Chapter 2 Theoretical Background
2.4 Band Structure
The band gaps of both silicon and germanium are indirect with values of 1.12 eV and 0.66 eV at room temperature, respectively. In both of these semiconductors the conduction and valence bands consist of a number of sub-bands. In silicon, the con- duction band minima lie along the<100>crystal direction and are six-fold degenerate. They are often referred to as the∆6 minima, since the ∆ axis in the Brillouin zone is
along the <100> direction and the subscript refers to the degeneracy. In germanium, the conduction band minima lie at the zone boundaries along the<111> direction of
[001] [001] [010] [100] [100] [111] Silicon Germanium
Figure 2.3: Schematic of the conduction band minima for bulk germanium and bulk silicon. After Paul [2004].
the Brillouin zone (L points) and are eight-fold degenerate [Sze, 1981] (Figure 2.3). The valence band edge of both silicon and germanium occurs at the zone centre (k = 0). At this point, the valence band consists of two degenerate bands and a split-off band separated in energy from the two degenerate bands due to the spin-orbit interaction. The spin-orbit splitting energy is 44 meV in silicon and 296 meV in germanium. The two degenerate bands are defined as the light-hole band (the narrower band) and the heavy-hole band (the wider band). The effective mass,m∗, is given in tensor form to account for crystal anisotropy, by [Kittel, 1996]:
1 m∗ ij = 1 ¯ h2 ∂2E(k) ∂ki∂kj (2.11) The mobility is related to the effective mass according to:
µ= qτ
m∗ (2.12)
whereτ is the relaxation time. The lower hole effective mass, in addition to the lower density of states in the valence band, and longer relaxation times in germanium compared to silicon is directly responsible for the higher bulk hole mobility observed in germanium, making it an attractive alternative channel material to silicon in pMOSFETs [Sze, 1981].
In addition to taking advantage of the intrinsically higher hole mobility offered by germanium, it is also possible to take advantage of the larger lattice constant of germanium and incorporate it into silicon to form a silicon-germanium (Si1−xGex) alloy to form a strained channel. The effects of strain on the Si1−xGex band structure and carrier mobility are well documented Fischetti and Laux [1996], Paul [1999], Sch¨aeffler, F. [1997], Whall and Parker [1998]. Consequently, the discussion presented will focus on how strain effects the valence band structure in a pseudomorphic Si1−xGex layer structure, relevant to the hole mobility for pMOSFETs.
Silicon and germanium are both group IV elements that are entirely miscible such that it is possible to form a Si1−xGex random alloy with properties that vary gradually from those that are Si-like to those which are Ge-like across the entire composition range. The unstrained alloy, like both silicon and germanium, crystallises in the diamond structure with a lattice constant that varies almost linearly with composition. The band structure of the unstrained Si1−xGex is almost the same as that of silicon up to a germanium composition of 85%.
When athinSi1−xGex(withx>0) layer is grownpseudomorphically on a silicon substrate such that the Si1−xGex layer takes on the lattice constant of the underlying silicon substrate, the Si1−xGex layer will be compressively strained in the plane and under tensile strain in the growth direction, resulting in a tetragonal distortion. This is depicted in Figure 2.4.
The effect of strain on the valence band is twofold and is shown schematically in Figure 2.5. Firstly, there is a splitting of the sub-bands. Secondly, there is a change in the effective mass of the holes that populate the sub-bands [Sch¨aeffler, F., 1997].
The band-splitting results in the light and heavy holes becoming non-degenerate at the zone centre, with the heavy hole band being lowered in energy (remembering that the hole energy increases downwards in Figure 2.5), and the light hole band being raised in energy. This has the effect of reducing phonon scattering events between the two
Si substrate Si substrate relaxed Si Ge (x>0)1-x x strained Si Ge (x>0)1-x x
Figure 2.4: Schematic showing the pseudomorphic strained Si1−xGex layer (below the critical thickness) on a silicon substrate.
sub-bands. The probability of a scattering event is strongly dependent on the number of available states for a carrier to scatter into. For the unstrained case, the degeneracy of the light and heavy hole bands at the zone centre yields a high number of available states with similar energies for a carrier to scatter into. The strain-splitting of the valence band significantly reduces the number of scattering events between the two sub-bands, thus helping to increase the effective hole mobility. Furthermore, the split-off band is raised even further in energy from the now non-degenerate light and heavy hole bands, further reducing the likelihood of holes scattering into this sub-band [Xie, 1999].
In addition to the band-splitting, the effective hole mass of both bands is changed. The effective mass of the heavy-hole band is lowered, whilst that of the light-hole band is increased. Depending upon the strain conditions, it is possible for the effective mass in the heavy hole band to be lower than that of the light hole band, thus resulting in mass inversion. The lower effective mass also reduces the density of states and increases the acceleration of holes between scattering events, which can lead to a higher hole mobility.
k split-off band light-hole band heavy-hole band E (a) k split-off band light-hole band heavy-hole band E (b)
Figure 2.5: Schematic of the (a) unstrained and (b) biaxial compressively strained Si1−xGex valence band showing the band splitting and change of effective mass.
The addition of germanium and the lowering of the heavy hole sub-band reduces the strained Si1−xGex bandgap. The difference in the band gap between the strained Si1−xGexand silicon substrate leads to discontinuities (or offsets) in both the conduction band and valence band [Sch¨aeffler, F., 1997]. However, the conduction band offset is small, typically less than around 20 meV, and is often neglected, such that we can speak of a discontinuity in the valence band only. The valence band offset varies linearly with germanium composition and is given by [Galdin et al., 2000]:
∆Ev ≈0.74x (2.13)
The discontinuity in the valence band results in the confinement of holes in a quantum well in the high-mobility Si1−xGex layer, with higher germanium compositions resulting in a larger valence band offset and a greater degree of confinement.
MISFIT (%)
0 0.2 0.4 0.6 0.8 1.0
MECHANICAL EQUILIBRUM THEORY
C R IT IC A L L A Y E R T H IC K N E S S (h c ) 1nm 10nm 100nm 1000nm 0 1 2 3 4 GERMANIUM FRACTION (X) EXPERIMENT (Bean et al )
(Matthews and Blakeslee) van der Merwe
PRESENT WORK EXPERIMENT (BEKV et al. )
Figure 2.6: Critical thickness of Si1−xGex layers grown on silicon substrate as a function of germanium composition. After People and Bean [1985].
maximum thickness of the silicon-germanium layer, such that it remains strained, the so-calledcritical thickness [People and Bean, 1985]. The critical thickness is shown as a function of germanium composition in Figure 2.6. Layers exceeding the critical thickness will relax by the formation of misfit dislocations. This emphasises that whilst a pure germanium layer might be best for quantum confinement of holes, only 3 monolayers of strained Ge can be grown on a silicon substrate before the layer starts to relax, which is of insufficient thickness to support a hole quantum well. It is possible to grow epitaxial layers that are thicker than the critical thickness by carefully control of the growth conditions. Such layers are said to bemetastable and care must be taken to ensure that these layers do not relax during processing.
When designing a strained Si1−xGex MOSFET, it is customary to grow a thin silicon cap on top of the strained Si1−xGex channel. The silicon cap serves three main purposes [Palmer, 2001]. Firstly, oxidation of Si1−xGex layer to form a gate dielectric tends to result in a snow-plough effect with a build-up of germanium at the interface and high densities of interface traps. Secondly, part of the silicon cap will be consumed during processing. Finally, it can provide extra stability against strain relaxation, particularly for layers that exceed the critical thickness.