Abstract
Transient spin grating experiments are used to investigate the electron spin diffusion in intrinsic (110) GaAs/AlGaAs multiple quantum well at room temperature. The measured spin diffusion length of optically excited electrons is about 4 μm at low spin density. Increasing the carrier density yields both a decrease of the spin relaxation time and the spin diffusion coefficient D_{s}.
Introduction
The interest in the spin properties of carriers in semiconductors has increased dramatically in the past 10 years due to potential application in the field of spintronics [1,2]. The design of practical spintronic devices usually requires efficient spin injection in the semiconductor, long carrier spin lifetimes, and long spin transport/diffusion lengths [37].
One of the key parameters describing the properties of carrier spin transport in semiconductors is the spin diffusion coefficient D_{s}, which is often assumed to be the same as charge diffusion coefficient D_{c }[8]. A direct optical measurement of the electron spin diffusion coefficient can be performed by creating electron spin grating in timeresolved fourwave mixing experiments [9]. This powerful transient spin grating (TSG) technique was used recently to study the spin transport properties and determine the spin diffusion coefficient D_{s }[911]. In particular it was demonstrated theoretically and experimentally that the spin diffusion coefficient D_{s }in ndoped (100)grown GaAs quantum wells can be smaller than the charge diffusion coefficient D_{c }due to Coulomb interaction among the electrons (the socalled Spin Coulomb Drag effect) [10,12]. In these (100)grown GaAs quantum wells, the electron spin lifetime is of the order of 100 ps at room temperature (RT) due to very efficient D'yakonovPerel (DP) spin relaxation mechanism [13]. In the classical twocomponent driftdiffusion model [14], the spin diffusion length L_{s }is determined by the spin lifetime and the spin diffusion coefficient D_{s }through . As a consequence, the spin diffusion length L_{s }at RT is smaller than 1 μm, limited by the short spin lifetime [10]. In (110)grown GaAs/AlGaAs QW, the DP spin relaxation mechanism is not efficient for electron spins parallel to the growth direction because the spin orientation of electrons is parallel to the direction of effective magnetic field induced by spinorbit coupling [15]. Spin relaxation times longer than 1 ns at RT in (110) GaAs QW have indeed been measured [16]. Long electron spin diffusion lengths can thus be expected at high temperature in these structures. In this report, the electron spin diffusion is measured by the TSG technique with heterodyne detection in (110) GaAs/AlGaAs QWs at RT. We find that the spin diffusion length L_{s }is about 4 μm at low carrier density. We also demonstrate that the spin diffusion coefficient D_{s }decreases when the carrier density increases.
Experimental procedure
The investigated sample was grown on (110)oriented semiinsulating GaAs substrate by molecular beam epitaxy. It consists of 20 planes of 8 nm thick GaAs QW with symmetric 27 nm Al_{0.28}Ga_{0.72}As barriers on both sides. The sample is nominally undoped. All the measurements are performed at RT. In the spin grating experiment, the laser pulses are generated by a modelocked Ti:sapphire laser with 120 fs pulse duration and 76 MHz repetition frequency and split into primary pump and probe beams. The center wavelength is set to 830 nm to get the maximum signal of Kerr rotation through the standard timeresolved Kerr rotation technique [17]. Both pump and probe beams are focused on a phase mask with a period d. The phase mask splits each of the primary beams by diffraction into the m = ± 1 orders. The geometry of the spin grating experiment in the socalled box geometry is schematically presented in Figure 1a[18,19]. For orthogonallinearly polarized pumps, the net polarization alternates between right and left circular polarization across the excitation spot while the total intensity of the incident light is uniform [9]. The period Λ of the TSG is simply: , where f_{1 }and f_{2 }are the focal lengths of two spherical mirrors. In our setup, the focal length of the first spherical mirror is fixed at f_{1 }= 30.4 cm. The focal length f_{2 }of the second spherical mirror can be changed to get a fine tuning of the period Λ. The spot sizes of both pump and probe beams are around 90 μm.
Figure 1. Schematic drawing of TSG setup and TSG signals. (a) k_{A }and k_{B }represent both the pump beams, k_{P }is the probe beam, and k_{R }is the reference beam. (b) TSG signal as a function of delay time at room temperature for two excitation powers: 2 and 18 mW.
According to the optical interband selection rules, this interference pattern will generate a periodical spin density in the sample. The delayed probe beam, diffracted from the grating, is monitored as a function of the delay time between the pump and the probe. In order to enhance the signaltonoise ratio, a reference beam is incident on the sample and its reflected beam is automatically collinear with the refracted probe beam. In this configuration, the spin grating signal (i.e., proportional to the electric field of the diffracted probe beam) is simply given by:
where A is a constant, Γ_{s }is the decay rate of the spin grating, and Δt is the delay time between pump and probe beams.
Results and discussion
Figure 1b presents the signal of TSGs as a function of the time delay for two typical pump powers, 2 and 18 mW, respectively. The wave vector q of the spin grating is equal to . It is clear that both curves exhibit different monoexponential decays. Using equation (1), we find Γ_{s }= 0.063 and 0.044 ps^{1 }for the pump powers 2 and 18 mW, respectively.
In the diffusion regime, the SG decay rate writes [8,9]:
where D_{s }is the spin diffusion coefficient, q is the spin grating wave vector, and is the spin lifetime which includes the effect of both the electron spin relaxation time τ_{s }and the recombination time τ_{r, }as expressed by . To separate the effects of spin diffusion and spin relaxation, the grating decay rate is measured as a function of the grating wave vector q by changing the phase mask with different periods (d = 5, 6, 7, and 8 μm) and/or the second spherical mirror with different focus lengths (f_{2 }= 15.2 and 30.4 cm). Figure 2a shows the grating decay rate as a function of q^{2 }for two excitation powers. Each set of data points can be fitted linearly, yielding the spin diffusion coefficient D_{s}. At low excitation power of 2 mW, which corresponds to an optical intensity of 30W/cm^{2}, we find D_{s }= ~102 cm^{2}/s. This value is in good agreement with the values obtained by other groups in (110)grown GaAs/AlGaAs QWs at RT [8,20]. It is also very close to the spin diffusion coefficient D_{s }measured in (100)grown GaAs/AlGaAs QWs at RT [9,10]. This result suggests that the spin diffusion coefficient D_{s }does not depend critically on the spinorbit coupling, which depends on the crystalline direction of the sample. Nevertheless, as shown in Figure 2a, it is very sensitive to the carrier density.
Figure 2. Spin diffusion coefficient and spin dynamics for two different powers. (a) Decay rate of spin grating as a function of q^{2 }for two excitation powers: 2 and 18mW. (b) Kerr rotation dynamics obtained from homogenous spin excitation.
In order to obtain the spin diffusion length L_{s}, the spin lifetime is measured independently by timeresolved Kerr rotation [17]. The excitation powers are the same as the ones used in the measurement of TSG. Figure 2b presents the Kerr rotation dynamics for two excitation powers. The spin lifetimes are extracted by monoexponential fits, which yield ~1220 ps and ~880 ps with excitation powers of 2 and 18 mW, respectively. As expected for (110)grown QWs, the spin lifetimes for both excitation powers are much longer than the ones ( ~ 50100 ps) measured in (100)grown GaAs/AlGaAs QWs at RT [9]. By combining the D_{s }measurement obtained with the spin grating technique and the electron spin lifetime probed by the Kerr rotation experiment, we find that the spin diffusion length decreases from L_{s }~ 3.5 μm down to 2.2 μm when the excitation power increases from 2 to 18 mW. To the best of our knowledge, these values are the longest electron spin diffusion lengths reported at room temperature in inorganic semiconductors.
In order to get further insights on this power dependence, we also measured the charge diffusion coefficient D_{c }with a concentration grating technique for different pump powers. We find that D_{c }remains constant with a typical value D_{c }~ 12.5 cm^{2}/s (data not shown here). This value is in good agreement with previous studies performed in nonintentionally doped (100)grown GaAs QWs which demonstrate that the concentration grating experiments are governed by the hole diffusion [9].
Our spin diffusion coefficient results obtained at RT on (110) QWs contrast with the previous measurements of the carrier density dependence of the spin diffusion obtained at low temperature in ndoped bulk GaAs or (100) quantum wells [11,21]. In ndoped QWs, Carter et al. observed that D_{s }increases by increasing the density of the optically excited carriers. This increase of the electron spin diffusion coefficient was interpreted in terms of heating of the excess electrons due to relaxation of energetic optically excited carriers. Remarkably, in nonintentionally doped GaAs (110)grown QWs, we observe at room temperature the opposite behavior. As displayed in Figure 3a, the spin diffusion coefficient D_{s }decreases abruptly for a pump power varying between 2 and 10 mW, and then remains almost coefficient up to 40 mW. In the same power range the spin lifetime (Figure 3b) has a different power dependence: it decreases monotonously as already observed by different groups, due to electron spin relaxation enhancement by the electronhole exchange interaction [16]. Since the sample was undoped, we can equate the electron spin diffusion coefficient D_{s }to the electron charge diffusion coefficient D_{e}. The spin diffusion coefficient D_{s }can thus be written [22]:
Figure 3. Powerdependence spin diffusion coefficient and spin lifetime. (a) Spin diffusion coefficient D_{s }versus pump power, i.e., spin density; the blue line is a simple fit according to . (b) Pump powerdependent spin lifetime through Kerr rotation measurement with a fixed probe power of 0.2 mW.
where <v^{2}> is the mean square velocity of electrons and τ_{p }is the momentum relaxation time. In a very simple approach, <v^{2}> in a QW can be approximated by . The momentum relaxation τ_{p }is strongly dependent on the density of photogenerated electrons n_{e}, with a typical power law [23]. In the low density regime below 2.5 × 10^{10 }cm^{2}, which corresponds to a pump power of 10 mW, the experimental data are well fitted by this power law as shown by the blue line in Figure 3a. In the high density regime above 2.5 × 10^{10 }cm^{2}, the spin diffusion coefficient is almost constant and the density dependence can no more be interpreted by the simple power law. In this density range, the above discussion is clearly oversimplified and we hope that these experimental results will stimulate theoretical investigations to elucidate the origin of the carrier density dependence of the spin diffusion coefficient.
Conclusions
We have measured optically the spin diffusion coefficient D_{s }in nonintentionally doped GaAs/AlGaAs (110) QWs at room temperature for different excitation powers. Under low excitation, the electron spin diffusion length L_{s }is around 4 μm; to the best of our knowledge, this is the largest reported value at T = 300 K in IIIV semiconductors. We also show that the spin diffusion coefficient of optically excited electrons decreases when the excitation density increases. These results could be useful to understand the spin transport properties in semiconductor structures, and possibly control/manipulate the spin transport by varying the excitation condition.
Abbreviations
DP: D'yakonovPerel; TSG: transient spin grating.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
CC, BL conceived and designed the experiments. CC, HQ carried out the experiments with contribution from GW and WQW. WXW and HT provided the samples. BL and XM supervised the work. CC, BL and XM wrote the manuscript. All authors read and approved the final manuscript.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Acknowledgements
We thank MingWei WU for useful discussions. We acknowledge the financial support of this study from National Science Foundation of China, Grant number: 10534030, 10774183, 10911130356, 10874212; also supported by Ministry of Finance and Chinese Academy of Sciences, National Basic Research Program of China (2006CB921300, 2009CB930500), the ANR project SpinMan.
References

Žutić I, Fabian J, Das Sama S: Spintronics: fundamentals and applications.

Datta S, Das B: Electronic analog of the electrooptic modulator.
Appl Phys Lett 1990, 56:665. Publisher Full Text

Kikkawa JM, Smorchkova IP, Samarth N, Awschalom DD: Roomtemperature spin memory in twodimensional electron gases.
Science 1997, 277:1284. Publisher Full Text

Vörös Z, Balili R, Snoke DW, Pfeiffer L, West K: Longdistance diffusion of excitons in double quantum well structures.
Phys Rev Lett 2005, 94:226401. PubMed Abstract  Publisher Full Text

Hägele D, Oestreich M, Rühle WW, Nestle N, Eberl K: Spin transport in GaAs.

Liu BL, Senes M, Couderc S, Bobo JF, Marie X, Amanda T, Fontaine C, Arnoult A: Optical and electrical spin injection in spinLEDs.
Physica E 2003, 17:358.
Zhu HJ, Ramsteiner M, Kostial H, Wassermeier M, Schönherr HP, Ploog KH: Roomtemperature spin injection from Fe into GaAs. Phys Rev Lett 2001, 87:016601
Publisher Full Text 
Couto ODD Jr, Iikawa F, Rudolph J, Hey R, Santos PV: Anisotropic spin transport in (110) GaAs quantum wells.
Phys Rev Lett 2007, 98:036603. PubMed Abstract  Publisher Full Text

Eldridge PS, Leyland WJH, Lagoudakis PG, Karimov OZ, Henini M, Taylor D, Phillips RT, Harley RT: Alloptical measurement of Rashba coefficient in quantum wells.
Phys Rev B 2008, 77:125344. Publisher Full Text

Cameron AR, Riblet P, Miller A: Spin gratings and the measurement of electron drift mobility in multiple quantum well semiconductors.
Phys Rev Lett 1996, 76:4793. PubMed Abstract  Publisher Full Text

Weber CP, Gedik N, Moore JE, Orenstein J, Stephens J, Awschalom DD: Observation of spin Coulomb drag in a twodimensional electron gas.
Nature 2005, 437:1330. PubMed Abstract  Publisher Full Text

Carter SG, Chen Z, Cundiff ST: Optical measurement and control of spin diffusion in ndoped GaAs quantum wells.
Phys Rev Lett 2006, 97:136602. PubMed Abstract  Publisher Full Text

D'Amico I, Vignale G: Theory of spin Coulomb drag in spinpolarized transport.

D'yakonov MI, Perel VI: Spin relaxation of conduction electrons in noncentrosymmetric semiconductors.

Zhang P, Wu MW: Spin diffusion in Si/SiGe quantum wells: Spin relaxation in the absence of D'yakonovPerel' relaxation mechanism.
Phys Rev B 2009, 79:075303. Publisher Full Text

D'Yakonov MI, Kachorovskii VYu: Spin relaxation of twodimensional electrons in noncentrosymmetric semiconductors.

Lombez L, Lagarde D, Renucci P, Amand T, Marie X, Liu BL, Wang WX, Xue QK, Chen DM: Optical spin orientation in (110) GaAs quantum wells at room temperature.
Phys Status Solidic 2007, 4:475.
Ohno Y, Terauchi R, Adachi T, Matsukura F, Ohno H: Spin relaxation in GaAs(110) quantum wells. Phys Rev Lett 1999, 83:4196; Döhrmann S, Hägele D, Rudolph J, Bichler M, Schuh D, Oestreich M: Anomalous spin dephasing in (110) GaAs quantum wells: Anisotropy and intersubband effects. Phys Rev Lett 2004, 93:147405
Publisher Full Text 
Liu BL, Zhao HM, Wang J, Liu LS, Wang WX, Chen DM: Electron density dependence of inplane spin relaxation anisotropy in GaAs/AlGaAs twodimensional electron gas.
Appl Phys Lett 2007, 90:112111. Publisher Full Text

Hu CC, Wang G, Ye HQ, Liu BL: Development of the transient spin grating system and its application in the study of spin transport.

Maznev AA, Nelson KA, Rogers JA: Optical heterodyne detection of laserinduced gratings.
Opt Lett 1998, 23:1319.
Gedik N, Orenstein J: Absolute phase measurement in heterodyne detection of transient gratings. Opt Lett 2004, 29:2109
PubMed Abstract  Publisher Full Text 
Eldridge PS, Leyland WJH, Mar JD, Lagoudakis PG, Winkler R, Karimov OZ, Henini M, Taylor D, Phillips RT, Harley RT: Absence of the Rashba effect in undoped asymmetric quantum wells.
Phys Rev B 2010, 82:045317. Publisher Full Text

Quast JH, Astakhov GV, Ossau W, Molenkamp LW, Heinrich J, Höfling S, Forchel A: Lateral spin diffusion probed by twocolor HanleMOKE technique.

Weng MQ, Wu MW, Cui HL: Spin relaxation in ntype GaAs quantum wells with transient spin grating.
J Appl Phys 2008, 103:063714. Publisher Full Text

Bigot JY, Portella MT, Schoenlein RW, Cunningham JE, Shank CV: Twodimensional carriercarrier screening in a quantumwell.
Phys Rev Lett 1991, 67:636. PubMed Abstract  Publisher Full Text