- Email: [email protected]
Si quantum wells
Physica B 314 (2002) 255–258
Femtosecond intersubband scattering of holes in Si1xGex =Si quantum wells R.A. Kaindla,1, M. Woernera,*, M. Wurma, K. Reimanna, T. Elsaessera, C. Miesnerb, K. Brunnerb, G. Abstreiterb a
Max-Born-Institut fur . Nichtlineare Optik und Kurzzeitspektroskopie, Max-Born-Straße 2A, 12489 Berlin, Germany b Walter-Schottky-Institut, Technische Universitat 85748 Garching, Germany . Munchen, .
Abstract The intersubband relaxation of holes in p-type modulation-doped Si1x Gex =Si multiple quantum wells is studied both experimentally and theoretically. Pump–probe experiments with 150-fs midinfrared pulses yield a lifetime of holes in the second heavy-hole subband of only 250 fs for 4.4-nm wide wells with x ¼ 0:5: This short lifetime is caused by interaction with phonons via the optical deformation potential. Calculations of hole–phonon scattering agree very well with the experimental results. The calculations show that longer heavy-hole lifetimes are possible by increasing the Ge content and the well widths. r 2002 Elsevier Science B.V. All rights reserved. PACS: 63.20.Ls; 78.47.+p; 73.21.Fg; 71.20.b Keywords: SiGe quantum wells; Intersubband relaxation; Optical deformation potential
In semiconductor quantum wells, the twodimensional confinement leads to the emergence of subbands within the conduction and valence bands. An important relaxation process is intersubband scattering: carriers promoted into a higher subband are scattered into lower subbands by phonon emission or by interacting with other charge carriers. For electrons, numerous studies of intersubband dynamics have been performed [1]. In contrast, only scarce information is available about the dynamics of holes. The valence band *Corresponding author. Tel.: +49-30-6392-1470; fax: +4930-6392-1489. E-mail address: [email protected] (M. Woerner). 1 Present address: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.
structure, which in tetrahedral semiconductors is considerably more complicated than the conduction band, allows additional scattering processes. For intersubband excitations of holes the most important scattering process is optical deformation potential scattering [2,3]. An understanding of this fundamental interaction is important not only from a basic-research point of view, but also for the accurate engineering of semiconductor devices, especially for the proposed silicon-based quantum cascade laser [4], which uses intersubband transitions of holes. While first measurements show electroluminescence [5] in such a structure, lasing has not yet been observed. Here, we present a femtosecond time-resolved study of hole plasmas in quantum wells using direct intersubband excitation and probing of
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 4 4 0 - 5
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R.A. Kaindl et al. / Physica B 314 (2002) 255–258
heavy holes (HH) in p-type modulation-doped Si0:5 Ge0:5 =Si multiple quantum wells [6]. We demonstrate that holes in the HH2 subband have a short lifetime of only 250 fs: Subsequent thermalization of backscattered holes with unexcited carriers via intraband optical phonon and Coulomb scattering occurs on a time scale of several hundreds of femtoseconds depending on the excitation density. Thermalization results in an ultrafast heating of the HH1 distribution, which subsequently cools down to lattice temperature on a 25-ps time scale. Furthermore, we have performed model calculations of hole–phonon scattering using the optical deformation potential tensor. The calculated lifetime of holes in the second heavy-hole subband is in good agreement with the experiment and shows that cascaded intersubband scattering involving the intermediate light-hole-split-off (LHSO1) subband is significantly faster than the scattering between the different heavy-hole subbands. We have investigated high-quality p-type Si1x Gex =Si multiple quantum wells grown by solid-source molecular beam epitaxy. The structure consists of ten periods of modulation-doped ðn ¼ 1:2 1012 cm2 Þ 4.4-nm-wide Si0:5 Ge0:5 quantum wells separated by 18-nm Si barriers. This sample shows an intersubband absorption line due to HH1-HH2 transitions centered at 167 meV with a spectral width FWHM of 25 meV: In our femtosecond experiments, we have used midinfrared pump and probe pulses of 150-fs duration, generated from the output of a 1-kHz optical parametric amplifier by difference frequency mixing in GaSe [7]. Broad pulse spectra allow probing across the whole intersubband line by spectral selection after the sample. Figs. 1(a)– (c) show pump-induced changes of transmission as a function of the time delay between pump and probe pulses at two characteristic spectral positions (maximum of bleaching at 168 meV and maximum of induced absorption between 140 and 150 meV) for different excitation densities Nex : At low excitation densities [Figs. 1(a) and (b)], both bleaching (positive DT=T0 ) and enhanced absorption (negative DT=T0 ) show a delayed rise on a time scale of several 100 fs; which becomes faster with increasing Nex : However, for the highest Nex
(a)
(c)
(b)
(d)
Fig. 1. (a)–(c) Time evolution of nonlinear transmission changes probed at the spectral positions of strongest induced bleaching (dashed lines) and of strongest enhanced absorption (solid lines). Results are shown for three different excitation densities Nex =N0 ¼ (a) 0.01, (b) 0.03, and (c) 0.3. (d) Fast bleaching signal obtained from the data, as explained in the text. It reflects IS relaxation in the time-range of sequential pump–probe interaction (filled circles, Dt > 70 fs). Solid line: single exponential decay with 250 fs: Inset: optical excitation and IS scattering in the band structure.
[Fig. 1(c)], the time evolution of the bleaching changes drastically. It displays a fast rise within the time resolution of the experiment, a maximum at early time delays and a pronounced decay within the first 2 ps: By contrast, the induced absorption still exhibits the delayed rise (neglecting the coherent artifact due to pump–probe coupling for Dto70 fs), thus significantly differing from the bleaching dynamics. All transmission changes decay completely on a time scale of 50–100 ps (not shown). Because of the nonparallel in-plane dispersions of the two optically coupled subbands (HH1 and HH2), the transition energy within the IS absorption line is correlated with the in-plane momen-
R.A. Kaindl et al. / Physica B 314 (2002) 255–258
tum. Hence, the transmission changes in our timeresolved experiments reflect the transient changes of the hole distributions in both optically coupled subbands. Specifically, enhanced transmission (bleaching) around the center of the IS absorption line can occur either due to stimulated HH2-HH1 emission of holes excited into the HH2 subband, or due to a depletion of originally occupied HH1 states close to k~jj ¼ 0: In contrast, enhanced absorption in the wings of the IS absorption line stems exclusively from holes in the HH1 subband which transiently populate states at initially unpopulated large in-plane momentum k~jj : At low excitation densities [Figs. 1(a) and (b)], the transmission changes are dominated by redistribution of holes in the HH1 subband. The E1 ps rise and subsequent slow decay on a E25 ps time scale correspond to rapid heating and subsequent cooling of the hole gas, respectively. In contrast, the additional fast component that is observed at high excitation [Fig. 1(c)] is due to a nonequilibrium population in the HH2 band. To isolate the IS dynamics, the contribution of HH1 heating at early times has to be subtracted. For this, we use a thermalized HH1 distribution, as suggested by our data for lower excitation density. For each temporal delay, the areas of enhanced absorption and of bleaching were spectrally integrated. The bleaching areas of thermalized spectra ðt > 2 psÞ were subtracted from each spectrum with the same area of enhanced absorption, which gives the excess bleaching (filled circles) plotted in Fig. 1(d) as a function of time delay. The resulting transient [Fig. 1(d)] essentially represents the population dynamics of the HH2 band and yields an intersubband relaxation time of tIS ¼ 250 fs: A microscopic analysis of the hole dynamics is based on calculations of the valence band structure and scattering processes. Intersubband scattering of holes in (strained) quantum wells occurs predominantly via emission of optical phonons via the deformation potential [2,3]. The hole– ~~ phonon matrix element /ijD ujf S contains hole wave functions jiS and jf S of initial and final states from a k~ ~ p band structure calculation [in the basis of cell-periodic wave functions ðSm; X m; Y m; Zm; Sk; X k; Y k; ZkÞT ], the wave
257
functions of three-dimensional optical phonons with different polarizations (LO, TO) and corresponding atomic displacements ~ u; and the defor~ with correct symmetry mation potential tensor D [3]. Each phonon polarization component couples to different types of valence band states: for example, the basis functions jX mS and jY mS are coupled when both fcc sublattices of the diamond lattice are displaced relative to each other in the z direction. After summation over the phonon modes (in SiGe quantum wells there exist Si–Si [_o ¼ 60 meV], Si–Ge ½50 meV ; and Ge–Ge ½36 meV modes [8]) and over the possible final states (Fermi’s Golden Rule), for an optical deformation potential constant of dO ¼ 36 eV [2] and a lattice temperature T ¼ 12 K; we obtain a HH2 lifetime of 290 fs; which agrees very well with the experimental result of 250 fs. Our calculations show that because of the symmetry of the deformation potential tensor the cascaded indirect process via the LHSO1 subband [arrow B in Fig. 1(d)] is significantly faster than direct HH2-HH1 intersubband scattering (arrow A). In Fig. 2, we present calculated scattering rates as a function of the HH2-HH1 optical transition energy, which is changed by varying the well width, for Si1x Gex wells with x ¼ 0:3; 0.5, and 0.7. Since optical-phonon emission is only possible if initial and final state are separated by one phonon energy, the HH2-LHSO1 scattering rate shows pronounced steps at the onsets for emission of the different optical phonons. One sees that the rate of HH2-LHSO1 scattering, if energetically possible, is nearly constant. In contrast, the HH2-HH1 scattering rate increases with increasing HH2-HH1 energy. The reason for this is that the final state has increasingly larger wave vector, which results in an increase of the LH and SO admixtures to the HH1 wave function (optical deformation potential scattering is only allowed because of these admixtures). For a SiGe quantum cascade laser one needs longer upper-state lifetimes than the 250 fs observed here. Our results suggest that considerably longer HH2 lifetimes are possible if one chooses HH2-HH1 transition energies below 100 meV; since then the dominant HH2-LHSO1 scattering is energetically forbidden. The same effect occurs if the Ge
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R.A. Kaindl et al. / Physica B 314 (2002) 255–258
concentration in the wells is increased, but because of the built-in strain such structures soon become impossible to grow pseudomorphic on Si substrates. In conclusion, we have presented the first femtosecond study of hole relaxation after resonant intersubband excitation in a quasi-twodimensional semiconductor. The midinfrared pump–probe studies on p-type Si0:5 Ge0:5 =Si multiple quantum wells reveal an intersubband relaxation time of only 250 fs from the second heavy-hole back to the first heavy-hole subband in good agreement with calculations considering cascaded emission of optical phonons via the deformation potential interaction.
References [1] [2] [3] [4] Fig. 2. Solid lines: calculated total scattering rates (left scale) and lifetimes (right scale) due to the optical-phonon deformation-potential scattering in Si1x Gex =Si multiple quantum wells as a function of the HH2-HH1 transition energy (obtained by varying the well width) for different Ge contents x: The gray areas show the part of the total scattering rate that is caused by direct scattering from HH2 to HH1, the remainder (white area) is caused by scattering from HH2 to LHSO1.
[5] [6] [7] [8]
T. Elsaesser, M. Woerner, Phys. Rep. 321 (1999) 253. . P. Vogl, Phys. Rev. B 24 (1981) 2025. W. Potz, M. Woerner, T. Elsaesser, Phys. Rev. B 51 (1995) 17490. L. Friedman, R.A. Soref, G. Sun, J. Appl. Phys. 83 (1998) 3480. G. Dehlinger, et al., Science 290 (2000) 2277. R.A. Kaindl, et al., Phys. Rev. Lett. 86 (2001) 1122. R.A. Kaindl, et al., J. Opt. Soc. Am. B 17 (2000) 2086. M.I. Alonso, K. Winer, Phys. Rev. B 39 (1989) 10056.
Femtosecond intersubband scattering of holes in Si1xGex =Si quantum wells R.A. Kaindla,1, M. Woernera,*, M. Wurma, K. Reimanna, T. Elsaessera, C. Miesnerb, K. Brunnerb, G. Abstreiterb a
Max-Born-Institut fur . Nichtlineare Optik und Kurzzeitspektroskopie, Max-Born-Straße 2A, 12489 Berlin, Germany b Walter-Schottky-Institut, Technische Universitat 85748 Garching, Germany . Munchen, .
Abstract The intersubband relaxation of holes in p-type modulation-doped Si1x Gex =Si multiple quantum wells is studied both experimentally and theoretically. Pump–probe experiments with 150-fs midinfrared pulses yield a lifetime of holes in the second heavy-hole subband of only 250 fs for 4.4-nm wide wells with x ¼ 0:5: This short lifetime is caused by interaction with phonons via the optical deformation potential. Calculations of hole–phonon scattering agree very well with the experimental results. The calculations show that longer heavy-hole lifetimes are possible by increasing the Ge content and the well widths. r 2002 Elsevier Science B.V. All rights reserved. PACS: 63.20.Ls; 78.47.+p; 73.21.Fg; 71.20.b Keywords: SiGe quantum wells; Intersubband relaxation; Optical deformation potential
In semiconductor quantum wells, the twodimensional confinement leads to the emergence of subbands within the conduction and valence bands. An important relaxation process is intersubband scattering: carriers promoted into a higher subband are scattered into lower subbands by phonon emission or by interacting with other charge carriers. For electrons, numerous studies of intersubband dynamics have been performed [1]. In contrast, only scarce information is available about the dynamics of holes. The valence band *Corresponding author. Tel.: +49-30-6392-1470; fax: +4930-6392-1489. E-mail address: [email protected] (M. Woerner). 1 Present address: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.
structure, which in tetrahedral semiconductors is considerably more complicated than the conduction band, allows additional scattering processes. For intersubband excitations of holes the most important scattering process is optical deformation potential scattering [2,3]. An understanding of this fundamental interaction is important not only from a basic-research point of view, but also for the accurate engineering of semiconductor devices, especially for the proposed silicon-based quantum cascade laser [4], which uses intersubband transitions of holes. While first measurements show electroluminescence [5] in such a structure, lasing has not yet been observed. Here, we present a femtosecond time-resolved study of hole plasmas in quantum wells using direct intersubband excitation and probing of
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 4 4 0 - 5
256
R.A. Kaindl et al. / Physica B 314 (2002) 255–258
heavy holes (HH) in p-type modulation-doped Si0:5 Ge0:5 =Si multiple quantum wells [6]. We demonstrate that holes in the HH2 subband have a short lifetime of only 250 fs: Subsequent thermalization of backscattered holes with unexcited carriers via intraband optical phonon and Coulomb scattering occurs on a time scale of several hundreds of femtoseconds depending on the excitation density. Thermalization results in an ultrafast heating of the HH1 distribution, which subsequently cools down to lattice temperature on a 25-ps time scale. Furthermore, we have performed model calculations of hole–phonon scattering using the optical deformation potential tensor. The calculated lifetime of holes in the second heavy-hole subband is in good agreement with the experiment and shows that cascaded intersubband scattering involving the intermediate light-hole-split-off (LHSO1) subband is significantly faster than the scattering between the different heavy-hole subbands. We have investigated high-quality p-type Si1x Gex =Si multiple quantum wells grown by solid-source molecular beam epitaxy. The structure consists of ten periods of modulation-doped ðn ¼ 1:2 1012 cm2 Þ 4.4-nm-wide Si0:5 Ge0:5 quantum wells separated by 18-nm Si barriers. This sample shows an intersubband absorption line due to HH1-HH2 transitions centered at 167 meV with a spectral width FWHM of 25 meV: In our femtosecond experiments, we have used midinfrared pump and probe pulses of 150-fs duration, generated from the output of a 1-kHz optical parametric amplifier by difference frequency mixing in GaSe [7]. Broad pulse spectra allow probing across the whole intersubband line by spectral selection after the sample. Figs. 1(a)– (c) show pump-induced changes of transmission as a function of the time delay between pump and probe pulses at two characteristic spectral positions (maximum of bleaching at 168 meV and maximum of induced absorption between 140 and 150 meV) for different excitation densities Nex : At low excitation densities [Figs. 1(a) and (b)], both bleaching (positive DT=T0 ) and enhanced absorption (negative DT=T0 ) show a delayed rise on a time scale of several 100 fs; which becomes faster with increasing Nex : However, for the highest Nex
(a)
(c)
(b)
(d)
Fig. 1. (a)–(c) Time evolution of nonlinear transmission changes probed at the spectral positions of strongest induced bleaching (dashed lines) and of strongest enhanced absorption (solid lines). Results are shown for three different excitation densities Nex =N0 ¼ (a) 0.01, (b) 0.03, and (c) 0.3. (d) Fast bleaching signal obtained from the data, as explained in the text. It reflects IS relaxation in the time-range of sequential pump–probe interaction (filled circles, Dt > 70 fs). Solid line: single exponential decay with 250 fs: Inset: optical excitation and IS scattering in the band structure.
[Fig. 1(c)], the time evolution of the bleaching changes drastically. It displays a fast rise within the time resolution of the experiment, a maximum at early time delays and a pronounced decay within the first 2 ps: By contrast, the induced absorption still exhibits the delayed rise (neglecting the coherent artifact due to pump–probe coupling for Dto70 fs), thus significantly differing from the bleaching dynamics. All transmission changes decay completely on a time scale of 50–100 ps (not shown). Because of the nonparallel in-plane dispersions of the two optically coupled subbands (HH1 and HH2), the transition energy within the IS absorption line is correlated with the in-plane momen-
R.A. Kaindl et al. / Physica B 314 (2002) 255–258
tum. Hence, the transmission changes in our timeresolved experiments reflect the transient changes of the hole distributions in both optically coupled subbands. Specifically, enhanced transmission (bleaching) around the center of the IS absorption line can occur either due to stimulated HH2-HH1 emission of holes excited into the HH2 subband, or due to a depletion of originally occupied HH1 states close to k~jj ¼ 0: In contrast, enhanced absorption in the wings of the IS absorption line stems exclusively from holes in the HH1 subband which transiently populate states at initially unpopulated large in-plane momentum k~jj : At low excitation densities [Figs. 1(a) and (b)], the transmission changes are dominated by redistribution of holes in the HH1 subband. The E1 ps rise and subsequent slow decay on a E25 ps time scale correspond to rapid heating and subsequent cooling of the hole gas, respectively. In contrast, the additional fast component that is observed at high excitation [Fig. 1(c)] is due to a nonequilibrium population in the HH2 band. To isolate the IS dynamics, the contribution of HH1 heating at early times has to be subtracted. For this, we use a thermalized HH1 distribution, as suggested by our data for lower excitation density. For each temporal delay, the areas of enhanced absorption and of bleaching were spectrally integrated. The bleaching areas of thermalized spectra ðt > 2 psÞ were subtracted from each spectrum with the same area of enhanced absorption, which gives the excess bleaching (filled circles) plotted in Fig. 1(d) as a function of time delay. The resulting transient [Fig. 1(d)] essentially represents the population dynamics of the HH2 band and yields an intersubband relaxation time of tIS ¼ 250 fs: A microscopic analysis of the hole dynamics is based on calculations of the valence band structure and scattering processes. Intersubband scattering of holes in (strained) quantum wells occurs predominantly via emission of optical phonons via the deformation potential [2,3]. The hole– ~~ phonon matrix element /ijD ujf S contains hole wave functions jiS and jf S of initial and final states from a k~ ~ p band structure calculation [in the basis of cell-periodic wave functions ðSm; X m; Y m; Zm; Sk; X k; Y k; ZkÞT ], the wave
257
functions of three-dimensional optical phonons with different polarizations (LO, TO) and corresponding atomic displacements ~ u; and the defor~ with correct symmetry mation potential tensor D [3]. Each phonon polarization component couples to different types of valence band states: for example, the basis functions jX mS and jY mS are coupled when both fcc sublattices of the diamond lattice are displaced relative to each other in the z direction. After summation over the phonon modes (in SiGe quantum wells there exist Si–Si [_o ¼ 60 meV], Si–Ge ½50 meV ; and Ge–Ge ½36 meV modes [8]) and over the possible final states (Fermi’s Golden Rule), for an optical deformation potential constant of dO ¼ 36 eV [2] and a lattice temperature T ¼ 12 K; we obtain a HH2 lifetime of 290 fs; which agrees very well with the experimental result of 250 fs. Our calculations show that because of the symmetry of the deformation potential tensor the cascaded indirect process via the LHSO1 subband [arrow B in Fig. 1(d)] is significantly faster than direct HH2-HH1 intersubband scattering (arrow A). In Fig. 2, we present calculated scattering rates as a function of the HH2-HH1 optical transition energy, which is changed by varying the well width, for Si1x Gex wells with x ¼ 0:3; 0.5, and 0.7. Since optical-phonon emission is only possible if initial and final state are separated by one phonon energy, the HH2-LHSO1 scattering rate shows pronounced steps at the onsets for emission of the different optical phonons. One sees that the rate of HH2-LHSO1 scattering, if energetically possible, is nearly constant. In contrast, the HH2-HH1 scattering rate increases with increasing HH2-HH1 energy. The reason for this is that the final state has increasingly larger wave vector, which results in an increase of the LH and SO admixtures to the HH1 wave function (optical deformation potential scattering is only allowed because of these admixtures). For a SiGe quantum cascade laser one needs longer upper-state lifetimes than the 250 fs observed here. Our results suggest that considerably longer HH2 lifetimes are possible if one chooses HH2-HH1 transition energies below 100 meV; since then the dominant HH2-LHSO1 scattering is energetically forbidden. The same effect occurs if the Ge
258
R.A. Kaindl et al. / Physica B 314 (2002) 255–258
concentration in the wells is increased, but because of the built-in strain such structures soon become impossible to grow pseudomorphic on Si substrates. In conclusion, we have presented the first femtosecond study of hole relaxation after resonant intersubband excitation in a quasi-twodimensional semiconductor. The midinfrared pump–probe studies on p-type Si0:5 Ge0:5 =Si multiple quantum wells reveal an intersubband relaxation time of only 250 fs from the second heavy-hole back to the first heavy-hole subband in good agreement with calculations considering cascaded emission of optical phonons via the deformation potential interaction.
References [1] [2] [3] [4] Fig. 2. Solid lines: calculated total scattering rates (left scale) and lifetimes (right scale) due to the optical-phonon deformation-potential scattering in Si1x Gex =Si multiple quantum wells as a function of the HH2-HH1 transition energy (obtained by varying the well width) for different Ge contents x: The gray areas show the part of the total scattering rate that is caused by direct scattering from HH2 to HH1, the remainder (white area) is caused by scattering from HH2 to LHSO1.
[5] [6] [7] [8]
T. Elsaesser, M. Woerner, Phys. Rep. 321 (1999) 253. . P. Vogl, Phys. Rev. B 24 (1981) 2025. W. Potz, M. Woerner, T. Elsaesser, Phys. Rev. B 51 (1995) 17490. L. Friedman, R.A. Soref, G. Sun, J. Appl. Phys. 83 (1998) 3480. G. Dehlinger, et al., Science 290 (2000) 2277. R.A. Kaindl, et al., Phys. Rev. Lett. 86 (2001) 1122. R.A. Kaindl, et al., J. Opt. Soc. Am. B 17 (2000) 2086. M.I. Alonso, K. Winer, Phys. Rev. B 39 (1989) 10056.