To further investigate the Sn effects on the SOC strength in the GeSn QW, the k-cubic Rashba coefficients (𝛼3) and the spin-splitting energy (𝛥𝑠𝑜) were estimated by the expression ℏ𝛺3 = 𝛥𝑠𝑜 = 𝛼3𝐸𝑧𝑘𝐹3 [14]. 𝛼3 can be derived with the average electric field (𝐸𝑧) calculated by Gauss’s law and the Fermi wavevetor by the Hall density (𝑘𝐹 =
√2𝜋𝑝2𝐷). The spin degeneracy in the HH band is broken due to the SOC effect with energy shifts of ±𝛥𝑠𝑜, so the energy difference between two spin subbands at a given 𝑘𝐹 is 2𝛥𝑠𝑜 (Fig. 4-1). Fig. 4-13 shows the spin-splitting energy as a function of carrier
density, and the red-dashed line represents the theoretical prediction (𝛥𝑠𝑜 ∝ 𝑝2𝐷5/2) for the k-cubic Rashba SOC [14]. For all GeSn/Ge heterostructures, as the carrier density increases by gating, the spin splitting energy is increased due to the enhanced Rashba SOC effect with the slopes close to the theoretical prediction.
Fig. 4-13 Spin splitting energy (𝛥𝑠𝑜) versus carrier density for three GeSn/Ge heterostructures and Ge QW [14]. The red-dashed line represents a theoretical curve predicted by the k-cubic relationship [14].
heterostructures used in this work, the lattice mismatch leads to strains in the epitaxially grown films. For III-V ternary or quaternary materials, the lattice mismatch is mitigated easily by tuning the compositions in the compounds, so spin-orbit energy splitting in the hole bands induced by strains can be avoided. For group-IV epitaxy, strains occur commonly due to the different lattice constants of the constituents (C, Si, Ge, and Sn).
The strains result in band splitting and deformations of the LH and HH subbands. In addition, the deeper confinement potential for a GeSn QW with a higher Sn fraction can further enhance this energy splitting. For the GeSn/Ge heterostructures, due to the large lattice mismatch between Ge and Sn (∼ 15% [116]), the strain effects are expected to be significant. The strain effects on SOC have been reported for III-V materials both experimentally [115] and theoretically [10, 15], but not yet demonstrated for any group-IV materials and their compounds.
The physics of the suppressed Rashba SOC effects for the GeSn devices with a higher Sn fraction is described as follows. For 2DHG in the GeSn/Ge heterostructures,
the total angular momentum ( 𝐽 = 𝐿⃗ + 𝑆 ) is mainly quantized along the growth (z) direction of the GeSn/Ge heterostructures [10, 15, 117], since only the z-axis maintains
the rotational symmetry. The z-component of angular momentum of carriers is ±3/2ℏ for the HH states (𝑚𝑗± 3/2) and ±1/2ℏ for the LH states (𝑚𝑗± 1/2) [15]. Fig. 4-14
(a) shows the spin orientations of holes in the HH state. Since the motion of the 2DHG is confined in the x-y plane and the electric field is along the z-direction, the resulting effective magnetic field (𝐵𝑅𝑎𝑠ℎ𝑏𝑎) created by the Rashba SOC also lies in the x-y plane [63] with the SOC Hamiltonian of H𝑠𝑜 = ( ℏ
4𝑚02𝑐2) 𝜎 ⋅ (−∇𝑉 × 𝑃⃗ ) [10]. If, for example, the effective magnetic field 𝐵𝑅𝑎𝑠ℎ𝑏𝑎 points to x-axis, the spin would tend to precess along this axis due to the quantization of the angular momentum in the x direction (Fig.
4-14 (b)). Under the uncertainty principle, a spin cannot be measured with two definite components of angular momentum along two orthogonal axes. This would result in two
Fig. 4-14 (a) The spin orientation of HH states in a GeSn QW due to the quantum confinement effect. (b) The spin orientation in the HH states under an induced magnetic field by SOC (𝐵𝑅𝑎𝑠ℎ𝑏𝑎).(c) Competition between the quantum confinement (𝐽𝑧 quantization) and the SOC effects (𝐽𝑥 quantization).
of the HH-LH energy splitting in the strained GeSn layer is relatively large (~ 10 meV) [118, 119] compared to our experimental results of the spin-splitting energy, which is only on the order of 0.1 meV. Thus, the quantization along the z-axis is dominant. Moreover, as the Sn fraction is higher, the larger compressive strain and deeper valence band potential further enhances the splitting of the HH and LH bands, resulting in a larger 𝛥ℎ𝑙. This makes the effects of 𝐽𝑧 quantization even stronger than 𝐽𝑥, suppressing the Rashba SOC strength further. A similar mechanism has been reported in prior works showing the anisotropy of Zeeman splitting [120, 121].
From the discussion above, the increased energy splitting of the HH and LH bands (Δℎ𝑙) due to the larger compressive strain and confinement potential results in the stronger quantization of 𝐽𝑧 than that of 𝐽𝑥 and simultaneously reduces the spin-splitting energy (𝛥𝑠𝑜). The negative dependency of 𝛥𝑠𝑜 on 𝛥ℎ𝑙 is reflected on the 𝛥ℎ𝑙 dependence of the k-cubic term in the Rashba coefficient (α3 ). This coefficient 𝛼3 for the first HH subbands is given by the third-order Löwdin perturbation theory [15, 117, 122],
𝛼3 =𝑒ℏ4 where c is a constant depending on the QW structure and is 64/9π2 for an infinitely deep
rectangular QW [15], 𝛾𝑖 are the L
ü
ttinger parameter, and 𝛥𝑖𝑗ℎ𝑙 denotes the energy difference between the ith HH subband and jth LH subband energy at 𝑘 = 0. The extracted Rashba coefficients of the k-cubic terms versus electric field (𝐸𝑧) are shown in Fig. 4-15.As predicted by Eq. 4-5, for the GeSn/Ge heterostructure with a higher Sn fraction, its HH-LH splitting energy is larger, leading to the smaller Rashba coefficient.
For all GeSn QWs, 𝛼3 is larger than that in the Ge QW, showing the potentials of the GeSn-based devices for spintronic applications. Note that for Ge0.94Sn0.06 and Ge devices, there is a negative dependency of 𝛼3 on the electric field while for Ge0.91Sn0.09
and Ge0.89Sn0.11 devices, 𝛼3 is insensitive to 𝐸𝑧. The negative trend has been observed in Al0.3Ga0.7As/GaAs structures [117], which was due to a further splitting between the HH and LH bands by gating. This effect is even more pronounced in a hetero-junction triangular QW because the confinement potential of a triangular QW depends on the band
Fig. 4-15Spin-orbit coefficient (𝛼3) versus vertical electric field (𝐸𝑧).
bands becomes larger, resulting in a decrease of Rashba coefficient ( α3 ) with the triangular potential approximation (α3 ∝ 𝐸𝑧−𝑠, s = 4/3) [117]. For a square QW, the confinement potential mainly depends on the band offsets at the heterojunction (Fig. 4-16).
The external electric field only causes minor effects on the splitting of HH and LH bands.
The dependence of the Rashba coefficient on the electric field should be weaker [14, 122], and indeed a weak dependence (s = 0.5) for the Ge0.94Sn0.06 and Ge heterostructures was observed. On the other hand, if the energy splitting of the HH and LH bands is dominated by the strain effects, the effects of HH-LH splitting through the electric field would be even weaker. Hence, Ge0.91Sn0.09 and Ge0.89Sn0.11 heterostructures show a much weaker dependence of 𝛼3 on the electric field.
Fig. 4-16 Splitting energy between the HH and LH states by gating a triangular and square QW.
4-5 Temperature dependence of weak localization and weak-anti localization in GeSn/Ge heterostructures
We investigate the temperature dependence of the WL/WAL patterns in this section.
Fig. 4-17 shows the magneto-conductivity at different temperatures (4, 3, 2, 1.6, and 1.2 K) for the Ge0.94Sn0.06/Ge,Ge0.91Sn0.09/Ge, and Ge0.89Sn0.11/Ge heterostructures. Those patterns could be divided into three types based on the regime of carrier density. At a low carrier density regime (Fig. 4-17 (a)-(c)) where only WL patterns are observed, the WL patterns are more pronounced as the temperature is reduced. At an intermediate carrier density regime (Fig. 4-17 (d)-(f)), a transition from WL to WAL occurs as the temperature is reduced. The curves in Fig. 4-17 (d)-(f) have been shifted vertically in order to show the transition curves (orange curves) clearly. In a high density regime (Fig. 4-17 (g)-(i)), the WAL patterns are more pronounced at lower temperatures. To explain this trend, we need to understand how the phase-coherence length (𝐿𝜙) and spin-relaxation length (𝐿𝑠𝑜)
change with the temperature because the WL/WAL patterns highly depend on the relative length between 𝐿𝜙 and 𝐿𝑠𝑜.
The spin-relaxation length is proportional to the product of spin-relaxation time (𝜏𝑠𝑜)
and diffusion coefficient (D) with the relationship 𝐿𝑠𝑜 = √𝐷𝜏𝑠𝑜 [63]. Based on the DP spin-relaxation mechanism (Fig. 4-11 (a)), there are two factors influencing the spin-
Fig. 4-17 The temperature dependence of WL/WAL patterns. (a)-(c) show the WL patterns at a low density regime. (d)-(f) show the transitions from WL to WAL at 𝑇 = 2 K at an intermediate density regime. The data are shifted vertically to show the transition curves clearly. (g)-(i) show the WAL patterns at a high density regime.
relaxation time: the transport lifetime (𝜏𝑡𝑟) and the strength of effective magnetic field (𝐵𝑅𝑎𝑠ℎ𝑏𝑎 ). The transport lifetime is inversely proportional to the spin-relaxation time, which barely changes (less than 3 %) from T = 4 K to 1.2 K for all heterostructures in this work. The effective magnetic field is determined by the z-direction electric field (𝐸𝑧) and the Fermi wavevector ( 𝑘𝐹 ). Both of them depend on the carrier density (𝐸𝑧 =
𝑒𝑝2𝐷/ϵ and 𝑘𝐹 = √2𝜋𝑝2𝐷) and have no dependence on temperature. The diffusion coefficient (𝐷 = 𝑉𝐹2× 𝜏𝑡𝑟/2) also varies slightly at this temperature range. Therefore, we can deduce the spin-relaxation length without too much impact by temperature difference.
On the other hand, a phase-coherence length depends on the temperature exponentially. A phase-coherence length is given by 𝐿𝜙 = √𝐷𝜏𝜙 [14]. However, the
phase-coherence time decreases exponentially as the temperature increases, which could be attributed to the hole-phonon scattering or hole-hole scattering [52]. As the temperature increases, more phonon modes are activated and the hole-phonon scattering rate (1/𝜏𝑝ℎ) becomes higher with a power law relation of 𝜏𝑝ℎ ∝ 𝑇−2 [123]. In addition,
a higher temperature also leads to an increase of the hole density because more holes have energies higher than the Fermi level, leading to a stronger hole-hole scattering rate (𝜏ℎℎ ∝
𝑇−1) [123, 124]. Both processes can cause dephasing and the effective dephasing rate should be proportional to the sum ( 1/𝜏𝜙= 1/𝜏𝑝ℎ+ 1/𝜏ℎℎ ). Therefore, the
phase-coherence length is enhanced as the temperature is reduced.
We use Fig. 4-18 to explain the trend observed in Fig. 4-17 In a low density regime (Fig. 4-18 (a)) where the SOC effect is weak, the spin-relaxation length is much longer than the phase-coherence length. Therefore, no WAL pattern is observed and the WL pattern is more pronounced as the phase-coherence length becomes longer at lower temperatures. At an intermediate density regime (Fig. 4-18 (b)), the spin-relaxation length is slightly longer than the phase-coherence length. As the temperature decreases, the spin-relaxation length barely changes, but the phase-coherence length is increased exponentially. As a result, a transition occurs when the phase-coherence length becomes longer than the spin-relaxation length. Finally, in a high density regime (Fig. 4-18 (c))
Fig. 4-18 Temperature dependence of WL/WAL patterns for the Ge0.91Sn0.09 structureat (a) a low, (b) an intermediate, and (c) a high density regimes with the relation between 𝐿𝜙 and 𝐿𝑠𝑜 at different temperatures.
where the SOC effect is very strong, the spin-relaxation length is smaller than phase-coherence length. As the phase-phase-coherence length becomes longer, more loops satisfy the relation of 𝐿𝑠𝑜 < 𝐿𝑙𝑜𝑜𝑝< 𝐿𝜙, and hence the WAL effect becomes stronger. Furthermore,
since the spin-relaxation length does not change with temperature, the value of 𝐵𝑡𝑢𝑟𝑛 remains the same.
By fitting with the HLN formula, we can derive the characteristic times (𝜏𝜙(𝑠𝑜)) and characteristic lengths (𝐿𝜙(𝑠𝑜)) at different temperatures (Fig. 4-19). The spin-relaxation
time (length) has weak dependence on the temperature, which is consistent to our previous arguments. The phase-coherence time shows a relation of 𝜏𝜙 ∝ 𝑇−1 (equivalent to 𝐿𝜙∝ 𝑇−0.5 because of 𝐿𝜙 ∝ √𝜏𝜙 ), indicating the hole-hole scattering
mechanism is dominant. The hole-hole scattering rate can be described by [123, 124].
1 Boltzmann constant. In our case, the transport lifetime (𝜏𝑡𝑟) is at the order of 1 ps (Fig.
4-12 (a)), and the corresponding temperature by Eq. 4-6 is 7.6 K. Since the data was taken
Fig. 4-19 Characteristic times (phase-coherence time (𝜏𝜙) and spin-relaxation time (𝜏𝑠𝑜)) and (b) characteristic lengths (phase-coherence length (𝐿𝜙) and spin-relaxation length (𝐿𝑠𝑜)) versus temperature for three GeSn/Ge heterostructures.
hole-hole scattering can be depicted as a hole being influenced by the electric field generated by all other holes in the valence band. Therefore, the dephasing mechanism would be suppressed when the carrier density increases due to the stronger screening effect [107].
4-6 Summary
In summary, we analyzed the Rashba spin-orbit coupling in the GeSn/Ge heterostructures with three different Sn fractions of 6%, 9%, and 11% by fitting the WL/WAL patterns with the HLN formula. We observe the strongest Rashba SOC among all group-IV materials in the GeSn/Ge heterostructures. The strength of the Rashba SOC can be modulated by tuning the carrier density via gating. Transitions from WL to WAL were observed for all GeSn/Ge heterostructures, and the spin-relaxation and phase-coherence times were extracted by the magneto-conductivity measurement results. As the carrier density increases, the spin-relaxation time decreases due to the stronger Rashba SOC effect while the phase-coherence time increases slightly. Experimental results show that the GeSn/Ge heterostructures are in the spin-diffusive regime.
As the Sn fraction in GeSn increases, the spin-orbit coupling becomes weaker, which is attributed to the effects of compressive strains on the Rashba SOC coefficients. The k-cubic term (α3) by the quantum confinement effect results in a z-direction quantization while the SOC-induced magnetic field leads to an in-plane (x-y plane) quantization. Two effects compete, and the stronger compressive strain and confinement potential enhance the z-direction quantization and further reduces the spin-splitting energy and thus, the Rashba SOC effect. Finally, the power law relation of 𝜏𝜙 ∝ 𝑇−1 suggests that the
hole-hole interaction is the dominant mechanism for the dephasing . Our results suggest undoped GeSn/Ge heterostructures are promising for group-IV based spintronics and fast qubit devices because of its strong SOC effects, gate tunability, and compatibility to the Si VLSI technology.
This thesis demonstrates the first 2DHGs in the undoped GeSn/Ge heterostructures with different Sn fractions of 6, 9, and 11%. The electrical and magneto-transport properties including SdH oscillations and quantum Hall plateaus are investigated by Hall measurements at cryogenic temperatures. Non-parabolicity effect on the hole effective mass and the SOC effect in the undoped GeSn/Ge heterostructures are also studied. The gated Hall bar devices of the GeSn/Ge heterostructures show transistor characteristics.
The highest mobility measured by Hall measurements at 1.2 K is 20,000 cm2/Vs with a
power law relation of 𝜇 = 𝑝2𝐷𝛼 with α = 0.4~0.7, which suggests that the mobility is limited by the background impurity scattering. Clear SdH oscillations and integer quantum Hall plateaus are observed under a magnetic field of up to 5 T. We extract the
hole effective mass and quantum lifetime through the temperature-dependent SdH oscillations. The calculated Dingle ratios are close to one, indicating the large-angle scattering effect is dominant. The effective mass shows good linearity with the carrier density, which suggests the non-parabolicity effects are dominant in the GeSn QWs. The effective mass ranges from 0.07 m0to 0.10 m0with the carrier density varying from 2.7 ×
1011 cm-2 to 6.1 × 1011 cm-2. The effective mass is smaller in the GeSn/Ge
heterostructure with a higher Sn fraction due to the stronger compressive strain. The non-parabolicity factor is 8.0 eV-1, 4.9 eV-1, and 4.0 eV-1 for Ge0.94Sn0.06/Ge,Ge0.91Sn0.09/Ge, Ge0.89Sn0.11/Ge heterostructures,respectively. The smaller non-parabolicity factor for a higher Sn fraction is attributed to the weaker band-mixing between the HH and LH bands with a larger HH-LH splitting.
We also investigate the Rashba SOC effects by fitting the magneto-conductivity to the HLN formula. Transitions from WL to WAL are observed for all GeSn/Ge heterostructures. Phase-coherence time (𝜏𝜙), spin-relaxation time (𝜏𝑠𝑜), spin-precession time ( 𝜏𝑝𝑟𝑒 ), spin-splitting energy ( 𝛥𝑠𝑜 ), and k-cubic Rashba coefficients ( 𝛼3 ) are
extracted. Strongest Rashba SOC effects among all group-IV materials are observed in the GeSn/Ge heterostructure with a Sn fraction of 6 %, and the SOC strength can be modulated by gating. The spin-relaxation time varies up to 85% and the spin-splitting energy is enhanced by a factor of 2 by increasing the carrier density. Experimental results show that the mean free path is shorter than the spin-precession length, indicating the GeSn/Ge heterostructures are in the spin-diffusive regime. As the Sn fraction in GeSn increases, the spin-orbit coupling becomes weaker, which is attributed to the effects of compressive strains on the k-cubic Rashba coefficients.
1. Distinguish the Sn effect and strain effect on Rashba SOC
The Rashba SOC effect is stronger when adding Sn into Ge crystals, but weaker under a larger compressive strain. The Sn effect and strain effect are intertwined in the GeSn/Ge heterostructures in this thesis. As the Sn fraction increases, the strain is also enhanced. By a careful design of the epitaxial structure in the future, the Sn effect and strain effect should be able to probe separately.
2. Investigation of SOC effects in bulk Ge(Sn)
There are many surface states at the Ge(Sn) surface or the Ge(Sn)/oxide interface [125], which would create surface electric fields and lead to SIA and WAL close to the surface [62]. We can apply gate biases on a junctionless FET to balance the electric field induced by those surface defects to reach a flat band condition, where the electric field becomes zero and the WAL effects should be suppressed significantly. By doing so, we can study the influence of surface states on the Rashba SOC in Ge or GeSn.
3. g-factor in the GeSn/Ge heterostructures.
In spin-based quantum computing devices, a constant magnetic field is required to break the spin-degeneracy, and the resulting two energy states serve as the qubit “1”
and “0”. The splitting energy is called the Zeeman splitting energy (Δ𝑍𝑒𝑒𝑚𝑎𝑛 =
𝑔𝜇𝐵𝐵), where 𝑔 is the g-factor and 𝜇𝐵 is the Bohr magneton. Therefore, a larger g-factor means that a relatively small magnetic field can be used to break the degeneracy. The reported g-factor of Ge is 28 in Ref. [33]. However, there is no work on the g-factors in a GeSn-based quantum well yet. By extracting the g-factors in the GeSn/Ge heterostructure, we can have a deeper understanding of the magneto-properties of the GeSn/Ge heterostructures for spin-based quantum computing applications.
In summary, those three directions focus on the quantum transport properties in the GeSn/Ge heterostructures, which have not been well-explored so far. Since GeSn has great potential for future electronic application, a thorough understanding of this material is important. The results of those projects will be very useful for both condensed matter physics and device applications.
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