To quantitatively analyze the WL/WAL features, we evaluate the SOC strength by fitting the magnetoresistance data to the HLN model. The Rashba SOC Hamiltonian in a HH band is k-cubic dependent and written as 𝑯𝑠𝑜𝐻𝐻 = α3𝐸𝑧𝑖(𝑘−3𝜎+− 𝑘+3𝜎−) while the Hamiltonian in a LH band is linearly dependent on k by 𝑯𝑠𝑜𝐿𝐻 = α1𝐸𝑧𝑖(𝑘−𝜎+− 𝑘+𝜎−) [36]. α3(1) represent the k-cubic (linear) Rashba coefficient and 𝐸𝑧 represents the average vertical electric field along the z-direction. 𝑘±= 𝑘𝑥± 𝑖𝑘𝑦 are the two 𝑘∥ components, and 𝜎±= (𝜎𝑥± 𝜎𝑦)/2 are the combinations of Pauli spin matrices. The
series of 𝐵𝜙 and 𝐵𝑠𝑜. If we only consider the contributions of the Rashba SOC effects
from the HH band (i.e. only the k-cubic contribution), ILP model could be simplified to a closed form [63], which is known as the HLN model [108]. The formula is written as follows [14, 63]:
To justify this assumption, we need to check whether the carriers only occupy in the HH band by taking the Fourier transform of the longitudinal resistivity (ρ𝑥𝑥). The inset in Fig. 4-5 (a)~(c) shows the periodical oscillations of ρ𝑥𝑥 versus the reciprocal magnetic field. The presence of the periodicity is due to the Fermi level sweeping over consecutive Landau levels (Fig. 2-4). Each Landau level has a degeneracy of 𝑒𝐵/ℎ, so
Fig. 4-5 Fast Fourier transformation of Δρ𝑥𝑥 for (a) Ge0.94Sn0.06/Ge, (b) Ge0.91Sn0.09/Ge, and (c) Ge0.89Sn0.11/Ge heterostructures.
we have the relationship of ν × 𝑒𝐵/ℎ = 𝑝2𝐷. The periodicity of reciprocal magnetic field
can be calculated as [21] the reciprocal magnetic field domain. The transformed results of the longitudinal resistivity are shown in Fig. 4-5. There are two peaks observed and one peak has twice the value of the other. The major peak corresponds to the carrier density and the second
peak is the harmonic term [109]. We used the first peak to calculate 𝑝𝐹𝐹𝑇 and compared it to the Hall measurement results (𝑝𝐻𝑎𝑙𝑙). Those carrier densities match with each other very well and no extra peak of 𝐵𝐹𝐹𝑇 in the SdH oscillations shows that only the lowest HH subband is occupied.
There are two other criteria to meet when using the HLN model. Since the HLN model is only valid in the spin-diffusive regime [14], the spin-precession length (𝐿𝑝𝑟𝑒) (the distance over which holes can travel during one period of Larmor precession) needs
Fig. 4-6 (a) A physical picture of the transport magnetic field. (b) The transport magnetic field versus carrier density for different GeSn/Ge heterostructures.
to be longer than the mean free path (𝐿𝑡𝑟) as the applied magnetic field (B) is smaller than the corresponding transport field (i.e. 𝐵 < 𝐵𝑡𝑟 = ℏ/(2𝑒𝐿2𝑡𝑟)). The physical picture of the transport field (𝐵𝑡𝑟 ) is shown in Fig. 4-6 (a). For a magnetic field smaller than the transport field, the orbital bending due to the magnetic field can be ignored compared to the scale of mean free path (𝐿𝑡𝑟) [68]. On the other hand, for a magnetic field larger than the transport field, we need to consider the orbital bending and the more general formalism is described by the Glazov and Golub (G&G) model [110]. The calculated results of the transport field are shown in Fig. 4-6 (b). Since the smallest transport field is 10 mT, we focus our fitting regime within ±10 mT.
We used the HLN model (Eq. 4-3) with 𝐵𝜙 and 𝐵𝑠𝑜 as the fitting parameters to fit
the WL/WAL patterns. Fig. 4-7 (a) and Fig. 4-8 (a) show the fitting results when only WL or WAL patterns are observed, respectively. To evaluate the fitting results, we calculate the coefficient of determination (R2) [111], which reflects how “good” the fitting function is. For a set of data points {(𝑥1, 𝑦1), (𝑥2, 𝑦2), (𝑥3, 𝑦3)…(𝑥𝑁, 𝑦𝑁)}, we can find a fitting
function 𝑓(𝑥) with 𝑓𝑖 = 𝑓(𝑥𝑖) . The coefficient of determination is defined as 𝑅2 ≡ 1 −𝑆𝑆𝑟𝑒𝑠
𝑆𝑆𝑡𝑜𝑡 , where 𝑆𝑆𝑟𝑒𝑠 = ∑(𝑦𝑖 − 𝑓𝑖)2 and 𝑆𝑆𝑡𝑜𝑡 = ∑(𝑦𝑖 − 𝑦̅)2 . 𝑦̅ represents the average of 𝑦𝑖. The closer to one for 𝑅2 is, the more precise this fitting function would be. Fig. 4-7 (b)-(e) show the color mapping of 𝑅2. For 𝑅2 greater than 0.8 (0.9, 0.95, or 0.98), we set the color to be skin color, while the other cases are set to be black. At a low density regime with only WL patterns, 𝐵𝑠𝑜 is not converged, which expands from 10-4 to 10-12 with high 𝑅2 values. On the other hand, at a high density regime when both WL and WAL patterns are observed (Fig. 4-8 (b)-(d)), the highest 𝑅2 converges for a limited region of 𝐵𝑠𝑜.
Fig. 4-7 (a) Fitting results of Δσ𝑥𝑥 for the Ge0.89Sn0.11/Ge heterostructure with only WL patterns observed. (b)-(e) Color mapping of 𝐵𝑠𝑜 and 𝐵𝜙 for different R2.
Fig. 4-8 (a) Fitting results of Δσ𝑥𝑥 for the Ge0.89Sn0.11/Ge heterostructure with both WL/WAL patterns observed. (b)-(e) Color mapping of 𝐵𝑠𝑜 and 𝐵𝜙 for different R2.
The fitting results for the GeSn/Ge heterostructures under various carrier densities are shown in Fig. 4-9 (a)-(c). For all structures, as the density increases, a transition from WL to WAL is observed. The magneto-conductivity near the cross-over density is shown as the orange curve in each plot. By increasing the gate voltage to induce more holes in the GeSn QW, a larger electric field perpendicular to the hetero-interface is created, which results in stronger SIA and enhances the Rashba SOC effects [112]. The strength of Rashba SOC in the HH band is expected to be proportional to 𝑘𝐹3 [10, 14, 36]. A higher carrier density leads to a larger 𝑘𝐹 and stronger SOC effects. Similar to our prior work
Fig. 4-9 (a)-(c) Magneto-conductivity for different GeSn/Ge heterostructures. Black lines are the fitting curves by the HLN formula, and the experimental data are shifted vertically for a better view. (d) The carrier density of the WL-WAL transition and the corresponding electric field (𝐸𝑧) versus Sn fraction. The inset shows the corresponding band diagrams.
on SOC in Ge [14], tuning the SOC strength is enabled by top gating the undoped GeSn/Ge heterostructures. The transition density and the corresponding electric field versus the Sn fraction are plotted in Fig. 4-9 (d). For the device with a higher Sn fraction, a larger electric field is required to observe WAL. The relationship between SIA and SOC have been reported in III-V materials [112], in which the strength of SOC is the largest for the devices with the largest electric field. Intuitively, the device with a higher Sn fraction should present stronger WAL effects due to its larger atomic potential variations.
The required electric fields for the transition from WL to WAL should be smaller.
However, our results show an opposite trend, suggesting that there are other mechanisms