Spin noise spectroscopy in GaAs (110) quantum wells: Access to intrinsic spin lifetimes and equilibrium electron dynamics
Georg M. Müller, Michael Römer, Dieter Schuh, Werner Wegscheider, Jens Hübner, Michael Oestreich
aa r X i v : . [ c ond - m a t . o t h e r] S e p Spin noise spectroscopy in GaAs (110) quantum wells:Access to intrinsic spin lifetimes and equilibrium electron dynamics
Georg M. M¨uller, ∗ Michael R¨omer, Dieter Schuh, Werner Wegscheider, Jens H¨ubner, and Michael Oestreich Institut f¨ur Festk¨orperphysik, Leibniz Universit¨at Hannover, Appelstraße 2, 30167 Hannover, Germany Institut f¨ur Experimentelle und Angewandte Physik,Universit¨at Regensburg, 93040 Regensburg, Germany (Dated: November 21, 2018)In this letter, the first spin noise spectroscopy measurements in semiconductor systems of reducedeffective dimensionality are reported. The non-demolition measurement technique gives access to theotherwise concealed intrinsic, low temperature electron spin relaxation time of n-doped GaAs (110)quantum wells and to the corresponding low temperature anisotropic spin relaxation. The Brownianmotion of the electrons within the spin noise probe laser spot becomes manifest in a modificationof the spin noise line width. Thereby, the spatially resolved observation of the stochastic spinpolarization uniquely allows to study electron dynamics at equilibrium conditions with a vanishingtotal momentum of the electron system.
The dream of spin quantum information processingand spin based optoelectronic devices drives the currentintense research on spin physics in semiconductors. Es-pecially GaAs quantum wells (QW) with their quanti-zation axis oriented in (110)-direction attract great at-tention since the electron spin relaxation times in thesestructures are extremely long even at room temperature.In bulk semiconductors with zinc blende structure, likeGaAs, the spin dephasing of electrons is dominated overa wide temperature and doping range by the Dyakonov-Perel (DP) mechanism [1]: The lack of crystal inversionsymmetry leads to a precession of the electron spin inthe wave vector dependent Dresselhaus field B ( k ) [2]. In(110) oriented GaAs QWs, the in-plane components of B ( k ) vanish to all orders in the quasi momentum due tothe special crystallographic symmetry and the quantumconfinement in growth direction [3]. Thereby, the life-time τ z of electron spins aligned in growth direction ofthe (110) QW is considerably larger than in a (100) QWas experimentally shown first by Ohno et al. [4]. The spinrelaxation time is not infinite but limited at high temper-atures by intersubband electron scattering induced spinrelaxation (ISR) [5]. At low temperatures, the efficiencyof ISR is negligible but the well known Bir Aronov Pikus(BAP) mechanism obviates in nearly all photolumines-cence based experiments the observation of long τ z . TheBAP mechanism is caused by the spin interaction of thephoto-created holes with the electrons in the conductionband which increases with decreasing temperature, i.e.,photoluminescence (PL) measurements yield shorter τ z with decreasing temperature. The BAP mechanism canbe avoided in (110) QWs by spatially separating electronand holes by surface acoustic waves [6]. However, theinfluence of the surface acoustic waves on τ z is not clearyet. In other words, the intrinsic, undisturbed τ z at lowtemperatures is unknown in (110) QWs.In this letter, we will show that the intrinsic, low tem-perature τ z in modulation n-doped GaAs (110) QWs canbe measured by the non-demolition measurement tech- nique of spin noise spectroscopy and that the intrinsic τ z is by more than one order of magnitude longer thanmeasured by Ohno et al. [4] and D¨ohrmann et al. [5]by time resolved PL. Spin noise spectroscopy (SNS) isa measurement technique known in quantum optics [7]and has been transferred to semiconductor systems justrecently [8, 9]. In SNS, the statistical fluctuations of thespin polarization are mapped via Faraday rotation ontothe light polarization of a linear polarized, continuous-wave laser. The temporal dynamics of the spin fluctu-ations are characterized by the electron spin lifetime τ and, additionally, in the case of a magnetic field in Voigtgeometry, by the precessional frequency ω . In the fre-quency domain, these temporal spin fluctuations trans-late to a Lorentzian line shape centered at ω/ π with afull width at half maximum of 1 /πτ [9, 10]. In SNS ex-periments, the energy of the probing laser light is usuallychosen to lie well below the energy gap where excitationof the semiconductor system is negligible [9]. Therefore,SNS allows experimental access to the electron spin dy-namics near equilibrium without evoking parasitic spinrelaxation by BAP.The light source for the SNS measurements is a lownoise, tunable diode laser in Littman configuration [11].A Faraday isolator avoids disturbing feedback and a spa-tial filter ensures a Gaussian beam profile. The laserlight is focused to a beam waist of w ≈ . µ m on thesample which is mounted in a He cold finger cryostat.Magnetic fields up to µ H ext = 14 mT can be appliedin Voigt geometry. The transmitted light is recollimatedand the rotation of the linear light polarization is resolvedby a combination of a polarizing beam splitter and a highbandwidth balanced photo receiver. The detected elec-trical signal is amplified by a low noise amplifier (40 dB)and sent through a low pass filter (-3dB at 67 MHz). Thefluctuation signal is digitized with 180 MHZ in the timedomain and Fourier transformed in real time.In previous SNS measurements on bulk GaAs, the spinnoise signal is shifted from zero frequency by applying amagnetic field in Voigt geometry. In (110)-oriented GaAsQWs, this method becomes difficult since spin relaxationis dominated in (110) QWs at finite magnetic fields bythe DP dominated spin relaxation of the in-plane spincomponent. Therefore, small applied magnetic fields inVoigt geometry only lead to a broadening and, conse-quently, the spin noise spectra in this work are all cen-tered around zero frequency. For the majority of spectra,large spurious background noise is most reliably removedby a double difference method: a) The polarization ofthe laser light is switched between linear and circularpolarization by a motorized Soleil Babinet compensatorin front of the sample. Circular polarized light is notsensitive to changes in the Faraday rotation angle and,therefore, contains no spin noise information. b) Foreach polarization the applied magnetic field is changedbetween µ H ext = 14 mT, at which the spin noise powerbecomes negligible in the observed frequency span, andzero or small magnetic fields. Subsequently, the noisepower spectra for linear and circular polarized light aresubtracted from each other, once for the applied mag-netic field of µ H ext = 14 mT and once for a vanishingor smaller magnetic field. These two curves are on theirpart again subtracted from each other and the result isdivided by a background noise spectrum to account forfrequency dependent amplification.The investigated sample consists of ten identical, sym-metrically grown, nominally 16.8 nm thick GaAs QWsseparated by 80 nm Al . Ga . As barriers grown bymolecular beam epitaxy and separated from the undoped(110) GaAs substrate by a 150 nm Al . Ga . As lift-offsacrifice layer. The QWs are symmetrically modulationdoped by Si δ -layers in the middle of the barriers andtransport measurements under illumination yield a dop-ing sheet density of n = 1 . · cm − at 1.5 K. Pho-togalvanic experiments by Belkov et al. [12] have shownthat spin dephasing due to structure inversion asymme-try is minimal in such symmetrically doped (110) QWsin contrast to symmetrically doped (100) QWs. The sub-strate is removed for the transmission SNS measurementsfollowing the lift-off recipe of Yablonovitch et al. [13],and the multi QW layer is van der Waals bonded to ac-cut Sapphire substrate. White light transmission mea-surements identify the optical absorption edge (interbandtransition to the Fermi level energy) of the QWs between813 and 814 nm.The inset of Fig. 1 shows a typical SNS spectrum mea-sured at a laser wavelength of about 814.25 nm and atemperature of 20 K. The measured spin noise spectra arefitted with a Lorentz function centered at zero frequency.The area under the Lorentz curve gives the integratedspin noise power and the width determines the spin life-time or spin decay rate γ , respectively. Figure 1 showsboth the measured integrated spin noise power (blackdots) and the spin relaxation rate (blue triangles) in de-pendence on laser wavelength for a lattice temperature
20 40 60
814 815 816 817 818012345 wavelength (nm) i n t. no i s e po w e r P ( a r b . u . ) s p i n de c a y r a t e z ( M H z ) s p i n no i s e po w e r S ( a r b . u . ) f (MHz) FIG. 1: (Color online) Spin decay rate γ z = 1 /τ z and relativeintegrated spin noise power at T = 20 K as a function the laserwavelength. Solid line: Integrated spin noise power accordingto model [9] (details in text). The inset shows a typical spinnoise spectrum. of 20 K. The spin relaxation rate increases sharply whenthe laser wavelength approaches the optical absorptionedge. This observation is consistent with the fact that inthis temperature regime traditional spin dephasing mea-surements based on optical excitation yield results whichare completely obstructed by the BAP mechanism. Mea-surements of this increase of γ z by SNS for wavelengthsshorter than 813.7 nm are hindered by the fact that theintegrated spin noise power approaches zero at the opti-cal absorption edge and by the fixed electrical frequencybandwidth of the detection setup. For wavelengths longerthan 815 nm, optical absorption becomes negligible andthe measured spin decay rate is in good approximationconstant indicating that the residual spin life time is de-termined by other spin relaxation mechanisms than BAP.Further experiments presented below show that this spindecay rate is not yet the intrinsic spin relaxation timebut limited by time of flight broadening.The black dots and and the solid black line in Fig. 1depict the measured and the calculated integrated spinnoise power, respectively. The integrated spin noisepower P is calculated by a phenomenological modelwhich is based on the change of the refractive index dueto the fluctuating imbalance of electron spins at the Fermienergy [9]: P ∝ ( dn/dα | α · N fluc /N bleach α ) , where n is the real part of the refractive index, N fluc the rootmean square imbalance between the two spin directions, N bleach the critical imbalance that would bleach the opti-cal transition for one spin direction, and α the absorptionconstant. The transition is set to 813.5 nm in accordancewith the optical transmission measurements. The cal-culated wavelength dependence with the minimum of P at the optical absorption edge agrees perfectly with theexperimental data. Figure 2 shows a more detailed com-parison of the measured and calculated P in dependenceon temperature and wavelength. The following sample P ( V ) ( n m ) T ( K ) P ( V ) ( n m ) T ( K ) FIG. 2: (Color online) Absolute integrated Spin noise poweras a function of laser wavelength and temperature, measured(left panel) and calculated (right panel, details in text). parameters are used for the calculation: The polarizabil-ity due to the optical selection rules of the optical tran-sition is set to β = 0 . α = 2 . · m − is assumed [20]. A linewidth of the optical resonance of k B ·
20 K for T ≤
20 Kand of k B · T for T >
20 K is used within the calculationssince the width of the measured white light transmis-sion change is constant below 20 K and increases linearlyabove. The energy of the resonance shifts with the tem-perature dependent Fermi level energy. Also this two-dimensional dependence of the calculated SNS on wave-length and temperature is in good agreement with ourmeasurements, demonstrating that the origin of the spinnoise signal is well understood.Next, we observe the influence of the spin relaxationanisotropy on the spin noise signal, which has previ-ously been examined at relatively high magnetic fields[5], where the spin relaxation anisotropy slows down theeffective Larmor frequency. However for very small mag-netic fields, with ω L < ( γ ⊥ − γ z ) /
2, there is no preces-sional motion of the spins as the spin decays due to thelarge in-plane spin relaxation rate γ ⊥ before even half arotation is carried out. The effective spin relaxation rateis in this case γ eff = γ ⊥ + γ z γ ⊥ − γ z s − ω ( γ ⊥ − γ z ) . (1)Figure 3 depicts the measured γ eff (black squares) as afunction of applied magnetic field at T=20 K. We havemeasured ω L , i.e., an electron g-factor g ∗ = 0 .
29, on thesame sample by time-resolved PL at B=4 T and fittedthe spin noise data by a least square fit correspondingto Eq. 1 (top blue solid curve). The fit directly yieldsthe anisotropy factors η = γ ⊥ /γ z which is shown as filledsquares in the inset of Fig. 3 in dependence on the laserwavelength. The anisotropy factor is at 815 nm smallerthan at longer wavelengths since BAP is in contrast toDP at most weakly dependent on the crystallographic di-rection, i.e., an efficient isotropic spin relaxation lowersthe spin relaxation anisotropy. In agreement with thewavelength dependent data (Fig. 1), the anisotropy fac-
815 816 8170369 s p i n de c a y r a t e e ff ( M H z ) applied field H ext (mT) an i s o t r op y wavelength (nm) FIG. 3: (Color online) Effective spin decay rate γ eff = 1 /τ eff ( T = 20 K) as a function of the applied magnetic field with fitsaccording to Eq. (1). Inset: Anisotropy determined by thefits as function of laser wavelength ( T = 20 K). Open symbolsindicate measurements with an enlarged focus in which timeof flight broadening is strongly reduced. tor is constant for long wavelengths since BAP is switchedoff.Anisotropy measurements at 815 nm with a defo-cused, i.e., enlarged, laser spot on the sample reduce thespin relaxation by BAP and yield an anisotropy factor η = 7 . .
0) which is a almost factor of two larger thanin the focused case. This anisotropy factor is of the samemagnitude as η measured in a similar sample at roomtemperature [5] and in an undoped GaAs (110) QW atlow temperature [6]. However, the physical origins aredifferent. In the room temperature case, the anisotropyis limited by ISR and, in the latter case, the anisotropyis probably dominated by the yet unclear influence ofthe surface acoustic waves. As it will be argued laterin this letter, the anisotropy measured in this work is incontrast given by the intrinsic low temperature spin life-times of the sample. Still, the strong measured increaseof τ z after defocusing can not be solely explained by thereduction of BAP since η is only equal to about 6 forlaser wavelengths of 816 and 817 nm where BAP is neg-ligible (see Fig. 3). In fact, the strong dependence on thelaser spot diameter indicates diffusion of electrons out ofthe laser spot which is equivalent to time of flight broad-ening. Figure 4 depicts the measured spin decay ratein dependence on the defocusing distance z which is thedistance between the focus of the Gaussian laser beamand the sample. The filled squares show γ z measured at815 nm where BAP can not be neglected and the filledcircles show γ z measured at 816 nm where BAP is unim-portant. Mesurements at 816 nm with twice the laserpower (filled triangles) rule out that the observations canbe completely attributed to excitation density dependentspin dephasing mechanisms as BAP. To model this timeof flight broadening, we extend the existing spin noisemodel [9] by taking into account the classical position of -80 -60 -40 -20 0 20 40 60 80406080100 s p i n de c a y r a t e z ( M H z ) z ( m) FIG. 4: (Color online) Effective Spin decay rate γ z = 1 /τ z ( T = 20 K) as a function of sample position z for a laserwavelength of λ = 815 nm (squares), λ = 816 nm (circles) and λ = 816 nm with doubled laser power (triangles). The z = 0position is set to the maxima of the measured curves. Thelines show calculations according to the model given by Eq.(2) for D eff = 100 cm / s (straight line), 200 cm / s (dot anddash line), 400 cm / s (dashed line) and 1000 cm / s (dottedline). the electron within the laser beam: S ( ω ) = Z d r (cid:12)(cid:12)(cid:12)(cid:12) F (cid:26)Z d r exp( − γ intr z t ) · P ( r , r ) · I ( r ) (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) . (2)Hereby, the two dimensional vector r gives the positionof an individual electron at the beginning of the mea-surement, exp( − γ intr z t ) describes the intrinsic spin decay, I ( r ) = I exp( − r /w ( z ) ) weights the position of theelectron in the laser spot, and P ( r , r ) gives the proba-bility distribution for classical Brownian motion in twodimensions with respect to a diffusion constant D [15]: P ( r , r ) = 1 / πD t · exp( − ( r − r ) / D t ). The depen-dence of the effective spin decay rate on the sample posi-tion z is calculated numerically with Eq. (2) and is plot-ted in Fig. 4 for different values of D . The experimentallydetermined beam parameters are: waist w = 3 . µ m andRayleigh range z R = 3 µ m. As intrinsic spin decay rate,we determine γ intr z = 42(2 .
5) MHz. This value is the av-erage decay rate to which the measured values convergewhen strongly defocused ( T = 20 K and λ = 815 nm)and corresponds to a spin lifetime of τ intr z = 24(2) nswhich is to our knowledge the longest measured spin life-time in n-doped GaAs (110) QWs. Best agreement be-tween model and experiment is obtained for values of D between 100 and 1000 cm / s (see Fig. 4). The mea-sured mobility at low temperature under illumination of µ = 3 . · cm / Vs gives together with the Einsteinrelation D = µE F e ≈ / s ( E F = 6 . τ ee = 1 / [ π ( k B T ) / ¯ hE F ln E F /k B T ] ≈
700 fs for oursample system [16, 17]. Calculating a diffusion constantfrom the Einstein relation together with the Drude modelgives D eff = 118 cm / s. Thus, this coarse estimate yieldsan effective diffusion constant of the correct order of mag-nitude and shows that the experiment is sensitive enoughto study electron motion near thermal equilibrium.We attribute the residual intrinsic spin dephasing rateof γ intr z = 42(2 .
5) MHz to a DP mechanism due to ran-dom Rashba fields caused by a fluctuating donor densityin the symmetric doping layers [18]. The relevant areaon which these donor density fluctuations between thelayers have to be considered is given by A = π ( v F τ ee ) .Assuming a Poisson distribution, the donor fluctuationsare of the order of 1 / √ A n ≈ ∗ Electronic mail: [email protected][1] M. I. D’yakonov and V. I. Perel’, Sov. Phys. Solid State , 3023 (1972).[2] G. Dresselhaus, Phys. Rev. , 580 (1955).[3] R. Winkler, Phys. Rev. B , 045317 (2004).[4] Y. Ohno et al. , Phys. Rev. Lett. , 4196 (1999).[5] S. D¨ohrmann et al. , Phys. Rev. Lett. , 147405 (2004).[6] O. D. D. Couto, Jr. et al. , Phys. Rev. Lett. , 036603(2007).[7] S. A. Crooker et al. , Nature , 49 (2004).[8] M. Oestreich et al. , Phys. Rev. Lett. , 216603 (2005).[9] M. R¨omer, J. H¨ubner, and M. Oestreich, Rev. Sci. In-strum. , 103903 (2007).[10] M. Braun and J. K¨onig, Phys. Rev. B , 085310 (2007). [11] M. G. Littman and H. J. Metcalf, Appl. Opt. , 2224(1978).[12] V. V. Bel’kov et al. , Phys. Rev. Lett. , 176806 (2008).[13] E. Yablonovitch et al. , Appl. Phys. Lett. , 2419 (1990).[14] S. Pfalz et al. , Phys. Rev. B , 165305 (2005).[15] G. Bergmann, Phys. Rev. B , 2914 (1983).[16] G. Fasol, Appl. Phys. Lett. , 2430 (1991).[17] H. Fukuyama and E. Abrahams, Phys. Rev. B , 5976(1983). [18] E. Y. Sherman, Appl. Phys. Lett. , 209 (2003).[19] P. S. Eldridge et al. , Phys. Rev. B , 125344 (2008).[20] The optical absorption constant is not measured by an in-dependent experiment but has been adjusted in the SNScalculations to the measured integrated spin noise powerat 815 nm and 20 K in Fig. 2. Nonetheless, the adjusted α0