Electronic structure of SiSi1−xGex and SiSi1−xSnx strained layer superlattices

Superlattices and Microstructures, Vol. 2, No. 4, 1986

329

ELECTRONIC STRUCTURE OF Si/Si1_xGex SUPERLATTICES

AND Si/Si1_xSn x STRAINED LAYER

Ian Morrison and M. Jaros Department of Theoretical Physics, T h e U n i v e r s i t y , Newcastle upon Tyne, United Kingdom. ( recieved August

17 1986

)

We present a quantitative study of the electronic structure and zone folding in Si/SiGe and Si/SISn s t r a i n e d layer superlattices. We find t h a t the e f f e c t of s t r a i n is to increase the degree of e l e c t r o n i c c o n f i n e m e n t a n d to enhance the optical matrix element a c r o s s the fundamental superlattice gap, although this l a t t e r e f f e c t is s m a l l in the systems studied.

~.

INTRODUCTION

It is now possible using modern crystal growing techniques to manufacture multilayer systems from constituents with quite different lattice constants. These strained structures offer interesting opportunities in band structure engineering, with applications in optoelectronics (11 Silicon/germanium and silicon/tin are examples of such strained layer superlattices. Because of the particular form of the bulk band structures, these systems promise to give rise to enhanced optical transitions across the superlattice gap. The purpose of this study is to evaluate the strength of this effect. Until very recently, the only Si-based systems considered in the literature have been those consisting of Si and SiGe layers. However, the work of Grossmann et al (2) shows that vigorous efforts are being made to use ~ - t i n in order to widen the scope for optimisation. Although there are considerable growth problems with G-tin, we believe than an exploratory calculation involving tin is desirable. 2.

superlattice period. This means that the strain is taken up by both constituent materials (tensile strain in silicon and compressive strain in the silicon alloy).

NO I I I I

STRAIN

Si

c.b. I ! I I I

I

I

'Si 6e I

'I

I

I

l,

1

I I

I i I I

i I

STRAINED

Si

Si 6e

cb

METHOD OF CALCULATION

It has been shown (3), for sufficiently thin superlattice layers, that in directions parallel to the interfaces the constituent materials will take up the lattice constant of the buffer layer upon which they are grown. The systems modeled here are chosen so that the buffer layer is matched to the

0749-6036/86/040329+05 $02.00/0

I

I

I

Figure I - A schematic diagram showing the conduction band offsets of Si/Si0.5Ge0. 5 as described in the text. The way the offsets change with the inclusion of strain is shown.

© 1986 Academic Press Inc. (London) Limited

Superlattices and Microstructures, VoL 2, No. 4, 1986

330

Energy (eV) ! I I

1.0

- _ _

0.95

3'

- -

2'

51J &l

1.00

~

- -

3

61

- - 2 3'

,--p

,,-Q- - I I

0 9O - -

3' 2'

"P

UNSTRAINED

~P ~Q

STRAINED

l&O ~ PERIOD

UNSTRAINED STRAINED UNSTRAINED STRAINED 70 ~ PERIOD 20 ~ PERIOD

Figure 2 - The energy levels of the first few conduction band states of the Si/SiGe superlattices described in the text. All energies are measured from the valence band edge of Si0.5Ge0.5 as

are

The calculations based on the

described here pseudopotential

approach of Jaros et al (4). The Hamiltonian used is of the form H=Ho+V , where in this case H is the Hamilo tonian of the overall alloy concentration of the superlattice (i.e. of the buffer layer) and V is the difference between the actual potential of the superlattice and the potential of the buffer layer. The calculation proceeds by expanding the superlattice wavefunctions, ~,in terms of the eigenfunctions o f the buffer, ~n,k, giving an equation of the form:(Ho-E+V) ~ = 0

(I)

with ~: ~ An, k ~n,k n,k

Here k is the reduced wavevector and n the band index. For the calculation performed at ~ of the superlattice the ~'s needed in the expansion are lim-

seen by the unstrained calculation. Also shown are the conduction band edges of Si, P, and Si0.5Ge0.5" Q, as seen by the unstrained calculation in each case.

ited, by symmetry, to ones equal to superlattice reciprocal lattice vectors. Equation (I) is then multiplied from the left by ~n,k and integrated over all space to give the secular equation:-

An,,k,(E~, k' -E

) +

~ An,k I ~ , ~ V ~ n , k d 3 ~

=0

(2)

This is solved by a direct diagonalisation procedure to give the eigenvalues, E, and eigenfunctions, An,k. The potential introduced in (I) is constructed from atomic pseudopotentials. No account of non locality or spin-orbit coupling is included and all alloys are treated using the virtual crystal approximation. In setting up the potential it is necessary to specify the atomic positions in the superlattice unit cell. Two calculations are performed here. In the "unstrained" case

Superlattices and Microstructures, Vol. 2, No. 4, 1986 all atomic positions are the same as in the buffer layer. In the "strained" calculation the atomic positions are shifted in directions perpendicular to the interfaces, from the positions in the "unstrained" calculation, such that the lattice spacing of the two constituents is the same as the corresponding bulk materials. The lattice spacing in directions parallel to the interfaces is left unchanged. This realistically models a superlattice grown on a buffer layer matched to the superlattice period.

3.

RESULTS

1.2 1.1 3 2 1

1.0 0.9

!

0.4. 0.3

The results of our calculations of superlattices made from Si/Si0.5Ge0.5 and Si/Si0.75Sn0.25 ' with approximately equal well widths are described here. A more detailed account of Si/Si0.5Ge0.5 can be found in reference

331

0.2 0.1

m

(5).

The results of Si0.5Ge0.5 superlattices with three different periods (22A, 71A and 137A) are shown here. Without the inclusion of strain in determining the band offsets for such superlattices it is expected that the first conduction band states will be confined in the alloy layers. However, the experimental results of Abstreiter et al. (6) show that the confinement is in the silicon layers. The inclusion of strain is expected to shift the band alignment so that the confinement might occur in silicon, see figure (1). This view is supported by our calculations for in the "unstrained" cases the first conduction band states are alloy confined whereas in the "strained" calculation the confinement shifts to the silicon layers. This view is also supported by the results of People and Bean (7) based on deformation potential theory. The energy eigenvalues, of the conduction band states, of our calculations are shown in figure (2). It is worth mentioning that the valence band splittings, not shown here, are in good agreement with calculations performed by People(8). Calculations were also performed on a Si/Si0.75Sn0.25" of 75A period. The results here show a great deal of similarity with the Si/Si0.5Ge0.5 superlattice. Again the confinement of the first conduction band state shifts from the alloy to silicon between the

0.0

LI -0.1

UNSTRAINED

STRAINED

Figure 3 - The energy levels of the states near the band edges of the Si/SiSn superlattice described in the text. All energies (in eV) are measured. from the valence band edge of Si0.75Sn0.25 as seen by the unstrained calculation. States I-III are valence band states and states I-4 are conduction band states.

"unstrained" and the "strained" calculations. The energies of these calculations are displayed in figure (3) and the charge densities associated with the first two "strained" conduction band states are shown in figure (4). Compared to the "unstrained" results the wavefunctions are much more highly confined, indicating that the strain in the system creates a very large effective barrier between the constituents. 4.

OPTICA~ PROPERTIES AND ZONE FOLDING

In our method of calculation the optical matrix element across the superlattice gap can be easily estimated from the contribution of the ~ s t a t e in the first conduction band of

332

Superlattices and Microstructures, Vol. 2, No. 4, 1986

]

SISn

SJ

Figure 4 - The charge densities associated with the first two conduction band states in the strained Si/Si0.75Sn0.25_

calculation. The charge densities are plotted along a line in the <001> direction going through bond centres.

the buffer layer to the first superlattice conduction band state. This is because the uppermost valence band states are very highly localised near ~. The introduction of strain into the calculation is expected to increase this contribution (the zone folding effect). However even in the strained calculations this contribution is still very low. A plot of the IAn,kl2's asso-

a very high tin content and hence a large degree of strain in the constituents.

ciated with the first conduction band state in the Si/Si0.75Sn0.25 strained calculation is shown in figure (5). It is seen that this state is highly localised around the first conduction band minima near X with very little contribution from In . A similar situation is seen in the Si/Si0.5Ge0. 5 systems studied. It transpires that the zone folding effect is weak unless the p minima of the first conduction band of one of the constituents is very close to the X minima. This would mean growing superlattices from alloys with

~.

CONCLUSIONS

The method described in this study can accurately model the electronic structure of the Si/SiGe and Si/SiSn superlattices modeled here. No adjustable parameters are used except the input band structures of Si, Ge a n d ~ Sn and the atomic positions of the strained layers. We believe that the technical simplifications adopted in our calculation (i.e. the virtual crystal approximation and the neglect of non-locality, spin orbit coupling and high order strain effects) will not affect the assessment of the zone folding effect significantly. The strain increases the degree of confinement, which in effect means an increase in the height of the confining barriers. The zone folding in such systems is

Superlattices and Microstructures, Vol. 2, No. 4, 1986

333 q u i t e w e a k and to p r o d u c e a s y s t e m w i t h a large optical matrix element across the s u p e r l a t t i c e g a p would require a layer w i t h a h i g h t i n c o n t e n t .

ACKNOWLEDGEMENTS We S.E.R.C. port.

would for

like to thank providing financial

the sup-

REFERENCES

X

~

Figure the

5 - A plot

first

of the

conduction

Si/Si0.75Sn0.25

-X band

strained

IAn,kl2's of state

in the

calculation.

Only the contribution f r o m the first c o n d u c t i o n band of the substrate is shown as all o t h e r b a n d s h a v e n e g l i g i ble c o n t r i b u t i o n s . The k vector runs from -X to X t h r o u g h ~ i n the B r i l l o u i n zone of the s u b s t r a t e .

I - J.C. Bean, L.C. Feldman, A.T. Fiory, S. N a k a h a r a and I.K. R o b i n s o n , J.Vac.Sci.Technol. A2(2) 436 (1984). 2 - H.J. G r o s s m a n n and L.C. Feldman, Appl.Phys.Lett. 48(17) 1141 (1986). 3 - F. C e r d e i r a , A. Pinczuk, J.C. Bean, B. Batlogg and B.A. Wilson, Appl. P h y s . L e t t . 45(10) 1138 (1984). 4 - M. Jaros, K.B. W o n g and M.A. Gell, P h y s . R e v . B 31 1205 (1985). 5 - I. Morrison, M. Jaros and K.B. Wong, J.Phys.C 19 239 (1986) and P h y s . R e v . B (to be p u b l i s h e d ) . 6 - G. A b s t r e i t e r , H. B r u g g e r , T. Wolf, H. Jorke and H.J. Herzog, P h y s . R e v . L e t t . 54(22) 2441 (1985). 7 R. People and J.C. Bean, A p p l . P h y s . L e t t . 48(8) 538 (1986). 8 - R. People, Phys.Rev.B 32 1405 (1985).