VALENCE BAND STRUCTURE OF STRAINED-LAYER Si-Si0.5Ge0.5 SUPERLATTICES

Transcription

1 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-c < /jphyscol: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1987 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 JOURNAL DE PHYSIQUE Colloque C5, supplement au n0ll, Tome 48, novembre 1987 VALENCE BAND STRUCTURE OF STRAINED-LAYER Si-Sio.,Geo., SUPERLATTICES U. EKENBERG, W. BATTY* and E.P. O'REILLY* Department of Theoretical Physics, University of Oxford, GB-Oxford, OX1 3NP, Great-Britain *~epartment 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 and on a Si0,,5Ge0,25 buffer. We use the 6x6 Luttinger-Kohn Hamiltonian, including the effects both of strain and of the split-off band. The heavy-hole zone centre states are separated by over 60meV from the light-hole states and the highest valence subband has a low zone-centre effective mass (m*.l5 -.18), suggesting such structures to be useful for high hole mobility applications. Away from the zone centre, the bands are strongly anisotropic. We predict little difference in electronic properties for p-type Sio,5Geo,,-Si superlattices when grown on Si substrates or Sio~,,Geo~,, buffers. Introduction : There is increasing interest in 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. In particular, much effort 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 are being probed and some of the most promising results to date suggest that high hole mobilities can be achieved in 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, and consider the two cases of growth on a Si substrate and growth on a Sio,,,Ge,,,, buffer. In the first case the Si barriers are unstrained while the Si.o,5Geo,5 wells are under biaxial compression to accommodate the 2% lattice mismatch wlth the substrate. For the latter case the Si layers are under biaxial tension while the Si0,,Geo,, layers are under reduced biaxial compression and the lattice mismatch for both layers relative to the buffer is about 1%. We consider 50a and I O O ~ wells, both within the experimentally determined limits of good quality growth [I]. The built-in biaxial strain splits the degeneracy of the bulk valence band maximum, and mixes the zone-centre light-hole and spin-split-off states [3]. Our calculations are the first to include the strain-induced band splitting and mixing, both of which play a major role in determining the subband structure. We begin by describing the method used and then present oar results and conclusions. Method: The band structure is calculated using the Luttinger-Kohn (LK) 6x6 Hamiltonian [4]. This incorporates the heavy-hole (HH), light-hole (LH) and spin-split-off (SO) bands. It is essential to include all three types of bands in 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 to the quantum well confinement energies. The Hamiltonian can be decoupled into two independent 3x3 matrices, using a transformation similar to that of Broido and Sham [5] for the LK 4x4 Hamiltoni:.n. The Hamiltonian then takes the form Article published online by EDP Sciences and available at

3 JOURNAL DE PHYSIQUE where k. is the wavevector along the growth direction, k, the magnitude of the wavevector in the well plane, and 0 the angle between the direction of k, and the (100) direction. S describes the strain-induced splitting of the valence band maximum, and the spin-orbit splitting is given by A We have taken li = m = 1. The valence band dispersion is determined by the three material parameters Y,, 7, and y3 which are related to inverse effective masses at the Brillouin zone centre. We use the experimentally determined 7 values for Si while the alloy values were chosen by linear interpolation of Si and Ge effective masses 161, and are shown in Table 1. The average valence band offset AEV and strain-induced splittings were taken from the work of van de Walle and Martin [7]. The valence subband dispersion was calculated using a modified variational method to ensure proper matching of the wavefunctions at the interface [8]. Table 1: 7 parameters, and valence band energies (in mev) yi E" S (Si substrate) S (buffer) A Si Results Fig. ](a) and (b) show the valence subband dispersion in the well plane for strained layer structures with 100a Sio,bGeo.5 wells between Si barriers, grown on (a) a Si (001) substrate and (b) a Sio,,Ge0,,, buffer. The results for a 50a well grown on a Si (001) substrate are shown in Fig. ](c). Fig. l(d) shows the dispersion for a loor well if strain effects are neglected. This last figure is included to model the case of poor quality growth, where strain is relieved by dislocations, and also to emphasise the importance of strain in determining the band structure of the remaining cases considered. The zero of energy in Fig. 1 is taken at the Sio,,Ge,,, heavy-hole band edge. The figure gives information on the zone centre energies, the valence subband dispersion and its anisotropy, and the effects of differing substrates on the band structure. We consider first the zone centre energies. For a given well thickness, the heavy-hole zone centre confinement energies are practically independent of the substrate material. If strain were neglected, the highest light-hole state LHI would be only 7meV below the highest heavy-hole state HHI (Fig. I(d)), but because of the strain LHI is in all cases at 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 - da2-2sa +9S" and equals 84meV and 64meV for growth on a SI (001) substrate and on a Sio,5Geo,25 buffer respectively. The splitting between the subbands HHI and LHI is within 2meV of these values in Fig. l(a)-(c). Because of the large valence band offset, there is little interaction between neighbouring alloy layers. We find for instance that even with a barrier as thin as 508 the dispersion of the HH2 zone centre state in Fig. l(c) along the growth direction is negligible. The calculations presented in Fig. l(c) for isolated quantum wells should therefore also be appropriate for superlattices with barriers as thin as 50%. The highest subband has a comparatively low zone centre effective mass, m*, varying from.l5 in Fig. ](a) to.l8 in Fig. l(c). This is as expected in structures with a significant strain-induced splitting between the heavy-hole and light-hole states [9]. Such low effective masses give the possibility of enhanced hole mobilities. High hole mobilities

4 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 in modulation doped Sio,,Geo,, strained layers wherc conduction takes place at a heterojunction interface with an experimentally determined effective mass m* = 0.32 *.03 [l]. We have not yet treated a heterojunction in our calculations, as the confining potential must be determined self-consistently. We calculate that m* = 0.22 in a 100% Sio,,Geg, quantum well. For the heterojunction, the valence subbands are spin-split for k, different from zero, with one Fermi-level dependent effective mass for each subband. It is difficult therefore to compare between the two cases but we predict the effective mass to decrease with increasing Ge content. As the mobility typically increases with decreasing effective mass, we then expect that even higher hole mobilities could be achieved in Si0.5Ge05 layers, making them good candidates for fast hole-based devices [IO]. The valence subband dispersion is highly anisotropic away from the zone centre, with the highest subband being considerably heavier along the (11) than the (10) direction. Interestingly, the situation is reversed for some of the lower subbands, with (11) bands having greater dispersion than (10) bands due to the stronger band mixing along the (11) direction [9]. The in-plane anisotropy is much greater than that typically found in 111-V heterostructures [Ill. We note for example that in Fig. I(b) the highest valence band along the (10) direction lies below the second band in the (11) direction by the time k = 6x10-~ a-'. Such warped bands may affect the mobility in high applied fields, giving it a directional dependence, or allowing increased intersubband scattering. The dependence of Landau levels on applied magnetic field B, when B is perpendicular to the layers, should be even more complicated than for 111-V heterostructures, as the valence band anisotropy leads to strong anti-crossing effects between different Landau levels We also expect that an angular dependence may be seen in the response of the structure to an applied field B in the plane of the wells but this effect is more difficult to interpret because of the interplay of magnetic and quantum well confinement effects. When we compare growth on a Si (001) substrate with growth on a buffer, we see that the chief differences in band structure occur at large energies and wavevectors. For 100%

5 C5-556 JOURNAL DE PHYSIQUE wells, some of the bands have electron-like effective masses (e.g. HH4 in Fig. l(a) or HH3 and LH3 in Fig. I(b)) and interesting anti-crossing effects are seen (e.g. HH3-LHI). For p-type conduction, we are chiefly interested in the highest subband. The dispersion of this band varies less between different substrates than it does between different directions on a given substrate. We conclude that there will be little difference in electronic properties for growth of p-type Si0,,Geo,, alloys on Si substrates or Si0,,5GepS5 buffers. We note however that 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 for growth on a Si substrate compared to a buffer. The highest band can then have a lower effective mass over a significant energy range, leading to an enhanced mobility on a Si substrate. Conclusions: We have presented the first calculations of the valence subband structure of a Si-SiGe superlattice which incorporate the influence both of strain and of the spin-split-off bands on the subband structure. For Si0,,Geo., wells, strain shifts the light-hole states down in energy almost rigidly with respect to the heavy-hole states, giving a splitting of over 60meV between HHl and LHl. The highest hole band then has a low effective mass in the well plane, with m* This effective mass is markedly lower than any in conventional silicon devices. Strained p-type SiGe could therefore provide the highest mobilities of any Si-based structures. Growth on a Si (001) substrate will be generally better than growth on an alloy buffer for high hole mobility applications, due to the greater strain-induced splitting, but this effect is not expected to be very significant for wells with a high Ge content, such as we consider here. The band structure is highly anisotropic away from the zone centre. This anisotropy should be reflected in the Landau level behaviour, for applied magnetic fields B both perpendicular to and parallel to the growth direction. Acknowledgements: We thank the European Research Office of the U.S. and the Science and Engineering Research Council (WB) for financial support. Army (UE) References 1. People R., IEEE J. Quantum Electronics, QE-22, (1986) People R., Bean J.C., and Lang D.V., J. Vac. Sci. Technol. A3, (1985), Pikus G.E. and Bir G.L. Sov. Phys. Solid State 1, (1959), Luttinger J.M. and Kohn W., Phys. Rev. 97, (1955), Broido D.A. and Sham L.J., Phys. Rev. B 31, (1985), Hensel J.C. and Feher G., Phys. Rev. 129, (1963), 1041; Hensel J.C. and Suzuki K., Phys. Rev. B 9, (1974), Van de Walle C.G. and Martin R.M., Phys.Rev. B 34, (1986), Altarelli M., Phys. Rev. B 28, (1983), O'Reilly E.P. and Witchlow G.P., Phys. Rev. B 34, (1986), Osbourn G.C., J. Vac. Sci. Tech. A3, (1985), Altarelli M., Ekenberg U. and Fasolino A., Phys. Rev. B 32 (1985), Bangert E. and Landwehr G., Surf. Sci. 170, (1986), 593.