Cubic Rashba Spin-Orbit Interaction of a Two-Dimensional Hole Gas in a Strained-Ge/SiGeQuantum Well

Abstract

The spin-orbit interaction (SOI) of a two-dimensional hole gas in the inversion symmetric semiconductor Ge is studied in a strained-Ge=SiGe quantum well structure. We observe weak antilocalization (WAL) in the magnetoconductivity measurement, revealing that the WAL feature can be fully described by the k-cubic Rashba SOI theory. Furthermore, we demonstrate electric field control of the Rashba SOI. Our findings reveal that the heavy hole (HH) in strained Ge is a purely cubic Rashba system, which is consistent with the spin angular momentum m j ¼ AE3=2 nature of the HH wave function. DOI: 10.1103/PhysRevLett.113.086601 PACS numbers: 72.25.Dc, 73.20.Fz, 73.21.-b The spin-orbit interaction (SOI) in a two-dimensional system is a subject of considerable interest because the SOI induces spin splitting at a zero magnetic field, which is important in both fundamental research and electronic device applications [1]. Recent developments of SOI-induced phenomena in the solid state demonstrate many possibilities utilizing spin current and the emergence of new physics such as the spin interferometer [2,3], persistent spin helix [4,5], spin Hall effect [6][7][8], and quantum spin Hall effect [9,10]. Up to now, there have been two well-known SOIs existing in solids: the Dresselhaus SOI [11] due to bulk inversion asymmetry (BIA) in the crystal structure and the Rashba SOI [12,13] due to spatial inversion asymmetry (SIA).In low-dimensional systems, the Rashba SOI becomes more important because it is stronger at the heterointerface and can be controlled by an external electric field. Many of the pioneering studies on the SOI-induced phenomena mentioned above were performed in two-dimensional electron systems, where the Rashba SOI is described by the k-linear Rashba term. In the Hamiltonian, the k-linear Rashba term can be written aswhere σ AE ¼ 1=2ðσ x AE iσ y Þ denote combinations of Pauli spin matrices, k AE ¼ k x AE ik y , and k x , k y are the components of the in-plane wave vector k ∥ . The effective magnetic field Ω 1 ðk ∥ Þ acting on the transport carrier due to the k-linear Rashba term is illustrated in Fig. 1(a).Recently, a higher-order contribution of the Rashba SOI, the so-called k 3 (k-cubic) Rashba SOI, has received more attention [14,15]. The Hamiltonian for the k-cubic Rashba SOI is expressed asand the effective magnetic field Ω 3 ðk ∥ Þ in k space is illustrated in Fig. 1(b) [15]. There is a significant difference in the effective field symmetry between the k-linear and the k-cubic Rashba SOI with one and three rotations in k space, respectively. The k 3 symmetry of the SOI is an interesting subject because it influences all of the SOI-induced phenomena as opposed to the k-linear Rashba term. For example, in case of the spin Hall effect, the k-cubic Rashba term is predicted to give rise to a larger spin Hall conductivity [17][18][19].

Strained germanium for applications in spintronics

“…Figure 7 shows schematics of typical Ge QW heterostructures exhibiting the Rashba S-O interaction. The following is a summary of the key recent findings in this area, focusing on detection and quantification using three methods: analysis of 'beating' patterns in Shubnikov-de Haas (SdH) oscillations [57,59], analysis of weak antilocalisation (WAL) [56,59,60] and cyclotron resonance using terahertz excitation [61]. One of the key advances necessary to observe the Rashba S-O interaction using SdH oscillations is the significant enhancement of 2D hole mobility at low temperatures.…”

“…known for many years, and evidence was detected using cyclotron resonance techniques [54,55], it is only recently that the strength of the interaction has been quantified. Also, the interaction in strained Ge QW has been identified as the cubic Rashba S-O interaction, due to quantum confinement in the heavy hole valence band only [56,57] The energy term arising from the Rashba interaction is cubic in k-space, and must be treated with a different analysis to the linear interaction in electron and light hole QW. The S-O interaction is an important component in future spintronic technologies, with applications in areas as diverse as spin transistors, quantum computing (S-O qubits), S-O torque, the spin Hall effect, chiral magnonics and to create band inversion for the formation of topologically insulating states to generate the quantum spin Hall effect [58].…”

“…Firstly, Moriya et al studied a Ge QW grown by SS-MBE on a Si 0.5 Ge 0.5 buffer, resulting in a high degree of biaxial compressive strain (2.1%), and with a low heavy hole mobility in the 2DHG (5000 cm 2 V À1 s À1 ) at low temperatures [56]. By fitting magnetoconductivity data to the Iordanskii-Lyanda-Geller-Pikus model they determined a cubic Rashba coefficient of 1.5 Â 10 À29 eVm 3 , with a corresponding spin splitting energy of 0.36 meV, at zero gate Figure 12 Simulated beating pattern in Shubnikov-de Haas oscillations, due to the Rashba spin-orbit interaction.…”

“…To date, measurement of WAL has been used in two Ge QW systems, grown by different techniques, to quantify the cubic Rashba interaction [56,60]. Firstly, Moriya et al studied a Ge QW grown by SS-MBE on a Si 0.5 Ge 0.5 buffer, resulting in a high degree of biaxial compressive strain (2.1%), and with a low heavy hole mobility in the 2DHG (5000 cm 2 V À1 s À1 ) at low temperatures [56].…”

Optical conductivity of a 2DEG with anisotropic Rashba interaction at the interface of LaAlO3/SrTiO3

“…In general, there are two different types of symmetry dependent SOI, Rashba 3,4 and Dresselhaus 5 SOIs, in various condensed matter systems. In twodimensional electron gas (2DEG) formed at the III-V semiconductor heterostructures 6 and in various topological insulating systems 7 , the Rashba SOI (RSOI) is linear in momentum and of the form H R = iαk − σ + +h.c., where α is the strength of RSOI, σ ± = σ x ± iσ y with σ x and σ y are the Pauli's spin matrices and k ± = k x ± ik y with k x and k y the components of the wave vector k. Besides, the Rashba SOI in two-dimensional hole gas formed at the interface of p-type GaAs/AlGaAs heterostructures 8,9 , 2DEG on the surface of SrTiO 3 single crystals 10 and in 2D hole gas formed in a strained Ge/SiGe quantum well 11 is cubic in momentum and is of the form H iso R = iαk 3 − σ + + h.c. The spin splitting energy due to this RSOI is always isotropic and hereafter we will mention this as isotropic cubic RSOI.…”