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Interstitial hydrogen in silicon: a theoretical study
I,1,1
JOURNAL 0£
Journal of Non-Crystalline Solids 164-166 (1993) 293-296 North-Holland
Interstitial hydrogen
in s i l i c o n : a t h e o r e t i c a l
study
R. Colle ~'b and K. Stavrev b ~Dipartimento di Chimica Applicata, Facolta di lngegneria, Universita di Bologna, 401:t6 Bologna, Italy bScuola Normale Superiore, 56126 Pisa, Italy Relative stability and geometrical deformations of charged states of interstitial hydrogen in silicon are studied by semiempirical (MNDO-PM3) and ab i,lio (tlF and MP2) methods within a cluster approach. Stable geometries are found only for positively charged ll-Si systems. The incorporation of correlation effects markedly favours the tl-Si association. Long range Si-Si interactions are observed even at high bond distances and the energy behaviour of the first excited state with ~he St-St distance is studied. 1. I N T R O D U C T I O N Interstitial hydrogen in crystaline and amorphous silicon has been extensively studied bolh theoretically and experimentally [1-3]. Among tile unresolved problems in this field, the relative stablity of charged interstitial hydrogen states is of importance to reveal the bonding patterns and mechanism of H-St interactions ill the lattice. Clusters of small size and relatively unsophislicated computational methods have been used so far in a number of studies [1,3]. Nevertheless, the problem of the relative stabilil.y of interstitial ti °, 1t + and H - in silicon has not been solved ewm at. the Hartree-Fock (HE) level of approximation [4]. D a t a related to geometry changes in the lo" ca.l Si environment due to interstitial hydrogen are also scarce [1]. The aim of this work is to provide new results that shed further light, on these t.opics using a cluster approach and quantum-mechanical methods within reasonable COml)utational limits, 2. M O D E L D E S C R I P T I O N Clusters of different size have been already used to model tim tetrahedral interstices [1,a]. It is well known that at. least. 4 Si atoms are needed to describe tile nearest environment of interstitial atoms in the Si lattice. Clusters of minimal size present only a poor replica of ttle solid state problem and more distant silicon atoms are needed to correctly model tile intersite [5]. Thus a Si10H16 cluster has been constructed to simulate tile crystalline environment with tile external (16) dangling bonds typically saturated by lI atoms [1].
Further extention to more distant coordination spheres leads to Si_~c,It4s and SissHs4 clusters that involve interactions up to the fifth nearest neighbours in the Si lattice. Tile cluster geometries are available upon request from the authors. 3. C O M P U T A T I O N A L
DETAILS
Becent quantum-chenfical packages have been used to perform lhe calcnlations. Tile geometry of each cluster has been optimized by MNDOPM3 method using MOPAC-6 program [6]. The PM.3 method has already been used in silicon studies [7]; a comparison with conventional MNDO was published elsewhere [8] tlF and MoilerPlesset (MP2) calculations have been done using Gaussian92 program [9] with 2-( valence basis sets and effective core potentials (ECP) [10]. All electron IIF calculat.ions ]lave been performed on small prototype clusters to verify the validity of the ECP results. Tile calculations were done by using a D95 basis set which has the same valence orbitals as the ECPs. These tests have substantially confirmed tile ECP results. Slight overestimation of i he energy differences up to 8 % was observed when using ECP, but this discrepancy is of minor importauce for our calculations. 4. R E S U L T S
AND DISCUSSION
The methods used were frst tested with respect to reproducing c-St proI)erties - lattice constant and band offset.. The results are given ill Table 1. It is seen that the geometries well agree with experimental data. At. the same time, as one
0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved.
294
R. Colle, K. Stavrev / Interstitial hydrogen in silicon: a theoretical study
F i g u r e 1: St-St bond stretch curves for the ground 1Alg state and the first excited state 3A2u calculated by the QCISD(T) method using ECP parametrized SislIls cluster of D3d symmetry. -580.95
~
,
l
,
l
\,
t ' iAlg'
\ \
'3A2u'
-~-"
l
-581.05
-581.1
O m
< [-~ o 5~
""k,
• ',
-581.15
-581.2
"+"- --~..4_.....
-581.25
-581.3
-581.35
t
t
I
2
3
4 SI-SI BOND
I
5 LENGTH
I
t
6
7
[A]
Table 1: Interatomic distances (A) as reproduced by PM3 (first row entry) and HF (second row entry, where available). PM3 band gap offsets (eV) eslimated through HOMO-LUMO (first row entry) and through CI singlet-triplet differences (second row entry).
Value
Sil0
Si26
Si66
Exp.
Rsi-si
2.38
2.41
2.36
2.35
Eg~p (eV)
6.80 2.72
5.58 2.33
4.81 2.09
1.12
should expect, the band gap is poorly matched by the energy difference between tile highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO, respectively) even using ab initio methods with large basis sets. Tlle results can be markedly improved (see Table 1) by estimating the gap as the energy difference between the ground state and first excited state which, in this cases, is a triplet. An important feature of
this state is tile repulsive character of tile potential energy curve obtained by elongating a St-St bond. Fig. 1 displays tile results of ab initio scan for singlet and triplet states of the SisHls cluster along the (H3Si)3Si...Si(SiH3)3 bond. These results indicate that populating the repulsive state may result in St-St dissociation. The direct singlet-triplet transition is, however, strongly
R. Colle, K. Stavrev I Interstitial hydrogen in silicon: a theoretical study
F i g u r e 2: The H:Sil0Hx6 cluster with negative charge after PM3 geometry optimization. saturating the dangling bonds are omitted for clarity.
295 H atoms
5~
S~
"--" 3 ~
forbidden by the spin rules and a possible conlribution of the triplet to the formalion of defects in tile silicon lattice should be attributed to other excitation mechanisms,
T a b l e 2: Population analysis of frontier orhitals as a function of Si-Si bond distance (~). Equal population of H O M O and LUMO reflects the dissociation limit, Bond length 2.00
HOMO 1.99382
LU.MO .00618
2.50
1.97786
.02214
3.00 3.50 4.00 4.50 5.00
1.92908 1.81036 1.60974 1.39268 1.22891
.07092 .18964 .39026 .60732 .77109
5.50 6.00 6.50
1.12734 1.06964 1.03728
.87266 .93036 .96272
7.00
1.01972
.98028
-..
Starting from t.he consideration that bond distortions are typical ill the amorphous solid where Si-Si bond lengths vary in a wide range, we have examined the population of the frontier orbitals as a flmction of the Si-Si distance in the ground state. We have used a 2-configuration waveflmction opt.infized using t.he MCSCF procedure and an extended basis set (6-311G++**) for" the silicon atoms. Tile orbital exponents used for the H atoms were scaled to give zero charge on the Si atoms. The Si-Si bond thus modelled has been found lo have a spring constant close to the experimental value. We observe (see Tahie 2) that the population of the antibonding orbital (LUMO in a IIF scheme) increases along with the Si-Si stretch. Furthermore, the LUMO population becomes practically equal to that of HOMO only at. Si-Si distances sufficienty higher than the twice the Van der Waals radius of Si (2.1 A). This fact indicates that non-negligible Si-Si interactions carl exsist well beyond this limit. In order to study properties of the interstitial hydrogen in silicon, we have first optimized lhe geometry of clusters like [tt:Sil0Hl~;] 6 with different global charge 6 and H placed initially at the centre of the interstitial hole. The
296
R. Colle, K. Stavrev / Interstitial hydrogen in silicon." a theoretical study
T a b l e 3: Differences between the energy (eV) of the l[:Sil0Ha6 cluster at the optimized geometry and
that of the separated fragments calculated for different dissociation patterns Associate [H:Sil0] +
[H:Sil0] °
Dissociate
t'M 3
HF
MP2
Sil0 ° +H +
-8.04
-4.35
-4.95
Sit0 + + l l °
-1.64
-2.73
-3.24
Sil0+++lI -
-13.90
-15.41
-15.45
St10- +ll +
-12.30
-11.29
-11.72
,qilO0 +11 °
-1.57
4.19
3.52
St10 + + I 1 [H:Sia0]-
-9.97
-10.62
-10.66
Sil0--+ll +
-17.65
-10.68
-12.27
Sil0- + l I ° St10 ° +11-
-3.27 -7.32
9.34 8.76
6.90 6.25
main results are: (i) tlle cluster with 6 = - I is strongly contracted with respect to the inilial configuration - see Fig. 2 - and the interstitial hydrogen is found in the bonding region of 3 silicons; 0/) similar but sufl]ciently less pronounced distortions are observed when 5 = +1 , while for 5 = 0 the geometry remains practically unchanged. Analogous results have been ohtained also for H:Si26H4s and ll:Si,~c,lI64 clusters optimized using PM3 method, Further, we have calculated energy differences between clusters with lI inside and the separated fragments - see Table 3. One can observe that PM3 favours the Si-lI complexes at, any charge. At the same time, the ab ini1io calculations confirm the PM3 results only for the positively charged [H:Si1(~]+ cluster. For the other charged states (0 and -1), l)osii.ive energy differences are detected that may result in dissoelation to silicon and non-bonded hydrogen. The MP2 results do not change the trend in the relative stability of the charged states but strengthen sufficiently the association effect. The estimated MP2 contribution is of the order of 15-25 % of the calculated energy differences. To conclude we can say that our results seem to guarantee stable geometries only for positively charged H-St systelns.
REFEI1,ENCES I. C.G. Van der Walle, in: Semiconductors and Semimetals, 34 (19f)l) 585, and references therei 11.
2. C.G. Van de Walle, P.J.H. Denteneer, Y. BarYam and S.T. Pantelides, Phys.Rev. B 39 (1989) 10791. 3. G.G. DeLeo and W.B. Fowler, in: Semiconductors and Semimetals, 34 (1991) 511, and references therein. 4. S.M. Myers,M.1. Baskes, tI.K. Birnbaum, J.W. Corbett, G.G. DeLeo, S.K. Estreicher, E.E. Hailer, P. Jena, N.M. Johnson, 1%. Kirchheim, S.J. Pearton and M.J. Stavola, Rev. Modern Phys. 64 (1992)559. 5. C. Pisani, R. Orlando and R. Nada, Rev. Solid State Sci. 5 (1991) 177. 6. J.J.P. Stewart, MOPAC 6.00 Q C P E 455, QCPE, Indiana University, (1990). 7. S. Kugler and (;. Naray-Szabo, J. Non-Cryst. Solids 137£138 ( 1991 ) 295. 8. K. Stavrev, Q C P E Bulletin, Indiana University 12 (1992) 59. 9. Gaussian92 Revision A, Gaussian Inc., Pittsburgh PA 1992. 10. P.J. Hay and W.R. Wadt, J. Chem. Phys. 85 (1985) 270; 299.
JOURNAL 0£
Journal of Non-Crystalline Solids 164-166 (1993) 293-296 North-Holland
Interstitial hydrogen
in s i l i c o n : a t h e o r e t i c a l
study
R. Colle ~'b and K. Stavrev b ~Dipartimento di Chimica Applicata, Facolta di lngegneria, Universita di Bologna, 401:t6 Bologna, Italy bScuola Normale Superiore, 56126 Pisa, Italy Relative stability and geometrical deformations of charged states of interstitial hydrogen in silicon are studied by semiempirical (MNDO-PM3) and ab i,lio (tlF and MP2) methods within a cluster approach. Stable geometries are found only for positively charged ll-Si systems. The incorporation of correlation effects markedly favours the tl-Si association. Long range Si-Si interactions are observed even at high bond distances and the energy behaviour of the first excited state with ~he St-St distance is studied. 1. I N T R O D U C T I O N Interstitial hydrogen in crystaline and amorphous silicon has been extensively studied bolh theoretically and experimentally [1-3]. Among tile unresolved problems in this field, the relative stablity of charged interstitial hydrogen states is of importance to reveal the bonding patterns and mechanism of H-St interactions ill the lattice. Clusters of small size and relatively unsophislicated computational methods have been used so far in a number of studies [1,3]. Nevertheless, the problem of the relative stabilil.y of interstitial ti °, 1t + and H - in silicon has not been solved ewm at. the Hartree-Fock (HE) level of approximation [4]. D a t a related to geometry changes in the lo" ca.l Si environment due to interstitial hydrogen are also scarce [1]. The aim of this work is to provide new results that shed further light, on these t.opics using a cluster approach and quantum-mechanical methods within reasonable COml)utational limits, 2. M O D E L D E S C R I P T I O N Clusters of different size have been already used to model tim tetrahedral interstices [1,a]. It is well known that at. least. 4 Si atoms are needed to describe tile nearest environment of interstitial atoms in the Si lattice. Clusters of minimal size present only a poor replica of ttle solid state problem and more distant silicon atoms are needed to correctly model tile intersite [5]. Thus a Si10H16 cluster has been constructed to simulate tile crystalline environment with tile external (16) dangling bonds typically saturated by lI atoms [1].
Further extention to more distant coordination spheres leads to Si_~c,It4s and SissHs4 clusters that involve interactions up to the fifth nearest neighbours in the Si lattice. Tile cluster geometries are available upon request from the authors. 3. C O M P U T A T I O N A L
DETAILS
Becent quantum-chenfical packages have been used to perform lhe calcnlations. Tile geometry of each cluster has been optimized by MNDOPM3 method using MOPAC-6 program [6]. The PM.3 method has already been used in silicon studies [7]; a comparison with conventional MNDO was published elsewhere [8] tlF and MoilerPlesset (MP2) calculations have been done using Gaussian92 program [9] with 2-( valence basis sets and effective core potentials (ECP) [10]. All electron IIF calculat.ions ]lave been performed on small prototype clusters to verify the validity of the ECP results. Tile calculations were done by using a D95 basis set which has the same valence orbitals as the ECPs. These tests have substantially confirmed tile ECP results. Slight overestimation of i he energy differences up to 8 % was observed when using ECP, but this discrepancy is of minor importauce for our calculations. 4. R E S U L T S
AND DISCUSSION
The methods used were frst tested with respect to reproducing c-St proI)erties - lattice constant and band offset.. The results are given ill Table 1. It is seen that the geometries well agree with experimental data. At. the same time, as one
0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved.
294
R. Colle, K. Stavrev / Interstitial hydrogen in silicon: a theoretical study
F i g u r e 1: St-St bond stretch curves for the ground 1Alg state and the first excited state 3A2u calculated by the QCISD(T) method using ECP parametrized SislIls cluster of D3d symmetry. -580.95
~
,
l
,
l
\,
t ' iAlg'
\ \
'3A2u'
-~-"
l
-581.05
-581.1
O m
< [-~ o 5~
""k,
• ',
-581.15
-581.2
"+"- --~..4_.....
-581.25
-581.3
-581.35
t
t
I
2
3
4 SI-SI BOND
I
5 LENGTH
I
t
6
7
[A]
Table 1: Interatomic distances (A) as reproduced by PM3 (first row entry) and HF (second row entry, where available). PM3 band gap offsets (eV) eslimated through HOMO-LUMO (first row entry) and through CI singlet-triplet differences (second row entry).
Value
Sil0
Si26
Si66
Exp.
Rsi-si
2.38
2.41
2.36
2.35
Eg~p (eV)
6.80 2.72
5.58 2.33
4.81 2.09
1.12
should expect, the band gap is poorly matched by the energy difference between tile highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO, respectively) even using ab initio methods with large basis sets. Tlle results can be markedly improved (see Table 1) by estimating the gap as the energy difference between the ground state and first excited state which, in this cases, is a triplet. An important feature of
this state is tile repulsive character of tile potential energy curve obtained by elongating a St-St bond. Fig. 1 displays tile results of ab initio scan for singlet and triplet states of the SisHls cluster along the (H3Si)3Si...Si(SiH3)3 bond. These results indicate that populating the repulsive state may result in St-St dissociation. The direct singlet-triplet transition is, however, strongly
R. Colle, K. Stavrev I Interstitial hydrogen in silicon: a theoretical study
F i g u r e 2: The H:Sil0Hx6 cluster with negative charge after PM3 geometry optimization. saturating the dangling bonds are omitted for clarity.
295 H atoms
5~
S~
"--" 3 ~
forbidden by the spin rules and a possible conlribution of the triplet to the formalion of defects in tile silicon lattice should be attributed to other excitation mechanisms,
T a b l e 2: Population analysis of frontier orhitals as a function of Si-Si bond distance (~). Equal population of H O M O and LUMO reflects the dissociation limit, Bond length 2.00
HOMO 1.99382
LU.MO .00618
2.50
1.97786
.02214
3.00 3.50 4.00 4.50 5.00
1.92908 1.81036 1.60974 1.39268 1.22891
.07092 .18964 .39026 .60732 .77109
5.50 6.00 6.50
1.12734 1.06964 1.03728
.87266 .93036 .96272
7.00
1.01972
.98028
-..
Starting from t.he consideration that bond distortions are typical ill the amorphous solid where Si-Si bond lengths vary in a wide range, we have examined the population of the frontier orbitals as a flmction of the Si-Si distance in the ground state. We have used a 2-configuration waveflmction opt.infized using t.he MCSCF procedure and an extended basis set (6-311G++**) for" the silicon atoms. Tile orbital exponents used for the H atoms were scaled to give zero charge on the Si atoms. The Si-Si bond thus modelled has been found lo have a spring constant close to the experimental value. We observe (see Tahie 2) that the population of the antibonding orbital (LUMO in a IIF scheme) increases along with the Si-Si stretch. Furthermore, the LUMO population becomes practically equal to that of HOMO only at. Si-Si distances sufficienty higher than the twice the Van der Waals radius of Si (2.1 A). This fact indicates that non-negligible Si-Si interactions carl exsist well beyond this limit. In order to study properties of the interstitial hydrogen in silicon, we have first optimized lhe geometry of clusters like [tt:Sil0Hl~;] 6 with different global charge 6 and H placed initially at the centre of the interstitial hole. The
296
R. Colle, K. Stavrev / Interstitial hydrogen in silicon." a theoretical study
T a b l e 3: Differences between the energy (eV) of the l[:Sil0Ha6 cluster at the optimized geometry and
that of the separated fragments calculated for different dissociation patterns Associate [H:Sil0] +
[H:Sil0] °
Dissociate
t'M 3
HF
MP2
Sil0 ° +H +
-8.04
-4.35
-4.95
Sit0 + + l l °
-1.64
-2.73
-3.24
Sil0+++lI -
-13.90
-15.41
-15.45
St10- +ll +
-12.30
-11.29
-11.72
,qilO0 +11 °
-1.57
4.19
3.52
St10 + + I 1 [H:Sia0]-
-9.97
-10.62
-10.66
Sil0--+ll +
-17.65
-10.68
-12.27
Sil0- + l I ° St10 ° +11-
-3.27 -7.32
9.34 8.76
6.90 6.25
main results are: (i) tlle cluster with 6 = - I is strongly contracted with respect to the inilial configuration - see Fig. 2 - and the interstitial hydrogen is found in the bonding region of 3 silicons; 0/) similar but sufl]ciently less pronounced distortions are observed when 5 = +1 , while for 5 = 0 the geometry remains practically unchanged. Analogous results have been ohtained also for H:Si26H4s and ll:Si,~c,lI64 clusters optimized using PM3 method, Further, we have calculated energy differences between clusters with lI inside and the separated fragments - see Table 3. One can observe that PM3 favours the Si-lI complexes at, any charge. At the same time, the ab ini1io calculations confirm the PM3 results only for the positively charged [H:Si1(~]+ cluster. For the other charged states (0 and -1), l)osii.ive energy differences are detected that may result in dissoelation to silicon and non-bonded hydrogen. The MP2 results do not change the trend in the relative stability of the charged states but strengthen sufficiently the association effect. The estimated MP2 contribution is of the order of 15-25 % of the calculated energy differences. To conclude we can say that our results seem to guarantee stable geometries only for positively charged H-St systelns.
REFEI1,ENCES I. C.G. Van der Walle, in: Semiconductors and Semimetals, 34 (19f)l) 585, and references therei 11.
2. C.G. Van de Walle, P.J.H. Denteneer, Y. BarYam and S.T. Pantelides, Phys.Rev. B 39 (1989) 10791. 3. G.G. DeLeo and W.B. Fowler, in: Semiconductors and Semimetals, 34 (1991) 511, and references therein. 4. S.M. Myers,M.1. Baskes, tI.K. Birnbaum, J.W. Corbett, G.G. DeLeo, S.K. Estreicher, E.E. Hailer, P. Jena, N.M. Johnson, 1%. Kirchheim, S.J. Pearton and M.J. Stavola, Rev. Modern Phys. 64 (1992)559. 5. C. Pisani, R. Orlando and R. Nada, Rev. Solid State Sci. 5 (1991) 177. 6. J.J.P. Stewart, MOPAC 6.00 Q C P E 455, QCPE, Indiana University, (1990). 7. S. Kugler and (;. Naray-Szabo, J. Non-Cryst. Solids 137£138 ( 1991 ) 295. 8. K. Stavrev, Q C P E Bulletin, Indiana University 12 (1992) 59. 9. Gaussian92 Revision A, Gaussian Inc., Pittsburgh PA 1992. 10. P.J. Hay and W.R. Wadt, J. Chem. Phys. 85 (1985) 270; 299.