Direct measurement of the spin gaps in a gated GaAs two-dimensional electron gas
© Huang et al.; licensee Springer. 2013
Received: 12 January 2013
Accepted: 9 March 2013
Published: 25 March 2013
We have performed magnetotransport measurements on gated GaAs two-dimensional electron gases in which electrons are confined in a layer of the nanoscale. From the slopes of a pair of spin-split Landau levels (LLs) in the energy-magnetic field plane, we can perform direct measurements of the spin gap for different LLs. The measured g-factor g is greatly enhanced over its bulk value in GaAs (0.44) due to electron–electron (e-e) interactions. Our results suggest that both the spin gap and g determined from conventional activation energy studies can be very different from those obtained by direct measurements.
where kT and Γ are the thermal and level broadening, respectively .
For practical applications, it is highly desirable that the generation of the spin currents can be accomplished without requiring the use of extremely high B. Therefore, an accurate measurement of the spin gap and g-factor would allow one to ensure that only a moderate B is required so that Equation 1 holds. Moreover, the precise measurement of the g-factor  would shed light on the predicted divergence of spin susceptibility χ ∝ g m* and ferromagnetic ground state , where the system exhibits the unexpected metal-insulator transition . Here m* represents the effective mass of electron (or hole). Given that the spin gap is the most important energy scale in any spin system and the g-factor is the central quantity characterizing the response of an electron or hole spin to an applied B, there have been many attempts to measure the spin gap in the literature. A standard method of obtaining the spin gap is to perform activation energy measurements at the minimum of the longitudinal resistivity , where Δs is the spin gap . However, such a measurement is rather restrictive as ρ xx must be very low and has to vary over at least an order of magnitude as a function of T. Moreover, Δs has to be much greater than the thermal energy kT over the whole measurement range. Most importantly, activation energy measurements yield the ‘mobility gap’, the width of the localized states in the energy spectrum. This may be quite different from the real spin gap which corresponds to the energy difference between the two maxima densities of neighboring extended states [4, 8].
In this paper, we report a method to directly measure the spin gaps in two-dimensional electron gases (2DEGs), in which the electrons are usually confined in layers of the nanoscale. We can change the applied gate voltage Vg to vary the electron density n2D and hence the local Fermi energy E in our system. By studying the peak positions of ρ xx at various n2D and B, we can construct the Landau levels in the E-B diagram. As shown later, from the difference between the slopes of a pair of spin-split Landau levels in the E-B plane, we are able to measure the g-factors for different Landau level indices n in the zero disorder limit. We find that the measured g-factors (approximately 10) are greatly enhanced over their bulk value (0.44). Most importantly, our results provide direct experimental evidence that both the spin gap and g-factor determined from the direct measurements are very different from those obtained by the conventional activation energy studies. A possible reason is that our method is conducted in the zero disorder limit, whereas activation studies are performed under the influence of the disorder within the quantum Hall system.
where ωC is the cyclotron frequency, and n = 0, 1, 2, 3…, respectively. According to early experimental work , it was established that in 2D systems in a magnetic field the g-factor is greatly enhanced over its bulk value due to exchange interactions [10, 11]. The precise measurement of the g-factor in 2D systems is a highly topical issue  since it has been predicted to be enhanced in strongly interacting 2D systems that exhibit the unexpected zero-field metal-insulator transition .
Magnetoresistance measurements were performed on three gated Hall bars (samples A, B and C) made from modulation-doped GaAs/Al0.33Ga0.67As heterostructures. For sample A, the structure consists of a semi-insulating (SI) GaAs (001) substrate, followed by an undoped 20-nm GaAs quantum well, an 80-nm undoped Al0.33Ga0.67As spacer, a 210-nm Si-doped Al0.33Ga0.67As, and finally a 10-nm GaAs cap layer. For sample B, the structure consists of an SI GaAs (001) substrate, followed by an undoped 20-nm GaAs quantum well, a 77-nm undoped Al0.33Ga0.67As spacer, a 210-nm Si-doped Al0.33Ga0.67As, and finally a 10-nm GaAs cap layer. Sample C is a modulation-doped GaAs/AlGaAs heterostructure in which self-assembled InAs quantum dots are inserted into the center of the GaAs well . The following sequence was grown on an SI GaAs (001) substrate: 40-nm undoped Al0.33Ga0.67As layer, 20-nm GaAs quantum well inserted with 2.15 monolayer of InAs quantum dots in the center, a 40-nm undoped Al0.33Ga0.67As spacer, a 20-nm Si-doped Al0.33Ga0.67As, and finally a 10-nm GaAs cap layer. Because of the lack of inversion symmetry and the presence of interface electric fields, zero-field spin splitting may be present in GaAs/AlGaAs heterostructures. However, it is expected that the energy splitting will be too small (0.01 K) to be important in our devices . For sample A, at Vg = 0 the carrier concentration of the 2DEG was 1.14 × 1011 cm-2 with a mobility of 1.5 × 106 cm2/Vs in the dark. For sample B, at Vg = 0 the carrier concentration of the 2DEG was 9.1 × 1010 cm-2 with a mobility of 2.0 × 106 cm2/Vs in the dark. The self-assembled InAs dots act as scattering centers in the GaAs 2DEG [12, 14]; thus, the 2DEG has a mobility much lower than those for samples A and B. For sample C, at Vg = 0 the carrier concentration of the 2DEG was 1.48 × 1011 cm-2 with a mobility of 1.86 × 104 cm2/Vs in the dark. Experiments were performed in a He3 cryostat and the four-terminal magnetoresistance was measured with standard phase-sensitive lock-in techniques.
Results and discussion
where we consider the effective Lande g-factor g*. We can see that Equation 3 corresponds to two straight line fits through the origin for a pair of spin-split Landau levels in the E-B plane as shown in Figure 2a,b. Such an approach was applied to a GaN-based 2DEG in our previous work . We note that our method does depend on the exact functional form of the Landau band since the peak positions of the Landau level is only related to the carrier density in our system.
In conclusion, we have performed direct measurements of the spin gaps in gated GaAs 2DEGs by studying the slopes of spin-split Landau levels in the energy-magnetic field plane. The measured g-factor is greatly enhanced over its bulk value (0.44). Since disorder exists in any experimentally realized system, conventional activation energy studies always measure the mobility gap due to disorder which is different from the real spin gap as shown in our results. As the spin gap is one of the most important energy scales and governs the electron spin degree of freedom, our experimental results provide useful information in the field of spintronics, spin-related phenomena, and quantum computation applications.
TYH, CTL and YFC were supported by the NSC, Taiwan and National Taiwan University (grant no. 102R890932 and grant no. 102R7552-2). The work at Cambridge was supported by the EPSRC, UK. This research was supported by the World Class University program funded by the Ministry of Education, Science and Technology through the National Research Foundation of Korea (R32-10204).
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