SOI in 1D systems

This can be translated into a mathematical relation: A node in 𝜌𝑥 𝑥(𝐵)appears when the

ratio 𝛿𝑠 𝑠/~𝜔𝑐becomes half-integer.

Via the amplitude modulation of the Shubnikov-de Haas oscillation, the Rashba SOI- strength 𝛼 can be determined. As was shown by Das et al. [83], to obtain a good estimate for the SOI strength 𝛼, the total spin-splitting of the eigenstates (3.16) can be approximated as

𝛿_{𝑠 𝑠} ≈ Δ𝐸_𝑠𝑜+ 𝑠𝑔∗𝜇_𝐵𝐵 .

The modulation of the amplitude 𝐴 of the Shubnikov-de Haas oscillation in a MT measurement of 𝜌𝑥 𝑥 is given by 𝐴∝ 𝑐𝑜𝑠  𝜋 𝛿 ~𝜔𝑐  .

Hence, the condition for a node-position reads as 𝛿𝑠 𝑠

~𝜔𝑐

= 𝑖 − 1 2 , with 𝑖 ∈ R+ _{being the node index of the beating in 𝜌}

𝑥 𝑥. Inserting 𝜇𝐵 = 𝑒_~

2𝑚0 into the upper

equation yields 𝑖 = 2𝑚 ∗ 𝛼 𝑘𝐹 ~𝑒 1 𝐵  + 𝑔 ∗ 𝑚∗ 2𝑚0 + 1 2  . (3.17)

Thus, from the slope of the linear dependence of the node-index 𝑖 on 1/𝐵 the Zeeman spin-splitting-independent estimation of the Rashba spin-orbit parameter 𝛼 can be obtained [79].

Yet, to give a meaningful estimation of 𝛼 by means of this beating-node analysis at least three clear nodes in 𝜌𝑥 𝑥(𝐵)have to be present. This often impedes the implementation of

this experimental evaluation method in reality.

Furthermore, from the intersection of the linear fit with the y-axis, the effective g-factor, 𝑔∗, can be determined.

3.3 SOI in 1D systems

Additionally to the above described SOI terms in 2D, arising from the periodic microscopic crystal potential (BIA term), as well as from electric fields, generated by the symmetry breaking along the growth-direction (𝑧-axis) due to heterointerfaces, doping and/or the application of an external electric field (SIA term), a further contribution to SOI can be identified in gate-defined 1D narrow constrictions. The electronic confinement potential of a QPC introduces a lateral inversion asymmetry in the transversal direction (here 𝑦-direction), generating electric fields at the opposing potential walls of the narrow constrictions [63, 88, 89]. This may lead to a spin accumulation at the opposing sidewalls

3 Fundamental concepts of spin-orbit interaction

of the QPC in a spin-orbit coupled system. For a symmetric confinement potential, the spin polarizations inside the 1D channel cancel each other. Yet, for an asymmetric confinement one particular spin polarization predominates. This effect was already implemented in a proof-of-concept all-electric spin-polarizer device by Debray et al. [19].

In asymmetric QW systems the contribution of this lateral SOI (LSOI) is generally up to three orders of magnitude smaller than SIA-type SOI for symmetric biasing of the QPC [79, 88]. However, introducing a pronounced asymmetry in the confinement potential of the 1D channel by means of a gate electric field between the QPC-defining finger-gate electrodes leads to an enlarged LSOI, which is likely to dominate the SOI-terms of the system [13]. Such an experimental situation is shown in figure 3.2(a), in which a split-gate (SG) defined QPC is introduced in a 2DEG with high intrinsic SOI.

Figure 3.2: (a)Sketch of a SG-defined QPC, asymmetrically biased with 𝑉𝑆𝐺1 ≠

𝑉_𝑆𝐺2. An electron moving with 𝑘_𝑥 through the 1D constriction experiences the electric field E between SG1 and SG2 as an effective magnetic field 𝐵𝑆𝑂

𝑒 𝑓 𝑓,

pointing in 𝑧-direction in their restframe. (b) Dispersion relation of electrons inside a QPC, in which the 1D subbands are laterally shifted due to LSOC. The green dashed line indicates the Fermi energy 𝐸𝐹.

Biasing the SG-electrodes SG1 and SG2 asymmetrically with 𝑉𝑆𝐺1 ≠ 𝑉𝑆𝐺2 < 0𝑉

introduces an electric field E = −∇𝑉 (r) in the channel. For simplicity, we assume a linear potential gradient between SG1 and SG2 with 𝑉 (r) = 𝑐 · ey. By means of relation (3.5),

the corresponding SOI Hamiltonian for electrons moving with momentum 𝑘𝑥 in the 1D

channel can be expressed as

H_𝑅1𝐷 = 𝛼∇𝑉 (r) · k × 𝝈 = 𝛼𝑐𝜎𝑧𝑘𝑥 . (3.18)

The corresponding effective magnetic field 𝐵1𝐷

𝑒 𝑓 𝑓 ≡ Ω𝐿 𝑆𝑂𝐶 reads as Ω𝐿 𝑆𝑂𝐶 = 𝛼 © ­ « 0 0 𝑐 𝑘_𝑥 ª ® ¬ .

Accordingly, the conduction electrons inside the QPC experience an effective magnetic field 𝐵1𝐷

𝑒 𝑓 𝑓, directed along the 𝑧-axis, perpendicular to their direction of motion 𝑘𝑥 and to

3.3 SOI in 1D systems

The LSOI results in a lateral shift of the 1D subbands for spin-up and spin-down electrons against each other as displayed in figure 3.2(b). The strength of the LSOI, and thus of the shift of the two spin-branches can be tuned by the applied offset-voltage Δ𝑉𝑆𝐺 = |𝑉𝑆𝐺1− 𝑉𝑆𝐺2| between the SG-electrodes. Tuning the Fermi energy 𝐸𝐹 under

the crossing point of the two branches and by applying a current between the source and drain 2D electron reservoirs leads to a spin-polarized current in 𝑥-direction. Thus, the QPC acts as an all-electrical spin filter with a theoretical spin-polarization efficiency of 100%. This concept of spin polarization can be employed in an all-electric spin-transistor device [13], which is schematically depicted in figure 3.3.

Figure 3.3:Sketch of an all-electric spin-transistor concept, composed of two

serial QPCs, QPC1 with 𝑆𝐺1

1and 𝑆𝐺12and QPC2 with 𝑆𝐺12and 𝑆𝐺22, having a

distance 𝑑 < 𝑙𝑚 𝑓 𝑝in 𝑥-direction. Via Rashba-type SOI, the middle gate-electrode

𝑀 𝐺 controls the spin precession of the conduction electrons moving in 𝑘 by modifying the strength of Rashba-type SOI in the 2DEG underneath.

Therein, two serial SG-defined QPCs are formed in a 2DEG with large intrinsic SOI. The SG-electrodes of QPC1 and QPC2 are biased asymmetrically with the corresponding offset voltages being defined as Δ𝑉1

𝑆𝐺 = |𝑉 1 𝑆𝐺1− 𝑉 1 𝑆𝐺2|at QPC1 and Δ𝑉 2 𝑆𝐺 = |𝑉 2 𝑆𝐺1− 𝑉 2 𝑆𝐺2|

at QPC2. The distance 𝑑 between the QPCs in 𝑥-direction has to be smaller than the elastic mean free path of the conduction electrons travelling inside the device. Thereby, the spin-polarized electrons flow ballistically from QPC1 to QPC2 and thus spin relaxation can be neglected. In this device, QPC1 acts as a spin polarizer, whereas QPC2 serves as a spin analyser. When 𝐵1𝐷

𝑒 𝑓 𝑓 in QPC1 and in QPC2 point into the same direction, the

conduction electrons can pass through both constrictions. For the opposite sign of 𝐵1𝐷 𝑒 𝑓 𝑓

in QPC1 and QPC2, the resistance of the device is increased.

The current flow through the constrictions can be further modified by an additional middle-gate (MG) situated in between the two QPCs. Applying an external electric field along the 𝑧-direction via the MG yields an effective magnetic field 𝐵2𝐷

𝑒 𝑓 𝑓, pointing

along the 𝑦-direction for electrons with momentum 𝑘𝑥. By tuning the voltage 𝑉𝑀 𝐺 at the

MG-electrode, the resulting spin precession angle of the conduction electrons impinging on QPC2 can be adjusted, which eventually controls the current through the device.

II

4 Experimental methods . . . 37 4.1 Molecular-beam epitaxy . . . 37 4.2 Postgrowth crystal characterisation methods . 39 4.3 Electric transport measurements. . . 40 5 Design and characterisation of In-

GaAs/InAlAs systems. . . 45 5.1 Buffer layer growth . . . 45 5.2 Design of InGaAs/InAlAs active layer systems . 49 5.3 Discussion and conclusion . . . 55

Epitaxy and methods

Experimental methods

4

This chapter shortly presents the applied structural and electrical characterisation methods of the heterostructure devices, which we employ during the course of this thesis. We further give technical details to the used molecular-beam epitaxy system, as well as to our low-temperature setups.