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Pillow-shape motion in antidot-arrays
Physica E 6 (2000) 507–509
www.elsevier.nl/locate/physe
Pillow-shape motion in antidot-arrays M. Hochgrafe ∗ , R. Krahne, Ch. Heyn, D. Heitmann Institut fur Angewandte Physik und Zentrum fur Mikrostrukturforschung, Universitat Hamburg, Jungiusstrae 11, 20355 Hamburg, Germany
Abstract Theory predicts a rich mode spectrum for the dynamic excitations in antidot arrays. In addition to the dominating high + !1;+ 0 - and low !EMP -modes, there are higher frequency !n;+ m - and in particular !n;−m -modes with pillow-shaped anticyclotron motions in a magnetic eld B. We have prepared antidot arrays with di erent potentials and were able to detect these modes. c 2000 Published by Elsevier Science B.V. All rights reserved.
PACS: 73.20.Mf; 78.30.Fs; 71.10.-w; 78.30.-j Keywords: Antidots; FIR-spectroscopy
Far infrared (FIR) experiments on antidot arrays [1– 6] show that the dynamic response in a perpendicular magnetic eld B is dominated by a high-frequency + !1; 0 -mode and a low-frequency edge magnetoplas+ which are both cyclotron polarized mon mode !EMP (+). This was con rmed by theory [4,7,8]. However, Refs. [4,8] also predict additional higher frequency ± modes !m; n . Some of them exhibit an anticyclotron polarization (−). We have prepared antidot arrays with di erent potentials in gated and etched Ga0:67 Al0:33 As–GaAs heterostructures and were able to detect these modes. The theoretical mode spectrum according to the e ective-medium model of [7] is given by the implicite expression 1− ∗
! !0
1−f − !c ! ± !0 !0
Corresponding author.
! !0
f ! !0
∓
!c !0
= 0:
(1)
Fig. 1. (a) Theoretical dispersion of antidot modes for f1; 0 = 0:3 (full lines) and f1; 0 = 0:05 (dashed lines). (b) The normalized +(−) oscillator strength for the !1; 0 -modes.
It is shown in Fig. 1a. In this model the antidot array is characterized by a geometrical lling factor f = R2 =a2 , where R is the radius of the depleted area, i.e. the ‘antidot’ in the original two-dimensional electron system (2DES), a is the period of the array and !c is the cyclotron resonance frequency. The frequen-
c 2000 Published by Elsevier Science B.V. All rights reserved. 1386-9477/00/$ - see front matter PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 0 9 5 - 8
508
M. Hochgrafe et al. = Physica E 6 (2000) 507–509
Fig. 2. Experimental dispersions of an antidot array for di erent gate voltages Vg . In (a) the gate voltage is Vg = 0 mV, (b) Vg = 600 mV. The threshold voltage was Vg = −100 mV. The variation of the gate voltage leads to di erent densities NS which are shown in the gure. ± cies of the modes at B = 0 T, !m; n , depend on the shape of the antidot potential [8]. Of particular interest − is the !1; 0 -mode, which has an anticyclotron polarization. This mode has, as shown in Fig. 1b, intrinsically a very weak oscillator strength. It exists practically only at small B and is weaker than the already weak + !1; 0 -mode at small B. For the sample preparation we start from a 2DES in a AlGaAs–GaAs-heterostructure with typical 2D densities of NS = 3 × 1011 cm−2 . All structures were dry etched through the original 2DES [1]. Onto some structures we evaporated an additional Ti-Gate. FIR transmission experiments were performed in superconducting magnets, which were connected to a Fourier transform spectrometer. In gated structures we evaluate the relative transmission T (NS )=T (NS = 0) and in structures without a gate T (B)=T (Bref ) where T (Bref ) has a at response in the relevant frequency regime. The temperature was 1.8 K. The experimental dispersions for a gated structure with period a = 600 nm are shown in Fig. 2. Fits to the dispersions demonstrate that one can change the lling factor f with the gate voltage Vg , e.g. from f = 0:3 to f = 0:46 for the given Vg . (For these f the + -mode is expected at such low frequencies that !EMP it is not within the spectral range of our spectrometer.) However, in these gated structures the intensity + of the !1; 0 -mode decreases so strongly with decreasing B that it was not possible to detect the even weaker − !1; 0 -mode. In Fig. 3 we show experimental results for another antidot array with a period a = 600 nm and a geometrical hole diameter d = 400 nm. The spectra in
Fig. 3. (a) Experimental spectra at di erent magnetic elds B for another sample without gate. One can observe higher frequency modes marked by arrows. (b) shows the experimental dispersion. The full line is the calculated dispersion with f1; 0 = 0:27.
Fig. 4.(a) Experimental relative transmission of a di erent antidot array at various magnetic elds B. Spectra for B ¿ 0 have been − shifted vertically for clarity. The !1; 0 -mode has an anticyclotron
− polarisation. (b) Experimental dispersions of the !1; 0 - and the + !1; -modes. Full lines are the theoretical dispersions according to 0 Eq. (1) for f1; 0 = 0:31.
Fig. 3a demonstrate the decreasing oscillator strength + of the dominating !1; 0 -mode with decreasing B, however, at higher B additional modes appear. The dispersions of the modes are depicted in Fig. 3b. The + !1; 0 -mode can be described by f1; 0 = 0:27. Comparisons to the theoretical results in [8] lead us to identify + these higher-frequency modes with the !1; 1 - and the + + !2; 0 -modes. The anticrossing of the !1; 0 -mode with the dashed line 2!C is the well-known interaction with Bernstein modes [1,6,9]. The experimental results in Fig. 3b indicate that this interaction is even stronger for the higher frequency modes. Similarly as for the + gated structures above, the intensity of the !1; 0 -mode at B = 0 T is so weak (see Fig. 3a) that it is not pos− sible to resolve the !1; 0 -mode in this sample. In Fig. 4 we show experimental results of another etched sample without gate with a = 1000 nm and d = 500 nm. In this sample we observe, as shown in
M. Hochgrafe et al. = Physica E 6 (2000) 507–509 − Fig. 4a, an additional resonance, labeled !1; 0 , which decreases in intensity and increases in energy with increasing B. The dispersions in Fig. 4b indicate that both modes nicely follow the theoretical dispersion for f1; 0 = 0:31. We have also performed experiments with circular polarization and could con rm the anticyclotron polarization of this mode. All FIR excitations are clearly collective excitations. Nevertheless, − in a simple model one can visualize this !1; 0 -mode by a classical single-particle motion, which is a pillow-shape trace between adjacent antidots and which indeed leads to an anticyclotron polarization.
Acknowledgements We acknowledge nancial support by the German Science Foundation through SFB 508 ‘Quantum Materials’.
509
References [1] K. Kern et al., Phys. Rev. Lett. 66 (1991) 1618. [2] A. Lorke, J.P. Kotthaus, K. Ploog, Superlattices Microstruct. 9 (1991) 103. [3] Y. Zhao et al., Appl. Phys. Lett. 60 (1992) 1510. [4] G.Y. Wu, Y. Zhao, Phys. Rev. Lett. 71 (1993) 2114. [5] K. Bollweg et al., Phys. Rev. B 52 (1995) 8379. [6] K. Bollweg et al., Phys. Rev. Lett. 76 (1996) 2774. [7] S.A. Mikhailov, V.A. Volkov, Phys. Rev. B 52 (1995) 17 260. [8] S.A. Mikhailov, Phys. Rev. B 54 (1996) R14 293. [9] V. Gudmundsson et al., Physical Rev. B 51 (1995) 17 744.
www.elsevier.nl/locate/physe
Pillow-shape motion in antidot-arrays M. Hochgrafe ∗ , R. Krahne, Ch. Heyn, D. Heitmann Institut fur Angewandte Physik und Zentrum fur Mikrostrukturforschung, Universitat Hamburg, Jungiusstrae 11, 20355 Hamburg, Germany
Abstract Theory predicts a rich mode spectrum for the dynamic excitations in antidot arrays. In addition to the dominating high + !1;+ 0 - and low !EMP -modes, there are higher frequency !n;+ m - and in particular !n;−m -modes with pillow-shaped anticyclotron motions in a magnetic eld B. We have prepared antidot arrays with di erent potentials and were able to detect these modes. c 2000 Published by Elsevier Science B.V. All rights reserved.
PACS: 73.20.Mf; 78.30.Fs; 71.10.-w; 78.30.-j Keywords: Antidots; FIR-spectroscopy
Far infrared (FIR) experiments on antidot arrays [1– 6] show that the dynamic response in a perpendicular magnetic eld B is dominated by a high-frequency + !1; 0 -mode and a low-frequency edge magnetoplas+ which are both cyclotron polarized mon mode !EMP (+). This was con rmed by theory [4,7,8]. However, Refs. [4,8] also predict additional higher frequency ± modes !m; n . Some of them exhibit an anticyclotron polarization (−). We have prepared antidot arrays with di erent potentials in gated and etched Ga0:67 Al0:33 As–GaAs heterostructures and were able to detect these modes. The theoretical mode spectrum according to the e ective-medium model of [7] is given by the implicite expression 1− ∗
! !0
1−f − !c ! ± !0 !0
Corresponding author.
! !0
f ! !0
∓
!c !0
= 0:
(1)
Fig. 1. (a) Theoretical dispersion of antidot modes for f1; 0 = 0:3 (full lines) and f1; 0 = 0:05 (dashed lines). (b) The normalized +(−) oscillator strength for the !1; 0 -modes.
It is shown in Fig. 1a. In this model the antidot array is characterized by a geometrical lling factor f = R2 =a2 , where R is the radius of the depleted area, i.e. the ‘antidot’ in the original two-dimensional electron system (2DES), a is the period of the array and !c is the cyclotron resonance frequency. The frequen-
c 2000 Published by Elsevier Science B.V. All rights reserved. 1386-9477/00/$ - see front matter PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 0 9 5 - 8
508
M. Hochgrafe et al. = Physica E 6 (2000) 507–509
Fig. 2. Experimental dispersions of an antidot array for di erent gate voltages Vg . In (a) the gate voltage is Vg = 0 mV, (b) Vg = 600 mV. The threshold voltage was Vg = −100 mV. The variation of the gate voltage leads to di erent densities NS which are shown in the gure. ± cies of the modes at B = 0 T, !m; n , depend on the shape of the antidot potential [8]. Of particular interest − is the !1; 0 -mode, which has an anticyclotron polarization. This mode has, as shown in Fig. 1b, intrinsically a very weak oscillator strength. It exists practically only at small B and is weaker than the already weak + !1; 0 -mode at small B. For the sample preparation we start from a 2DES in a AlGaAs–GaAs-heterostructure with typical 2D densities of NS = 3 × 1011 cm−2 . All structures were dry etched through the original 2DES [1]. Onto some structures we evaporated an additional Ti-Gate. FIR transmission experiments were performed in superconducting magnets, which were connected to a Fourier transform spectrometer. In gated structures we evaluate the relative transmission T (NS )=T (NS = 0) and in structures without a gate T (B)=T (Bref ) where T (Bref ) has a at response in the relevant frequency regime. The temperature was 1.8 K. The experimental dispersions for a gated structure with period a = 600 nm are shown in Fig. 2. Fits to the dispersions demonstrate that one can change the lling factor f with the gate voltage Vg , e.g. from f = 0:3 to f = 0:46 for the given Vg . (For these f the + -mode is expected at such low frequencies that !EMP it is not within the spectral range of our spectrometer.) However, in these gated structures the intensity + of the !1; 0 -mode decreases so strongly with decreasing B that it was not possible to detect the even weaker − !1; 0 -mode. In Fig. 3 we show experimental results for another antidot array with a period a = 600 nm and a geometrical hole diameter d = 400 nm. The spectra in
Fig. 3. (a) Experimental spectra at di erent magnetic elds B for another sample without gate. One can observe higher frequency modes marked by arrows. (b) shows the experimental dispersion. The full line is the calculated dispersion with f1; 0 = 0:27.
Fig. 4.(a) Experimental relative transmission of a di erent antidot array at various magnetic elds B. Spectra for B ¿ 0 have been − shifted vertically for clarity. The !1; 0 -mode has an anticyclotron
− polarisation. (b) Experimental dispersions of the !1; 0 - and the + !1; -modes. Full lines are the theoretical dispersions according to 0 Eq. (1) for f1; 0 = 0:31.
Fig. 3a demonstrate the decreasing oscillator strength + of the dominating !1; 0 -mode with decreasing B, however, at higher B additional modes appear. The dispersions of the modes are depicted in Fig. 3b. The + !1; 0 -mode can be described by f1; 0 = 0:27. Comparisons to the theoretical results in [8] lead us to identify + these higher-frequency modes with the !1; 1 - and the + + !2; 0 -modes. The anticrossing of the !1; 0 -mode with the dashed line 2!C is the well-known interaction with Bernstein modes [1,6,9]. The experimental results in Fig. 3b indicate that this interaction is even stronger for the higher frequency modes. Similarly as for the + gated structures above, the intensity of the !1; 0 -mode at B = 0 T is so weak (see Fig. 3a) that it is not pos− sible to resolve the !1; 0 -mode in this sample. In Fig. 4 we show experimental results of another etched sample without gate with a = 1000 nm and d = 500 nm. In this sample we observe, as shown in
M. Hochgrafe et al. = Physica E 6 (2000) 507–509 − Fig. 4a, an additional resonance, labeled !1; 0 , which decreases in intensity and increases in energy with increasing B. The dispersions in Fig. 4b indicate that both modes nicely follow the theoretical dispersion for f1; 0 = 0:31. We have also performed experiments with circular polarization and could con rm the anticyclotron polarization of this mode. All FIR excitations are clearly collective excitations. Nevertheless, − in a simple model one can visualize this !1; 0 -mode by a classical single-particle motion, which is a pillow-shape trace between adjacent antidots and which indeed leads to an anticyclotron polarization.
Acknowledgements We acknowledge nancial support by the German Science Foundation through SFB 508 ‘Quantum Materials’.
509
References [1] K. Kern et al., Phys. Rev. Lett. 66 (1991) 1618. [2] A. Lorke, J.P. Kotthaus, K. Ploog, Superlattices Microstruct. 9 (1991) 103. [3] Y. Zhao et al., Appl. Phys. Lett. 60 (1992) 1510. [4] G.Y. Wu, Y. Zhao, Phys. Rev. Lett. 71 (1993) 2114. [5] K. Bollweg et al., Phys. Rev. B 52 (1995) 8379. [6] K. Bollweg et al., Phys. Rev. Lett. 76 (1996) 2774. [7] S.A. Mikhailov, V.A. Volkov, Phys. Rev. B 52 (1995) 17 260. [8] S.A. Mikhailov, Phys. Rev. B 54 (1996) R14 293. [9] V. Gudmundsson et al., Physical Rev. B 51 (1995) 17 744.