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Simultaneous investigation of vertical transport and intersubband absorption in a superlattice: Continuum Wannier–Stark ladders and next-nearest-neighbor tunneling
Physica B 272 (1999) 194}197
Simultaneous investigation of vertical transport and intersubband absorption in a superlattice: Continuum Wannier}Stark ladders and next-nearest-neighbor tunneling M. Helm!,*, W. Hilber!, G. Strasser", R. DeMeester#, F.M. Peeters#, A. Wacker$ !Institut fu( r Halbleiter- und Festko( rperphysik, Universita( t Linz, A-4040 Linz, Austria "Institut fu( r Festko( rperelektronik, TU Wien, A-1040 Wien, Austria #Department of Physics, University of Antwerp, B-2610 Antwerpen, Belgium $Institut fu( r Theoretische Physik, TU Berlin, D-10623 Berlin, Germany
Abstract We report the "rst experiment which relates electronic transport and intraband (intersubband) absorption in a biased superlattice. Zener-coupled Wannier}Stark ladders far in the continuum are observed in an n-type GaAs/AlGaAs superlattice using infrared spectroscopy. Under the same conditions, transport reveals the formation of electric-"eld domains, related to tunneling of electrons to the next-nearest superlattice period. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Superlattice; Wannier}Stark ladder; Intersubband absorption; Tunneling
The application of an electric "eld perpendicular to the layers of a superlattice (SL) is known to lead to the remarkable phenomena of Bloch oscillations and the formation of a Wannier}Stark ladder (WSL), which re#ect two complementary views of the same phenomenon, i.e., the break-up of the minibands together with the localization of electrons. Depending on the experimental technique (each of which usually requires a slightly di!erent sample structure), one or another aspect can be observed; e.g. optical spectroscopy on undoped SLs
* Corresponding author. Tel.: #43-70-2468-9602; fax: #43732-2468-650. E-mail address: [email protected] (M. Helm)
directly reveals the WSL states [1], ultrafast spectroscopy can directly follow the Bloch oscillation [2], and transport (in doped SLs) exhibits a negative di!erential resistance (NDR) [3]. Infrared (IR) experiments on biased SLs [4], which have the potential of probing the dynamical conductivity, are rather scarce (apart from measurements of the photocurrent response [5]). The IR absorption should reveal contributions from transitions between WSLs (at mid-IR energies) as well as from intra-WSL transitions (at far IR energies). For the latter ones optical gain below the Bloch frequency has been predicted [6]. For a more thorough understanding of the behavior of a SL under bias, it would appear useful to study the same sample with di!erent experimental
0921-4526/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 2 6 9 - 0
M. Helm et al. / Physica B 272 (1999) 194}197
195
techniques [7]. Due to the high current densities of 103}104 A/cm2 involved in SL transport, it is however di$cult to use the same sample also for IR absorption measurements, which usually require a relatively large sample size. Therefore, we designed a SL (300 periods of 50 As GaAs and 80 As Al Ga As) with a relatively narrow "rst 0.29 0.71 miniband (width D "1.2 meV), to keep the cur1 rent density low, but a rather wide second miniband (D "30 meV), to achieve strong coup2 ling between the adjacent wells via the excited states. The doping was n"2.25]1011 cm~2/period. The second miniband is located right above the barriers in the continuum. Due to the relatively narrow width of the "rst miniband, we can expect sequential tunneling rather than miniband transport to be the transport mechanism at low bias [8]. The mid-IR absorption spectrum without bias [9] shows the usual interminiband transitions, that is a peak at 162 meV and a shoulder at 180 meV, resulting from the singularities at the center and the edge of the mini-Brillouin zone, respectively [10]. The current}voltage characteristic, measured on a 200-lm square mesa at ¹"10 K (like all measurements reported here) is shown in Fig. 1. After a linear rise of the current the SL breaks up into low- and high-"eld domains. A detailed inspection shows that the spacing of the sawtooth-like NDR
structure is approximately 90 mV, consistent with the "nal steep rise of the current at 90 mV]300 periods"27 V. In the low-"eld domain, electrons tunnel from each ground state to the ground state in the adjacent well, but in the high-"eld domain transport does not proceed via tunneling into the "rst excited state in the adjacent well (as is usually observed [11]), since this would require a voltage drop of approximately (E !E )/e"170 mV. The 2 1 only possible explanation for the observed voltage drop of 90 mV is to assume that in the high-"eld domain the electrons tunnel directly to the xrst excited state in the next-nearest SL period [9] (see inset of Fig. 1). The IR absorption experiments under bias were performed on 1-mm2-size etched mesas with a Bruker IFS66 Fourier-transform spectrometer operated in a gated step-scan mode [9]. The transmission was measured during the 10-ls voltage pulse and some 10 ls afterwards as a reference. The resulting transmission change ¹(F)/¹(F"0) is plotted in Fig. 2 for a series of bias voltages. Note that minima in the signal correspond to decreased transmission, i.e. they essentially correspond to absorption lines induced by the electric "eld (marked
Fig. 1. Current}voltage characteristic of the superlattice. The inset shows the conduction-band edge near the boundary of the low- and high-"eld domain together with a schematic view of the transport process.
Fig. 2. Di!erential transmission spectrum (¹(F)/¹(0)) of the superlattice for a series of bias voltages between 8 and 30 V. Minima essentially correspond to electric-"eld-induced absorption lines (marked by arrows).
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M. Helm et al. / Physica B 272 (1999) 194}197
by arrows). Remarkably, the positions of the maxima and minima hardly change with bias, but their size increases and reaches up to 30% at the highest voltage. Yet this is completely consistent with the observations in the I}< characteristics, since the electric-"eld values in the low- and high-"eld domains are "xed to (nearly) zero and approximately F"(E )/ed"85 mV/ 21 130 As "65 kV/cm, respectively. Increasing the voltage only increases the relative length of the high-electric-"eld domain, resulting in a larger optical signal. Understanding the transmission spectra of Fig. 2 requires a calculation of the absorption coef"cient, a, under bias. This has been performed using a "nite system of 19 SL periods and a phenomenological Lorentzian broadening of C"7 meV (HWHM) of each transition. The result is plotted in Fig. 3 for electric "elds from F"0 up to 70 kV/cm. Only up to 20 kV/cm the splitting of the interminiband absorption into a regular WSL can be observed. At higher "elds, strong mixing of several continuum WSLs occurs due to Zener coupling [12]. For direct comparison with the experiment, the quantity a(0)!a(F), which is proportional to the measured transmission change ¹(F)/¹(0), is plotted in the upper panel of Fig. 3. Most of the experimental features can be reasonably well reproduced for a "eld of F"60 kV/cm (drawn by thicker lines for clarity); only the maximum at +180 meV is larger than predicted. The above "eld corresponds to a voltage drop of 80 meV per SL period, which is very close to half the energy separation E , and thus con"rms the tunneling to 21 the next-nearest SL period. A more graphical understanding of the transitions dominating the absorption spectrum can be obtained by using a classi"cation of the energy levels in terms of pure WSL quantum numbers, E "e #peFd (neglecting the interaction due to m,p m Zener coupling for the moment). Such an assignment can be done by inspecting in detail the wavefunctions and oscillator strengths over several SL periods and is shown in Fig. 4. (Transitions from a state Dm"1, pT to Dm@, p@T are written as (m@, p@!p) for brevity.) The main part of the relevant wave functions are drawn bold and also the calculated absorption coe$cient is shown for
Fig. 3. Absorption coe$cient calculated for electric "elds from 0 to 70 kV/cm. In addition, the transmission change is shown for 50, 60, and 70 kV/cm in the upper panel to facilitate comparison with the experiment (see text). The experimentally relevant 60 kV/cm spectra are plotted with thicker lines (compare with Fig. 2).
Fig. 4. Conduction-band pro"le, energy levels and squared wavefunctions drawn for four superlattice periods with an electric "eld of 60 kV/cm. The experimentally observed transitions are indicated by arrows and the main parts of the respective wavefunctions are emphasized by thicker lines. The numbers on the right represent an (approximate) assignment in terms of WS states (see text). In addition, the calculated absorption coe$cient is shown on the left for comparison, including the peak photon energies.
M. Helm et al. / Physica B 272 (1999) 194}197
comparison. The main observed transitions (indicated by arrows) are, from low to high energies, (3, !1), (2, 0) (this is the main vertical intersubband transition), (4, !2), and (3, 0). As seen in Fig. 4, the continuum WSL wavefunctions display the character of both localized WSLand free-electron-like states. Note also that due to the "niteness of the structure used in the calculation, no perfect periodicity of the solution for the wave functions under a translation zPz#d and EPE!eFd is obtained. In the absorption coe$cient (which is the observable quantity) this symmetry is however recovered, when calculated with a "nite C. This shows the relation of C to the coherence length; when C is very small, the wavefunctions more easily feel the edges of the "nite SL [13]. In summary, we have reported the "rst combined measurement of transport and IR absorption of a biased SL, and observed Wannier}Stark ladders in the continuum as well as next-nearest-neighbor tunneling. Extension of the present technique to the far-IR spectral region may allow to measure the spectral response of a Bloch oscillator.
197
Acknowledgements This work was supported by the FWF, the GMe (both Austria), and the FWO-VI (Belgium). R.D.M. is supported by IWT (Belgium).
References [1] E.E. Mendez, F. Agullo-Rueda, J.M. Hong, Phys. Rev. Lett. 60 (1988) 2426. [2] K. Leo, Semicond. Sci. Technol. 13 (1998) 249 and references therein. [3] A. Sibille et al., Superlattices Microstruct. 13 (1993) 247. [4] G. Scamarcio et al., Phys. Rev. B 57 (1998) R6811. [5] K. Unterrainer et al., Phys. Rev. Lett. 76 (1996) 2973. [6] S.J. Allen et al., Semicond. Sci. Technol. 7 (1992) B1. [7] A. Sibille, J.F. Palmier, F. Mollot, Appl. Phys. Lett. 60 (1992) 457. [8] A. Wacker, A.-P. Jauho, Phys. Rev. Lett. 80 (1998) 369. [9] M. Helm et al., Phys. Rev. Lett. 82 (1999) 3120. [10] M. Helm, Semicond. Sci. Technol. 10 (1995) 557. [11] S.H. Kwok et al., Phys. Rev. B 51 (1995) 9943. [12] A. Sibille, J.F. Palmier, F. Laruelle, Phys. Rev. Lett. 80 (1998) 4506. [13] C. Rauch et al., Phys. Rev. Lett. 81 (1998) 3495 in these Proceedings.
Simultaneous investigation of vertical transport and intersubband absorption in a superlattice: Continuum Wannier}Stark ladders and next-nearest-neighbor tunneling M. Helm!,*, W. Hilber!, G. Strasser", R. DeMeester#, F.M. Peeters#, A. Wacker$ !Institut fu( r Halbleiter- und Festko( rperphysik, Universita( t Linz, A-4040 Linz, Austria "Institut fu( r Festko( rperelektronik, TU Wien, A-1040 Wien, Austria #Department of Physics, University of Antwerp, B-2610 Antwerpen, Belgium $Institut fu( r Theoretische Physik, TU Berlin, D-10623 Berlin, Germany
Abstract We report the "rst experiment which relates electronic transport and intraband (intersubband) absorption in a biased superlattice. Zener-coupled Wannier}Stark ladders far in the continuum are observed in an n-type GaAs/AlGaAs superlattice using infrared spectroscopy. Under the same conditions, transport reveals the formation of electric-"eld domains, related to tunneling of electrons to the next-nearest superlattice period. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Superlattice; Wannier}Stark ladder; Intersubband absorption; Tunneling
The application of an electric "eld perpendicular to the layers of a superlattice (SL) is known to lead to the remarkable phenomena of Bloch oscillations and the formation of a Wannier}Stark ladder (WSL), which re#ect two complementary views of the same phenomenon, i.e., the break-up of the minibands together with the localization of electrons. Depending on the experimental technique (each of which usually requires a slightly di!erent sample structure), one or another aspect can be observed; e.g. optical spectroscopy on undoped SLs
* Corresponding author. Tel.: #43-70-2468-9602; fax: #43732-2468-650. E-mail address: [email protected] (M. Helm)
directly reveals the WSL states [1], ultrafast spectroscopy can directly follow the Bloch oscillation [2], and transport (in doped SLs) exhibits a negative di!erential resistance (NDR) [3]. Infrared (IR) experiments on biased SLs [4], which have the potential of probing the dynamical conductivity, are rather scarce (apart from measurements of the photocurrent response [5]). The IR absorption should reveal contributions from transitions between WSLs (at mid-IR energies) as well as from intra-WSL transitions (at far IR energies). For the latter ones optical gain below the Bloch frequency has been predicted [6]. For a more thorough understanding of the behavior of a SL under bias, it would appear useful to study the same sample with di!erent experimental
0921-4526/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 2 6 9 - 0
M. Helm et al. / Physica B 272 (1999) 194}197
195
techniques [7]. Due to the high current densities of 103}104 A/cm2 involved in SL transport, it is however di$cult to use the same sample also for IR absorption measurements, which usually require a relatively large sample size. Therefore, we designed a SL (300 periods of 50 As GaAs and 80 As Al Ga As) with a relatively narrow "rst 0.29 0.71 miniband (width D "1.2 meV), to keep the cur1 rent density low, but a rather wide second miniband (D "30 meV), to achieve strong coup2 ling between the adjacent wells via the excited states. The doping was n"2.25]1011 cm~2/period. The second miniband is located right above the barriers in the continuum. Due to the relatively narrow width of the "rst miniband, we can expect sequential tunneling rather than miniband transport to be the transport mechanism at low bias [8]. The mid-IR absorption spectrum without bias [9] shows the usual interminiband transitions, that is a peak at 162 meV and a shoulder at 180 meV, resulting from the singularities at the center and the edge of the mini-Brillouin zone, respectively [10]. The current}voltage characteristic, measured on a 200-lm square mesa at ¹"10 K (like all measurements reported here) is shown in Fig. 1. After a linear rise of the current the SL breaks up into low- and high-"eld domains. A detailed inspection shows that the spacing of the sawtooth-like NDR
structure is approximately 90 mV, consistent with the "nal steep rise of the current at 90 mV]300 periods"27 V. In the low-"eld domain, electrons tunnel from each ground state to the ground state in the adjacent well, but in the high-"eld domain transport does not proceed via tunneling into the "rst excited state in the adjacent well (as is usually observed [11]), since this would require a voltage drop of approximately (E !E )/e"170 mV. The 2 1 only possible explanation for the observed voltage drop of 90 mV is to assume that in the high-"eld domain the electrons tunnel directly to the xrst excited state in the next-nearest SL period [9] (see inset of Fig. 1). The IR absorption experiments under bias were performed on 1-mm2-size etched mesas with a Bruker IFS66 Fourier-transform spectrometer operated in a gated step-scan mode [9]. The transmission was measured during the 10-ls voltage pulse and some 10 ls afterwards as a reference. The resulting transmission change ¹(F)/¹(F"0) is plotted in Fig. 2 for a series of bias voltages. Note that minima in the signal correspond to decreased transmission, i.e. they essentially correspond to absorption lines induced by the electric "eld (marked
Fig. 1. Current}voltage characteristic of the superlattice. The inset shows the conduction-band edge near the boundary of the low- and high-"eld domain together with a schematic view of the transport process.
Fig. 2. Di!erential transmission spectrum (¹(F)/¹(0)) of the superlattice for a series of bias voltages between 8 and 30 V. Minima essentially correspond to electric-"eld-induced absorption lines (marked by arrows).
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M. Helm et al. / Physica B 272 (1999) 194}197
by arrows). Remarkably, the positions of the maxima and minima hardly change with bias, but their size increases and reaches up to 30% at the highest voltage. Yet this is completely consistent with the observations in the I}< characteristics, since the electric-"eld values in the low- and high-"eld domains are "xed to (nearly) zero and approximately F"(E )/ed"85 mV/ 21 130 As "65 kV/cm, respectively. Increasing the voltage only increases the relative length of the high-electric-"eld domain, resulting in a larger optical signal. Understanding the transmission spectra of Fig. 2 requires a calculation of the absorption coef"cient, a, under bias. This has been performed using a "nite system of 19 SL periods and a phenomenological Lorentzian broadening of C"7 meV (HWHM) of each transition. The result is plotted in Fig. 3 for electric "elds from F"0 up to 70 kV/cm. Only up to 20 kV/cm the splitting of the interminiband absorption into a regular WSL can be observed. At higher "elds, strong mixing of several continuum WSLs occurs due to Zener coupling [12]. For direct comparison with the experiment, the quantity a(0)!a(F), which is proportional to the measured transmission change ¹(F)/¹(0), is plotted in the upper panel of Fig. 3. Most of the experimental features can be reasonably well reproduced for a "eld of F"60 kV/cm (drawn by thicker lines for clarity); only the maximum at +180 meV is larger than predicted. The above "eld corresponds to a voltage drop of 80 meV per SL period, which is very close to half the energy separation E , and thus con"rms the tunneling to 21 the next-nearest SL period. A more graphical understanding of the transitions dominating the absorption spectrum can be obtained by using a classi"cation of the energy levels in terms of pure WSL quantum numbers, E "e #peFd (neglecting the interaction due to m,p m Zener coupling for the moment). Such an assignment can be done by inspecting in detail the wavefunctions and oscillator strengths over several SL periods and is shown in Fig. 4. (Transitions from a state Dm"1, pT to Dm@, p@T are written as (m@, p@!p) for brevity.) The main part of the relevant wave functions are drawn bold and also the calculated absorption coe$cient is shown for
Fig. 3. Absorption coe$cient calculated for electric "elds from 0 to 70 kV/cm. In addition, the transmission change is shown for 50, 60, and 70 kV/cm in the upper panel to facilitate comparison with the experiment (see text). The experimentally relevant 60 kV/cm spectra are plotted with thicker lines (compare with Fig. 2).
Fig. 4. Conduction-band pro"le, energy levels and squared wavefunctions drawn for four superlattice periods with an electric "eld of 60 kV/cm. The experimentally observed transitions are indicated by arrows and the main parts of the respective wavefunctions are emphasized by thicker lines. The numbers on the right represent an (approximate) assignment in terms of WS states (see text). In addition, the calculated absorption coe$cient is shown on the left for comparison, including the peak photon energies.
M. Helm et al. / Physica B 272 (1999) 194}197
comparison. The main observed transitions (indicated by arrows) are, from low to high energies, (3, !1), (2, 0) (this is the main vertical intersubband transition), (4, !2), and (3, 0). As seen in Fig. 4, the continuum WSL wavefunctions display the character of both localized WSLand free-electron-like states. Note also that due to the "niteness of the structure used in the calculation, no perfect periodicity of the solution for the wave functions under a translation zPz#d and EPE!eFd is obtained. In the absorption coe$cient (which is the observable quantity) this symmetry is however recovered, when calculated with a "nite C. This shows the relation of C to the coherence length; when C is very small, the wavefunctions more easily feel the edges of the "nite SL [13]. In summary, we have reported the "rst combined measurement of transport and IR absorption of a biased SL, and observed Wannier}Stark ladders in the continuum as well as next-nearest-neighbor tunneling. Extension of the present technique to the far-IR spectral region may allow to measure the spectral response of a Bloch oscillator.
197
Acknowledgements This work was supported by the FWF, the GMe (both Austria), and the FWO-VI (Belgium). R.D.M. is supported by IWT (Belgium).
References [1] E.E. Mendez, F. Agullo-Rueda, J.M. Hong, Phys. Rev. Lett. 60 (1988) 2426. [2] K. Leo, Semicond. Sci. Technol. 13 (1998) 249 and references therein. [3] A. Sibille et al., Superlattices Microstruct. 13 (1993) 247. [4] G. Scamarcio et al., Phys. Rev. B 57 (1998) R6811. [5] K. Unterrainer et al., Phys. Rev. Lett. 76 (1996) 2973. [6] S.J. Allen et al., Semicond. Sci. Technol. 7 (1992) B1. [7] A. Sibille, J.F. Palmier, F. Mollot, Appl. Phys. Lett. 60 (1992) 457. [8] A. Wacker, A.-P. Jauho, Phys. Rev. Lett. 80 (1998) 369. [9] M. Helm et al., Phys. Rev. Lett. 82 (1999) 3120. [10] M. Helm, Semicond. Sci. Technol. 10 (1995) 557. [11] S.H. Kwok et al., Phys. Rev. B 51 (1995) 9943. [12] A. Sibille, J.F. Palmier, F. Laruelle, Phys. Rev. Lett. 80 (1998) 4506. [13] C. Rauch et al., Phys. Rev. Lett. 81 (1998) 3495 in these Proceedings.