- Email: [email protected]
Cyclotron resonance in asymmetric double quantum wells
Physica E 2 (1998) 116 — 120
Cyclotron resonance in asymmetric double quantum wells Yu.B. Vasilyev!,",*, K.V. Klitzing!, K. Eberl! ! Max-Planck-Institut fu( r Festko( rperforschung, D-70569, Stuttgart, Germany " Ioffe Physical Technical Institute, 194021 St. Petersburg, Russian Federation
Abstract Far-infrared magnetotransmission measurements in magnetic fields are carried out on asymmetric coupled double wells. We observe a splitting in the cyclotron resonance (CR) line for a wide range of intermediate magnetic fields and only one line at high magnetic fields. Two peaks observed in the CR spectra correspond to transitions between Landau levels in individual wells. We propose that phase transition between weak and strong coupling regimes may be responsible for the features. The characteristics of the transition are studied via an analysis of CR masses, CR splitting and line widths as a function of the magnetic field. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Double quantum wells; Cyclotron resonance; Phase transition
1. Introduction Cyclotron resonance (CR) is a powerful technique to study subtle physical effects in low-dimensional systems. Of particular current interest is the CR in multi-component systems when the well-known Kohn’s theorem is violated and electron—electron interactions may affect the CR. Recently, such effects have been investigated with respect to the spin-splitting CR in GaAs/AlGaAs heterostructures in the extreme quantum limit [1] and it was shown that CR can be used successfully to probe electron—electron interactions. Another relevant system where electron—electron interac* Correspondence address: Max-Planck-Institut fu¨r Festko¨rperforschung, Heisenbergstr. 1, D-70569, Stuttgart, Germany. Fax: #049 0711 6891572; e-mail: [email protected].
tions play a crucial role is a system consisting of two parallel two-dimensional electron gas (2DEG) layers. Intensive experimental efforts were directed to study transport properties and there have been very few far-infrared transmission studies of these systems [2,3]. In this paper we report our investigation of CR in a bilayer high-mobility 2D EG in a GaAs/Al Ga As double quantum well (DQW) 0.33 0.67 structure. The difference of electron densities in individual wells allows us to resolve responses from each layer due to band nonparabolicity and extract the effective masses, relaxation times and electron densities for electrons in each well. The data reveal that CR measurements can be used to study interlayer coupling in double quantum well systems. We observe two regimes of weak and strong interlayer coupling with the phase transition controlled by the magnetic field. The dependence of this phase transition on a small in-plane component of the
1386-9477/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved PII: S 1 3 8 6 - 9 4 7 7 ( 9 8 ) 0 0 0 2 6 - 5
Yu.B. Vasilyev et al. / Physica E 2 (1998) 116—120
117
magnetic field was discovered. Some possible mechanisms of the coupling such as tunnelling and Coulomb interaction are discussed.
2. Experiment The DQW samples were grown by molecularbeam epitaxy and each consists of two GaAs quantum wells with the top well width of 30 nm and bottom well of 25 nm separated by 5 nm Al Ga As barrier. An important feature of the 0.33 0.67 structure is asymmetric doping on both sides of the wells. The consequence of such a doping is the different electron densities in the individual layers. Electron densities obtained from Shubnikov—de Haas oscillation measurements are 1.2]1011 cm~2 and 4.2]1011 cm~2 for top and bottom wells, respectively. Cyclotron resonance measurements were performed using Fourier transform spectroscopy with a resolution of 0.25 cm~1. The sample transmission was always normalised to a reference spectrum taken at B"0. The samples were wedged to avoid interference. All experiments are done at a device temperature of ¹"2.2 K.
3. Results and discussions Fig. 1 shows the transmission spectra for a series of magnetic fields covering the range 2.36—12.69 T. At low magnetic fields (B(4 T) a single cyclotron resonance is observed. At intermediate fields a second resonance appears on the higher energy side while the total absorption strength seems to remain constant. Finally, at high magnetic fields (B'10.5 T) only the main peak (with low energy) remains. Small angle tilted magnetic field experiments indicate that the intersubband-Landau level coupling (because of a small inevitable tilt angle) can be ruled out as a reason for the splitting. We attribute the appearance of the second peak in the CR spectra to the presence of two closely spaced electron layers [4]. Each peak corresponds to CR transitions in the individual wells. As a result of non-parabolicity of the conduction band, the CR effective mass for the higher density well are larger
Fig. 1. A typical series of cyclotron resonance spectra in perpendicular magnetic fields in a double quantum well structure.
than that in the lower density well. This interpretation is confirmed by the data in Fig. 2 where we plot the magnetic field dependence of the cyclotron effective mass m/m and resonance full—width at 0 half— maximum (FWHM) obtained from the fit of the experimental spectra by two Lorentzians. Asymmetric doping of the structure above and below the double layers leads to the difference in the electron densities in the wells [5]. The low-energy peak originates from the bottom layer with a higher electron density (4]1011 cm~2) and its effective mass remains approximately constant, roughly corresponding to the Fermi energy at B"0. For this well the quantum limit (filling factor l"2) is reached at around B"10.5 T. In the top layer the electron density is considerably lower and the filling factor is smaller than 2 at B'2.5 T. In this case, as it should be for a single independent quantum well, the effective mass increases approximately linearly with magnetic field [6]. The CR line width dependences (Fig. 2b) for both peaks also correspond their electron densities. The line width of the CR in the bottom well shows a broad maximum at a filling factor close to 4 (B"5 T). This is explained by a filling-factor dependence of the screening of the impurity scattering. At the same time for the top well the line width slightly decreases with magnetic field (l(2). So the magnetic field dependences of both the effective mass and FWHM have characteristics of single quantum wells. This allows us to conclude that the
118
Yu.B. Vasilyev et al. / Physica E 2 (1998) 116—120
Fig. 3. Phase diagram showing the transition between weak and strong coupling regimes
Fig. 2. Magnetic field dependences of the CR effective mass (a) and resonance line width (b) for the top (circles) and bottom (squares) wells. Crosses correspond to measurements after illuminating the sample with a red LED.
two quantum wells are independent and coupling between them is weak at intermediate magnetic fields. In this case the electron wave functions are localised only in the individual wells. The most striking feature of CR spectra in Fig. 1 is that the two peaks collapse in one very symmetric CR peak at a strong magnetic field around 10.5 T. Note that around this field the filling factor in the bottom well becomes equal to two. At first glance this may be explained by transition of all electrons from the top well to the bottom as it should be in the case of the single electron consideration (analog is a single well with two populated subbands). However, tilted magnetic field experiments seem inconsistent with this explanation. We observe that a small tilt angle between the direction of the magnetic field and the
normal to the plane, effects the CR splitting. A small in-plane component of the magnetic field results in a collapse of the two CR peaks. There is a correlation between the in-plane and perpendicular magnetic fields: the higher parallel component corresponds to the smaller perpendicular component when the collapse of the two peaks takes place. This dependence is shown in Fig. 3. Here the solid line divides the plane into two parts; the lower one where two peaks are observed and the upper one with only one peak. The dashed line indicates the maximum magnetic field used in our experiments. We can conclude from this observation that the physical origin of the collapse of the two peaks at a perpendicular magnetic field and the processes in tilted magnetic fields are related. We believe that the splitting behaviour reflects the occurrence of different regimes of interlayer coupling. Appearance and disappearance of the additional CR peak correspond to regimes of weak and strong interlayer coupling, respectively. Within this picture of the phase transition the question arises as to what circumstance is responsible for the interlayer coupling. There are two assumptions which can be related to the effect. First is that the phase transition can be defined by interlayer Coulomb interactions. This interpretation is similar to the conclusions of the works on spinsplitting in pure GaAs/AlGaAs single—layer heterostructures [1], where it was shown that strong electron—electron interactions can effectively
Yu.B. Vasilyev et al. / Physica E 2 (1998) 116—120
couple two modes arising from the different spin states. In a number of theoretical publications [7] it was predicted that at strong magnetic fields DL 2DES can form an unusual broken-symmetry state with spontaneous interlayer phase coherence. The system is described by a pseudospin language in which the layer index is mapped onto a pseudospin degree of freedom. In this mapping pseudospin up refers to an electron which is definitely in one layer whereas pseudospin down refers to an electron in the other layer. This approach reveals a clear analogy with spin-splitting effects at the extreme quantum limit. In both cases the internal electrostatic interactions in the 2DEG couple the motion of the two cyclotron transitions, which can be considered as two sets of particles with slightly different effective masses. An alternative explanation is based on the fact that the filling factor in the bottom well becomes an integer and equal to 2 at the boundary of the transition. It means that the Fermi energy coincides with the lowest Landau level in the bottom well (we ignore the Zeeman energy). As far as in the top well only the lowest Landau level is filled, there is an alignment of Landau levels with the same quantum numbers in both wells. Consequently, such alignment allows the start of interwell resonant tunnelling which is forbidden at smaller fields when Landau levels with different quantum numbers align. In this case interlayer coupling occurs through tunnelling between Landau levels. We believe that it is interlayer tunnelling (hopping) which is responsible for the observed features. Indeed, Coulomb interaction also as the charge transfer between the layers cannot be used to explain the origin of the tilted magnetic field dependence. In contrast, the interlayer hopping is relevant because the in-plane magnetic field only effects closed loops of electron paths between the layers [7]. Further examination of the effect showed that disorder (impurities) plays an important role. Additional measurements were performed after illuminating the sample with a red light-emitting diode. After a short time illumination no changes in the spectra have been seen. Increasing the time of illumination we find that after critical exposure the spectra are dramatically changed and instead of two peaks only one single CR peak is observed at
119
all magnetic fields. From the CR absorption we defined that the total electron density in the sample increases simultaneously. The corresponding effective mass is shown in Fig. 2 by crosses. The result of illumination may be attributed to smoothing of the disorder potential and consequently to a decrease of the disorder during illumination. Disorder may be especially important for interpretation of our tilted magnetic field results. Recently, it was anticipated that in double-layer quantum wells there is the Kosterlitz-Thouless-type metal—insulator phase transition [8] which is controlled by an inplane magnetic field, coupling between the wells and disorder. A similar explanation can probably be applied to understand our results. In a system without disorder (or with small enough disorder) there is only one regime of strong interlayer coupling (as we assume through interlayer Coulomb interaction). This case corresponds to CR spectra with only one CR peak observed in all magnetic fields after illumination. For a disordered system (with fixed disorder) the phase transition between strong and weak coupling regimes is defined by interlayer hopping and by the magnitude of the in-plane magnetic field. This is what we observe experimentally. Both the increases of the interlayer tunnelling by changing perpendicular magnetic field and the increase of the parallel component of magnetic field result in the phase transition from weak to strong interlayer coupling. In conclusion, we studied cyclotron resonance in asymmetric double quantum well structures and found that CR measurements can be used to study interlayer coupling. We interpret the behaviour of the CR line in double-layer structures as evidence for a phase transition between weak and strong interlayer coupling in such structures. We observe that the phase transition is defined by disorder, interlayer tunnelling and the in-plane magnetic field.
Acknowledgements We are grateful to D. Bertram for his participation in some stages of this work. We acknowledge useful discussions with L. Brey, A. Efros, S. Suchalkin, A. Dmitriev, and V. Kocharovskii.
120
Yu.B. Vasilyev et al. / Physica E 2 (1998) 116—120
Yu. Vasilyev thanks the MPI for financial support and the hospitality during his stay in Stuttgart.
References [1] R.J. Nicholas, in: D.J. Lockwood (Ed.), Proc. 22nd ICPS, Vancouver, 1994, p. 1440. [2] K. Ensslin et al., Phys. Rev. B 39 (1989) 11179. [3] A. Lorke et al., Phys. Rev. B 42 (1990) 1321. [4] Yu. Vasilyev et al., in: G. Landwehr, W. Ossau (Eds.), 12th Int. Conf. High Magnetic Fields in the Physics of Semiconductors II, Wuerzburg, 1996, p. 781.
[5] Electron densities in each well were also defined from the CR spectra by the Drude model. These values are in good agreement with those obtained from Shubnikov—de Haas measurements for the top well (1.2]1011 cm~2) while slightly lower for the bottom well (3.4]1011 cm~2). The densities remain approximately constant with increase of the magnetic field. [6] S. Huant, A. Mandray, Phys. Rev. B 46 (1992) 2613. [7] Kun Yang et al., Phys. Rev. B 54 (1996) 11644. [8] D.Z. Liu, X.C. Xie, Phys. Rev. B 55 (1997) 15824.
Cyclotron resonance in asymmetric double quantum wells Yu.B. Vasilyev!,",*, K.V. Klitzing!, K. Eberl! ! Max-Planck-Institut fu( r Festko( rperforschung, D-70569, Stuttgart, Germany " Ioffe Physical Technical Institute, 194021 St. Petersburg, Russian Federation
Abstract Far-infrared magnetotransmission measurements in magnetic fields are carried out on asymmetric coupled double wells. We observe a splitting in the cyclotron resonance (CR) line for a wide range of intermediate magnetic fields and only one line at high magnetic fields. Two peaks observed in the CR spectra correspond to transitions between Landau levels in individual wells. We propose that phase transition between weak and strong coupling regimes may be responsible for the features. The characteristics of the transition are studied via an analysis of CR masses, CR splitting and line widths as a function of the magnetic field. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Double quantum wells; Cyclotron resonance; Phase transition
1. Introduction Cyclotron resonance (CR) is a powerful technique to study subtle physical effects in low-dimensional systems. Of particular current interest is the CR in multi-component systems when the well-known Kohn’s theorem is violated and electron—electron interactions may affect the CR. Recently, such effects have been investigated with respect to the spin-splitting CR in GaAs/AlGaAs heterostructures in the extreme quantum limit [1] and it was shown that CR can be used successfully to probe electron—electron interactions. Another relevant system where electron—electron interac* Correspondence address: Max-Planck-Institut fu¨r Festko¨rperforschung, Heisenbergstr. 1, D-70569, Stuttgart, Germany. Fax: #049 0711 6891572; e-mail: [email protected].
tions play a crucial role is a system consisting of two parallel two-dimensional electron gas (2DEG) layers. Intensive experimental efforts were directed to study transport properties and there have been very few far-infrared transmission studies of these systems [2,3]. In this paper we report our investigation of CR in a bilayer high-mobility 2D EG in a GaAs/Al Ga As double quantum well (DQW) 0.33 0.67 structure. The difference of electron densities in individual wells allows us to resolve responses from each layer due to band nonparabolicity and extract the effective masses, relaxation times and electron densities for electrons in each well. The data reveal that CR measurements can be used to study interlayer coupling in double quantum well systems. We observe two regimes of weak and strong interlayer coupling with the phase transition controlled by the magnetic field. The dependence of this phase transition on a small in-plane component of the
1386-9477/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved PII: S 1 3 8 6 - 9 4 7 7 ( 9 8 ) 0 0 0 2 6 - 5
Yu.B. Vasilyev et al. / Physica E 2 (1998) 116—120
117
magnetic field was discovered. Some possible mechanisms of the coupling such as tunnelling and Coulomb interaction are discussed.
2. Experiment The DQW samples were grown by molecularbeam epitaxy and each consists of two GaAs quantum wells with the top well width of 30 nm and bottom well of 25 nm separated by 5 nm Al Ga As barrier. An important feature of the 0.33 0.67 structure is asymmetric doping on both sides of the wells. The consequence of such a doping is the different electron densities in the individual layers. Electron densities obtained from Shubnikov—de Haas oscillation measurements are 1.2]1011 cm~2 and 4.2]1011 cm~2 for top and bottom wells, respectively. Cyclotron resonance measurements were performed using Fourier transform spectroscopy with a resolution of 0.25 cm~1. The sample transmission was always normalised to a reference spectrum taken at B"0. The samples were wedged to avoid interference. All experiments are done at a device temperature of ¹"2.2 K.
3. Results and discussions Fig. 1 shows the transmission spectra for a series of magnetic fields covering the range 2.36—12.69 T. At low magnetic fields (B(4 T) a single cyclotron resonance is observed. At intermediate fields a second resonance appears on the higher energy side while the total absorption strength seems to remain constant. Finally, at high magnetic fields (B'10.5 T) only the main peak (with low energy) remains. Small angle tilted magnetic field experiments indicate that the intersubband-Landau level coupling (because of a small inevitable tilt angle) can be ruled out as a reason for the splitting. We attribute the appearance of the second peak in the CR spectra to the presence of two closely spaced electron layers [4]. Each peak corresponds to CR transitions in the individual wells. As a result of non-parabolicity of the conduction band, the CR effective mass for the higher density well are larger
Fig. 1. A typical series of cyclotron resonance spectra in perpendicular magnetic fields in a double quantum well structure.
than that in the lower density well. This interpretation is confirmed by the data in Fig. 2 where we plot the magnetic field dependence of the cyclotron effective mass m/m and resonance full—width at 0 half— maximum (FWHM) obtained from the fit of the experimental spectra by two Lorentzians. Asymmetric doping of the structure above and below the double layers leads to the difference in the electron densities in the wells [5]. The low-energy peak originates from the bottom layer with a higher electron density (4]1011 cm~2) and its effective mass remains approximately constant, roughly corresponding to the Fermi energy at B"0. For this well the quantum limit (filling factor l"2) is reached at around B"10.5 T. In the top layer the electron density is considerably lower and the filling factor is smaller than 2 at B'2.5 T. In this case, as it should be for a single independent quantum well, the effective mass increases approximately linearly with magnetic field [6]. The CR line width dependences (Fig. 2b) for both peaks also correspond their electron densities. The line width of the CR in the bottom well shows a broad maximum at a filling factor close to 4 (B"5 T). This is explained by a filling-factor dependence of the screening of the impurity scattering. At the same time for the top well the line width slightly decreases with magnetic field (l(2). So the magnetic field dependences of both the effective mass and FWHM have characteristics of single quantum wells. This allows us to conclude that the
118
Yu.B. Vasilyev et al. / Physica E 2 (1998) 116—120
Fig. 3. Phase diagram showing the transition between weak and strong coupling regimes
Fig. 2. Magnetic field dependences of the CR effective mass (a) and resonance line width (b) for the top (circles) and bottom (squares) wells. Crosses correspond to measurements after illuminating the sample with a red LED.
two quantum wells are independent and coupling between them is weak at intermediate magnetic fields. In this case the electron wave functions are localised only in the individual wells. The most striking feature of CR spectra in Fig. 1 is that the two peaks collapse in one very symmetric CR peak at a strong magnetic field around 10.5 T. Note that around this field the filling factor in the bottom well becomes equal to two. At first glance this may be explained by transition of all electrons from the top well to the bottom as it should be in the case of the single electron consideration (analog is a single well with two populated subbands). However, tilted magnetic field experiments seem inconsistent with this explanation. We observe that a small tilt angle between the direction of the magnetic field and the
normal to the plane, effects the CR splitting. A small in-plane component of the magnetic field results in a collapse of the two CR peaks. There is a correlation between the in-plane and perpendicular magnetic fields: the higher parallel component corresponds to the smaller perpendicular component when the collapse of the two peaks takes place. This dependence is shown in Fig. 3. Here the solid line divides the plane into two parts; the lower one where two peaks are observed and the upper one with only one peak. The dashed line indicates the maximum magnetic field used in our experiments. We can conclude from this observation that the physical origin of the collapse of the two peaks at a perpendicular magnetic field and the processes in tilted magnetic fields are related. We believe that the splitting behaviour reflects the occurrence of different regimes of interlayer coupling. Appearance and disappearance of the additional CR peak correspond to regimes of weak and strong interlayer coupling, respectively. Within this picture of the phase transition the question arises as to what circumstance is responsible for the interlayer coupling. There are two assumptions which can be related to the effect. First is that the phase transition can be defined by interlayer Coulomb interactions. This interpretation is similar to the conclusions of the works on spinsplitting in pure GaAs/AlGaAs single—layer heterostructures [1], where it was shown that strong electron—electron interactions can effectively
Yu.B. Vasilyev et al. / Physica E 2 (1998) 116—120
couple two modes arising from the different spin states. In a number of theoretical publications [7] it was predicted that at strong magnetic fields DL 2DES can form an unusual broken-symmetry state with spontaneous interlayer phase coherence. The system is described by a pseudospin language in which the layer index is mapped onto a pseudospin degree of freedom. In this mapping pseudospin up refers to an electron which is definitely in one layer whereas pseudospin down refers to an electron in the other layer. This approach reveals a clear analogy with spin-splitting effects at the extreme quantum limit. In both cases the internal electrostatic interactions in the 2DEG couple the motion of the two cyclotron transitions, which can be considered as two sets of particles with slightly different effective masses. An alternative explanation is based on the fact that the filling factor in the bottom well becomes an integer and equal to 2 at the boundary of the transition. It means that the Fermi energy coincides with the lowest Landau level in the bottom well (we ignore the Zeeman energy). As far as in the top well only the lowest Landau level is filled, there is an alignment of Landau levels with the same quantum numbers in both wells. Consequently, such alignment allows the start of interwell resonant tunnelling which is forbidden at smaller fields when Landau levels with different quantum numbers align. In this case interlayer coupling occurs through tunnelling between Landau levels. We believe that it is interlayer tunnelling (hopping) which is responsible for the observed features. Indeed, Coulomb interaction also as the charge transfer between the layers cannot be used to explain the origin of the tilted magnetic field dependence. In contrast, the interlayer hopping is relevant because the in-plane magnetic field only effects closed loops of electron paths between the layers [7]. Further examination of the effect showed that disorder (impurities) plays an important role. Additional measurements were performed after illuminating the sample with a red light-emitting diode. After a short time illumination no changes in the spectra have been seen. Increasing the time of illumination we find that after critical exposure the spectra are dramatically changed and instead of two peaks only one single CR peak is observed at
119
all magnetic fields. From the CR absorption we defined that the total electron density in the sample increases simultaneously. The corresponding effective mass is shown in Fig. 2 by crosses. The result of illumination may be attributed to smoothing of the disorder potential and consequently to a decrease of the disorder during illumination. Disorder may be especially important for interpretation of our tilted magnetic field results. Recently, it was anticipated that in double-layer quantum wells there is the Kosterlitz-Thouless-type metal—insulator phase transition [8] which is controlled by an inplane magnetic field, coupling between the wells and disorder. A similar explanation can probably be applied to understand our results. In a system without disorder (or with small enough disorder) there is only one regime of strong interlayer coupling (as we assume through interlayer Coulomb interaction). This case corresponds to CR spectra with only one CR peak observed in all magnetic fields after illumination. For a disordered system (with fixed disorder) the phase transition between strong and weak coupling regimes is defined by interlayer hopping and by the magnitude of the in-plane magnetic field. This is what we observe experimentally. Both the increases of the interlayer tunnelling by changing perpendicular magnetic field and the increase of the parallel component of magnetic field result in the phase transition from weak to strong interlayer coupling. In conclusion, we studied cyclotron resonance in asymmetric double quantum well structures and found that CR measurements can be used to study interlayer coupling. We interpret the behaviour of the CR line in double-layer structures as evidence for a phase transition between weak and strong interlayer coupling in such structures. We observe that the phase transition is defined by disorder, interlayer tunnelling and the in-plane magnetic field.
Acknowledgements We are grateful to D. Bertram for his participation in some stages of this work. We acknowledge useful discussions with L. Brey, A. Efros, S. Suchalkin, A. Dmitriev, and V. Kocharovskii.
120
Yu.B. Vasilyev et al. / Physica E 2 (1998) 116—120
Yu. Vasilyev thanks the MPI for financial support and the hospitality during his stay in Stuttgart.
References [1] R.J. Nicholas, in: D.J. Lockwood (Ed.), Proc. 22nd ICPS, Vancouver, 1994, p. 1440. [2] K. Ensslin et al., Phys. Rev. B 39 (1989) 11179. [3] A. Lorke et al., Phys. Rev. B 42 (1990) 1321. [4] Yu. Vasilyev et al., in: G. Landwehr, W. Ossau (Eds.), 12th Int. Conf. High Magnetic Fields in the Physics of Semiconductors II, Wuerzburg, 1996, p. 781.
[5] Electron densities in each well were also defined from the CR spectra by the Drude model. These values are in good agreement with those obtained from Shubnikov—de Haas measurements for the top well (1.2]1011 cm~2) while slightly lower for the bottom well (3.4]1011 cm~2). The densities remain approximately constant with increase of the magnetic field. [6] S. Huant, A. Mandray, Phys. Rev. B 46 (1992) 2613. [7] Kun Yang et al., Phys. Rev. B 54 (1996) 11644. [8] D.Z. Liu, X.C. Xie, Phys. Rev. B 55 (1997) 15824.