- Email: [email protected]
Far-infrared spectroscopy of tailored quantum wires, quantum dots and antidot arrays
Physica E 14 (2002) 37 – 44
www.elsevier.com/locate/physe
Far-infrared spectroscopy of tailored quantum wires, quantum dots and antidot arrays D. Heitmanna; ∗ , V. Gudmundssonb , M. Hochgr,afea , R. Krahnea , D. Pfannkuchec a Institut
fur Angewandte Physik und Zentrum fur Mikrostrukturforschung, Universitat Hamburg, Jungiusstra e 11, D-20355 Hamburg, Germany b Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland c Institut f ur Theoretische Physik, Universitat Hamburg, Jungiusstra e 9, D-20355 Hamburg, Germany
Abstract We review recent far-infrared transmission experiments on modulation-doped quantum dots, quantum wires and antidot arrays with tailored potentials. In contrast to the simple two-mode behavior that is observed, in a magnetic 3eld, in many experiments on dots with a parabolic potential, a tailored nonparabolic potential breaks the Kohns theorem and allows one to observe detailed many-body e4ects from internal electron motion in the lateral quantum structures. Among these are high-frequency magnetoplasmon modes which show interesting anticrossing behaviors, below-Kohn modes and anticyclotron motion in antidot arrays. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 73.20.Mf; 78.30.Fs; 71.10.−w; 78.30.−j Keywords: Quantum dots; Antidots; Far-infrared spectroscopy; Many-body e4ects
1. Introduction With sophisticated semiconductor technology, in particular lithography and etching techniques, it is possible to fabricate quantum dots of arbitrary shape, starting from two-dimensional electron systems (2DES) in modulation-doped Ga1−x Alx As-GaAs heterostructures. In these systems, electrons are con3ned in all three dimensions. The excitation spectrum of such quantum dots can be investigated most conveniently with far-infrared (FIR) spectroscopy [1–3].
∗
Corresponding author. Tel.: +49-40-428-385-672; fax: +4040-428-386-332. E-mail address: [email protected] (D. Heitmann).
The simplest situation is a circular-shaped dot with parabolic lateral con3nement V (x; y) = 12 m? !02 (x2 + y2 ), where x and y are the coordinates in the plane of the original 2DES. At magnetic 3eld B = 0, one observes one mode. With increasing B the resonance splits into two, one with a positive dispersion (!+1 ) and one with a negative dispersion (!−1 ). Since for a parabolic external potential the center-of-mass motion is totally decoupled from all relative motion of the electrons, the FIR dipole excitation, which couples to the center-of-mass, occurs at the transition frequencies of the bare external potential, independent of how many electrons are con3ned in the dots. This behavior is known as the ‘Generalized Kohn Theorem’ [4]. Because of this, the spectra show no sign of the internal electron–electron interaction. In this review we will discuss some recent experiments,
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 3 5 7 - 0
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D. Heitmann et al. / Physica E 14 (2002) 37 – 44
where we have tailored the lateral con3ning potential in such a way, that it is not parabolic [5 –11]. We will show that then a rich excitation spectrum is observed that demonstrates di4erent aspects of the internal electron–electron interaction. Within the allotted space we cannot discuss all details and cite all references. Here we refer to the original papers.
2. Sample design In Fig. 1 we show some examples for di4erently designed lateral quantum structures. Fig. 1a sketches a 3eld-e4ect con3ned quantum-dot array. Here, a photoresist-dot array is prepared by holographic lithography on top of the heterostructure. A thin Ni or Ti gate, which is semitransparent for the FIR radiation, is evaporated onto the sample. With a negative gate voltage applied to the gate, the electrons are depleted in the regions between the photoresist dots and are thus laterally con3ned. A semitransparent Si- -doped backgate, in a distance of about 200 nm from the 2DES, is inserted during the MBE growth and allows one to charge the dots. The typical period of the array is 200 nm and the electronic diameter of the dot is 50 nm. In such a device, as demonstrated by Meurer et al. [3], it is possible to charge each dot of the array simultaneously with N = 1; 2; 3; : : : ; electrons. These well de3ned electron numbers are controlled by the large Coulomb charging energies in the dots, which requires an increase of the gate voltage by typical 10 meV to transfer the next electron into the individual dot. In Fig. 1b we show an AFM image of a quantum-wire array that has been etched through the active layer of the heterostructure. A thin metal layer evaporated on the top and side walls of the wires serves as a gate and allows one to tune the electron density in the wire and eventually deplete the systems at the narrow constriction, thus that elliptically shaped dots are formed (for more details see Ref. [11]). Fig. 1(c) shows an AFM image of an antidot array, where an array of geometrical holes is etched into a heterostructure [10]. In our experiments the FIR transmission experiments were usually performed at T = 2 K in perpendicular magnetic 3elds of a superconducting magnet, which was connected to a Fourier transform spectrometer.
Fig. 1. (a) Sketch of a 3eld-e4ect con3ned quantum-dot array. With a negative gate voltage electrons in a heterostructure are depleted except under the photoresist dot (light gray regions) and are thus laterally con3ned. (b) AFM image of a quantum-wire array after dry etching and evaporating of a Ti gate. In the narrow regions of the lithographically de3ned laterally modulated wires the electrons can be completely depleted, thus that elliptical dots are formed. The lithographic width of the wire, at maximum, is wx =500 nm. (c) AFM image of an antidot array after dry etching. The white rings are an artefact of the AFM tip.
3. FIR spectroscopy on tunable elliptical quantum dots As an example for experimental spectra we show in Fig. 2 measurements for a sample as shown in Fig. 1b where elliptically shaped dots with tunable ellipticity are formed [11]. We have plotted the normalized
D. Heitmann et al. / Physica E 14 (2002) 37 – 44
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Fig. 2. Spectra of elliptically shaped quantum dots (sample Fig. 1(b) at VG = 0:8 V) corresponding to N = 320 electrons per dot from B = 0 to B = 1:8 T incremented by 0:2 T. Spectra are shifted vertically for B ¿ 0 for clarity. The resonance positions are marked by arrows. The regime where an anticrossing of the modes occurs is marked by thick arrows [11].
transmission T (VG )=T (VTh ), VTh is the threshold voltage where the electron system is completely depleted. At VG = 0:8 V (VTh = −0:2 V) and at B = 0 there are two sets of closely separated resonances. These resonances separate increasingly in a magnetic 3eld and eventually undergo several anticrossings. At high magnetic 3elds (B = 9 T) nearly all oscillator strength is concentrated in one cyclotronlike mode. This allows one, since the transition matrix elements are known, to extract from the integrated intensity the number of electrons per dot N [12]. At VG = 0:8 V we have N = 320 electrons per dot. In Fig. 3 we show the experimental dispersion for the resonances of Fig. 2. The dominant intensity is in the branches labelled !+1 and !−1 . In contrast to circular or quadratically shaped dots [2], there is a splitting into two resonances at B = 0. These two resonances correspond to oscillations along the long and short axis of the ellipse, which can be experimentally con3rmed from the measured linear polarization along the respective axis. The low-frequency branch decreases in intensity with increasing B and can, due to the decreasing sensibility of our experimental setup at low frequencies, no longer be resolved at larger B. The high-frequency branch increases in intensity with increasing B, similar as it is known for circular dots.
Fig. 3. Experimental dispersions of elliptically shaped dots for different VG with corresponding electron number per dot (a) N =320, (b) N = 140 and (c) N = 90. Modes which have a linear polarization along the short (long) axis at small B are marked by full (open) symbols.
It approaches the cyclotron resonance frequency !c . The cyclotron resonance itself is not observed in the experiment as expected for isolated dots. These two dominant modes, !+1 and !−1 , can be perfectly described by transitions between
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D. Heitmann et al. / Physica E 14 (2002) 37 – 44
single-particle energies of the external potential of an elliptical quantum dot [13,14]: 1 2 !± = [!x2 + !y2 + !c2 2 ± !c4 + 2!c2 (!x2 + !y2 ) + (!x2 − !y2 )2 ];
(1)
where !c = eB=m? is the cyclotron resonance frequency and m? the e4ective mass. According to the Kohn theorem, these two modes would be the only modes if the external potential would be parabolic both in x and y direction. In Fig. 3 we observe not only the two fundamental Kohn modes, but sets of modes which undergo di4erent types of anticrossings in a magnetic 3eld. This directly indicates that the external potential is not parabolic. Before we discuss below a full theoretical treatment of the dynamic response in quantum structures we 3rst would like to present an intuitive picture to explain the observed resonances in terms of con3ned magnetoand edge magneto-plasmon modes. In a 3nite-sized 2DES of radius R one gets, at B = 0, the energies of the magneto-plasmon as [15]: !i2 = f1
i Ne2 = fi !02 i ? ? 2m 0 R3
i = 1; 2; 3; : : :
(2)
? is the e4ective dielectric constant and f1 ≈ 1 is a constant which depends slightly on the shape of the potential. In this model the modes are 2D plasmon modes which are ‘con3ned’, with a di4erent number of nodes, in the 3nite-sized dots. For an elliptically shaped dot we expect, replacing !0 in Eq. (2) by !x and !y in the respective direction: 1 2 2 2 !±i = [!xi + !yi + !c2 2 2 + !2 ) + (!2 − !2 )2 ] (3) ± !c4 + 2!c2 (!xi yi xi yi 2 2 = fi !x2 i and !yi = fi !y2 i. Within this apwith !xi proximation we assume that fi is the same in x as well as in y direction. Indeed, except for the anticrossing, the experimental dispersions in Fig. 3 are quite well described by this expression. This suggests that the higher frequency modes i = 2; 3; : : : correspond to higher con3ned plasmon modes. In particular the !−i modes represent con3ned edge magnetoplasmon modes, which exhibit a characteristic negative B
dispersion. In a hard wall potential, these modes can be described microscopically by a collective electron motion, where the individual electrons perform skipping orbits along the circumference of the dot. We would now like to discuss that the dispersions shown in Fig. 3 exhibit two di4erent types of anticrossings. One type occurs close to 2!c and resembles the interaction of plasmons in a 2DES with the Bernstein modes. In this interaction regime we have a complex internal excitation that has been discussed in detail in Refs. [16,17]. Another type of anticrossing occurs if a higher order !−i mode intersects with an !+j mode (i ¿ j). Model calculations for few electron systems [18] have shown that such an anticrossing does not occur for circularly shaped dots, even if the potential is not parabolic, rather it was shown that a noncircular, for example, a quadratic shape, is required to break this degeneracy. It was discussed in Ref. [18] that besides the symmetry and geometry, the electron– electron interaction determines the strength of the interaction and the resulting splitting in the anticrossing regime. We believe that the same parameters, the geometrical shape and the electron–electron interaction, are responsible for the anticrossing at B ¿ 0 that we observe for our elliptically shaped dots. A very interesting behavior occurs if we go to low gate voltages, i.e., VG = 0:2 V in Fig. 3(c), because then the two lowest modes both show a negative B dispersion. Such a situation is not possible for circular or quadratic dots. Obviously, we have increased the ellipticity of the dots with decreasing gate voltage so strongly that the frequency of the !−2 mode becomes smaller than the !+1 mode. 4. Internal motion in quantum dots manifested in below-Kohn modes In Fig. 4 we show the experimental dispersion measured on a 3eld-e4ect con3ned quantum-dot array as sketched in Fig. 1a [9]. The dominant modes, the lower branch with the negative B dispersion and the high-frequency branch, can be well 3tted with Eq. (1). They represent the Kohn’s modes. The interesting 3nding is that we observe a new mode which has, in contrast to the con3ned plasmon modes described above, a frequency that is below the Kohn mode but
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Fig. 5. Calculated dipole absorption for a quantum dot with 5 electrons in a Mattened potential described in the text. In addition to the strong Kohn modes new modes below the high-frequency Kohn mode are found also in the calculation. The half-linewidth is 0:3 meV and T = 1 K. Fig. 4. Experimental dispersion for quantum dots de3ned by 3eld e4ect with a corrugated gate (Fig. 1(a)) for (a) 30 electrons and (b) 6 electrons. Full lines are 3ts with the Kohn modes (Eq. (1)), the dotted lines are !c and 2!c extracted from this 3t. A new mode, the below-Kohn mode, is observed below the upper Kohn mode but clearly above !c [9].
de3nitely larger than !c . The observation of a new mode, in addition to the Kohn modes, implies again that the external potential is not parabolic. Since the external potential is formed by the 3eld e4ect of the modulated gate, one would expect that it should be Mattened at higher energies where it eventually overlaps with the neighboring dots of the array. This should thus a4ect the excitation spectrum. To con3rm this explanation and to get a deeper microscopic insight into this new type of excitation we have performed self-consistent Hartree calculations for di4erently shaped potentials and calculated the dynamic response within the RPA. The details of the applied formalism have been described for example in Ref. [16]. We have modelled an external potential, with a Mattened bottom and a step that softens for higher energy, by the expression V (x) = ax2 + bx4 + W (x) where x = r=a∗0 is the radial coordinate scaled by the e4ective Bohr radius a∗0 = 9:77 nm in GaAs and W (x) = c[1 − f(3:9x − 12)], with f(x) = 1=(exp(x) + 1). Typical values, used for the calculation are a = 0:48 meV, b = −1:8 × 10−3 meV and
c = 6 meV. The calculated FIR absorption is plotted in a 3D plot in Fig. 5. We see from this calculation that the absorption is dominated by the strong Kohn modes. In the regime between 1 and 2 T we have obtained in the calculation a very complex mode spectrum. In this regime again interaction with Bernstein modes occurs. However, with increasing B the new mode is clearly de3ned. To get a microscopic insight into this new mode we have calculated in Fig. 6 the equilibrium density and the induced density. The latter exhibits for the !+ and !− modes indeed a nearly perfect rigid displacement which justi3es that we call these modes still ‘Kohn modes’. However, the new mode represents a complex charge oscillation with several nodes. This is clearly an internal relative motion of the involved electrons. The ‘con3ned’ plasmons at higher frequencies, that we have discussed above, can actually also be seen in the Hartree=RPA calculations. If one looks into the calculated absorption in Fig. 5, one 3nds, very faintly, but clearly resolved on an enlarged scale, modes which start at about 8 meV at B = 0 and split with increasing B, as expected from Eq. (3). Obviously, both experiments and calculations show that such excitations are very weak in potential with a Mattened pro3le. We have also calculated the induced density for the ‘con3ned’ plasmon modes, see Fig. 6(d). These
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Fig. 6. Equilibrium density (full curves) and the density induced by the FIR radiation for two di4erent moments in time (dotted, dashed) with a phase di4erence of (B = 5 T). The low-frequency Kohn mode in (a) and the high-frequency Kohn mode in (c) represent a nearly perfect center-of-mass motion. The below-Kohn mode in (b) and the high-frequency ‘con3ned’ plasmon mode in (d) involve relative electron motion with several nodes.
modes of course also involve relative electron motion. However, this motion is di4erent from the one of the new mode, with 9:27 meV in Fig. 6(b), it has additional nodes in the lower part of the Manks. 5. Anticyclotron motion in antidot arrays Antidots are a reversed structure as compared to dots. An antidot potential is formed if an array of
repulsive potential peaks is superimposed onto a 2DES. Such an array can be realized, as sketched in Fig. 1(c), by starting from a high-mobility modulation-doped Alx Ga1−x As-GaAs heterostructure and etching an array of periodic geometrical holes through the active electron layer. These systems have attracted broad theoretical and experimental interest, in particular due to their commensurability e4ects in magnetic 3elds between the antidot array period and the magnetic length. Experiments [19 –24], in agreement with theory [22,25,26], show that the FIR response of antidot arrays in a magnetic 3eld B is dominated by two modes. In Fig. 7 we show the experimental dispersion of an antidot array [6]. The dominant modes are the + + modes labelled !EMP and !1; 0. We 3rst would like to give an intuitive picture for this mode behavior as presented in Ref. [19]. At high magnetic 3elds the dispersions are very similar to the dispersions in quantum dots. As sketched in + Fig. 8, electrons perform at large B for the !1; 0 mode cyclotron orbits in between the antidots and for the + low-frequency magneto-plasmon mode, !EMP , skipping orbits around the outer edge of the antidots. Both modes are thus cyclotron polarized, i.e., circularly polarized in the same sense as the cyclotron resonance of a 2DES [23]. Similarly, the low-frequency branch for dots is a collective motion of the electrons, where the individual electrons perform skipping orbits along the inner circumference of the dots. The low-frequency mode in a dot is thus anticyclotron polarized. For decreasing magnetic 3eld, and thus increasing cyclotron orbit the electron in dots ‘feel an increasing con3nement’ and the frequency increases. For antidots, however, at low enough 3elds, the electrons can eventually perform closed cyclotron orbits around the antidots. Then the frequency decreases and approaches the cyclotron frequency. If we use the same type of picture one could also imagine that there should be an excitation as sketched for the − pillow-shaped trajectory labelled !1; 0 . We see that this antidot mode has an anticyclotron polarization. This mode has actually been predicted by theory [26], however, it was a challenge to observe this mode, because of its intrinsically weak oscillator strength at B =0, which decreases even further with increasing B. We have prepared antidot arrays with optimized potentials and were able to detect this mode [6]. The
D. Heitmann et al. / Physica E 14 (2002) 37 – 44
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experimental dispersion is plotted in Fig. 7 and compared with theory. Mikhailov and Volkov [25] have presented an e4ective medium approach, which gives an analytical expression for the resonance frequencies ! for an antidot array: 1−
1−f !=!0 (!=!0 ± !c =!0 )
−
Fig. 7. Experimental dispersion (a), normalized oscillator strength I (b), and circular polarization degree (c) of an antidot array (Fig. 1(c)). The full lines in (a) are the theoretical dispersions for an areal 3lling factor f = 0:31, the full lines in (b) are the theoretical normalized oscillator strengths. At B ≈ 1:4 T interaction with Bernstein modes near 2!c occurs, which is not included in the theory for the dispersion and oscillator strength. At B ≈ 0:8 T an interaction with 3!c is visible in the oscillator strength [6].
Fig. 8. Sketch to visualize in a one-particle picture the cy+H clotron motion of the !1; 0 mode at large B, the skipping +H (L)
orbit motion of the !EMP mode, and the pillow-shape trace of − an anticyclotron motion, !1; 0.
f = 0: !=!0 (!=!0 ∓ !c =!0 )
(4)
The antidot array is characterized by an areal 3lling factor f = R2 =a2 , where R is the radius of the depleted area under the geometrical hole and !0 is the ± energy of the !1; 0 modes at B = 0. This equation can be solved easily, and allows one to calculate the dispersion that reproduces the characteristic features of our observations. More sophisticated theories [22,26] require a detailed knowledge of the shape of the antidot potential. Fig. 7 shows that for f = 0:31 the dis− + persions of both the !1; 0 and the !1; 0 mode are well reproduced (f ¿ 0:33 or f ¡ 0:29 does not 3t the data). For f = 0:31 we expect the edge-magneto-plas+ mon mode, !EMP , to be at very low frequencies, so that it is not accessible with our spectrometer. This mode, however, has been clearly observed in antidots with di4erent potentials, for example in Refs. [19,22,24]. − The mode !1; 0 is the new mode with the anticyclotron polarization. To con3rm this in detail we show in Fig. 7(a) the dispersion, in (b) the intensity, and in (c) the polarization of the two modes. I + and I − are, respectively, the intensities of the cyclotron and the anticyclotron resonance. In Fig. + 7(b) the !1; 0 mode dominates the spectrum at large B and decreases in intensity with decreasing B to a − 3nite value I + (B = 0). The !1; 0 mode starts at the − + same value I = I at B = 0 and then decreases in intensity. Theory says that the ratio I1;+0 (B = 0)=IS+ = I1;−0 (B = 0)=IS+ , where IS+ is the saturated value at large B, depends on the shape of the antidot potential. An explicit expression for the normalized +(−) oscillator strengths Sm; n (B)=Sm; n (B = 0) for the two modes is given in Eq. (17) of Ref. [26]. We compare in Fig. 7(b) our experimental intensities with this theoretical expression and 3nd, within the experimental accuracy, a reasonable agreement. Fig. 7(c) shows the experimental polarization. Although the scatter of the data (within an accuracy of 20%)
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D. Heitmann et al. / Physica E 14 (2002) 37 – 44
is relatively large, due to the diOcult determination and the interaction with the Bernstein modes, the experimental data clearly demonstrate the anticyclotron polarization. 6. Conclusions FIR spectroscopy on quantum structures with tailored nonparabolic potential give detailed insight into complex many-body e4ects of quantum wires, quantum dots and antidot arrays. Acknowledgements We gratefully acknowledge support by the “Deutsche Forschungsgemeinschaft” (through SFB 508 “Quantum Materials” and the Graduiertenkolleg “Nanostrukturierte Festk,orper”), the Research Fund of the University of Iceland, and the Icelandic Natural Science Council. References [1] C. Sikorski, U. Merkt, Phys. Rev. Lett. 62 (1989) 2164. [2] T. Demel, D. Heitmann, P. Grambow, K. Ploog, Phys. Rev. Lett. 64 (1990) 788. [3] B. Meurer, D. Heitmann, K. Ploog, Phys. Rev. Lett. 68 (1992) 1371. [4] P. Maksym, T. Chakraborty, Phys. Rev. Lett. 65 (1990) 108. [5] M. Hochgr,afe, R. Krahne, Ch. Heyn, D. Heitmann, Phys. Rev. B 60 (1999) R13974.
[6] M. Hochgr,afe, R. Krahne, Ch. Heyn, D. Heitmann, Phys. Rev. B 60 (1999) 10680. [7] R. Krahne, M. Hochgr,afe, Ch. Heyn, D. Heitmann, Phys. Rev. B 61 (2000) R16319. [8] R. Krahne, M. Hochgr,afe, Ch. Heyn, D. Heitmann, M. Hauser, K. Eberl, Phys. Rev. B 62 (2000) 15345. [9] R. Krahne, V. Gudmundsson, Ch. Heyn, D. Heitmann, Phys. Rev. B 63 (2001) 195303-1. [10] M. Hochgr,afe, B.P. van Zyl, Ch. Heyn, D. Heitmann, E. Zaremba, Phys. Rev. B 63 (2001) 033316-1. [11] M. Hochgr,afe, Ch. Heyn, D. Heitmann, Phys. Rev. B 63 ( 2001) 35303-1. [12] U. Merkt, in: U. R,ossler (Ed.), Advances in Solid State Physics, Vol. 30, Vieweg, Braunschweig, 1990, p. 77. [13] S.K. Yip, Phys. Rev. B 43 (1991) 1707. [14] Q.P. Li, K. Karrai, S.K. Yip, S. Das Sarma, H.D. Drew, Phys. Rev. B 43 (1991) 5151. [15] A.L. Fetter, Phys. Rev. B 32 (1985) 7676; A.L. Fetter, Phys. Rev. B 33 (1986) 3717. [16] V. Gudmundsson, A. Brataas, P. Grambow, B. Meurer, T. Kurth, D. Heitmann, Phys. Rev. B 51 (1995) 17744. [17] C. Steinebach, R. Krahne, G. Biese, C. Sch,uller, D. Heitmann, K. Eberl, Phys. Rev. B 54 (1996) R14281. [18] D. Pfannkuche, R. Gerhardts, Phys. Rev. B 44 (1991) 13132. [19] K. Kern, D. Heitmann, P. Grambow, Y.H. Zhang, K. Ploog, Phys. Rev. Lett. 66 (1991) 1618. [20] A. Lorke, J.P. Kotthaus, K. Ploog, Superlattices Microstruct. 9 (1991) 103. [21] Y. Zhao, D.C. Tsui, M. Santos, M. Shayegan, R.A. Ghanbari, D.A. Antonias, H.I. Smith, Appl. Phys. Lett. 60 (1992) 1510. [22] G.Y. Wu, Y. Zhao, Phys. Rev. Lett. 71 (1993) 2114. [23] K. Bollweg, T. Kurth, D. Heitmann, Phys. Rev. B 52 (1995) 8379. [24] K. Bollweg, T. Kurth, D. Heitmann, V. Gudmundsson, E. Vasiliadou, P. Grambow, K. Eberl, Phys. Rev. Lett. 76 (1996) 2774. [25] S.A. Mikhailov, V.A. Volkov, Phys. Rev. B 52 (1995) 17260. [26] S.A. Mikhailov, Phys. Rev. B 54 (1996) R14293.
www.elsevier.com/locate/physe
Far-infrared spectroscopy of tailored quantum wires, quantum dots and antidot arrays D. Heitmanna; ∗ , V. Gudmundssonb , M. Hochgr,afea , R. Krahnea , D. Pfannkuchec a Institut
fur Angewandte Physik und Zentrum fur Mikrostrukturforschung, Universitat Hamburg, Jungiusstra e 11, D-20355 Hamburg, Germany b Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland c Institut f ur Theoretische Physik, Universitat Hamburg, Jungiusstra e 9, D-20355 Hamburg, Germany
Abstract We review recent far-infrared transmission experiments on modulation-doped quantum dots, quantum wires and antidot arrays with tailored potentials. In contrast to the simple two-mode behavior that is observed, in a magnetic 3eld, in many experiments on dots with a parabolic potential, a tailored nonparabolic potential breaks the Kohns theorem and allows one to observe detailed many-body e4ects from internal electron motion in the lateral quantum structures. Among these are high-frequency magnetoplasmon modes which show interesting anticrossing behaviors, below-Kohn modes and anticyclotron motion in antidot arrays. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 73.20.Mf; 78.30.Fs; 71.10.−w; 78.30.−j Keywords: Quantum dots; Antidots; Far-infrared spectroscopy; Many-body e4ects
1. Introduction With sophisticated semiconductor technology, in particular lithography and etching techniques, it is possible to fabricate quantum dots of arbitrary shape, starting from two-dimensional electron systems (2DES) in modulation-doped Ga1−x Alx As-GaAs heterostructures. In these systems, electrons are con3ned in all three dimensions. The excitation spectrum of such quantum dots can be investigated most conveniently with far-infrared (FIR) spectroscopy [1–3].
∗
Corresponding author. Tel.: +49-40-428-385-672; fax: +4040-428-386-332. E-mail address: [email protected] (D. Heitmann).
The simplest situation is a circular-shaped dot with parabolic lateral con3nement V (x; y) = 12 m? !02 (x2 + y2 ), where x and y are the coordinates in the plane of the original 2DES. At magnetic 3eld B = 0, one observes one mode. With increasing B the resonance splits into two, one with a positive dispersion (!+1 ) and one with a negative dispersion (!−1 ). Since for a parabolic external potential the center-of-mass motion is totally decoupled from all relative motion of the electrons, the FIR dipole excitation, which couples to the center-of-mass, occurs at the transition frequencies of the bare external potential, independent of how many electrons are con3ned in the dots. This behavior is known as the ‘Generalized Kohn Theorem’ [4]. Because of this, the spectra show no sign of the internal electron–electron interaction. In this review we will discuss some recent experiments,
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 3 5 7 - 0
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D. Heitmann et al. / Physica E 14 (2002) 37 – 44
where we have tailored the lateral con3ning potential in such a way, that it is not parabolic [5 –11]. We will show that then a rich excitation spectrum is observed that demonstrates di4erent aspects of the internal electron–electron interaction. Within the allotted space we cannot discuss all details and cite all references. Here we refer to the original papers.
2. Sample design In Fig. 1 we show some examples for di4erently designed lateral quantum structures. Fig. 1a sketches a 3eld-e4ect con3ned quantum-dot array. Here, a photoresist-dot array is prepared by holographic lithography on top of the heterostructure. A thin Ni or Ti gate, which is semitransparent for the FIR radiation, is evaporated onto the sample. With a negative gate voltage applied to the gate, the electrons are depleted in the regions between the photoresist dots and are thus laterally con3ned. A semitransparent Si- -doped backgate, in a distance of about 200 nm from the 2DES, is inserted during the MBE growth and allows one to charge the dots. The typical period of the array is 200 nm and the electronic diameter of the dot is 50 nm. In such a device, as demonstrated by Meurer et al. [3], it is possible to charge each dot of the array simultaneously with N = 1; 2; 3; : : : ; electrons. These well de3ned electron numbers are controlled by the large Coulomb charging energies in the dots, which requires an increase of the gate voltage by typical 10 meV to transfer the next electron into the individual dot. In Fig. 1b we show an AFM image of a quantum-wire array that has been etched through the active layer of the heterostructure. A thin metal layer evaporated on the top and side walls of the wires serves as a gate and allows one to tune the electron density in the wire and eventually deplete the systems at the narrow constriction, thus that elliptically shaped dots are formed (for more details see Ref. [11]). Fig. 1(c) shows an AFM image of an antidot array, where an array of geometrical holes is etched into a heterostructure [10]. In our experiments the FIR transmission experiments were usually performed at T = 2 K in perpendicular magnetic 3elds of a superconducting magnet, which was connected to a Fourier transform spectrometer.
Fig. 1. (a) Sketch of a 3eld-e4ect con3ned quantum-dot array. With a negative gate voltage electrons in a heterostructure are depleted except under the photoresist dot (light gray regions) and are thus laterally con3ned. (b) AFM image of a quantum-wire array after dry etching and evaporating of a Ti gate. In the narrow regions of the lithographically de3ned laterally modulated wires the electrons can be completely depleted, thus that elliptical dots are formed. The lithographic width of the wire, at maximum, is wx =500 nm. (c) AFM image of an antidot array after dry etching. The white rings are an artefact of the AFM tip.
3. FIR spectroscopy on tunable elliptical quantum dots As an example for experimental spectra we show in Fig. 2 measurements for a sample as shown in Fig. 1b where elliptically shaped dots with tunable ellipticity are formed [11]. We have plotted the normalized
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Fig. 2. Spectra of elliptically shaped quantum dots (sample Fig. 1(b) at VG = 0:8 V) corresponding to N = 320 electrons per dot from B = 0 to B = 1:8 T incremented by 0:2 T. Spectra are shifted vertically for B ¿ 0 for clarity. The resonance positions are marked by arrows. The regime where an anticrossing of the modes occurs is marked by thick arrows [11].
transmission T (VG )=T (VTh ), VTh is the threshold voltage where the electron system is completely depleted. At VG = 0:8 V (VTh = −0:2 V) and at B = 0 there are two sets of closely separated resonances. These resonances separate increasingly in a magnetic 3eld and eventually undergo several anticrossings. At high magnetic 3elds (B = 9 T) nearly all oscillator strength is concentrated in one cyclotronlike mode. This allows one, since the transition matrix elements are known, to extract from the integrated intensity the number of electrons per dot N [12]. At VG = 0:8 V we have N = 320 electrons per dot. In Fig. 3 we show the experimental dispersion for the resonances of Fig. 2. The dominant intensity is in the branches labelled !+1 and !−1 . In contrast to circular or quadratically shaped dots [2], there is a splitting into two resonances at B = 0. These two resonances correspond to oscillations along the long and short axis of the ellipse, which can be experimentally con3rmed from the measured linear polarization along the respective axis. The low-frequency branch decreases in intensity with increasing B and can, due to the decreasing sensibility of our experimental setup at low frequencies, no longer be resolved at larger B. The high-frequency branch increases in intensity with increasing B, similar as it is known for circular dots.
Fig. 3. Experimental dispersions of elliptically shaped dots for different VG with corresponding electron number per dot (a) N =320, (b) N = 140 and (c) N = 90. Modes which have a linear polarization along the short (long) axis at small B are marked by full (open) symbols.
It approaches the cyclotron resonance frequency !c . The cyclotron resonance itself is not observed in the experiment as expected for isolated dots. These two dominant modes, !+1 and !−1 , can be perfectly described by transitions between
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single-particle energies of the external potential of an elliptical quantum dot [13,14]: 1 2 !± = [!x2 + !y2 + !c2 2 ± !c4 + 2!c2 (!x2 + !y2 ) + (!x2 − !y2 )2 ];
(1)
where !c = eB=m? is the cyclotron resonance frequency and m? the e4ective mass. According to the Kohn theorem, these two modes would be the only modes if the external potential would be parabolic both in x and y direction. In Fig. 3 we observe not only the two fundamental Kohn modes, but sets of modes which undergo di4erent types of anticrossings in a magnetic 3eld. This directly indicates that the external potential is not parabolic. Before we discuss below a full theoretical treatment of the dynamic response in quantum structures we 3rst would like to present an intuitive picture to explain the observed resonances in terms of con3ned magnetoand edge magneto-plasmon modes. In a 3nite-sized 2DES of radius R one gets, at B = 0, the energies of the magneto-plasmon as [15]: !i2 = f1
i Ne2 = fi !02 i ? ? 2m 0 R3
i = 1; 2; 3; : : :
(2)
? is the e4ective dielectric constant and f1 ≈ 1 is a constant which depends slightly on the shape of the potential. In this model the modes are 2D plasmon modes which are ‘con3ned’, with a di4erent number of nodes, in the 3nite-sized dots. For an elliptically shaped dot we expect, replacing !0 in Eq. (2) by !x and !y in the respective direction: 1 2 2 2 !±i = [!xi + !yi + !c2 2 2 + !2 ) + (!2 − !2 )2 ] (3) ± !c4 + 2!c2 (!xi yi xi yi 2 2 = fi !x2 i and !yi = fi !y2 i. Within this apwith !xi proximation we assume that fi is the same in x as well as in y direction. Indeed, except for the anticrossing, the experimental dispersions in Fig. 3 are quite well described by this expression. This suggests that the higher frequency modes i = 2; 3; : : : correspond to higher con3ned plasmon modes. In particular the !−i modes represent con3ned edge magnetoplasmon modes, which exhibit a characteristic negative B
dispersion. In a hard wall potential, these modes can be described microscopically by a collective electron motion, where the individual electrons perform skipping orbits along the circumference of the dot. We would now like to discuss that the dispersions shown in Fig. 3 exhibit two di4erent types of anticrossings. One type occurs close to 2!c and resembles the interaction of plasmons in a 2DES with the Bernstein modes. In this interaction regime we have a complex internal excitation that has been discussed in detail in Refs. [16,17]. Another type of anticrossing occurs if a higher order !−i mode intersects with an !+j mode (i ¿ j). Model calculations for few electron systems [18] have shown that such an anticrossing does not occur for circularly shaped dots, even if the potential is not parabolic, rather it was shown that a noncircular, for example, a quadratic shape, is required to break this degeneracy. It was discussed in Ref. [18] that besides the symmetry and geometry, the electron– electron interaction determines the strength of the interaction and the resulting splitting in the anticrossing regime. We believe that the same parameters, the geometrical shape and the electron–electron interaction, are responsible for the anticrossing at B ¿ 0 that we observe for our elliptically shaped dots. A very interesting behavior occurs if we go to low gate voltages, i.e., VG = 0:2 V in Fig. 3(c), because then the two lowest modes both show a negative B dispersion. Such a situation is not possible for circular or quadratic dots. Obviously, we have increased the ellipticity of the dots with decreasing gate voltage so strongly that the frequency of the !−2 mode becomes smaller than the !+1 mode. 4. Internal motion in quantum dots manifested in below-Kohn modes In Fig. 4 we show the experimental dispersion measured on a 3eld-e4ect con3ned quantum-dot array as sketched in Fig. 1a [9]. The dominant modes, the lower branch with the negative B dispersion and the high-frequency branch, can be well 3tted with Eq. (1). They represent the Kohn’s modes. The interesting 3nding is that we observe a new mode which has, in contrast to the con3ned plasmon modes described above, a frequency that is below the Kohn mode but
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Fig. 5. Calculated dipole absorption for a quantum dot with 5 electrons in a Mattened potential described in the text. In addition to the strong Kohn modes new modes below the high-frequency Kohn mode are found also in the calculation. The half-linewidth is 0:3 meV and T = 1 K. Fig. 4. Experimental dispersion for quantum dots de3ned by 3eld e4ect with a corrugated gate (Fig. 1(a)) for (a) 30 electrons and (b) 6 electrons. Full lines are 3ts with the Kohn modes (Eq. (1)), the dotted lines are !c and 2!c extracted from this 3t. A new mode, the below-Kohn mode, is observed below the upper Kohn mode but clearly above !c [9].
de3nitely larger than !c . The observation of a new mode, in addition to the Kohn modes, implies again that the external potential is not parabolic. Since the external potential is formed by the 3eld e4ect of the modulated gate, one would expect that it should be Mattened at higher energies where it eventually overlaps with the neighboring dots of the array. This should thus a4ect the excitation spectrum. To con3rm this explanation and to get a deeper microscopic insight into this new type of excitation we have performed self-consistent Hartree calculations for di4erently shaped potentials and calculated the dynamic response within the RPA. The details of the applied formalism have been described for example in Ref. [16]. We have modelled an external potential, with a Mattened bottom and a step that softens for higher energy, by the expression V (x) = ax2 + bx4 + W (x) where x = r=a∗0 is the radial coordinate scaled by the e4ective Bohr radius a∗0 = 9:77 nm in GaAs and W (x) = c[1 − f(3:9x − 12)], with f(x) = 1=(exp(x) + 1). Typical values, used for the calculation are a = 0:48 meV, b = −1:8 × 10−3 meV and
c = 6 meV. The calculated FIR absorption is plotted in a 3D plot in Fig. 5. We see from this calculation that the absorption is dominated by the strong Kohn modes. In the regime between 1 and 2 T we have obtained in the calculation a very complex mode spectrum. In this regime again interaction with Bernstein modes occurs. However, with increasing B the new mode is clearly de3ned. To get a microscopic insight into this new mode we have calculated in Fig. 6 the equilibrium density and the induced density. The latter exhibits for the !+ and !− modes indeed a nearly perfect rigid displacement which justi3es that we call these modes still ‘Kohn modes’. However, the new mode represents a complex charge oscillation with several nodes. This is clearly an internal relative motion of the involved electrons. The ‘con3ned’ plasmons at higher frequencies, that we have discussed above, can actually also be seen in the Hartree=RPA calculations. If one looks into the calculated absorption in Fig. 5, one 3nds, very faintly, but clearly resolved on an enlarged scale, modes which start at about 8 meV at B = 0 and split with increasing B, as expected from Eq. (3). Obviously, both experiments and calculations show that such excitations are very weak in potential with a Mattened pro3le. We have also calculated the induced density for the ‘con3ned’ plasmon modes, see Fig. 6(d). These
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Fig. 6. Equilibrium density (full curves) and the density induced by the FIR radiation for two di4erent moments in time (dotted, dashed) with a phase di4erence of (B = 5 T). The low-frequency Kohn mode in (a) and the high-frequency Kohn mode in (c) represent a nearly perfect center-of-mass motion. The below-Kohn mode in (b) and the high-frequency ‘con3ned’ plasmon mode in (d) involve relative electron motion with several nodes.
modes of course also involve relative electron motion. However, this motion is di4erent from the one of the new mode, with 9:27 meV in Fig. 6(b), it has additional nodes in the lower part of the Manks. 5. Anticyclotron motion in antidot arrays Antidots are a reversed structure as compared to dots. An antidot potential is formed if an array of
repulsive potential peaks is superimposed onto a 2DES. Such an array can be realized, as sketched in Fig. 1(c), by starting from a high-mobility modulation-doped Alx Ga1−x As-GaAs heterostructure and etching an array of periodic geometrical holes through the active electron layer. These systems have attracted broad theoretical and experimental interest, in particular due to their commensurability e4ects in magnetic 3elds between the antidot array period and the magnetic length. Experiments [19 –24], in agreement with theory [22,25,26], show that the FIR response of antidot arrays in a magnetic 3eld B is dominated by two modes. In Fig. 7 we show the experimental dispersion of an antidot array [6]. The dominant modes are the + + modes labelled !EMP and !1; 0. We 3rst would like to give an intuitive picture for this mode behavior as presented in Ref. [19]. At high magnetic 3elds the dispersions are very similar to the dispersions in quantum dots. As sketched in + Fig. 8, electrons perform at large B for the !1; 0 mode cyclotron orbits in between the antidots and for the + low-frequency magneto-plasmon mode, !EMP , skipping orbits around the outer edge of the antidots. Both modes are thus cyclotron polarized, i.e., circularly polarized in the same sense as the cyclotron resonance of a 2DES [23]. Similarly, the low-frequency branch for dots is a collective motion of the electrons, where the individual electrons perform skipping orbits along the inner circumference of the dots. The low-frequency mode in a dot is thus anticyclotron polarized. For decreasing magnetic 3eld, and thus increasing cyclotron orbit the electron in dots ‘feel an increasing con3nement’ and the frequency increases. For antidots, however, at low enough 3elds, the electrons can eventually perform closed cyclotron orbits around the antidots. Then the frequency decreases and approaches the cyclotron frequency. If we use the same type of picture one could also imagine that there should be an excitation as sketched for the − pillow-shaped trajectory labelled !1; 0 . We see that this antidot mode has an anticyclotron polarization. This mode has actually been predicted by theory [26], however, it was a challenge to observe this mode, because of its intrinsically weak oscillator strength at B =0, which decreases even further with increasing B. We have prepared antidot arrays with optimized potentials and were able to detect this mode [6]. The
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experimental dispersion is plotted in Fig. 7 and compared with theory. Mikhailov and Volkov [25] have presented an e4ective medium approach, which gives an analytical expression for the resonance frequencies ! for an antidot array: 1−
1−f !=!0 (!=!0 ± !c =!0 )
−
Fig. 7. Experimental dispersion (a), normalized oscillator strength I (b), and circular polarization degree (c) of an antidot array (Fig. 1(c)). The full lines in (a) are the theoretical dispersions for an areal 3lling factor f = 0:31, the full lines in (b) are the theoretical normalized oscillator strengths. At B ≈ 1:4 T interaction with Bernstein modes near 2!c occurs, which is not included in the theory for the dispersion and oscillator strength. At B ≈ 0:8 T an interaction with 3!c is visible in the oscillator strength [6].
Fig. 8. Sketch to visualize in a one-particle picture the cy+H clotron motion of the !1; 0 mode at large B, the skipping +H (L)
orbit motion of the !EMP mode, and the pillow-shape trace of − an anticyclotron motion, !1; 0.
f = 0: !=!0 (!=!0 ∓ !c =!0 )
(4)
The antidot array is characterized by an areal 3lling factor f = R2 =a2 , where R is the radius of the depleted area under the geometrical hole and !0 is the ± energy of the !1; 0 modes at B = 0. This equation can be solved easily, and allows one to calculate the dispersion that reproduces the characteristic features of our observations. More sophisticated theories [22,26] require a detailed knowledge of the shape of the antidot potential. Fig. 7 shows that for f = 0:31 the dis− + persions of both the !1; 0 and the !1; 0 mode are well reproduced (f ¿ 0:33 or f ¡ 0:29 does not 3t the data). For f = 0:31 we expect the edge-magneto-plas+ mon mode, !EMP , to be at very low frequencies, so that it is not accessible with our spectrometer. This mode, however, has been clearly observed in antidots with di4erent potentials, for example in Refs. [19,22,24]. − The mode !1; 0 is the new mode with the anticyclotron polarization. To con3rm this in detail we show in Fig. 7(a) the dispersion, in (b) the intensity, and in (c) the polarization of the two modes. I + and I − are, respectively, the intensities of the cyclotron and the anticyclotron resonance. In Fig. + 7(b) the !1; 0 mode dominates the spectrum at large B and decreases in intensity with decreasing B to a − 3nite value I + (B = 0). The !1; 0 mode starts at the − + same value I = I at B = 0 and then decreases in intensity. Theory says that the ratio I1;+0 (B = 0)=IS+ = I1;−0 (B = 0)=IS+ , where IS+ is the saturated value at large B, depends on the shape of the antidot potential. An explicit expression for the normalized +(−) oscillator strengths Sm; n (B)=Sm; n (B = 0) for the two modes is given in Eq. (17) of Ref. [26]. We compare in Fig. 7(b) our experimental intensities with this theoretical expression and 3nd, within the experimental accuracy, a reasonable agreement. Fig. 7(c) shows the experimental polarization. Although the scatter of the data (within an accuracy of 20%)
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is relatively large, due to the diOcult determination and the interaction with the Bernstein modes, the experimental data clearly demonstrate the anticyclotron polarization. 6. Conclusions FIR spectroscopy on quantum structures with tailored nonparabolic potential give detailed insight into complex many-body e4ects of quantum wires, quantum dots and antidot arrays. Acknowledgements We gratefully acknowledge support by the “Deutsche Forschungsgemeinschaft” (through SFB 508 “Quantum Materials” and the Graduiertenkolleg “Nanostrukturierte Festk,orper”), the Research Fund of the University of Iceland, and the Icelandic Natural Science Council. References [1] C. Sikorski, U. Merkt, Phys. Rev. Lett. 62 (1989) 2164. [2] T. Demel, D. Heitmann, P. Grambow, K. Ploog, Phys. Rev. Lett. 64 (1990) 788. [3] B. Meurer, D. Heitmann, K. Ploog, Phys. Rev. Lett. 68 (1992) 1371. [4] P. Maksym, T. Chakraborty, Phys. Rev. Lett. 65 (1990) 108. [5] M. Hochgr,afe, R. Krahne, Ch. Heyn, D. Heitmann, Phys. Rev. B 60 (1999) R13974.
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