Optical properties of perfect and imperfect SiGe superlattices

Thin Solid Films, 183 (1989) 49-55

49

OPTICAL PROPERTIES OF PERFECT A N D I M P E R F E C T S i - G e SUPERLATTICES K. B. WONG, R. J. TURTON AND M. JAROS

Physics Department, The Unioersity, Newcastle upon Tyne ( U.K. ) (Received May 30, 1989)

We report fresh pseudopotential calculations of the electronic structure and optical properties of S i - G e superlattices. We assess the effect of the finite character of heterostructures, and of the breakdown of the translational symmetry in the interface plane due to interdiffusion and defects. We show that intersubband optical transitions can be used as a useful alternative to band gap luminescence for interface characterization. We also evaluate the key electronic and optical properties of related systems, e.g. Si-SixSn~ -x and Ge-GexSn~ -x.

1. INTRODUCTION

The purpose of this study is in essence threefold. Firstly, we consider the effect upon electronic structure and optical spectra of interdiffusion across the interface and presence of impurities and defects. All these imperfections lead to the breakdown of the translational symmetry invoked in idealized models 1-4 and to deviations from the corresponding selection rules for electronic transitions. Furthermore, in most cases of practical interest we have to deal with a finite structure along the growth direction, for instance with a single interface, double barrier structure, or a superlattice with a few periods only such as when the structure is grown on silicon or germanium buffers. We find that the consequence of these deviations from ideal condition is a change in localization of confined states and in some cases new levels. Secondly, it is essential to identify those features in the electronic structure that might provide alternative means for identifying a signature of the condition of the interface. It transpires that one such alternative is to study the intersubband transitions in such structures. We find that in some structures these transitions are very strong and occur in the important 5-13 ~tm wavelength band. Thirdly, we study strained layer heterostructures analogous to S i - G e (e.g. G e - G e 1 _xSn x, Si-Sil _xSnx). Such systems are of interest because their band gaps span a wider range of energies. 0040-6090/89/$3.50

© ElsevierSequoia/Printedin The Netherlands

50

K. B. WONG, R. J. TURTON, M. JAROS

2. COMPUTATIONAL PROCEDURE

Our method of calculation has been described in earlier studies of o u r s 2 ' 5 - 7 concerning the electronic structure and optical properties of strained layer semiconductor superlattices. The scheme is a natural extension of the semiempirical pseudopotential method we have used to model lattice-matched superlattices 8-x°. The input into these calculations are the lattice constants of the heterostructure configuration and the bulk band structure of the constituents. In an idealized structure, the lattice constant in the interface plane, all, is determined by the substrate. This means that a thin layer of germanium grown on silicon has the same all as that in bulk silicon. The longitudinal lattice constant a I perpendicular to the interface) is then determined by the thickness of the germanium layer or, in a thin layer quantum well or superlattice, by the Poisson ratio. The interface spacing is taken simply as an average of the separation of layers in the silicon and germanium regions. We begin by choosing a suitable bulk crystal hamiltonian Ha (e.g. a pseudopotential for a silicon crystal under appropriate strain) such that the eigenfunction ~b(n,k) and the band structure energies E(n, k) can be generated in the relevant range of energies in good agreement with experiment. We then expand the superlattice wavefunction ~bin terms of the complete set of q~(n,k). The Schroedinger equation for the heterostructure is (H o - E + V) ~ A(n, k)c~(n, k) = 0 n,k

where V is the microscopic potential representing the difference between the atoms forming the superlattice layer and that of the starting material. In modelling the structures grown on silicon or germanium, the starting hamiltonian we use has tetragonal symmetry (point group D2h ). In order to model deviations from the bulk translational symmetry in the interface plane, we enlarge the unit cell in the interface plane into a rectangular block whose dimensions are 16 and eight atomic layers along the (100) and (010) directions respectively (Fig. 1). LARGE UNIT CELL IN THE INTERFACE PLANE

16 otoms per celt

<010) 8 Layers

(100) 16 tayers Fig. 1. The enlarged unit cell in the interface plane.

In the case of an ideal infinite Si-Ge superlattice consisting of four monolayers of silicon and four monolayers of germanium our calculations show, in agreement

OPTICAL PROPERTIES OF S i - G e SUPERLATTICES

51

with other authors, that the fundamental gap is indirect at about 0.9 eV with the minimum lying at the non-folded All conduction band minimum. The first direct transition at the zone centre is found at about 1.2 eV. This zone folded state (C2 in Fig. 2) is derived from the longitudinal valleys and is located mainly in silicon. It lies well above the absolute minimum of the conduction band because of the large barrier potential (Fig. 3) separating the conduction states of silicon and germanium layers in the idealized defect-free superlattice model. The same procedure has been used to generate the electronic structures of related systems discussed in the forthcoming paragraphs. We shall not address here the important question 6 as to whether the nuclear configuration used in this and other models is a realistic one in the presence of defects since there is at present no means of settling it. We shall also confine ourselves to purely electronic processes and leave out the question of e l e c t r o n - p h o n o n interactions.

C~ 2 0

~

30

~

A - mdirect 8 - direct

1 5

A±

20-

u~a

A 05

w10

0

P

0.25

050 kz

075

B

A

rv ae

0

I

| Pv

-c5[

6.

si

s,~ -ae~

100 P

Fig. 2. The electronic band structure of an ideal SQGe4 superlattice. The lattice separation in the interfaceplane is that of bulk silicon. The state V1 is the top of the valenceband and state C1 is the bottom of the conduction band. The first finite optical transition at k = 0 (F) is between V1 and C2. Fig. 3. The band offset diagram for Si-Ge heterostructures on silicon. The lowest direct and indirect transitions are indicated.

3. IMPERFECTS i - G e INTERFACES The effect of interdiffusion upon the electronic structure of a S i - G e heterostructure, can be modelled by replacing silicon atoms from the silicon layer by those of germanium and vice versa. The best studied structure t ' is that ofSi4Ge 4 described in the above paragraph. In the ideal structure the purely electronic transition across the fundamental gap (A in Fig. 3) is forbidden (i.e. it can occur only via phonons). The existence of foreign atoms in the layer breaks the translational symmetry in the

52

K . B . WONG, R. J. TURTON, M. JAROS

interface plane and relaxes the selection rule. The magnitude of the optical matrix element is a good measure of the strength of this effect. In particular, we can compare the strength of transition A with the first direct transition (B in Fig. 3) at the centre of the Brillouin zone, involving the zone-folded state. Six different nuclear configurations have been considered. (1) One silicon atom in the interface layer is replaced by a germanium atom. (2) One germanium atom is replaced by a silicon atom. (3) As in (1) but also with an additional displacement of the protruding germanium atom by 0.05/~ in the (001) direction. (4) As in (2) but also with an additional displacement of the protruding silicon atom by 0.05/~ along (001). (5) One germanium atom in the interface layer replaced by a tin atom. (6) The ideal SiaGe 4 structure which serves as a reference point. The results are summarized in Table I. We can see that when every sixteenth atom is replaced in the interface layer by the species from the adjacent material the optical matrix element for the indirect electronic transition is no longer zero. However, the effect is not strong enough to make A and B comparable, In fact, we found that when we replaced two silicon atoms with germanium ones and vice versa there was no significant increase in the matrix element for the indirect transition A compared with the case when only one atom is replaced! This is because the difference between the silicon and germanium atoms is small and the corresponding difference potential is shallow. It means that the interdiffusion would have to involve more than one out of eight atoms in the interface layers to make A and B comparable. TABLE I THE MODULUSSQUAREDOF THE OPTICAL MATRIXELEMENT(IN ATOMICUNITS)FOR THE LOWESTDIRECT AND INDIRECTTRANSITIONSIN THE Si~Ge~ SUPERLATTICE (1) Every sixteenth Si atom in the interface layer replaced with Ge (2) Every sixteenth Ge atom in the interface layer replaced with Si (3) Protruding Ge atom in (1) displaced by 0.05/~ (4) Every sixteenth Ge atom in the interface layer of Ge replaced with Sn (5) The result for a perfect Si4Ge 4 superlattice. The magnitude of the indirect transition quoted here is the effective zero in the numerical calculations. A and B are shown in Fig. 3 Transition

1

2

3

4

5

Indirect(A) Direct(B)

0.97×10 -5 0.10x 10 -2

0.77 x 10 -5 0.15 x 10 -2

0.54x 10 -5 0.10x 10 -2

0.30x 10 -3 0.60 x 10 -3

0.11×10 -12 0.14x 10 -2

The difference between silicon (or germanium) and tin is larger than that between silicon and germanium. When tin is used to replace one of the interface atoms, the corresponding enhancement of transition probability for A is also larger. 4.

INTERSUBBAND TRANSITIONS

S i - G e heterostructures are plagued by extrinsic (point and line defect) optical lines which are not readily reproducible and which make the transitions across the gap difficult to identify and process. We propose that an alternative way to obtain

OPTICAL PROPERTIES OF S i - G e SUPERLATTICES

53

the interface signature is to concentrate on the optical transitions between the lowest two conduction subbands (C1 and C2 in Fig. 2). The separation of these states is also a sensitive function of the layer width, strain and light polarization. The distribution of the strain at the interface affects the dispersion of the subbands and consequently changes the line width. Our calculations show 12 that these transitions are comparable in strength with direct transitions across the band gap of direct gap bulk semiconductors. They occur in the range 5-13~tm which also makes them technologically important. For such transitions to be easily observable, we must use a structure in which it is possible to populate the lowest conduction subband (for example by doping gently a thick portion of the substrate or by homogeneously doping the whole of the superlattice system) so as to take advantage of the allowed transitions. If, for example, we took a structure grown on a silicon buffer, the lowest conduction band valley is that in the interface plane. The minimum of the conduction band in the growth direction lies much higher in energy. However, the conduction valley in the growth direction lies lower in a structure grown on germanium or on a suitably chosen Si x_xGex alloy. This is because the strain splits the degenerate (bulk) X and A conduction band states of silicon and germanium in such a way that germanium on a smaller lattice constant substrate has its Xll pushed above X± and vice versa. The best structure to use is one with a symmetric strain (i.e. strain is equally distributed between silicon and Si~ _xGex layers). This makes it possible to grow a large number of periods without exceeding the critical width of the microstructure for dislocation formation. 5. Si-Sil _xSn~, AND G e - G e z -xSnx HETEROSTRUCTURES The Si-Sil_xSnx on silicon superlattice with the 4% lattice mismatch characteristic of S i - G e is that with x = 0.22. The value of the valence band offset is not accurately known. The best estimate based on a critical assessment of our theory 5"13 yields 0.5 eV. Our results show that the band gap of the four-monolayer superlattice analogous to Si4Ge 4 is a sensitive function of the valence band offset, in contrast to the S i - G e case. The material with the 4% mismatch (x = 0.22) is similar to the S i - G e case in that the principal gap of the ideal structure is indirect. The matrix element, involving the lowest energy direct transition at the zone centre from the zone-folded conduction state, is smaller by about an order of magnitude. The electronic wave functions also greatly resemble the case available in the literature 1-6. The band gaps of unstrained Gel _xSnx alloys span a large range of energies. Also, the lattice constants of germanium and ct-Sn are more similar than those of silicon and cx-Sn. We have therefore considered Ge-Gez_xSnx heterostructures grown on germanium. The concentration at which we obtain 49/0 strain for comparison with S i - G e is x = 0.3 and the valence band offset is 0.5 eV. We predict that in its ideal form this system is an indirect gap material whose conduction band is dominated by the germanium layer. It is a type two structure in that electrons are localized in the germanium and holes in the alloy. However, the separation in the conduction band between the F and L and X valleys is small in germanium and the

54

K . B . WONG, R. J. TURTON, M. JAROS

effective mass is also very small so that in thin period superlattices the conduction band wavefunction penetrates easily into the alloy layer through the shallow F barrier. Hence the overlap with the confined valence levels in the alloy layer is strong and the optical transition at the centre of the superlattice Brillouin zone is that characteristic of a direct gap material. Since both the conduction and the valence band offsets are small for 30~o of tin in the alloy layer, the gap can only be changed by a couple of tenths of an electronvolt depending on the well and barrier widths. 6. INTERFACES IN FINITE

Si-Ge

STRUCTURES

The best example of a finite system is an S i - G e superlattice grown on silicon or germanium. Then the total thickness of the strained material reduces to about 20 monolayers so that it is possible to grow, for example, five germanium layers each one cubic lattice constant (four atomic monolayers) thick separated by silicon layers of equal thickness. This is the structure studied by Pearsall et al.1 ~ The five-period structures are separated by thick layers of silicon. An analogous result is obtained if the layers are deposited on germanium. In order to model this finite structure, we set up a supercell which includes all five periods and also enough of the surrounding silicon material so as to isolate the five-period structure from the rest of the sample. Our method of calculation is unique in that it is quite capable of dealing with such large blocks of atoms. The details of these calculations have been described elsewhere ?. The lowest conduction level with a finite optical transition probability for a jump to the valence band in the five-period structure is the eighth state C8. If we line up state C8 with the corresponding state in the infinite superlattice system (see state C2 at F of the dispersion diagram for the S i - G e infinite superlattice model in Fig. 2),

( Si413et.)S SO0

Si4Oe4 supertottice

C8__

C2 ( r )

40O C 5 - -

bQnds C1, C2

>. 300 C4 C3

200

C2__ CI

100

Si conduction bond edge J 0

Fig. 4. This diagram shows the distribution of the conduction states C 1 - C 8 of the five-period S i - G e superlattice grown on silicon relative to the corresponding states in the infinite superlattice model whose electronic structure is presented in Fig. 2. The lowest conduction states (C8 and C2 respectively) giving direct transitions are lined up. The shaded area indicates the width of the conduction subbands in the periodic structure.

OPTICAL PROPERTIESOF S i - G e SUPERLATTICES

55

we can see without invoking any details of the calculation that the lowest conduction b a n d levels in the five-period structure lie below the b o t t o m of the conduction band of the infinite superlattice (Fig. 4). In particular, the lowest states C 1 - C 4 are largely localized on the outer side of the five-period system. These states are really resonances or simply regions of higher density of states c o m p a r e d with that expected of a bare substrate, and their position indicated in the diagram should be understood in that way. These states are to some degree analogous to the resonances created at certain defects in semiconductors 14 in the range of energies coinciding with the host crystal c o n t i n u u m states. The present study is still very m u c h of a preliminary account, merely indicating the kind of novel p h e n o m e n a we encounter when some of the c o m m o n l y accepted idealizations in modelling strained layer systems are removed. However, in spite of the sketchy character of the results outlined above, this study shows that a realistic description of S i - G e heterostructures can only be achieved when the microscopic potential characteristic of these systems and their finite and imperfect character are fully taken into account. ACKNOWLEDGMENTS This work has been supported in part by the Science and Engineering Research Council (U.K.), British Telecom, the Procurement Executive o f M O D / R S R E Malvern, and by the O N R (contract n u m b e r 00014-88-J-1003). REFERENCES 1 2 3 4 5 6

C.G. Van de Walle and R. M. Martin, Phys. Rev. B, 34 (1986) 5621. I. Morrison, M. Jaros and K. B. Wong, Phys. Rev. B, 35 (1987) 9693. S. Froyen, D. M. Wood and A. Zunger, Phys. Rev. B, 36 (I 987) 4547; 37 (1988) 6893. M.S. Hybertson and M. Schluter, Phys. Rev. B, 36 (1987) 9683. I. Morrison and M. Jaros, Phys. Rev. B, 37 (1988) 916. K.B. Wong, M. Jaros, I. Morrison and J. P. Hagon, Phys. Rev. Lett., 60 (1988) 2221.

7 K.B. WongandM. Jaros, AppI. Phys. Lett.,53(1988)657. 8 M. Jaros, Rep. Progr. Phys., 48 (1985) 1091. 9 M.A. Gell, D. Ninno, M. Jaros and D. C. Herbert, Phys. Rev. B, 34 (1986) 2416.

10 M.A. Gell, D. Ninno, M. Jaros, D. J. Wolford, T. F. Kuech and J. A. Bradley, Phys. Rev. B, 35 (1987) 1196. 11 T.P. Pearsall, J. Bevk, L. C. Feldman, J. M. Bonar, J. P. Manaerts and A. Ourmazd, Phys. Rev. Lett., 58 (1987) 729. 12 R.J. Turton and M. Jaros, Appl. Phys. Lett., 54 (1989) 1986. 13 M. Jaros, Phys. Rev. B, 37 (1988) 7112; see also M. Jaros, Physics and Applications of Semiconductor Microstructures, Oxford University Press, Oxford, 1989. 14 M. Jaros, Deep Levels in Semiconductors, Hilger, Bristol, 1982.