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Electronic band structures of silicon–germanium (SiGe) alloys
2
Electronic band structures of silicon–germanium (SiGe) alloys
N. M o r i, osaka university, japan
Abstract: Electronic band structures of SiGe systems are described mainly from a theoretical point of view. The electronic band structures of bulk Si, bulk Ge, and SiGe alloys are briefly reviewed. Strain effects on the electronic band structures of pseudomorphic strained SiGe heterostructures are discussed in detail. Band offset and effective masses in those heterostructures are also presented. Key words: pseudomorphic SiGe, band structure, strain, band offset, effective mass.
2.1
Band structures
In this chapter, electronic band structures of SiGe systems are described mainly from a theoretical point of view. The electronic band structures of bulk Si, bulk Ge, and SiGe alloys are first reviewed (Yu and Cardona, 1999; Hamaguchi, 2010). Then, strain effects on the electronic band structures of pseudomorphic strained SiGe heterostructures are discussed. Finally, effective masses in those heterostructures are presented. The discussions in this chapter focus mainly on the most commonly studied SiGe heterostructures of (100)-strained Si1–xGex on a Si substrate and (100)-strained Si on a relaxed virtual Si1–yGey substrate. In the following, we include some numerical values of material parameters, which are cited from Madelung (2004) unless otherwise stated. Note, however, that widely scattered values of some material parameters have been reported and our choice of numerical values here is not comprehensive. For more comprehensive reviews, see, e.g., Fischetti and Laux (1996), Schaffler (1997), Brunner (2002), Cressler and Niu (2003), Yang et al. (2004), Paul (2004) and Lee et al. (2005).
2.1.1 Bulk silicon (Si) and germanium (Ge) Si and Ge are both group IV elemental semiconductors and form the diamond crystal structure with a face-centered cubic arrangement of the atoms (see Fig. 1.1 of Chapter 1). The lattice constant of Ge is aGe = 0.5658 nm, and is larger by 4.2% than the Si lattice constant, aSi = 0.5431 nm. There is a trend that semiconductors with a larger lattice constant are more likely to 26 © Woodhead Publishing Limited, 2011
3
3
2
2
1
Bandgap
VBM (G)
CBM (D)
0
Energy (eV)
Energy (eV)
Electronic band structures of silicon–germanium (SiGe) alloys
–1
27
1 VBM (G)
CBM (L)
0
Bandgap
–1
–2
–2 G Wave vector (a)
L
X
G Wave vector (b)
L
X
2.1 Electronic band structure of (a) Si and (b) Ge calculated by the empirical tight-binding method (Jancu et al., 1998). kz
kz
X W K
kx
X W
U L G
U
K
ky
kx
(a)
L G
ky
(b)
2.2 Many-valley structure of (a) Si and (b) Ge.
have a smaller bandgap. This is the case for Si and Ge as shown in Fig. 2.1, in which the electronic band structures of Si and Ge are plotted. Both Si and Ge are indirect gap semiconductors. Si has the fundamental bandgap of Eg = 1.12 eV at T = 300 K. The conduction-band minima (CBM) of Si are located at the D point, 15% from the X point, along the ·100Ò axes, and thus there are six equivalent conduction valleys in Si (see Fig. 2.2(a)). When a ·100Ò axis is chosen as the principal axis of a newly defined k-vector and the origins of k and E are set at the bottom of the conduction band, the dispersion relation is written as 2 2 E (k ) = (kx2 + ky2 ) + kz2 2 mt 2m
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2.1
28
Silicon–germanium (SiGe) nanostructures
For Si, the transverse effective mass is mt = 0.19m0 and the longitudinal effective mass is m = 0.92m0, and Eq. 2.1 describes ellipsoidal constant energy surfaces as schematically shown in Fig. 2.2(a). Compared to Si, Ge has a significantly smaller bandgap of Eg = 0.66 eV at T = 300 K. The conduction-band minima of Ge are located at the L point along the ·111Ò axes and there are four equivalent conduction valleys (or eight half-valleys) in Ge (see Fig. 2.2(b)). The E–k relation of the Ge conduction band is described also by Eq. 2.1 with mt = 0.82m0 and m = 1.6m0. unlike Si, the second conduction-band minimum, which is located at the G point, has relatively lower energy in Ge (see Fig. 2.1(b)) and can play a role in electron transport. The energy gap of this direct band-edge is Eg = 0.81 eV. For the valence band, the valence-band maxima (VBM) is located at the G point in both Si and Ge (see Fig. 2.1). There are three principal valence bands of heavy-hole (HH), light-hole (LH), and split-off (SO) bands in Si and Ge. Ge has a significantly larger spin-orbit splitting (DGe = 0.30 eV) compared to Si (DSi = 0.044 eV), which is primarily due to the heavier atom of Ge. The HH and LH bands are degenerated at the G point, so the interactions are strong for k ≠ 0 and the band shapes become non-parabolic and anisotropic, usually referred to as warped shapes. However, for simplicity, it is often expedient to assume that the valence-band effective masses are isotropic. In such a case, average hole masses are defined by averaging an anisotropic mass over all possible directions of k. The averaged experimental hole masses are mHH = 0.54m0, mLH = 0.15m0, and mSo = 0.23m0 for Si, and mHH = 0.34m0, mLH = 0.043m0, and mSo = 0.095m0 for Ge (Cardona and Pollak, 1966; Yu and Cardona, 1999). We notice that Ge has smaller effective masses, especially for the LH band.
2.1.2
SiGe alloys
Si and Ge are miscible in all proportions, forming a continuous series of solid substitutional solutions of a fixed crystal structure over the entire composition range. The lattice constant aSiGe(x) of Si1–xGex alloys shows minor departures from the linear dependence of Vegard’s rule as discussed in Chapter 1. The fundamental bandgap of Si1–xGex alloys is Si-like between the G-valence and ∆-conduction valleys up to a Ge content of about x = 0.85. The material has a Ge-like character with a conduction-band minimum at the L point for x > 0.85 (Braunstein et al., 1958). Figure 2.3 shows the experimental lowtemperature excitonic bandgap fitted by analytical expressions (Weber and Alonso, 1989): EgDx (x ) = 1.155 – 0.43x + 0.206x 2 eV EgLx (x ) = 2.010 – 1.270x eV
(Si-llike) ike)
(Ge-like)
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2.2 2.3
Electronic band structures of silicon–germanium (SiGe) alloys
29
1.2 Unstrained SiGe
1.1
Valleys
Energy gap (eV)
D
L
1 Strained SiGe on Si substrate
0.9
0.8
0.7
0.6
0
0.2
0.4 0.6 0.8 Germanium fraction x
1
2.3 Low-temperature bandgap vs composition x for Si1–xGex alloys (solid line), compared with the bandgap of a strained Si1–xGex layer on a Si substrate (dashed line).
2.2
Strain effects
High-quality epitaxial interfaces can be produced between materials with different lattice constants. When the difference in lattice constants is small enough, the lattice mismatch is accommodated only by the tetragonal deformation of the epilayer. This so-called pseudomorphic layer is characterized by an in-plane lattice constant which remains the same throughout the structure. The strain present in such a structure dramatically alters its optical and electronic properties, in comparison to the bulk constituents. For example, the energy gap of strained Si1–xGex on a Si substrate is substantially smaller than that of unstrained Si1–xGex alloys as shown in Fig. 2.3, where the experimental low-temperature excitonic bandgap of strained Si1–xGex fitted by an analytical expression (Robbins et al., 1992)
Egx(x) = 1.155 – 0.874x + 0.376x2 eV (x < 0.24)
2.4
is plotted by the dashed line. In this section, strain effects are first described in detail for (100)-strained Si1–xGex on a Si substrate (People, 1985; Van de Walle, 1989). The energy levels of (100)-strained Si1–xGex layers on a Si1–yGey substrate are then briefly discussed (Rieger and Vogl, 1993) followed by a discussion on the conduction and valence band offsets in strained-Si1–xGex/ Si and strained-Si/Si1–yGey heterostructures.
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2.2.1
Silicon–germanium (SiGe) nanostructures
Strained Si1–xGex on Si substrate
To understand strain effects on electronic band structures in strained Si1–xGex on a Si substrate (see Fig. 2.4), we first describe the band alignment between unstrained Si1–xGex and bulk Si by introducing an ‘absolute’ energy scale (Van de Walle, 1989). Then strain in the pseudomorphic Si1–xGex layer is calculated within an elastic theory. After that, we discuss strain effects on the electronic band structure by calculating the energy shift and splitting of the band-edges of pseudomorphic Si1–xGex layers on a Si substrate. For heterostructures consisting of different semiconductors, the band lineups of the host semiconductors play an essential role in determining the electronic band structure. In principle it is possible to perform firstprinciples calculations of the band offsets at a semiconductor interface. However, a simple and reliable model to predict the band lineups is very useful for practical purposes. A theory based on the model-solid theory of Van de Walle and Martin gives us a simple and reliable means to examine not only lattice-matched but also strained layers (Van de Walle, 1989). The theory provides an absolute energy level for each semiconductor, which can be used to predict the band lineups of semiconductors. The absolute energy scale Ev,av is defined as an average of three uppermost valence bands at the G point (Ev, av = 1 (EHH + E LH + E So ))) and is parametrized in the literature 3 (Van de Walle, 1989, Table II) for common IV and III–V semiconductors. Relaxed SiGe
aSiGe
aSiGe
Strained SiGe
aSiGe
a||
a||
a^ aSi
aSi
aSi
aSi z y Si substrate (a)
aSi x
aSi Si substrate (b)
2.4 (a) The lattice constant of a Si1–xGex film to be grown on top of a Si substrate is larger than the Si lattice constant (aSiGe > aSi). (b) The in-plane lattice constant remains the same throughout the structure for the pseudomorphic boundary condition (a ÍÍ = aSi < aSiGe). The biaxial compressive strain in the x–y plane leads to an extension along the z-direction (a^ > aSiGe).
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Electronic band structures of silicon–germanium (SiGe) alloys
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The absolute energy scale Ev,av of Si is Ev,av = –7.03 eV and that of Ge is Ev,av = –6.35 eV. For alloys, linear interpolation between the pure materials is appropriate when considering a strain contribution (Cardona and Christensen, 1988; Van de Walle, 1989), since in a lattice mismatched alloy one material is effectively expanded, whereas the other is compressed. Rieger and Vogl (1993) have proposed a formula for Ev,av differences in a general system of a strained Si1–xGex layer on a Si1–yGey substrate: DEv,av = (0.47 – 0.06y)(x – y) eV
2.5
where positive values refer to higher energies in the strained layer. In a pseudomorphic system, the in-plane lattice constant a|| is the same throughout the structure. For Si1–xGex on a Si substrate, the in-plane lattice constant is thus equal to the bulk Si lattice constant (a|| = aSi). Since the unstrained Si1–xGex lattice constant is larger than the Si lattice constant (aSiGe > aSi), the strained Si1–xGex layer is subjected to biaxial compressive strain in the x–y plane perpendicular to the growth direction (z-direction). This leads to an extension normal to the plane so as to minimize the elastic energy. The lattice constant a^ of the strained layer in the direction perpendicular to the interface is then given by a^ = aSiGe[1 + D001(1 – a||/aSiGe)]
2.6
where the coefficient D001 is related to the elastic constants C11 and C12 as D001 = 2C12/C11 (see Table 2 of Chapter 1 for the numbers of C11 and C12). The resulting strain tensor, which has only diagonal components, is given by
e xx = e yy = e zz =
a|| – aSiGe aSi – aSiGe = aSiGe aSiGe
a^ – aSiGe = –D 001e xx aSiGe
2.7 2.8
Figure 2.5 shows the lattice constants a|| and a^ of a strained Si1–xGex layer, and the strain tensor components by the hydrostatic dilation DW/W = Tr(e) = exx + eyy + ezz = 2exx + ezz and the deviatoric uniaxial component ezz – exx = (a^ – a||)/aSiGe. The strain-induced energy change is originated in a hydrostatic component and a traceless deviatoric component. These components cause a shift and a splitting of degenerate band-edge levels, respectively (see Figs 2.6 and 2.7). The energy shift is proportional to the fractional volume change ∆W/W and is given by DEc = acTr(e) = ac(2exx + ezz) for the conduction band and
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2.9
32
Silicon–germanium (SiGe) nanostructures
Lattice constant (nm)
0.60 0.58
a^
aGe
0.56
a|| = aSi
0.54 0.52
(a)
0.10 ezz – exx
Strain
0.05 0.00
Tr(e)
–0.05 –0.10 0
0.2
0.4 0.6 0.8 Germanium fraction x (b)
1
2.5 (a) Lattice constants a ÍÍ and a ^ of a strained Si1–x Gex layer on a Si substrate. (b) Strain tensor components represented by the hydrostatic relative volume change Tr(e) and the uniaxial component ezz – exx.
DEv = avTr(e) = av(2exx + ezz)
2.10
for the valence band, where av (ac) is the dilation deformation potential for the valence (conduction) band. The bandgap deformation potential is equal to a = ac – av. Their values are av = 2.46 eV, ac∆ = 4.18 eV, and a = 1.72 eV, for Si, and av = 1.24 eV, acL = – 1.54 eV, and a = –2.78 eV for Ge (Van de Walle, 1989). The traceless part of the strain splits the six-fold degenerate ∆ conduction valleys into four-fold D|| valleys within the layer plane and two-fold D^ valleys along the growth direction. The splitting of the bands is proportional to the uniaxial strain component, ezz – exx, and is given by DE D Ec|| = – 1 X uD (e zz – e xx ) 3
2.11
DE D Ec^ = – 2 X uD (e zz – e xx ) 3
2.12
Since the deformation potential of Si is XuD = 8.7 eV > 0 (Balslev, 1966), the four-fold D|| valleys move downward in energy and become lower than the two-fold D^ valleys for the biaxial compressive strain in a strained
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Electronic band structures of silicon–germanium (SiGe) alloys
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Hydrostatic Deviatoric deformation deformation D^
D
D
D||
HH LH
HH, LH SO
HH, LH SO SO
(a)
(b)
Unstrained SiGe
(c)
(d)
Strained SiGe
Si substrate
2.6 Schematic band lineups in a strained SiGe layer on a Si substrate. (a) The valence band maximum (VBM) in a SiGe alloy is located above the VBM of a Si substrate. (b) The hydrostatic compressive strain reduces the band-edges. (c) The deviatoric strain components cause a splitting of degenerate band-edges. The heavyhole (HH) valence band moves upward in energy, resulting in a large valence-band offset with respect to the Si substrate. (d) The four-fold D ÍÍ conduction valleys move downward with a quite small conduction band offset. Hydrostatic Deviatoric deformation deformation D||
D
D
D^
HH, LH HH HH, LH SO
SO
LH SO Unstrained Si
Strained Si
Relaxed SiGe
2.7 Schematic band lineups in a strained Si layer on a SiGe substrate.
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Silicon–germanium (SiGe) nanostructures
SiGe layer on a Si substrate (see Fig. 2.6). For the biaxial tensile strain in a strained Si layer on a SiGe substrate, the two-fold D^ valleys become lower in energy than the four-fold D|| valleys (see Fig. 2.7). Under the tetragonal deformation, the four L points remain equivalent and are conventionally labeled by N in a tetragonal lattice. The traceless part has no influence on their energy levels. For valence bands of a strained SiGe alloy, the band structures can be described by the following 6 ¥ 6 k·p Hamiltonian (we follow the notation of Chao and Chuang (1992)): È P+Q Í Í –S† Í † Í R H (k ) = – Í 0 Í Í – 12 S † Í † ÍÎ 2 R
1 S 2
–S
R
0
–
P–Q
0
R
– 2R
0 R† – 2Q 3 † 2S
3 † P–Q S 2S S† P + Q – 2 R† 3 2S
– 2R
22Q Q
1 S 2
–
P+D 0
˘ ˙ 3 ˙ 2S ˙ 2Q ˙ – 12 S † ˙ ˙ 0 ˙ ˙ P+D ˙ ˚ 2R
2.13 with P = Pk + Pe, Q = Qk + Qe, R = Rk + Re, S = Sk + Se, and 2 Pk = g 1 (kx2 + ky2 + kz2 ) 2m0
2.14
2 Qk = g 2 (kx2 + ky2 + 2kz2 ) 2m0
2.15
2 Rk = 3[– g 2 (kx2 – ky2 ) + 2ig 3 kx ky ) 2m0
2.16
2 Sk = 2 3g 3 (kx – ik iky )kz 2m0
2.17
Pe = – avTr(e)
2.18
Qe = – b (e xx + e yy – 2e zz ) 2
2.19
Re =
3 b( b e xx – e yy ) – ide xy 2
Se = – d(exx – eyz)
2.20 2.21
where ∆ is the spin-orbit splitting, g1, g2, and g3 are the Luttinger parameters,
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Electronic band structures of silicon–germanium (SiGe) alloys
35
and b and d are the Bir–Pikus shear deformation potentials. The band-edge levels can be obtained by setting k = 0 in det[H(k) − E] = 0 and we have EHH = – Pe – Qe
2.22
ELH = – Pe – 1 [[Q Q –D+ 2 e
D 2 + 2DQe + 9Qe2 ]
2.23
Q –D– ESo = – Pe – 1 [[Q 2 e
D 2 + 2DQe + 9Qe2 ]
2.24
where use has been made of exx = eyy and eij = 0 for i ≠ j. The first terms on the right-hand sides of these equations (– Pe = avTr(e)) correspond to the energy shift due to the hydrostatic strain component as already given in Eq. 2.10. The second terms, which depend on – Qe = b(exx – ezz), represent effects of the deviatoric strain component and lift the degeneracy between the HH and LH bands. Figure 2.8 shows a Ge fraction dependence of the band-edge levels of strained Si1–xGex on a Si substrate. The shaded region corresponds to the Si bandgap. The conduction-band minima consist of the four-fold D|| valleys. The bandgap of strained Si1–xGex is significantly smaller than that of the unstrained Si1–xGex alloy. The valence band offset, DEv, is the most substantial in this system, while the conduction-band offset, DEc, is quite small. Note that DEc depends on the parameter set used for the calculation and can have the opposite sign to that shown in Fig. 2.8 (Rieger and Vogl, 1993).
kz
Energy (eV)
1.5
kz
D^
D||
1
HH
0.5
DEv
LH
0 SO 0
0.2
0.4 0.6 Germanium fraction x
0.8
2.8 Band-edge levels of a strained Si1–xGex layer on a Si substrate as a function of Ge fraction x.
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Silicon–germanium (SiGe) nanostructures
2.2.2 Strained Si1–xGex on Si1–yGey substrate A complete set of band parameters for pseudomorphic (100)-strained Si1–xGex on a Si1–yGey substrate was presented by Rieger and Vogl (1993). The band parameters needed for device applications (bandgaps, band offsets, effective masses, and deformation potentials) are parametrized from nonlocal pseudopotential calculations with spin-orbit interaction. Figure 2.9 shows a contour plot of the fundamental bandgap of a strained Si1–xGex layer on a Si1–yGey substrate. The bandgap becomes largest in unstrained Si (x = y = 0) and smallest in a strained Si layer on a pure-Ge substrate (x = 0 and y = 1). When x < y, the strained Si1–xGex layer is subjected to biaxial compressive strain because aSiGe(x) < aSiGe(y). This results in the Si-like conduction-band minimum on the four-fold D|| axis for most compositions. On the other hand, when x > y, the strained Si1–xGex layer is subjected to biaxial tensile strain, resulting in the Si-like conduction-band minimum on the two-fold D^ axis. Only for weakly strained alloys with high Ge content does the conduction-band minimum lie at the N point, which corresponds to the L point for pure Ge.
2.2.3 Band offsets The band alignments between heterointerfaces are one of the most important parameters for both transport and optical device applications. The precise 1 0.5
0.6
0.7
0.8
N
Germanium fraction y
0.8 D^ 0.6
0.4
D||
0.2 1.0 0
0
0.2
0.9
0.8
0.4 0.6 0.8 Germanium fraction x
0.7 1
2.9 Fundamental bandgap in eV of a (100)-strained Si1–xGex layer on a Si1–yGey substrate as a function of x and y (Rieger and Vogl, 1993). The thick solid line represents the boundary between different conduction-band minima.
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Electronic band structures of silicon–germanium (SiGe) alloys
37
determination of the band offsets is, however, generally difficult in both experiments and theory. A strained Si1–xGex layer on a Si substrate has a large valence band offset DEv for holes being confined in the strained Si1–xGex layer (see Fig. 2.8). On the other hand, the conduction-band offset is very small, suggesting marginal type-I (DEc = Ec(Si) – Ec(Si1–xGex) > 0) or type-II (DEc < 0) heterostructures. The experimental values for the band offsets are somewhat uncertain, but show a clear trend of a near-linear dependence of DEv on a Ge fraction, x, for low x. The valence-band offset can be approximated by (Yang et al., 2004)
DEv = 0.71x eV
2.25
For a strained Si layer on a Si1–yGey substrate, the system is a type-II heterostructure with larger band offsets for both the conduction and valence bands (see Fig. 2.10(b)). The valence-band and conduction-band offsets can be approximated by (Yang et al., 2004)
DEv = 0.238y – 0.03y2 eV
2.26
DEc = 0.35y + 0.35y2 – 0.12y3 eV
2.27
and
For a general system of a strained Si1–xGex layer on a Si1–yGey substrate, the valence band maximum occurs always in the layer with the higher Ge content (Schaffler, 1997). A type-I alignment is predicted for a Ge-rich strained layer on a Ge-rich substrate (Rieger and Vogl, 1993).
Ec
DEc
Relaxed Si
Strained Si Strained Si1–xGex DEv
Relaxed Si1–yGey DEv
Ev
Ev Strained Si1–xGex
Strained Si
Relaxed Si
Relaxed Si1–yGey
(a)
(b)
2.10 Schematic band alignments for (a) a biaxial compressively strained Si1–xGex layer on a Si substrate and (b) a biaxial tensile strained Si layer on a Si1–yGey substrate.
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2.3
Silicon–germanium (SiGe) nanostructures
Effective mass
In this section, we first consider electron effective masses in strained SiGe layers. As discussed in the previous section, the six-fold degenerate ∆ conduction valleys in unstrained Si split into the four-fold D|| and the two-fold D^ conduction valleys in a strained Si1–xGex layer on a Si1–yGey substrate. This results in a change in the electron effective mass. Let us consider a change in the conductivity mass, which is particularly important for transport applications, for example. For the six-fold ∆ valleys, the conductivity mass can be written as mc =
3 m–1 + 2mt–1
2.28
For the four-fold D|| valleys, this will change into mc|| =
2 , mc^ = mt m–1 + mt–1
2.29
in the in-plane (x–y) and perpendicular (z) directions, respectively. For the two-fold D^ valleys, mc|| = mt and mc^ = m. These types of changes in the effective mass, which originate in a change in the location of the conductionband minimum, significantly affect the transport and optical properties of the SiGe heterostructures. A most significant change in the effective mass will occur when the L minimum sinks energetically below the ∆ minimum for weakly strained alloys with high Ge content. Note, however, that we have to take into account all the energetically close lying conduction-band minima at finite temperatures, which will smooth out discontinuities in the effective masses associated with an abrupt change in the conduction-band minima. In addition to the above-mentioned changes in the electron effective mass, the strain and Ge fraction cause a change in the effective mass value relative to the bulk Si value. Figure 2.11 shows a Ge fraction dependence of the conductivity mass in strained Si1–xGex on a Si substrate and in strained Si on a Si1–yGey substrate (Rieger and Vogl, 1993). The effective masses weakly depend on the Ge fraction. For example, the parallel conductivity mass, mc|| ( yy), in strained Si on a Si1–yGey substrate increases slightly with the Ge fraction y from the bulk Si value of mt at y = 0. We next consider hole effective masses in strained SiGe layers. A simple effective-mass description is not usually sufficient for describing the valenceband structure of strained SiGe heterostructures, since it is highly non-parabolic and anisotropic. once an appropriate k·p band-parameter set is known, the six-band k·p Hamiltonian of Eq. 2.13 can be used for accurately describing the valence-band structure of strained SiGe alloys. Rieger and Vogl (1993) reported the k·p band parameters L, M, and N of the Dreselhaus notation,
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0
0.195
0.200
0.205
0.210
0.320
0.325
0.330
0.2
0.4 0.6 0.8 Germanium fraction x (a)
Perpendicular
Parallel
1
0.860
0.880
0.900
0.920
0.195
0.200
0.205
0.210
0
0.2
0.4 0.6 0.8 Germanium fraction y (b)
Perpendicular
Parallel
1
2.11 Ge fraction dependence of the conductivity masses in (a) a strained Si1–xGex layer on a Si substrate and (b) a strained Si layer on a Si1–yGey substrate calculated from the parametrized effective masses of Rieger and Vogl (1993).
Electron conductivity mass (m0)
0.335
Electron conductivity mass (m0)
40
Silicon–germanium (SiGe) nanostructures
which fit remarkably well with the pseudopotential band structure, as (P standing equally for L, M, or N) P(x) = P(0) + a ln(1 – Sxb)
2.30
S = 1 – exp{[P(1) – P(0)]/a}
2.31
with α = 6.7064, β = 1.35, L(0) = −6.69, M(0) = −4.62, N(0) = −8.56, L(1) = −21.65, M(1) = −5.02, and N(1) = −23.48 (units are 2/2m0). The L, M, and N parameters can be converted to the Luttinger parameters through g1 = –(L + 2M)/3 – 1, g2 = –(L – M)/6, and g3 = –N/6, where L, M, and N are in units of 2/2m0. As an example of the valence-band structure calculation based on the six-band k·p Hamiltonian, Fig. 2.12 shows a Ge fraction dependence of hole density-of-states (DOS) masses, md, at the room-temperature thermal energy Eth = 23 kkT in strained Si1–xGex on a Si substrate and in strained Si on a Si1–yGey substrate (Fischetti and Laux, 1996; Yang et al., 2004). The DoS mass md is defined through 1 Ê 2 md ˆ 2p 2 Ë 2 ¯
3/2
1/2 E1/2 th = 2 S d (Eth – E kin (k )))
2.32
k
where Ekin(k) (= E(0) − E(k)) is the hole kinetic energy. The substantial reduction in md of the HH band (the band coming from the Si HH band) can be seen for a finite Ge fraction in both strained Si1–xGex on a Si substrate
Hole DOS mass (m0)
1
Strained Si on Si1–yGey
Strained Si1–xGex on Si HH
0.8
0.6
0.4
LH
0.2 SO 0 1
x
0.5
0 0.5 y Germanium fraction
1
2.12 Ge fraction dependence of hole density-of-states masses at T = 300 K for a strained Si1–xGex layer on a Si substrate (left half) and a strained Si layer on a Si1–yGey substrate (right half).
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Electronic band structures of silicon–germanium (SiGe) alloys
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and strained Si on a Si1–yGey substrate. Note that the HH band is the topmost valence band in strained Si1–xGex on a Si substrate, while the LH band is at the top of the valence band in strained Si on a Si1–yGey substrate (see Figs 2.6 and 2.7).
2.4
Conclusion
In a pseudomorphic SiGe layer, the lattice mismatch is accommodated by the tetragonal deformation of the layer. The strain present in such a layer dramatically alters its optical and electronic properties, in comparison to the bulk constituents. The strain effects on the electronic band structures can be understood by considering (1) the band alignment between unstrained host semiconductors, (2) strain present in a pseudomorphic layer, which can be estimated from an elastic theory, and (3) the energy shift and splitting of the band-edges due to the strain. The fundamental bandgap of a strained Si1–xGex layer on a Si1–yGey substrate becomes largest in unstrained Si (x = y = 0) and smallest in a strained Si layer on a pure-Ge substrate (x = 0 and y = 1). Effective mass variations originating from a change in the location of the band extrema have a strong impact on the transport properties of the SiGe heterostructures.
2.5
References
Balslev I (1966), ‘Influence of uniaxial stress on indirect absorption edge in silicon and germanium’, Physical Review, 143, 636–647. Braunstein R, Moore AR and Herman F (1958), ‘Intrinsic optical absorption in germanium–silicon alloys’, Physical Review, 109, 695–710. Brunner K (2002) , ‘Si/Ge nanostructures’, Reports on Progress in Physics, 65, 27–72. Cardona M and Christensen NE (1988), ‘Comment on “Spectroscopy of excited states in In0.53Ga0.47As-InP single quantum wells grown by chemical-beam epitaxy” ’, Physical Review B, 37, 1011–1012. Cardona M and Pollak FH (1966), ‘Energy-band structure of germanium and silicon: The k·p method’, Physical Review, 142, 530–543. Chao CYP and Chuang SL (1992), ‘Spin-orbit-coupling effects on the valence-band structure of strained semiconductor quantum-wells’, Physical Review B, 46, 4110–4122. Cressler JD and Niu G (2003), Silicon–Germanium Heterojunction Bipolar Transistors, Boston, MA, Artech House. Fischetti MV and Laux SE (1996), ‘Band structure, deformation potentials, and carrier mobility in strained Si, Ge, and SiGe alloys’, Journal of Applied Physics, 80, 2234–2252. Hamaguchi C (2010), Basic Semiconductor Physics, Berlin, Splinger-Verlag. Jancu JM, Scholz R, Beltram F and Bassani F (1998), ‘Empirical spds* tight-binding calculation for cubic semiconductors: General method and material parameters’, Physical Review B, 57, 6493–6507. Lee ML, Fitzgerald EA, Bulsara MT, Currie MT and Lochtefeld A (2005), ‘Strained
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Silicon–germanium (SiGe) nanostructures
Si, SiGe, and Ge channels for high-mobility metal-oxide-semiconductor field-effect transistors’, Journal of Applied Physics, 97, 011101 (1–27). Madelung O (2004), Semiconductors: Data Handbook, Berlin, Springer-Verlag. Paul DJ (2004), ‘Si/SiGe heterostructures: from material and physics to devices and circuits’, Semiconductor Science and Technology, 19, R75–R108. People R (1985), ‘Indirect band-gap of coherently strained GexSi1–x bulk alloys on (001) silicon substrates’, Physical Review B, 32, 1405–1408; Erratum: ibid., 33, 1451 (1986). Rieger MM and Vogl P (1993), ‘Electronic-band parameters in strained Si1–xGex alloys on Si1–yGey substrates’, Physical Review B, 48, 14276–14287; Erratum: ibid., 50, 8183 (1994). Robbins DJ, Canham LT, Barnett SJ, Pitt AD and Calcott P (1992), ‘Near-band-gap photoluminescence from pseudomorphic Si1–xGex single layers on silicon’, Journal of Applied Physics, 71, 1407–1414. Schaffler F (1997), ‘High-mobility Si and Ge structures’, Semiconductor Science and Technology, 12, 1515–1549. Van de Walle CG (1989), ‘Band lineups and deformation potentials in the model-solid theory’, Physical Review B, 39, 1871–1883. Weber J and Alonso MI (1989), ‘Near-band-gap photoluminescence of Si-Ge alloys’, Physical Review B, 40, 5683–5693. Yang LF, Watling JR, Wilkins RCW, Borici M, Barker JR, Asenov A and Roy S (2004), ‘Si/SiGe heterostructure parameters for device simulations’, Semiconductor Science and Technology, 19, 1174–1182. Yu PY and Cardona M (1999), Fundamentals of Semiconductors, Berlin, SpringerVerlag.
© Woodhead Publishing Limited, 2011
Electronic band structures of silicon–germanium (SiGe) alloys
N. M o r i, osaka university, japan
Abstract: Electronic band structures of SiGe systems are described mainly from a theoretical point of view. The electronic band structures of bulk Si, bulk Ge, and SiGe alloys are briefly reviewed. Strain effects on the electronic band structures of pseudomorphic strained SiGe heterostructures are discussed in detail. Band offset and effective masses in those heterostructures are also presented. Key words: pseudomorphic SiGe, band structure, strain, band offset, effective mass.
2.1
Band structures
In this chapter, electronic band structures of SiGe systems are described mainly from a theoretical point of view. The electronic band structures of bulk Si, bulk Ge, and SiGe alloys are first reviewed (Yu and Cardona, 1999; Hamaguchi, 2010). Then, strain effects on the electronic band structures of pseudomorphic strained SiGe heterostructures are discussed. Finally, effective masses in those heterostructures are presented. The discussions in this chapter focus mainly on the most commonly studied SiGe heterostructures of (100)-strained Si1–xGex on a Si substrate and (100)-strained Si on a relaxed virtual Si1–yGey substrate. In the following, we include some numerical values of material parameters, which are cited from Madelung (2004) unless otherwise stated. Note, however, that widely scattered values of some material parameters have been reported and our choice of numerical values here is not comprehensive. For more comprehensive reviews, see, e.g., Fischetti and Laux (1996), Schaffler (1997), Brunner (2002), Cressler and Niu (2003), Yang et al. (2004), Paul (2004) and Lee et al. (2005).
2.1.1 Bulk silicon (Si) and germanium (Ge) Si and Ge are both group IV elemental semiconductors and form the diamond crystal structure with a face-centered cubic arrangement of the atoms (see Fig. 1.1 of Chapter 1). The lattice constant of Ge is aGe = 0.5658 nm, and is larger by 4.2% than the Si lattice constant, aSi = 0.5431 nm. There is a trend that semiconductors with a larger lattice constant are more likely to 26 © Woodhead Publishing Limited, 2011
3
3
2
2
1
Bandgap
VBM (G)
CBM (D)
0
Energy (eV)
Energy (eV)
Electronic band structures of silicon–germanium (SiGe) alloys
–1
27
1 VBM (G)
CBM (L)
0
Bandgap
–1
–2
–2 G Wave vector (a)
L
X
G Wave vector (b)
L
X
2.1 Electronic band structure of (a) Si and (b) Ge calculated by the empirical tight-binding method (Jancu et al., 1998). kz
kz
X W K
kx
X W
U L G
U
K
ky
kx
(a)
L G
ky
(b)
2.2 Many-valley structure of (a) Si and (b) Ge.
have a smaller bandgap. This is the case for Si and Ge as shown in Fig. 2.1, in which the electronic band structures of Si and Ge are plotted. Both Si and Ge are indirect gap semiconductors. Si has the fundamental bandgap of Eg = 1.12 eV at T = 300 K. The conduction-band minima (CBM) of Si are located at the D point, 15% from the X point, along the ·100Ò axes, and thus there are six equivalent conduction valleys in Si (see Fig. 2.2(a)). When a ·100Ò axis is chosen as the principal axis of a newly defined k-vector and the origins of k and E are set at the bottom of the conduction band, the dispersion relation is written as 2 2 E (k ) = (kx2 + ky2 ) + kz2 2 mt 2m
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2.1
28
Silicon–germanium (SiGe) nanostructures
For Si, the transverse effective mass is mt = 0.19m0 and the longitudinal effective mass is m = 0.92m0, and Eq. 2.1 describes ellipsoidal constant energy surfaces as schematically shown in Fig. 2.2(a). Compared to Si, Ge has a significantly smaller bandgap of Eg = 0.66 eV at T = 300 K. The conduction-band minima of Ge are located at the L point along the ·111Ò axes and there are four equivalent conduction valleys (or eight half-valleys) in Ge (see Fig. 2.2(b)). The E–k relation of the Ge conduction band is described also by Eq. 2.1 with mt = 0.82m0 and m = 1.6m0. unlike Si, the second conduction-band minimum, which is located at the G point, has relatively lower energy in Ge (see Fig. 2.1(b)) and can play a role in electron transport. The energy gap of this direct band-edge is Eg = 0.81 eV. For the valence band, the valence-band maxima (VBM) is located at the G point in both Si and Ge (see Fig. 2.1). There are three principal valence bands of heavy-hole (HH), light-hole (LH), and split-off (SO) bands in Si and Ge. Ge has a significantly larger spin-orbit splitting (DGe = 0.30 eV) compared to Si (DSi = 0.044 eV), which is primarily due to the heavier atom of Ge. The HH and LH bands are degenerated at the G point, so the interactions are strong for k ≠ 0 and the band shapes become non-parabolic and anisotropic, usually referred to as warped shapes. However, for simplicity, it is often expedient to assume that the valence-band effective masses are isotropic. In such a case, average hole masses are defined by averaging an anisotropic mass over all possible directions of k. The averaged experimental hole masses are mHH = 0.54m0, mLH = 0.15m0, and mSo = 0.23m0 for Si, and mHH = 0.34m0, mLH = 0.043m0, and mSo = 0.095m0 for Ge (Cardona and Pollak, 1966; Yu and Cardona, 1999). We notice that Ge has smaller effective masses, especially for the LH band.
2.1.2
SiGe alloys
Si and Ge are miscible in all proportions, forming a continuous series of solid substitutional solutions of a fixed crystal structure over the entire composition range. The lattice constant aSiGe(x) of Si1–xGex alloys shows minor departures from the linear dependence of Vegard’s rule as discussed in Chapter 1. The fundamental bandgap of Si1–xGex alloys is Si-like between the G-valence and ∆-conduction valleys up to a Ge content of about x = 0.85. The material has a Ge-like character with a conduction-band minimum at the L point for x > 0.85 (Braunstein et al., 1958). Figure 2.3 shows the experimental lowtemperature excitonic bandgap fitted by analytical expressions (Weber and Alonso, 1989): EgDx (x ) = 1.155 – 0.43x + 0.206x 2 eV EgLx (x ) = 2.010 – 1.270x eV
(Si-llike) ike)
(Ge-like)
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2.2 2.3
Electronic band structures of silicon–germanium (SiGe) alloys
29
1.2 Unstrained SiGe
1.1
Valleys
Energy gap (eV)
D
L
1 Strained SiGe on Si substrate
0.9
0.8
0.7
0.6
0
0.2
0.4 0.6 0.8 Germanium fraction x
1
2.3 Low-temperature bandgap vs composition x for Si1–xGex alloys (solid line), compared with the bandgap of a strained Si1–xGex layer on a Si substrate (dashed line).
2.2
Strain effects
High-quality epitaxial interfaces can be produced between materials with different lattice constants. When the difference in lattice constants is small enough, the lattice mismatch is accommodated only by the tetragonal deformation of the epilayer. This so-called pseudomorphic layer is characterized by an in-plane lattice constant which remains the same throughout the structure. The strain present in such a structure dramatically alters its optical and electronic properties, in comparison to the bulk constituents. For example, the energy gap of strained Si1–xGex on a Si substrate is substantially smaller than that of unstrained Si1–xGex alloys as shown in Fig. 2.3, where the experimental low-temperature excitonic bandgap of strained Si1–xGex fitted by an analytical expression (Robbins et al., 1992)
Egx(x) = 1.155 – 0.874x + 0.376x2 eV (x < 0.24)
2.4
is plotted by the dashed line. In this section, strain effects are first described in detail for (100)-strained Si1–xGex on a Si substrate (People, 1985; Van de Walle, 1989). The energy levels of (100)-strained Si1–xGex layers on a Si1–yGey substrate are then briefly discussed (Rieger and Vogl, 1993) followed by a discussion on the conduction and valence band offsets in strained-Si1–xGex/ Si and strained-Si/Si1–yGey heterostructures.
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2.2.1
Silicon–germanium (SiGe) nanostructures
Strained Si1–xGex on Si substrate
To understand strain effects on electronic band structures in strained Si1–xGex on a Si substrate (see Fig. 2.4), we first describe the band alignment between unstrained Si1–xGex and bulk Si by introducing an ‘absolute’ energy scale (Van de Walle, 1989). Then strain in the pseudomorphic Si1–xGex layer is calculated within an elastic theory. After that, we discuss strain effects on the electronic band structure by calculating the energy shift and splitting of the band-edges of pseudomorphic Si1–xGex layers on a Si substrate. For heterostructures consisting of different semiconductors, the band lineups of the host semiconductors play an essential role in determining the electronic band structure. In principle it is possible to perform firstprinciples calculations of the band offsets at a semiconductor interface. However, a simple and reliable model to predict the band lineups is very useful for practical purposes. A theory based on the model-solid theory of Van de Walle and Martin gives us a simple and reliable means to examine not only lattice-matched but also strained layers (Van de Walle, 1989). The theory provides an absolute energy level for each semiconductor, which can be used to predict the band lineups of semiconductors. The absolute energy scale Ev,av is defined as an average of three uppermost valence bands at the G point (Ev, av = 1 (EHH + E LH + E So ))) and is parametrized in the literature 3 (Van de Walle, 1989, Table II) for common IV and III–V semiconductors. Relaxed SiGe
aSiGe
aSiGe
Strained SiGe
aSiGe
a||
a||
a^ aSi
aSi
aSi
aSi z y Si substrate (a)
aSi x
aSi Si substrate (b)
2.4 (a) The lattice constant of a Si1–xGex film to be grown on top of a Si substrate is larger than the Si lattice constant (aSiGe > aSi). (b) The in-plane lattice constant remains the same throughout the structure for the pseudomorphic boundary condition (a ÍÍ = aSi < aSiGe). The biaxial compressive strain in the x–y plane leads to an extension along the z-direction (a^ > aSiGe).
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Electronic band structures of silicon–germanium (SiGe) alloys
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The absolute energy scale Ev,av of Si is Ev,av = –7.03 eV and that of Ge is Ev,av = –6.35 eV. For alloys, linear interpolation between the pure materials is appropriate when considering a strain contribution (Cardona and Christensen, 1988; Van de Walle, 1989), since in a lattice mismatched alloy one material is effectively expanded, whereas the other is compressed. Rieger and Vogl (1993) have proposed a formula for Ev,av differences in a general system of a strained Si1–xGex layer on a Si1–yGey substrate: DEv,av = (0.47 – 0.06y)(x – y) eV
2.5
where positive values refer to higher energies in the strained layer. In a pseudomorphic system, the in-plane lattice constant a|| is the same throughout the structure. For Si1–xGex on a Si substrate, the in-plane lattice constant is thus equal to the bulk Si lattice constant (a|| = aSi). Since the unstrained Si1–xGex lattice constant is larger than the Si lattice constant (aSiGe > aSi), the strained Si1–xGex layer is subjected to biaxial compressive strain in the x–y plane perpendicular to the growth direction (z-direction). This leads to an extension normal to the plane so as to minimize the elastic energy. The lattice constant a^ of the strained layer in the direction perpendicular to the interface is then given by a^ = aSiGe[1 + D001(1 – a||/aSiGe)]
2.6
where the coefficient D001 is related to the elastic constants C11 and C12 as D001 = 2C12/C11 (see Table 2 of Chapter 1 for the numbers of C11 and C12). The resulting strain tensor, which has only diagonal components, is given by
e xx = e yy = e zz =
a|| – aSiGe aSi – aSiGe = aSiGe aSiGe
a^ – aSiGe = –D 001e xx aSiGe
2.7 2.8
Figure 2.5 shows the lattice constants a|| and a^ of a strained Si1–xGex layer, and the strain tensor components by the hydrostatic dilation DW/W = Tr(e) = exx + eyy + ezz = 2exx + ezz and the deviatoric uniaxial component ezz – exx = (a^ – a||)/aSiGe. The strain-induced energy change is originated in a hydrostatic component and a traceless deviatoric component. These components cause a shift and a splitting of degenerate band-edge levels, respectively (see Figs 2.6 and 2.7). The energy shift is proportional to the fractional volume change ∆W/W and is given by DEc = acTr(e) = ac(2exx + ezz) for the conduction band and
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2.9
32
Silicon–germanium (SiGe) nanostructures
Lattice constant (nm)
0.60 0.58
a^
aGe
0.56
a|| = aSi
0.54 0.52
(a)
0.10 ezz – exx
Strain
0.05 0.00
Tr(e)
–0.05 –0.10 0
0.2
0.4 0.6 0.8 Germanium fraction x (b)
1
2.5 (a) Lattice constants a ÍÍ and a ^ of a strained Si1–x Gex layer on a Si substrate. (b) Strain tensor components represented by the hydrostatic relative volume change Tr(e) and the uniaxial component ezz – exx.
DEv = avTr(e) = av(2exx + ezz)
2.10
for the valence band, where av (ac) is the dilation deformation potential for the valence (conduction) band. The bandgap deformation potential is equal to a = ac – av. Their values are av = 2.46 eV, ac∆ = 4.18 eV, and a = 1.72 eV, for Si, and av = 1.24 eV, acL = – 1.54 eV, and a = –2.78 eV for Ge (Van de Walle, 1989). The traceless part of the strain splits the six-fold degenerate ∆ conduction valleys into four-fold D|| valleys within the layer plane and two-fold D^ valleys along the growth direction. The splitting of the bands is proportional to the uniaxial strain component, ezz – exx, and is given by DE D Ec|| = – 1 X uD (e zz – e xx ) 3
2.11
DE D Ec^ = – 2 X uD (e zz – e xx ) 3
2.12
Since the deformation potential of Si is XuD = 8.7 eV > 0 (Balslev, 1966), the four-fold D|| valleys move downward in energy and become lower than the two-fold D^ valleys for the biaxial compressive strain in a strained
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Electronic band structures of silicon–germanium (SiGe) alloys
33
Hydrostatic Deviatoric deformation deformation D^
D
D
D||
HH LH
HH, LH SO
HH, LH SO SO
(a)
(b)
Unstrained SiGe
(c)
(d)
Strained SiGe
Si substrate
2.6 Schematic band lineups in a strained SiGe layer on a Si substrate. (a) The valence band maximum (VBM) in a SiGe alloy is located above the VBM of a Si substrate. (b) The hydrostatic compressive strain reduces the band-edges. (c) The deviatoric strain components cause a splitting of degenerate band-edges. The heavyhole (HH) valence band moves upward in energy, resulting in a large valence-band offset with respect to the Si substrate. (d) The four-fold D ÍÍ conduction valleys move downward with a quite small conduction band offset. Hydrostatic Deviatoric deformation deformation D||
D
D
D^
HH, LH HH HH, LH SO
SO
LH SO Unstrained Si
Strained Si
Relaxed SiGe
2.7 Schematic band lineups in a strained Si layer on a SiGe substrate.
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Silicon–germanium (SiGe) nanostructures
SiGe layer on a Si substrate (see Fig. 2.6). For the biaxial tensile strain in a strained Si layer on a SiGe substrate, the two-fold D^ valleys become lower in energy than the four-fold D|| valleys (see Fig. 2.7). Under the tetragonal deformation, the four L points remain equivalent and are conventionally labeled by N in a tetragonal lattice. The traceless part has no influence on their energy levels. For valence bands of a strained SiGe alloy, the band structures can be described by the following 6 ¥ 6 k·p Hamiltonian (we follow the notation of Chao and Chuang (1992)): È P+Q Í Í –S† Í † Í R H (k ) = – Í 0 Í Í – 12 S † Í † ÍÎ 2 R
1 S 2
–S
R
0
–
P–Q
0
R
– 2R
0 R† – 2Q 3 † 2S
3 † P–Q S 2S S† P + Q – 2 R† 3 2S
– 2R
22Q Q
1 S 2
–
P+D 0
˘ ˙ 3 ˙ 2S ˙ 2Q ˙ – 12 S † ˙ ˙ 0 ˙ ˙ P+D ˙ ˚ 2R
2.13 with P = Pk + Pe, Q = Qk + Qe, R = Rk + Re, S = Sk + Se, and 2 Pk = g 1 (kx2 + ky2 + kz2 ) 2m0
2.14
2 Qk = g 2 (kx2 + ky2 + 2kz2 ) 2m0
2.15
2 Rk = 3[– g 2 (kx2 – ky2 ) + 2ig 3 kx ky ) 2m0
2.16
2 Sk = 2 3g 3 (kx – ik iky )kz 2m0
2.17
Pe = – avTr(e)
2.18
Qe = – b (e xx + e yy – 2e zz ) 2
2.19
Re =
3 b( b e xx – e yy ) – ide xy 2
Se = – d(exx – eyz)
2.20 2.21
where ∆ is the spin-orbit splitting, g1, g2, and g3 are the Luttinger parameters,
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Electronic band structures of silicon–germanium (SiGe) alloys
35
and b and d are the Bir–Pikus shear deformation potentials. The band-edge levels can be obtained by setting k = 0 in det[H(k) − E] = 0 and we have EHH = – Pe – Qe
2.22
ELH = – Pe – 1 [[Q Q –D+ 2 e
D 2 + 2DQe + 9Qe2 ]
2.23
Q –D– ESo = – Pe – 1 [[Q 2 e
D 2 + 2DQe + 9Qe2 ]
2.24
where use has been made of exx = eyy and eij = 0 for i ≠ j. The first terms on the right-hand sides of these equations (– Pe = avTr(e)) correspond to the energy shift due to the hydrostatic strain component as already given in Eq. 2.10. The second terms, which depend on – Qe = b(exx – ezz), represent effects of the deviatoric strain component and lift the degeneracy between the HH and LH bands. Figure 2.8 shows a Ge fraction dependence of the band-edge levels of strained Si1–xGex on a Si substrate. The shaded region corresponds to the Si bandgap. The conduction-band minima consist of the four-fold D|| valleys. The bandgap of strained Si1–xGex is significantly smaller than that of the unstrained Si1–xGex alloy. The valence band offset, DEv, is the most substantial in this system, while the conduction-band offset, DEc, is quite small. Note that DEc depends on the parameter set used for the calculation and can have the opposite sign to that shown in Fig. 2.8 (Rieger and Vogl, 1993).
kz
Energy (eV)
1.5
kz
D^
D||
1
HH
0.5
DEv
LH
0 SO 0
0.2
0.4 0.6 Germanium fraction x
0.8
2.8 Band-edge levels of a strained Si1–xGex layer on a Si substrate as a function of Ge fraction x.
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Silicon–germanium (SiGe) nanostructures
2.2.2 Strained Si1–xGex on Si1–yGey substrate A complete set of band parameters for pseudomorphic (100)-strained Si1–xGex on a Si1–yGey substrate was presented by Rieger and Vogl (1993). The band parameters needed for device applications (bandgaps, band offsets, effective masses, and deformation potentials) are parametrized from nonlocal pseudopotential calculations with spin-orbit interaction. Figure 2.9 shows a contour plot of the fundamental bandgap of a strained Si1–xGex layer on a Si1–yGey substrate. The bandgap becomes largest in unstrained Si (x = y = 0) and smallest in a strained Si layer on a pure-Ge substrate (x = 0 and y = 1). When x < y, the strained Si1–xGex layer is subjected to biaxial compressive strain because aSiGe(x) < aSiGe(y). This results in the Si-like conduction-band minimum on the four-fold D|| axis for most compositions. On the other hand, when x > y, the strained Si1–xGex layer is subjected to biaxial tensile strain, resulting in the Si-like conduction-band minimum on the two-fold D^ axis. Only for weakly strained alloys with high Ge content does the conduction-band minimum lie at the N point, which corresponds to the L point for pure Ge.
2.2.3 Band offsets The band alignments between heterointerfaces are one of the most important parameters for both transport and optical device applications. The precise 1 0.5
0.6
0.7
0.8
N
Germanium fraction y
0.8 D^ 0.6
0.4
D||
0.2 1.0 0
0
0.2
0.9
0.8
0.4 0.6 0.8 Germanium fraction x
0.7 1
2.9 Fundamental bandgap in eV of a (100)-strained Si1–xGex layer on a Si1–yGey substrate as a function of x and y (Rieger and Vogl, 1993). The thick solid line represents the boundary between different conduction-band minima.
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Electronic band structures of silicon–germanium (SiGe) alloys
37
determination of the band offsets is, however, generally difficult in both experiments and theory. A strained Si1–xGex layer on a Si substrate has a large valence band offset DEv for holes being confined in the strained Si1–xGex layer (see Fig. 2.8). On the other hand, the conduction-band offset is very small, suggesting marginal type-I (DEc = Ec(Si) – Ec(Si1–xGex) > 0) or type-II (DEc < 0) heterostructures. The experimental values for the band offsets are somewhat uncertain, but show a clear trend of a near-linear dependence of DEv on a Ge fraction, x, for low x. The valence-band offset can be approximated by (Yang et al., 2004)
DEv = 0.71x eV
2.25
For a strained Si layer on a Si1–yGey substrate, the system is a type-II heterostructure with larger band offsets for both the conduction and valence bands (see Fig. 2.10(b)). The valence-band and conduction-band offsets can be approximated by (Yang et al., 2004)
DEv = 0.238y – 0.03y2 eV
2.26
DEc = 0.35y + 0.35y2 – 0.12y3 eV
2.27
and
For a general system of a strained Si1–xGex layer on a Si1–yGey substrate, the valence band maximum occurs always in the layer with the higher Ge content (Schaffler, 1997). A type-I alignment is predicted for a Ge-rich strained layer on a Ge-rich substrate (Rieger and Vogl, 1993).
Ec
DEc
Relaxed Si
Strained Si Strained Si1–xGex DEv
Relaxed Si1–yGey DEv
Ev
Ev Strained Si1–xGex
Strained Si
Relaxed Si
Relaxed Si1–yGey
(a)
(b)
2.10 Schematic band alignments for (a) a biaxial compressively strained Si1–xGex layer on a Si substrate and (b) a biaxial tensile strained Si layer on a Si1–yGey substrate.
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2.3
Silicon–germanium (SiGe) nanostructures
Effective mass
In this section, we first consider electron effective masses in strained SiGe layers. As discussed in the previous section, the six-fold degenerate ∆ conduction valleys in unstrained Si split into the four-fold D|| and the two-fold D^ conduction valleys in a strained Si1–xGex layer on a Si1–yGey substrate. This results in a change in the electron effective mass. Let us consider a change in the conductivity mass, which is particularly important for transport applications, for example. For the six-fold ∆ valleys, the conductivity mass can be written as mc =
3 m–1 + 2mt–1
2.28
For the four-fold D|| valleys, this will change into mc|| =
2 , mc^ = mt m–1 + mt–1
2.29
in the in-plane (x–y) and perpendicular (z) directions, respectively. For the two-fold D^ valleys, mc|| = mt and mc^ = m. These types of changes in the effective mass, which originate in a change in the location of the conductionband minimum, significantly affect the transport and optical properties of the SiGe heterostructures. A most significant change in the effective mass will occur when the L minimum sinks energetically below the ∆ minimum for weakly strained alloys with high Ge content. Note, however, that we have to take into account all the energetically close lying conduction-band minima at finite temperatures, which will smooth out discontinuities in the effective masses associated with an abrupt change in the conduction-band minima. In addition to the above-mentioned changes in the electron effective mass, the strain and Ge fraction cause a change in the effective mass value relative to the bulk Si value. Figure 2.11 shows a Ge fraction dependence of the conductivity mass in strained Si1–xGex on a Si substrate and in strained Si on a Si1–yGey substrate (Rieger and Vogl, 1993). The effective masses weakly depend on the Ge fraction. For example, the parallel conductivity mass, mc|| ( yy), in strained Si on a Si1–yGey substrate increases slightly with the Ge fraction y from the bulk Si value of mt at y = 0. We next consider hole effective masses in strained SiGe layers. A simple effective-mass description is not usually sufficient for describing the valenceband structure of strained SiGe heterostructures, since it is highly non-parabolic and anisotropic. once an appropriate k·p band-parameter set is known, the six-band k·p Hamiltonian of Eq. 2.13 can be used for accurately describing the valence-band structure of strained SiGe alloys. Rieger and Vogl (1993) reported the k·p band parameters L, M, and N of the Dreselhaus notation,
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0
0.195
0.200
0.205
0.210
0.320
0.325
0.330
0.2
0.4 0.6 0.8 Germanium fraction x (a)
Perpendicular
Parallel
1
0.860
0.880
0.900
0.920
0.195
0.200
0.205
0.210
0
0.2
0.4 0.6 0.8 Germanium fraction y (b)
Perpendicular
Parallel
1
2.11 Ge fraction dependence of the conductivity masses in (a) a strained Si1–xGex layer on a Si substrate and (b) a strained Si layer on a Si1–yGey substrate calculated from the parametrized effective masses of Rieger and Vogl (1993).
Electron conductivity mass (m0)
0.335
Electron conductivity mass (m0)
40
Silicon–germanium (SiGe) nanostructures
which fit remarkably well with the pseudopotential band structure, as (P standing equally for L, M, or N) P(x) = P(0) + a ln(1 – Sxb)
2.30
S = 1 – exp{[P(1) – P(0)]/a}
2.31
with α = 6.7064, β = 1.35, L(0) = −6.69, M(0) = −4.62, N(0) = −8.56, L(1) = −21.65, M(1) = −5.02, and N(1) = −23.48 (units are 2/2m0). The L, M, and N parameters can be converted to the Luttinger parameters through g1 = –(L + 2M)/3 – 1, g2 = –(L – M)/6, and g3 = –N/6, where L, M, and N are in units of 2/2m0. As an example of the valence-band structure calculation based on the six-band k·p Hamiltonian, Fig. 2.12 shows a Ge fraction dependence of hole density-of-states (DOS) masses, md, at the room-temperature thermal energy Eth = 23 kkT in strained Si1–xGex on a Si substrate and in strained Si on a Si1–yGey substrate (Fischetti and Laux, 1996; Yang et al., 2004). The DoS mass md is defined through 1 Ê 2 md ˆ 2p 2 Ë 2 ¯
3/2
1/2 E1/2 th = 2 S d (Eth – E kin (k )))
2.32
k
where Ekin(k) (= E(0) − E(k)) is the hole kinetic energy. The substantial reduction in md of the HH band (the band coming from the Si HH band) can be seen for a finite Ge fraction in both strained Si1–xGex on a Si substrate
Hole DOS mass (m0)
1
Strained Si on Si1–yGey
Strained Si1–xGex on Si HH
0.8
0.6
0.4
LH
0.2 SO 0 1
x
0.5
0 0.5 y Germanium fraction
1
2.12 Ge fraction dependence of hole density-of-states masses at T = 300 K for a strained Si1–xGex layer on a Si substrate (left half) and a strained Si layer on a Si1–yGey substrate (right half).
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Electronic band structures of silicon–germanium (SiGe) alloys
41
and strained Si on a Si1–yGey substrate. Note that the HH band is the topmost valence band in strained Si1–xGex on a Si substrate, while the LH band is at the top of the valence band in strained Si on a Si1–yGey substrate (see Figs 2.6 and 2.7).
2.4
Conclusion
In a pseudomorphic SiGe layer, the lattice mismatch is accommodated by the tetragonal deformation of the layer. The strain present in such a layer dramatically alters its optical and electronic properties, in comparison to the bulk constituents. The strain effects on the electronic band structures can be understood by considering (1) the band alignment between unstrained host semiconductors, (2) strain present in a pseudomorphic layer, which can be estimated from an elastic theory, and (3) the energy shift and splitting of the band-edges due to the strain. The fundamental bandgap of a strained Si1–xGex layer on a Si1–yGey substrate becomes largest in unstrained Si (x = y = 0) and smallest in a strained Si layer on a pure-Ge substrate (x = 0 and y = 1). Effective mass variations originating from a change in the location of the band extrema have a strong impact on the transport properties of the SiGe heterostructures.
2.5
References
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