Spin-orbit coupling (SOC) arises from the relative motion between electrons and nuclei. Under the relativistic effect, the moving electron would view an electric field as an effective magnetic field by Lorentz transformation (Fig. 2-6) [10]. An electron in the solid feels a k-dependent magnetic field that interacts with the electron’s spin, which is similar to the Zeeman effect, so the electron’s spin and its motion couple together. In general, the associated Hamiltonian is called the Pauli SO term and given by − ℏ
4𝑚02𝑐2σ ∙ p × (∇V0), where σ is the Pauli spin matrix, p is the momentum of the electron, and V0 is the atomic potential.
Fig. 2-6 Physical picture of SOC effects.
The valence band splits into heavy hole (HH), light hole (LH), and split-off (SO) bands [10] when the Pauli SO term is included in the Hamiltonian. Fig. 2-7 (a) shows the band structure with the isolated electron motion and spin as the SOC effect is ignored.
The orbital quantum number (l) and spin (s) are two individual quantum numbers, and each band can be expressed by four quantum numbers |𝑙, 𝑚𝑙, 𝑠, 𝑚𝑠⟩. Fig. 2-7 (b) shows the band structure by considering the SOC effects. The electron motion and spin couple,
so we need to use J (defined as 𝐽 = 𝐿⃗ + 𝑆 ) to describe the total angular momentum. This results in a further splitting in the valence band. The HH, LH, and SO bands are expressed in terms of |𝑗, 𝑚𝑗⟩ = |3/2, ±3/2⟩ , |3/2, ±1/2⟩ , and |1/2, ±1/2⟩ , respectively. 𝛥 denotes the energy difference between the SO and HH bands at = 0. The term 𝑉0 in
the Pauli SO term represents the atomic potential. For an atom with a larger atomic number, the potential variation in space is larger (more protons in a nucleus), and hence the SOC effect is stronger. The value of 𝛥 is expected to be 44 meV for Si, 290 meV for Ge, and 800 meV for α-Sn. To break the spin-degeneracy, either the time-reversal
Fig. 2-7 Band structures as the SOC effect is (a) ignored or (b) included [10].
symmetry or the space-reversal symmetry needs to be broken [10]. One example of breaking the time-reversal symmetry is to apply an external magnetic field. The spin degeneracy is lifted and this is known as the Zeeman-splitting. On the other hand, there are two ways to break the space-reversal symmetry: bulk inversion asymmetry (BIA) and structure inversion asymmetry (SIA). The SOC effect due to the BIA is called the Dresselhaus SOC [53], while SIA leads to the Rashba SOC effects by an electric field [54].
The Dresselhaus SOC effect has been observed in III-V materials such as GaAs [55-57], InAs [58], InGaAs [41] and GaN [38]. The Zinc-Blende structure in III-V materials lacks of an inversion symmetric point in the crystal structure. This crystal asymmetry acts like a built-in electric field (Fig. 2-8), which cannot be eliminated or controlled by external modulation. On the other hand, the diamond structure of group-IV materials
Fig. 2-8 The built-in electric field (𝐸⃗ ) in a Zinc-Blend structure [59].
possesses symmetry points. Therefore, no Dresselhaus SOC effect is observed in group-IV materials. In this thesis, we focus on the analysis of SOC effect in GeSn/Ge heterostructures, so the Dresselhaus SOC is not considered.
The Rashba SOC effect can be controlled by external gating through the modulation of the electric fields [54]. Fig. 2-9 shows the band diagram of an undoped Ge/GeSi heterostructure as an example. Under a flat band condition, there is no SIA and the Rashba SOC effect vanishes. Through gating, we can change the extent of SIA and adjust the strength of Rashba SOC. The advantage of the Rashba SOC over the Dresselhaus SOC is the tunability of the SOC strength through gating, which is critical for device applications.
In III-V materials, the Dresselhaus and Rashba SOC effects are mixed. There has been much work focused on disentangling their contributions [60] and the interplay between them [61]. On the other hand, group-IV materials only exhibit the Rashba SOC effect due to the absence of BIA [36]. Prior work on Si [31] and Ge quantum wells [14, 36, 37], bulk Ge [62], and GeSn quantum well [16] showed Rashba SOC effects. The Rashba SOC
Fig. 2-9 Structure inversion asymmetry induced by gating.
effects in Ge is much stronger than in Si. Our group has demonstrated a ballistic transport in a Ge/GeSi heterostructure [14], which is a necessary to realize a working spin FET. In 2020, we demonstrated a strongest Rashba SOC in GeSn/Ge heterostructures [16].
We now investigate the Rashba SOC Hamiltonian in the valence band. The Hamiltonian of the LH band is linearly dependent on k and written as 𝑯𝑠𝑜𝐿𝐻=
α1𝐸𝑧𝑖(𝑘−𝜎+− 𝑘+𝜎−), while for the HH band, it is k-cubic dependent given by 𝑯𝑠𝑜𝐻𝐻 = α3𝐸𝑧𝑖(𝑘−3𝜎+− 𝑘+3𝜎−) [36]. α3(1) represent the k-cubic (linear) Rashba coefficient and 𝐸𝑧 represents the average electric field along the z-direction. 𝑘±= 𝑘𝑥± 𝑖𝑘𝑦 are the combination of two 𝑘∥ components, and 𝜎± = (𝜎𝑥± 𝜎𝑦)/2 are the combinations of
Pauli spin matrices. The resulting effective magnetic field has different orientations in the
𝑘∥ plane for the LH and HH bands (Fig. 2-10) [63]. For the LH band, the SOC-induced magnetic field is given by 𝐵⃗⃗⃗⃗⃗⃗⃗ (𝑘𝐿𝐻 ∥) = |𝐵𝐿𝐻|(± sin 𝜃 , ∓ cos 𝜃) while for the HH band, it is 𝐵⃗⃗⃗⃗⃗⃗⃗ (𝑘𝐻𝐻 ∥) = |𝐵𝐻𝐻|(± sin 3𝜃 , ∓ cos 3𝜃) . 𝜃 is the angle between the wavevector
𝑘//
⃗⃗⃗⃗⃗ = (𝑘𝑥, 𝑘𝑦) and the 𝑘𝑥 axis. For a given k-state, the directions of effective magnetic
fields are different for the HH and LH bands.
Fig. 2-10 The effective magnetic field orientations in k-space for (a) the LH band and (b) the HH band.
Finally, the spin-splitting energy due to SOC for the LH band has a k-linear dependence ( 𝐸𝐿𝐻 = ±𝛼1𝐸𝑧𝑘 ) and a k-cubic dependence for the HH band ( 𝐸𝐻𝐻 =
±𝛼3𝐸𝑧𝑘3) [10]. The splitting energy is proportional to 𝑘 and Ez because the SOC effect arises from the Lorentz transformation of an electric field for a moving carrier, and hence
it depends on the electric field (Ez) and the carrier’s momentum (𝑘⃗ ). This also shows the tunability of the Rashba SOC strength by changing the electric field (as well as SIA) and the carrier density via gating.
phase-coherence time (𝜏𝜙), spin-relaxation time (𝜏𝑠𝑜), and spin-precession time (𝜏𝑝𝑟𝑒).
All of them are important parameters for the weak-localization and weak-anti localization effect, which have been used to identify the Rashba SOC effects (section 2-5). (i) The transport lifetime is an average time between two consecutive scattering events (Fig. 2-11 (a)). The corresponding characteristic length is the mean free path defined as 𝐿𝑡𝑟 = 𝜏𝑡𝑟 ×
𝑉𝐹. (ii) The phase-coherence time (𝜏𝜙) represents the time scale of carriers in phase with its corresponding length (𝐿𝜙 = √𝐷𝜏𝜙), where D is the diffusion coefficient. Fig. 2-11 (b) shows the physical picture of the phase-coherence time. For a carrier moving clockwise (green path) or counter-clockwise (blue path), the associated wave is originally in-phase during a time scale of 𝜏𝜙 , resulting in a constructive interference pattern (red line).
However, beyond this duration, the phase difference gradually accumulates and the carriers become out-of-phase, so the interference pattern fades away as well. The carriers cannot maintain in-phase for infinitely long due to the inelastic scattering events [52].
Qualitatively, the wavefunction of a propagating electron can be expressed as
𝜓(𝑟 , 𝑡) = 𝐴(𝑟 , 𝑡)𝑒𝑥𝑝[𝑖(𝑘⃗ ⋅ 𝑟 − 𝐸𝑡/ℏ)] Eq. 2-1
Fig. 2-11 Physical picture of (a) transport lifetime and (b) phase-coherence time.
The phase is proportional to the term (𝑘⃗ ⋅ 𝑟 − 𝐸𝑡/ℏ). The phase of the electron wave accumulates as the electron moves through the spatial term (𝑘⃗ ⋅ 𝑟 ) and temporal term (𝐸𝑡/ℏ). For electrons moving clockwise and counter-clockwise, the phase accumulation
of spatial part is the same: ∫ 𝑘𝑎𝑏⃗ ⋅ 𝑟 (𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒) = ∫ (−𝑘⃗ ) ⋅𝑎𝑏
(−𝑟 ) (𝑐𝑜𝑢𝑛𝑡𝑒𝑟𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒) since both the direction of 𝑘⃗ and the integral path have opposite direction. If only elastic scattering events occur when the electron moves along the paths, the energy of the electron remains constant and the phase accumulation of temporal term is also the same. Hence, the carriers for different paths remain in phase.
However, inelastic scattering events change the electron’s energy via scattering, so the phase temporal term (𝐸𝑡/ℏ) would not be conserved and finally the carriers become out-of-phase. A single inelastic scattering may not be strong enough to cause de-phasing, so
(iii) The spin-relaxation time (𝜏𝑠𝑜) represents the time scale of carriers holding the
same spin orientations with a corresponding spin-relaxation length (𝐿𝑠𝑜= √𝐷𝜏𝑠𝑜). For a semiconductor lack of inversion symmetry (BIA or SIA), the SOC-induced effective magnetic field can randomize the spin-orientations. This is called D’yakonov-Perel (DP) spin-relaxation mechanism [64] and illustrated in Fig. 2-12 (a) (left). Due to the SOC effect, the carriers feel a k-dependent effective magnetic field and the direction of magnetic field changes with the momentum. The spin orientation changes between two collision events and finally randomized. If the collisions happen too frequently, the time interval between two collisions is too short for a spin to follow the fast changing magnetic field. Thus, the DP spin-relaxation rate is inversely proportional to the scattering rate.
Besides DP mechanism, there is another mechanism called Elliott-Yafet mechanism [65, 66] to interpret the spin-relaxation in metals and semiconductors with inversion symmetry.
The physical picture of EY mechanism is shown in the right of Fig. 2-12 (b). The EY mechanism states that when the carriers collide with impurities, there is a probability to flip over the spin state [67]. Therefore, the more collision events, the higher chance to
Fig. 2-12 Physical picture of (a) spin-relaxation mechanism and (b) spin-precession mechanism.
observe a different spin states. The spin-relaxation rate for the EY effect is proportional to the scattering rate, which is the major difference from the DP mechanism. Finally, (iv) The spin-precession time (𝜏𝑝𝑟𝑒) describes the time scale for a spin to complete a full
precession under a magnetic field (i.e Larmor precession), and the corresponding length
𝐿𝑝𝑟𝑒 = 𝑉𝐹 × 𝜏𝑝𝑟𝑒 is the travelling distance within 𝜏𝑝𝑟𝑒 . All those characteristic times and their corresponding characteristic lengths are important parameters for the analysis of WL/WAL patterns and Rashba SOC effect.
conductivity. Both of them represent the interference effects of a carrier traveling a closed loop clockwise and counter-clockwise [68]. Those two paths possess time reversal symmetry as the trajectory is exactly the same and the moving direction is opposite. The perimeter of the loop is defined as 𝐿𝑙𝑜𝑜𝑝. For large loops, as long as their perimeters are longer than the phase-coherence length (𝐿𝑙𝑜𝑜𝑝> 𝐿𝜙), the carrier wavefunctions are
out-of-phase, so neither WL nor WAL will be observed no matter what the spin-relaxation length is. For the loops whose perimeters are smaller than the phase-coherence length (𝐿𝑙𝑜𝑜𝑝 < 𝐿𝜙), the carrier wavefunction remains in-phase, so WL or WAL will be observed depending on the length of 𝐿𝑠𝑜. If the spin direction remains the same (𝐿𝑙𝑜𝑜𝑝 < 𝐿𝜙 < 𝐿𝑠𝑜 or 𝐿𝑙𝑜𝑜𝑝 < 𝐿𝑠𝑜< 𝐿𝜙), the constructive interference will occur, leading to a decrease of
the magneto-conductivity at B = 0, which refers to the WL effect (Fig. 2-13 (a)).
If the spin-relaxation length is so small that the relationship of 𝐿𝑠𝑜< 𝐿𝑙𝑜𝑜𝑝 < 𝐿𝜙 is
satisfied, the spin orientation will alter along the loop (Fig. 2-13 (b)). The constructive interference turns into a destructive one due to a phase difference of 𝜋 [68]. The physical reason is described as follows. The initial spin state of a carrier is |𝜓⟩ , and the spin rotation operator is given as [69]:
𝑅̂ = [ 𝑒𝑖(𝜙+𝛾)/2cos (𝜃/2) 𝑖𝑒𝑖(𝛾−𝜙)/2sin (𝜃/2)
𝑖𝑒−𝑖(𝛾−𝜙)/2sin (𝜃/2) 𝑒−𝑖(𝜙+𝛾)/2cos (𝜃/2)] Eq. 2-2 For a carrier moving clockwise, the state becomes |𝜓𝐶𝑊⟩ = 𝑅̂|𝜓⟩. For a carrier moving counter-clockwise, its movement is opposite. The direction of the SOC-induced magnetic field is opposite as well since the effective magnetic field is proportional to the
cross-product of wavevector (𝑘⃗ ) and electric field (𝐸𝑧). Therefore, the spin rotates in an opposite way, and |𝜓𝐶𝐶𝑊⟩ can be written as |𝜓𝐶𝐶𝑊⟩ = 𝑅̂−1|𝜓⟩ . The interference term is
Fig. 2-13 Mechasnism of (a) WL, (b) WAL, and (c) Aharonov–Bohm effect.
without applying a magnetic field and this is called WAL effect.
When applying a magnetic field, an additional phase difference will emerge for a carrier travelling clockwise and counter-clockwise due to Aharonov–Bohm (AB) effect (Fig. 2-13 (c)) [69]. We use different colors in Fig. 2-13 (c) to represent the accumulated phase difference induced by the AB effect. This phase difference (AB phase) is proportional to the magnetic flux (𝛷𝐵) of a closed loop given by 2𝑒𝛷𝐵/ℏ, so the larger loops acquire more phase difference under the same magnetic field. The AB phase can destroy the in-phase condition and smear out the interference pattern.
The effects of a combination of the WL/WAL patterns on the magneto-resistance are shown in Fig. 2-14. We use “Random” with the gray color to represent the carriers being out-of-phase for larger loops satisfying 𝐿𝑙𝑜𝑜𝑝> 𝐿𝜙. The “Constructive” label with the blue color represents the in-phase carriers for the loops satisfying 𝐿𝑙𝑜𝑜𝑝 < 𝐿𝜙 . The
“Destructive” label with the green color represents the carriers being in-phase and their spin states change if the condition of 𝐿𝑠𝑜 < 𝐿𝑙𝑜𝑜𝑝< 𝐿𝜙 is satisfied. The “AB phase”
label with the red color represents the loops that are influenced by a magnetic field. We
Fig. 2-14 The relationship between 𝐿𝜙, 𝐿𝑠𝑜, 𝐿𝑙𝑜𝑜𝑝 and the magnetic field for (a) WL and (b) WAL first focus on the WL pattern in Fig. 2-14 (a). A WL pattern occurs as the SOC effect is
weak, and the spin-relaxation length is longer than the phase-coherence length (𝐿𝑠𝑜 >
𝐿𝜙). For the loops with the perimeters smaller than 𝐿𝜙, they contribute to constructive interference and the WL effect is observed because the electron is “localized”. The WL
pattern is more pronounced (larger Δσ𝑥𝑥 defined in Fig. 2-14 (a)) when more loops contribute to the constructive interference. Hence, the length “Constructive” label is
proportional to Δσ𝑥𝑥. By applying a magnetic field, those in-phase carriers become out-of-phase as the AB effect becomes stronger. The constructive interference disappears and the magneto-conductivity increases because the electron waves are delocalized. Finally,
The WAL pattern occurs if the SOC effect is strong enough that the spin-relaxation length is shorter than the phase-coherence length ( 𝐿𝑠𝑜 < 𝐿𝜙 ). The loops satisfying
𝐿𝑠𝑜 < 𝐿𝑙𝑜𝑜𝑝< 𝐿𝜙 contribute to destructive interference, leading to the WAL effect. The window range of “Destructive” label is proportional to the WAL signal (Δσ𝑥𝑥 defined in Fig. 2-14 (b)). The dip of WAL pattern is proportional to 𝐿𝜙
𝐿𝑠𝑜= 𝜏𝜙
𝜏𝑠𝑜 [70]. The external magnetic field suppresses the WAL feature as well. The carriers have a phase difference of π (Eq. 2-4) due to the changing of spin states, but the additional AB phase changes this value and destroys the destructive interference. When the external small magnetic field is small (𝐵 = 𝐵1 < 𝐵𝑡), the larger loops have more magnetic fluxes due to their larger enclosed areas and hence acquire more AB phases. On the other hand, the smaller loops are almost not affected by the AB effect. Therefore, the AB phase destroys the destructive interference first, and the magneto-conductivity decreases. At 𝐵 = 𝐵𝑡, all the destructive interferences are washed away by the AB phase, resulting in a dip of the magneto-conductivity. 𝐵𝑡𝑢𝑟𝑛 is inversely proportional to 𝐿𝑠𝑜. For a smaller 𝐿𝑠𝑜, more small loops contribute to destructive interference and a larger magnetic field is required to gain enough AB phases. At 𝐵 = 𝐵2 > 𝐵𝑡, the AB phase starts to destroy constructive
interference and an increase in magneto-conductivity will be observed. Finally, the magneto-conductivity saturates since all interference patterns disappear because of the AB effect.
2-6 Summary
In this chapter, the physical background of SOC and WL/WAL effects are introduced.
The SOC effects arise from the relative motion between electrons and nuclei, and there are two types of SOC effects: Dresselhaus and Rashba SOC due to BIA and SIA, respectively. The Rashba SOC strength can be adjusted by external gating, which is of great interest for device applications. Four characteristic times such as lifetime (𝜏𝑡𝑟 ), phase-coherence time (𝜏𝜙), spin-relaxation time (𝜏𝑠𝑜), and spin-precession time (𝜏𝑝𝑟𝑒),
and their corresponding lengths are introduced and the dephasing mechanism and spin-relaxation mechanism are also discussed. The relationship between the phase coherence length (𝐿𝜙), spin-relaxation length (𝐿𝑠𝑜), and the loop length (𝐿𝑙𝑜𝑜𝑝) and the resulting
interference patterns are described and accounted for the WL and WAL effects. We also described the AB effect and how it suppresses the WL/WAL features.