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VALENCE BAND STRUCTURE OF
STRAINED-LAYER Si-Si0.5Ge0.5 SUPERLATTICES
U. Ekenberg, W. Batty, E. O’Reilly
To cite this version:
U. Ekenberg, W. Batty, E. O’Reilly. VALENCE BAND STRUCTURE OF STRAINED-LAYER Si-Si0.5Ge0.5 SUPERLATTICES. Journal de Physique Colloques, 1987, 48 (C5), pp.C5-553-C5-556.
�10.1051/jphyscol:19875119�. �jpa-00226702�
JOURNAL DE PHYSIQUE
Colloque C5, supplement au n 0 l l , Tome 48, novembre 1987
VALENCE BAND STRUCTURE O F STRAINED-LAYER S i - S i o . , G e o . , SUPERLATTICES
U. EKENBERG, W. BATTY* and E.P. O'REILLY*
Department of Theoretical Physics, University of Oxford, GB-Oxford, OX1 3NP, Great-Britain
* ~ e p a r t m e n t of Physics, University of Surrey, GB-Guildford, GU2 5XH, Great-Britain
Abstract : We calculate the valence subband dispersion of Sio,5Ge,,5
-
Si strained layer structures grown on a Si (001) substrate a n d on a Si0,,5Ge0,25 buffer.We use the 6x6 Luttinger-Kohn Hamiltonian, including t h e effects both of strain and of t h e split-off band. T h e heavy-hole zone centre states a r e separated by over 60meV f r o m the light-hole states a n d the highest valence subband has a low zone-centre effective mass (m* . l 5 - .18), suggesting such structures t o be useful f o r high hole mobility applications. Away f r o m t h e zone centre, the bands a r e strongly anisotropic. We predict little d i f f e r e n c e in electronic properties f o r p-type Sio,5Geo,,-Si superlattices when grown on Si substrates or Sio~,,Geo~,, buffers.
Introduction : T h e r e is increasing interest i n the growth of lattice-mismatched epilayers on a Si substrate [l]. Such layers may combine the advantages of low dimensional structures with the well-established Si technology. I n particular, much e f f o r t has centred on the growth of Sil-xGex alloys on Si and high quality growth has been confirmed over a wide range of alloy compositions. Many optical and electronic applications a r e being probed and some of the most promising results to d a t e suggest t h a t high hole mobilities can be achieved i n modulation-doped structures [2].
In this paper we present the first calculations of the valence subband structure of Sil-xGex-Si superlattices. We choose X = 0.5, a n d consider the two cases of growth on a Si substrate a n d growth on a Sio,,,Ge,,,, buffer. I n the f i r s t case t h e Si barriers are unstrained while the Si.o,5Geo,5 wells a r e under biaxial compression to accommodate the 2% lattice mismatch wlth the substrate. F o r the latter case t h e Si layers a r e under biaxial tension while the Si0,,Geo,, layers a r e under reduced biaxial compression a n d the lattice mismatch f o r both layers relative to the b u f f e r is about 1%. We consider 5 0 a and I O O ~ wells, both within t h e experimentally determined limits of good quality growth [I].
The built-in biaxial strain splits the degeneracy of the bulk valence b a n d maximum, a n d mixes the zone-centre light-hole and spin-split-off states [3] . Our calculations a r e the f i r s t t o include t h e strain-induced band splitting a n d mixing, both of which play a major role i n determining the subband structure. We begin by describing t h e method used and then present oar results a n d conclusions.
Method: T h e band structure is calculated using the Luttinger-Kohn (LK) 6x6 Hamiltonian [4]. This incorporates the heavy-hole (HH), light-hole (LH) a n d spin-split-off (SO) bands.
It is essential to include all three types of bands i n studies of Si-Si,-xGe as the magnitude of the spin-orbit splitting is comparable both to the strain-induced spfitting of the valence band maximum and t o the quantum well confinement energies. The Hamiltonian c a n be decoupled i n t o two independent 3x3 matrices, using a transformation similar to t h a t of Broido a n d Sham [5] f o r the L K 4x4 Hamiltoni:.n. T h e Hamiltonian then takes the f o r m
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19875119
JOURNAL DE PHYSIQUE
where
k. is t h e wavevector along the growth direction, k, t h e magnitude of t h e wavevector in the well plane, a n d 0 t h e angle between t h e direction of k, a n d t h e (100) direction. S describes t h e strain-induced splitting of the valence band maximum, a n d the spin-orbit splitting is given by A We have taken li = m = 1. T h e valence b a n d dispersion is determined by the three material parameters Y,, 7, a n d y 3 which a r e related to inverse effective masses a t the Brillouin zone centre. We use t h e experimentally determined 7 values f o r Si while the alloy values were chosen by linear interpolation of Si a n d Ge effective masses 161, and are shown in Table 1. T h e average valence band offset AEV and strain-induced splittings were taken f r o m the work of van d e Walle a n d Martin [7]. The valence subband dispersion was calculated using a modified variational method to ensure proper matching of t h e wavefunctions a t the interface [8].
Table 1: 7 parameters, and valence band energies (in meV)
y i 7 2 7 3 E" S (Si substrate) S ( b u f f e r ) A
Si 4.285 0.339 1.446 -270 0 39 44
Results Fig. ](a) a n d (b) show the valence subband dispersion in the well plane f o r strained layer structures with 1 0 0 a Sio,bGeo.5 wells between Si barriers, grown on (a) a Si (001) substrate a n d (b) a Sio,,Ge0,,, buffer. T h e results f o r a 5 0 a well grown on a Si (001) substrate a r e shown i n Fig. ](c). Fig. l(d) shows the dispersion f o r a lOOR well if strain effects are neglected. T h i s last f i g u r e is included t o model t h e case of poor quality growth, where strain is relieved by dislocations, a n d also to emphasise the importance of strain i n determining the band structure of the remaining cases considered.
T h e zero of energy i n Fig. 1 is taken a t t h e Sio,,Ge,,, heavy-hole band edge. T h e f i g u r e gives information on the zone centre energies, the valence subband dispersion and its anisotropy, a n d the effects of d i f f e r i n g substrates on the band structure.
We consider f i r s t the zone centre energies. For a given well thickness, the heavy-hole zone centre confinement energies a r e practically independent of the substrate material. If strain were neglected, the highest light-hole state L H I would be only 7meV below the highest heavy-hole state H H I (Fig. I(d)), but because of the strain LHI is i n all cases a t least 60meV below HHI. The strain-induced splitting AELH between the heavy-hole and light-hole bulk band edges is given by = S + %A + S
-
d a 2 - 2SA +9S" a n d equals 84meV a n d 64meV f o r growth on a SI (001) substrate a n d on a Sio,5Geo,25 buffer respectively. T h e splitting between the subbands H H I and L H I i s within 2meV of these values i n Fig. l(a)-(c). Because of t h e large valence band offset, there is little interaction between neighbouring alloy layers. We f i n d f o r instance t h a t even with a barrier as thin as 5 0 8 the dispersion of the H H 2 zone centre state i n Fig. l(c) along the growth direction is negligible. T h e calculations presented i n Fig. l(c) f o r isolated q u a n t u m wells should therefore also be appropriate f o r superlattices with barriers a s t h i n a s 50%.The highest subband has a comparatively low zone centre effective mass, m*, varying from . l 5 i n Fig. ](a) t o . l 8 i n Fig. l(c). This is as expected i n structures with a significant strain-induced splitting between the heavy-hole a n d light-hole states [9]. Such low e f f e c t i v e masses give t h e possibility of enhanced hole mobilities. High hole mobilities
Wavevec tar k (IO-'
W -' )
Figure 1: Valence subband structure for Sio.sGe,., quantum wells between Si barriers:
(a), (b) 100 A wide wells lattice-matched to Si (001) substrate and Sio.,5Geo.,5 buffer respectively; (c) 50 A well on Si (001) substrate; (d) 100 A well ignoring strain effects.
Solid lines: dispersion along (10) direction; dotted lines: dispersion along (11) direction.
have already been observed i n modulation doped Sio,,Geo,, strained layers wherc conduction takes place a t a heterojunction interface with a n experimentally determined effective mass m* = 0.32
*
.03 [l]. We have not yet treated a heterojunction i n our calculations, a s the confining potential must be determined self-consistently. We calculate t h a t m* = 0.22 i n a 100% Sio,,Geg, quantum well. For the heterojunction, t h e valence subbands a r e spin-split f o r k, d i f f e r e n t f r o m zero, with one Fermi-level dependent effective mass f o r each subband. I t is d i f f i c u l t therefore to compare between the two cases but we predict the effective mass to decrease with increasing G e content. As the mobility typically increases with decreasing effective mass, we then expect t h a t even higher hole mobilities could be achieved in Si0.5Ge05 layers, making them good candidates f o r f a s t hole-based devices [IO].The valence subband dispersion is highly anisotropic away from t h e zone centre, with the highest subband being considerably heavier along the (11) t h a n the (10) direction.
Interestingly, the situation is reversed f o r some of the lower subbands, with (11) bands having greater dispersion than (10) bands d u e to the stronger band mixing along the (11) direction [9]. T h e in-plane anisotropy is much greater than that typically f o u n d i n 111-V heterostructures [ I l l . We note f o r example t h a t i n Fig. I(b) the highest valence band along the (10) direction lies below the second band i n the (11) direction by the time k =
6 x 1 0 - ~
a-'.
Such warped bands may a f f e c t the mobility i n high applied fields, giving i t a directional dependence, or allowing increased intersubband scattering. T h e dependence of L a n d a u levels on applied magnetic field B, when B is perpendicular to the layers, should be even more complicated than f o r 111-V heterostructures, as the valence band anisotropy leads t o strong anti-crossing effects between d i f f e r e n t Landau levels 1121. We also expect t h a t a n angular dependence may be seen i n the response of the structure to a n applied field B i n t h e plane of the wells but this e f f e c t is more d i f f i c u l t to interpret because of the interplay of magnetic a n d quantum well confinement effects.When we compare growth on a Si (001) substrate with growth on a buffer, we see that the chief differences i n band structure occur a t large energies a n d wavevectors. For 100%
C5-556 JOURNAL DE PHYSIQUE
wells, some of the bands have electron-like effective masses (e.g. HH4 i n Fig. l(a) or HH3 a n d LH3 i n Fig. I(b)) a n d interesting anti-crossing effects a r e seen (e.g.
HH3-LHI). F o r p-type conduction, we a r e chiefly interested i n t h e highest subband. T h e dispersion of this band varies less between d i f f e r e n t substrates than i t does between d i f f e r e n t directions on a given substrate. We conclude t h a t there will be little difference i n electronic properties f o r growth of p-type Si0,,Geo,, alloys on Si substrates or Si0,,5GepS5 buffers. We note however t h a t this is not a conclusion general to all alloy composlttons. For alloys with small Ge content, the strain- induced light-hole -
heavy-hole splitting will be twice as large f o r growth on a Si substrate compared to a buffer. T h e highest b a n d can then have a lower effective mass over a significant energy range, leading to a n enhanced mobility on a Si substrate.
Conclusions: We have presented the f i r s t calculations of t h e valence subband structure of a Si-SiGe superlattice which incorporate the influence both of strain a n d of the spin-split-off bands on the subband structure. F o r Si0,,Geo., wells, strain shifts the light-hole states down i n energy almost rigidly with respect to the heavy-hole states, giving a splitting of over 60meV between H H l and L H l . The highest hole band then has a low effective mass i n the well plane, with m*
-
.16. This effective mass is markedly lower than any i n conventional silicon devices. Strained p-type SiGe could therefore provide the highest mobilities of a n y Si-based structures. Growth on a Si (001) substrate will be generally better than growth on a n alloy b u f f e r f o r high hole mobility applications, d u e t o the greater strain-induced splitting, but this e f f e c t is not expected to be very significant f o r wells with a high Ge content, such a s we consider here. T h e band structure is highly anisotropic away from the zone centre. T h i s anisotropy should be reflected i n the Landau level behaviour, f o r applied magnetic fields B both perpendicular to a n d parallel to t h e growth direction.Acknowledgements: We thank the European Research O f f i c e of the U.S. Army (UE) and t h e Science a n d Engineering Research Council (WB) f o r financial support.
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