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Dependence of Effective Mass on Spin and Valley Degrees of Freedom

We measure the effective mass (m*) of interacting two-dimensional electrons confined to an AlAs quantum well while we change the conduction-band valley occupation and the spin polarization via the application of strain and magnetic field, respectively. Compared to its band value, m* is enhanced unless the electrons are fully valley and spin polarized. Incidentally, in the fully spin- and valley-polarized regime, the electron system exhibits an insulating behavior.

In Fermi liquid theory, interacting electrons can be treated as non-interacting quasi-particles with a renormalized effective mass (m * ). The parameter m * has been studied extensively [1,2,3,4,5] for various twodimensional electron systems (2DESs) as a function of interaction strength, which is characterized by the ratio r s of the Coulomb energy to Fermi energy. In the highly interacting, dilute regime (r s > 3), m * is typically significantly enhanced compared to its band value (m b ) and tends to increase with increasing r s [1,2,3,4,5,6,7,8,9,10]. A question of particular interest is the dependence of m * on the 2D electrons' spin and valley degrees of freedom as these affect the exchange interaction. Measurements on 2DESs in Si-MOSFETs (metal-oxide-semiconductor field effect transistors), have indicated that the m * enhancement is independent of the degree of spin-polarization [3]. On the other hand, recent m * measurements for 2D electrons confined to highly strained AlAs quantum wells revealed that, when the 2DES is fully spin-polarized, m * is suppressed down to values near or even slightly below m b [5]. The reason for this apparent discrepancy might be that the 2D electrons in the Si-MOSFET case occupy two conduction-band valleys [9] while the 2DES studied in Ref. [5] is a single-valley system. Here we experimentally study m * in an AlAs 2DES while we tune the valley and spin degrees of freedom in a single sample. The results, summarized in Fig. 1, provide direct and conclusive evidence that the valley and spin degrees of freedom are indeed both important in m * re-normalization. If the electrons are only partially valley and/or spin polarized, then their m * is enhanced with respect to m b . But if they are fully valley and spin polarized, then m * is suppressed. The fully spin and valley polarized regime is incidentally also the regime where the 2DES shows an insulating behavior.
We studied a high-mobility 2DES confined to an 11 nm-thick, modulation-doped AlAs layer, grown by molecular beam epitaxy on a (001) GaAs substrate [11]. In this sample, the 2D electrons occupy two energetically degenerate conduction-band valleys with elliptical Fermi contours [11], each centered at an X point of the Brillouin zone, and with an anisotropic mass (longitudinal   [11]. We control the valley occupation using ǫ and the spin occupation (polarization) via the application of magnetic field. We studied several samples; here we focus on data from one sample patterned with a 60×60µm 2 square mesa whose edges are aligned along the [100] and [010] directions, and the current is passed along [010]. The sample is glued to the side of a stacked piezo actuator allowing us to apply in-plane strain which can be controlled and measured in situ [11]. The magneto-resistance measurements were performed in a dilution refrigerator with a base temperature (T ) of 0.02 K and equipped with a tilting stage, allowing the angle θ between the sample normal and the magnetic field to be varied in situ.
The B vs. ǫ phase diagram of Fig. 1 highlights the important features of our results (B is the magnetic field applied parallel to the 2D plane). In this figure, the thick, solid lines mark the boundaries between the four possible spin and valley subband occupations for our sample, which had a density n = 3.8 × 10 11 cm −2 ; we denote the occupations by (s=2, v=2), (s=2, v=1), (s=1, v=2) and (s=1, v=1), where s and v stand for spin and valley, and 1 and 2 denote the number of spin/valley subbands that are occupied [12]. We determined these boundaries from measurements of sample resistance (R) vs. either B or ǫ while the other parameter is kept fixed. Examples of such data are shown in Fig. 2 at two temperatures. In each set of traces, R increases and shows a kink at a field or strain, B P or ǫ P , which marks the onset of full spin or valley polarization, respectively [13]. The B P and ǫ P values are shown in Fig. 2 by open square symbols. Note in Fig. 1 that the boundaries are not simple horizontal or vertical lines. This is because the Fermi energy (E F ) depends on the size of the (partial) occupation of the valleys or spins. Moreover, in a 2DES with finite layer thickness there is a shift, with B , of ǫ at which the valleys are equally occupied, or are "balanced", as indicated by the thin grey line going through ǫ = 0 in Fig. 1 [14]. We emphasize that, although we can measure and explain the boundaries shown in Fig. 1 accurately, their precise locations are not important for the conclusions of our manuscript. As we describe below, we carefully choose the conditions of our sample (B , ǫ, and θ) when we perform our m * measurements in different quadrants of Fig. 1. The stars show the coordinates where we measured m * in these four quadrants, and the deduced m * values are indicated next to the stars. Finally, the closed squares and the dashed lines in Fig. 1 mark the boundary for the apparent metal-insulator transition observed at this density in our sample (see Fig. 2 data). The 2DES shows an insulating behavior only as it becomes sufficiently spin and valley polarized [13].
In the remainder of the paper we describe our determination of m * , from the T -dependence of the Shubnikovde Haas (SdH) oscillations, in different quadrants of Fig.  1. Figure 3 shows the data for the (s=1, v=2) case which were taken at θ = 74.8 • and ǫ = −0.16 × 10 −4 . These values for θ and ǫ were chosen carefully so that the lowest eight Landau levels (LLs) are spin polarized and the two valleys have equal densities at the LL filling factor ν = 6, as shown in Fig. 3(b). Figure 3(a) shows the SdH oscillation near ν = 6 at several T. To deduce m * , we analyzed the T -dependence of the am- plitude of SdH oscillations using the standard Dingle expression [15]: ∆R/R o = 8exp(−π/ω c τ q )ξ/sinh(ξ), where the factor ξ/sinh(ξ) represents the T -induced damping, ξ = 2π 2 k B T /hω c , and ω c = eB ⊥ /m * is the cyclotron frequency, R o is the non-oscillatory component of the resistance, and τ q is the single-particle (quantum) lifetime. Here we make the common assumption that both R o and τ q are T -independent, and will return to its justification later in the Letter. Figure 3(c) shows that the data points (red circles) fit the Dingle expression (red line) quite well, yielding m * = 1.58m b , where m b = √ m l m t = 0.46m e . In Fig. 3(d) we show an alternative set of data from which we determine m * for the (s=1, v=2) case. In contrast to the standard SdH oscillations shown in Fig. 3(a), where we vary the magnetic field at constant ǫ, in Fig.  3(d) we fix the value of field so that the 2DES remains at ν = 6 and continuously change ǫ (see Ref. [16] for details). With applied ǫ, the LLs for the [100] and [010] valleys cross each other, as the fan diagram in Fig. 3(e) indicates, and R exhibits minima as E F goes through consecutive energy gaps, and maxima as it coincides with the LL crossings. We analyzed the T -dependence of the amplitudes of these resistance oscillations to deduce m * . In Fig. 3(c) we show a plot of ∆R/T vs. T (blue squares) for the oscillation centered around ǫ = −0.16 × 10 −4 ; the data fit the Dingle expression quite well (blue line), and yield m * = 1.46m b which is close to the value obtained from the field-sweep data of Fig. 3(a). In a similar fashion, we analyzed the T -dependence of the other resistance oscillations observed in Fig. 3(d); the data fit the Dingle expression well, and provide the m * values plotted in Fig.  3(f). Evidently, m * is smaller when we "valley polarize" the 2DES: note that the valley polarization (P V ), defined as the difference between the [010] and [100] valley populations divided by the total 2DES density, is equal to -0.33, 0, 0.33, and 0.67 for the four data points shown in Fig. 3(f) (from left to right).
The data for the (s=1, v=1) case are shown in Fig.  4(a). Here ǫ is large enough so that only the [010] valley is occupied. Moreover, by tilting the sample to a sufficiently large θ we ensure that the lowest seven LLs are spin polarized, as indicated in Fig. 4(a) energy level diagram. Note the qualitative similarity of the field traces in Figs. 4(a) and 2(b): R shows a pronounced increase with field and then saturates above ∼ 11T once the 2DES is completely spin polarized. Since the sample is not completely parallel to the field in Fig. 4(a), however, there are SdH oscillations which become well resolved after the full spin polarization; these are the oscillations from which we deduce m * . The inset to Fig. 4(a) shows the T -dependence of the SdH oscillation centered around ν = 5. From the fit to the Dingle expression we obtain m * =0.62m b at this ν = 5; data at ν = 6 yield m * =0.60m b . Figure 4(b) shows data for the (s=2, v=1) case. Here ǫ is chosen large enough so that the system is completely valley polarized and θ is adjusted such that the LLs at odd ν are at coincidence. The corresponding LL energy diagram is shown in Fig. 4(b) inset. Consistent with this diagram, in Fig. 4(b) traces, resistance minima at even ν are strong while the minima at odd ν are either weak or entirely absent (at high ν), or are accompanied by spikes (e.g., at ν = 3) [17]. By fitting the amplitude of the SdH oscillations near ν = 8, 10, and 12 to the Dingle expression, we obtain m * =1.16, 1.22, and 1.22m b , respectively (see Fig. 4(b) inset as an example). This observation implies that m * does not depend on the spin polarization (P S ), defined as the difference between the spin up and down populations divided by the total electron density; such independence is not surprising since in the range of 8 ≤ ν ≤ 12, P S changes only from 0.17 to 0.25.
In Fig. 4(c), we show data for (s=2, v=2). Here ǫ = 0 and θ is chosen so that the spin-up and spin-down LLs are at coincidence, as illustrated in the energy diagram in Fig. 4(c). The strongest resistance minima are observed at ν = 8, 12, and 16. We obtain m * = 1.58m b from the fit of SdH oscillation amplitude around ν = 12 to the Dingle expression, as shown in Fig. 4(c) inset.
In our determination of m * from the SdH oscillations, we have assumed that R o and τ q are T independent. It is apparent in Figs. 3-4, however, that R o has some dependence on T . To take this into account, we also analyzed our data by defining R o as the average resistance value near a SdH oscillation, and assumed that the relative Tdependence of τ q is similar to but has half the size of the relative T -dependence of R o [18]. The deduced m * from such analysis are about 10% smaller than the values reported above, indicating that the main conclusions of our study remain intact.
Our results presented in Figs. 3 and 4, and summarized in Fig. 1, provide direct experimental evidence for the dependence of m * on valley and spin polarization of the 2DES. This is intuitively plausible since m * renormalization in an interacting 2DES partly stems from the exchange interaction which depends on the spin and valley degrees of freedom. Several features of our data are noteworthy. First, we observe the largest m * enhancement for the (s=2, v=2) case. As we fully spin polarize the 2DES while keeping v=2, we observe little dependence of m * on spin-polarization. This observation is consistent with the experimental results of Shashkin et al. [3] who reported that m * does not depend on the degree of spin-polarization in Si-MOSFET 2DESs where the electrons occupy two valleys. Theoretical justification for this independence is provided in Ref. [9] where it is argued that in a two-valley 2DES the dependence of m * on the spin polarization is likely to be too weak to be experimentally measurable. On the other hand, when we keep s=2 and fully valley polarize the 2DES, we observe about 20% reduction in m * . The reason for this reduction is not clear, but we note that the spin and valley degrees of freedom are not identical. Second, data of Fig. 3(f) suggest that, if s=1, then there is reduction of m * with increasing valley polarization [19]. If one treats the spin and valley degrees of freedom as qualitatively similar, this observation is consistent with the theoretical work of Ref. [10] where, for the majority spin electrons, a reduction of m * with increasing spin polarization has been reported in a single-valley 2DES. Third, we find that m * is significantly reduced, to values even below m b when the 2DES is fully spin and valley polarized [5], implying that the exchange interaction is notably different when the spin and valley degrees of freedom are both frozen out.
Finally, our data imply that there may be a link between the m * suppression and the metal-insulator (MIT) transition observed in our 2DES. As seen in Fig. 1, the strong m * suppression and the insulating phase both occur in the regime where the 2DES is spin and valley polarized. It is important to note, however, that our m * data were all taken in the presence of a perpendicular component of the magnetic field. Such a field could sup-press the insulating behavior; this can be seen, e.g., in Fig. 4(a) where the MIT observed around B = 9T in the upper traces of Fig. 2(a) is no longer visible.
In summary, measurements in an AlAs 2DES reveal variations of m * with spin and valley polarizations. Some of the variations we observe can be qualitatively explained by the existing calculations [9,10]. A quantitative understanding of our observations, including the apparent link between the m * suppression and the insulating phase in the fully spin and valley polarized regime, awaits future work.