- Email: [email protected]
Si superlattices
246
Thin Solid Films, 222 (1992) 246 250
Optical transitions in strained Ge/Si superlattices U. Schmid*, J. Humli~ekt, F. Lukegt and M. Cardona Max-Planck-lnstitut fiir FestkgrperJbrschung, Heisenbergstr. 1, W-7000 Stuttgart 80 (Germany)
H. Presting, H. Kibbel and E. Kasper Daimler Benz Research Centre at Ulm, W-7900 UIm (Germany)
K. Eberl~:, W. Wegscheider and G. Abstreiter Walter Schottky lnstitut, Technische Universitiit Miinchen, W-8046 Garching (Germany)
Abstract We present spectroellipsometric measurements of the linear optical response of ultrathin Ge/Si strained-layer superlattices (SLSs) grown by molecular beam epitaxy with varying strain state and periodicity. The experiments were performed at 300 K in the energy range 1.6 5.7 eV. As predicted from ab initio calculations, the Ej transitions of the bulk split into various components in the SLSs. From the fitted critical point (CP) energy of the most pronounced of these components, the composition of the SLSs can be estimated. Compared with alloys of the same composition, the onset of the absorption in the SLSs is shifted towards lower energies than in compositionally equivalent alloys. The E 2 transitions can be fitted to two CPs. Both exhibit a shift due to the hydrostatic component associated with the internal strain that is about one-half of what would be expected from bulk properties. We also present evidence of confinement effects.
I. Introduction Ge/Si strained-layer superlattices (SLSs) offer exciting perspectives from both the technological and the physical point of view [ 1, 2]. The large lattice mismatch of about 4% has drastic effects on the band structure, and thus on the optical properties of the SLSs [3, 4] and can be used to "tailor" the electronic and optical properties to specific needs. The internal strain can be varied by growing the SLSs on different "virtual" substrates, such as GexSi~_~ allows on an Si(001) substrate (concept of "strain symmetrization") [5] or on a germanium substrate [6]. In this contribution we will demonstrate that the hydrostatic component of the internal strain results in a shift in the E2 transitions that can be measured ellipsometrically, while the E~ transition of the SLSs is spread out by about 1 eV and starts to absorb the light at lower energies, if compared with the sharp El peak of G e , Si~ x alloys of the same composition.
*Present address: Digital Equipment Corporation G.m.b.H., Campus-based Engineering Centre, Favoritenstr. 7, A-1040 Vienna, Austria. ?Permanent address: Department of Solid State Physics, Faculty of Science, Masaryk University,Kotlfi2skfi,61137 Brno, Czechoslovakia. ++Present address: Max-Planck-Institut ffir Festk6rperforschung, Heisenbergstr. 1, W-7000 Stuttgart 80, Germany.
The tetragonal deformation of the SLSs also introduces an optical anisotropy for polarizations parallel and perpendicular to the main axis. While this effect has been studied theoretically [3, 7], it is at present impossible to measure the polarization along the growth direction (001) owing to the small total thickness of the samples. (Ge)n/(Si)m SLSs with both n and m even have orthorhombic and not tetragonal symmetry [8]. For this reason their in-plane dielectric function should differ along the main axes ((110) and ( l i 0 ) of the bulk). Although this anisotropy has been calculated with first principles methods [3], we have not been able to detect it experimentally in our samples. This is because the requirements for measuring these effects are extremely high (no monoatomic steps and deviations from the layer thickness of less than one atomic layer over hundreds of periods) and cannot be guaranteed under the present growth conditions. During the last decade, spectroscopic ellipsometry has been used successfully for the simultaneous determination of both the real and the imaginary part of the dielectric function ~(e))=~l(CO)+i~2(co) of most elemental and compound semiconductors [9] and of GexSi~-x alloys [10]. The critical points (CPs) of the dielectric function can be related to interband transitions at certain regions of the Brillouin zone and thus to the electronic band structure of the materials, an example of which is
Elsevier Sequoia
U. Schmid et al. [ Optical transitions in strained Ge/Si
shown in detail in ref. 11. A comprehensive theoretical description of the linear optical properties of Ge/Si SLSs, based on ab initio band structure calculations, is given in ref. 3, while a rigorous experimental discussion is given in ref. 12. The Ge/Si SLSs studied here were grown by low temperature molecular beam epitaxy at about 300350 °C on two different substrates: (a) the symmetrically strained SLSs were grown on Si(001) substrates with a homogeneous Ge, Si~ , buffer layer, so that both the silicon and the germanium layers were strained [1, 5]; (b) in pseudomorphic growth on germanium substrates, all the strain is taken up by the silicon layers [6], resulting in a small critical thickness. For case (a), changes in the design of the buffer, such as the germanium content x and its thickness, allow for the variation in the lateral lattice constant and thus the distribution of the internal strain. The individual layer thicknesses n, m of these SLSs were in the range from 2 to 12, and the nominal thickness was 200 nm, with one exception: we also had a 1/zm thick (Ge)7/(Si)3 sample. The series of (Ge)3k/(Si)k (k = 2, 3, 4) samples grown on germanium substrates (case (b)) were much thinner (33 nm). A more detailed description of the samples is given in ref. 12. Oxide overlayers were taken into account by assuming a multiphase model in the analysis. The individual layer thicknesses and strain distributions of the samples were measured by Raman spectroscopy [13, 14]. The optical spectra of the SLSs were taken at room temperature in the energy range between 1.7 and 5.7 eV with a rotating analyser spectrometer as described in ref. 15. Ellipsometry measures the complex ratio p of the reflection coefficients re and r~ (parallel and perpendicular to the plane of incidence, respectively). This can be expressed in terms of the amplitude ratio tan ~ and the phase angle A:
rp_
p = -- - tan ~, e ia
(1)
247
where A is the amplitude, E the CP energy, and F the broadening parameter. The phase angle q5 = 0, rt/2, rt corresponds to a minimum, saddle point, and maximum respectively. For a discussion of other types of CPs see ref. 18.
2. Results and discussion
Figure 1 shows the real and imaginary parts of the dielectric function of the 1/~m thick, fully symmetrically strained (Ge)7/(Si)3 sample (lateral in-plane strain E~I~ of the silicon layers 2.9%). The effects of the oxide overlayer on e(~o) have been removed numerically using a three-phase model (SLS plus overlayer). In the low energy region (E < 2.4 eV), the absorption E2(o)) of the SLS seems to be much stronger than that of a compositionally equivalent GexSi,-x (x = 0.7) alloy (broken lines in Fig. 1). For an alloy film of the same thickness as that of the SLS (1 #m), it would suffice to evaluate ~(~o) with a three-phase model down to E ~ 2.2 eV. Below this energy, the penetration depth of the incident light would exceed the film thickness, causing oscillatory patterns in the measured spectra. As the absorption of the SLS in the low energy region is even stronger than that of the alloy, we can expect it to be sufficient to evaluate the SLS spectra within a three-phase model over most of the energy region (2.05.7 eV). In contrast to the alloy, the SLS does not exhibit a well-defined, single E, transition at about 2.46 eV, but broadens into a wide band and has a distinctive peak in dZE2(~o)/d~o2 at 2.49eV, with a weaker satellite at 2.64 eV. Such "multiplet" structures seem to be a fingerprint of each single Ge/Si SLS [3, 7] and have previously been observed with other experimental techniques, such as electroreflectance [19] and resonant Raman scattering [20].
rs
The complex dielectric function e(co) = e, (co) + ie2(co ) can readily be derived using the two-phase model [16] (ambient and sample). While this is the "true" function for bulk material in the absence of an overlayer, it is called the "pseudodielectric" function for multilayer (including just bulk plus overlayer) systems. In such cases, the genuine values can only be obtained using multiphase models [ 16]. The structures observed in the e(~o) spectra are attributed to CPs and can be conveniently analysed in the second derivative spectrum. In the case of a two-dimensional (2D) CP, which is of importance in our analysis, its line shape can be written as [ 17, 18] d2e dro 2
-
A ei¢(h~ - E + iF) -2
(2)
30
~J 0
x~"~
-10 I
2
l
I
3
I
' ~ ~-, 4 5 I
6
Energy (eV) Fig. 1. ~(~0) o f the (Ge)7/(Si)3 SLS resulting from a three-phase model ( ), and ~(0)) of a Geo.7Sio.3 alloy ( - - -).
248
U. Schmid et al. / Optical transitions in strained Ge/Si
TABLE 1. Value of the parameters
E, E~ E2
A (eV)
E (eV)
F (eV)
~b (deg)
5.0 (7) 1.3 (5) 1.3 (2)
2.49 (1) 2.64 (1) 3.19 (1)
0.16 (I) 0.10(1) 0.13 (1)
78 (9) 50(19) 2 (6)
7.1 (1.3) 8.7 (1.2)
4.21 (1) 4.36 (1)
0.16 (1) 0.15 (1)
41 (11) 182 (8)
A, E, F and ~b were obtained by fitting d%(to)/do92 of the (Ge)7/(Si)3 SLS to a 2D CP line shape (eqn. (2)).
These two pronounced peaks can be well fitted to 2D CPs (Table 1), and both their dimensions (2D) and their phase angles qS, which display a mixture of a minimum and a saddle point, are in excellent agreement with those obtained for bulk germanium [15]. As in this case, a fit to an excitionic line shape gives somewhat worse agreement. If we compare a series of Ge/Si SLSs with varying composition, we find a large variation in the energetic positions of the most pronounced E~ peaks with their silicon content y, similar to those known from Ge~ y Si~. alloys, as described by a best-fit parabola (eqn. (4) in ref. 10) (Fig. 2). The open square symbols in this figure denote the results of ab initio calculations [3, 11]. Figure 2 shows that the dependence of the alloy can only give an initial indication of the SLSs' values; the exact values differ by up to 0.17 eV, i.e. the corresponding average composition x of the SLSs can only be determined within an uncertainty of about 12% from the positions of the E~ peaks. Such a large deviation is not surprising considering the complexity and splitting of the E~ transitions in the SLSs. The transition at E = 3.19 eV is very close to the E~ values interpolated from bulk germanium and silicon (E~ ~ 3.17 eV for 70% Ge contribution [15, 18]). In ad-
,3.5
30
tS 2.5
o
2'v.0
i
I 0.2
I 0.4
, 0.16
,
I 0.8
1.0
Si fraction Fig. 2. Dominant E 1 peak vs. silicon fraction x = m / ( n + m ) in (Ge),/(Si),,, SLS ( O ) compared with the E l transitions in Ge I _ ~Si, alloys ( ; ref. 10). D, results from ab initio linearized muffin tin orbital calculations.
dition, its line shape fit yields a 2D minimum, in accordance with both germanium and silicon results [15, 18]. We thus have good reason to assume that we have actually observed the average E~ transition. The most pronounced peaks in e2(~o) are found around 4.3 eV in analogy to the E2 transitions of the bulk materials. While a splitting in the spectra can only be observed for large periods (n + m ~> 15) at room temperature, a fit of these peaks to CP line shapes is only satisfactory when two CPs are considered, even for SLSs with small periods. We find that the data fit best to two 2D CPs in all cases. Whereas the phase of the lower E2 transition E~ gives a mixture of minimum and saddle point (40°~<~b ~<60°), the energetically higher E b transition is best fitted to a mixture of a saddle point and a maximum or simply a pure maximum (140°~< ~b ~< 180°), with negligible variations over the period and composition in both cases (see also Table 1). This is similar to the E 2 gaps in the bulk materials that are also represented by a mixture of 2D minima and saddle points [ 15, 18]. The energy of the E2 transition in bulk silicon (4.26 eV at T = 300 K [18]) is somewhat smaller than that in germanium (4.37eV [15]) and rather close to E~. To examine any possible confinement effects, we plot in Fig. 3 the 2D CP energies E~ and E b over the silicon period and the germanium period. Three different sets of samples with comparable composition and strain are shown. Both the silicon-like E~ and the E b energies decrease somewhat with increasing period. In the (Ge)3k/(Si)k set (Fig. 3(c)), the germanium layers are unstrained, and we have thus indicated the position of the E2 transition of the bulk with an arrow. The E2 transition of silicon (corrected for the shift due to the hydrostatic expansion when bulk silicon is matched to the lattice constant of germanium), is also drawn in this figure (ESi/Ge). For both transitions there is a considerable difference between the extrapolated energies E~ "b for n, rn --. ~ and the bulk values. This is also found for the other two sets of samples presented in Fig. 3, after correction for the hydrostatic shift. The decrease in energy of both E~ and E b when going from Fig. 3(a) to Fig. 3(c) is due to the hydrostatic component of E~Ii, as explained later. Further support for confinement effects of the E~ transitions is based on the behaviour of its amplitude A and broadening F; both exhibit a drastic decrease with increasing superlattice period n + m in contrast to the E2b transitions, whose values remain almost constant [12]. The strain in both slabs of the SLSs can be decomposed into a traceless uniaxial and a hydrostatic component; only the latter shifts the average energies of various indirect and direct gaps, such as E2. On the basis of the hydrostitic strain in the germanium and
U. Schmid et al. / Optical transitions in strained Ge/Si
. . . .
,~ . . . .
19
. . . .
n (Ge period)
n (Ge period)
n (Ge period)
4.41
i
....
~ ....
19 . . . .
249
15
. . . .
,~ . . . .
154. 4
1,o . . . .
.~
E2c4.3
t" - --,I_ _ z
43
~'~4.2
4.2
E2m/Ge -
4.1
~Ge)2k/(Si)3k ~,%'2~ 4.0 0 . . . . ,~ . . . . I'0. . . . 15 m (Si period)
(ce)./(si).
E,s'-3~
. . . . + . . . . I'0. . . . 15
4.1
Ge).sk/(Si)k
~,s;"4.2sI
....
;b . . . .
m (Si period)
~
. . . .
;5 4.0
rn (Si period)
Fig. 3. Both E 2 transitions as obtained from a 2D CP line shape fit (Eqn. (2)) for three different sets of samples with the same nominal composition and approximately the same strain. The lower E~ transitions are plotted against m, and the higher E b against n. , guides for the eye.
silicon layers, the shift in the E2 transitions can be expressed in terms of the bulk deformation potentials (DPs) a~ e and a m by means of the following ansatz:
4.5
.
.
.
.
.
(Ge)./(Si),.
(3a)
d E 2 = a h ~ ~1i ÷ g c
.
4.4
.
.
.
.
.
.
.
.
.
.
.
.
.
(n+m)"10 • -, n , m
=
(4. I),(6.
omn,m
=
7,3
I
with ah=
2[n~m
aSi(1-
Cs~]ClS~']÷
a Ge( 1 _ :: 11 - nn+ m \ C~I//] v
(3b) Here, ai are the equilibrium lattice constants of the two materials (i = Ge, Si) and C~ and C~2 the elastic stiffness constants of the strained material. With the values from ref. 21 we obtain for the "effective" hydrostatic DP ah ~ --4.2 eV. We compare this DP ansatz in Fig. 4 (broken line, with E 2 taken to be the weighted average of the bulk E 2 values) with the experimental data for a series of (Ge)4/(Si)6 SLSs (full symbols), i.e. a welldefined set of samples with approximately the same composition and period, but only different e~l+. A linear fit of the data yields
L7
DP ansatz
4.2
4.1
, . . . , . . , , i , , , l i , , , ,
.o
I.+
2.0
+,,+% )
+.5
3.0
Fig+ 4. Variations with in-plane strain e~li in the E 2 transitions in a series of (Ge)4/(Si)6 SLSs. - - , linear fit to the experimental points; , obtained from a linear deformation potential ansatz based on bulk properties, as given in eqn. (3b); ©, [], experimental results from a (Ge)7/(Si)3 SLS.
E ~ = ( 4 " 1 6 ( 3 ) - l ' 7 ( 9 ) e ~ ' i ÷ 0 " l l n + m n ) ev
(4) E b = ( 4 . 3 2 ( 2 ) - l 5 ( 6 ) e ~ l i + 0 . 0 9 n + m n ) ev as indicated in Fig. 4 by the full lines. The compositional dependence en/(n + m) was calculated from the experimental data of the (Ge)7/(Si)3 SLS (open symbols in Fig. 4). Its values are in excellent agreement with the difference in the E 2 energies from bulk germanium and silicon ( A E 2 = c ~0.1 eV). Owing to the large uncertainty of these values, all we can say at this point is that the experimental DPs for both parts of the E2 transition are approximately one-half of what would be expected
from a linear DP ansatz, based on the bulk properties (ah ~ - 4 . 2 eV). This finding is in agreement with predictions based on ab initio calculations [3].
3. Conclusion
We thus have presented dielectric functions e(tn) of strained Ge/Si SLSs, covering measurements and analysis over a three-dimensional parameter space (composition, strain, and period).
250
U. Schmid et al. / Optical transitions in strained Ge/Si
The Ej transitions split up into a broad band, as predicted by first principles calculations [3, 7, 11], but nevertheless their dominant component can be used to estimate the composition of the SLS. The E 2 transitions exhibit a doublet structure. A quantitative description of both its shift due to the hydrostatic component of e~li as well as its compositional dependence has been given. We also have evidence of confinement effects in these states.
Acknowledgments It is a pleasure to thank N. E. Christensen, M. Garriga, S. Zollner, and M. Kelly for fruitful discussions. Parts of this work have been supported financially by ESPRIT Basic Research Action 3174.
6 K. Eberl, W. Wegscheider and G. Abstreiter, J. Cryst. Growth. I l I (1991) 882, and references cited therein. 7 E. Ghahramani, D. J. Moss and J. E. Sipe, Phys. Rev. B, 41 (1990) 5112; 42(19 ) 9193. 8 M. I. Alonso, M. Cardona and G. Kanellis, Solid State Commun., 69 (1989) 479; 70 ( 1 9 ) i (Corrigendum). 9 D. E. Aspnes and A. A. Studna, Phys. Rev. B, 27 (1983) 985. 10 J. Humlif:ek, M. Garriga, M. I. Alonso and M. Cardona, J. Appl. Phys., 65 (1989) 2827. 1l U. Schmid, F. Lukeg, N. E. Christensen, M. Alouani, M. Cardona, E. Kasper, H. Kibbel and H. Presting, Phys. Rev. Lett., 65 (1990) 1933. 12 U. Schmid, J. Humli~ek, F. Lukeg, M. Cardona, H. Presting, E. Kasper, H. Kibbel, K. Eberl, W. Wegscheider and G. Abstreiter, Phys. Rev. B, 45 (1992) 6793. 13 M. I. Alonso, F. Cerdeira, D. Niles, M. Cardona, E. Kasper and H. Kibbel, J. Appl. Phys. 66 (1989) 5645. 14 E. Friess, K. Eberl, U. Menczigar and G. Abstreiter, Solid State Commun., 73 (1990) 203. 15 L. Vifia, S. Logothetidis and M. Cardona, Phys. Rev. B, 30(1984) 1979.
References 1 E. Kasper, in T. P. Pearsall (ed.), Strained-Layer Superlattices: Materials" Science and Technology, Semiconductors and Semimetals', Vol. 33, Academic Press, New York, 1991, Chap. 4. 2 T. P. Pearsall, CRC Crit. Rev. Solid State Mater. Sei., 15 (1989) 551. 3 U. Schmid, N. E. Christensen, M. Alouani and M. Cardona, Phys. Rev. B, 43 (1991) 14597. 4 U. Schmid, N. E. Christensen and M. Cardona, Phys. Rev. Lett., 65 (1990) 2610. 5 E. Kasper, H. Kibbel, H. Jorke, H. Brugger, E. Friess and G. Abstreiter, Phys. Rev. B, 38 (1988) 3599.
16 R. M. A. Azzam and N. M. Bashara, Ellipsomet,T and Polarized Light, North-Holland, Amsterdam, 1977. 17 D. E. Aspnes, in M. Balkanski (ed.), Handbook on Semiconductors, Vol. 2, North-Holland, Amsterdam, 1980, p. 109. 18 P. Lautenschlager, M. Garriga, L. Vifia and M. Cardona, Phys. Rev. B, 36 (1987) 4821. 19 T. P. Pearsall, J. Bevk, J. C. Bean, J. Bonar, J. P. Mannaerts and A. Ourmazd, Phys. Rer. B, 39(1989) 3741. 20 F. Cerdeira, M. I. Alonso, D. Niles, M. Garriga, M. Cardona, E. Kasper and H. Kibbel, Phys. Rev. B, 40(1989) 1361. 21 O. Madelung (ed.), Landoh-B6rnstein: Numerical Data and Functional Relationships in Science and Technology, New Series. Gp. lII, Vol. 17a, Springer, Berlin, 1982.
Thin Solid Films, 222 (1992) 246 250
Optical transitions in strained Ge/Si superlattices U. Schmid*, J. Humli~ekt, F. Lukegt and M. Cardona Max-Planck-lnstitut fiir FestkgrperJbrschung, Heisenbergstr. 1, W-7000 Stuttgart 80 (Germany)
H. Presting, H. Kibbel and E. Kasper Daimler Benz Research Centre at Ulm, W-7900 UIm (Germany)
K. Eberl~:, W. Wegscheider and G. Abstreiter Walter Schottky lnstitut, Technische Universitiit Miinchen, W-8046 Garching (Germany)
Abstract We present spectroellipsometric measurements of the linear optical response of ultrathin Ge/Si strained-layer superlattices (SLSs) grown by molecular beam epitaxy with varying strain state and periodicity. The experiments were performed at 300 K in the energy range 1.6 5.7 eV. As predicted from ab initio calculations, the Ej transitions of the bulk split into various components in the SLSs. From the fitted critical point (CP) energy of the most pronounced of these components, the composition of the SLSs can be estimated. Compared with alloys of the same composition, the onset of the absorption in the SLSs is shifted towards lower energies than in compositionally equivalent alloys. The E 2 transitions can be fitted to two CPs. Both exhibit a shift due to the hydrostatic component associated with the internal strain that is about one-half of what would be expected from bulk properties. We also present evidence of confinement effects.
I. Introduction Ge/Si strained-layer superlattices (SLSs) offer exciting perspectives from both the technological and the physical point of view [ 1, 2]. The large lattice mismatch of about 4% has drastic effects on the band structure, and thus on the optical properties of the SLSs [3, 4] and can be used to "tailor" the electronic and optical properties to specific needs. The internal strain can be varied by growing the SLSs on different "virtual" substrates, such as GexSi~_~ allows on an Si(001) substrate (concept of "strain symmetrization") [5] or on a germanium substrate [6]. In this contribution we will demonstrate that the hydrostatic component of the internal strain results in a shift in the E2 transitions that can be measured ellipsometrically, while the E~ transition of the SLSs is spread out by about 1 eV and starts to absorb the light at lower energies, if compared with the sharp El peak of G e , Si~ x alloys of the same composition.
*Present address: Digital Equipment Corporation G.m.b.H., Campus-based Engineering Centre, Favoritenstr. 7, A-1040 Vienna, Austria. ?Permanent address: Department of Solid State Physics, Faculty of Science, Masaryk University,Kotlfi2skfi,61137 Brno, Czechoslovakia. ++Present address: Max-Planck-Institut ffir Festk6rperforschung, Heisenbergstr. 1, W-7000 Stuttgart 80, Germany.
The tetragonal deformation of the SLSs also introduces an optical anisotropy for polarizations parallel and perpendicular to the main axis. While this effect has been studied theoretically [3, 7], it is at present impossible to measure the polarization along the growth direction (001) owing to the small total thickness of the samples. (Ge)n/(Si)m SLSs with both n and m even have orthorhombic and not tetragonal symmetry [8]. For this reason their in-plane dielectric function should differ along the main axes ((110) and ( l i 0 ) of the bulk). Although this anisotropy has been calculated with first principles methods [3], we have not been able to detect it experimentally in our samples. This is because the requirements for measuring these effects are extremely high (no monoatomic steps and deviations from the layer thickness of less than one atomic layer over hundreds of periods) and cannot be guaranteed under the present growth conditions. During the last decade, spectroscopic ellipsometry has been used successfully for the simultaneous determination of both the real and the imaginary part of the dielectric function ~(e))=~l(CO)+i~2(co) of most elemental and compound semiconductors [9] and of GexSi~-x alloys [10]. The critical points (CPs) of the dielectric function can be related to interband transitions at certain regions of the Brillouin zone and thus to the electronic band structure of the materials, an example of which is
Elsevier Sequoia
U. Schmid et al. [ Optical transitions in strained Ge/Si
shown in detail in ref. 11. A comprehensive theoretical description of the linear optical properties of Ge/Si SLSs, based on ab initio band structure calculations, is given in ref. 3, while a rigorous experimental discussion is given in ref. 12. The Ge/Si SLSs studied here were grown by low temperature molecular beam epitaxy at about 300350 °C on two different substrates: (a) the symmetrically strained SLSs were grown on Si(001) substrates with a homogeneous Ge, Si~ , buffer layer, so that both the silicon and the germanium layers were strained [1, 5]; (b) in pseudomorphic growth on germanium substrates, all the strain is taken up by the silicon layers [6], resulting in a small critical thickness. For case (a), changes in the design of the buffer, such as the germanium content x and its thickness, allow for the variation in the lateral lattice constant and thus the distribution of the internal strain. The individual layer thicknesses n, m of these SLSs were in the range from 2 to 12, and the nominal thickness was 200 nm, with one exception: we also had a 1/zm thick (Ge)7/(Si)3 sample. The series of (Ge)3k/(Si)k (k = 2, 3, 4) samples grown on germanium substrates (case (b)) were much thinner (33 nm). A more detailed description of the samples is given in ref. 12. Oxide overlayers were taken into account by assuming a multiphase model in the analysis. The individual layer thicknesses and strain distributions of the samples were measured by Raman spectroscopy [13, 14]. The optical spectra of the SLSs were taken at room temperature in the energy range between 1.7 and 5.7 eV with a rotating analyser spectrometer as described in ref. 15. Ellipsometry measures the complex ratio p of the reflection coefficients re and r~ (parallel and perpendicular to the plane of incidence, respectively). This can be expressed in terms of the amplitude ratio tan ~ and the phase angle A:
rp_
p = -- - tan ~, e ia
(1)
247
where A is the amplitude, E the CP energy, and F the broadening parameter. The phase angle q5 = 0, rt/2, rt corresponds to a minimum, saddle point, and maximum respectively. For a discussion of other types of CPs see ref. 18.
2. Results and discussion
Figure 1 shows the real and imaginary parts of the dielectric function of the 1/~m thick, fully symmetrically strained (Ge)7/(Si)3 sample (lateral in-plane strain E~I~ of the silicon layers 2.9%). The effects of the oxide overlayer on e(~o) have been removed numerically using a three-phase model (SLS plus overlayer). In the low energy region (E < 2.4 eV), the absorption E2(o)) of the SLS seems to be much stronger than that of a compositionally equivalent GexSi,-x (x = 0.7) alloy (broken lines in Fig. 1). For an alloy film of the same thickness as that of the SLS (1 #m), it would suffice to evaluate ~(~o) with a three-phase model down to E ~ 2.2 eV. Below this energy, the penetration depth of the incident light would exceed the film thickness, causing oscillatory patterns in the measured spectra. As the absorption of the SLS in the low energy region is even stronger than that of the alloy, we can expect it to be sufficient to evaluate the SLS spectra within a three-phase model over most of the energy region (2.05.7 eV). In contrast to the alloy, the SLS does not exhibit a well-defined, single E, transition at about 2.46 eV, but broadens into a wide band and has a distinctive peak in dZE2(~o)/d~o2 at 2.49eV, with a weaker satellite at 2.64 eV. Such "multiplet" structures seem to be a fingerprint of each single Ge/Si SLS [3, 7] and have previously been observed with other experimental techniques, such as electroreflectance [19] and resonant Raman scattering [20].
rs
The complex dielectric function e(co) = e, (co) + ie2(co ) can readily be derived using the two-phase model [16] (ambient and sample). While this is the "true" function for bulk material in the absence of an overlayer, it is called the "pseudodielectric" function for multilayer (including just bulk plus overlayer) systems. In such cases, the genuine values can only be obtained using multiphase models [ 16]. The structures observed in the e(~o) spectra are attributed to CPs and can be conveniently analysed in the second derivative spectrum. In the case of a two-dimensional (2D) CP, which is of importance in our analysis, its line shape can be written as [ 17, 18] d2e dro 2
-
A ei¢(h~ - E + iF) -2
(2)
30
~J 0
x~"~
-10 I
2
l
I
3
I
' ~ ~-, 4 5 I
6
Energy (eV) Fig. 1. ~(~0) o f the (Ge)7/(Si)3 SLS resulting from a three-phase model ( ), and ~(0)) of a Geo.7Sio.3 alloy ( - - -).
248
U. Schmid et al. / Optical transitions in strained Ge/Si
TABLE 1. Value of the parameters
E, E~ E2
A (eV)
E (eV)
F (eV)
~b (deg)
5.0 (7) 1.3 (5) 1.3 (2)
2.49 (1) 2.64 (1) 3.19 (1)
0.16 (I) 0.10(1) 0.13 (1)
78 (9) 50(19) 2 (6)
7.1 (1.3) 8.7 (1.2)
4.21 (1) 4.36 (1)
0.16 (1) 0.15 (1)
41 (11) 182 (8)
A, E, F and ~b were obtained by fitting d%(to)/do92 of the (Ge)7/(Si)3 SLS to a 2D CP line shape (eqn. (2)).
These two pronounced peaks can be well fitted to 2D CPs (Table 1), and both their dimensions (2D) and their phase angles qS, which display a mixture of a minimum and a saddle point, are in excellent agreement with those obtained for bulk germanium [15]. As in this case, a fit to an excitionic line shape gives somewhat worse agreement. If we compare a series of Ge/Si SLSs with varying composition, we find a large variation in the energetic positions of the most pronounced E~ peaks with their silicon content y, similar to those known from Ge~ y Si~. alloys, as described by a best-fit parabola (eqn. (4) in ref. 10) (Fig. 2). The open square symbols in this figure denote the results of ab initio calculations [3, 11]. Figure 2 shows that the dependence of the alloy can only give an initial indication of the SLSs' values; the exact values differ by up to 0.17 eV, i.e. the corresponding average composition x of the SLSs can only be determined within an uncertainty of about 12% from the positions of the E~ peaks. Such a large deviation is not surprising considering the complexity and splitting of the E~ transitions in the SLSs. The transition at E = 3.19 eV is very close to the E~ values interpolated from bulk germanium and silicon (E~ ~ 3.17 eV for 70% Ge contribution [15, 18]). In ad-
,3.5
30
tS 2.5
o
2'v.0
i
I 0.2
I 0.4
, 0.16
,
I 0.8
1.0
Si fraction Fig. 2. Dominant E 1 peak vs. silicon fraction x = m / ( n + m ) in (Ge),/(Si),,, SLS ( O ) compared with the E l transitions in Ge I _ ~Si, alloys ( ; ref. 10). D, results from ab initio linearized muffin tin orbital calculations.
dition, its line shape fit yields a 2D minimum, in accordance with both germanium and silicon results [15, 18]. We thus have good reason to assume that we have actually observed the average E~ transition. The most pronounced peaks in e2(~o) are found around 4.3 eV in analogy to the E2 transitions of the bulk materials. While a splitting in the spectra can only be observed for large periods (n + m ~> 15) at room temperature, a fit of these peaks to CP line shapes is only satisfactory when two CPs are considered, even for SLSs with small periods. We find that the data fit best to two 2D CPs in all cases. Whereas the phase of the lower E2 transition E~ gives a mixture of minimum and saddle point (40°~<~b ~<60°), the energetically higher E b transition is best fitted to a mixture of a saddle point and a maximum or simply a pure maximum (140°~< ~b ~< 180°), with negligible variations over the period and composition in both cases (see also Table 1). This is similar to the E 2 gaps in the bulk materials that are also represented by a mixture of 2D minima and saddle points [ 15, 18]. The energy of the E2 transition in bulk silicon (4.26 eV at T = 300 K [18]) is somewhat smaller than that in germanium (4.37eV [15]) and rather close to E~. To examine any possible confinement effects, we plot in Fig. 3 the 2D CP energies E~ and E b over the silicon period and the germanium period. Three different sets of samples with comparable composition and strain are shown. Both the silicon-like E~ and the E b energies decrease somewhat with increasing period. In the (Ge)3k/(Si)k set (Fig. 3(c)), the germanium layers are unstrained, and we have thus indicated the position of the E2 transition of the bulk with an arrow. The E2 transition of silicon (corrected for the shift due to the hydrostatic expansion when bulk silicon is matched to the lattice constant of germanium), is also drawn in this figure (ESi/Ge). For both transitions there is a considerable difference between the extrapolated energies E~ "b for n, rn --. ~ and the bulk values. This is also found for the other two sets of samples presented in Fig. 3, after correction for the hydrostatic shift. The decrease in energy of both E~ and E b when going from Fig. 3(a) to Fig. 3(c) is due to the hydrostatic component of E~Ii, as explained later. Further support for confinement effects of the E~ transitions is based on the behaviour of its amplitude A and broadening F; both exhibit a drastic decrease with increasing superlattice period n + m in contrast to the E2b transitions, whose values remain almost constant [12]. The strain in both slabs of the SLSs can be decomposed into a traceless uniaxial and a hydrostatic component; only the latter shifts the average energies of various indirect and direct gaps, such as E2. On the basis of the hydrostitic strain in the germanium and
U. Schmid et al. / Optical transitions in strained Ge/Si
. . . .
,~ . . . .
19
. . . .
n (Ge period)
n (Ge period)
n (Ge period)
4.41
i
....
~ ....
19 . . . .
249
15
. . . .
,~ . . . .
154. 4
1,o . . . .
.~
E2c4.3
t" - --,I_ _ z
43
~'~4.2
4.2
E2m/Ge -
4.1
~Ge)2k/(Si)3k ~,%'2~ 4.0 0 . . . . ,~ . . . . I'0. . . . 15 m (Si period)
(ce)./(si).
E,s'-3~
. . . . + . . . . I'0. . . . 15
4.1
Ge).sk/(Si)k
~,s;"4.2sI
....
;b . . . .
m (Si period)
~
. . . .
;5 4.0
rn (Si period)
Fig. 3. Both E 2 transitions as obtained from a 2D CP line shape fit (Eqn. (2)) for three different sets of samples with the same nominal composition and approximately the same strain. The lower E~ transitions are plotted against m, and the higher E b against n. , guides for the eye.
silicon layers, the shift in the E2 transitions can be expressed in terms of the bulk deformation potentials (DPs) a~ e and a m by means of the following ansatz:
4.5
.
.
.
.
.
(Ge)./(Si),.
(3a)
d E 2 = a h ~ ~1i ÷ g c
.
4.4
.
.
.
.
.
.
.
.
.
.
.
.
.
(n+m)"10 • -, n , m
=
(4. I),(6.
omn,m
=
7,3
I
with ah=
2[n~m
aSi(1-
Cs~]ClS~']÷
a Ge( 1 _ :: 11 - nn+ m \ C~I//] v
(3b) Here, ai are the equilibrium lattice constants of the two materials (i = Ge, Si) and C~ and C~2 the elastic stiffness constants of the strained material. With the values from ref. 21 we obtain for the "effective" hydrostatic DP ah ~ --4.2 eV. We compare this DP ansatz in Fig. 4 (broken line, with E 2 taken to be the weighted average of the bulk E 2 values) with the experimental data for a series of (Ge)4/(Si)6 SLSs (full symbols), i.e. a welldefined set of samples with approximately the same composition and period, but only different e~l+. A linear fit of the data yields
L7
DP ansatz
4.2
4.1
, . . . , . . , , i , , , l i , , , ,
.o
I.+
2.0
+,,+% )
+.5
3.0
Fig+ 4. Variations with in-plane strain e~li in the E 2 transitions in a series of (Ge)4/(Si)6 SLSs. - - , linear fit to the experimental points; , obtained from a linear deformation potential ansatz based on bulk properties, as given in eqn. (3b); ©, [], experimental results from a (Ge)7/(Si)3 SLS.
E ~ = ( 4 " 1 6 ( 3 ) - l ' 7 ( 9 ) e ~ ' i ÷ 0 " l l n + m n ) ev
(4) E b = ( 4 . 3 2 ( 2 ) - l 5 ( 6 ) e ~ l i + 0 . 0 9 n + m n ) ev as indicated in Fig. 4 by the full lines. The compositional dependence en/(n + m) was calculated from the experimental data of the (Ge)7/(Si)3 SLS (open symbols in Fig. 4). Its values are in excellent agreement with the difference in the E 2 energies from bulk germanium and silicon ( A E 2 = c ~0.1 eV). Owing to the large uncertainty of these values, all we can say at this point is that the experimental DPs for both parts of the E2 transition are approximately one-half of what would be expected
from a linear DP ansatz, based on the bulk properties (ah ~ - 4 . 2 eV). This finding is in agreement with predictions based on ab initio calculations [3].
3. Conclusion
We thus have presented dielectric functions e(tn) of strained Ge/Si SLSs, covering measurements and analysis over a three-dimensional parameter space (composition, strain, and period).
250
U. Schmid et al. / Optical transitions in strained Ge/Si
The Ej transitions split up into a broad band, as predicted by first principles calculations [3, 7, 11], but nevertheless their dominant component can be used to estimate the composition of the SLS. The E 2 transitions exhibit a doublet structure. A quantitative description of both its shift due to the hydrostatic component of e~li as well as its compositional dependence has been given. We also have evidence of confinement effects in these states.
Acknowledgments It is a pleasure to thank N. E. Christensen, M. Garriga, S. Zollner, and M. Kelly for fruitful discussions. Parts of this work have been supported financially by ESPRIT Basic Research Action 3174.
6 K. Eberl, W. Wegscheider and G. Abstreiter, J. Cryst. Growth. I l I (1991) 882, and references cited therein. 7 E. Ghahramani, D. J. Moss and J. E. Sipe, Phys. Rev. B, 41 (1990) 5112; 42(19 ) 9193. 8 M. I. Alonso, M. Cardona and G. Kanellis, Solid State Commun., 69 (1989) 479; 70 ( 1 9 ) i (Corrigendum). 9 D. E. Aspnes and A. A. Studna, Phys. Rev. B, 27 (1983) 985. 10 J. Humlif:ek, M. Garriga, M. I. Alonso and M. Cardona, J. Appl. Phys., 65 (1989) 2827. 1l U. Schmid, F. Lukeg, N. E. Christensen, M. Alouani, M. Cardona, E. Kasper, H. Kibbel and H. Presting, Phys. Rev. Lett., 65 (1990) 1933. 12 U. Schmid, J. Humli~ek, F. Lukeg, M. Cardona, H. Presting, E. Kasper, H. Kibbel, K. Eberl, W. Wegscheider and G. Abstreiter, Phys. Rev. B, 45 (1992) 6793. 13 M. I. Alonso, F. Cerdeira, D. Niles, M. Cardona, E. Kasper and H. Kibbel, J. Appl. Phys. 66 (1989) 5645. 14 E. Friess, K. Eberl, U. Menczigar and G. Abstreiter, Solid State Commun., 73 (1990) 203. 15 L. Vifia, S. Logothetidis and M. Cardona, Phys. Rev. B, 30(1984) 1979.
References 1 E. Kasper, in T. P. Pearsall (ed.), Strained-Layer Superlattices: Materials" Science and Technology, Semiconductors and Semimetals', Vol. 33, Academic Press, New York, 1991, Chap. 4. 2 T. P. Pearsall, CRC Crit. Rev. Solid State Mater. Sei., 15 (1989) 551. 3 U. Schmid, N. E. Christensen, M. Alouani and M. Cardona, Phys. Rev. B, 43 (1991) 14597. 4 U. Schmid, N. E. Christensen and M. Cardona, Phys. Rev. Lett., 65 (1990) 2610. 5 E. Kasper, H. Kibbel, H. Jorke, H. Brugger, E. Friess and G. Abstreiter, Phys. Rev. B, 38 (1988) 3599.
16 R. M. A. Azzam and N. M. Bashara, Ellipsomet,T and Polarized Light, North-Holland, Amsterdam, 1977. 17 D. E. Aspnes, in M. Balkanski (ed.), Handbook on Semiconductors, Vol. 2, North-Holland, Amsterdam, 1980, p. 109. 18 P. Lautenschlager, M. Garriga, L. Vifia and M. Cardona, Phys. Rev. B, 36 (1987) 4821. 19 T. P. Pearsall, J. Bevk, J. C. Bean, J. Bonar, J. P. Mannaerts and A. Ourmazd, Phys. Rer. B, 39(1989) 3741. 20 F. Cerdeira, M. I. Alonso, D. Niles, M. Garriga, M. Cardona, E. Kasper and H. Kibbel, Phys. Rev. B, 40(1989) 1361. 21 O. Madelung (ed.), Landoh-B6rnstein: Numerical Data and Functional Relationships in Science and Technology, New Series. Gp. lII, Vol. 17a, Springer, Berlin, 1982.