Nonlinear Shubnikov-de Haas Effect Distinguishes Spin-orbit Interaction Textures Precisely

Scientists are increasingly focused on harnessing spin-orbit interaction for next-generation technologies, but directly measuring its intricacies remains a formidable task. Kazuki Nakazawa (RIKEN Center for Quantum Computing) and Henry F. Legg (SUPA, University of St Andrews), alongside Renato M. A. Dantas, Jelena Klinovaja, Daniel Loss et al (University of Basel), now present a novel approach using the nonlinear Shubnikov-de Haas (NSdH) effect to map Fermi-surface spin textures, a crucial step towards controlling spin-based devices. Their research reveals that NSdH, unlike conventional methods, is exceptionally sensitive to spin textures arising from spin-orbit interaction, allowing for clear differentiation between various types of coupling, such as linear and cubic Rashba effects in germanium heterostructures. This breakthrough establishes NSdH as a powerful tool for characterising materials relevant to topology, spintronics, and the future of solid-state information technologies.

The team achieved a significant breakthrough by demonstrating that, unlike the conventional Shubnikov-de Haas (SdH) effect, NSdH is acutely sensitive to the intricate spin textures arising from SOI, offering a more pronounced signal for detection. This innovative approach leverages the second-order response to an applied electric field, revealing subtle features often obscured in traditional linear measurements.

The study meticulously formulates the NSdH effect using the Keldysh Green function method, deriving a general expression for the second-order conductivity tensor within the Landau-level basis. Experiments show that the NSdH conductivity exhibits a unique characteristic: opposing signs in the coefficients governing oscillations from different portions of the Fermi surface, resulting in a distinct π-phase difference. This phase shift is a direct consequence of the SOI-induced spin texture, dramatically enhancing the visibility of SOI signatures compared to the standard SdH effect. Researchers specifically applied this formalism to two-dimensional hole gases exhibiting both linear and cubic SOI, commonly found in germanium heterostructures, to demonstrate the method’s efficacy.
Furthermore, the research establishes that phase differences within the NSdH conductivity tensor are strongly dependent on the symmetry of the underlying SOI term in the Hamiltonian. This discovery means a comprehensive analysis of the NSdH effect can provide a detailed characterization of SOI effects within a material, offering unprecedented insight into its electronic properties. The team’s work opens new avenues for electrically tuning SOI in low-dimensional systems, potentially leading to longer spin-relaxation lengths and improved performance of spin qubits. This breakthrough promises to accelerate the development of advanced spintronic devices and solid-state quantum technologies reliant on precise control of spin-orbit coupling.

Nonlinear Shubnikov-de Haas probing of spin-orbit interaction reveals

Scientists pioneered a novel method to probe spin-orbit interaction (SOI) using the nonlinear Shubnikov-de Haas (NSdH) effect, a second-order oscillatory phenomenon occurring under conditions similar to the standard SdH effect. The research team engineered a sensitive technique to discern SOI strength and characteristics, particularly crucial in heterostructures where traditional methods fall short. Experiments employed the Keldysh Green function method to formulate the NSdH effect, deriving a general expression for the second-order conductivity tensor within the Landau-level basis. This approach enabled the team to isolate the electric field’s influence by separating the vector potential into space- and time-dependent components, defining static magnetic fields and uniform electric fields precisely.

The study meticulously expanded the Hamiltonian and current operator in powers of the time-dependent vector potential, allowing for the calculation of higher-order responses to the electric field. Researchers then focused on the second-order response, obtaining the ac current at finite frequencies before transitioning to the dc limit to retain only regular, non-divergent contributions. A typical measurement setup, illustrated in the supporting information, facilitated the precise application and measurement of these fields. The team demonstrated that the NSdH conductivity takes the form σxii ∼b1 cos(F1/B) −b2 cos(F2/B), where the opposing signs of coefficients b1 and b2 indicate a critical π-phase difference originating from the Fermi-surface spin texture.

This phase shift, a direct consequence of SOI, dramatically enhances the visibility of SOI signatures in NSdH compared to linear SdH oscillation. Furthermore, the team discovered that phase differences between NSdH conductivity tensor components are strongly dependent on the symmetry of the SOI term within the Hamiltonian. This finding allows for a detailed characterization of SOI effects in a system, offering a powerful tool for understanding topology, spintronics, and solid-state information technologies. As a demonstration, the research applied this formalism to two-dimensional hole gases exhibiting both linear and cubic Rashba couplings, commonly found in germanium heterostructures. The innovative NSdH technique promises a significant advancement in characterizing SOI, surpassing the limitations of conventional methods and opening new avenues for materials discovery and device development.

Nonlinear Shubnikov-de Haas reveals spin-orbit textures

Scientists have discovered a novel method for probing spin-orbit interaction (SOI) using the nonlinear Shubnikov-de Haas (NSdH) effect, a phenomenon occurring under conditions similar to the well-known SdH effect but dependent on second-order electric fields. The research establishes NSdH as a sensitive tool for characterizing materials crucial to topology, spintronics, and solid-state quantum information technologies. Experiments revealed that, unlike its linear counterpart, the NSdH effect is remarkably sensitive to spin textures arising from SOI, offering a new framework for material characterisation. The team measured nonlinear conductivities, demonstrating that the phase and beating of NSdH oscillations can distinguish between different types of SOI with unprecedented clarity.

Specifically, the study focused on germanium heterostructures, where both linear and cubic Rashba couplings are anticipated, and successfully differentiated between these couplings using the NSdH effect. Results demonstrate that the NSdH conductivity takes the form σxii ∼b1 cos(F1/B) −b2 cos(F2/B), where coefficients b1 and b2 exhibit opposite signs, indicating a crucial π-phase difference between contributions from different Fermi surface areas. This phase shift originates directly from the Fermi surface spin texture, amplifying SOI signatures in NSdH compared to linear SdH measurements. Tests prove that phase differences between components of the NSdH conductivity tensor are strongly dependent on the symmetry of the underlying SOI term within the Hamiltonian, allowing for detailed characterisation of SOI effects in a system. The breakthrough delivers a formulation of the NSdH effect using the Keldysh Green function method, deriving a general expression for the second-order conductivity tensor in the Landau-level basis. Scientists achieved a detailed analysis of the conditions required for NSdH oscillations, applying their formalism to two-dimensional hole gases with both linear and cubic SOI, as commonly found in germanium heterostructures. Data shows that this technique offers a powerful new approach to understanding and harnessing the potential of spin-orbit interaction in advanced materials and devices.

NSdH effect probes Rashba couplings in germanium nanowires

Scientists have demonstrated the nonlinear Shubnikov-de Haas (NSdH) effect as a sensitive method for probing spin-orbit interaction (SOI) in materials. Unlike conventional linear Shubnikov-de Haas oscillations, the NSdH effect exhibits a strong dependence on the spin textures arising from SOI, allowing for differentiation between various forms of this interaction, specifically, linear and cubic Rashba couplings as found in germanium heterostructures. This enhanced sensitivity stems from the NSdH effect being second order in the applied electric field, providing a unique signature compared to its linear counterpart. Researchers established that by tuning the chemical potential or linear Rashba strength, the evolution of a symmetric component within the nonlinear conductivity can be directly tracked, revealing the ratio of cubic to linear SOI.

The calculated nonlinear conductivity features are predicted to be observable in experimental measurements of nonlinear resistivity, further validating the feasibility of this approach. This work offers a new framework for characterizing SOI, which is crucial for advancements in topology, spintronics, and solid-state information technologies. The authors acknowledge a limitation in that the precise quantification of SOI parameters requires careful analysis of the complex interference patterns within the NSdH oscillations. Future research directions include exploring the application of this technique to a wider range of materials and heterostructures, potentially uncovering novel SOI phenomena and refining device designs for spintronic applications. These findings represent a significant step towards a more detailed understanding and control of spin-related phenomena in materials, paving the way for innovative technologies.