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Work function of K and Rb submonolayers adsorbed on Al(111) and Mg(0001)
Vacuum~volume41/numbers 1-3/pages 580 to 582/1990
0042-207X/9053.00 + .00 © 1990 Pergamon Press plc
Printed in Great Britain
W o r k function of K and Rb submonolayers adsorbed on A1(111 ) and Mg(0001 ) A K i e j n a , Institute of Experimental Physics, University of Wroclaw, ul. Cybulskiego 36, 50-205 Wroclaw, Poland and Sektion Physik der Universita't M~inchen, Theresienstr. 37, 8000 M~inchen 2, FRG
The work function variation due to K and Rb atom adsorption on AI(111) and Mg(O001) has been calculated self-consistently within the density functional formalism. To go beyond the jellium model for the substrate the effect of metallic ions on the conduction electrons is represented by a parametrized form of the perturbing potential. The adsorbed alkafi atoms are represented by a jellium slab whose thickness varies with coverage. The computed work functions vs coverage curves, the work function minimum and initial dipole moments show good agreement with recent experimental data on work function change.
The adsorption of alkali metals on metallic substrates leads to characteristic changes in the measured work function. It decreases rapidly at low coverages, reaches a minimum and increases to saturate at coverages greater than one monolayer, reaching the value of the work function of solid adsorbate. The application of the density functional formalism has stimulated a greater understanding of alkali metal adsorption L2. In the pioneering work of Lang ~ both substrate and adsorbate were represented by the positive charge background. The calculations have been performed self-consistently, giving work function changes with a minimum for coverages of O < 1 in agreement with experimental data. The experiments imply that the shape of curves representing the changes of work function vs coverage are more sensitive to the kind of adsorbate than to the substrate 2. Nonetheless, the jellium with r, = 2 (density of positive charge background fi = 3/(4nr3)) does not represent properly such substrates as transition metals for which the experimental data on adsorption was available. For the bare metal surface the work functions calculated with the same formalism are in good agreement with experimental data for higher density metals like AI and Mg. Therefore the recent experimental measurements a-5 of alkali metal adsorption on AI(I 11) and Mg(0001) provide an ideal basis for the test of the theory. We present here a self-consistent calculation of work function changes due to alkali adsorption on AI(111) and Mg(0001) in the spirit of Lang's model. However, in our calculations we take into account the discrete atomic structure of the substrate. The adsorbed atoms are represented by a positively charged slab of jellium whose thickness changes with coverage. To improve agreement of calculated surface energies and work functions of bare surfaces with experimental results, Lang and Kohln 6"7 took into account the difference between the pseudopotentiai of a semi-infinite lattice of ions and the electrostatic potential ofjellium, treating it as a first order perturbation. The limitations of such an approach were discussed by Monnier and Perdew 8. They presented a variational self-consistent method in which the discrete lattice perturbation is treated variationally. In their approach the three-dimensional discrete-lattice potential is replaced by a step potential (with the step in the vicinity of the 580
surface) and added to the external potential which enters the Schrrdinger equation. The step height is varied giving different electron density profiles. The electron density which is the solution to the problem is the one which gives the lowest calculated surface energy. The calculated density profiles show a noticeable variation among different crystal planes. A detailed description of this method is given in ref 8. In the present calculation we make use of the approach of Monnier and Perdew. The discrete lattice potential of the substrate, just as for a bare surface, is replaced by a step potential of the form: V ( x ) = C H ( - x),
(1)
where H(x) is the Heaviside step function and the optimum value of the variational parameter C was determined from the minimum of surface energy of the bare metallic substrate. F o r AI(111) and Mg(0001) surface we have used the values of C given by Perdew 9. The adsorbed alkali atoms are represented by the jellium slab of density fi,~. In Lang's calculation the thickness of the slab was chosen to have the fixed value dr, of the interplanar spacing of the most densely packed planes in the bulk alkali metals. However, during the process of adsorption there is a transition from an ionic to atomic character, of adsorption at higher coverages. This leads to the change in the distance between the first substrate lattice plane and alkali adatoms. The analysis of experimental data a'5 implies that the bonding distance of alkali adatoms varies between the ionic bond length dio, (twice the the ionic radius) and d m (interplanar distance in the bulk alkali) but is closer to d~o.. These facts suggest approximating the overlayer thickness by the linear relation 1° d ( O ) = dio. +
O(drn
--
dio,)p
(2)
where ® is the coverage. F o r the calculations presented in this work we took p = 1.0, but we have made also some trial computations with p = 0.5 which gives a value of d(®) closer to dio.. The height of the jellium slab was also varied with coverage. The electron density profiles n(x), were determined from the self-consistent solution of the K o h n - S h a m equations with the V(x) term included in the effective potential. The exchange-
A Kiejna: Work function of K and Rb submonolayers
correlation potential is treated in the local density approximation with the Wigner expression for the correlation energy. The jellium slab of adsorbed alkali metal modifies the electrostatic potential q~(x) which is calculated from the integral equation 8 and iterated along with the effective potential. The self-consistency procedure led to the satisfied charge neutrality condition*
fff[n(x)-n+(x)]dx--O
(3)
1.2"
1.0
-"
.a"
X
\ ........ !
~
i
.6"
r .4
and a well-converged work function and surface dipole moment
.2
Ad~ = 4 ~
x[n(x)
- n÷(x)]dx.
(4) 0.0
Here n ÷ (x) = h H ( - x) + h,d[H(x) - H ( x - d)]. Another criterion of self-consistency imposed was the Budd-Vannimenus theorem ~~.a, modified by Bigun 12 for the case of an adsorbate: 4,(0) -
4,( -
oo) + [,t(d)
[3
-
s Io is x ( o . u.) Figure 1. Electron density profiles for the jellium Slab representing a full monolayer of K atoms absorbed on jellium (r, = 2.07) and real AI(I l 1) surface (full line). -Io
-s
o
4,(O)l~,~.~/ff
= iid ~ eF (h) + e ~ ( ~ )
]/
d h + C n (O)fii
(5) .S N\
where er is the Fermi energy and e~ is the exchange-correlation energy per particle in an uniform electron gas of density ~i. The electron density profiles for the alkali slabs on A l ( l l l ) Mg(0001) substrates calculated by us differ from those calculated for alkali slabs on jellium substrate. The difference is illustrated in Figure 1 where the density profiles for K-AI(I 11) and K-AI(jellium) are drawn for comparison. As is seen for the jellium substrate the density profile is more diffuse and the first peak in the Friedel oscillations of the electron density is much reduced, compared to the real metal substrate. For the bare metal surface the work functions were calculated from the change in self-consistent-field expression~ 3: da (I)=--
dE
1.0
<~ I
.5
\\\
2.0
2.5
0.0
.2
.4 .6 Covero9e 0
.a
1.0
Figure 2. Work function change for different coverages of K on AI(I 1I). The full curve was calculated for p = 1.0 whereas the broken one for p = 0.5 (see equation 2). The dotted curve shows the experimental data 3.
(6)
where the derivative of the surface energy tr with respect to the surface charge density E was evaluated at E = 0. The calculated work functions for the bare AI(111) and Mg(0001) are 4.2 eV and 4.0eV respectively, in a good agreement with the measured values which are 4.24eV for A I ( l l l ) 14 and 3.84eV Is for Mg(0001). The change in the work function for different adatom coverage is calculated from the change in the dipole moment. In Figures 2-4 we present the calculated changes in the work function AO vs coverage O, for potasium and rubidium atoms on A I ( l l l ) and K-Mg(0001). In Figure 2 the dependence AO(O) is presented for K-AI(111) for two different choices of parameter p appearing in (2). The broken curve is for p = 0.5 while the full curve is for p = 1.0, with dr, = 7.13 au and d~o. = 5.03 au. As one can see better agreement with experimental data 3 is achieved for the choice p = 1.0. For this curve the work function attains a minimum a t O m i n "~' 0.5 and A(l)mi n = 2.39 eV is in a good agreement with the experimentally measured ®m~, = 0 . 4 2 and AOm~. = 2.17 eV. The inital dipole moment (the slope of the A~(®) curve to zero coverage) obtained for our curve is 3.9 debye (D) in very good agreement with experimental value 4.1 D. *Atomic units (au) are used unless otherwise stated.
~
The results for changes in work function for R b - A I ( l l l ) are displayed in Figure 3, for d,, = 7.51 au, dis. = 5.56 au and p = 1.0 used in (2). The overall shape of the calculated AS vs (9 curve agrees well with the experimental curve 5. Compared to the measured value (®m~, = 0.42) the position of AOm~, is, similar to K-AI(111), shifted towards higher coverages and gives Omi,. "" 0.5. The calculated value of AOm=. is 2.52 eV compared to the measured of 2.25 eV. The calculated initial dipole moment, 4.0 D is in very good agreement with the experimental value of 4.1 D. For higher coverages the experimental and theoretical curves agree well. For the K-Mg(0001) system, the changes in the work function vs coverage are presented in Figure 4. The agreement between the calculated and experimental curve 3 is worse compared to that for the AI substrate. The calculated A~m~. = 1.76 eV appears for ®,,5. ~- 0.45, whereas the corresponding measured value for AOm~n= 1.49 eV at ~m~, = 0.35. The calculated initial dipole moment is 2.7 D compared to experimental 3.8 D. The worse agreement for K-Mg(0001) is presumably connected with the less accurate determination of the surface properties of the bare Mg in the variational self-consistent method of Monnier and Perdew. This is suggested for example by the difference between the calculated and measured work function for the bare Mg(0001) surface. 581
A Kiejna." Work function of K and Rb submonolayers .0
.0
\k
.5
\\\
.5 1.0
03
~ 1.5
\kk\k\
1.0
\\
e .4
2.(1
1.5 2.5
3.0
0.0
.2
.* .6 Coveroge @
.8
.0
2.0
0.0
,2
.4
.6
.8
Coveroge 8
Figure 3. Calculated work function changes for different coverages of Rb on AI(I I1)(full curve), compared with the measured values a. 5 (broken curve).
Figure 4. The same as in Figure 3 but for K on Mg(000l).
In conclusion, the presented self-consistent calculations which take into account the discrete-lattice structure of the substrate provide a good basis for explanation of the characteristic variations of work function during the process of adsorption of alkali metals on AI(111) and Mg(0001). The calculated depth of AOmi., its position and initial dipole m o m e n t for K and Rb on A I ( l l l ) , are in a good agreement with measured values. To achieve a better agreement for alkali metal adsorption on Mg(0001) a better description of M g substrate would be desirable.
References
Acknowledgements The author wishes to thank Prof J P Perdew for the c o m p u t e r program for selfconsistent calculation of bare metal surface. This work was partially supported by the Polish Ministry of National Education ( M E N ) within the project C P B P 01.08.A2. Financial support from the Alexander von H u m b o l d t F o u n d a t i o n during author's stay at the University of Munich is gratefully acknowledged.
582
t N D Lang, Phys Rev, 134,4234 (1971). 2 A Kiejna and K F Wojciechowski, Progress Surface Sci, I1,293 (1981). 3 A Hohlfeld, Ph D thesis, Freie Universidit, Berlin (1986). 4 A Hohlfeld, M Sunjic and K Horn, J Vac Sei Technol, AS, 679 (1987). 5 A Hohlfeld and K Horn, Surface Sci, 211/212, 844 (1989). 6 N D Lang and W Kohln, Phys Rev, BI, 4555 (1970). 7 N D Lang and W Kohln, Phys Rev, B3, 1215 (1971). s R Monnier and J P Perdew, Phys Rev, BI7, 2595 (1978). 9 j p Perdew, Phys Rev, B25, 6291 (1982). ~0 p A Serena, J M Soler, N Garcia and I P Batra, Phys Rev, B36, 3452 (1987). ~ H F Budd and J Vannimenus, Phys Rev Lett, 31, 1218 (1973); 31, 1430(E) (1973). lz G I Bigun, Ukrahr Fiz Zh, 24, 1313 (1979). ~3 R Monnier, J P Perdew, D C Langreth and J W Wilkins, Phys Rev, B18, 656 (1978). ~4j K Grepstad, P O Gartland and B J Slogsvold, Surface Sci, 57, 348 (1976). 15 B E Hayden, E Schweizer, R K6tz and A M Bradshaw, Surface Sci, 111, 26 (1981).
0042-207X/9053.00 + .00 © 1990 Pergamon Press plc
Printed in Great Britain
W o r k function of K and Rb submonolayers adsorbed on A1(111 ) and Mg(0001 ) A K i e j n a , Institute of Experimental Physics, University of Wroclaw, ul. Cybulskiego 36, 50-205 Wroclaw, Poland and Sektion Physik der Universita't M~inchen, Theresienstr. 37, 8000 M~inchen 2, FRG
The work function variation due to K and Rb atom adsorption on AI(111) and Mg(O001) has been calculated self-consistently within the density functional formalism. To go beyond the jellium model for the substrate the effect of metallic ions on the conduction electrons is represented by a parametrized form of the perturbing potential. The adsorbed alkafi atoms are represented by a jellium slab whose thickness varies with coverage. The computed work functions vs coverage curves, the work function minimum and initial dipole moments show good agreement with recent experimental data on work function change.
The adsorption of alkali metals on metallic substrates leads to characteristic changes in the measured work function. It decreases rapidly at low coverages, reaches a minimum and increases to saturate at coverages greater than one monolayer, reaching the value of the work function of solid adsorbate. The application of the density functional formalism has stimulated a greater understanding of alkali metal adsorption L2. In the pioneering work of Lang ~ both substrate and adsorbate were represented by the positive charge background. The calculations have been performed self-consistently, giving work function changes with a minimum for coverages of O < 1 in agreement with experimental data. The experiments imply that the shape of curves representing the changes of work function vs coverage are more sensitive to the kind of adsorbate than to the substrate 2. Nonetheless, the jellium with r, = 2 (density of positive charge background fi = 3/(4nr3)) does not represent properly such substrates as transition metals for which the experimental data on adsorption was available. For the bare metal surface the work functions calculated with the same formalism are in good agreement with experimental data for higher density metals like AI and Mg. Therefore the recent experimental measurements a-5 of alkali metal adsorption on AI(I 11) and Mg(0001) provide an ideal basis for the test of the theory. We present here a self-consistent calculation of work function changes due to alkali adsorption on AI(111) and Mg(0001) in the spirit of Lang's model. However, in our calculations we take into account the discrete atomic structure of the substrate. The adsorbed atoms are represented by a positively charged slab of jellium whose thickness changes with coverage. To improve agreement of calculated surface energies and work functions of bare surfaces with experimental results, Lang and Kohln 6"7 took into account the difference between the pseudopotentiai of a semi-infinite lattice of ions and the electrostatic potential ofjellium, treating it as a first order perturbation. The limitations of such an approach were discussed by Monnier and Perdew 8. They presented a variational self-consistent method in which the discrete lattice perturbation is treated variationally. In their approach the three-dimensional discrete-lattice potential is replaced by a step potential (with the step in the vicinity of the 580
surface) and added to the external potential which enters the Schrrdinger equation. The step height is varied giving different electron density profiles. The electron density which is the solution to the problem is the one which gives the lowest calculated surface energy. The calculated density profiles show a noticeable variation among different crystal planes. A detailed description of this method is given in ref 8. In the present calculation we make use of the approach of Monnier and Perdew. The discrete lattice potential of the substrate, just as for a bare surface, is replaced by a step potential of the form: V ( x ) = C H ( - x),
(1)
where H(x) is the Heaviside step function and the optimum value of the variational parameter C was determined from the minimum of surface energy of the bare metallic substrate. F o r AI(111) and Mg(0001) surface we have used the values of C given by Perdew 9. The adsorbed alkali atoms are represented by the jellium slab of density fi,~. In Lang's calculation the thickness of the slab was chosen to have the fixed value dr, of the interplanar spacing of the most densely packed planes in the bulk alkali metals. However, during the process of adsorption there is a transition from an ionic to atomic character, of adsorption at higher coverages. This leads to the change in the distance between the first substrate lattice plane and alkali adatoms. The analysis of experimental data a'5 implies that the bonding distance of alkali adatoms varies between the ionic bond length dio, (twice the the ionic radius) and d m (interplanar distance in the bulk alkali) but is closer to d~o.. These facts suggest approximating the overlayer thickness by the linear relation 1° d ( O ) = dio. +
O(drn
--
dio,)p
(2)
where ® is the coverage. F o r the calculations presented in this work we took p = 1.0, but we have made also some trial computations with p = 0.5 which gives a value of d(®) closer to dio.. The height of the jellium slab was also varied with coverage. The electron density profiles n(x), were determined from the self-consistent solution of the K o h n - S h a m equations with the V(x) term included in the effective potential. The exchange-
A Kiejna: Work function of K and Rb submonolayers
correlation potential is treated in the local density approximation with the Wigner expression for the correlation energy. The jellium slab of adsorbed alkali metal modifies the electrostatic potential q~(x) which is calculated from the integral equation 8 and iterated along with the effective potential. The self-consistency procedure led to the satisfied charge neutrality condition*
fff[n(x)-n+(x)]dx--O
(3)
1.2"
1.0
-"
.a"
X
\ ........ !
~
i
.6"
r .4
and a well-converged work function and surface dipole moment
.2
Ad~ = 4 ~
x[n(x)
- n÷(x)]dx.
(4) 0.0
Here n ÷ (x) = h H ( - x) + h,d[H(x) - H ( x - d)]. Another criterion of self-consistency imposed was the Budd-Vannimenus theorem ~~.a, modified by Bigun 12 for the case of an adsorbate: 4,(0) -
4,( -
oo) + [,t(d)
[3
-
s Io is x ( o . u.) Figure 1. Electron density profiles for the jellium Slab representing a full monolayer of K atoms absorbed on jellium (r, = 2.07) and real AI(I l 1) surface (full line). -Io
-s
o
4,(O)l~,~.~/ff
= iid ~ eF (h) + e ~ ( ~ )
]/
d h + C n (O)fii
(5) .S N\
where er is the Fermi energy and e~ is the exchange-correlation energy per particle in an uniform electron gas of density ~i. The electron density profiles for the alkali slabs on A l ( l l l ) Mg(0001) substrates calculated by us differ from those calculated for alkali slabs on jellium substrate. The difference is illustrated in Figure 1 where the density profiles for K-AI(I 11) and K-AI(jellium) are drawn for comparison. As is seen for the jellium substrate the density profile is more diffuse and the first peak in the Friedel oscillations of the electron density is much reduced, compared to the real metal substrate. For the bare metal surface the work functions were calculated from the change in self-consistent-field expression~ 3: da (I)=--
dE
1.0
<~ I
.5
\\\
2.0
2.5
0.0
.2
.4 .6 Covero9e 0
.a
1.0
Figure 2. Work function change for different coverages of K on AI(I 1I). The full curve was calculated for p = 1.0 whereas the broken one for p = 0.5 (see equation 2). The dotted curve shows the experimental data 3.
(6)
where the derivative of the surface energy tr with respect to the surface charge density E was evaluated at E = 0. The calculated work functions for the bare AI(111) and Mg(0001) are 4.2 eV and 4.0eV respectively, in a good agreement with the measured values which are 4.24eV for A I ( l l l ) 14 and 3.84eV Is for Mg(0001). The change in the work function for different adatom coverage is calculated from the change in the dipole moment. In Figures 2-4 we present the calculated changes in the work function AO vs coverage O, for potasium and rubidium atoms on A I ( l l l ) and K-Mg(0001). In Figure 2 the dependence AO(O) is presented for K-AI(111) for two different choices of parameter p appearing in (2). The broken curve is for p = 0.5 while the full curve is for p = 1.0, with dr, = 7.13 au and d~o. = 5.03 au. As one can see better agreement with experimental data 3 is achieved for the choice p = 1.0. For this curve the work function attains a minimum a t O m i n "~' 0.5 and A(l)mi n = 2.39 eV is in a good agreement with the experimentally measured ®m~, = 0 . 4 2 and AOm~. = 2.17 eV. The inital dipole moment (the slope of the A~(®) curve to zero coverage) obtained for our curve is 3.9 debye (D) in very good agreement with experimental value 4.1 D. *Atomic units (au) are used unless otherwise stated.
~
The results for changes in work function for R b - A I ( l l l ) are displayed in Figure 3, for d,, = 7.51 au, dis. = 5.56 au and p = 1.0 used in (2). The overall shape of the calculated AS vs (9 curve agrees well with the experimental curve 5. Compared to the measured value (®m~, = 0.42) the position of AOm~, is, similar to K-AI(111), shifted towards higher coverages and gives Omi,. "" 0.5. The calculated value of AOm=. is 2.52 eV compared to the measured of 2.25 eV. The calculated initial dipole moment, 4.0 D is in very good agreement with the experimental value of 4.1 D. For higher coverages the experimental and theoretical curves agree well. For the K-Mg(0001) system, the changes in the work function vs coverage are presented in Figure 4. The agreement between the calculated and experimental curve 3 is worse compared to that for the AI substrate. The calculated A~m~. = 1.76 eV appears for ®,,5. ~- 0.45, whereas the corresponding measured value for AOm~n= 1.49 eV at ~m~, = 0.35. The calculated initial dipole moment is 2.7 D compared to experimental 3.8 D. The worse agreement for K-Mg(0001) is presumably connected with the less accurate determination of the surface properties of the bare Mg in the variational self-consistent method of Monnier and Perdew. This is suggested for example by the difference between the calculated and measured work function for the bare Mg(0001) surface. 581
A Kiejna." Work function of K and Rb submonolayers .0
.0
\k
.5
\\\
.5 1.0
03
~ 1.5
\kk\k\
1.0
\\
e .4
2.(1
1.5 2.5
3.0
0.0
.2
.* .6 Coveroge @
.8
.0
2.0
0.0
,2
.4
.6
.8
Coveroge 8
Figure 3. Calculated work function changes for different coverages of Rb on AI(I I1)(full curve), compared with the measured values a. 5 (broken curve).
Figure 4. The same as in Figure 3 but for K on Mg(000l).
In conclusion, the presented self-consistent calculations which take into account the discrete-lattice structure of the substrate provide a good basis for explanation of the characteristic variations of work function during the process of adsorption of alkali metals on AI(111) and Mg(0001). The calculated depth of AOmi., its position and initial dipole m o m e n t for K and Rb on A I ( l l l ) , are in a good agreement with measured values. To achieve a better agreement for alkali metal adsorption on Mg(0001) a better description of M g substrate would be desirable.
References
Acknowledgements The author wishes to thank Prof J P Perdew for the c o m p u t e r program for selfconsistent calculation of bare metal surface. This work was partially supported by the Polish Ministry of National Education ( M E N ) within the project C P B P 01.08.A2. Financial support from the Alexander von H u m b o l d t F o u n d a t i o n during author's stay at the University of Munich is gratefully acknowledged.
582
t N D Lang, Phys Rev, 134,4234 (1971). 2 A Kiejna and K F Wojciechowski, Progress Surface Sci, I1,293 (1981). 3 A Hohlfeld, Ph D thesis, Freie Universidit, Berlin (1986). 4 A Hohlfeld, M Sunjic and K Horn, J Vac Sei Technol, AS, 679 (1987). 5 A Hohlfeld and K Horn, Surface Sci, 211/212, 844 (1989). 6 N D Lang and W Kohln, Phys Rev, BI, 4555 (1970). 7 N D Lang and W Kohln, Phys Rev, B3, 1215 (1971). s R Monnier and J P Perdew, Phys Rev, BI7, 2595 (1978). 9 j p Perdew, Phys Rev, B25, 6291 (1982). ~0 p A Serena, J M Soler, N Garcia and I P Batra, Phys Rev, B36, 3452 (1987). ~ H F Budd and J Vannimenus, Phys Rev Lett, 31, 1218 (1973); 31, 1430(E) (1973). lz G I Bigun, Ukrahr Fiz Zh, 24, 1313 (1979). ~3 R Monnier, J P Perdew, D C Langreth and J W Wilkins, Phys Rev, B18, 656 (1978). ~4j K Grepstad, P O Gartland and B J Slogsvold, Surface Sci, 57, 348 (1976). 15 B E Hayden, E Schweizer, R K6tz and A M Bradshaw, Surface Sci, 111, 26 (1981).