Dynamics of the collective excitations of the quantum Hall system

Dynamics of the collective excitations of the quantum Hall system

ARTICLE IN PRESS Physica E 34 (2006) 206–209 www.elsevier.com/locate/physe Dynamics of the collective excitations of the quantum Hall system Keshav ...

280KB Sizes 2 Downloads 71 Views

ARTICLE IN PRESS

Physica E 34 (2006) 206–209 www.elsevier.com/locate/physe

Dynamics of the collective excitations of the quantum Hall system Keshav M. Dania,b,, Jerome Tignonc, Michael Breitb, Daniel S. Chemlaa,b, Eleftheria G. Kavousanakid, Ilias E. Perakisd a Department of Physics, University of California at Berkeley, Berkeley, CA 94720, USA Materials Sciences Division, E.O. Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA c Laboratoire Pierre Aigrain, Ecole Normale Supe´rieure, F-75005 Paris, France d Department of Physics, University of Crete and Institute of Electronic Structure and Laser, Foundation of Research and Technology, -Hellas, Heraklion, Crete, Greece

b

Available online 27 April 2006

Abstract Using the non linear optical technique of 3-pulse 4-wave mixing, we study the dynamics of the collective excitations of the quantum Hall system. We excite the system with 100 fs pulses propagating in directions k1 and k3 and then probe its time evolution with a delayed pulse k2. We measure the non-linear optical response from the lowest Landau level along the direction k1+k2k3. As function of the time delay of pulse k2, this signal shows striking beats for short time delays (500 fs), followed by a rise (20 ps) and then a decay (100 ps). We identify the microscopic origin of this dynamics by extending the standard theory of ultra fast nonlinear optics to include the effects of the correlations. r 2006 Published by Elsevier B.V. PACS: 78.66.Fd; 73.20.Dx Keywords: Non linear spectroscopy; Quantum wells; Two-dimensional electron gas

A 2-dimensional electron gas (2DEG) in a large perpendicular magnetic field exhibits quantum Hall (QH) effects [1]. The strongly correlated ground state of this system is an incompressible electron fluid with zero longitudinal resistance under certain conditions. Transport and optical experiments [2,3] have accessed the collective excitations of the QH system, such as the magnetoplasmon and the magnetoroton states [4,5]. However, they are unable to access the dynamics of these excited quasiparticles. On the other hand, ultrafast non linear spectroscopy is well suited for studying the non equilibrium Coulomb correlation effects that occur during time-scales shorter than the dephasing times of the optical coherences [6]. Four-wave mixing (FWM) non linear spectroscopy has been used extensively to study the ultrafast response of undoped semiconductors [7]. Here, the photoexcited quasiparticles are electron–hole pairs between the conduction and valence bands [8] that bind to form excitons. The Corresponding author. Tel.: +1 510 486 5265; fax: +1 510 486 6695.

E-mail address: [email protected] (K.M. Dani). 1386-9477/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.physe.2006.03.113

signatures of strong exciton–exciton correlations were identified by analyzing the FWM profile [8]. Recently, 2pulse FWM experiments were used to study for the first time the ultra fast response of the QH system [9–12]. Unlike undoped semiconductors, the response is determined by the interactions between the photoexcited carriers and the 2DEG. The 2-pulse FWM experiments revealed an unusual temporal and spectral behavior due to such interactions [9–12]. However, they do not directly access the time evolution of the 2DEG itself. Here we use 3-pulse FWM experiments to study the evolution of the 2DEG in response to photoexcitation. In a 2-pulse FWM experiment, two optical pulses, separated by a time delay, are focused on the sample in different directions (k1 and k3 in Fig. 1a). The third-order optical response is diffracted in the background-free direction 2k1k3. In a 3-pulse FWM experiment, three pulses are focussed on the sample in directions k1, k2 and k3 (Fig. 1a). The third-order optical response is now diffracted in the direction k1+k2k3. Pulses k1 and k2 (k3) are separated by a time delay Dt12 (Dt13), where pulse k1

ARTICLE IN PRESS K.M. Dani et al. / Physica E 34 (2006) 206–209

Fig. 1. (a) Experimental scheme: pulses k1, k2 and k3 excite the sample. The non-linear response is in the directions 2k1k3 (2-pulse FWM) and k1+k2k3 (3-pulse FWM). The time delays between pulses k1 and k2 (k3) are Dt12 (Dt13). k1 is first for negative delay. (b) Interaction process for negative Dt12 axis: k1 and k3 arrive simultaneously and create an MP coherence, which evolves for |Dt12| and is then probed by k2 to give the 3pulse FWM signal. (c) Interaction process for positive Dt13 axis: k3 arrives first to create a LL1 polarization, which evolves for |Dt13|; k1 and k2 then arrive simultaneously and interact with the LL1 polarization via the creation and annihilation of a magnetoplasmon to give the 3-pulse FWM signal.

arrives first for negative values of the delay. We measure the time integrated (TI) FWM signal, i.e. the average power of this third-order response, as a function of the delays Dt12 and Dt13. The signal along the Dt13 axis (Dt12 ¼ 0) corresponds to 2-pulse FWM experiments, while the signal along the Dt12 axis provides new information. To see this, let us first recall the FWM signal of a two-level system. For positive Dt13 and Dt12 ¼ 0, k3 arrives first and creates an inter-band polarization that decays on a timescale T2. Pulses k1 and k2 arrive simultaneously Dt13 later and diffract off the decaying polarization to give the FWM signal. The decay of the TI FWM signal as a function of Dt1340 with Dt12 ¼ 0 thus gives the polarization dephasing time T2. On the other hand, when measuring along the negative Dt12 axis (Dt13 ¼ 0), the two pulses k1 and k3 arrive simultaneously to create a population grating, which

207

relaxes on a time-scale T1. When the third pulse k2 arrives at a later time 9Dt129, it diffracts off the decaying population grating in direction k1+k2k3. Thus, the TI FWM signal as a function of Dt12o0 with Dt13 ¼ 0 gives the T1 relaxation time of the photoexcited carrier population. Hence, measuring along both axes provides complementary information. In the case of the QH system, one still measures the polarization dephasing time along the positive Dt13 axis. However, the FWM signal along the negative Dt12 axis also gives access to the evolution of the 2DEG. We investigated a modulation-doped QW structure consisting of 10 periods of 12 nm GaAs wells and 42 nm AlGaAs barrier layers with Si doped at their centers. The 2DEG density due to this doping is n ¼ 2.1  1011 cm2. We used 100 fs pulses with sufficiently low intensity so that the density of photoexcited carriers (5  109 cm2) is much smaller than the 2DEG density. The sample was kept at 4 1K in a split-coil magneto-optical-cryostat. A field of 7 T perpendicular to the sample set the filling fraction to 1.3. Thus, in the ground state, all electrons reside in the lowest Landau level (LL0) with most of their spins pointing up. With s+ light, we excite only spin-down electrons. We suppress the Pauli blocking effects of the photoexcited carrier populations by tuning the laser to excite largely into the second Landau level (LL1) and measure the FWM signal at the LL0 energy. The LL0 signal is then dominated by interactions with the 2DEG. Such interaction processes are shown in Figs. 1b and c for measurements along the negative Dt12 axis and the positive Dt13 axis, respectively. In Fig. 1b, the two pulses k1 and k3 arrive together to create optical polarizations in the sample at LL0 and LL1, respectively. Via the Coulomb interaction, the two polarizations interact with the 2DEG, which evolves by exciting a long wavelength magnetoplasmon (MP) coherence. This coherence oscillates with the MP frequency, close to resonance with the inter LL energy spacing. When the third pulse k2 arrives a short while |Dt12| later, it creates an optical polarization that interacts with the MP coherence and governs the third-order FWM signal at LL0. Thus the TI FWM signal from LL0 along the negative Dt12 axis reflects the evolution of the MP coherence of the excited 2DEG. Fig. 2a shows the TI FWM signal along the Dt12 axis for short time delays (o1 ps) from both LL1 and LL0. In spite of the low LL0 excitation (LL1:LL0420:1 as back panel shows, excitation pulse in red and the linear absorption in yellow), one sees an unexpectedly strong signal from LL0 as compared to LL1. Under identical excitation of a similar undoped quantum well (not 2DEG), almost no LL0 signal is observed. This confirms that the transfer of signal from LL1 to LL0 is due to 2DEG effects. In addition to the transfer of signal strength, one sees striking oscillations from LL0 but not from LL1. These oscillations are close to the inter-LL spacing and show a linear dependence on the magnetic field. Additionally, as the density of the photoexcited carriers approaches the 2DEG density, these

ARTICLE IN PRESS 208

K.M. Dani et al. / Physica E 34 (2006) 206–209

Fig. 2. (a) TI FWM signal from LL1 and LL0 along Dt12 axis for short time delays. We have large LL1 excitation as compared to LL0 (see back panel, excitation pulse in red and linear absorption spectrum showing both LLs). (b) TI FWM signal from LL0 along Dt12 axis for increasing total photoexcitation.

Fig. 3. (a) TI FWM signal from LL1 and LL0 along Dt12 axis for large time delays. We have large LL1 excitation as compared to LL0 (LL1:LL0 15:1) similar to conditions in Fig. 2. (b) TI FWM signal from LL0 along Dt12 axis for long time delays for large and small LL1:LL0 excitation.

oscillations disappear (Fig. 2b). Differences in the ultrafast reponse between the doped and undoped QWs diminish with increasing photoexcitation intensity. We also studied the TI FWM signal along the Dt12 axis for large time delays. One sees a slow rise in the FWM signal from both LL0 and LL1 that lasts for several picoseconds before slowly decaying over 100 ps (Fig. 3a). This slow rise occurs only for low photoexcitation intensities. In comparable measurements in the undoped QW, one does not see such a rise. Also, if we excite just LL0 in the doped system, the TI FWM signal for large time delays is similar to that of a two-level system (Fig. 3b). We analyzed the experiment in the coherent regime using a model extracted from a recent theory [13] that extends the ‘dynamics-controlled truncation scheme’ approach to the case of a strongly correlated ground state. In addition to terms similar to an undoped QW, we included the contributions due to the MP-induced intra-band coherences between the ground state and the MP states as well as

between exciton and MP+exciton states. Our calculations, to be published in detail elsewhere, show that the interference of the contributions to the nonlinear optical response from these various MP related coherences leads to the coherent oscillations. The slow rise over several picoseconds is due to incoherent exciton populations and their MP-mediated scattering from LL1 to LL0. In conclusion, we used FWM non-linear optical spectroscopy to directly access the evolution of the photoexcited quantum Hall system. We see unusual spectral and temporal features in the non linear response that can be attributed to the dynamics and interactions of the magnetoplasmon 2DEG excitations. We thank N. Fromer and S. Cundiff for useful discussions. This work was supported by the Office of Basic Energy Sciences of the US Department of Energy, the FranceBerkeley Foundation, and by the EU Research Training Network HYTEC.

ARTICLE IN PRESS K.M. Dani et al. / Physica E 34 (2006) 206–209

References [1] See articles in: S. Das Sarma, A. Pinczuk, (Eds.), Perspectives Quantum Hall Effects, Wiley, New York, 1997. [2] A. Pinczuk, et al., Phys. Rev. Lett 70 (1993) 3938. [3] J.P. Eisenstein, Solid State Commun 117 (2001) 123. [4] S.M. Girvin, et al., Phys. Rev. Lett. 54 (1985) 581. [5] A.H. MacDonald, et al., Phys. Rev. Lett. 55 (1985) 2211. [6] D.S. Chemla, et al., Nature 411 (2001) 549.

209

[7] D.S. Chemla In: R.K. Willardson, A.C. Beers, (Eds.), Nonlinear Optics in Semiconductors, Academic Press, New York, 1999. [8] W. Schafer, M. Wegener, Optics and Transport Phenomena, Springer, Berlin, 2002. [9] N.A. Fromer, et al., Phys. Rev. Lett. 89 (2002) 067401. [10] N.A. Fromer, et al., Phys. Rev. B 66 (2002) 205314. [11] N.A. Fromer, et al., Phys. Rev. Lett. 83 (1999) 4646. [12] T. Karathanos, et al., Phys. Rev. B 67 (2003) 035316. [13] I.E. Perakis, et al., Chem. Phys. 318 (2005) 118.