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Resonant spin ampliﬁcation of hole spin dynamics in two-dimensional hole systems: experiment and simulation T. Korn∗, M.

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Resonant spin amplification of hole spin dynamics in twodimensional hole systems: experiment and simulation T. Korn, M. Kugler, M. Hirmer, D. Schuh, W. Wegscheider et al. Citation: AIP Conf. Proc. 1399, 645 (2011); doi: 10.1063/1.3666542 View online: http://dx.doi.org/10.1063/1.3666542 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1399&Issue=1 Published by the American Institute of Physics.

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Resonant spin ampliﬁcation of hole spin dynamics in two-dimensional hole systems: experiment and simulation T. Korn∗ , M. Kugler∗ , M. Hirmer∗ , D. Schuh∗ , W. Wegscheider† and C. Schüller∗ ∗

Institut für Experimentelle und Angewandte Physik, Universität Regensburg, D-93040 Regensburg, Germany † Solid State Physics Laboratory, ETH Zürich, 8093 Zurich, Switzerland

Abstract. Spins in semiconductor structures may allow for the realization of scalable quantum bit arrays, an essential component for quantum computation schemes. Speciﬁcally, hole spins may be more suited for this purpose than electron spins, due to their strongly reduced interaction with lattice nuclei, which limits spin coherence for electrons in quantum dots. Here, we present resonant spin ampliﬁcation (RSA) measurements, performed on a p-modulation doped GaAs-based quantum well at temperatures below 500 mK. The RSA traces have a peculiar, butterﬂy-like shape, which stems from the initialization of a resident hole spin polarization by optical orientation. The combined dynamics of the optically oriented electron and hole spins are well-described by a rate equation model, and by comparison of experiment and model, hole spin dephasing times of more than 70 ns are extracted from the measured data. Keywords: Optical orientation, hole spin dynamics, GaAs heterostructures PACS: 78.67.De, 78.55.Cr

Spin dynamics in semiconductor hetero- and nanostructures based on the GaAs material system have been studied intensely in recent years, driven by possible applications in the ﬁelds of semiconductor spintronics [1, 2] and quantum information processing [3]. However, electron spin dynamics have been investigated in many more studies than hole spin dynamics. One of the reasons is the rapid dephasing of hole spins in bulk GaAs: here, the heavy-hole (HH) and light-hole (LH) valence bands, which have different angular momentum, are degenerate at k = 0, hence any momentum scattering can lead to hole spin dephasing [4]. This degeneracy is lifted in quantum well systems due to the different conﬁnement energies for light and heavy holes. Due to valence band mixing for k > 0, which again allows for hole spin relaxation during momentum scattering [5], long-lived hole spin coherence has only been observed at low temperatures [6, 7, 8], where holes can become localized at quantum well thickness ﬂuctuations. In our study of hole spin dynamics, we use the resonant spin ampliﬁcation (RSA) technique [9]. In RSA, the interference of spin polarizations created by subsequent laser pulses leads to pronounced maxima in the Faraday rotation angle, which is probed at a ﬁxed time delay between pump and probe pulses, as a function of an in-plane magnetic ﬁeld. Our sample is a 4 nm wide p-modulation-doped GaAs quantum well (QW) embedded in Al0.3 Ga0.7 As barriers, with a doping density p = 1.1 × 1011 cm−2 . For measurements in transmission, the sample is glued to a sapphire support and thinned by selective etching to remove the substrate and leave only the MBE-grown layers. It is mounted in an optical cryostat

with a 3 He insert, allowing for sample temperatures below 500 mK and magnetic ﬁelds of up to 10 Tesla in the sample plane. Optical experiments are performed using a pulsed Ti-Sapphire laser system, details of the experiment are published elsewhere [10]. Figure 1 (a) shows a typical RSA trace measured at 400 mK (open dots), compared to simulation (solid line). No RSA maximum is observed in zero ﬁeld, and for ﬁnite ﬁelds, the amplitude of the RSA maxima ﬁrst increases, then decreases again, leading to a butterﬂy-like shape of the signal, which stems from the complex initialization process of a resident hole spin polarization using optical orientation [10]. The combined dynamics of the optically oriented electrons and holes, as well as the resident holes in the QW, are modelled using two coupled differential equations: de dt dh dt

e ge μB (B × e) + h¯ τR h ez z gh μB = − ∗+ (B × h) + T2 τR h¯ =

−

(1) (2)

Here, e and h are the electron and hole spin polarization vectors, τR is the photocarrier recombination time, T2∗ is the hole spin dephasing time (SDT), ge and gh are the electron and hole g factors. Numerical simulation using these equations allows us to model the RSA measurements precisely and capture all salient features (see simulated trace in Figure 1 (a)), if all relevant parameters are carefully tuned to the measurement. The electron g factor can be precisely measured using timeresolved Faraday rotation (TRFR), and the hole g factor is extracted from the spacing ΔB of the RSA max-

Physics of Semiconductors AIP Conf. Proc. 1399, 645-646 (2011); doi: 10.1063/1.3666542 © 2011 American Institute of Physics 978-0-7354-1002-2/$30.00

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FIGURE 1. (a) Experimental RSA trace measured at 400 mK (black circles) and simulated trace (solid line). (b) FWHM of the RSA maxima extracted from the measurement as a function of magnetic ﬁeld (black dots) and hole SDT extracted from these values (red stars).

ima: gh = h frep /(μB ΔB), where frep is the laser repetition frequency. Below, we will demonstrate how the hole SDT can be extracted from the full width at half maximum (FWHM) of the RSA maxima. The FWHM of the measured RSA maxima (as deﬁned in Figure 2 (a)) increases monotonously with the applied magnetic ﬁeld, as Figure 1 (b) shows. This corresponds to a signiﬁcant decrease of the hole SDT, which is caused by the hole g factor inhomogeneity,Δg∗h . The inhomogeneous hole spin ensemble dephases more rapidly with increasing precession frequency [11]: 1 2π Δg∗h μB B −1 T2∗ = + . (3) T2 h This decrease of the measured ensemble hole SDT with magnetic ﬁeld makes it difﬁcult to extract the single hole spin coherence time T2 from high-ﬁeld measurements like TRFR. In order to investigate the correlation between the RSA maxima FWHM and the hole SDT T2∗ , we simulate RSA traces using hole SDT calculated with equation 3. Figure 2 (a) shows such a trace for T2 = 80 ns, Δg∗h = 0.0025. The FWHM of the RSA peaks, starting at the second peak, is extracted from this trace. In Figure 2 (b), we plot the inverse square of the FWHM as a function of magnetic ﬁeld (black dots). In the same ﬁgure, the hole SDT T2∗ is plotted as a function of magnetic ﬁeld (blue circles), calculated using equation 3 with the same parameters as the RSA trace. Both sets of data show the same, 1/B-like dependence on magnetic ﬁeld. This allows us to determine the proportionality constant α = 7.36 × 10−3 ns T2 , T2∗ = α /(FWHM)2 . Using this value of α , the hole SDT data shown in Figure 1 (b) are calculated. We observe a maximum hole SDT T2∗ = 74 ± 15 ns for a magnetic ﬁeld of B≈ 0.2 Tesla. This represents a lower bound for the single hole spin coherence time T2 . From the decay

FIGURE 2. (a) Simulated RSA trace. The deﬁnition of the FWHM is indicated by the grey lines. (b) Comparison of 1/(FWHM)2 of the simulated RSA maxima (black dots) to the hole spin dephasing time T2∗ (blue circles) used in the simulation as a function of magnetic ﬁeld.

of the hole SDT with magnetic ﬁeld, we ﬁnd a g factor inhomogeneity Δg∗h = 0.003. In conclusion, we have investigated hole spin dynamics in a p-doped QW by using the RSA technique. The RSA traces are closely modelled by a system of differential equations. By analysis of the simulated RSA traces, we demonstrate that the hole SDT, T2∗ , at a given magnetic ﬁeld is inversely proportional to the square of the FWHM of the RSA peak at that magnetic ﬁeld. Using this relation, we extract hole SDTs in excess of 70 ns from the measured RSA traces.

ACKNOWLEDGMENTS The authors would like to thank E.L. Ivchenko, M.M. Glazov and M.W. Wu for fruitful discussion. Financial support by the DFG via SPP 1285 and SFB 689 is gratefully acknowledged.

REFERENCES 1. 2. 3.

J. Fabian, et al., Acta Physica Slovaca 57, 565 (2007). M. W. Wu, et al., Physics Reports, in press (2010). D. Awschalom, D. Loss, and N. Samarth, Semiconductor Spintronics and Quantum Computation, Springer, 2002. 4. D. J. Hilton, et al., Phys. Rev. Lett. 89, 146601 (2002). 5. T. C. Damen, et al., Phys. Rev. Lett. 67, 3432 (1991). 6. B. Baylac, et al., Solid State Communications 93, 57 (1995). 7. M. Syperek, et al., Phys. Rev. Lett. 99, 187401 (2007). 8. M. Kugler, et al., Phys. Rev. B 80, 035325 (2009). 9. J. M. Kikkawa, et al.,Phys. Rev. Lett. 80, 4313 (1998). 10. T. Korn, et al.,New J. Phys. 12, 043003 (2010). 11. D. Yakovlev, and M. Bayer, Spin Physics in Semiconductors, Springer,Berlin, 2008, p. 135.

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