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Dynamics of partial dislocations in silicon
MATERIALS SClENCE & EMIWEERIWG ELSEVIER
B
Materials Scienceand EngineeringB37 (1996) 185-158
Dynamics of partial dislocations in silicon L.B. Hansena, K. Stokbrob, B.I. Lundqvist”, K.W. Jacobsenc ‘Department
of Applied Physics, Chalmers University of Techology bScuola Internazionale Superior di Studi Avamati, ‘CAMP, The Techical University of Demark,
and Goteborg hiversity, S-41296 via Beirut 4, I-34014 Trieste Italy DK-2800, Lyngby, Denmark
Goteborg,
Sweden
Abstract Atomic-scale calculations for the dynamics of the 90” partial glide dislocation in silicon are made using the effective-medium tight-binding theory. Kink formation and migration energies for the reconstructed partial dislocation are compared with experimental results for the mobility of this dislocation. The results confirm the theory that the partial moves in the dissociated state via the formation of stable kinks. The correlation between glide activation energy and band gap in semiconducting systems is discussed. Keywords:
Silicon; Partial dislocations
1. Introduction Metallic systems are often ductile even at low temperature, which is due to the high mobility of dislocations in these systems. In semiconducting systems, on the contrary there are larger barriers which must be overcome in order to move a dislocation, which results in the brittle behaviour of these systems. The large barrier is a result of the electronic structure in the semiconducting system, and a direct proportionality between the band gap and the dislocation glide activation energy has been observed by Gilman [l]. Experimental observations [2] indicate that dislocations in Si belonging to the glide sets are dissociated into partial dislocations bounding stacking faults on the order of 30-50 A. The perfect 60” dislocation in Si dissociates into a 90” partial followed by a 30” partial (see Section 3). The 90”-30” dislocation has been shown to move in the dissociated state and the mobility of the individual partial has been measured [3]. Dislocation glide is for a relatively small applied stress believed to proceed via the formation of stable kink pairs and the subsequent spreading of these along the dislocation line [4]. Recently a calculation by Huang et al., using a related total energy method, found the migration energy for the 30” partial dislocation to be in good agreement with experimental results [6]. In this work Elsevier ScienceS.A.
the kink formation and migration energies are found for the 90” reconstructed dislocation, using the effective-medium tight-binding (EMTB) [7] total energy method to calculate the energetics. 2. The effective-medium method
tight-binding
total energy
The EMTB total energy method is based on the effective-medium concept of a reference system and is described in detail in Ref. [7]. Instead of calculating the total energy of a collection of interacting Si atoms, the energy of each atom is associated with a reference system that is close to the system considered. The small difference between the energy of the reference system and the real system can then be calculated approximately. The two main approximations are the use of a transferable charge density [8] and a first-order linearized muffin tin orbital (LMTO) [9] basis set. The total energy can be written E = EC+ AE,, + AE,,, = T eref(si) + (AG+ AV) + AE,,,
(1)
The first term, the cohesive function, is the energy of the Si atom in the reference system, written as a function of the neutral atomic sphere radius si. With only
186
L.B.
(4
Hansen
et al. I Materials
Science
@I
Fig. 1. Atomic structure projected on the (111) slip plane. Atoms immediately above (0) and below (0) the slip plane are shown. (a) Symmetric reconstruction; (b) asymmetric reconstruction.
this term, the solid is viewed as a collection of neutral atomic sphere, i.e. in the atomic sphere approximation (ASA). The second term, the atomic sphere correction, corrects the ASA calculation with respect to electrostatic and exchange-correlation energies. The last term is the energy correction due to the difference between the band structures in the reference and real systems. The band structure is calculated using an LMTO parametrized tight-binding hamiltonian [7].
3. Dislocations in silicon The dislocation structure of Si has been discussed in many excellent reviews (see, for example, Refs. [lo] and [ 111). Because the diamond cubic structure is formed by two f.c.c. lattices dislocations are expected to be similar to the f.c.c. dislocations. In Si the main slip plane is the (111) plane and the major slip direction is [llO]. The diamond lattice allows for two distinct locations of the slip plane, placed between atomic planes that are separated by either (a) a nearest-neighbour (NN) dis-
(4
and Btgineering
837
(1996)
185- I8S
tance (planes with the same index), where a slip breaks one NN bond, the so-called LLshufIle” slip plane, or (b) one-third of an NN distance (planes of different indices), breaking three NN bonds, the “glide” slip plane. Because of the low intrinsic stacking fault energy the perfect glide dislocation is known to dissociate, following the reaction
t(ioi/)-t~(Zii)+4(ii2)
(2)
The 90” edge dislocation is formed by a glide &iT2) (Fig. 1). Several models for the core structure have been proposed. Hirch [12] and Jones [ 131 have proposed a symmetric reconstruction, where the atoms with the dangling bonds (Fig. l(b)) move closer together and form “quasi-fivefold” [ 141 coordinated atoms. The bonds for the atoms with “quasi-fivefold” coordination in the core are shown with broken lines. An asymmetric reconstruction was proposed by Duesbery et al. [14] and by Bigger et al. [15], where the symmetry along the dislocation line is broken and the atoms in the core has retain fourfold coordination. The latter reconstruction has been found by Bigger et al. [15] to be energetically more favourable than the symmetric reconstruction, The EMTB model exhibits the same reconstruction with the energy difference between the asymmetric and the symmetric reconstruction being 0.18 eV A-‘, in very good agreement with the ab initio result by Bigger et al. [15] of 0.2 eV A-‘. The unit cell for the atomic simulation contains two 90” partial edge dislocations with opposite Burgers vector (b = 2.18 A), such that periodic boundary conditions can be used. Fig. 2(a) shows a side view (close to the [ii01 direction) of the two partials introduced by
Q
-
Cb) 9-Q I
I
stacking fault region Fig, 2. The 8 x 9 x 3 atom unit cell used for the EMTB calculation. (a) Side view of the two partials: 0 indicate (111) glide plane. (b) Top view of the (111) glide plane: --, dislocation lines; ---, size of the unit cell. (c) The quadrupolar lattice of dislocations formed by the periodic repetition of the unit cell.
LB. Hamerz
el al. / Materials
Science
and Engineering
B37 (1996)
185-188
imposing the isotropic elastic fields from the dislocation array. The black atoms indicate the (111) glide plane, which is seen from above in Fig. 2(b). The total number of atoms in the unit cell is 232. As pointed out by Bigger et al. [15] care has to be taken when using periodic boundary conditions in order to avoid a tilt boundary by stacking dislocations with the same Burgers vectors below each other. This tilt boundary can create a grain boundary along the horizontal border of the unit cell. The quadrupolar array introduced by Bigger et al. [15] that avoids the tilt boundary is used. The stacking of the partial dislocations in the quadrupolar array is illustrated in Fig. 2(c).
4. Dislocation
dynamics
Dislocation glide for dislocation parallel to a (110) Peierls valley is for a relatively small applied stress believed to proceed via the formation of stable kink pairs and the subsequent spreading of these along the dislocation line [4]. The velocity ud of the steady state motion of the dislocation is in this theory given by vd cc exp
(
- Ulik12 f w,, k,T
1
(3)
Here U,, and W,, are the formation energy for a stable double kink and the migration energy for a single kink respectively. Using the intermittent loading technique combined with transmission electron microscopy the two contributions to the activation energy can be individually determined experiment gives Fi161. The U,, = 2.0 & 0.3 eV and FY, = 1.2 f 0.3 eV for Si. The apparent activation energy for dislocation motion is then from Eq. (3) given as Ue,=
U,,/2 + Wm = U, + W, = 2.2 eV
(4)
Fig. 3(a) shows the perfect reconstructed dislocation. The kinks are created (see also Ref. [17]) by moving the shaded atoms in Fig. 3(a) antiparallel in the direction of the arrows. The reaction coordinate is chosen as the difference between the coordinates of the two moving atoms in the direction of the arrows. All other degrees of freedom in the unit cell have been allowed to relax. The stable single atom kink is shown in Fig. 3(c). From this process the energy for creating a single atomic kink is found to be 1.2 eV. This single atomic kink has no dangling bond, as seen in Fig. 3(c), and is therefore a relatively low energy defect. We estimate this energy to be 0.3 eV. The barrier W, = 1.45 eV for kink migration is found from the process in Figs. 3(c) and 3(d). The barrier U,, for creating a double kink is then from this calculation 0.3 eV + 1.45 eV = 1.75 eV.
Fig. 3 (a) The perfect reconstructed dislocation: 0 atoms moved to create single atomic kink. (b) Transition state for forming a single atom kink pair. (c) Single atomic kink pair. (b) Transition state for forming a stable double kink. The unit cell is repeated twice in the vertical direction, and only part of the unit cell is shown in the horizontal direction.
Our estimate for W, is consistent with the experimental values, and also the calculated effective activation energy U,, = 1.75 eV/2 + 1.45 eV = 2.32 eV compares well with the experimental value of 2.2 eV. The migration energy W, for the 30” partial dislocation has recently been found by Huang et al. [5]. Here W, was found to be 2.1 + 0.3 eV. The 90” partial is experimentally known to be more mobile, compared with the 30” partial [3]. Although the difference in apparent activation energy has the same magnitude as the experimental resolution, the trend seen in the two calculations is correct. 5. Discussion and conclusions The migration and formation energies for kinks on the 90” partial dislocation are in agreement with the experimental findings. This supports the theory that the 90”-30” dislocation moves in the dissociated state via the formation of stable kinks. One important aspect of the dynamics of dislocations is the close correlation found by Gilman [I] between the highest occupied molecular orbital-lowest unoccupied molecular orbital gap I$,, and the glide activation energy. For homopolar semiconductors the activation energy equals twice the I&, gap. In this theory a bondbreaking process takes an energy of Eh,/2. Gilman considers the “shuffle” type of dislocation, where kink movement involves the breaking of one bond.
188
L.B. Hansen et al. / Materials Science and Engineering 837 (1996) 185-158
In our case the kink migration requires two broken bonds. Consider the transition states in Figs. 3(b) and 3(d). In both cases two bonds are broken, but instead four half-bonds are formed, giving the observed energy barriers. The energy associated with these half-bonds is not included in the theory of Gilman. The fact that the 30”-90” dislocation has been observed experimentally to move in the dissociated state and that the barriers found in this work and in Ref. [5] are in agreement with the barriers deduced from experiment indicate that the processes we are considering are the important processes for the dislocation mobility of this glide partial. A detailed study of the correlation between band structure and energy barrier for this type of process and the relation to the work of Gilman is in progress
[181* Acknowledgements Lars B. Hansen acknowleges support from The Swedish National Board for Industrial and Technical Development and the Swedish Natural Science Research Council and from the EED Contract CHRX-CT93-0134. Kurt Stokbro acknowledges EEC Contract ERBCHBGCT 920180 and Contract ERBCHRXCT 930342 and CNR project Supaltemp.
PI J.J. Gilman,Science, 261 (1993) 1436. P.Grosbras,H. GaramandJ.C. Dcsoyer,Philos. PI J.L. Demenet, Mag. A, 59 (1989) 501. N. Hiller,S.Sauerland, P.SpechtandH. Alexan[31 H. Gottschalk, der, Phys. Status S’olidi, 138 (1993)547. 141J.P. Hirth and J. Lothe, Theory of Dislocntiorrs, Wiley, New
York, 2ndedn.,1982. and0-F. Sankey,Phys. Reo. Lett., 74 151Y.M. Huang,J.C. Spence (1995)3392. PI V.I. Nikitenko,B.Ya. FaberandYu.L, Iunin,Sac.Ply. JETP, 66(4)(1987)738. andJ.K. Norskov,Phys. 171 K. Stokbro,N. Chetty,K.W. Jacobsen Rev. B, 50 (1994)10727, and references citedtherein. andJ.K. Narskov,J. P/Q-S.: Cor~&ru. PI N. Chetty,K.W. Jacobsen Matter, 3 (1991) 5437. and0. Jepsen, in M. Yussouff(cd.), Electrouic [91 O.K. Andesen Bandstrucwe and its Applications, SpringerLecture Notes,
Springer,Berlin1987. and G.Y. Richardson,CRC Crit. Ret!, Solid 1101 MS. Duesbeery State Mater. Sci., 17 (1991). 1111A. GeorgeandJ. Rabier,Rev. Phys. Appl., 22 (1987) 207. WI P.B.Hirch, J. Phys. (Paris), Colloq. C6, 40 (1979) 27. El31 R. Jones,J. Phys. (Puris), Colloq. CG, 40 (1979) 33. B.JoosandD.J. Michel,Plys. Rec. B, 43 (1991) 1141 MS. Duesbery, 5143. [I51 J.R.K. Bigger,D.A. McInnes,A.P. Sutton,M.C. Payne,I. Stich, R.D. King-Smsith, D.M. Bird andL.J. Clarke,Ply. Rer. Lett.,
69(1992)2224. El61B.Ya.Farber,Yu.L. IuninandV.I. Nikitenko,Soo.P1iJzJETP, (1986). El71R. Jones,Philas. Mag. B, 42 (1980) 213. K. Stokbro,B.I. LundqvistandK.W. Jacobsen, to El81L.B. Hansen, bepublished.
B
Materials Scienceand EngineeringB37 (1996) 185-158
Dynamics of partial dislocations in silicon L.B. Hansena, K. Stokbrob, B.I. Lundqvist”, K.W. Jacobsenc ‘Department
of Applied Physics, Chalmers University of Techology bScuola Internazionale Superior di Studi Avamati, ‘CAMP, The Techical University of Demark,
and Goteborg hiversity, S-41296 via Beirut 4, I-34014 Trieste Italy DK-2800, Lyngby, Denmark
Goteborg,
Sweden
Abstract Atomic-scale calculations for the dynamics of the 90” partial glide dislocation in silicon are made using the effective-medium tight-binding theory. Kink formation and migration energies for the reconstructed partial dislocation are compared with experimental results for the mobility of this dislocation. The results confirm the theory that the partial moves in the dissociated state via the formation of stable kinks. The correlation between glide activation energy and band gap in semiconducting systems is discussed. Keywords:
Silicon; Partial dislocations
1. Introduction Metallic systems are often ductile even at low temperature, which is due to the high mobility of dislocations in these systems. In semiconducting systems, on the contrary there are larger barriers which must be overcome in order to move a dislocation, which results in the brittle behaviour of these systems. The large barrier is a result of the electronic structure in the semiconducting system, and a direct proportionality between the band gap and the dislocation glide activation energy has been observed by Gilman [l]. Experimental observations [2] indicate that dislocations in Si belonging to the glide sets are dissociated into partial dislocations bounding stacking faults on the order of 30-50 A. The perfect 60” dislocation in Si dissociates into a 90” partial followed by a 30” partial (see Section 3). The 90”-30” dislocation has been shown to move in the dissociated state and the mobility of the individual partial has been measured [3]. Dislocation glide is for a relatively small applied stress believed to proceed via the formation of stable kink pairs and the subsequent spreading of these along the dislocation line [4]. Recently a calculation by Huang et al., using a related total energy method, found the migration energy for the 30” partial dislocation to be in good agreement with experimental results [6]. In this work Elsevier ScienceS.A.
the kink formation and migration energies are found for the 90” reconstructed dislocation, using the effective-medium tight-binding (EMTB) [7] total energy method to calculate the energetics. 2. The effective-medium method
tight-binding
total energy
The EMTB total energy method is based on the effective-medium concept of a reference system and is described in detail in Ref. [7]. Instead of calculating the total energy of a collection of interacting Si atoms, the energy of each atom is associated with a reference system that is close to the system considered. The small difference between the energy of the reference system and the real system can then be calculated approximately. The two main approximations are the use of a transferable charge density [8] and a first-order linearized muffin tin orbital (LMTO) [9] basis set. The total energy can be written E = EC+ AE,, + AE,,, = T eref(si) + (AG+ AV) + AE,,,
(1)
The first term, the cohesive function, is the energy of the Si atom in the reference system, written as a function of the neutral atomic sphere radius si. With only
186
L.B.
(4
Hansen
et al. I Materials
Science
@I
Fig. 1. Atomic structure projected on the (111) slip plane. Atoms immediately above (0) and below (0) the slip plane are shown. (a) Symmetric reconstruction; (b) asymmetric reconstruction.
this term, the solid is viewed as a collection of neutral atomic sphere, i.e. in the atomic sphere approximation (ASA). The second term, the atomic sphere correction, corrects the ASA calculation with respect to electrostatic and exchange-correlation energies. The last term is the energy correction due to the difference between the band structures in the reference and real systems. The band structure is calculated using an LMTO parametrized tight-binding hamiltonian [7].
3. Dislocations in silicon The dislocation structure of Si has been discussed in many excellent reviews (see, for example, Refs. [lo] and [ 111). Because the diamond cubic structure is formed by two f.c.c. lattices dislocations are expected to be similar to the f.c.c. dislocations. In Si the main slip plane is the (111) plane and the major slip direction is [llO]. The diamond lattice allows for two distinct locations of the slip plane, placed between atomic planes that are separated by either (a) a nearest-neighbour (NN) dis-
(4
and Btgineering
837
(1996)
185- I8S
tance (planes with the same index), where a slip breaks one NN bond, the so-called LLshufIle” slip plane, or (b) one-third of an NN distance (planes of different indices), breaking three NN bonds, the “glide” slip plane. Because of the low intrinsic stacking fault energy the perfect glide dislocation is known to dissociate, following the reaction
t(ioi/)-t~(Zii)+4(ii2)
(2)
The 90” edge dislocation is formed by a glide &iT2) (Fig. 1). Several models for the core structure have been proposed. Hirch [12] and Jones [ 131 have proposed a symmetric reconstruction, where the atoms with the dangling bonds (Fig. l(b)) move closer together and form “quasi-fivefold” [ 141 coordinated atoms. The bonds for the atoms with “quasi-fivefold” coordination in the core are shown with broken lines. An asymmetric reconstruction was proposed by Duesbery et al. [14] and by Bigger et al. [15], where the symmetry along the dislocation line is broken and the atoms in the core has retain fourfold coordination. The latter reconstruction has been found by Bigger et al. [15] to be energetically more favourable than the symmetric reconstruction, The EMTB model exhibits the same reconstruction with the energy difference between the asymmetric and the symmetric reconstruction being 0.18 eV A-‘, in very good agreement with the ab initio result by Bigger et al. [15] of 0.2 eV A-‘. The unit cell for the atomic simulation contains two 90” partial edge dislocations with opposite Burgers vector (b = 2.18 A), such that periodic boundary conditions can be used. Fig. 2(a) shows a side view (close to the [ii01 direction) of the two partials introduced by
Q
-
Cb) 9-Q I
I
stacking fault region Fig, 2. The 8 x 9 x 3 atom unit cell used for the EMTB calculation. (a) Side view of the two partials: 0 indicate (111) glide plane. (b) Top view of the (111) glide plane: --, dislocation lines; ---, size of the unit cell. (c) The quadrupolar lattice of dislocations formed by the periodic repetition of the unit cell.
LB. Hamerz
el al. / Materials
Science
and Engineering
B37 (1996)
185-188
imposing the isotropic elastic fields from the dislocation array. The black atoms indicate the (111) glide plane, which is seen from above in Fig. 2(b). The total number of atoms in the unit cell is 232. As pointed out by Bigger et al. [15] care has to be taken when using periodic boundary conditions in order to avoid a tilt boundary by stacking dislocations with the same Burgers vectors below each other. This tilt boundary can create a grain boundary along the horizontal border of the unit cell. The quadrupolar array introduced by Bigger et al. [15] that avoids the tilt boundary is used. The stacking of the partial dislocations in the quadrupolar array is illustrated in Fig. 2(c).
4. Dislocation
dynamics
Dislocation glide for dislocation parallel to a (110) Peierls valley is for a relatively small applied stress believed to proceed via the formation of stable kink pairs and the subsequent spreading of these along the dislocation line [4]. The velocity ud of the steady state motion of the dislocation is in this theory given by vd cc exp
(
- Ulik12 f w,, k,T
1
(3)
Here U,, and W,, are the formation energy for a stable double kink and the migration energy for a single kink respectively. Using the intermittent loading technique combined with transmission electron microscopy the two contributions to the activation energy can be individually determined experiment gives Fi161. The U,, = 2.0 & 0.3 eV and FY, = 1.2 f 0.3 eV for Si. The apparent activation energy for dislocation motion is then from Eq. (3) given as Ue,=
U,,/2 + Wm = U, + W, = 2.2 eV
(4)
Fig. 3(a) shows the perfect reconstructed dislocation. The kinks are created (see also Ref. [17]) by moving the shaded atoms in Fig. 3(a) antiparallel in the direction of the arrows. The reaction coordinate is chosen as the difference between the coordinates of the two moving atoms in the direction of the arrows. All other degrees of freedom in the unit cell have been allowed to relax. The stable single atom kink is shown in Fig. 3(c). From this process the energy for creating a single atomic kink is found to be 1.2 eV. This single atomic kink has no dangling bond, as seen in Fig. 3(c), and is therefore a relatively low energy defect. We estimate this energy to be 0.3 eV. The barrier W, = 1.45 eV for kink migration is found from the process in Figs. 3(c) and 3(d). The barrier U,, for creating a double kink is then from this calculation 0.3 eV + 1.45 eV = 1.75 eV.
Fig. 3 (a) The perfect reconstructed dislocation: 0 atoms moved to create single atomic kink. (b) Transition state for forming a single atom kink pair. (c) Single atomic kink pair. (b) Transition state for forming a stable double kink. The unit cell is repeated twice in the vertical direction, and only part of the unit cell is shown in the horizontal direction.
Our estimate for W, is consistent with the experimental values, and also the calculated effective activation energy U,, = 1.75 eV/2 + 1.45 eV = 2.32 eV compares well with the experimental value of 2.2 eV. The migration energy W, for the 30” partial dislocation has recently been found by Huang et al. [5]. Here W, was found to be 2.1 + 0.3 eV. The 90” partial is experimentally known to be more mobile, compared with the 30” partial [3]. Although the difference in apparent activation energy has the same magnitude as the experimental resolution, the trend seen in the two calculations is correct. 5. Discussion and conclusions The migration and formation energies for kinks on the 90” partial dislocation are in agreement with the experimental findings. This supports the theory that the 90”-30” dislocation moves in the dissociated state via the formation of stable kinks. One important aspect of the dynamics of dislocations is the close correlation found by Gilman [I] between the highest occupied molecular orbital-lowest unoccupied molecular orbital gap I$,, and the glide activation energy. For homopolar semiconductors the activation energy equals twice the I&, gap. In this theory a bondbreaking process takes an energy of Eh,/2. Gilman considers the “shuffle” type of dislocation, where kink movement involves the breaking of one bond.
188
L.B. Hansen et al. / Materials Science and Engineering 837 (1996) 185-158
In our case the kink migration requires two broken bonds. Consider the transition states in Figs. 3(b) and 3(d). In both cases two bonds are broken, but instead four half-bonds are formed, giving the observed energy barriers. The energy associated with these half-bonds is not included in the theory of Gilman. The fact that the 30”-90” dislocation has been observed experimentally to move in the dissociated state and that the barriers found in this work and in Ref. [5] are in agreement with the barriers deduced from experiment indicate that the processes we are considering are the important processes for the dislocation mobility of this glide partial. A detailed study of the correlation between band structure and energy barrier for this type of process and the relation to the work of Gilman is in progress
[181* Acknowledgements Lars B. Hansen acknowleges support from The Swedish National Board for Industrial and Technical Development and the Swedish Natural Science Research Council and from the EED Contract CHRX-CT93-0134. Kurt Stokbro acknowledges EEC Contract ERBCHBGCT 920180 and Contract ERBCHRXCT 930342 and CNR project Supaltemp.
PI J.J. Gilman,Science, 261 (1993) 1436. P.Grosbras,H. GaramandJ.C. Dcsoyer,Philos. PI J.L. Demenet, Mag. A, 59 (1989) 501. N. Hiller,S.Sauerland, P.SpechtandH. Alexan[31 H. Gottschalk, der, Phys. Status S’olidi, 138 (1993)547. 141J.P. Hirth and J. Lothe, Theory of Dislocntiorrs, Wiley, New
York, 2ndedn.,1982. and0-F. Sankey,Phys. Reo. Lett., 74 151Y.M. Huang,J.C. Spence (1995)3392. PI V.I. Nikitenko,B.Ya. FaberandYu.L, Iunin,Sac.Ply. JETP, 66(4)(1987)738. andJ.K. Norskov,Phys. 171 K. Stokbro,N. Chetty,K.W. Jacobsen Rev. B, 50 (1994)10727, and references citedtherein. andJ.K. Narskov,J. P/Q-S.: Cor~&ru. PI N. Chetty,K.W. Jacobsen Matter, 3 (1991) 5437. and0. Jepsen, in M. Yussouff(cd.), Electrouic [91 O.K. Andesen Bandstrucwe and its Applications, SpringerLecture Notes,
Springer,Berlin1987. and G.Y. Richardson,CRC Crit. Ret!, Solid 1101 MS. Duesbeery State Mater. Sci., 17 (1991). 1111A. GeorgeandJ. Rabier,Rev. Phys. Appl., 22 (1987) 207. WI P.B.Hirch, J. Phys. (Paris), Colloq. C6, 40 (1979) 27. El31 R. Jones,J. Phys. (Puris), Colloq. CG, 40 (1979) 33. B.JoosandD.J. Michel,Plys. Rec. B, 43 (1991) 1141 MS. Duesbery, 5143. [I51 J.R.K. Bigger,D.A. McInnes,A.P. Sutton,M.C. Payne,I. Stich, R.D. King-Smsith, D.M. Bird andL.J. Clarke,Ply. Rer. Lett.,
69(1992)2224. El61B.Ya.Farber,Yu.L. IuninandV.I. Nikitenko,Soo.P1iJzJETP, (1986). El71R. Jones,Philas. Mag. B, 42 (1980) 213. K. Stokbro,B.I. LundqvistandK.W. Jacobsen, to El81L.B. Hansen, bepublished.