- Email: [email protected]
Twinning superlattices
Solid State Communications, Vol. 86, No. 12, pp. 799-802, 1993. Printed in Great Britain.
003%1098/93$6.00+.00 Pergamon Press Ltd
TWINNING SUPERLATTICES Z. Ikonic*, G.P.Srivastava and J.C.Inkson Semiconductor Physics Group, Department of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, UK (Received 25 March 1993 by R. T. Phillips) (acceptedforpubfication
2 1 April 1993)
A new type of superlattice based on a periodic array of twinned stacking faults is proposed. Such superlattices based on Si and Ge are predicted to be character&d by zero miniband energy gaps, evanescent state derived Bloch minibands and minibands with anomalous dispersion. They can be direct band gap materials and offer the significant benefits of simple chemistry, lack of stress and a favoured technological environment.
structures,^.cubic and . hexagonal, . . . .are interleaved but no change ot orientation is involved. At present no clear reports exist of growth of twinning superlattices in semiconductors. However, ribbons of hexagonal material (corresponding to a twinning sequence) have been produced6 when silicon is indented in the temperature range 400-650’. Similarly, in stress-deformed monocrystals large area single stacking faults have been indicated, not only in silicon but in III-V crystals as we117, although this method is probably too crude for making twinning superlattices. There has also been a report8 on a more refined method of fabrication of a single, large-area, low-angle twist boundarv in silicon that in nrincinle allows for making multiple stacking faults, but-it is not certain if the method could be extended to growing ultra thin layer structures. As stacking fault formation &ergies are very low, growing twinning superlattices may be possible, perhaps by applying the indentation technique used by Pirouz et al6 or an appropriate stress (variable in direction and magnitude) during the crystal growth that would favour particular stacking sequences. The 1800 rotation of the crystal on either side of a stacking fault has nontrivial consequences for its electronic structure. The interface between the two orientations is perfectly lattice-matched but at the same time highly wavefunction-symmetry-mismatched. This gives rise to large electron (intervalley) scattering at the interface, especially for energies close to the band edge9. In the case of a periodic structure, as proposed here, we show that this leads to the appearance of prominent miniband structures. Twinning superlattices thus form a completely new class of systems very different from commonly studied heterojunction based superlattices, either strained or unstrained, where the wavefunctions in the two layers are “symmetry-similar”. They have the advantage of chemical simplicity and operation in a favoured environment compatible with standard semiconductor technology. To understand how the scattering comes about in these superlattices, consider the interface Brillouin zone reflecting the two dimensional symmetry (Fig.1). The bulk Brillouin zone I- point and two of the L (“directed” --_ along the [l_l l] and [ 11l] directions) project into r’. Each of the six M points has projected onto it an L (at kL say)
In this Communication we investigate electronic properties of a new type of superlattice - twinning superlattices, based on purposely built-in periodic reversals of atomic plane stacking sequences in semiconductors such as silicon and germanium. Although at present there is no means of fabricating such superlattices, our work presents some interesting ideas on band structure engineering. Stacking faults are one of the most common types of defects in crystalline diamond-type semiconductors such as silicon and germanium. They correspond to changes of atomic plane stacking sequence in the [ 11 l] direction, as compared to the perfect crystal, without breaking any bonds, and have very low formation energies. Occurring unintentionally, e.g. due to strain, they may have a For considerable effect on semiconductor devices. example, stacking faults are usually observed in silicon near silicon-silicon-dioxide interfaces bridging two partial dislocationsl. The most elementary stacking fault is the twin stacking fault (or twin boundary, 1800 twist boundary) and comprises the reversal of the stacking sequence at some plane2, i.e.. AA’BB’CC’AIA’CC’BB’AA. It may be constructed by cutting the [ 11l] directed bonds in AA’ layer, rotating half of the crystal by 1800 about a (bond) axis, and then reconnecting all cut bonds of the two crystal halves. The other two common types of stacking faults, intrinsic and extrinsic, having one missing or one extra layer, respectively, in an otherwise unperturbed crystal, may also be viewed as two twin stacking faults separated by a single (intrinsic) or double (extrinsic) monolayer of reverse oriented material. Twinning is commonly observed in crystals with well-developed faces3. It is important to note that twinning superlattices, also known as polysynthetic twins (not necessarily strictly periodic), exist in a number of insulating minerals3 in nature, notably the plagioclase feldspars and albite. Similarly, pure single crystals of calcium mesodigermanate grown from melt have been found to display regular twinning4. It is well known that Sic and ZnS show naturally occuring examples of where two types of crystal polytype superlattices
1
* Permanent address: Faculty of Engineering, University of Belgrade, Bulevar Revoluije 73, 11000 Belgrade, Yugoslavia 799
TWINNING
800
Vol. 86, No.
SUPERLATTICES
Gi(X,L) -
0
/
K
I I/ I // 1,’ I F u-9L)
I
(a) 1
2
3
4(“,n;
6
I
I
7
*
Fig. 1. The (111) interface Brillouin zone, with mapping of some important points of bulk Brillouin zone.
and an X (at kX=-2kL) point, making f and M the most important k points to be exploredlO. In the two reverseoriented layers of material forming the stacking fault, due to the 1800 rotation, the signs of kL and kX at the same l?l point are reversed. Consequently the X (L) states of the same energy and parallel wavevector on either side of the fault are very different states, say Xa (La) and Xh (Lb) because they have quite different cell-periodic parts arising from their origin in different valleys. There is no direct match for the Xa state in the b layer, and vice versa. The electronic states which span both sides of the fault will, of necessity, then be formed from an admixtures of states, including Bloch and evanescent modes, just as at the junction of two completely different crystalsll. This mixing is expected to be more significant in germanium than in silicon because the energy difference between the two lowest conduction band states (X,L) is much smaller in the former12. At the F point the symmetry is higher, and the band structure is symmetric, so the effect will be smaller. From the above we see why this type of interface can not be understood within a simple model. Because neither the potential nor the effective mass changes across the fault, methods like the effective-mass theory, which do not recognise the cell-periodic part of the wavefunction, see no defect at all. Only microscopic methods can be used for this system. Most calculations1S-20 performed on stacking faults have considered their formation energy, phonon spectra, existence of interface bound states, though Stiles and Hamanno have dealt with electron transmission through them. In calculating the electronic structure of twinning we have used the superlattice superlattices, modification21 of the S-matrix semi-empirical pseudopotential layer method1 l, that guarantees high stability even for large superlattice periods. This is expected to be quite accurate since ab - initio calculationsl6-18 of atomic relaxation and self-consistent charge redistribution in stacking faults show that these effects do not have very large influence on their electronic properties. Such a calculation gives all the band structure properties of twinning superlattices arising from band mixing and bulk Brillouin zone folding. The formfactors for silicon and germanium are taken from Refs. 22 and 23, respectively. Consider now the low energy miniband electronic structure of the silicon-based (m,n) superlattices. A (m,n) twinning superlattice period in diamond-type materials has m and nbilayers (two atomic monolayers) of reverseoriented material along [ill]. (Special cases (n.1) with n>l, and (n.2) with n>2 may be, in accordance with the
Fig.2. (a) Allowed minibands in siJcon-based (n,n) twinning superlattices, close to the M point (X-valley bottom), from which the energy is measured. The first four minibands (I - IV) are shown. Solid lines connect miniband edges at kSL=O, and broken lines show the zero energy gap points at ksLd=n. (b) Dispersion of the lowest two minibands in silicon (6,6) twinning superlattice. (c) Envelope wavefunctions and current j components for the (6,6) superlattice resolved into bulk L, Xl, X3 contributions (solid, broken and dotted lines respectively) at the k point A, close to the fist miniband edge. The ticks along the x-axis denote atomic bilayers, and the vertical line indicates the interface between the two orientations of the material; (d) Minibands in silicon (n.1) intrinsic stacking fault superlattice, exhibiting no zero energy gaps.
existing terminology, called intrinsic and extrinsic stacking fault superlattices, respectively.) For n=l-8, the electronic structure at the bulk conduction band minimum (close to the M point) is given in Fig.2(a). The feature to note is the occurrence of zero energy gaps at the superlattice Brillouin zone boundary, i.e.. pairs of minibands are joined back-to-back but separated by a finite gap at zone centre. This is a general feature throughout the interface Brillouin zone, and is a consequence of a screw symmetry characterising the twinning superlattice unit cell. Zero energy gaps may also be found in conventional superlattices, especially in effective mass theory calculations24, but are usually restricted to some specific values of electron transverse wavevector, and have different physical properties. The
Vol. 86, No. 12
TWINNING SUPERLATTICES
dispersion along the superlattice axis of the lowest pair of minibands in the (6,6) twin superlattice is displayed in Fig.2(b). It is mirror symmetric with the lowest point at the Brillouin zone centre, and so corresponds to a Cauasijdirect band eao material. The wavefunctions. &en in Fig.Z(c), indicate that the Xt state dominates this miniband, as expected, but X3 and L states, though highly evanescent, are by no means negligible, due to the symmetry mismatch at the interface. The same applies to current components. Other superlattice configurations, like (n,l), lacking the screw symmetry, do not have zero energy gaps, but separated- minibands with normal disoersion. as seen in Fin.2(d). We note. however, that m&band extrema for (m$),.m#n, SLs are positioned off the high symmetry points of the superlattice Brillouin zone. Since bulk germanium is an L-type indirect gap semiconductor, both F and z points are equally important, and experimentally accessible, in twinning superlattices. The miniband structure of germanium
(n,n)
(dl
10)
Fig.3. (a) Miniband structure of germanium based (n,n) twinning superlattices at the F point. Energy is measured from the conduction band edge at F, which coincides with the L valley bottom, and the notation is given in Fig.2. (b,c) Dispersion of the lowest two minibands i,n germanium (5.5) and (6.6) twinning superlattices at the M poin_t.Energy is measured from the conduction band edge at M, which coincides with the L valley bottom. (d,e) Envelope wavefunctions and current components resolved into bulk L, Xl, X3 contributions (solid, broken and dotted lines respectively), calculated at points A and C indicated in (b) (slightly off the miniband extrema) in (6,6) twinning superlattice. The ticks along the x-axis denote atomic bilavers. and the vertical line indicates the interface between*the’two orientations of the material. The situation at point B is quite similar to that at point A.
801
twinning superlattices at r is given in Fig.3(a). Similar to that in silicon-based twinning superlattices, the minibands in these superlattices are characterised by zero energy gaps. In addition, note that there also exist minibands (I and II) which are derived from bound (evanescent) states. Miniband I lies below the conduction band minimum for all n. Miniband II has a larger band width, and for n I 5 covers energies both above and below the conduction band minimum. As the superlattice period increases, these minibands tend towards a single bound state leve125. Exhibiting almost purely L character, the associated wavefunctions are composed of pairs of growing and decaying evanescent states, the admixture of I being significant for higher energies only. Silicon-based superlattices have essentially the same features as germanium-has@ ones at r, but their minibands are -1 eV above those at M, so may not be expected to have a large influence on most electronic and optical properties of these structures. The electronic structure of the germanium-based twinning superlattices at the M point is remarkably different from the one in silicon, as shown in the two examples in Fig.3(b,c). From the bulk bandstructure, it is the L:derived bands which are expected to be dominant at the fl noint. The minibands form pairs, one with anomalous dispersion joined via a zero energy gap at the zone boundary with one with normal dispersion. Of the two adjacent minibands, it is the lower one that has anomalous dispersion for (odd, odd), and the higher one in (even, even) superlattices (Fig.3(b) and 3(c), respectively). Analysis of the wavefunctions at some characteristic points, Fig.3(d,e), show that X states, although evanescent, are very highly excited, contributing a larger part of the overall charge density than the propagating L states, and are also responsible for a considerable fraction of total current. The extrema of the minibands with anomalous dispersion are character&d by large current counter-flows derived from X and L states, that cancel at the exact extremum point (on the other hand, the current at the Brillouin zone centre vanishes because all its components vanish). Thus the anomalous dispersion is due to the existence of X- related interface bands close to the energy of the L derived Bloch states. We have found these states over almost the whole interface Brillouin zone for the isolated stacking fault. The wavefunctions off the zone center and the total current also have very different compositions in the two layers caused by asymmetry of the underlying single layer bulk dispersion. The effect of anomalous dispersion disappears at higher energies, and minibands with higher indices behave normally, just _ as in silicon (Fig.2(b)). The twinning SLs considered here offer possibilities of unusual electron transport and optical properties over conventional SLs. For example, in -contrast with conventional SLs the (n.1) SLs construct ed from Si show a decrease in the effective mass of the first miniband with increase in SL period (m*/m = 2.0,0.73,0.53 and 0.41 for n=3,5,6 and 8, respectively). [The effective mass of Si (n-n) SLs shows oscillatory behaviour with increasing n.] In’general the energies and widths of minibands in the twinning SLs are such that thev would enable (for n-doped materials) intraband abso&on (i.e. inter-miniband) in the infrared, as can be appreciated from Fig. 2. As this behaviour is due to superlattice miniband formation, it differs from the situation in a bulk crystal of say silicon. Finally, in spite of the fact that the constituent bulk semiconductor is an indirect band gap material, interband (between valence band and conduction band minibands) transitions are possible in twinning SLs because the resulting folded conduction band is direct. In conclusion, we have proposed growth of twinning
802
TWINNING
superlattices and shown that in silicon and germanium such structures offer almost as much versatility in producing useful electronic miniband structure as there exists in ordinary heterostructure-based superlattices. This information combined with the two principles of building superlattice periodicity (viz. variation of material
SUPERLATTICES
Vol. 86. No. 12
composition and of orientation) may be used to extend the possibilities of “band structure engineering” even more. Acknowledgements - The authors would like to thank the SERC (UK) for computational facilities through the CSI scheme. One of authors (Z.I.) is grateful to the Royal Society (UK) for support
REFERENCES
1. L F Ray and D J H Cockayne, Philos Mag 22, 853 (1970) 2. J P Stark, Phys Rev B 38, 1139 (1988) 3. C A Sorrel and G F Sandstrom, “The rocks and minerals of the world” (Collins, London, 1977) 4. P J Born, D S Robertson, P W Smith and I M Young, J Mater Sci Lett 6,635 (1987) 5. G B Dubrovokij and A A Lepneva, Fizika Tverdogo Tela 19, 1252 (1977) 6. P Pirouz, R Cl&m,-U Dahmen and K H Westmacott, Acta metal1 mater 38.3 13 (1990) 7. A.Lefebvre, Y.Androussi and G.Vanderschaeve. Philos.Mag.Lett. 56, 135 (1987) 8. G.C.Perreault. S.L.Hvland and D.G.Ast. Philos.Mae. A64,587 (1991) 9. M.D.Stiles and D.R.Hamann, Phys. Rev. B38 2021 (1988); ibid Phys.Rev., B41,5280 (1990) 10. More precisely, kL=(1/243)(2rc/a), with either + or sign, depending on the M point chosen, where a is the cubic lattice constant. 11. D.Y.K.Ko and J.C.Inkson, Phys. Rev. B38, 9945 (1988) 12. In Si the X. Land I- states are as much as -1eV apart (at the exact M point X1 and X3 states are degenerate, but at thesore important point of Xl valley minimum, close to M, the X3 state is -0.5 eV above and also highly evanescent). On the other hand, F and X states in Ge are only -0.2 eV above the lowest (L) states. 13. L.J.Chen and L.M.Falicov, PhilosMag. 29, 1 (1974)
14. E.Chacon, C.Teiedor and F.Flores, Phvs.Stat.Sol b98, K117 (1980) ” 15. S.Marklund, Phys.Stat.Sol. blO8.97 (1981) 16. L.F.Mattheiss and J.R.Patel, Phys.Rev. B23, 5384 (1981) 17. J.Sanches-Dehesa, J.A.Verges and CTejedor, PhysRev. B24,1006 (1981) 18. M.Y.Chou, M.L.Cohen and S.G.Louie, Phys.Rev. B32.7979 (1985) 19. Z.Zhao-bo and L.Jin-line. J.Phvs.C: Solid State Phvs. . 19,6739 (1986) 20. A.Gross and H.Teichler, Philos.Mag. B64,413 (1991) 21. Z.Ikonic, G.P.Srivastava and J.C.Inkson, Phys Rev B46,15150 (1992) 22. J.R.Chelikowski and M.L.Cohen, Phys.Rev. B14,556 (1976) 23. M.L.Cohen and T.K.Bergstresser, PhysRev. 141,789 (1966) 24. V.Milanovic, and Z.Ikonic, Phys.Rev., B37, 7125 (1988) 25. We have also made calculations of transmission and interface bound states in single-twin stacking faults. Results for germanium are generally similar to those in silicon (given in Refs.16 and 17), with an interface bound state 39 meV (89 meV) below the conduction band ed&e of germanium (silicon) at F but none at the exact M point (however, the interface state in germanium persists from r’ almost to fi). In both cases the bound states are very purely L in character.
003%1098/93$6.00+.00 Pergamon Press Ltd
TWINNING SUPERLATTICES Z. Ikonic*, G.P.Srivastava and J.C.Inkson Semiconductor Physics Group, Department of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, UK (Received 25 March 1993 by R. T. Phillips) (acceptedforpubfication
2 1 April 1993)
A new type of superlattice based on a periodic array of twinned stacking faults is proposed. Such superlattices based on Si and Ge are predicted to be character&d by zero miniband energy gaps, evanescent state derived Bloch minibands and minibands with anomalous dispersion. They can be direct band gap materials and offer the significant benefits of simple chemistry, lack of stress and a favoured technological environment.
structures,^.cubic and . hexagonal, . . . .are interleaved but no change ot orientation is involved. At present no clear reports exist of growth of twinning superlattices in semiconductors. However, ribbons of hexagonal material (corresponding to a twinning sequence) have been produced6 when silicon is indented in the temperature range 400-650’. Similarly, in stress-deformed monocrystals large area single stacking faults have been indicated, not only in silicon but in III-V crystals as we117, although this method is probably too crude for making twinning superlattices. There has also been a report8 on a more refined method of fabrication of a single, large-area, low-angle twist boundarv in silicon that in nrincinle allows for making multiple stacking faults, but-it is not certain if the method could be extended to growing ultra thin layer structures. As stacking fault formation &ergies are very low, growing twinning superlattices may be possible, perhaps by applying the indentation technique used by Pirouz et al6 or an appropriate stress (variable in direction and magnitude) during the crystal growth that would favour particular stacking sequences. The 1800 rotation of the crystal on either side of a stacking fault has nontrivial consequences for its electronic structure. The interface between the two orientations is perfectly lattice-matched but at the same time highly wavefunction-symmetry-mismatched. This gives rise to large electron (intervalley) scattering at the interface, especially for energies close to the band edge9. In the case of a periodic structure, as proposed here, we show that this leads to the appearance of prominent miniband structures. Twinning superlattices thus form a completely new class of systems very different from commonly studied heterojunction based superlattices, either strained or unstrained, where the wavefunctions in the two layers are “symmetry-similar”. They have the advantage of chemical simplicity and operation in a favoured environment compatible with standard semiconductor technology. To understand how the scattering comes about in these superlattices, consider the interface Brillouin zone reflecting the two dimensional symmetry (Fig.1). The bulk Brillouin zone I- point and two of the L (“directed” --_ along the [l_l l] and [ 11l] directions) project into r’. Each of the six M points has projected onto it an L (at kL say)
In this Communication we investigate electronic properties of a new type of superlattice - twinning superlattices, based on purposely built-in periodic reversals of atomic plane stacking sequences in semiconductors such as silicon and germanium. Although at present there is no means of fabricating such superlattices, our work presents some interesting ideas on band structure engineering. Stacking faults are one of the most common types of defects in crystalline diamond-type semiconductors such as silicon and germanium. They correspond to changes of atomic plane stacking sequence in the [ 11 l] direction, as compared to the perfect crystal, without breaking any bonds, and have very low formation energies. Occurring unintentionally, e.g. due to strain, they may have a For considerable effect on semiconductor devices. example, stacking faults are usually observed in silicon near silicon-silicon-dioxide interfaces bridging two partial dislocationsl. The most elementary stacking fault is the twin stacking fault (or twin boundary, 1800 twist boundary) and comprises the reversal of the stacking sequence at some plane2, i.e.. AA’BB’CC’AIA’CC’BB’AA. It may be constructed by cutting the [ 11l] directed bonds in AA’ layer, rotating half of the crystal by 1800 about a (bond) axis, and then reconnecting all cut bonds of the two crystal halves. The other two common types of stacking faults, intrinsic and extrinsic, having one missing or one extra layer, respectively, in an otherwise unperturbed crystal, may also be viewed as two twin stacking faults separated by a single (intrinsic) or double (extrinsic) monolayer of reverse oriented material. Twinning is commonly observed in crystals with well-developed faces3. It is important to note that twinning superlattices, also known as polysynthetic twins (not necessarily strictly periodic), exist in a number of insulating minerals3 in nature, notably the plagioclase feldspars and albite. Similarly, pure single crystals of calcium mesodigermanate grown from melt have been found to display regular twinning4. It is well known that Sic and ZnS show naturally occuring examples of where two types of crystal polytype superlattices
1
* Permanent address: Faculty of Engineering, University of Belgrade, Bulevar Revoluije 73, 11000 Belgrade, Yugoslavia 799
TWINNING
800
Vol. 86, No.
SUPERLATTICES
Gi(X,L) -
0
/
K
I I/ I // 1,’ I F u-9L)
I
(a) 1
2
3
4(“,n;
6
I
I
7
*
Fig. 1. The (111) interface Brillouin zone, with mapping of some important points of bulk Brillouin zone.
and an X (at kX=-2kL) point, making f and M the most important k points to be exploredlO. In the two reverseoriented layers of material forming the stacking fault, due to the 1800 rotation, the signs of kL and kX at the same l?l point are reversed. Consequently the X (L) states of the same energy and parallel wavevector on either side of the fault are very different states, say Xa (La) and Xh (Lb) because they have quite different cell-periodic parts arising from their origin in different valleys. There is no direct match for the Xa state in the b layer, and vice versa. The electronic states which span both sides of the fault will, of necessity, then be formed from an admixtures of states, including Bloch and evanescent modes, just as at the junction of two completely different crystalsll. This mixing is expected to be more significant in germanium than in silicon because the energy difference between the two lowest conduction band states (X,L) is much smaller in the former12. At the F point the symmetry is higher, and the band structure is symmetric, so the effect will be smaller. From the above we see why this type of interface can not be understood within a simple model. Because neither the potential nor the effective mass changes across the fault, methods like the effective-mass theory, which do not recognise the cell-periodic part of the wavefunction, see no defect at all. Only microscopic methods can be used for this system. Most calculations1S-20 performed on stacking faults have considered their formation energy, phonon spectra, existence of interface bound states, though Stiles and Hamanno have dealt with electron transmission through them. In calculating the electronic structure of twinning we have used the superlattice superlattices, modification21 of the S-matrix semi-empirical pseudopotential layer method1 l, that guarantees high stability even for large superlattice periods. This is expected to be quite accurate since ab - initio calculationsl6-18 of atomic relaxation and self-consistent charge redistribution in stacking faults show that these effects do not have very large influence on their electronic properties. Such a calculation gives all the band structure properties of twinning superlattices arising from band mixing and bulk Brillouin zone folding. The formfactors for silicon and germanium are taken from Refs. 22 and 23, respectively. Consider now the low energy miniband electronic structure of the silicon-based (m,n) superlattices. A (m,n) twinning superlattice period in diamond-type materials has m and nbilayers (two atomic monolayers) of reverseoriented material along [ill]. (Special cases (n.1) with n>l, and (n.2) with n>2 may be, in accordance with the
Fig.2. (a) Allowed minibands in siJcon-based (n,n) twinning superlattices, close to the M point (X-valley bottom), from which the energy is measured. The first four minibands (I - IV) are shown. Solid lines connect miniband edges at kSL=O, and broken lines show the zero energy gap points at ksLd=n. (b) Dispersion of the lowest two minibands in silicon (6,6) twinning superlattice. (c) Envelope wavefunctions and current j components for the (6,6) superlattice resolved into bulk L, Xl, X3 contributions (solid, broken and dotted lines respectively) at the k point A, close to the fist miniband edge. The ticks along the x-axis denote atomic bilayers, and the vertical line indicates the interface between the two orientations of the material; (d) Minibands in silicon (n.1) intrinsic stacking fault superlattice, exhibiting no zero energy gaps.
existing terminology, called intrinsic and extrinsic stacking fault superlattices, respectively.) For n=l-8, the electronic structure at the bulk conduction band minimum (close to the M point) is given in Fig.2(a). The feature to note is the occurrence of zero energy gaps at the superlattice Brillouin zone boundary, i.e.. pairs of minibands are joined back-to-back but separated by a finite gap at zone centre. This is a general feature throughout the interface Brillouin zone, and is a consequence of a screw symmetry characterising the twinning superlattice unit cell. Zero energy gaps may also be found in conventional superlattices, especially in effective mass theory calculations24, but are usually restricted to some specific values of electron transverse wavevector, and have different physical properties. The
Vol. 86, No. 12
TWINNING SUPERLATTICES
dispersion along the superlattice axis of the lowest pair of minibands in the (6,6) twin superlattice is displayed in Fig.2(b). It is mirror symmetric with the lowest point at the Brillouin zone centre, and so corresponds to a Cauasijdirect band eao material. The wavefunctions. &en in Fig.Z(c), indicate that the Xt state dominates this miniband, as expected, but X3 and L states, though highly evanescent, are by no means negligible, due to the symmetry mismatch at the interface. The same applies to current components. Other superlattice configurations, like (n,l), lacking the screw symmetry, do not have zero energy gaps, but separated- minibands with normal disoersion. as seen in Fin.2(d). We note. however, that m&band extrema for (m$),.m#n, SLs are positioned off the high symmetry points of the superlattice Brillouin zone. Since bulk germanium is an L-type indirect gap semiconductor, both F and z points are equally important, and experimentally accessible, in twinning superlattices. The miniband structure of germanium
(n,n)
(dl
10)
Fig.3. (a) Miniband structure of germanium based (n,n) twinning superlattices at the F point. Energy is measured from the conduction band edge at F, which coincides with the L valley bottom, and the notation is given in Fig.2. (b,c) Dispersion of the lowest two minibands i,n germanium (5.5) and (6.6) twinning superlattices at the M poin_t.Energy is measured from the conduction band edge at M, which coincides with the L valley bottom. (d,e) Envelope wavefunctions and current components resolved into bulk L, Xl, X3 contributions (solid, broken and dotted lines respectively), calculated at points A and C indicated in (b) (slightly off the miniband extrema) in (6,6) twinning superlattice. The ticks along the x-axis denote atomic bilavers. and the vertical line indicates the interface between*the’two orientations of the material. The situation at point B is quite similar to that at point A.
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twinning superlattices at r is given in Fig.3(a). Similar to that in silicon-based twinning superlattices, the minibands in these superlattices are characterised by zero energy gaps. In addition, note that there also exist minibands (I and II) which are derived from bound (evanescent) states. Miniband I lies below the conduction band minimum for all n. Miniband II has a larger band width, and for n I 5 covers energies both above and below the conduction band minimum. As the superlattice period increases, these minibands tend towards a single bound state leve125. Exhibiting almost purely L character, the associated wavefunctions are composed of pairs of growing and decaying evanescent states, the admixture of I being significant for higher energies only. Silicon-based superlattices have essentially the same features as germanium-has@ ones at r, but their minibands are -1 eV above those at M, so may not be expected to have a large influence on most electronic and optical properties of these structures. The electronic structure of the germanium-based twinning superlattices at the M point is remarkably different from the one in silicon, as shown in the two examples in Fig.3(b,c). From the bulk bandstructure, it is the L:derived bands which are expected to be dominant at the fl noint. The minibands form pairs, one with anomalous dispersion joined via a zero energy gap at the zone boundary with one with normal dispersion. Of the two adjacent minibands, it is the lower one that has anomalous dispersion for (odd, odd), and the higher one in (even, even) superlattices (Fig.3(b) and 3(c), respectively). Analysis of the wavefunctions at some characteristic points, Fig.3(d,e), show that X states, although evanescent, are very highly excited, contributing a larger part of the overall charge density than the propagating L states, and are also responsible for a considerable fraction of total current. The extrema of the minibands with anomalous dispersion are character&d by large current counter-flows derived from X and L states, that cancel at the exact extremum point (on the other hand, the current at the Brillouin zone centre vanishes because all its components vanish). Thus the anomalous dispersion is due to the existence of X- related interface bands close to the energy of the L derived Bloch states. We have found these states over almost the whole interface Brillouin zone for the isolated stacking fault. The wavefunctions off the zone center and the total current also have very different compositions in the two layers caused by asymmetry of the underlying single layer bulk dispersion. The effect of anomalous dispersion disappears at higher energies, and minibands with higher indices behave normally, just _ as in silicon (Fig.2(b)). The twinning SLs considered here offer possibilities of unusual electron transport and optical properties over conventional SLs. For example, in -contrast with conventional SLs the (n.1) SLs construct ed from Si show a decrease in the effective mass of the first miniband with increase in SL period (m*/m = 2.0,0.73,0.53 and 0.41 for n=3,5,6 and 8, respectively). [The effective mass of Si (n-n) SLs shows oscillatory behaviour with increasing n.] In’general the energies and widths of minibands in the twinning SLs are such that thev would enable (for n-doped materials) intraband abso&on (i.e. inter-miniband) in the infrared, as can be appreciated from Fig. 2. As this behaviour is due to superlattice miniband formation, it differs from the situation in a bulk crystal of say silicon. Finally, in spite of the fact that the constituent bulk semiconductor is an indirect band gap material, interband (between valence band and conduction band minibands) transitions are possible in twinning SLs because the resulting folded conduction band is direct. In conclusion, we have proposed growth of twinning
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superlattices and shown that in silicon and germanium such structures offer almost as much versatility in producing useful electronic miniband structure as there exists in ordinary heterostructure-based superlattices. This information combined with the two principles of building superlattice periodicity (viz. variation of material
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Vol. 86. No. 12
composition and of orientation) may be used to extend the possibilities of “band structure engineering” even more. Acknowledgements - The authors would like to thank the SERC (UK) for computational facilities through the CSI scheme. One of authors (Z.I.) is grateful to the Royal Society (UK) for support
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14. E.Chacon, C.Teiedor and F.Flores, Phvs.Stat.Sol b98, K117 (1980) ” 15. S.Marklund, Phys.Stat.Sol. blO8.97 (1981) 16. L.F.Mattheiss and J.R.Patel, Phys.Rev. B23, 5384 (1981) 17. J.Sanches-Dehesa, J.A.Verges and CTejedor, PhysRev. B24,1006 (1981) 18. M.Y.Chou, M.L.Cohen and S.G.Louie, Phys.Rev. B32.7979 (1985) 19. Z.Zhao-bo and L.Jin-line. J.Phvs.C: Solid State Phvs. . 19,6739 (1986) 20. A.Gross and H.Teichler, Philos.Mag. B64,413 (1991) 21. Z.Ikonic, G.P.Srivastava and J.C.Inkson, Phys Rev B46,15150 (1992) 22. J.R.Chelikowski and M.L.Cohen, Phys.Rev. B14,556 (1976) 23. M.L.Cohen and T.K.Bergstresser, PhysRev. 141,789 (1966) 24. V.Milanovic, and Z.Ikonic, Phys.Rev., B37, 7125 (1988) 25. We have also made calculations of transmission and interface bound states in single-twin stacking faults. Results for germanium are generally similar to those in silicon (given in Refs.16 and 17), with an interface bound state 39 meV (89 meV) below the conduction band ed&e of germanium (silicon) at F but none at the exact M point (however, the interface state in germanium persists from r’ almost to fi). In both cases the bound states are very purely L in character.