gives the total crystal potential v(xyr) at this point.
together with the distance (1' + +' + n2)'" = rr. Due Kohn and coworkers [41, have indicated that to the of the cubic lattice, points lying corrections 1251 due to the rapidly changing p(xyz) within the basic 1/48 of the unit cell were considered.
should be included in calculating Vx,(xyz). value This leads to a total number of points of 286. Thus
of is then changed to D' = + 2, which is equi- for each of these 286 points, m, n we calculate those valent to moving to the next unit cell. At this position, atoms which will contribute to the potential and the
the list of is scanned for those values for which distances of these atoms. m > +- D. These represent empty lattice sites and As this data was obtained for a FCC lat- the potential contribution V,,,,, is subtracted from tice, it can be used for any FCC structure surface
VO(xyz) (also p,,, is subtracted from pz,,,). When
summation.
Now consider a face-centered cubic lattice with all m values have been tested, the remaining potential V(xyz ; D') is added to the new exchange potential cell edges parallel to co-ordinate axes X, Y , 2. We
obtained from p O ( ~ ' ) to give V(xyz ; D') the potential assume that the lattice exists only for Y < 0 and that
of the point [xyz] in the unit cell where the origin is . . -
(*) This work was carried out in part while one of the authors at a distance D' from the surface. This process is
(JTW) was a visiting professor at the Centre EuropCen de Calcul repeated until D' = - (R f 1 ) i. e. we include the
Atomique et Molkculaire, Orsay. potential outside the last row of ion cores for a dis-
C3-112 J. T. WABER, E. KENNARD AND YU-PING TSUI
tance equivalent to that inside. For unit cells which have lost potential contributions due to the surface, the potential distribution no longer has full cubic symmetry and thus the potential at the point [xyz]
will no longer be equal to the potential at the 48 points which are equivalent in the bulk. Thus corresponding to each point [xyz], the potential must be calculated separately for those equivalent points which are not produced by a rotation around Y. The procedure is
FIG. 13. - An isometric drawing of the one-electron potential (including exchange) near the surface of platinum. The poten- tials near ion-cores which are less than - 5 Ry were set to
- 5 Ry and appear as straight lines.
FIG. 12. - Map showing the relative coordinates of atoms in a Face centered cubic metal such as platinum. Three vacant atom sites are indicated. Various planes which are perpendicular to the X axis are indicated by the intercept 1, 6, 11. The numbers along the Z axis for the vacancies should be 11 (mod 20). The
top four numbers are inadvertently 10 too high.
FIG. 14. - A perspective drawing of the potential illustrating the non-uniformity in the cc mufin tin >> region. Truncated values were not connected and appear as dots in the ion-core
region.
FIG. 15. - A contour map of the potential variation on the X = 1 plane near the free surface of thorium.
SURFACE ENERGY AND SURFACE POTENTIALS OF TRANSITION METALS C3-113 thus to calculate V(xyz) for each value of D and then
repeat this for each equivalent position before pro- ceeding to the next [xyz] value. We confirm that the potential converged by increasing R several fold.
Let us cite a few examples of the crystal potentials calculated. In order to interpret the notations on the subsequent figures, the coordinate positions of lattice sites are indicated in figure 12 in terms of integers times (a,/20). Three vacancies are mentioned in this figure and indicated as squares. Discussion of them will be deferred. Lattice planes can also be identified by the X-intercepts such as 1, 6 and 1 1 .
FIG. 16. - Contour map of the potential variation on the (331) plane of platinum.
An isometric drawing of the potential near the surface is presented in figure 13. The deep potentials of the ion-cores were truncated at - 5.0 Ry are shown as straight lines. An alternative technique has recently been developed by Mui and Waber [26]. Figure 14 is a perspective drawing, in which the truncated values are represented as points. The drawing shows that the muffin-tin potential of a metal like platinum is signi- ficantly warped. A third way of representing a complex potential surface is in terms of isopotential lines.
Figure 15 is the two-dimensional potential variation over the plane x = I ; this cutting plane passes through the centers of the atoms.
The method outlined above (generating surface potentials) can easily be used for V on the (hkl) plane.
Figure 16 is the potential contours for the (311) plane of platinum.
A similar contour plot is presented in figure 17 for the platinum crystal in which the 3 atom sites indi- cated in figure 12 are vacant. This indicates the potential variation near a step on the surface.
A similar picture is presented for BCC tungsten in figure 18, three sites in the uppermost row are also vacant. Another significant difference is that the cutting plane is X = 6 and consequently it passes between the atoms. The steep gradients in V,,, just beyond the atoms is clearly indicated in this figure.
Another type of surface defect is illustrated in figure 19. For this, the cutting plane y = 71 is parallel to the free surface. For this drawing the vacancy is
(( split )) by an atom which is at the saddle point bet- ween lattice sites. This can be regarded as an atom in
the <( act of diffusing D. The distorted region extends
over several interatomic distances. It is instructive
FIG. 17. - Contour map for the 3 vacant sites indicated in figure 12. Relates to the variation near a lattice step.
Cutting plane is X = 1.
J. T. WABER, E. KENNARD AND YU-PING TSUI
FIG. 18. - A similar contour map for 3 vacant sites in Body Centered Cubic Tungsten. The cutting plane X = 6 passes between atoms and the variation of potential is smaller.
FIG. 19. - A contour map showing the large region of distortion in the crystal potential associated with the migration of an atoms midway between two vacant lattice sites.
SURFACE ENERGY AND SURFACE POTENTIALS OF TRANSITION METALS
FIG. 20. - A companion contour to figure 19 map showing that the distorted region is detectable in free space.
FIG. 21. - A contour map cutting through the surface at X = 1 showing a monolayer of oxygen atoms added at points [20 h + 11, 78.66, 20 1 + 11 ] where h and I are the integers 0, 1, 2.. .
C3-116 J. T. WABER, E. KENNARD AND YU-PING TSUI
to show the potential variation over another cutting plane which is parallel to the free surface and 20 units (or a,) above the plane in figure 19. Most of the geometric features of the contour lines in the latter figure are present in figure 20.
Thus a charged particle outside the metal will experience a potential which reveals any surface defects present in the metal.
Another situation is indicated in the next four figures.
A monolayer of oxygen atoms were added with their
centers on the plane y = 78.68. Typical values for X and Z coordinates would be 1, 21, 41, etc. ; that is, the foreign atoms were placed at the centers of the square of surface atoms. Local distortion of the surface potential by these ad-atoms is shown in figure 21. The pattern parallel to the surfaces as shown in figure 22 is interesting. The cutting plane is y = 75. The poten- tial from the oxygen atoms is a small, roughly circular region. The potential has been further and <<bow ties )) occur in 4-fold symmetry between the thorium
FIG. 24. - The companion drawing to figure 23 showing the FIG. 22. - A contour map parallel to the free surface at Y = 75 complex potential variation along a surface plane Y = 71. The for the situation described in figure 21. Note the ((bow ties )> potential between surface atoms is deeped by the absorbed
lying along fourfold < 110 > directions. layer.
FIG. 23. - A contour map for a monolayer of carbon afoms added at the points [20 h t 11, 72.48,201 + 111
that is between four thorium neighbors. Note the partial bow ties )) between neighboring surface atoms.
