Evaluation of SOI strength

contribution clearly dominates over BIA. Furthermore, studying the magnetotransport in a symmetrically doped, 10𝑛𝑚 In0.53Ga0.47As QW, Faniel et al. [78] were able to

experimentally demonstrate that the BIA contribution to SOI is much smaller than the contribution from SIA. The MT measurements of Schäpers et al. [79] found that the BIA-term in their In0.77Ga0.23As/InP heterostructure can be neglected as compared to the

SIA contribution.

Based on these literature reports, we can summarize that for structures having a higher In content and a larger QW thickness, the Rashba-mechanism is expected to clearly dominate over the Dresselhaus contribution. Considering the comparably large thickness of our 20𝑛𝑚 In0.75Ga0.25As QWs, we consequently infer that the PSH-state, as well as BIA spin

splitting can be neglected in the analysis of SOI-related effects in our heterostructure.

3.2 Evaluation of SOI strength

An experimental method to estimate the strength of Rashba-type SOI in a 2D system is given by the evaluation of the magnetooscillations, arising in the longitudinal resistivity in a MT measurement. In the following, we briefly summarize the two commonly employed evaluation methods of the Shubnikov-de Haas oscillations to obtain the Rashba coefficient 𝛼of the system.

3.2.1 FFT analysis of the magnetooscillations

The most commonly utilised approach to determine the SOI strength is by a fast Fourier transformation (FFT) of the longitudinal resistivity 𝜌𝑥 𝑥(𝐵), acquired in a MT measurement.

Therein, the coefficient 𝛼 is calculated on the basis of the determined 2D electron densities in the FFT of 𝜌𝑥 𝑥. Here, we shortly recapitulate the main steps towards the description

of Rashba-type SOI based on the electron densities of the 2DEG, wherein we follow an envelope function approach as presented in [80] and [79]. Considering the k·p-interaction of the Γ6conduction band with the remote valence bands to which the coupling is strongest,

i.e. the Γ7and Γ8valence bands, the corresponding Hamiltonian for a 2D electron system

can be written as H_2𝐷 =  𝐸𝑠𝑢 𝑏 𝑧 − ~2 2𝑚∗Δ 2 k+ 𝜑(𝑟k))  · 1 + 1 ~ 𝛼·  𝜎× ~ 𝑖 ∇ . (3.8) 𝐸𝑠𝑢 𝑏

𝑧 is the energy of the first size-quantized subband of the QW and 𝑧 points along the

(001) growth direction. The effective mass 𝑚∗_{in the above equation takes the form}

𝑚∗−1 =Ψ_𝑧   𝑚∗−1(𝑧)  Ψ𝑧  ,

with |Ψ𝑧ibeing the wave function of the unperturbed electrons of the first subband. By

expressing 𝑚∗_{in this form, the leakage of the wave function into the barrier material is}

3 Fundamental concepts of spin-orbit interaction

of the above equation (3.8) can be written as 𝛼 = ~ 2𝐸_𝑝 6𝑚0 {𝑎 0_hE 𝐴i + 𝑏 0_(E 𝐵𝑢 + E𝐵𝑙) − 1 2𝑏(|Ψ𝑧𝑢| 2_{− |Ψ} 𝑧𝑙| 2_{)} . (3.9)}

𝐸_𝑝is a so-called k · p-interaction parameter, which describes the coupling strength of the Γ6 conduction band to the coupled valence bands Γ7 and Γ8. hE𝐴i represents the

expectation value of the electric field inside the QW, i.e. material 𝐴, E𝐵𝑢

and E𝐵𝑙

 describe the expectation values of the electric field in the upper and lower barriers of the QW, i.e. material 𝐵. |Ψ𝑧𝑢|

2_{and |Ψ} 𝑧𝑙|

2_{are the wave function probabilities at the upper and}

lower QW barriers. The prefactor 𝑎0_{, 𝑏}0_{and 𝑏 contain material-dependent band structure}

parameters, as well as the energy level 𝐸 of the electronic state inside the QW. These prefactors can be expressed as

𝑎0=  1 (𝐸 − 𝐸𝐴 Γ₇)2 − 1 (𝐸 − 𝐸𝐴 Γ₈)2  (3.10) 𝑏0=  1 (𝐸 − 𝐸𝐴 Γ₇ − Δ𝐸Γ7)2 − 1 (𝐸 − 𝐸𝐴 Γ₈− Δ𝐸Γ8)2  (3.11) 𝑏 =  Δ𝐸_Γ 7 (𝐸 − 𝐸𝐴 Γ₇− Δ𝐸Γ7)2 + Δ𝐸Γ7 (𝐸 − 𝐸𝐴 Γ₇)2 − Δ𝐸Γ8 (𝐸 − 𝐸𝐴 Γ₈ − Δ𝐸Γ8)2 − Δ𝐸Γ8 (𝐸 − 𝐸𝐴 Γ₈)2  . (3.12)

The terms Δ𝐸Γ₇ and Δ𝐸Γ₈ present the valence band offsets of materials 𝐴 and 𝐵.

By means of equation (3.9) with the prefactors as described by the above relations (3.10) to (3.12), we deduce that the Rashba SOI is based on the coupling of the conduction band with the valence bands. By partitioning the Rashba term as in expression (3.9), it becomes clearly visible that the resulting Rashba-coupling coefficient 𝛼 is comprised of three contributing terms:

(1) The first term containing the prefactor 𝑎0_{parameterizes the SOI strength due to an}

electric field inside the QW, i.e. material 𝐴, acting on the conduction electrons. It can be simply written as the expectation value of the field inside the well, i.e. hE𝐴i, multiplied

by a material- and energy-dependent prefactor, 𝑎0_{. For an infinitely deep QW, this would}

be the only contribution to the Rashba spin-splitting.

(2) The second term takes the extension of the wave function |Ψ𝑧iinto the barrier material

𝐵 into account. Thus, this contribution is proportional to the expectation value of the electric field inside the barrier layers, i.e. E𝐵𝑢

and E𝐵𝑙

 .

(3) The third contribution to 𝛼 arises from the abrupt change of material the wave function experiences at the interfaces between materials 𝐴 and 𝐵. This contribution can be described as the difference of the probability densities of the conduction electrons at the upper and lower QW interfaces, multiplied by a band structure-dependent parameter 𝑏, which contains the corresponding band offsets [81, 82]. For a correct calculation of the Rashba parameter the last term in equation (3.9) is of vital importance as was explicitly shown in various publications in literature [79–85].

3.2 Evaluation of SOI strength

Thus, we find that an external electric field, which is for example generated by a gate electrode, enters into all of the three terms contributing to the expression (3.9) of 𝛼, since it modulates the band profile, as well as the wave function probability distribution. As already introduced in the preceding section, the resulting energy dispersion 𝐸 (𝑘k)of

the conduction electrons inside the QW splits into two spin-resolved parabolas, which are shifted horizontally in 𝑘-space according to

𝐸(𝑘_k) = 𝐸𝑠𝑢 𝑏

𝑧 +

~𝑘2k

2𝑚∗ ± 𝛼|𝑘k| . (3.13)

The horizontal shift of the two spin-split parabolas is determined by the above described Rashba-coefficient 𝛼. The energy eigenstates for a specific value of 𝑘𝑥are separated in

energy by Δ𝐸𝑘 =2𝛼𝑘. Commonly, one refers to this energy difference as the spin-splitting

energy Δ𝐸𝑠 𝑠 for a specific 𝑘.

Furthermore, the effective mass of an electron in a spin-resolved eigenstate depends on the particular energy branch: Electrons on the outer branch exhibit a larger effective mass than electrons on the inner branch. Consequently, the energy spin-splitting manifests itself in the 2D DOS, which is given by

D±(𝐸) = 1 2 𝑚∗ 𝜋_~2 1 ∓ 1 q 1 + [2(𝐸 − 𝐸𝑠𝑢 𝑏 𝑧 )~2]/(𝛼𝑚 ∗₎ ! . (3.14)

The individual DOS of each branch is found to resemble half the one of the spin-degenerate 2D system, i.e. D2𝐷 =2𝑚∗/(𝜋~2), which is now further modified by an energy-dependent

correction factor. This factor takes the altered effective electron masses on the inner and outer parabolas into account. Looking at a particular Fermi energy 𝐸𝐹, we see by means

of equation (3.14) that an imbalance of the charge carrier densities in the two spin-splitted energy branches is present. By integrating over the DOS D+(𝐸)and D−(𝐸)up to 𝐸𝐹

yields the corresponding spin-degenerate electron densities, which are now labeled as 𝑛+

and 𝑛−.

In a MT measurement, these two spin-split branches in the 2DEG with the densities 𝑛+

and 𝑛− both contribute to the Shubnikov-de Haas oscillations, described by equation

(2.21). If the SOI lifts the spin degeneracy in this manner, a FFT of the longitudinal resistivity 𝜌𝑥 𝑥(𝐵)over 1/𝐵, which spectrally decomposes the signal, yields a prominent

double-peak structure, which thus serves as an experimental signature of the spin-split dispersion. The two frequencies obtained in the FFT spectrum correspond to the densities 𝑛₊and 𝑛₋. The FFT process is described in detail in subsection 9.2.4.

As was first pointed out by Luo et al. [86], measuring 𝑛+ and 𝑛− in a MT measurement

therefore yields access to the SOC coefficient 𝛼 via the relation 𝛼= Δ𝑛~ 2 𝑚∗ r 𝜋 2(𝑛 − Δ𝑛) . (3.15)

3 Fundamental concepts of spin-orbit interaction

Δ𝑛 = 𝑛+− 𝑛−, whereas 𝑛 is the average value of the charge carrier densities 𝑛 = 𝑛++ 𝑛−.

If no further conducting channel contributes to the transport in the system, 𝑛 can be substituted by the measured Hall density 𝑛𝐻 𝑎𝑙 𝑙.

We want to point out that for the derivation of formula (3.15) Zeeman spin-splitting, as well as contributions from BIA are neglected. Since we expect the Dresselhaus contribution to spin-splitting to be small as compared to the SIA contribution in our system, the effect of Zeeman spin-splitting will dominate potential deviations of the calculated 𝛼 from reality in our case.

3.2.2 Beating-node analysis of the magnetooscillations

Based on the analysis of the magnetooscillations in the longitudinal resistivity, there is a second evaluation method to determine the SOI strength, which we will present in the following. In contrast to the above presented method, however, Zeeman spin-splitting is not incorporated in the estimation of the Rashba coefficient.

The total Hamiltonian describing a Rashba spin-orbit coupled system in an external magnetic field can be expressed as

H = H𝑂 + H𝑍 + H𝑅 .

Therein, H𝑂 is the Hamiltonian of a quasi-free electron in a magnetic field, H𝑍 is the

contributing Zeeman term and H𝑅takes the structure-induced inversion asymmetry into

account. The corresponding eigenenergy spectrum is given by [70, 87]: 𝐸𝑛_± = ~𝜔𝑐(𝑛 + 1 2 ± 1 2) ∓ r (~𝜔𝑐− 𝑔∗𝜇𝐵𝐵)2+ 8𝛼2𝑒 𝐵 ~ (𝑛 + 1 2 ± 1 2) (3.16) (+) and (−) denote the two spin-resolved eigenstates. Whereas the DOS of the Landau levels and the Zeeman spin-splitting go linearly with the applied external magnetic field 𝐵 - if B is orthogonal to the 2DEG - the Rashba SOI causes a B-field-dependent enhancement of the spin-splitting in the system. Hence, the ratio of the total spin-splitting energy 𝛿𝑠 𝑠

and the Landau level separation Δ𝐸𝐿 𝐿 = ~𝜔𝑐, i.e. 𝛿𝑠 𝑠/Δ𝐸𝐿 𝐿, is not constant anymore

as it would be for an inversion-symmetric system subjected to a perpendicular magnetic field. In this case, the ratio 𝛿𝑠 𝑠/Δ𝐸𝐿 𝐿 = 𝑔

∗

𝜇𝐵𝐵/~𝜔𝑐is constant. Owing to the presence

of Rashba SOI, the ratio becomes 𝐵-field-dependent.

The modification of 𝛿𝑠 𝑠due to SOI becomes more dominant in the limit of small magnetic

field values. Here, Landau levels, described by equation (3.16), cross when the ratio 𝛿_{𝑠 𝑠}/Δ𝐸_{𝐿 𝐿}takes an integer value. The alteration of the energy separation of the eigenstates by sweeping the external magnetic field manifests itself in an amplitude modulation of the Shubnikov-de Haas oscillations: a beating in 𝜌𝑥 𝑥(𝐵) arises. A node in 𝜌𝑥 𝑥(𝐵)

corresponds to the situation where the energy separations of the neighboring eigenstates from equation (3.16) exhibit equivalent distances, whereas a peak in the amplitude of the envelope of 𝜌𝑥 𝑥(𝐵)is observed when the eigenstates are not equally spaced in energy.