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A model pseudopotential calculation of the electronic structure of Si (111)-1 × 1 surface
Solid State Communications, Vol. 33, pp. 1209—1212. Pergamon Press Ltd. 1980. Printed in Great Britain. A MODEL PSEUDOPOTENTIAL CALCULATION OF THE ELECTRONIC STRUCTURE OF Si (111)—! x 1 SURFACE G.P. Srivastava Physics Department, New University of Ulster, Coleraine, N. Ireland (Received 14 November 1979 by F. Bassani) We report a model pseudopotential calculation of the electronic structure of Si (11 1)—I x 1 unrelaxed, relaxed and expanded surfaces. It is shown that our results for the surface states on these faces are in good agreement with results from other self-consistent pseudopotential and semi-empirical tight-binding calculations. for Si~4is the starting point in our calculation. SchlUter eta!. [2] have used a local, “on Fermi sphere” approximation to fit the Fourier transform to the following expression
TO DATE there are a number of theoretical approaches available to calculate the electronic structure of semiconductor surfaces (see Schlüter [1] for references). Although the main features regarding bona fide surface states and resonances on such faces from almost all theoretical approaches are similar, their details differ.
v 10~(q)=
One of[21, theismost successful approaches, due to Schlüter eta!. the application of the pseudopotential method to a periodic slab configuration perpendicular to the surface (forming “supercells”). These authors have asserted that for such a periodic system there is a strong case for making the pseudopotential calculation self-consistent in the sense of electronic response to a given structural model. However, these authors [2—41 point out that self-consistency is hard to achieve and in some cases not very satisfactory. The need for self-consistency in such calculations owes much to the fact that while the empirical pseudopotential method for bulk calculations [5] demands fitting atomic form factors (pseudopotential) corresponding only to a few reciprocal lattice vectors (q), in the case of “supercells” many more atomic form factors are to be included. Form factors for short q are crucial foryield surface state calculations and must be chosen so as to good results for the work-function and ionization potential for the material under study. The need for self-consistency can therefore be relaxed [6] or regarded, to some degree, less important if a reasonably “good” pseudopotential is employed in a surface state calculation. Such an argument is also presented by Flores
0.992 2 [cos (0.791q) 0.352] e~0I8Q4(1) q which is normalised to an atomic volume in the supercell comprising 12 atomic layers plus a gap as explained in their paper. We screen the ionic potential in equation (1) by the Heine—Abarenkov [10] dielectric factor HA(q) to get our model pseudopotential —
—
v~ 0~(q) vmo~e1(q)= eHA(q)
(2)
Figure 1 gives a comparison of Vmo~ with the empirical pseudopotential Vemp used by Schluter eta!, as their starting atomic potential. We see that Vm(yJel is deeper than Vemp for short q and Vm(gjeI (q = 0) is close to the value reported by Cohen and Heine [11]. In the usual local pseudopotential method one has to solve the deterininant (in Ry units) 2 —Efl(k)}SGG’+ V(G—G’)I = 0 (3) detl{(k+G) where n represents the band index, k is the electron wave vector in the first Brillouin zone, and the pseudopotential coefficient V(G G’) is given by —
V(G
—
G’)
=
V(q)
=
S(q)v~~~ 1(q)
eta!. [7]. With the above arguments in view we have repeated the slab calculation for Si (111)—i x 1 surfaces (ideal, relaxed and expanded) with a model pseudopotential. The results of our calculation are compared with the self-consistent pseudopotential calculation of Schluter et aL [2] and also with other calculations [7—91. The Heine—Abarenkov [101 ionic model potential
(4)
with 5(q)
=
~ e~
(5)
1209
as the structure factor describing theM atomic positions R1 in the supercell. The real space potential corresponding to V(q) is
1210
ELECTRONIC STRUCTURE OF Si (111)—i xl SURFACE
Vol. 33, No. 12
0.2 0.2 CI 0
2
3
4 —0.2
—0 I —0.2
a
-042
U
Si Si
Si Si
Si Si
11
11
-06 >~
—0.4
> —0.8
/
-0.5
-I.0
,/
—0.6 —0.7
12
10
8
6
z, Fig. 1. The model pseudopotential V~~gjel for Si used in the present calculations. For comparison is plotted the starting empirical pseudopotentia1~v~~ of Vemp Schlulter 0moiel and are et aL Vemp. Both normalised to the supercell volume as explained in Schlüter eta!. — — —
Table 1. Dangling bond surface state energies on the unrelaxed Si (111)—i x 1 face. Energies in eV refer to the top ofthe bulk valence band edge k 11
Present
AH [13]
Flores eta!. Schluter eta!. [7] [2]
F
0.3 0.61
0.56
M
0.2 0.03 Width 0.1 0.64
—0.01 0.58
No dispersion observed Nil
1~. (6) V(r) = ~ci V(q) e Figure 2 gives V(z) averaged parallel to the unrelaxed surface and plotted as a function of the coordinate z perpendicular to the surface. We see that V(z) rises quite high at the outermost atomic layer and gives an ionization potential of about 4eV which is in good agreement* with the experimental value of 5.15 eV [12]. It can also be noticed that V(z) compares well with the fully self-consistent potential of Sch.lüter eta!. We have solved the matrix in equation (3) by considering plane waves exactly up to a cut off in energy at 3.5 Ry and including more plane waves up to an energy cut off at 4.2 Ry via Löwdin’s perturbation method [111. An effort to include more plane waves, up to a cut off at 6 Ry, via Löwdin’s method did not make _________________
*
As remarked by Schlüter et a!. ionization potentials are difficult to detennine precisely by this method. The calculated value is only approximate ( 1 eV).
4
2
0
~
Fig. 2. Plot of the real space pseudopotential V(z) averaged parallel to the unrelaxed (Ill) surface of Si. zthe = 0 is the centre of the slab geometry. The top of bulk valence-band edge is indicated by E~.An ionization potential of about 4eV is calculated. significant changes in our results at I’, M and K points). The computed band structures for the unrelaxed and relaxed surfaces are shown in Figs. 3 and 4 respectively. As displayed in Table 1 our results for the unrelaxed case show no dispersion and are in complete agreement with Schluter eta!. but disagree with the findings of Appelbaum and Hamann [13] and Flores et al. [7] who observe a dangling-bond width of 0.64 and 0.58 eV respectively. With relaxation of surface atoms [2] the dangling-bond band becomes dispersive, though we observe less dispersion than Schluter et a!. [2] Pandey— Phillips [9] and Appelbaum—Hamann [13]. Figure 5 geometry. results andrelaxed those gives a plotThe of difference the densitybetween of statesour curve for the of Schluter et aL should be accounted for by the selfconsistency in their calculation (Table 2). in Table 3 we have compared our band structure for surface states on the relaxed and expanded Si (111)—i x I surfaces with those of Appelbaum and Hamann [8].Our results compare reasonably well with Appelbaum—Hamann (AH) for the relaxed geometry, but lie significantly lower for the expanded case. We conclude that the use of a “good” model pseudopotential, or for that matter, a better empirical pseudopotential can yield satisfactory results in surface state calculations, for semiconductor faces, when applied to a slab geometry and that self-consistency may be regarded as less important than previously quoted. The results presented here for relaxed and expanded Si (111)—i x 1 ,
surfaces should prove helpful in understanding the electronic structure and the photoelectron spectroscopy of a realistic 2 x I surface. The importance of these studies is further strengthened by the observation [14] that the
Vol. 33, No.12
ELECTRONICSTRUCFUREOFSi(III)—1 xl SURFACE
4
1211
~0 C
~ 0 —2 ~
-4
~ —6 —8
—12 —10 —8
—4 E,
-10 —12
r
M
—2
0
2
4
eV
Fig. 5. Density of states curve for the Si (111)—I x I relaxed geometry of Schlüter et al. Prominent surface states are indicated by arrows. E= 0 is the top of the bulk valence-band edge. The Fermi level lies at 0.5 eV.
K
Fig. 3. Two-dimensional band structure of a slab of 12 atomic layers plus a gap (unrelaxed surface model of Schluter et aL). The energy referred to the top of the bulk valence-band edge is plotted as a function of k11 ifl the irreducible two-dimensional surface Brillouin zone. The labeffing follows Schluter et al. _______________________________ 4 2
—6
~
Table 2. Calculatedsurface state energies on a relaxed Si (111)—I x 1 face. The surface geometry and the surface Brillouin zone labelling are as given in Schlüter et al. Energies are in eV and refer to the top ofthe bulk va!ence-band edge k11
Present cal.
Ref. [2]
Ref. [9]
Ref. [13]
1’
0.6 —1.4 —12.6 0.2 —3.0 0.4 —4.8 7.8 93
1.2 —1.5 —12.7 0.5 —3.1 0.5 —4.2 —8.5 —9.8
1.04 —1.71 —12.9 0.17 —3.78 0.11 —5.65 —8.35 —9.6
0.88 —1.95 —12.87 0.04 —3.55
0 —2
M
>
K
LU
—6 —8 —10 —12 —14
r
M
K
I
Fig. 4. Same as Fig.3 for the relaxed geometry of Schluter eta!, transformation upon annealing of the metastable 2 x 1 surface to a stable 7 x 7 structure takes place via a I x I phase transition. The intermediate phase with a 1 x 1 structure can be impurity stabilised and we have found that the present calculations can explain recent angle-resolved UPS data [15] for Si (111)—Al with Al coverage of less than 10% of an monolayer, 3atomcm2. making less than 7 xlO’ Acknowledgements I would like to express my sincere thanks to Drs R.ll. Williams and M. Schluter for encouragement and very useful discussions at various stages of this work. —
Table 3. Calculated dangling dond surface state energies on the relaxed and expanded Si (111)—i x 1 faces as considered by Appelbaum and Hamann [8]. Energies in eV are referred to the top of the bulk valence-band edge Re
Relaxed
All [8]
Present
0.8 0.01 0.6 0.24
0.42 0.06 0.25 0.32 0.18
Expanded k11 F M ~K K
Present 0.03 11.03 .0 —0.87 0.77 — —
—
AH [8] 0.26 0.28 —0.06 —0.18 —
1212
ELECTRONICSTRUCTUREOFSi(lIl)—1 xl SURFACE REFERENCES
10.
M. Schluter, Festkörperprobleme (Adv. Solid State Phys.) 18, 155 (1978). 2. M. Schluter, J.R. Chelikowsky, S.G. Louie & M.L. Cohen,Phys. Rev. B12, 4200 (1975). 3. J.R. Chelikowsky,Phys. Rev. B16, 3618 (1977). 4. H.I. Zhang&M. SchlUter,Phys. Rev. B18, 1923 (1978). 5. M.L. Cohen,&T.K. Bergstresser,Phys. Rev. 141,789 (1966). 6. M. SchlUter (private communication). 7. F. Flores, F. Garcia-Moliner, E. Louis & C. Tejedor,J. Phys. C9, L429 (1976). 8. J.A. Appelbaum & D.R. Hamann, Phys. Rev. Bl 2 1410 (1975). 9. K.C. Pandy & J.C. Phillips, Phys. Rev. Lett. 32, 1433, (1974).
ii.
1.
12. 13. 14. 15.
Vol. 33, No.12
V. Heine & IV. Abarenkov, Phil. Mag. 9,451 (1964). M.L. Cohen & V. Heine, Solid State Physics (Edited by H. Ehrenreich, F. Seitz & D. Tumbull). Vol. 24, p. 37. Academic Press, New York (1970). J.C. Riviere, Solid State Surface Science (Edited by M. Green), p. 179. Marcell Dekker, New York (1969). J. Appelbaum & D.R. Hamann,Phys. Rev. Lett. 31,106(1973). W. Monch.FestkOrperprobleme (Advances in Physics) (Edited by H.J. Queisser),XII, p. 241. Pergamon Vieweg, New York (1973). RB. Williams. A.W. Parke, A. McKinley & G.P. Srivastava (to be published).
TO DATE there are a number of theoretical approaches available to calculate the electronic structure of semiconductor surfaces (see Schlüter [1] for references). Although the main features regarding bona fide surface states and resonances on such faces from almost all theoretical approaches are similar, their details differ.
v 10~(q)=
One of[21, theismost successful approaches, due to Schlüter eta!. the application of the pseudopotential method to a periodic slab configuration perpendicular to the surface (forming “supercells”). These authors have asserted that for such a periodic system there is a strong case for making the pseudopotential calculation self-consistent in the sense of electronic response to a given structural model. However, these authors [2—41 point out that self-consistency is hard to achieve and in some cases not very satisfactory. The need for self-consistency in such calculations owes much to the fact that while the empirical pseudopotential method for bulk calculations [5] demands fitting atomic form factors (pseudopotential) corresponding only to a few reciprocal lattice vectors (q), in the case of “supercells” many more atomic form factors are to be included. Form factors for short q are crucial foryield surface state calculations and must be chosen so as to good results for the work-function and ionization potential for the material under study. The need for self-consistency can therefore be relaxed [6] or regarded, to some degree, less important if a reasonably “good” pseudopotential is employed in a surface state calculation. Such an argument is also presented by Flores
0.992 2 [cos (0.791q) 0.352] e~0I8Q4(1) q which is normalised to an atomic volume in the supercell comprising 12 atomic layers plus a gap as explained in their paper. We screen the ionic potential in equation (1) by the Heine—Abarenkov [10] dielectric factor HA(q) to get our model pseudopotential —
—
v~ 0~(q) vmo~e1(q)= eHA(q)
(2)
Figure 1 gives a comparison of Vmo~ with the empirical pseudopotential Vemp used by Schluter eta!, as their starting atomic potential. We see that Vm(yJel is deeper than Vemp for short q and Vm(gjeI (q = 0) is close to the value reported by Cohen and Heine [11]. In the usual local pseudopotential method one has to solve the deterininant (in Ry units) 2 —Efl(k)}SGG’+ V(G—G’)I = 0 (3) detl{(k+G) where n represents the band index, k is the electron wave vector in the first Brillouin zone, and the pseudopotential coefficient V(G G’) is given by —
V(G
—
G’)
=
V(q)
=
S(q)v~~~ 1(q)
eta!. [7]. With the above arguments in view we have repeated the slab calculation for Si (111)—i x 1 surfaces (ideal, relaxed and expanded) with a model pseudopotential. The results of our calculation are compared with the self-consistent pseudopotential calculation of Schluter et aL [2] and also with other calculations [7—91. The Heine—Abarenkov [101 ionic model potential
(4)
with 5(q)
=
~ e~
(5)
1209
as the structure factor describing theM atomic positions R1 in the supercell. The real space potential corresponding to V(q) is
1210
ELECTRONIC STRUCTURE OF Si (111)—i xl SURFACE
Vol. 33, No. 12
0.2 0.2 CI 0
2
3
4 —0.2
—0 I —0.2
a
-042
U
Si Si
Si Si
Si Si
11
11
-06 >~
—0.4
> —0.8
/
-0.5
-I.0
,/
—0.6 —0.7
12
10
8
6
z, Fig. 1. The model pseudopotential V~~gjel for Si used in the present calculations. For comparison is plotted the starting empirical pseudopotentia1~v~~ of Vemp Schlulter 0moiel and are et aL Vemp. Both normalised to the supercell volume as explained in Schlüter eta!. — — —
Table 1. Dangling bond surface state energies on the unrelaxed Si (111)—i x 1 face. Energies in eV refer to the top ofthe bulk valence band edge k 11
Present
AH [13]
Flores eta!. Schluter eta!. [7] [2]
F
0.3 0.61
0.56
M
0.2 0.03 Width 0.1 0.64
—0.01 0.58
No dispersion observed Nil
1~. (6) V(r) = ~ci V(q) e Figure 2 gives V(z) averaged parallel to the unrelaxed surface and plotted as a function of the coordinate z perpendicular to the surface. We see that V(z) rises quite high at the outermost atomic layer and gives an ionization potential of about 4eV which is in good agreement* with the experimental value of 5.15 eV [12]. It can also be noticed that V(z) compares well with the fully self-consistent potential of Sch.lüter eta!. We have solved the matrix in equation (3) by considering plane waves exactly up to a cut off in energy at 3.5 Ry and including more plane waves up to an energy cut off at 4.2 Ry via Löwdin’s perturbation method [111. An effort to include more plane waves, up to a cut off at 6 Ry, via Löwdin’s method did not make _________________
*
As remarked by Schlüter et a!. ionization potentials are difficult to detennine precisely by this method. The calculated value is only approximate ( 1 eV).
4
2
0
~
Fig. 2. Plot of the real space pseudopotential V(z) averaged parallel to the unrelaxed (Ill) surface of Si. zthe = 0 is the centre of the slab geometry. The top of bulk valence-band edge is indicated by E~.An ionization potential of about 4eV is calculated. significant changes in our results at I’, M and K points). The computed band structures for the unrelaxed and relaxed surfaces are shown in Figs. 3 and 4 respectively. As displayed in Table 1 our results for the unrelaxed case show no dispersion and are in complete agreement with Schluter eta!. but disagree with the findings of Appelbaum and Hamann [13] and Flores et al. [7] who observe a dangling-bond width of 0.64 and 0.58 eV respectively. With relaxation of surface atoms [2] the dangling-bond band becomes dispersive, though we observe less dispersion than Schluter et a!. [2] Pandey— Phillips [9] and Appelbaum—Hamann [13]. Figure 5 geometry. results andrelaxed those gives a plotThe of difference the densitybetween of statesour curve for the of Schluter et aL should be accounted for by the selfconsistency in their calculation (Table 2). in Table 3 we have compared our band structure for surface states on the relaxed and expanded Si (111)—i x I surfaces with those of Appelbaum and Hamann [8].Our results compare reasonably well with Appelbaum—Hamann (AH) for the relaxed geometry, but lie significantly lower for the expanded case. We conclude that the use of a “good” model pseudopotential, or for that matter, a better empirical pseudopotential can yield satisfactory results in surface state calculations, for semiconductor faces, when applied to a slab geometry and that self-consistency may be regarded as less important than previously quoted. The results presented here for relaxed and expanded Si (111)—i x 1 ,
surfaces should prove helpful in understanding the electronic structure and the photoelectron spectroscopy of a realistic 2 x I surface. The importance of these studies is further strengthened by the observation [14] that the
Vol. 33, No.12
ELECTRONICSTRUCFUREOFSi(III)—1 xl SURFACE
4
1211
~0 C
~ 0 —2 ~
-4
~ —6 —8
—12 —10 —8
—4 E,
-10 —12
r
M
—2
0
2
4
eV
Fig. 5. Density of states curve for the Si (111)—I x I relaxed geometry of Schlüter et al. Prominent surface states are indicated by arrows. E= 0 is the top of the bulk valence-band edge. The Fermi level lies at 0.5 eV.
K
Fig. 3. Two-dimensional band structure of a slab of 12 atomic layers plus a gap (unrelaxed surface model of Schluter et aL). The energy referred to the top of the bulk valence-band edge is plotted as a function of k11 ifl the irreducible two-dimensional surface Brillouin zone. The labeffing follows Schluter et al. _______________________________ 4 2
—6
~
Table 2. Calculatedsurface state energies on a relaxed Si (111)—I x 1 face. The surface geometry and the surface Brillouin zone labelling are as given in Schlüter et al. Energies are in eV and refer to the top ofthe bulk va!ence-band edge k11
Present cal.
Ref. [2]
Ref. [9]
Ref. [13]
1’
0.6 —1.4 —12.6 0.2 —3.0 0.4 —4.8 7.8 93
1.2 —1.5 —12.7 0.5 —3.1 0.5 —4.2 —8.5 —9.8
1.04 —1.71 —12.9 0.17 —3.78 0.11 —5.65 —8.35 —9.6
0.88 —1.95 —12.87 0.04 —3.55
0 —2
M
>
K
LU
—6 —8 —10 —12 —14
r
M
K
I
Fig. 4. Same as Fig.3 for the relaxed geometry of Schluter eta!, transformation upon annealing of the metastable 2 x 1 surface to a stable 7 x 7 structure takes place via a I x I phase transition. The intermediate phase with a 1 x 1 structure can be impurity stabilised and we have found that the present calculations can explain recent angle-resolved UPS data [15] for Si (111)—Al with Al coverage of less than 10% of an monolayer, 3atomcm2. making less than 7 xlO’ Acknowledgements I would like to express my sincere thanks to Drs R.ll. Williams and M. Schluter for encouragement and very useful discussions at various stages of this work. —
Table 3. Calculated dangling dond surface state energies on the relaxed and expanded Si (111)—i x 1 faces as considered by Appelbaum and Hamann [8]. Energies in eV are referred to the top of the bulk valence-band edge Re
Relaxed
All [8]
Present
0.8 0.01 0.6 0.24
0.42 0.06 0.25 0.32 0.18
Expanded k11 F M ~K K
Present 0.03 11.03 .0 —0.87 0.77 — —
—
AH [8] 0.26 0.28 —0.06 —0.18 —
1212
ELECTRONICSTRUCTUREOFSi(lIl)—1 xl SURFACE REFERENCES
10.
M. Schluter, Festkörperprobleme (Adv. Solid State Phys.) 18, 155 (1978). 2. M. Schluter, J.R. Chelikowsky, S.G. Louie & M.L. Cohen,Phys. Rev. B12, 4200 (1975). 3. J.R. Chelikowsky,Phys. Rev. B16, 3618 (1977). 4. H.I. Zhang&M. SchlUter,Phys. Rev. B18, 1923 (1978). 5. M.L. Cohen,&T.K. Bergstresser,Phys. Rev. 141,789 (1966). 6. M. SchlUter (private communication). 7. F. Flores, F. Garcia-Moliner, E. Louis & C. Tejedor,J. Phys. C9, L429 (1976). 8. J.A. Appelbaum & D.R. Hamann, Phys. Rev. Bl 2 1410 (1975). 9. K.C. Pandy & J.C. Phillips, Phys. Rev. Lett. 32, 1433, (1974).
ii.
1.
12. 13. 14. 15.
Vol. 33, No.12
V. Heine & IV. Abarenkov, Phil. Mag. 9,451 (1964). M.L. Cohen & V. Heine, Solid State Physics (Edited by H. Ehrenreich, F. Seitz & D. Tumbull). Vol. 24, p. 37. Academic Press, New York (1970). J.C. Riviere, Solid State Surface Science (Edited by M. Green), p. 179. Marcell Dekker, New York (1969). J. Appelbaum & D.R. Hamann,Phys. Rev. Lett. 31,106(1973). W. Monch.FestkOrperprobleme (Advances in Physics) (Edited by H.J. Queisser),XII, p. 241. Pergamon Vieweg, New York (1973). RB. Williams. A.W. Parke, A. McKinley & G.P. Srivastava (to be published).