CALCULATION OF BOUND STATES IN A STRAINED Ge0.25Si0.75/Si/Ge0.25Si0.75 QUANTUM WELL

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CALCULATION OF BOUND STATES IN A

STRAINED Ge0.25Si0.75/Si/Ge0.25Si0.75 QUANTUM WELL

D. Hughes, S. Brand

To cite this version:

D. Hughes, S. Brand. CALCULATION OF BOUND STATES IN A STRAINED

Ge0.25Si0.75/Si/Ge0.25Si0.75 QUANTUM WELL. Journal de Physique Colloques, 1987, 48 (C5),

pp.C5-565-C5-568. �10.1051/jphyscol:19875122�. �jpa-00226706�

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CALCULATION OF BOUND STATES IN A STRAINED G e o . 2 5 S i , ~ 7 , / S i / G e o ~ 2 5 S i o 0 7 5 QUANTUM WELL

D.T. HUGHES and S. BRAND

Applied Physics Group, School of Engineering and Applied Science, South Road, GB-Durham, DH1 3LE, Great-Britain

Abstract:- We present results of a complex bandstructure matching technique applied to a calculation of the bound states of a Geo.z5Sio.75/Si/Geo.25Sio.75 strained quantum well. The effects of alloying and strain in the barrier and well regions are introduced into the bulk bandstructure by appropriate adjustment of the pseudopotential form factors. The form of the conduction band bound-state energy levels as a function of well width is shown to agree with the periodic small-energy splittings predicted by the envelope function approximation. Finally some averaged charge densities are plotted along the (100) direction.

The possibility of growing epitaxial structures consisting of lattice mismatched materials h a stimulated much interest in Si/Ge,~i~-, heterostructures. The strain produced by growing layers of these materials with large lattice mismatch (Si and Ge differ by 4.2%) can be utilised in modulation doped structures to cause the conduction electrons to reside and travel in the Si layers, with much higher mobilities than would be experienced in the alloy layers1. Also, these materials have bandgaps which span the 1.30pm-1.55pm range2, promising applications for optical fibre communications.

Despite the important technological possibilities there has been ccmparitively little theoretical work done on strained IV-IV structures compared t o the lattice matched 111-V structures, apart from the works of Morrison et a13 and de Sterke and Hall4 which are concerned with ~ e ~ . 2 ~ ~ i ~ . ~ ~ ( ~ i / ~ e ~ . ~ ~ i ~ . ~ ) superlattices. In this paper a complex bandstructure matching method is used t o calculate the confined states of a strained

~ e o . 2 5 ~ i o . 7 5 / ~ i / ~ e o . 2 5 ~ i o . 7 5 quantum well aligned in the (100) direction. Our matching technique5 is ideally suited to calculating energy states and wavefunctions of general low dimensional structures if the complex bandstructures of the constituent materials are known. In the present case the quantum well potential is introduced t o the problem by imposing the appropriate offset between the barriers and the well regions. The wavefunction, at a particular energy, is expanded in the well and barrier regions in terms of the appropriate Bloch functions:

= W i a i ( k j ) exp (ki

- +

^g).t

i +?

where the a, - are Bloch function expansion coefficients, the W i are the bound state expansion coefficients and the set of wavevectors { k i )

-

includes evanescent and real solutions in the well region but only decaying solutions in the barriers. A bound state exists if the wavefunction and derivative can be matched a t the two interfaces.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19875122

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C5-566 JOURNAL DE PHYSIQUE

The bandstructure was produced by means of a local pseudopotential calculation using 65 plane waves. The pseudopotential method provides a simple means of including the effects of strain and alloying in the calculations by treating the

0.14

-

pseudopotential atomic form factors

0.12

- -

as part of a continuous pseudopoten-

tial curve V(q), where 0.10

-

V(P) =

-

V(r) expi(q.3 dr (1)

0.08 -

^no

2 J

with n o =unit cell volume. Initially

0.06 -

the three symmetric pseudopotential

0.04 -

form factors V3, V8, V11 (PPF)

S

were chosen in order t o give the

0.02

- -

best bandstructure fits for unstrained

-

0 bulk Si and Ge. In Si both the

3 Omo io ' ' -&

^'

io

^'

&) ^{r - r}

^and

^r ^-

A bandgaps were fitted

e

to experimental results to an accu-

racy of better than 1%. In addition

-

the longtidudinal and transverse

a-

minimum effective masses are fitted

-

t o within 5%

.

The

r

- T, T - A and

r

- L band gaps are all fitted

-

to experimental data t o better than 5% in the Ge7. Similarly the variation of bandgaps with hydrostatic pressure is fitted t o the experimental results t o within 1% by taking into account the change in the size

0.020 -

of the unit cell, no, and the variation

in the crystal momentum p.

30 32 34 s 38

Having obtained what amounts to Width of Mll

(A)

a Taylor expansion of the pseudopotential curve about the P P F for bulk.

Fig. l(a) (upper) Energy levels plotted against well

width for lowest states. l(b)(lower) Magnified Si and Ge it was possible, by using portion with integer monolayers indicated open the virtual crystal approximation, circles and predicted crossing by arrows. t o caIculate the bandstructure of any Ge,Sil-, alloy. The calculations gave the correct bandgaps t o within 1.0% for both the bulk Si and Ge (1.lleV and 0.66eV respectively at 300K) and predict a crossover from a Si to a Ge-like bandgap a t x = 0.86. This result agrees well with more involved self consistent virtual crystal approximation calculations which predict the crossover at x = 0.88' and with experimental results, x = 0.85=

.

I t has been noted by other workers that the virtual crystal approximation may not be entirely appropriate for Ge-Si alloysg however, the agreement of the calculated crossover with experiment gives some extra confidence to the use of our method.

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used t o model the effects of strain on the Si well region. The Si experiences a tetrag- onal distortion, stretching in the lateral directions to take up the lattice constant of the unstrained Geo.~~Sio.76 and com- pressed in the (100) direction by an amount dictated by the Poisson ra- tio (for Si v

-

^0.28) giving values of aL = 5.485jl and a,, = 5.357A. The bandstructure obtqined is found to show the expected lifting of the six fold degeneracy of the silicon min- ima: four moving up in energy and the two corresponding to the (100) direction moving down more quickly hence giving a smaller bandgap in the (100) direction. The conduction band offset remained as a parame-

z

r, ter to be inserted into our model.

t

^Agreat deal of theoretical work has

0 been done on this problem but there

O) remains a lack of conclusive experi-

E

mental evidence. However ,by using

U Van de Walle and Martin'sl" values

B

for the valence band offsets we were

H

able to show that the lower con-

s!

duction band lies in the strained Si

4 rather than the alloy.

The above procedures were used to calculate the energy levels of the quantum well. Figure l(a) shows the variation of energy levels with well

-20

0

20 40

60

width. The most striking feature of the levels is the oscillatory nature of Fig. 2 Charge density averaged over unit cell face the small energy splittings. ~h~~~

for ground state and f i s t excited state of 16 mono-

layer quantum well. splittings, which are consistent, in

form, with the results of de Sterke and Hall4

,

are typically of the order of 2mev for the widths shown. The small size of this effect makes it unobservable at room temperature but the splitting may be resolvable a t low temperatures. Figure l ( b ) shows a magnified version of the 28.0A- 40.0A well width range. The open circles indicate integer monolayer widths. It should be noted that the calculations have also been carried out for fractional monolayer(i.e.

physically non-obtainable) widths in order to give a clearer indication of the oscillatory nature of the splitting effect. The arrows indicate the widths 'w' at which de Sterke and Hall predict zero splittings using an envelope-function approximation (EFA) for the ground states of an infinite quantum well:

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C5-568 JOURNAL DE PHYSIQUE

with UJ measured in numbers of unit cells, n is an integer and the Si minimum is 85%

of the way towards the zone edge. Figure 2 shows the charge density function averaged over the face of a unit cell perpendicular to the (100) direction for one of the ground state pair and one of the excited state pair of the energy solutions for a well width of 16 monolayers or 43.lA-the partner wavefunctions are virtually identical, all showing the conventional envelope form. Additional investigation of the charge densities reveals more complicated structure than would be predicted by de Sterke and Hall's simple model. This appears t o be analagous t o the extra structure which has been shown to occur in the valence band of 111-V systems5."

In conclusion we have used our matching technique with realistic complex bandstructure -including the effects of alloying and strain- to calculate the energy levels of bound states in a strained ~ e o . ~ 5 ~ i ~ . 7 ~ / ~ i / ~ e ~ . ~ 5 S i 0 . , 6 quantum well. Our results demonstrate that the EFA gives the correct form for the energy levels and their splittings in systems with indirect-gap constituents. However, initial calculations of the non-averaged charge densities indicate more structure than predicted by the EFA and consequently a more detailed analysis is in progress.

Acknowledgements

DTH would like to thank SERC and RSRE for financial support. The authors would like to thank Drs. MJ Kirton, S Monaghan and RA Abram for useful discussions.

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