Structure and dynamics at the Al(111)-surface

surface science ELSEVIER

Surface Science 324 (1995) 113-121

Structure and dynamics at the Al(111)-surface J. Sch6chlin a, K.P. Bohnen a,*, K.M. HO b a Kernforschungszentrum Karlsruhe, lnstitutfiir Nukleare Festkfrperphysik, P.O. Box 3640, D-76021 Karlsruhe, Germany b Ames Laboratory, US Department of Energy and Department of Physics, Iowa State University, Ames, IA 50011, USA Received 26 July 1994; accepted for publication 27 October 1994

Abstract

Using first-principles total-energy calculations the lattice relaxation, surface energy, work function and surface phonons have been determined for the Al(lll)-surface. The use of the Hellmann-Feynman theorem allows for a very efficient determination of equilibrium geometries and interplanar force constants. Results will be presented and compared with available experimental information as well as with other theoretical treatments. Keywords: Aluminum; Density functional calculations; Low index single crystal surfaces; Phonons; Surface electronic phenomena; Surface energy; Surface relaxation and reconstruction

1. Introduction

The first and foremost question in surface sience concerns the location of atoms at the surface. Atoms near the surface of a metal are under the influence of different forces from those in the bulk, and in most cases relaxations of the bulk lattice will occur in the topmost layers. Such changes in geometry can have significant effects on the physical properties of the metal surface. For example they have consequences for the surface phonon spectrum. Surface vibrations are involved in many processes on surfaces at ambient or elevated temperatures. A detailed knowledge of the surface phonon spectrum is essential in studies of surface diffusion, phase transitions on clean and adsorbate-covered surfaces, and desorption processes. The past few years have witnessed a rapid growth

* Corresponding author.

in experimental effort to measure surface atomic arrangements and vibration spectra. Mostly LEED and ion-scattering experiments gave insight into the atomic arrangements [1] while EELS and He-scattering experiments provided information about the surface phonon spectrum [2,3]. However with the development of a new generation of supercomputers it has become possible also theoretically to determine structure and dynamics of surfaces from first principles. It is well established that first-principles totalenergy calculations using local density formalism (LDA) are very successful in determining structural and vibrational properties of a large variety of bulk materials [4-6]. Recent studies have shown that these techniques can also successfully be applied to surfaces [7]. In this paper we present results for the A I ( l l l ) surface. Our studies show a small expansion of the top interlayer spacing of 1.18%. All other layer spacings deviate from bulk interplanar distances by less than 0.40%. These results are in excellent agree-

0039-6028/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 0 3 9 - 6 0 2 8 ( 9 4 ) 0 0 7 1 0 - 1

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J. Schrchlin et al. / Surface Science 324 (1995) 113-121

ment with LEED data of Noonan and Davis [8]. They are also in excellent agreement with a recent calculation by Feibelman [9] indicating the high degree of reliability which can be obtained theoretically. Calculation of interplanar force constants allowed the determination of surface pho_non modes at the high symmetry points F, M and K of the SBZ. Frequencies of the surface modes deviate from those obtained with bulk force constants. The effects are smaller than those predicted for the Al(ll0)-surface [10,11]. He-scattering experiments for A I ( l l l ) in ~ ( 1 1 2 ) and T ( l l 0 ) direction show only the Rayleigh mode [12]. This is in good agreement with our calculations. In the past A I ( l l l ) has been studied using the dielectric response approach [13]. This method allows the calculation of the surface phonons at arbitary wavevectors in the two-dimensional Brillouin zone, however this approach is limited to systems which can be described by weak local pseudopotentials. This limits the numerical accuracy even for a system like aluminium. A clear indication of this effect can be seen for AI(100) where our nonperturbative approach gives the Rayleigh mode frequency at X in excellent agreement with experiment [14] while the response approach needs an adjustment of force constants [15]. A similar adjustment is needed for A I ( l l l ) [13]. In view of this limitation we think that the interpretation of very weak structures in the He-atom scattering is very difficult [16]. At least at the high symmetry points M and K, our calculations do not support the picture of clear resonances. This is different for the noble metal ( I l l ) surfaces where our approach determines these resonances in excellent agreement with EELS measurements [17].

2. Method Self-consistent pseudopotential calculations are performed for the Al(lll)-surface with a periodic slab geometry (see Fig. 1). Slabs 12 layers thick are used, separated by 6 layers of vacuum. For a given trial geometry the total energy of the system is calculated within the local-density-functional formalism [18] using the Wigner interpolation formula [19] for the electronic exchange and correlation energy. The norm conserving pseudopotential [20] in the

a31 al

Fig. 1. Periodic slab geometry of AI(lll). The film contains 12 layers of atoms and 6 vacuum layers.

present calculations has been used in previous calculations of the bulk structural properties of aluminium and phonon frequencies for the AI(ll0) [10,11] and the AI(100) [21] surface with excellent results. Plane waves with kinetic energy up to E = 12.0 Ry are used in the expansion of the electronic wave functions. The sampling grid for the surface calculations of A I ( l l l ) was 37 points in the irreducible part of the SBZ. Partial occupancy of states near the Fermi level was taken into account by a Gaussian smearing scheme which broadens each energy level to calculate the Fermi energy and fractional occupancy for each state from the resultant density of states. This scheme has proven to be very useful in previous calculations [22]. In addition to the total energy, forces exerted on each atomic layer are calculated using the Hellmann-Feynman (HF) theorem [23]. The use of HF forces minimizes the number of trial geometries needed to determine the equilibrium geometry, especially when a number of layers are relaxed simultaneously. Furthermore the determination of forces is the first step to a calculation of interplanar force constants needed for the determination of phonon modes. In our calculation a modified Broyden scheme [24] is used to accelerate convergence to the selfconsistent solution. Iteration is carried out until the

J. Schfchlin et al. / Surface Science 324 (1995) I 13-121

total energy differences are stable to within 5 X 10 - 6 Ry. Having determined the equilibrium geometry an extension of the 'frozen phonon' method [25] was used to calculate surface phonon frequencies and phonon polarization in surface layers. The 'frozen phonon' method has proved very successful in providing a detailed and accurate description of bulk phonons. A straightforward application of this method is not possible for surfaces due to the fact that even at a high symmetry point in the SBZ the vibration pattern of the surface modes is not completely determined by symmetry. The decay of modes into the interior is not known. To avoid the ambiguity which would result from making assumptions about the decay we calculated intra- and interplanar force constant matrices for our slab at certain high symmetry points in the SBZ. These calculations were done by distorting the equilibrium geometry in an appropriate way. The force constants were obtained via the calculated force differences. Using these force constant matrices the phonon modes of the slab could be calculated [26].

3. Equilibrium geometry The results of our calculations for the relaxation of A1(111) are given in Table 1. These results were obtained starting from ideal geometry (truncated bulk) and deducing the equilibrium geometry via a force constant matrix. For our 12-layer slab the equilibrium geometry is given by Ad12 = + 1.18%, Ad23 : - 0 . 4 0 % and Ad34 : +0.20%, where Adij is the change in interlayer separation between layers i and j. A 'force flee' situation is reached when fluctuations in the interlayer spacing are less then

Table 1 Comparison of calculated data for the relaxation of the AI(lll)surface with LEED data and other theoretical results from LDA pseudopotential calculations Ad12(%)

Ad23(%)

Ad34(%)

Ref.

+ 0.9 ( 5: 0.5) + 1.7 ( 5: 0.3) + 1.00 + 1.18

+ 0.5 ( -+ 0.7) - 0.07 - 0.40

+ 0.22

[27] [8] [9] This work

115

Table 2 Comparison of calculated data for the work function (in eV) of the low indicated aluminium surfaces with several photoemission data AI(100)

AI(110)

AI(111)

Ref.

4.20 ( + 0.03) 4.41 (_+0.03) 4.51 (+_0.03)

4.06 ( +- 0.03) 4.28 (+0.02) 4.32 (+_0.03)

4.28 4.26 4.24 4.31

[281 [29] [29] [21];[11]; this work

(+0.01) ( + 0.03) (_+0.02) (___0.03)

Table 3 Comparison of calculated data for the surface energy (in erg/cm 2) of the low indicated aluminium surfaces with experimental data (see text) Al(100)

Al(ll0)

1081(+30)

1180 1169 1090(+30)

Al(lll)

Ref. [30] [31] [21];[11]; this work

939(+30)

0.01%. The calculated Adij are in good agreement with experimental and other theoretical results [8,9,27]. Calculated work function and surface energy are given in Tables 2 and 3. For comparison also recent theoretical results for the Al(110)- and the Al(100)surface are listed [11,21]. The calculated work function of 4.31 eV is in good agreement with new photoemission values of Dierkes and Winter [28] of 4.28 _+ 0.01 eV. For the surface energy we find a value of 939 e r g / c m 2. The experimental values of <011>

p.

~

<211>

Fig. 2. Surface Brillouin zone of a fcc (111) lattice with the high symmetry points.

116

J. Schi~chlin et al. / Surface Science 324 (1995) 113-121 (a)

a

.................

1180 e r g / c m 2 from Tyson and Miller [30] and of 1169 e r g / c m 2 from Wawra [31] are isotropic values and could not differentiate between single crystallographic surface directions. The first value is determined from measurements of the surface tension of liquid aluminium extrapolating through the liquidsolid phase transition. The second value is determined from elastic constants and measurements of the bulk modulus with ultrasound. The A I ( l l l ) surface shows the smallest surface energy as expected.

•

al

~

(34. Surface phonons

(b)

•

T a' (3-'-

•

e,a~

(3--

~

(3--

(3-.-

(c)

O •

a2 i

?

? I

/

I

//z

?

y//"

•

•

•

Fig. 3. The unit cell and the distortions of atoms in the surface layer at (a) F-point, (b) M-point and (c) K,-pointof the SBZ to calculate surface phonons.

As already described in Section 2 the phonon spectrum (bulk and surface modes) is calculated at high symmetry points (Fig. 2) in the SBZ. This was done by calculating the force constant matrices coupling the layers in the slab. To obtain reliable surface modes, surface resonances and surface phonon density of states it is well known that thick films are needed. We calculated the phonon modes with films containing 96 layers. The force constant matrices coupling these layers were determined in two ways. Microscopically we calculated the inter- and intraplanar force constant matrices coupling the outermost layers. The coupling of the inner layers was described by force constant matrices obtained from bulk atomic force constants. These were determined fitting inelastic-neutron-scattering data by an axially symmetric Born-von Kfirmfin model [32]. The microscopic force constants are determined via the calculation of force differences. The two geometries used for these calculations are always the equilibrium geometry and a situation where the whole layer of atoms has been moved in the direction of the basis vectors a i (see Figs. 3a-3c). Determining the self-consistent solution and obtaining the forces acting on each layer we can calculate the interplanar force constants. Using the equilibrium geometry as reference situation instead of the true force free geometry introduces only small errors. The unit cell and the type of distortions used are indicated in Figs. 3a-3c. The magnitude of the distortion was 1-2% of the bulk interlayer spacing. As tests showed this is big enough to get results beyond the numerical noise level and on the other hand small enough to avoid

117

J. Schfchlin et al. / Surface Science 324 (1995) 113-121 Table 4 Force constant matrices (in N / m ) at high symmetry points describing the coupling between the topmost layers Microscopic calculation

Bulk parameters

[7.72772 38.35] i-6.75 675 40541

?6 786 41.33] [_7.55 755 42781

F-point

M-point "131.26 121.03

39.67

] 36.71

41.251

28.50 [ -

2.28 + i3.94 0 10.96 - i18.98

0 - 3.61 - i6.26 0

- 7.66 O i13.26 ] - 5 . 1 7 - i8.95 J

2.95 + i5.11 0 - 11.42 - i19.78

0 - 4.08 - i7.06 0

-1142oi1978] - 6 . 1 9 - i10.72 ]

K,-point 0

127.10 -1.59

/-1.59 - 5.52 i5.52 8.54

i5.52 5.52 - i8.54

-

99.88

99.88

30.09]

41.67 - 7.90 i7.90 - 14.23

- 14.74 - i14.74 0

i7.90 7.90 - i14.23

- 14.23 ] - i14.23

o

For comparison force constants from microscopic calculation and the bulk parameter model are given, a and /3 refer to Carthesian coordinates. Force constants q~lx = ~ at ~,-point are not exactly zero because of broken symmetry at the surface. Directions of basis vectors: F-point: x = (121), y = (101) and z = (111). M-point: x = (211), y = (011) and z = (111). K,-point: x = (121), y = (10]) and z = (111).

anharmonic contributions. Distortions in the a3-direction (perpendicular to the surface) are done inward and outward to eliminate the anharmonic part. For the high symmetry point M the surface unit cell had to be doubled while the area of the surface unit

cell at K. must be a factor of three greater as in the case of the F-point. The calculated interplanar force constant matrices are given in Table 4. For comparison also values obtained with bulk atomic force constants are given. At the Al(lll)-surface only the

Table 5 Phonon frequencies and polarization vectors at the AI(111)-surface from microscopic calculation Microscopic calculation

Bulk parameters

Frequency (THz)

Polarization vector l s t / 2 n d layer

Frequency (THz)

Polarization vector l s t / 2 n d layer

~4.1

Maximum of a resonance (0.89, 0.00, 0.04) (0.07, 0.00, 0.25)

3.25

(0.05, 0.00, 0.89) (0.20, 0.00, 0.31) (0.80, 0.00, 0.06) ( - 0 . 2 0 , 0.00, -0.16)

4.1

(0.03, 0.02, 0.95) (0.21, -0.21, 0.01)

3.72

(0.00, 0.00, 0.97) (0.18, -0.18, 0.00)

4.5

8.5

4.5

8.72

He scattering Frequency

(Tnz)

Error bars of this results are less then 0.1 THz. For comparison results from the bulk parameter model are also given. Polarization vectors are normalized over the whole slab and are given in Carthesian coordinates (see Table 4). He-scattering data from Toennies are given as available [3].

J. Sch6chlin et al. / Surface Science 324 (1995) 113-121

118

2000 surface tnmeated bulk

(a) n

M-Point Polarization: x

1600

"~

1200

400

1

2

3

5

4

6

10

Frequency[THz] 2000 .....

(b)

~ n c a t e d bulk o

M-Point 1600

Polarization: y

' ~ 1200

400 I

;-,', 0 0

1

2

3

4

5

6

7

8

9

10

Frequency[THzl Fig. 4. Phonon density of states at M-point. The polarizations x, y and z are the same as given in Table 4. Intensity is given in arbitrary units.

J. Schi~chlin et aL / Surface Science 324 (1995) 113-121

119

2000 surface trunoated bulk

.....

(c)

I

M-Point 1600

Polarization: z

1200

i I i I i I

,,, I ii

II

e~

II

800

I'I II II 'I I,

400

I

. . . .

I 1

. . . .

1 . . . .

I

2

3

.

.

. 4

.

.

.

5

6

1 . . . .

I

7

8

'

~'='

'

I 9

T '

'

' 10

Frequency [THz] Fig. 4 (continued).

coupling between the two topmost layers differs substantially from the bulk values. Using the microscopically calculated surface force constant matrix elements as given in Table 4 together with matrices determined by bulk atomic force constants given in Ref. [32] for the inner layers we obtained the surface modes (see Table 5). For comparison also the results with bulk parameters up to the surface are indicated. The polarization direction in the outermost layers is also given. Our calculation demonstrates that changes in force constants at the surface can occur which are not negligible even for a very 'bulk'dike surface. The effect on the phonon frequencies can be seen clearly. In Figs. 4 and 5 we give the surface phonon density of states for the high symmetry points in the SBZ. It is obtained by weighting the density of states with the amplitude square in the outermost layer of our film. Different plots correspond to different polarizations. At M-point we have a strong longitudinal polarized surface mode with a phonon frequency of 8.5 THz. This is a softening of ~ 0.2 THz compared to the value of 8.72 THz obtained with bulk force

constants. In contrast to this softening behaviour for the longitudinal mode the Rayleigh mode frequency shifts upward from 3.25 THz (with bulk parameters only) to the border of the bulk continuum and shows up as a resonance with a maximum at 4.1 THz. This is in good agreement with He-scattering data of Toennies [3]. At K.-point we found the Rayleigh mode lying at 4.5 THz also in very good agreement with the experimental data [3]. The changes in surface intra- and interlayer coupling detected by microscopic calculation compared with bulk parameters are responsible for this strong shift in phonon frequencies.

5. Summary We have used first-principles self-consistent total-energy calculations to investigate successfully the surface structure and surface phonon spectrum of the Al(lll)-surface. As it has been demonstrated also for AI(ll0) and AI(100) this is a reliable method to obtain important information about surface properties. The results indicate that the densely packed

120

J. Schfchlin et a l . / Surface Science 324 (1995) 113-121

2500 SUI~aOC

.....

trunoated bulk

(a)

K-Point

2000

Pohtrt2ation: x

' ~ 1500

1000

500

I ii i I I I I I I

. . . .

i

. . . .

1

I

. . . .

2

I

. . . .

i

3

. . . .

I I ~

4

1':

. . . .

5

rl II

I

6

i

7

8

. . . .

9

10

Frequency [THz] 2500 surf~c .

.

.

.

~no.db.,k

.

-K-Point

D

iI

2000

(b)

Polari:ation: z

' ~ 1500

~ 10011

500

0

. . . .

0

I

l

. . . .

I

2

. . . .

I

3

'

'

'

I

4

'

'

'

I

'

'

I'

'

5

I

6

. . . .

I

7

. . . .

I

8

. . . .

I

. . . .

9

Frequency [THz] Fig. 5. Phonon density of states at g,-point. The polarizations x and y are equal. Intensity is given in arbitrary units.

10

J. Schbchlin et al. / Surface Science 324 (1995) 113-121

Al(lll)-surface is very bulk-like for regions outside the first layer of atoms. The surface phonon frequencies calculated with our method are in very good agreement with available He-scattering data [3]. Anomalous resonant modes found on the noble metals, the transition metals Ni, Pd and Pt and W(001) which are attributed to a reduction in sp-d hybridization at the surface [3] are not present at AI(lll). First-principles phonon and multi-scattering EELS studies of C u ( l l l ) and A g ( l l l ) have also shown that the resonances can be explained quite naturally without invoking large changes in charge densities at the (lll)-surface [17,33]. Calculated and measured EELS sprectra are in good agreement indicating that the lattice dynamics is correctly treated in the microscopic calculation. The remaining intensity problem for He scattering is thus probably related to the interaction potential between the He atoms and the surface [34,35].

References [1] M.A. Van Hove and S.Y. Young, The Structure of Surfaces (Springer, New York, 1985). [2] H. lbach and D.L. Miles, Electron Energy Loss Spectroscopy and Surface Vibrations (Academic Press, New York, 1982). [3] J.P. Toennies, Phys. Scr. T 19 (1987) 39. [4] P.K. Lain and M.L. Cohen, Phys. Rev. B 24 (1981) 4224. [5] P.K. Lain and M.L. Cohen, Phys. Rev. B 25 (1982) 6139. [6] V.L. Moruzzi, J.F. Janak and A.R. Williams, Calculated Electronic Properties of Metals (Pergamon, New York, 1978). [7] K.P. Bohnen and K.M. Ho, Surf. Sci. Rep. 19 (1993) 99. [8] J.R. Noonan and H.L. Davis, J. Vac. Sci. Technol. A 8 (1990) 2671. [9] P.J. Feibelman, Phys. Rev. B 46 (1992) 15416. [10] K.M. Ho and K.P. Bohnen, Phys. Rev. Lett. 56 (1986) 934. [11] K.M. Ho and K.P. Bohnen, Phys. Rev. B 32 (1985) 3446. [12] A. Lock et al., Phys. Rev. B 37 (1988) 7087.

121

[13] LA. Gaspar, A.G. Eguiluz, M. Gester, A. Loch and J.P. Toennies, Phys. Rev. Lett. 66 (1991) 337. [14] K.P. Bohnen and K.M. Ho, Surf. Sci. 207 (1988) 105. [15] J.A. Gaspar and A.G. Eguiluz, Phys. Rev. B 40 (1989) 11976. [16] A. Franchini, V. Bortolani, G. Santoro, V. Celli, A.G. Eguiluz, J.H. Gaspar, M. Gester, A. Lock and J.P. Toennies, Phys. Rev. B 47 (1993) 4691. [17] S.Y. Tong, Y. Chen, K.P. Bohnen, T. Rodach and K.M. Ho, Surf. Rev. Lett. 1 (1994) 97. [18] P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) 864; W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) 1133. [19] E. Wigner, Phys. Rev. 46 (1934) 1002. [20] D.R. Hamann, M. Schliiter and C. Chiang, Phys. Rev. Lett. 43 (1979) 1494. [21] K.P. Bohnen and K.M. Ho, Surf. Sci. 207 (1988) 105. [22] C.L. Fu and K.M. Ho, Phys. Rev. B 28 (1983) 5480; K.M. Ho, C.L. Fu and B.N. Harmon, Phys. Rev. B 29 (1984) 1575. [23] H. Hellmann, Einfiihrung in die Quantenchemie (Deuticke, Leipzig, 1937); R.P. Feynman, Phys. Rev. 56 (1939) 340. [24] C.G. Broyden, Math. Comput. 19 (1965) 577; D. Vanderbilt and S.G. Louie, Phys. Rev. B 30 (1984) 6118. [25] K.M. Ho, C.L. Fu and B.N. Harmon, Phys. Rev. B 29 (1984) 1575, and references therein. [26] F.W. de Wette and G.P. All&edge, in: Methods in Computational Physics, Vol. 15, Lattice Dynamics of Surfaces and Solids (New York, 1976) p. 163. [27] H.B. Nielsen and D.L. Adams, J. Phys. C 15 (1982) 615. [28] G. Dierkes and H. Winter, to be published. [29] J. HSlzl and F.K. Schulte, in: Solid Surface Physics, Vol. 85, Work Function of Metals (Berlin, 1979), and references therein. [30] W.R. Tyson and W.A. Miller, Surf. Sci. 62 (1977) 267. [31] H. Wawra, Z. Metallkd. 66 (1975) 395, and references therein. [32] G. Gilat and R.M. Nicklow, Phys. Rev. 143 (1966) 487. [33] Y. Chen, S.Y. Tong, K.P. Bohnen, T. Rodach and K.M. Ho, Phys. Rev. kett. 70 (1993) 603. [34] A. Francini, G. Santoro, V. Bortolani, A. Bellman, D. Cvetko, L. Floreano, A. Morgante, M. Peloi, F. Tommasini and T. Zambelli, Surf. Rev~ Lett. 1 (1994) 67. [35] C. Kaden, R. Ruggerone, J.P. Toennies, G. Zhan and G. Benedek, Phys. Rev. B 46 (1992) 13509.