Descripción general Millennials

Usually, in the literature, the FFT of the Shubnikov-de Haas oscillations is restricted to small magnetic fields. This limitation is mainly motivated by two effects: The Zeeman spin-splitting, as well as the breakdown of the semi-classical model description of the Landau level DOS as a sinusoidal function of 1/𝐵 both lead to a modification of the frequency spectrum of the longitudinal resistivity at sufficiently high magnetic fields. These two mechanisms are discussed in the following.

Limits of the semi-classical description

As introduced in subsection 2.1.2, we can describe the magnetoresistivity of a 2DEG in the limit of small magnetic fields by means of a semi-classical formula:

Δ𝜌𝑥 𝑥 = 𝜌0+4· 2𝜋𝑘𝐵𝑇 𝑚 ∗ ~𝑒𝐵 · 1 sinh2𝜋𝑘𝐵𝑇 𝑚 ∗ ~𝑒𝐵 ·exp−𝜋𝑚 ∗ 𝑒 𝜏𝑞 · 1 𝐵  ·cos2𝜋E𝑚 ∗ ~𝑒 ·1 𝐵 − 𝜋 , (9.2)

with E = 𝐸𝐹 − 𝐸𝑛, where 𝐸𝑛is the energy of the nth subband and 𝜏𝑞 is the quantum

lifetime [48]. This semi-classical description of 𝜌𝑥 𝑥(𝐵)is roughly valid for magnetic fields

for which 𝜔𝑐𝜏𝑞 5 1 still holds. For larger magnetic fields, however, the magnetoresistivity

starts to deviate from this behavior as 𝜌𝑥 𝑥(𝐵)vanishes over finite magnetic field intervals.

This effect, referred to as quantized Hall effect [53, 54], sets in when the Fermi level lies in between two consecutive Landau levels and only negligible scattering of charge carriers into neighboring extended Landau states takes place. We can determine 𝜏𝑞by means of

equation (9.2) from the linear slope of a Dingle plot, in which the exponential decay of the Shubnikov-de Haas oscillation amplitude as a function of 1/𝐵 is evaluated, as long as further modulations of the oscillation amplitude in 𝜌𝑥 𝑥(𝐵)are not present. Accordingly,

we cannot give an estimation of 𝜏𝑞 in gating area III since therein the magnetooscillation

amplitude is modulated by the additional envelope function (see figure 9.1(b)). To this end, we evaluate the amplitude decay of the Shubnikov-de Haas oscillations in gating area I. This yields an upper limit for 𝜏𝑞 in our samples, as the charge migration in gating areas

9.2 FFT of magnetooscillations

II and III will tend to reduce the electron mobility in the heterostructures (see section 7.5.1). Based on this evaluation method of 𝜌𝑥 𝑥(𝐵), we find 𝜏𝑞to decrease with increasing

𝑉_{𝑇 𝐺} at the end of gating area I and in gating area II. This is shown in the appendix B.2. At 𝑉𝑇 𝐺 = +1.4𝑉, 𝜏𝑞is as low as 1.3 · 10

−13

𝑠, which is about 30-times smaller than the corresponding transport scattering time 𝜏𝑡𝑟, pointing towards long-range scattering

potentials [43]. This value of 𝜏𝑞limits the legitimized FFT interval to 𝐵 ≤ 2𝑇. Since we

assume a further drop of 𝜏𝑞 in gating interval III due to effects from charge migration, we

do no not consider the above determined boundary value to be a hard limit. Zeeman spin-splitting

An external magnetic field, breaking the time reversal symmetry, lifts the spin degeneracy. For bulk semiconductors, in first order, the Zeeman spin-splitting Δ𝐸𝑍 is isotropic in

space and can be described with

Δ𝐸_Z = 𝑔∗𝜇𝐵𝐵 , (9.3)

where 𝑔∗ _{is the effective Landé g-factor and 𝜇}

𝐵 is the Bohr magneton. Thus, the spin-

splitting strength is not only proportional to the applied magnetic field 𝐵 but also to the prefactor 𝑔∗_{. Including the Zeeman spin-splitting from equation (9.3) into the description}

of the Landau level eigenstates of equation (2.18) simply yields 𝐸_𝑛 𝑥 𝑦 = ~𝜔𝑐  𝑛_{𝑥 𝑦} +1 2  ± 1 2𝑔 ∗ 𝜇_𝐵𝐵 (9.4)

in the case of B k z, with z being the growth direction. This energy spin-splitting enters the description of the corresponding DOS of the Landau levels, provoking a modification of the periodicity of the Shubnikov-de Haas oscillations at sufficiently strong magnetic fields. Since the factor of two in equation (9.1) is not included anymore in the spin-split case, a transition of the 1/𝐵-periodicity of the magnetoresistivity at a frequency 𝑓 to a harmonic at 2 𝑓 in the spectrum takes place if the Zeeman splitting is fully resolved [197]. Winkler [55] pointed out that Zeeman spin-splitting does not affect the determined frequency 𝑓 of the spin-degenerate oscillations in the Fourier power spectrum of 𝜌𝑥 𝑥. Yet, it alters

the amplitude and phase of the Shubnikov-de Haas oscillations. Consequently, for our experiments, we conclude from these considerations that while Zeeman spin-splitting will induce a modification of the FFT of the magnetooscillations, yet it cannot be responsible for the observed double-peak structure in the Fourier transformed power spectra in gating area III. Moreover, Zeeman splitting is not resolved in our MT measurements for 𝐵 < 3.5𝑇. Correspondingly, we expect Zeeman spin-splitting-induced features in the magnetooscillations to be less prominent as compared to samples, for which Zeeman-type spin-splitting is already resolved at low magnetic field values.

In the following, in order to test the dependence of the FFT spectra on the utilised B-field interval, we contrast the Fourier transformed spectra for two different upper magnetic field limits.

9 Signatures of Rashba-type spin-orbit interaction

Comparison of B-field intervals for FFT analysis Figure 9.3 displays two FFT spectra of the magnetoresistivity measurement of sample 𝐻 at 𝑉𝑇 𝐺 = +4.0𝑉, determined

in different magnetic field intervals. The cyan curve displays a FFT spectrum, determined in the magnetic field interval 𝐵 ≤ 2𝑇, legitimized by the semi-classical description in the framework of the QHE, for a corresponding quantum lifetime of 𝜏𝑞 ≈ 1.3 · 10

−13

𝑠. In contrast, the grey curve displays the Fourier transformed magnetooscillations in an extended magnetic field range of 𝐵 ≤ 4𝑇, which we transformed in the FFT process in figure 9.1. By comparing the Fourier transformed signals, we find both spectra to exhibit the equivalent spectral information. However, the double-peak structure is not fully resolved in the reduced magnetic field interval. Instead, the second peak appears as a shoulder at the right side of the main peak at 𝑓 ≈ 12𝑇. We assign the undulated background, which arises in both spectra, to be caused by the finite amount of Fourier transformed oscillations in 1/𝐵 and by the abrupt ending of the transformation interval, as well as by the residual background in 𝜌𝑥 𝑥, which has not been fully eliminated by the

polynomial fit (see figures 9.1(a) and (b)). Mathematically, an interaction between the two transport channels, which correspond to the observed main frequencies 𝑓𝑎and 𝑓𝑏, is

likely to generate sum- and difference-frequency peaks in the spectral decomposition of the signal. This will be further addressed in subsection 9.3.1.

Figure 9.3:Comparison of two FFT spectra of the magnetoresistivity of sample

𝐻 at 𝑉_{𝑇 𝐺} = +4𝑉, determined in different magnetic field intervals.

From the above analysis, we infer that using an extended Fourier transformed magnetic field interval (here up to 4𝑇) is not responsible for the generation of the observed double- peak structure in our spectra. Restricting the applied FFT magnetic field interval to 𝐵 ≤ 2𝑇 solely deteriorates the spectral resolution of the transformation due to the reduced number of included magnetooscillations.

In the following, we thus deliberately employ an extended magnetic field range for our FFT analysis of the MT measurements in gating regime III to improve our analysis of the observed frequency components.