I.V. Kukushkin, J.H. Smet, and K. v. Klitzing; W. Wegscheider (Universit¨at Regensburg) The introduction of suitable fictitious entities,
quasi-particles, occasionally permits to cast otherwise difficult strongly interacting many-body systems in a single particle form. We can then take the customary physical approach, using concepts and representations, which for-merly could only be applied to systems with weak interactions, and yet still capture the es-sential physics. A notable recent example oc-curs in the conduction properties of a two-dimensional electron system (2DES), when ex-posed to a strong perpendicular magnetic field B. They are governed by electron-electron interactions, that at low temperatures bring about the Nobel prize awarded fractional quan-tum Hall effect (FQHE). Composite fermions [O. Heinonen, Editor, ‘Composite Fermions’, World Scientific Publishing, 1998], each assembled from one electron and two (or more generally,
an even number) magnetic flux quanta, were identified as apposite quasi-particles that sim-plify considerably our understanding of the FQHE. The flux attachment is a natural way to minimize the energy of the 2DES, since the associated vortex expels other electrons from its neighborhood and decreases the re-pulsive Coulomb interaction between adjacent electrons. In many experiments, composite fermions may be viewed as non-interacting.
These composite fermions (CFs) behave as legitimate particles with well-defined charge, spin and statistics. They do not experience the external magnetic field B, but a drastically re-duced effective magnetic field B, that vanishes exactly when two flux quanta penetrate the sam-ple per electron in the system (filling ν= 1/2).
They precess, like electrons, along circular cy-clotron orbits, but with a diameter determined
by B rather than B. The Landau quantization of this cyclotron motion gives rise to the inte-ger quantum Hall effect of CFs and is equiva-lent to the FQHE of the original electrons. The frequency of their cyclotron motion remained hitherto enigmatic, since the effective mass is no longer related to the band mass of the orig-inal electrons and is entirely generated from electron-electron interactions. Here, we look for the enhanced absorption of a microwave field when it is in resonance with the frequency
of the CF circular motion in order to determine their effective mass[I.V. Kukushkin et al., Nature 415, 409 (2002)].
The search of the CF cyclotron resonance re-quires substantial sophistication over conven-tional methods, used to detect the electron cy-clotron resonance, since the consequences of Kohn’s theorem must be circumvented. This theorem states that in a translationally invariant system homogeneous radiation can only couple to the center-of-mass coordinate.
Figure 44: Illustration of the optical scheme to detect resonant microwave absorption for the electron cyclotron-magnetoplasmon hybrid mode at low B-fields. (a) Luminescence spectrum in the presence of (dotted line) and without (solid line) a 50µW microwave excitation of 18 GHz obtained on a disk-shaped 2DES with carrier densityn2D= 5.81010cm 2at a fixed magnetic field B = 22 mT. The difference reflects the increased electron temperatureTeldue to resonant microwave absorption for the spectrum obtained in the presence of microwaves. The bottom dashed curve is the differential spectrum obtained by subtracting both curves. (b) Top panel: The microwave absorption amplitude at 29 GHz and 39 GHz as a function of B. The peaks are identified as the dimensional magnetoplasma-cyclotron hybrid mode. The inset shows a conventional bolometer measurement. Bottom panel: Resonance position forn2D= 1.091011(open circles) and 1.11010cm 2(solid circles) as a function of incident microwave frequency. The dashed lines represents the theoretical dependence of the hybrid dimensional magnetoplasma-cyclotron resonance. The dotted line corresponds to the cyclotron mode only.
Phenomena originating from electron-electron interactions will thus not be reflected in the ab-sorption spectrum. An elegant way to bypass this theorem is to impose a periodic density modulation to break translational invariance.
The non-zero wavevectors defined by the ap-propriately chosen modulation may then offer access to the cyclotron transitions of CFs, even though they are likely to remain very weak.
Therefore, the development of an optical detec-tion scheme, that boosts the sensitivity to reso-nant microwave absorption by up to two orders of magnitude in comparison with traditional techniques, was a prerequisite for our studies.
Furthermore, we exploited to our benefit the ac-cidental discovery that microwaves, already in-cident on the sample, set up a periodic modu-lation through the excitation of surface acoustic waves (SAW).
The 2DES, patterned into a 1 mm diameter disk, is placed in a short-circuited microwave waveguide in the electric field maximum of the microwave excitation inside a He3-cryostat. At a fixed B-field, luminescence spectra with and without microwave excitation were recorded consecutively. Under resonant conditions heat-ing causes a significate difference between both spectra. This can be highlighted by integrating the absolute value of the difference across the entire energy range. This quantity is considered a measure of the microwave absorption ampli-tude. The same procedure is then repeated for different values of B.
To establish trustworthiness in this unconven-tional detection scheme, we apply it in Fig. 44 to the well-known case of the electron cyclotron resonance ωcr= eBm, with m the effective mass of GaAs (0.067 m0). Due to its limited size, the sample also supports a dimensional plasma mode that depends on both the den-sity and diameter of the sample. The plasma and cyclotron mode hybridize. The optical method indeed recovers this well-known hybrid modeωDMR. Additional support for the valid-ity of the detection method comes from a com-parison with measurements using a bolometer
(Fig. 44(b), top panel). Not only does one find the same resonance position, but also the same line shape. The only difference is the improved signal to noise ratio (30–100 times) for the op-tical scheme.
Disorder and the finite dimensions of the sam-ple in princisam-ple suffice to break translational invariance and outwit Kohn’s theorem, as at-tested by the interaction of the cyclotron and dimensional plasma mode. However, they pro-vide access to internal degrees of freedom either at poorly defined wavevectors or too small a wavevector for appropriate sample sizes.
Therefore, the imposition of a periodic den-sity modulation, that introduces larger and well-defined wavevectors, is desirable. Transport ex-periments in the Hall bar geometry disclosed that additional processing is not required, since the microwaves, already incident on the sample, concomitantly induce a periodic modulation at sufficiently high power. A clear signature is the appearance of commensurability oscillations in the magnetoresistance due to the interplay be-tween the B-dependent cyclotron radius of elec-trons and the length scale of the modulation.
Figure 45: Magnetotransport data in a Hall bar geometry without (top curve) and under 100µW of microwave radiation at 12 (middle curve) and 17 GHz (bottom curve). Curves are offset for clar-ity. Besides the well-known Shubnikov-de Haas os-cillations, additional magnetoresistance oscillations appear under microwave radiation. They are com-monly observed in 2DESs on which a static periodic modulation of the density has been imposed.
Examples are displayed in Fig. 45. The follow-ing scenario is conceivable: Owfollow-ing to the piezo-electric properties of the AlxGa1xAs-crystal, the radiation is partly transformed into SAWs with opposite momentum, so that both energy and momentum are conserved. Reflection from cleaved boundaries of the sample then produces
a standing wave with a periodicity determined by the sound wavelength. The involvement of sound waves can be deduced from the trans-port data, since from the minima we can extract the modulation period. The ratio of this period to the sound wavelength at these frequencies is about 1.1.
Figure 46: (a) Microwave absorption amplitude for n2D= 0.811011 and 1.151011cm 2and a frequency of 20 GHz. The response near ν= 1 and 1/3 does not shift with frequency. (b) Microwave absorption amplitude in the vicinity of ν= 1/2 at three different frequencies and n2D= 1.091011cm 2. (c) Posi-tion of the CF cyclotron mode as a funcPosi-tion of B= B – 2φ0n2D (φ0 is the elementary flux quantum) for n2D= 1.091011cm 2(circles) and 0.591011cm 2(squares).
Figure 46(a) depicts the microwave absorption amplitude up to high B-fields. Apart from the strong dimensional magnetoplasma-cyclotron resonance signal at low B-field discussed above, several peaks, that scale with a variation of the density, emerge near filling 1, 1/2 and 1/3. Those peak positions associated withν= 1 and 1/3 remain fixed when tuning the micro-wave frequency and are ascribed to heating in-duced by non-resonant absorption of micro-wave power. In contrast, the weak maxima sur-rounding filling 1/2 readily respond to a change in frequency as illustrated in Fig. 46(b). They are symmetrically arranged around half filling and their splitting grows with frequency.
The B-dependence is summarized in Fig. 46(c) for two densities. To underline the symmetry, B was chosen as the abscissa. The linear re-lationship between frequency and field extrapo-lates to zero at vanishing B. We conclude that the resonance in Fig. 46 is the long searched for cyclotron resonance of CFs, ωCFcr = eBmCFcr . The data, shown in Fig. 47, confirm qualita-tively the strong enhancement in comparison with the electron mass (more than 10 times) predicted in theory, however a fit to the square root dependence requires a prefactor that is four times larger.
The slope of the CF cyclotron frequency as a function of B in Fig. 46(c) defines the cy-clotron mass mCFcr . This mass is set by the electron-electron interaction scale, so that a
square root behavior on density or B-field is forecasted from a straightforward dimensional analysis. It varies from 0.7 m0 to 1.2 m0 as the density is tuned from 0.61011cm2 to 1.21011cm2 [I.V. Kukushkin et al., Nature 415, 409 (2002)].
Figure 47: Dependence of the CF effective mass nearν= 1/2 and 1/4 on the carrier densityn2D(solid triangles). The dashed line is a square root fit to the data. The solid curve is a prediction from theory not including finite width of the quasi-two-dimensional electron system and Landau level mixing effects.
In summary, the fortuitous breaking of trans-lational invariance induced by the microwave irradiation combined with the virtues of an optical detection scheme for resonant absorp-tion has enabled to unveil resonances, sym-metrically arranged around ν= 1/2 (and also ν= 1/4), that are most naturally interpreted as the cyclotron resonance of composite fermions.