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Origin of enhanced dynamic nuclear polarization and all-optical nuclear magnetic resonance in GaAs quantum wells

Time-resolved optical measurements of electron-spin dynamics in a (110) GaAs quantum well are used to study the consequences of a strongly anisotropic electron g-tensor, and the origin of previously discovered all-optical nuclear magnetic resonance. All components of the g-tensor are measured, and a strong anisotropy even along the in-plane directions is found. The amplitudes of the spin signal allow the study of the spatial directions of the injected spin and its precession axis. Surprisingly efficient dynamic nuclear polarization in a geometry where the electron spins are injected almost transverse to the applied magnetic field is attributed to an enhanced non-precessing electron spin component. The small absolute value of the electron g-factor combined with efficient nuclear spin polarization leads to large nuclear fields that dominate electron spin precession at low temperatures. These effects allow for sensitive detection of all-optical nuclear magnetic resonance induced by periodically excited quantum-well electrons. The mechanism of previously observed Delta m = 2 transitions is investigated and found to be attributable to electric quadrupole coupling, whereas Delta m = 1 transitions show signatures of both quadrupole and electron-spin induced magnetic dipole coupling.


I. INTRODUCTION
The coherent dynamics of electron spins in semiconductors is of fundamental interest for novel "spintronic" devices, 1 and can be investigated using time-resolved optical techniques. A well-known manifestation of spin coherence is the beating of timeand polarization-resolved photoluminescence due to precession of electron spins about an applied magnetic field. 2 Because this type of beating involves the spins of recombining charges, the observable duration is limited to the charge recombination time. Spin coherence detected using time-resolved absorption 3 or Faraday rotation 4 (FR) can in principle be monitored over arbitrary long time scales in doped semiconductors and heterostructures. 5 Using time-resolved FR, spin lifetimes exceeding the charge recombination times by four orders of magnitude have been observed in n-doped GaAs. 6 Besides enabling macroscopic transport of spin coherence 7 and potentially semiconductor-based quantum-computation devices, 8 such long spin lifetimes allow for precise measurement of the electron-spin precession frequency. Due to the contact hyperfine interaction with nuclear spins, time-resolved FR can be employed as sensitive magnetometer for nuclear spin polarizations ("Larmor magnetometry"). 9,10,11 In such experiments, dynamic nuclear polarization (DNP) is used to achieve nuclear spin polarizations exceeding the thermal equilibrium values by orders of magnitude. DNP has been demonstrated in bulk semiconductors, 12,13,14 quantum wells (QWs), 15,16,17 and quantum dots. 18 Combining DNP and Larmor magnetometry enables optical detection of nuclear magnetic resonance (NMR) with direct and accurate measurement of nuclear spin polarization at arbitrary magnetic fields. This is in contrast to the conventional approach of optically detected NMR, which relies on cw measurements of photoluminescence polarization reduced by the Hanle effect. 14 In NMR experiments, the resonance is usually induced by externally applied radio-frequency magnetic fields. It has been demonstrated in Ref. 19 that the radiofrequency tipping field can be replaced by modulated light interacting indirectly with the nuclei through the electron system. Such an "all-optical" NMR scheme has been realized recently in Refs. 9,10 using a mode-locked pump laser. In this approach, the resonance condition is obtained when the laser repetition rate matches the nuclear precession period in a static magnetic field. Resonant depolarization of the nuclear spin is observed through electron Larmor magnetometry. In conventional NMR experiments, the radio-frequency magnetic fields penetrate the whole semiconductor. Applying the all-optical scheme in a quantum-confined electronic structure 10 leads to spatial confinement of the nuclear excitation and therefore to unprecedented localization of all the NMR mechanismsinitial polarization, excitation, and detection.
Here, we discuss in more depth electron spin coherence and all-optical NMR measurements in a narrow GaAs QW grown on a GaAs (110) substrate. Electrons confined in such QWs feel fluctuating effective fields from spin-orbit coupling that are oriented along [110] to lowest order in perturbation theory. 20 This special symmetry significantly reduces D'yakonov-Perel spin scattering, 21 and spin lifetimes > 1 ns were measured at room temperature. 22 In order to characterize the electron-spin dynamics without contributions from nuclear effects, the samples are measured at intermediate temperatures (135 K), where DNP is inefficient. By varying the angle between the sample normal and the magnetic field B, we are able to measure the strongly anisotropic electron g tensor. Because the FR signal measures a projection of the electron spin, the relative orientation of the precession axis can be reconstructed, and we find an expected deviation of its direction from B. Furthermore the data suggests that electron spin is injected along the quantization axis of the QW, and not along the direction of light propagation. The g-factor anisotropy is found to be crucial for efficient nuclear spin polarization, leading to additional modifications of both electron-spin precession frequency and orientation of the precession axis. At liquid He temperatures, lowering the laser pump intensity reduces DNP, and contributions from the nuclear spin and the anisotropic electron g-tensor can be separated. This technique allows for a quantitative determination of the nuclear spin polarization.
Monitoring the nuclear spin polarization, we study resonant nuclear depolarization induced by a laser pulse train (all-optical NMR). In Ref. 10, such resonances were observed also at laser repetition rates matching twice the nuclear precession frequency, and were interpreted as ∆m = 2 transitions within the nuclear spin states with angular quantum number m. Using two pump beams, we find that for ∆m = 2, the depth of the resonance does not depend on the spin-polarization of the electrons periodically injected by the second pump beam. Together with the lack of observed sidebands in case of spin-modulation of the resonantly injected electrons, we rule out interaction with the electron spin as tipping mechanism for ∆m = 2, in agreement with selection rules for magnetic dipole coupling. Without injection of electrons, the resonance disappears, suggesting that the charge of the injected carriers depolarizes nuclear spin through interaction with nuclear electric quadrupolar moments. On the other hand, ∆m = 1 resonances are found to be deeper when mediated by spin-polarized electrons, and sidebands occur in case of spin-modulation. This suggests that the originally proposed hyperfine mechanism for nuclear spin tipping significantly contributes to all-optical NMR at ∆m = 1.
The paper is organized as follows: In section II, we describe the experimental setup and the measured samples. Experimental characterization of the spin-coherence, including measurements of the anisotropic electron g-tensor at 135 K and at 5 K is discussed in section III. A model for DNP describes the measured angle-dependence of the electron-spin precession frequency at low temperatures. Results on all-optical NMR are presented in section IV together with studies of nuclear tipping mechanisms responsible for ∆m = 1 and ∆m = 2 transitions. We conclude in section V.

II. EXPERIMENTAL SETUP
The samples studied are single, 7.5 nm wide GaAs QWs with Al 0.4 Ga 0.6 As barriers on both sides. The samples are grown on (110) GaAs wafers by molecular beam epitaxy and are modulation doped with a nominal Si density of 4⋅10 11 cm -2 . The mobility and electron density at 300 K are 1700 cm 2 V -1 s -1 and 9⋅10 10 cm -2 , respectively, as measured in a Hall bar configuration.
The samples are glued on fused silica, and the GaAs substrate is removed by mechanical polishing and selective etching, allowing for measurement of the FR in transmission. They are placed in an optical cryostat with a variable temperature insert, where temperatures between 2 K and 300 K can be achieved. The energy of a 100 fs Ti:sapphire laser with a pulse repetition rate of 76 MHz is tuned close to the heavy-hole absorption edge of the QW (1.572 eV at 5 K, 1.537 eV at 135 K). Pump and probe pulses are separated with a beam splitter. A delay line allows for a tunable time delay ∆t between pump and probe pulses ranging from -0.3 to 2.7 ns. The circularly polarized pump pulses and the linearly polarized probe pulses are focused to spatially overlap on the sample surface within a focal spot diameter of ~50 µm. Using an average pump power of 1.2 mW leads to injection of around 10 10 cm -2 electron-hole pairs per pulse, well below the doping density. The angle between the pump and probe beam after passing through the lens is less than 5°. We measure the FR, i.e. the rotation angle of the transmitted probe beam's linear polarization, using a balanced photo-diode bridge. The pump beam is chopped at frequencies between 1 and 6 kHz, allowing for lock-in detection of the FR, which is proportional to the carrier's spin component along the probe beam. Electrons and holes recombine on a time scale of 100 ps, as inferred from measured time-resolved absorption. Thereafter, a spin imprint in the doping electrons is detected, and varying ∆t reveals the precessional relaxation of such spins. We measure transverse spin lifetimes around 10 ns at low magnetic fields and 5 K.

III. ELECTRON SPIN COHERENCE
Due to quantum-confinement and penetration of the electron wave function into the AlGaAs barriers, the g-factor in a narrow QW can differ substantially from the bulk GaAs value 23,24 . Heavy-and light-hole splitting in a QW leads to different values of g for B oriented in-plane or along the quantization axis 24,25,26,27 . Generally, the g-factor can be expressed as a tensor ĝ . Due to crystal symmetry and the direction of the QW, the main axes of ĝ are expected to be the growth axis [110] (z), the [001]-direction (y) and the [ 10 1 ]-direction (x). With our high-resolution measurement of the electron-spin precession frequency, L Ω , we are able to determine all three components of ĝ , and find strong anisotropy even for the two in-plane directions 28 . The anisotropy gives rise to a dependence of L Ω on the orientation of B with respect to the coordinate system of the sample. Furthermore, the precession axis h / which can be observed in the FR measurements and has important consequences for the nuclear polarization at lower temperatures.
In order to identify g-factor anisotropies in the absence of nuclear polarization, we first discuss measurements at 135 K, where DNP is inefficient. 10 where g x and g z are the respective components of the g tensor. The solid circles in Another error arises from possible misalignment of the sample's in-plane directions with respect to the rotation axis. For this angle, we estimate a precision of ±10°, which gives rise to the errors included in Tab. I, which summarizes the components of the g-tensor at 135 K. The value for |g y | is more than a factor of two smaller than |g x | and almost eight times smaller than |g z |. Because the α-dependence of g is given by the squares of the gtensor components, we cannot determine the sign of the g-tensor components with this technique.
The precession axis Ω Ω Ω Ω is tilted away from the x or y axis by an angle γ, given by For an anisotropic g-tensor, Ω Ω Ω Ω is generally not collinear with B, and the electron spin S contains both a precessing (S ⊥ ) and a nonprecessing (S || ) component 29 , as shown in Fig.1(b). We fit the FR data to the sum of . The two parameters, T 1 and T 2 , are effective longitudinal and transverse spin lifetimes with respect to the precession axis. If one assumes that the FR measures the projection of S onto the probe beam direction, the amplitudes a 1 and a 2 are proportional to the respective projections of S || and S ⊥ , respectively.
The symbols in Fig. 2(b) show the obtained amplitude a 1 of the non-oscillating usually leads to an electron spin population further out of equilibrium and therefore to a larger nuclear spin polarization. 12 By tilting the sample by an angle α [ Fig. 1(b)], a component of S || along B is introduced, which dominates DNP even for small α. We find that DNP sensitively depends on the in-plane orientation of B, which reflects the dependence of S || on the anisotropic electron g-factor.
The nuclear spin polarization modifies coherent electron spin dynamics due to the contact hyperfine interaction, which can be described by an effective nuclear field B n acting on the electron spin. If B n has a component perpendicular to the injected electron spin, S, it leads to a change in the electron spin precession frequency, which we measure using time-resolved FR. Fig. 3(a) shows the apparent g-factors (hΩ L /µ B B) vs. α We obtain the real g factor by reducing the laser pump intensity, which decreases DNP. The squares in Fig. 3(a) show additional data for right-circularly polarized pump pulses at 90 µW. The asymmetry is strongly reduced, and we can fit the α-dependence with the model of an anisotropic g-tensor in order to obtain the g-tensor components at ⋅ where S is the time-averaged electron spin, I = 3/2 the spin of the nuclei, and f a leakage factor describing electron-spin relaxation other than through hyperfine-induced flipping of nuclear spins. Equation (2) is derived for I << I. For simplicity, we assume that S initially points along the z-direction, is of magnitude ½, and precesses about Ω. Ω. Ω. Ω.
Generally, S can be decomposed into || S and ⊥ S [ Fig. 1(b)]. The transverse component ⊥ S is reduced due to spin precession, and we find that even for small α,  Here, a good fit can only be obtained with same signs of g y and g z (dotted lines). The best value for ξ is 0.22, suggesting that DNP is more efficient for B along y than along x. The absolute signs of the g-tensor elements are obtained using the same arguments given for the 135 K data. We find that g z and g y are both negative, whereas g x must be positive. The data is summarized in Tab. I.
With this model, we can predict how the nuclear polarization would behave with an isotropic g-factor. The dash-dotted lines in Fig. 4 shows data obtained assuming g x,y = g z . For easier comparision, the nuclear contribution to Ω L was added onto an angledependent Zeeman background Ω corresponding to the respective anisotropic g-tensor.
The data shows that for B in the (x, z)-plane, the anisotropy enhances nuclear polarization for -10° < α < 7°, whereas for B in the (y, z)-plane, the enhancement extends from 10° to all negative angles. The increased nuclear polarization is due to the finite S || , and is advantageous for Larmor magnetometry in the quasi-transverse geometry.
In the following, we describe how the nuclear field, B n , can be obtained from the measurement. Using Eq. (4), we rewrite the definition of the nuclear field, 14 B n = A I /gµ B , as B n = h(Ω Ω Ω Ω tot -Ω Ω Ω Ω)/gµ B . From the experiment, we obtain Ω L = |Ω Ω Ω Ω tot | (symbols in Fig. 4) and Ω = |Ω Ω Ω Ω| (dashed line in Fig. 4). Because of the anisotropic g-factor, Ω Ω Ω Ω tot and Ω Ω Ω Ω are not collinear, and B n is generally not proportional to the difference of Ω tot and Ω. However, for small α, and if the nuclear field enhances Ω L, i.e. when Ω Ω Ω Ω tot and Ω Ω Ω Ω point approximately into the same direction, Ω tot ≈ Ω + Ω n and B n ≈ B(Ω L -Ω)/Ω is a good approximation. For GaAs, it was predicted 13 that the maximum nuclear field is 5.3 T for 100% nuclear polarization. This value scales with g GaAs /g for g-factors different from the bulk GaAs value of g GaAs = -0.44. This leads to comparably large nuclear fields for low electron g-factors. As an example, we measure Ω L = 50 GHz for α = -20°, as shown in Fig. 4(b). The Zeeman contribution is Ω ≈ 23 GHz, and B n ≈ B(Ω L -Ω)/Ω ≈ 4.7 T. If one considers that Ω Ω Ω Ω and Ω Ω Ω Ω n are at an angle of γα ≈ 35°, one gets B n ≈ 5.1 T.
Since at this angle, the absolute value of the g-factor is 0.067, the maximum nuclear field amounts to 35 T, and the measured B n corresponds to a nuclear polarization of 15%.

IV. ALL-OPTICAL NMR
In this section, we study the resonant depolarization of nuclear spin induced by the periodicity of laser pulse arrivals. The nuclear gyromagnetic ratios γ for the three different isotopes in GaAs are γ( 71 Ga) = 8.158⋅10 7 rad/Ts, γ( 69 Ga) = 6.421⋅10 7 rad/Ts, and Furthermore, the proposed effect was observed at small fields of a few Gauss and decreases with 1/B 2 , ruling out its occurrence at higher fields. However, the interaction of the nuclear electric quadrupole moment with modulated electric field gradients can induce transitions at 2ω excitation, as was experimentally observed in bulk GaAs by modulating an electric field across the semiconductor. 33 In the case of all-optical NMR, electric-field gradients might be modulated by periodically injected charges, 10,34 in contrast to the hyperfine scenario. We are able to experimentally separate the two mechanisms and find that the charge of the electrons, not the spin, is responsible for nuclear resonances at 2ω, and that both mechanisms play a role at ω excitations.  Fig. 5(b). The maximum relative depolarization for the ω excitation is 14%, whereas the 2ω excitation leads to a depolarization by 12%.
In the following, we make use of sidebands that occur in the excitation spectrum phase. An increase of the FR signal shown in Fig. 6(a) corresponds to a decrease of B n .
The quadrupolar features are not visible, and periodic oscillations appear instead. A Fourier transform of FR vs. B reveals that the field periods ∆B correspond to frequencies γ∆B/2π matching ν ch and 2ν ch [ Fig. 6(b)], indicating that the even and odd harmonics in the sideband spectrum are weighted differently, as depicted in the insert of Fig. 6(a). This is expected for a symmetric modulation that contains mainly odd higher harmonics.
In order to better resolve individual sideband peaks, we modulate the amplitude at This makes a quantitative analysis of the peaks difficult.
Alternatively, we can modulate the helicity of the circularly polarized pump pulses at 50.1 kHz using the same photoelastic modulator set at λ/4 retardance without the linear polarizer. This leads to periodically injected electrons with modulated spin orientation. Although much smaller in amplitude, DNP still takes place with this set-up, contrary to the general belief of Ref. 14. We enhance the nuclear polarization further by slightly detuning the balance between left-and right-circular polarization. Curve B in Fig. 7(a) shows the resonance spectrum. If the electron spin is not connected to the nuclear depolarization mechanism, then no sidebands should be observed in the resonance spectrum. However, we clearly resolve sidebands, suggesting that spin is  Unintentional amplitude modulation of the laser pulse train right before the cryostat window is measured to be smaller than 10 -3 . However, we can not exclude small charge modulations inside the sample. As a reference, Curve C shows the broadened resonance peak for mechanically chopped pump pulses.
A more direct way to study the tipping mechanism is to compare the resonance induced by a circularly polarized pump beam with that from a linearly polarized one. In the first case, the polarized electron spin allows for periodic contact hyperfine interaction with the nuclear spin, whereas in the second case, up and down spins are populated equally, eliminating efficient hyperfine coupling. Because Larmor magnetometry relies on circularly polarized pump pulses injecting spin-polarized carriers, we cannot tune the pump to linear polarization. Therefore, we focus a second pump beam onto the sample, whose pulse repetition rate, energy and polarization can be tuned independently.
In Fig. 8(a), data is shown for the 69 Ga resonance excited at ω. doublet is insensitive to the electron spin polarization, as suggested by the sideband experiments described above, then the decrease in resonance depth is entirely due to a decrease of the ∆m = 1 triplet, and the ∆m = 2 doublet should gain in relative strength.
However, the data is not conclusive enough to support this idea.   Circles are measured FR data. The angle-dependent oscillation frequency arises from an anisotropic electron g-factor. The data is fit (solid lines) by an exponentially decaying harmonic oscillation added to a second, non-precessing exponential decay.  Open diamonds are for right-circularly polarized pump pulses, filled symbols for leftcircular excitation at two different laser intensities. The electron precession frequency is largely affected by hyperfine coupling to dynamically polarized nuclear spins. Lowering the pump intensity (solid squares) decreases nuclear polarization and reveals the bare electron g factor, which is obtained by fitting the lowest-intensity data to the angledependence of an anisotropic g-factor, thus obtaining the three components |g x |, |g y | and |g z | at 5 K. diamonds represent the measured data already shown in Fig. 3. The fits indicate that g x and g z have different signs, whereas g y and g z have the same sign (which is negative).
The fit parameter ξ represents the maximum relative nuclear polarization and is weaker in (a).   suggests that no spin mechanism accounts for the resonant tipping at 2ω excitation.