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Electronic structure of lithium interstitials in Si
Solid State Communications, Vol. 37, pp. 341—343. Pergamon Press Ltd. 1981. Printed in Great Britain. ELECTRONIC STRUCTURE OF LITHIUM INTERSTITIAI~SIN Si* B. Szablak Department of Physics, University of Illinois, Urbana, IL 61801, U.S.A. M. Altarelli Department of Physics and Materials Research Laboratory, University of Illinois, Urbana, IL 61801, U.S.A. and Max-Planck—Institut für Festkorperforschung, Heisenbergstr. 1, 7000 Stuttgart 80, Fed. Rep. of Germanyt (Received l5May 1980 byM Cardona) The valley—orbit split ground state manifold of interstitial U donors in Si is characterized by very shallow levels and by an anomalous level order. A calculation based on a non-local pseudopotential description of the Li ion and on the inclusion of site-dependent umklapp terms in the intervalley interaction provided a quantitative explanation of the observed spectrum. AMONG shallow donors in Si, interstitial Li is remarkable for several features of its electronic ground state experimental ionization energy [1] of 32.8 meV, to be compared with [2] 45 meV for substitutional P, 53meV for As, 42 for Sb. Also, the usual valley—orbit pattern, with an s-like (A1) ground state and p- and
tentials is particularly appealing, because it does not suffer from the compatibility problems that arise for substitutional impurities. Indeed, the host pseudoBloch functions behave as “true” functions near the U core, irrespective of the choice of the host pseudopo-
d-like (7’2 and E respectively) excited states, is reversed for Li, and the A1 state is at 31.0meV, 1.8 meV above the degenerate ground state [1] (experiments cannot resolve any T2—E splitting). This is at odds not only with the other shallow donor patterns, but also with the ordering of the 2s and 2p levels in the Li atom, which must have the same nodal structure of the A1 and T2 states in Si. Furthermore, recent work [3] has shown that a purely point-charge impurity at an ainterstitial position in diamond semiconductorshas deep s ground state, as a result of a very strong attractive valley—orbit interaction, in agreement with experimental results on electrons trapped by positive muons [4] Therefore, .
tential. It is therefore legitimate to obtain the Si Bloch functions from Chelikowsky and Cohen’s empirical pseudopotential [8], and describe the bare Li~ion by a non-local pseudopotential of the Heine-—Abarenkov type [9]. The bare U~potential can then be described as V B1~r)
—
VLB~ ~J)
B VNL~T
,
where 2~ VBr\ e L’. I is the ordinary local bare Coulomb potential, and — —
2
2
V~L(r)= ~
—A 1 F,
the shallow ground state of Li must be a consequence of the deviations of the Li + potential from the pure Coulomb form. It is therefore of interest, partly as a test of recent progress in our understanding of donor states in multi-valley semiconductors [4—7], to see if a better description of the short-range Li + potential in Si can reproduce the observed level spectrum. In the present work, a pseudopotential description of the Li ion is adapted. It is worth remarking that for interstitial impurities the use of emperical pseudopo-
1=0 =
‘2
—
0
r
T
r > RM.
Here the A,’s are constants, the P,’s are angular momentum projection operators and RM is a limit radius. The parameters A, (1 = 0, 1, 2) and RM were fitted to atomic data and tabulated by Animalu and Heine [10], and for Li they are (in a.u.): A0 = 0.336, A1 = 0.504, A2 = 0.455, RM = 2.8. The physics implied by equation (1)—(3) is that the Li~ion appears as a point charge to states with 1> 3, as shown by the perfectly hydrogenic spectrum for JL and higher 1 states of the U atom, whereas appreciable deviations appear for more penetrating d- and p-states, as well as for s-states, for which
*Work at the University of illinois supported by NSF under grant DMR7~24155. tPresent address. 341
342
ELECTRONIC STRUCTURE OF LITHIUM INTERSTITIALS IN Si
Vol. 37, No.4
Table 1. Top: renormalization factors (as defined in [3] }for intervalley matrix elements of the Lit, s-, p-and d-wave pseudopotential [equation (7)J andfor a point c/large, at the interstitial position. Bottom, theoretical and experimental [1] binding energies (in meV) of the A1, T2 and E Li ground state multiplet components Valley pair
Li s(A1)
Li p(T2)
Li d(E)
~:~:
0.65
0.58
0.22
1.23
—0.32
0.68
0.39
1.08
~
Binding energy (th) (exp)
29.6 31.0
31.2 32.8
a strong orthogonality correction is also included in A0. The bare potential, equation (1), is screened when introduced into the Si crystal. We apply here the prescription of Animalu [111 for screening of non-local potentials, according to which, to the bare potential VB, equation (1), corresponds the screened potential Vs given by (in momentum space): s V (q) VLs(q) + VN~(q) (4) — —
where \ B1 \f ~ \ / \ V~~q) V~~q,/e~q,+1~q,
f~\
—
and Vs NL(~)
= —
B
j
~
(6)
Thus, the non-local portion of the screened potential is identical to the bare one, and the local portion corresponds to screening the local bare potential as usual, by the dielectric function [121 c(q), and adding 1(q), a (local!) correction, resulting from the charge rearrangement produced by VI~L,which can be numerically evaluated. We neglect this contribution here, because it is small in a uniform electron gas, and even more so in Si, where the Li ion sits in the tetrahedral interstitial position, a site with very low valence charge density. In fact, when the non-uniformity of the Si valence charge density (or in a different language, the local-field effect) is included [13] a substantial reduction in the effectiveness of screening is obtained for a potential with origin at the interstitial site. We therefore approximate: ,
S
B
B
V (q) = VL (q)/c(q) + VNL(q). (7) This is the potential used in the calculation of the energy levels. Since the A1, E and T2 components of the ground state multiplet have s- d-, and p-like syrnmetiy about the Li site, respectively, the corresponding angular momentum component in V~ is effective in each case.
Coulomb
30.7 32.8
When equation (7) is used as input in a multivalley calculation, according to the procedure of [6] the second, short range term drastically affects the large momentum, intervalley matrix elements. In Table 1 we list the dimensionless renormalization factor [3, 6], which represents the scaling of the intervalley matrix element magnitude with respect to the C = 0, pure Coulomb matrix element, due to the extra term in equation (7) and to consideration of all umklapp cornponents G ~0 in the conduction band wave-functions. The latter are obtained, as already mentioned, by a pseudopotential calculation [81with 88 plane waves. Notice the sign reversal of the s-wave matrix element, corresponding to a repulsive intervalley interaction, deriving from the orthogonalization contribution from the Li~s-core. The resulting binding energies are also ,
listed in Table 1. It is important to notice how the A1 level, which is deep for the interstitial Coulomb potential, is now pushed up above the T2 and E levels in agreement with the experimental results, because of the repulsive character of the valley—orbit interaction. It is however necessary to take the results of Table 1 with some caution, as the uncertainties involved in the calculation exceed the A1—7’2 splitting, which is a small fraction of the binding energy. Nonetheless, it is fair to say that the calculated binding energies for A1, T2 and E, all lie in the neighborhood of 30 meV, with a 10% uncertainty, as established by some numerical experimentation. In conclusion, it was shown that a non-local pseudopotential description of the screened Li~potential in Si, and the inclusion of multivalley effects account for the very shallow levels of this interstitial donor in Si, and clearly identify the physical origin of the anomalous level ordening. -~
Acknowledgement One of us (M. A.) is grateful to Professor V. Heine for an enhghtning discussion. -
.
.
.
.
Vol. 37, No.4
ELECTRONIC STRUCTURE OF LITHIUM INTERSTITIALS IN Si REFERENCES
1. 2. 3. 4. 5. 6. 7.
R. L. Aggarwal, P. Fischer, V. Mourzine & A. K. Ramdas,Phys. Rev. 138, A882 (1965). R. L. Aggarwal & A. K. Ramdas, Phys. Rev. 140, A1246 (1965). M. Altarelli & W. Y. Hsu, Phys. Rev. Lett. 43, 1346 (1979). K. Shindo & H. Nara, .1. Phys. Soc. Japan 40, 1346 (1979) R. Resta,J. Phys. dO, L179 (1977). M. Altarelli, W. Y. Hsu & R. A. Sabatini,J. Phys. dO, L605 (1977). D.C. Herbert&J. Inkson,J. Phys. dO, L695 (1977).
8. 9. 10. 11. 12. 13.
343
J. R. Chelikowsky & M. L. Cohen, Phys. Rev. BlO, 5095 (1974). V. Heine & I. Abarenkov, Phil. Mag. 9, 451 (1964). A. 0. E. Animalu & V. Heine, Phil. Mag. 12, 1249 (1965). A. 0. E. Animalu, Phil. Mag. 11, 379 (1965). P. K. W. Vinsome & D. Richardson, J. Phys. C4, 2650 (1971). R. Car & A. Selloni, Physics of Semiconductors 1978, Edited by B. L. H. Wilson, p.407. The Institute of Physics, London (1979).
tentials is particularly appealing, because it does not suffer from the compatibility problems that arise for substitutional impurities. Indeed, the host pseudoBloch functions behave as “true” functions near the U core, irrespective of the choice of the host pseudopo-
d-like (7’2 and E respectively) excited states, is reversed for Li, and the A1 state is at 31.0meV, 1.8 meV above the degenerate ground state [1] (experiments cannot resolve any T2—E splitting). This is at odds not only with the other shallow donor patterns, but also with the ordering of the 2s and 2p levels in the Li atom, which must have the same nodal structure of the A1 and T2 states in Si. Furthermore, recent work [3] has shown that a purely point-charge impurity at an ainterstitial position in diamond semiconductorshas deep s ground state, as a result of a very strong attractive valley—orbit interaction, in agreement with experimental results on electrons trapped by positive muons [4] Therefore, .
tential. It is therefore legitimate to obtain the Si Bloch functions from Chelikowsky and Cohen’s empirical pseudopotential [8], and describe the bare Li~ion by a non-local pseudopotential of the Heine-—Abarenkov type [9]. The bare U~potential can then be described as V B1~r)
—
VLB~ ~J)
B VNL~T
,
where 2~ VBr\ e L’. I is the ordinary local bare Coulomb potential, and — —
2
2
V~L(r)= ~
—A 1 F,
the shallow ground state of Li must be a consequence of the deviations of the Li + potential from the pure Coulomb form. It is therefore of interest, partly as a test of recent progress in our understanding of donor states in multi-valley semiconductors [4—7], to see if a better description of the short-range Li + potential in Si can reproduce the observed level spectrum. In the present work, a pseudopotential description of the Li ion is adapted. It is worth remarking that for interstitial impurities the use of emperical pseudopo-
1=0 =
‘2
—
0
r
T
r > RM.
Here the A,’s are constants, the P,’s are angular momentum projection operators and RM is a limit radius. The parameters A, (1 = 0, 1, 2) and RM were fitted to atomic data and tabulated by Animalu and Heine [10], and for Li they are (in a.u.): A0 = 0.336, A1 = 0.504, A2 = 0.455, RM = 2.8. The physics implied by equation (1)—(3) is that the Li~ion appears as a point charge to states with 1> 3, as shown by the perfectly hydrogenic spectrum for JL and higher 1 states of the U atom, whereas appreciable deviations appear for more penetrating d- and p-states, as well as for s-states, for which
*Work at the University of illinois supported by NSF under grant DMR7~24155. tPresent address. 341
342
ELECTRONIC STRUCTURE OF LITHIUM INTERSTITIALS IN Si
Vol. 37, No.4
Table 1. Top: renormalization factors (as defined in [3] }for intervalley matrix elements of the Lit, s-, p-and d-wave pseudopotential [equation (7)J andfor a point c/large, at the interstitial position. Bottom, theoretical and experimental [1] binding energies (in meV) of the A1, T2 and E Li ground state multiplet components Valley pair
Li s(A1)
Li p(T2)
Li d(E)
~:~:
0.65
0.58
0.22
1.23
—0.32
0.68
0.39
1.08
~
Binding energy (th) (exp)
29.6 31.0
31.2 32.8
a strong orthogonality correction is also included in A0. The bare potential, equation (1), is screened when introduced into the Si crystal. We apply here the prescription of Animalu [111 for screening of non-local potentials, according to which, to the bare potential VB, equation (1), corresponds the screened potential Vs given by (in momentum space): s V (q) VLs(q) + VN~(q) (4) — —
where \ B1 \f ~ \ / \ V~~q) V~~q,/e~q,+1~q,
f~\
—
and Vs NL(~)
= —
B
j
~
(6)
Thus, the non-local portion of the screened potential is identical to the bare one, and the local portion corresponds to screening the local bare potential as usual, by the dielectric function [121 c(q), and adding 1(q), a (local!) correction, resulting from the charge rearrangement produced by VI~L,which can be numerically evaluated. We neglect this contribution here, because it is small in a uniform electron gas, and even more so in Si, where the Li ion sits in the tetrahedral interstitial position, a site with very low valence charge density. In fact, when the non-uniformity of the Si valence charge density (or in a different language, the local-field effect) is included [13] a substantial reduction in the effectiveness of screening is obtained for a potential with origin at the interstitial site. We therefore approximate: ,
S
B
B
V (q) = VL (q)/c(q) + VNL(q). (7) This is the potential used in the calculation of the energy levels. Since the A1, E and T2 components of the ground state multiplet have s- d-, and p-like syrnmetiy about the Li site, respectively, the corresponding angular momentum component in V~ is effective in each case.
Coulomb
30.7 32.8
When equation (7) is used as input in a multivalley calculation, according to the procedure of [6] the second, short range term drastically affects the large momentum, intervalley matrix elements. In Table 1 we list the dimensionless renormalization factor [3, 6], which represents the scaling of the intervalley matrix element magnitude with respect to the C = 0, pure Coulomb matrix element, due to the extra term in equation (7) and to consideration of all umklapp cornponents G ~0 in the conduction band wave-functions. The latter are obtained, as already mentioned, by a pseudopotential calculation [81with 88 plane waves. Notice the sign reversal of the s-wave matrix element, corresponding to a repulsive intervalley interaction, deriving from the orthogonalization contribution from the Li~s-core. The resulting binding energies are also ,
listed in Table 1. It is important to notice how the A1 level, which is deep for the interstitial Coulomb potential, is now pushed up above the T2 and E levels in agreement with the experimental results, because of the repulsive character of the valley—orbit interaction. It is however necessary to take the results of Table 1 with some caution, as the uncertainties involved in the calculation exceed the A1—7’2 splitting, which is a small fraction of the binding energy. Nonetheless, it is fair to say that the calculated binding energies for A1, T2 and E, all lie in the neighborhood of 30 meV, with a 10% uncertainty, as established by some numerical experimentation. In conclusion, it was shown that a non-local pseudopotential description of the screened Li~potential in Si, and the inclusion of multivalley effects account for the very shallow levels of this interstitial donor in Si, and clearly identify the physical origin of the anomalous level ordening. -~
Acknowledgement One of us (M. A.) is grateful to Professor V. Heine for an enhghtning discussion. -
.
.
.
.
Vol. 37, No.4
ELECTRONIC STRUCTURE OF LITHIUM INTERSTITIALS IN Si REFERENCES
1. 2. 3. 4. 5. 6. 7.
R. L. Aggarwal, P. Fischer, V. Mourzine & A. K. Ramdas,Phys. Rev. 138, A882 (1965). R. L. Aggarwal & A. K. Ramdas, Phys. Rev. 140, A1246 (1965). M. Altarelli & W. Y. Hsu, Phys. Rev. Lett. 43, 1346 (1979). K. Shindo & H. Nara, .1. Phys. Soc. Japan 40, 1346 (1979) R. Resta,J. Phys. dO, L179 (1977). M. Altarelli, W. Y. Hsu & R. A. Sabatini,J. Phys. dO, L605 (1977). D.C. Herbert&J. Inkson,J. Phys. dO, L695 (1977).
8. 9. 10. 11. 12. 13.
343
J. R. Chelikowsky & M. L. Cohen, Phys. Rev. BlO, 5095 (1974). V. Heine & I. Abarenkov, Phil. Mag. 9, 451 (1964). A. 0. E. Animalu & V. Heine, Phil. Mag. 12, 1249 (1965). A. 0. E. Animalu, Phil. Mag. 11, 379 (1965). P. K. W. Vinsome & D. Richardson, J. Phys. C4, 2650 (1971). R. Car & A. Selloni, Physics of Semiconductors 1978, Edited by B. L. H. Wilson, p.407. The Institute of Physics, London (1979).