Los Millennials y el compromiso marcario hacia el medioambiente

All analysed top-gated Hall bar samples 𝐴 to 𝑁 from chapter 7 exhibit an equivalent gating response as sample 𝐻, which we have discussed in more detail: After the linear gating response of the charge carrier density 𝑛𝑠 in regime I, 𝑛𝑠 decreases with

increasing TG-voltage (regime II, III and IV). The density decrease in gating areas II, III and IV is attributed to a pronounced charge migration from the QW towards the semiconductor/dielectric interface via the deep level donor states inside the InAlAs spacer layer according to our charge transfer model. In gating area III, we find clear evidence of a large Rashba-type SOI in our MT measurements in the form of a beating in the longitudinal resistivity 𝜌𝑥 𝑥(𝐵). These undulations of the Shubnikov-de Haas oscillation

amplitude vanish when we enter the transitional regime IV (𝑉𝑇 𝐺 > +4.5𝑉) to the saturation

regime V, in which we have no longer capacitive coupling to the 2DEG due to the formed parasitic conductive layer at the semiconductor/dielectric interface. The assignment of our experimental observations in 𝜌𝑥 𝑥(𝐵)to SIA-induced SOI is not evident at first sight since

the total electric field at the QW is a superposition of the positive applied TG-voltage and the electric field, which is generated by the migrated electrons. Additionally, the ionized impurity potential of the InAlAs defect sites acts on the 2DEG. From this simple view, one would expect the electric field acting on the conduction electrons and thus the Rashba-type SOI to be reduced in regime III as compared to the end of the linear regime

3_{As described in 3.2.2, via the node analysis we acquire values for the 𝑔}∗_{-factor from the linear fitting of}

the node-positions. However, these values scatter strongly. They accumulate around values of 𝑔∗_{= −15,}

9 Signatures of Rashba-type spin-orbit interaction

I (in which no beating in 𝜌𝑥 𝑥(𝐵)is present).

A key observation in our experiments is that measurable SOI effects in our heterostructures solely arise in gating area III in which charge migration from the QW towards the interface takes place and a significant amount of electrons is located at InAlAs defect sites in the upper barrier layer. To visualize this situation, we self-consistently calculate the band structure of our gate stacking with a Schrödinger-Poisson solver [119–122]. In our simulations, we compare three exemplary electrostatic situations in the heterostructure, which are assigned to different gating areas in our MT measurement sequence. These band structure simulations are further evaluated by calculations in the envelope function approximation within the k · p -method, in which we use the total electric potential, calculated with the Schrödinger-Poisson solver, as input parameter. Performing a general folding-down procedure, the Rashba SOC parameter is calculated [55, 79, 200, 201].4 _We

want to emphasize that we do not intend to give an exact value of the Rashba coefficient with these calculations since we are aware that we lack an exact knowledge of the complex electrostatics in our gate stacking, i.e. the distribution and ionization of the deep level donor states inside the InAlAs barrier layers, as well as of the electron population of semiconductor/dielectric interface states. Instead, we aim to analyse the experimentally observed trend of the evolution of the Rashba-induced SOI strength in our heterostructure during gating. To this end, we employ simplified model assumptions of the density and ionization state of the InAlAs deep level donor states in the Schrödinger-Poisson simulations.

A sketch of the utilised layer system is shown in figure 7.1(a). The metal gate electrode is separated by a layer of 50𝑛𝑚 Al2O3 from the underlying semiconducting layers: a

2.5𝑛𝑚 InGaAs capping, a 130𝑛𝑚 InAlAs spacer and the 20𝑛𝑚 InGaAs QW. The electron accumulation at the semiconductor/dielectric interface, arising due to residual energy states at the interface (see chapter 7) is accounted for as a thin negatively charged layer in these simulations. Since we want to give a qualitative picture, we choose an InAlAs deep level donor density of 𝑛𝐷 ,1 = 𝑛𝐷 ,2=3 · 1016𝑐𝑚

−3_{, which is rather large compared to}

literature, being evenly distributed inside the InAlAs layers, for the calculations. The TG electric field inside the heterostructure is modeled by the applied bias at the metal gate electrode. Figure 9.9(a) displays the conduction band profile 𝐸𝐶 𝐵 at the Γ-point (black

solid line) as a function of the growth-direction 𝑧 for which the applied TG-voltage 𝑉𝑇 𝐺

is set to 0𝑉. The heterostructure is arranged in such a way that the InGaAs cap ends at 𝑧 = 0. Thus, the top InGaAs-QW interface is positioned at 𝑧 = 132.5𝑛𝑚, the bottom QW interface at 𝑧 = 152.5. In negative 𝑧-direction, we find the high-band gap insulator, followed by the metal gate electrode. The blue dashed line indicates the InAlAs deep level donor density in the heterostructure, the black dashed lines indicate the density of the ionized InAlAs deep level donor states. The negatively charged interface layer and the asymmetric ionization of InAlAs defect states above and beneath the QW lead to an asymmetric conduction band profile even in the zero-biased case. A zoom into the

4_{The calculations were performed by Dr. Paulo E. de Faria Junior from the group of Prof. Dr. Jaroslav}

9.4 SOI in undoped InGaAs/InAlAs systems

Figure 9.9:(a)−(c): Band structure simulation, corresponding to regime I, with

VTG= 0V, nD1= nD2= 3·1011cm-2evenly distributed in the InAlAs spacer layers.

(d)−(f): Band structure simulation, corresponding to the end of regime I, with VTG = +1V, nD1 = nD2 = 3·1011cm-2 evenly distributed in the InAlAs spacer

layers. (g)−(i): Band structure simulation, corresponding to regime II, III and IV, with VTG= +1V, nD1= nD2 = 3·1011cm-2evenly distributed in the InAlAs

spacer layers and artificially deionized in the z-interval [50nm, 120nm].

The plots (a), (d) and (g) display the conduction band edge 𝐸𝐶 𝐵 at the Γ-point as

solid line (a.u.) with the total density of the InAlAs defect states (cyan dotted line) and the ionized density of the InAlAs defect states (black dotted line). Figures (b), (e) and (h) show the conduction band edge (solid line) with the wave function probability |Ψ|2_{(dotted line). Figures (c), (f) and (i) present the}

conduction band edge (dotted line) with the derivative of the potential weighted by the wave function probability, i.e 𝑑𝜑/𝑑𝑧 |Ψ|2_{(solid line).}

9 Signatures of Rashba-type spin-orbit interaction

band profile in the vicinity of the QW, shown in figure 9.9(b), displays an asymmetric tilting of the QW towards the bottom QW interface. The wave function probability |Ψ(𝑧) |2is indicated as dotted line. Figure 9.9(c) shows a plot of the electric field in the heterostructure, i.e. 𝑑𝜑(𝑧)/𝑑𝑧, weighted by |Ψ(𝑧)|2_{. According to equation (3.9), the}

barrier contribution to the Rashba coefficient scales with the band offsets weighted by |Ψ(𝑧) |2 at the bottom and top barrier QW interfaces [80–82, 84]. For the electrostatic situation shown here in figures 9.9(a) to (c), we find the contributions of the top and bottom QW interfaces to almost cancel each other. Furthermore, the strength of the electric field inside the QW, giving rise to a non-interfacial contribution to the Rashba coefficient, is small. We thus may expect the Rashba-induced SOI in the case of 𝑉𝑇 𝐺 =0𝑉

to be below our limit of detection with MT measurements. From the calculations of the strength of SOI by means of the band structure simulations, we find a Rashba coefficient of 𝛼= −0.1·10−12𝑒𝑉 𝑚. This value is far below the resolution limit in our MT measurements. The calculation also tells us that the interface and the non-interface contribution terms (see section 3.2) are comparable in magnitude. Both, the value and the comparable small contributions of interface and non-interface, are consistent with the observed absence of SOI effects in the longitudinal resistance in gating regime I.

Figure 9.9(d) displays an electrostatic situation that corresponds to the end of the linear gating area I of our MT measurements, for which 𝑉𝑇 𝐺 = +1.0𝑉 is applied. As compared

to figure 9.9(a) with 𝑉𝑇 𝐺 =0𝑉, we find the conduction band profile to be tilted downwards

here. Figure 9.9(e) displays a zoom into the conduction band profile in the vicinity of the QW, for which the corresponding wave function probability is plotted as a dotted line. This configuration presents itself as more symmetric than the band profile in the zero-bias case. We assign this development to a compensation of internal electric fields and the external TG-field. As illustrated in figure 9.9(f), the symmetric arrangement of the wave function probability inside the QW manifests itself in an almost total balancing of the interface contributions to Rashba-type SOI. Consistently, the calculation of the Rashba coefficient shows a decrease of 𝛼 by a factor of four to 𝛼 = −0.024 · 10−12

𝑒𝑉 𝑚. The simulated situation in figures 9.9(d) - (f) corresponds to the end of gating regime I in our MT measurement sequence, in which we find no indications of SOI in the magnetooscillations in our experiments. This in line with the calculations of 𝛼.

Figure 9.9(g) displays the conduction band profile for 𝑉𝑇 𝐺 = +1.0𝑉, for which we

artificially deionize the deep level donor states inside the upper InAlAs barrier layer in the 𝑧-interval between 50𝑛𝑚 and 120𝑛𝑚. This corresponds to the electrostatic situation in gating areas II, III and IV in our MT measurement sequence, in which significant charge migration from the QW via InAlAs deep level donor states takes place. Evaluating the gained electron density in the QW in this situation yields an equivalent charge density as for the case depicted in figure 9.9(a) with 𝑉𝑇 𝐺 =0𝑉. A zoom into the band profile

in the vicinity of the QW in figure 9.9(h) reveals a pronounced structural asymmetry. The band profile is tilted towards the bottom QW interface, which leads to an equivalent shift of the wave function probability. Figure 9.9(i) depicts the wave function probability weighted with the electric field. The plot clearly shows the modification of the spatial symmetry due to charge transfer from the QW into the top InAlAs barrier. This leads to a

9.4 SOI in undoped InGaAs/InAlAs systems

significant change in the strength of the SOC. The calculations of the Rashba coefficient yield an increase of 𝛼 by around a factor of 12, as compared to the situation in figure 9.9(d) with 𝑉𝑇 𝐺 = +1.0𝑉 without the artificial deionization of InAlAs defect states. This

is in line with our experimental observation of a significant increase of the Rashba-type SOI, as we experimentally find it to rise above our experimental resolution limit (see figure 9.7) in gating area III. Even though, our calculated values of 𝛼 do not coincide with the experimentally determined values, the above band structure simulations together with the corresponding calculations of 𝛼 clearly reflect the evolution of the SOI strength in our heterostructure and underline the central role of intrinsic electric fields on the structural inversion asymmetry in the layer system.