- Email: [email protected]
Measuring the energy levels and wave functions in a single quantum dot
Physica E 13 (2002) 634 – 637
www.elsevier.com/locate/physe
Measuring the energy levels and wave functions in a single quantum dot R.J.A. Hilla; ∗ , A. Patan*ea , P.C. Maina , L. Eavesa , B. Gustafsona , M. Heninia , S. Taruchab; c , D.G. Austingc a School
of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK of Physics and ERATO Mesoscopic Correlation Project, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan c NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan b Department
Abstract We use an array of gate electrodes to select an individual self-assembled quantum dot from an ensemble. In combination with magneto-tunnelling spectroscopy, this allows us to measure the energy levels and wave functions associated with the ground and excited state of the selected quantum dot. ? 2002 Elsevier Science B.V. All rights reserved. Keywords: Wave function mapping; Resonant tunnelling; Quantum dots
In a recent paper [1] we demonstrated the use of magneto-tunnelling spectroscopy (MTS) to image the k-space probability density of the single electron energy eigenstates in self-assembled InAs quantum dots (QDs). However, in that original experiment it was not possible to identify which states belonged to which QD. In this paper, we present a new technique to overcome this limitation. Our sample is based on a vertical GaAs= Al0:2 Ga0:8 As=GaAs tunnel structure. We fabricate a novel sub-micron device that incorporates a set of independent gates. The thickness of the (AlGa)As barrier is 14 nm; 3 nm thick undoped spacer layers separate the barrier layer from n-doped GaAs regions in which the doping concentration is increased from 1×1017 cm−3 close to the barrier, to 2×1018 cm−3 . Buried within the centre of the
∗
Corresponding author. Fax: +44-115-951-5180. E-mail address: [email protected] (R.J.A. Hill).
(AlGa)As barrier layer is a single layer of InAs self-assembled QDs. The sample was grown by molecular beam epitaxy on a (3 1 1)B-oriented GaAs substrate and then processed into a cross-shaped mesa with a central 0:7 m square region [2– 4] (see Fig. 1). Metal is evaporated onto the structure to produce four independent gate electrodes which surround the central square. It is important to note that, due to side-wall depletion, the active area of the mesa is reduced to a small area within the central square that encloses fewer than 50 QDs. Applying a bias, Vsd , between the source and drain electrodes lowers the electrostatic potential of the QDs relative to the source electrode (emitter). When an electron state in the QD becomes resonant with the emitter, electrons tunnel through the QD and a resonance is observed in the I (Vsd ) characteristic [5 –9]. The eFect of applying a voltage, Vg , to one of the gates, is to oFset the electrostatic potential of a particular QD by an amount dependent on its proximity to the gate. This shifts the position of all the current
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 2 ) 0 0 2 0 5 - 9
R.J.A. Hill et al. / Physica E 13 (2002) 634 – 637
Fig. 1. Top-left: Schematic conduction band of our tunnelling diode for positive Vsd . Top-right: Schematic of the device showing gates 1– 4. Bottom: I (Vsd ) characteristic for three diFerent gate voltages applied to gates 1 and 2 at T = 0:3 K.
635
resonances associated with this particular QD to proportionally higher or lower source-drain bias. Fig. 1 shows three typical low temperature (0:3 K) I (Vsd ) traces. Each trace represents a diFerent gate voltage, Vg , which has been applied to two out of the four gates. Resonant features associated with tunnelling through single QD states are observed; some of these occur at diFerent values of Vsd for diFerent Vg . The resonances can be seen more clearly in greyscale plots of the diFerential conductance, G(Vg ; Vsd )= dI=dVsd , shown in Fig. 2. Lighter shading represents more positive values of G. The gate voltage is applied to the same two gates as in Fig. 1. These plots reveal a complicated array of resonances with widely varying sensitivity to Vg . For example, we see some resonances that are very sensitive to Vg , generated by QDs close to one of the active gates. When one or the other of these two gates is switched oF (Fig. 2b), a number of resonances become insensitive to Vg indicating that they are generated by QDs far away from the remaining active gate. We now focus our attention on the resonances labelled a1 and a2. Both resonances show an almost identical dependence upon gate voltage within experimental error. This dependence is not exhibited by any
Fig. 2. Grey-scale plots of the diFerential conductance, G(Vg ; Vsd ) = dI=dVsd at 0:3 K. Lighter shading represents more positive values of G. (a) Dependence of resonances on a magnetic Jeld, B, applied parallel to the barrier layer. The gate voltage is applied to gates 1 and 2; (b) Dependence of resonances on individual gates at B = 0 T.
R.J.A. Hill et al. / Physica E 13 (2002) 634 – 637
(a)
G (arb. un.)
90
180
a1
0
5
10
90
G (arb. un.)
180
5
(b)
10
0
90
o
o
d
8 B (T)
16
o
0 0
b2
0
o
o
(c)
o
0 0
o
15
B(T) 90
b1
0
15
a2
0
o
o
B(T)
G (a.u.)
other resonance within the voltage limits shown. This strongly suggests that a1 and a2 are due to tunnelling through diFerent quantum states of the same QD. We can examine resonances a1 and a2 further by using the MTS technique [1]. This involves applying a magnetic Jeld, B = (Bx ; By ; 0), parallel to the plane of the (AlGa)As barrier layer, which we will call the xy plane. The grey-scale plots in Fig. 2a show the effect on the resonance amplitudes of a magnetic Jeld aligned along the (arbitrarily deJned) y-axis direction of the xy plane. Plotting the amplitude, G, of a particular resonance against B reveals the k-space probability density of an electron in the associated QD state, | QD (kx ; ky )|2 ∼ G(Bx ; By ), where kx =eBy Ks=˜ and ky = −eBx Ks=˜ [1]. Ks is the tunnelling distance from the emitter to the QD layer (∼10 nm). Fig. 3a shows a plot of G(0; By ) ∼ | (eBx Ks=˜; 0)|2 for resonances a1 and a2. Resonance a1 shows a maximum amplitude at 0 T followed by a steady decay with increasing Jeld, behaviour which is characteristic of a ground state wave function. Resonance a2 shows a clearly diFerent behaviour: a minimum in the amplitude at 0 T, followed by a broad maximum at around 10 T (see also Fig. 2a). This behaviour is characteristic of a Jrst excited state wave function. The assignment of a1 and a2 to the ground and Jrst excited states is conJrmed if we now rotate the Jeld. The insets to Fig. 3a are polar plots which show how the amplitude of the resonances change as we rotate the direction of a 4 T magnetic Jeld within the xy-plane. Resonance a1 shows circular symmetry consistent with ground state behaviour whilst a2 displays two clear lobes characteristic of the Jrst excited state. We can determine the energy separation, KE, between the two states by re-scaling the voltage spacing, KVsd ∼114 mV, between them with a leverage factor, f=2:5±0:5, which we estimate from a simple electrostatic model. We Jnd KE = eKVsd =f = 50 ± 10 meV. Note that the measured probability densities are very similar to those of the two lowest energy states of a simple harmonic oscillator (SHO). Therefore, we can obtain another independent estimate of the energy separation by Jtting the data to these analytical forms. We Jnd that the two estimates are consistent if we use a value for the eFective mass of m∗ = 0:026 − 0:038me , a value somewhat larger than that of bulk InAs. However, an increase over the band edge mass is expected due to strain in the QDs [10], non-parabolicity, re-
G (a.u.)
636
8 B (T)
16
8 B (T)
16
o
e
8 B (T)
16
0
Fig. 3. Dependence of the conductance, G, of some representative resonances on a magnetic Jeld, B, applied parallel to the (AlGa)As barrier layer. The polar plots show the dependence of the conductance on the direction of a 4 T magnetic Jeld, rotated within the barrier plane. Continuous lines are Jts to the data by the two lowest energy states of a simple harmonic oscillator.
duced In content due to alloying [11] and the penetration of the wave function into the (AlGa)As barrier. Fourier transforming the Jtted SHO wave functions into real space provides us with an estimate of the spatial extent of our measured wave functions. We obtain a characteristic radius of ∼7 nm, which is comparable to the in-plane radius of our QDs [12]. In addition to resonances a1 and a2 we can also identify other possible groups from the grey-scale plots. The resonances labelled c1 and c2, for example, also shift in bias position, Vsd , at an identical rate
R.J.A. Hill et al. / Physica E 13 (2002) 634 – 637
with gate voltage, Vg . It is therefore possible that c1 and c2 also represent diFerent states of a single QD. It is not possible to verify this by the MTS technique, however, because the resonance amplitudes are too weak and are partially obscured by other resonances. On the other hand, the resonances labelled b1 and b2 are more clearly resolved and show ground-state and Jrst-excited state behaviour, respectively (see Fig. 3b). They also have a similar energy spacing to that of the a resonances. In this case, however, determining the angular dependence of b2 is made diMcult by the presence of another close resonance and both resonances are insensitive to gate voltage so we cannot be conJdent of their assignment to a single QD. Finally, we observe a few resonances which shift very rapidly with gate voltage, indicating that they originate from QDs close to a gate. Performing MTS on the fraction of these that are clearly resolved, we Jnd that the amplitudes fall continuously to zero with increasing magnetic Jeld aligned along the y-axis. This is characteristic of ground state behaviour. However, the fall is quite rapid in contrast to other observed ground state resonances (see Figs. 2 and 3). Two examples are shown in Fig. 3c. For one of these, resonance d, we can clearly resolve the angular dependence. We Jnd that the amplitude of this particular resonance is sharply enhanced when ◦ we rotate the Jeld toward the x-axis by about 40 (see Fig. 3c). This implies an elliptically shaped ground state. The explanation for this behaviour is not clear at the moment. We might speculate, however, that the side-wall potential close to a gate has a gradient steep enough to drop a signiJcant potential across the diameter of a QD. This would deform the circularly-symmetric ground state wave function of the QD into the elliptical shape which we observe.
637
In conclusion, we have employed a novel multiplegated technology as a selection mechanism that allows us to address an individual QD. Combined with the MTS technique, this allows us to probe the probability density and energy of an electron in a particular QD. This work is supported by the Engineering and Physical Sciences Research Council (United Kingdom). We gratefully acknowledge Dr. Hirayama for his support. References [1] E.E. Vdovin, A. Levin, A. Patan*e, L. Eaves, P.C. Main, Yu.N. Khanin, Yu.V. Dubrovskii, M. Henini, G. Hill, Science 290 (2000) 122. [2] M.W. Dellow, P.H. Beton, C.J.G.M. Langerak, T.J. Foster, P.C. Main, L. Eaves, M. Henini, S.P. Beaumont, C.D.W. Wilkinson, Phys. Rev. Lett. 68 (1992) 1754. [3] D.G. Austing, T. Honda, S. Tarucha, Semicond. Sci. Technol. 12 (1997) 631. [4] D.G. Austing, S. Tarucha, P.C. Main, M. Henini, S.T. Stoddart, L. Eaves, Appl. Phys. Lett. 75 (1999) 671. [5] P.C. Main, A.S.G. Thornton, R.J.A. Hill, S.T. Stoddart, T. Ihn, L. Eaves, K.A. Benedict, M. Henini, Phys. Rev. Lett. 84 (2000) 729. [6] M. Narihiro, G. Yusa, Y. Nakamura, T. Noda, H. Sakaki, Appl. Phys. Lett. 70 (1997) 105. [7] I.E. Itskevich, T. Ihn, A.S.G. Thornton, M. Henini, T.J. Foster, P. Moriarty, A. Nogaret, P.H. Beton, L. Eaves, P.C. Main, Phys. Rev. B 54 (1996) 16401. [8] I. Hapke-Wurst, U. Zeitler, H.W. Schumacher, R.J. Haug, K. Pierz, F.J. Ahlers, Semicond. Sci. Technol. 14 (1999) L41. [9] T. Suzuki, K. Nomoto, K. Taira, I. Hase, Jpn. J. Appl. Phys. Part 1 36 (1997) 1917. [10] D.E. Aspnes, M. Cardona, Phys. Rev. B 17 (1978) 726. [11] T. Surkova, A. Patan*e, L. Eaves, P.C. Main, M. Henini, A. Polimeni, A.P. Knights, C. Jeynes, J. Appl. Phys. 89 (2001) 6044. [12] A. Polimeni, A. Patan*e, M. Henini, L. Eaves, P.C. Main, S. Sanguinetti, M. Guzzi, J. Crystal Growth 201 (1999) 276.
www.elsevier.com/locate/physe
Measuring the energy levels and wave functions in a single quantum dot R.J.A. Hilla; ∗ , A. Patan*ea , P.C. Maina , L. Eavesa , B. Gustafsona , M. Heninia , S. Taruchab; c , D.G. Austingc a School
of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK of Physics and ERATO Mesoscopic Correlation Project, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan c NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan b Department
Abstract We use an array of gate electrodes to select an individual self-assembled quantum dot from an ensemble. In combination with magneto-tunnelling spectroscopy, this allows us to measure the energy levels and wave functions associated with the ground and excited state of the selected quantum dot. ? 2002 Elsevier Science B.V. All rights reserved. Keywords: Wave function mapping; Resonant tunnelling; Quantum dots
In a recent paper [1] we demonstrated the use of magneto-tunnelling spectroscopy (MTS) to image the k-space probability density of the single electron energy eigenstates in self-assembled InAs quantum dots (QDs). However, in that original experiment it was not possible to identify which states belonged to which QD. In this paper, we present a new technique to overcome this limitation. Our sample is based on a vertical GaAs= Al0:2 Ga0:8 As=GaAs tunnel structure. We fabricate a novel sub-micron device that incorporates a set of independent gates. The thickness of the (AlGa)As barrier is 14 nm; 3 nm thick undoped spacer layers separate the barrier layer from n-doped GaAs regions in which the doping concentration is increased from 1×1017 cm−3 close to the barrier, to 2×1018 cm−3 . Buried within the centre of the
∗
Corresponding author. Fax: +44-115-951-5180. E-mail address: [email protected] (R.J.A. Hill).
(AlGa)As barrier layer is a single layer of InAs self-assembled QDs. The sample was grown by molecular beam epitaxy on a (3 1 1)B-oriented GaAs substrate and then processed into a cross-shaped mesa with a central 0:7 m square region [2– 4] (see Fig. 1). Metal is evaporated onto the structure to produce four independent gate electrodes which surround the central square. It is important to note that, due to side-wall depletion, the active area of the mesa is reduced to a small area within the central square that encloses fewer than 50 QDs. Applying a bias, Vsd , between the source and drain electrodes lowers the electrostatic potential of the QDs relative to the source electrode (emitter). When an electron state in the QD becomes resonant with the emitter, electrons tunnel through the QD and a resonance is observed in the I (Vsd ) characteristic [5 –9]. The eFect of applying a voltage, Vg , to one of the gates, is to oFset the electrostatic potential of a particular QD by an amount dependent on its proximity to the gate. This shifts the position of all the current
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 2 ) 0 0 2 0 5 - 9
R.J.A. Hill et al. / Physica E 13 (2002) 634 – 637
Fig. 1. Top-left: Schematic conduction band of our tunnelling diode for positive Vsd . Top-right: Schematic of the device showing gates 1– 4. Bottom: I (Vsd ) characteristic for three diFerent gate voltages applied to gates 1 and 2 at T = 0:3 K.
635
resonances associated with this particular QD to proportionally higher or lower source-drain bias. Fig. 1 shows three typical low temperature (0:3 K) I (Vsd ) traces. Each trace represents a diFerent gate voltage, Vg , which has been applied to two out of the four gates. Resonant features associated with tunnelling through single QD states are observed; some of these occur at diFerent values of Vsd for diFerent Vg . The resonances can be seen more clearly in greyscale plots of the diFerential conductance, G(Vg ; Vsd )= dI=dVsd , shown in Fig. 2. Lighter shading represents more positive values of G. The gate voltage is applied to the same two gates as in Fig. 1. These plots reveal a complicated array of resonances with widely varying sensitivity to Vg . For example, we see some resonances that are very sensitive to Vg , generated by QDs close to one of the active gates. When one or the other of these two gates is switched oF (Fig. 2b), a number of resonances become insensitive to Vg indicating that they are generated by QDs far away from the remaining active gate. We now focus our attention on the resonances labelled a1 and a2. Both resonances show an almost identical dependence upon gate voltage within experimental error. This dependence is not exhibited by any
Fig. 2. Grey-scale plots of the diFerential conductance, G(Vg ; Vsd ) = dI=dVsd at 0:3 K. Lighter shading represents more positive values of G. (a) Dependence of resonances on a magnetic Jeld, B, applied parallel to the barrier layer. The gate voltage is applied to gates 1 and 2; (b) Dependence of resonances on individual gates at B = 0 T.
R.J.A. Hill et al. / Physica E 13 (2002) 634 – 637
(a)
G (arb. un.)
90
180
a1
0
5
10
90
G (arb. un.)
180
5
(b)
10
0
90
o
o
d
8 B (T)
16
o
0 0
b2
0
o
o
(c)
o
0 0
o
15
B(T) 90
b1
0
15
a2
0
o
o
B(T)
G (a.u.)
other resonance within the voltage limits shown. This strongly suggests that a1 and a2 are due to tunnelling through diFerent quantum states of the same QD. We can examine resonances a1 and a2 further by using the MTS technique [1]. This involves applying a magnetic Jeld, B = (Bx ; By ; 0), parallel to the plane of the (AlGa)As barrier layer, which we will call the xy plane. The grey-scale plots in Fig. 2a show the effect on the resonance amplitudes of a magnetic Jeld aligned along the (arbitrarily deJned) y-axis direction of the xy plane. Plotting the amplitude, G, of a particular resonance against B reveals the k-space probability density of an electron in the associated QD state, | QD (kx ; ky )|2 ∼ G(Bx ; By ), where kx =eBy Ks=˜ and ky = −eBx Ks=˜ [1]. Ks is the tunnelling distance from the emitter to the QD layer (∼10 nm). Fig. 3a shows a plot of G(0; By ) ∼ | (eBx Ks=˜; 0)|2 for resonances a1 and a2. Resonance a1 shows a maximum amplitude at 0 T followed by a steady decay with increasing Jeld, behaviour which is characteristic of a ground state wave function. Resonance a2 shows a clearly diFerent behaviour: a minimum in the amplitude at 0 T, followed by a broad maximum at around 10 T (see also Fig. 2a). This behaviour is characteristic of a Jrst excited state wave function. The assignment of a1 and a2 to the ground and Jrst excited states is conJrmed if we now rotate the Jeld. The insets to Fig. 3a are polar plots which show how the amplitude of the resonances change as we rotate the direction of a 4 T magnetic Jeld within the xy-plane. Resonance a1 shows circular symmetry consistent with ground state behaviour whilst a2 displays two clear lobes characteristic of the Jrst excited state. We can determine the energy separation, KE, between the two states by re-scaling the voltage spacing, KVsd ∼114 mV, between them with a leverage factor, f=2:5±0:5, which we estimate from a simple electrostatic model. We Jnd KE = eKVsd =f = 50 ± 10 meV. Note that the measured probability densities are very similar to those of the two lowest energy states of a simple harmonic oscillator (SHO). Therefore, we can obtain another independent estimate of the energy separation by Jtting the data to these analytical forms. We Jnd that the two estimates are consistent if we use a value for the eFective mass of m∗ = 0:026 − 0:038me , a value somewhat larger than that of bulk InAs. However, an increase over the band edge mass is expected due to strain in the QDs [10], non-parabolicity, re-
G (a.u.)
636
8 B (T)
16
8 B (T)
16
o
e
8 B (T)
16
0
Fig. 3. Dependence of the conductance, G, of some representative resonances on a magnetic Jeld, B, applied parallel to the (AlGa)As barrier layer. The polar plots show the dependence of the conductance on the direction of a 4 T magnetic Jeld, rotated within the barrier plane. Continuous lines are Jts to the data by the two lowest energy states of a simple harmonic oscillator.
duced In content due to alloying [11] and the penetration of the wave function into the (AlGa)As barrier. Fourier transforming the Jtted SHO wave functions into real space provides us with an estimate of the spatial extent of our measured wave functions. We obtain a characteristic radius of ∼7 nm, which is comparable to the in-plane radius of our QDs [12]. In addition to resonances a1 and a2 we can also identify other possible groups from the grey-scale plots. The resonances labelled c1 and c2, for example, also shift in bias position, Vsd , at an identical rate
R.J.A. Hill et al. / Physica E 13 (2002) 634 – 637
with gate voltage, Vg . It is therefore possible that c1 and c2 also represent diFerent states of a single QD. It is not possible to verify this by the MTS technique, however, because the resonance amplitudes are too weak and are partially obscured by other resonances. On the other hand, the resonances labelled b1 and b2 are more clearly resolved and show ground-state and Jrst-excited state behaviour, respectively (see Fig. 3b). They also have a similar energy spacing to that of the a resonances. In this case, however, determining the angular dependence of b2 is made diMcult by the presence of another close resonance and both resonances are insensitive to gate voltage so we cannot be conJdent of their assignment to a single QD. Finally, we observe a few resonances which shift very rapidly with gate voltage, indicating that they originate from QDs close to a gate. Performing MTS on the fraction of these that are clearly resolved, we Jnd that the amplitudes fall continuously to zero with increasing magnetic Jeld aligned along the y-axis. This is characteristic of ground state behaviour. However, the fall is quite rapid in contrast to other observed ground state resonances (see Figs. 2 and 3). Two examples are shown in Fig. 3c. For one of these, resonance d, we can clearly resolve the angular dependence. We Jnd that the amplitude of this particular resonance is sharply enhanced when ◦ we rotate the Jeld toward the x-axis by about 40 (see Fig. 3c). This implies an elliptically shaped ground state. The explanation for this behaviour is not clear at the moment. We might speculate, however, that the side-wall potential close to a gate has a gradient steep enough to drop a signiJcant potential across the diameter of a QD. This would deform the circularly-symmetric ground state wave function of the QD into the elliptical shape which we observe.
637
In conclusion, we have employed a novel multiplegated technology as a selection mechanism that allows us to address an individual QD. Combined with the MTS technique, this allows us to probe the probability density and energy of an electron in a particular QD. This work is supported by the Engineering and Physical Sciences Research Council (United Kingdom). We gratefully acknowledge Dr. Hirayama for his support. References [1] E.E. Vdovin, A. Levin, A. Patan*e, L. Eaves, P.C. Main, Yu.N. Khanin, Yu.V. Dubrovskii, M. Henini, G. Hill, Science 290 (2000) 122. [2] M.W. Dellow, P.H. Beton, C.J.G.M. Langerak, T.J. Foster, P.C. Main, L. Eaves, M. Henini, S.P. Beaumont, C.D.W. Wilkinson, Phys. Rev. Lett. 68 (1992) 1754. [3] D.G. Austing, T. Honda, S. Tarucha, Semicond. Sci. Technol. 12 (1997) 631. [4] D.G. Austing, S. Tarucha, P.C. Main, M. Henini, S.T. Stoddart, L. Eaves, Appl. Phys. Lett. 75 (1999) 671. [5] P.C. Main, A.S.G. Thornton, R.J.A. Hill, S.T. Stoddart, T. Ihn, L. Eaves, K.A. Benedict, M. Henini, Phys. Rev. Lett. 84 (2000) 729. [6] M. Narihiro, G. Yusa, Y. Nakamura, T. Noda, H. Sakaki, Appl. Phys. Lett. 70 (1997) 105. [7] I.E. Itskevich, T. Ihn, A.S.G. Thornton, M. Henini, T.J. Foster, P. Moriarty, A. Nogaret, P.H. Beton, L. Eaves, P.C. Main, Phys. Rev. B 54 (1996) 16401. [8] I. Hapke-Wurst, U. Zeitler, H.W. Schumacher, R.J. Haug, K. Pierz, F.J. Ahlers, Semicond. Sci. Technol. 14 (1999) L41. [9] T. Suzuki, K. Nomoto, K. Taira, I. Hase, Jpn. J. Appl. Phys. Part 1 36 (1997) 1917. [10] D.E. Aspnes, M. Cardona, Phys. Rev. B 17 (1978) 726. [11] T. Surkova, A. Patan*e, L. Eaves, P.C. Main, M. Henini, A. Polimeni, A.P. Knights, C. Jeynes, J. Appl. Phys. 89 (2001) 6044. [12] A. Polimeni, A. Patan*e, M. Henini, L. Eaves, P.C. Main, S. Sanguinetti, M. Guzzi, J. Crystal Growth 201 (1999) 276.