Anisotropic in-plane spin splitting in an asymmetric (001) GaAs/AlGaAs quantum well
© Ye et al; licensee Springer. 2011
Received: 2 June 2011
Accepted: 2 September 2011
Published: 2 September 2011
The in-plane spin splitting of conduction-band electron has been investigated in an asymmetric (001) GaAs/Al x Ga1-x As quantum well by time-resolved Kerr rotation technique under a transverse magnetic field. The distinctive anisotropy of the spin splitting was observed while the temperature is below approximately 200 K. This anisotropy emerges from the combined effect of Dresselhaus spin-orbit coupling plus asymmetric potential gradients. We also exploit the temperature dependence of spin-splitting energy. Both the anisotropy of spin splitting and the in-plane effective g-factor decrease with increasing temperature.
PACS: 78.47.jm, 71.70.Ej, 75.75.+a, 72.25.Fe,
Keywordsquantum beats spin-orbit coupling magnetic properties of nanostructures optical creation of spin polarized
The properties of spins in semiconductor materials have attracted much more attentions since the invention of spintronics and spin-based quantum information [1–3]. In those fields, the spin-orbit coupling (SOC) plays a key role on the properties of spin states in bulk and low-dimensional semiconductor materials. It not only results in the zero-magnetic field spin splitting, which is the main source of the spin relaxation through D'yakonov-Perel (DP) mechanism and novel phenomenon such as the generation of the spin current , but also significantly affects the spin splitting with an external magnetic field B > 0.
In general, the spin splitting of electron or hole at B > 0 in semiconductor is described by a finite Zeeman splitting energy and characterized by the effective g-factor, which is necessary for the spin manipulation and spin-based qubit with an external electrical/magnetic field in semiconductor. So far, the effective g-factor has been intensively investigated in many literatures during past few decades [5–13]. For conduction-band electron, it is found that the effective g-factor is strongly dependent on the point group symmetry in semiconductor materials . It is isotropic and independent on the orientation of applied magnetic field in GaAs bulk due to T d point symmetry group. On the contrast, the effective g-factor becomes anisotropic and significantly depends on the direction of magnetic field in quantum structures such as GaAs/AlGaAs heterostructures and quantum well (QW) due to the reducing symmetry . For example, where the point symmetry group is reduced to D 2d, in a rectangular/symmetric QW grown on the (001)-orientated substrate, the effective g-factor can have different values for B applied in the direction normal to the plane of QW and for B in the plane of the QW due to the additional potential confinement: g xx = g yy ≠ g zz (x//) [5–7, 10]. Furthermore, where the symmetry is reduced to C 2v in an asymmetric QW with the inversion-asymmetric confining potentials, the effective g-factor is dependent on the direction of an applied in-plane magnetic field, which results in the anisotropic Zeeman splitting . Up to now, the spin splitting (Zeeman splitting) at B > 0 is considered to be only characterized by the effective g-factor. In fact, two proposals [7, 14] have been predicted that the Dresselhaus SOC significantly affects the spin splitting of electron at B > 0 plus structure inversion asymmetry. A new term, defined as in Ref. , can result in the measurable anisotropy of the in-plane spin splitting, although it is not a Zeeman term. We call it as Zeeman-like term thereinafter. The anisotropic spin splitting was measured experimentally at B > 0 with an applied external electric field to reduce the symmetry of quantum film but interpreted in terms of anisotropic effective g-factor by Oestreich et al.  In this Letter, we use the time-resolved Kerr rotation (TRKR) [15, 16] technique to study the in-plane spin splitting via the accurate measurements of the Larmor procession frequency in a specially designed (001) GaAs/AlGaAs undoped QW with asymmetric confined barriers under an in-plane magnetic field. We show that the spin splitting is found to be obviously anisotropic for B parallel to  and [1-10].
The sample on investigation here is grown on (001) oriented semi-insulating GaAs substrate by molecular beam epitaxy. It contains a 50-nm-wide Al0.28Ga0.72As barrier layer, an 8-nm-wide GaAs quantum well, the other sloping barrier grown with content of Al changing from 4.28% to 28% on the length of approximately 9 nm, and the barrier layer of a width 50 nm. The upper part of the structure is covered with a 5-nm GaAs layer to avoid the oxidation of barrier. TRKR experiment was carried out in an Oxford magneto-optical cryostat supplied with a 7-T split-coil super-conducting magnet. The sample was excited near normal incidence with a degenerate pump and delayed probe pulses from a Coherent mode-locked Ti-sapphire laser (approximately 120 fs, 76 MHz). The center of the photon energy was tuned for the maximum Kerr rotation signal for each temperature setting. The laser beams were focused to a spot size of approximately 100 μm, and the pump and probe beams have an average power of 5.0 and 0.5 mW, respectively. The helicity of linearly polarized pump beam was modulated at 50 kHz by a photoelastic modulator for lock-in detection. The temporal evolution of the electron spins, which were generated by the circularly polarized pump pulse, was recorded by measuring Kerr rotation angle θ K(Δt) of the linearly polarized probe pulse while sweeping Δt.
Results and discussion
where S 0 is the initial spin density, τs is spin lifetime, and ω the Larmor frequency. By this way, we obtain the exact value of the Larmor frequency ω and then the splitting energy ΔE B// = 0.0263 meV and ΔE B//[1-10] = 0.0326 meV through the equation ΔE = ћω with in-plane magnetic field parallel to  and [1-10], respectively. Here, we use to denote the anisotropy of the in-plane spin splitting. We found that this anisotropy is more than 19% in this single asymmetric (001) GaAs/AlGaAs QW. We also checked the photogenerated spin concentration dependence of the spin splitting, which can be reached by changing the pump power. Figure 1b shows the pump power dependence of spin splitting with the magnetic fields along  and [1-10] at 1.5 K. The splitting energy slowly decreases with increasing pump power up to approximately 20 mW. The change of spitting energy is less than 7% for both curves and can be ignored. Therefore, the observed anisotropy is not relevant to the carrier concentration.
where g * is the effective g-factor, B // is the in-plane external magnetic field, ζ = +1 for B// [1-10] and ζ = -1 for B//, and γ is the cubic Dresselhaus constant. The Zeeman-like term , which is derived from first-order perturbation theory applied to the Dresselhaus term, emerges from the combined effect of BIA and SIA . It is clear that the term results in the anisotropic spin splitting at B > 0. As expected in Equation 2, the measured spin splitting is linearly dependent on the magnetic field with a prefactor for both directions of applied magnetic fields as shown in Figure 1c. The slope of the B linear dependence will allow us to obtain the value of G accurately, which are G  = 0.0130 meV/T and = 0.0162 meV/T for B along  and [1-10]. The difference of two values results from the opposite sign of prefactor ζ. According to Equation 3, is found to be equal to approximately 0.8 μeV/T. As discussed above, a proper anisotropic Zeeman term, described in Equation (7.4) in Ref. , also produces the anisotropic spin splitting at B > 0 in an asymmetric (001) GaAs/AlGaAs QW. However, the prefactor is about 0.039 μeV/T for realistic parameters with the assumed internal electric field of approximately 50 kV/cm induced by the asymmetric potential gradients. It is about 20 times smaller comparing to the value of term . We conclude that the Zeeman-like term is the main source of the anisotropy of spin splitting at B > 0 in an asymmetric QW. Additionally, the Rashba term also gives rise to a nontrivial splitting in the presence of a magnetic field, but the splitting is isotropic . In fact, the Rashba term is considered to be very small in this work because we did not observe significantly the anisotropy of in-plane spin relaxation  as shown in Figure 1a. It is consistent with the results of Ref. . As shown in Equation 3, the Zeeman-like term is proportional to the cubic Dresselhaus constant γ. Numerical calculations yields γ = 29.96 eV/Å3 at approximately 1.5 K. Here, we use the value of approximately 0.8 μeV/T of and an electron wave function calculated by the k p method  in this asymmetric QW. It is in agreement with the value of 27.58 eV/Å3 (see Table 6.3 in Ref. ). The remaining deviations of γ probably result from differences between the actual and the nominal sample structures which lead to uncertainties in the calculation of the wave function asymmetry.
We observed the anisotropic in-plane spin splitting of the conduction-band electron in an asymmetric (001) GaAs/AlGaAs quantum well using TRKR technique with applied magnetic fields. It is confirmed that Dresselhaus spin-orbit coupling can significantly affect the in-plane spin splitting at B > 0 combining the asymmetric confinement potential via the numerical comparison with the proper anisotropic Zeeman splitting.
bulk inversion asymmetry
structure inversion asymmetry
time-resolved Kerr rotation.
We acknowledge the financial support of this work from the Chinese-French NSFC-ANR project (grant number 10911130356), National Science Foundation of China (grant number 10774183, 10874212), and National Basic Research Program of China (2009CB930500). One of the authors (HQ) would like to thank Prof. R. Winkler, Prof. V. K. Kalevich, and Prof. V. L. Korenev for many fruitful discussions.
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